From rrosebru@mta.ca Wed Mar 1 09:16:06 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Mar 2006 09:16:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FER6i-0000Ek-Cr for categories-list@mta.ca; Wed, 01 Mar 2006 09:11:08 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) In-Reply-To: <440335A6.3050806@cs.stanford.edu> References: <440335A6.3050806@cs.stanford.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: Re: Undirected graphs citation Date: Wed, 1 Mar 2006 09:41:45 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk X-IMAPbase: 1144781894 151 Status: O X-Status: X-Keywords: X-UID: 1 Vaughan Pratt asked about: > undirected graphs ... as presheaves on the full subcategory 1 and > 2 of Set? It is the 2-truncation of "symmetric simplicial sets" as presheaves on finite cardinals, cf (*). Curiously, symmetric simplicial sets have been rarely considered. Even if simplicial complexes (well-known!) are a symmetric notion and have a natural embedding in symmetric simplicial sets. While simplicial sets are a directed notion, used as an undirected one in classical Algebraic Topology. (*) M. Grandis, Finite sets and symmetric simplicial sets, Theory Appl. Categ. 8 (2001), No. 8, 244-252. Marco Grandis From rrosebru@mta.ca Wed Mar 1 19:33:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Mar 2006 19:33:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FEanx-00029X-Hp for categories-list@mta.ca; Wed, 01 Mar 2006 19:32:25 -0400 To: categories@mta.ca Subject: categories: Re: Undirected graphs citation Date: Wed, 01 Mar 2006 16:29:08 -0500 From: wlawvere@buffalo.edu Message-ID: <1141248548.44061224070b2@mail2.buffalo.edu> References: <440335A6.3050806@cs.stanford.edu> In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 Why the "curious" omission of this topos from most discussions of combinatorial topology ? The introduction of ordered simplices by Eilenberg 60 years ago is usually explained in subjective terms like the annoying extra degeneracies that had to be eliminated in the previous theory . However there is a objective requirement clearly pointed out by Gabriel & Zisman. (They spoke of Kelley spaces but that was a mistake due to Kelley's excellent exposition of the k-spaces of Hurewicz). The requirement is that geometric realization r (left adjoint to "singular" s) be left exact, so that products and equations on fleshed out spaces can be the reflection simply of the same operations on the combinatorial models. If we construe spaces as Johnstone did (or in many other possible ways) as forming themselves a topos, the the above requirement is simply that r/s constitute a geometric morphism of toposes. To understand the qualitative distinction between the possible codomain combinatorial toposes, it is very helpful to note their role as CLASSIFYING toposes. That role is not always easy to grasp in the specific if one starts with primitives and axioms for a first order theory, tries to present the corresponding Lindenbaum category, then takes sheaves on that, etc. Fortunately in some cases one can bypass that presentation process because the resulting small category is already well known. Preheaves on the category of non-empty finite posets clearly classify arbitrary non-trivial distributive lattices in any Grothendieck topos. In other words an s/r teory can be based on an "interval" object in spaces that has a DL structure. If we want that to factor through the subtopos of simplicial sets, we note that the latter is the classifier for those special DLs that are totally ordered, which translates geometrically to a condition on an interval that the square is a union of two triangles; since "union" depends on the topology of space, that condition is often not true for DLs that at first glance look like intervals. There are other relevant subtoposes (ie positive classes of DLs) for example those for which the canonical map from the generic one I to the truth-object is actually a homomorphism with respect to both lattice operations. But the presheaves on non-empty finite sets is the subtopos that classifies Boolean algebras. The generic BA is the obvious one. It does not really describe well its relation to the total orderings to call it the "synmmetric version". As the one contains all small categories, this one analogously contains all groupoids. It can still receive an r/s pair as required if only a space with a BA structure is used as the "interval". The natural choice for that is the in finite dimensional sphere, which indeed has a continuous BA structure that contains the usual interval as a subDL. If only this had been known 60 years ago, we could have done without the simplicial sets, for this singular theory reads to the same homotopy category. Note that any Grothendieck topos (including ss!) has a canonical BA object, hence enjoys a canonical r/s theory valued here. Quoting Marco Grandis : > Vaughan Pratt asked about: > > > undirected graphs ... as presheaves on the full subcategory 1 > and > > 2 of Set? > > It is the 2-truncation of "symmetric simplicial sets" as presheaves > on finite cardinals, cf (*). > > Curiously, symmetric simplicial sets have been rarely considered. > Even if simplicial complexes (well-known!) are a symmetric notion > and > have a natural embedding in symmetric simplicial sets. While > simplicial sets are a directed notion, used as an undirected one in > classical Algebraic Topology. > > (*) M. Grandis, Finite sets and symmetric simplicial sets, Theory > Appl. Categ. 8 (2001), No. 8, 244-252. > > > Marco Grandis > > > > From rrosebru@mta.ca Wed Mar 1 19:33:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Mar 2006 19:33:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FEalK-000215-QJ for categories-list@mta.ca; Wed, 01 Mar 2006 19:29:42 -0400 Date: Wed, 01 Mar 2006 12:58:14 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu Organization: Stanford University MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Undirected graphs citation Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 3 Marco Grandis wrote: > It is the 2-truncation of "symmetric simplicial sets" as presheaves > on finite cardinals, cf (*). Marco, thanks for that, this is really nice. It hadn't occurred to me to extend undirected graphs to higher dimensions but ... of course! While "symmetric" is technically correct terminology here (and indeed graph theorists often define undirected graphs as the symmetric case of directed graphs), "undirected" conveys the appropriate intuition that the edges and higher-dimensional cells have no specific orientation. Whereas the automorphism group of a directed n-cell is the trivial group, that of an undirected n-cell is S_N where N=n+1, i.e. undirected n-cells are permitted to "flop around" in all N! possible ways. Moreover the group as a whole behaves like a single cell with regard to identification: if one of the N! copies of an undirected edge is identified with a copy of another undirected edge, all copies are identified bijectively, i.e. the two undirected cells are identified. So without taking issue with Marco's terminology "symmetric" here, since it is correct and natural, I would nevertheless like to suggest that in the context of simplicial complexes, and with ordinary graphs as a precedent, that "undirected" be considered an acceptable synonym for "symmetric". But that connection leads to another that hadn't previously occurred to me (though again this is unlikely to be news to at least some). This is the question of an appropriate language for the signature of simplicial complexes in general. Each operation can be named with a lambda-calculus term of the form \xyz.xyzzy, that is, a string of (distinct of course) variables followed by another string of the same variables with repetitions or omissions allowed. Dually to undirected simplicial complexes being a special case of (directed) simplicial complexes, the language for the latter is the special case of that for the former in which the body of the lambda term preserves the order of the formal parameter list; the smallest term thus disallowed is \xy.yx. In particular s and t (source and target) arise as respectively \xy.x and \xy.y: given an edge, bind x and y to its source and target respectively and return the designated vertex. Similarly \x.xx denotes the distinguished self-loop at a given vertex x (these being reflexive graphs since we allow contraction). The lambda terms with N=n+1 parameters have as domain the set of n-cells. The one operation that undirected graphs have that is absent in the general directed case is \xy.yx, which names the other member of the group of automorphic copies of an undirected edge. These two copies always travel together (literally as a group), justifying the intuition that the group of both of them constitutes a single edge (or n-cell). For general n these copies of a given cell are named by the linear lambda terms, those with exactly one occurrence of each formal parameter. Any given cell of a graph attaches to the rest of the graph at various points around that cell, but graph homomorphisms cannot disturb those points of attachment or incidence, though it can certainly map the cell to any of the N! isomorphic copies of itself. It should be pointed out that "undirected graph" as a "special case" of "directed graph" has its syntactic rather than semantic meaning here, in the sense that UGraph (undirected graphs) does not embed in DGraph (directed graphs), at least not in the expected way. Consider a graph with two vertices x,y, two edges from x to y, and two edges from y to x. If a graph homomorphism identifies the two edges from x to y, it need not identify the other two edges in DGraph, but it does need to identify them in UGraph. Unless I've overlooked some subtlety, 2-UGraph does however embed in the expected way in 2-DGraph, where 2 = {0,1} (= V in enriched parlance) are the possible cardinalities of "homsets", i.e. at most one edge in each direction. This is because the implicit pairing in 2-DGraph perfectly mimics the explicit pairing in 2-UGraph. This would explain why graph theorists, who usually work in 2-DGraph, encounter no ambiguity of the Set-UGraph < Set-DGraph kind when they define an undirected graph as simply a symmetric graph, one with no one-way streets. Vaughan Pratt Marco Grandis wrote: > Vaughan Pratt asked about: > >> undirected graphs ... as presheaves on the full subcategory 1 and >> 2 of Set? > > > > It is the 2-truncation of "symmetric simplicial sets" as presheaves > on finite cardinals, cf (*). > > Curiously, symmetric simplicial sets have been rarely considered. > Even if simplicial complexes (well-known!) are a symmetric notion and > have a natural embedding in symmetric simplicial sets. While > simplicial sets are a directed notion, used as an undirected one in > classical Algebraic Topology. > > (*) M. Grandis, Finite sets and symmetric simplicial sets, Theory > Appl. Categ. 8 (2001), No. 8, 244-252. > > > Marco Grandis > From rrosebru@mta.ca Wed Mar 1 19:33:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Mar 2006 19:33:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FEamT-00025B-U1 for categories-list@mta.ca; Wed, 01 Mar 2006 19:30:53 -0400 From: Thomas Streicher Message-Id: <200603011517.k21FHYEH009161@fb04209.mathematik.tu-darmstadt.de> Subject: categories: Re: intuitionism and disjunction To: categories@mta.ca Date: Wed, 1 Mar 2006 16:17:34 +0100 (CET) X-Mailer: ELM [version 2.4ME+ PL100 (25)] MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 4 W.r.t. to the discussion on intuitionism and disjunction I would like to remark that this relation depends heavily on whether the ambient setting is predicative or not. In an impredicative setting like toposes (or the Calculus of Constructions) one may define disjunction and existential quantification `a la Russell-Prawitz by quantification over \Omega (that's what (some) logicians call impredicative, i.e. the possibility to quantify over propositions and thus also predicates). However in a first order setting this is not possible. E.g. if one considers Heyting arithmetic with ist usual axioms and as logic the \neg,\wedge,->,\forall fragment of first order logic then all formulas are provably double negation closed. This shows that disjunction and existence are not definable from the rest via first order logic. So far the logical side. What Paul Levy had in mind was slightly different as I understand it. Categories of domains (cpos with \bot) are cartesian closed and have weak finite sums but certainly no proper finite sums (since every map has a fixpoint). For modelling his CBPV paradigm he needs in addition a category of predomains which is bicartesian closed. The former is an example of a ccc which is not bicartesian but can be embedded into the latter which is. So one doesn't get for free the finite sums from a ccc which is problematic both in logic and semantics of computation. Thomas From rrosebru@mta.ca Thu Mar 2 11:07:28 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Mar 2006 11:07:28 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FEpGy-0003LK-Sc for categories-list@mta.ca; Thu, 02 Mar 2006 10:59:20 -0400 Date: Thu, 02 Mar 2006 11:13:18 +0100 From: "Clemens.BERGER" Reply-To: cberger@math.unice.fr Organization: Laboratoire J.-A. =?ISO-8859-1?Q?Dieudonn=E9?= User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.6) Gecko/20040116 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Undirected graphs citation References: <440335A6.3050806@cs.stanford.edu> <1141248548.44061224070b2@mail2.buffalo.edu> In-Reply-To: <1141248548.44061224070b2@mail2.buffalo.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 5 wlawvere@buffalo.edu wrote: >Why the "curious" omission of this topos from most discussions of combinatorial topology ? > >The introduction of ordered simplices by Eilenberg 60 years ago is usually explained in >subjective terms like the annoying extra degeneracies that had to be eliminated in the >previous theory . However there is a objective requirement clearly pointed out by Gabriel & >Zisman. (They spoke of Kelley spaces but that was a mistake due to Kelley's excellent >exposition of the k-spaces of Hurewicz). The requirement is that geometric realization r (left >adjoint to "singular" s) be left exact, so that products and equations on fleshed out spaces >can be the reflection simply of the same operations on the combinatorial models. > >If we construe spaces as Johnstone did (or in many other possible ways) as forming >themselves a topos, the the above requirement is simply that r/s constitute a geometric >morphism of toposes. To understand the qualitative distinction between the possible >codomain combinatorial toposes, it is very helpful to note their role as CLASSIFYING toposes. >That role is not always easy to grasp in the specific if one starts with primitives and axioms >for a first order theory, tries to present the corresponding Lindenbaum category, then takes >sheaves on that, etc. Fortunately in some cases one can bypass that presentation process >because the resulting small category is already well known. > >Preheaves on the category of non-empty finite posets clearly classify arbitrary non-trivial >distributive lattices in any Grothendieck topos. In other words an s/r teory can be based on >an "interval" object in spaces that has a DL structure. If we want that to factor through the >subtopos of simplicial sets, we note that the latter is the classifier for those special DLs that >are totally ordered, which translates geometrically to a condition on an interval that the >square is a union of two triangles; since "union" depends on the topology of space, that >condition is often not true for DLs that at first glance look like intervals. > >There are other relevant subtoposes (ie positive classes of DLs) for example those for which >the canonical map from the generic one I to the truth-object is actually a homomorphism >with respect to both lattice operations. > >But the presheaves on non-empty finite sets is the subtopos that classifies Boolean algebras. >The generic BA is the obvious one. It does not really describe well its relation to the total >orderings to call it the "synmmetric version". As the one contains all small categories, this >one analogously contains all groupoids. It can still receive an r/s pair as required if only a >space with a BA structure is used as the "interval". The natural choice for that is the in finite >dimensional sphere, which indeed has a continuous BA structure that contains the usual >interval as a subDL. If only this had been known 60 years ago, we could have done without >the simplicial sets, for this singular theory reads to the same homotopy category. Note that >any Grothendieck topos (including ss!) has a canonical BA object, hence enjoys a canonical >r/s theory valued here. > >Quoting Marco Grandis : > >>Vaughan Pratt asked about: >> >>> undirected graphs ... as presheaves on the full subcategory 1 >>and 2 of Set? >>> >>It is the 2-truncation of "symmetric simplicial sets" as presheaves >>on finite cardinals, cf (*). >> >>Curiously, symmetric simplicial sets have been rarely considered. >>Even if simplicial complexes (well-known!) are a symmetric notion >>and >>have a natural embedding in symmetric simplicial sets. While >>simplicial sets are a directed notion, used as an undirected one in >>classical Algebraic Topology. >> >>(*) M. Grandis, Finite sets and symmetric simplicial sets, Theory >>Appl. Categ. 8 (2001), No. 8, 244-252. >> >> >>Marco Grandis > A comment on combinatorial models for spaces : there is the notion of a CW-poset (due to Bjoerner) which is a poset that can be identified with the poset of cells of a regular CW-complex; those posets have very special properties, but from a homotopy point of view, every space is weakly homotopy equivalent to a regular CW-complex, so every space has a ``CW-poset-model''. Taking the nerve of this poset yields a ``simplicial-set-model'' of the same space; geometrically, the passage from the poset to its nerve corresponds to barycentric subdivision. Now, there is an old paper by Daniel Kan (Amer. J. Math. 79 (1957) 449-476, section 10), where he shows that the barycentric subdivision of a simplicial set actually has the structure of a symmetric simplicial set. So, to some extent, symmetric simplicial sets are doubly subdivided regular CW-complexes. What I wanted to point out is that in the progression CW-poset->simplicial set->symmetric simplicial set, the combinatorial information about the space is not increasing, but rather decreasing, which is one possible argument in favour of simplicial sets versus symmetric simplicial sets. Another argument for the category of simplicial sets is that it is a topos which contains the category of small categories as a full reflective monoidal subcategory. A third argument is the ubiquitous occurence of simplicial cotriple resolutions. In his pursuit of stacks, Grothendieck raises the question of what is the structure on a small category A, which ensures that the presheaf topos A^ may be endowed with a homotopy theory equivalent to the homotopy theory of spaces (such a category is called a test-category) or with the homotopy theory of spaces above a given space (such a category is called a local-test-category). In the presence of an r/s adjunction like in Lavwere's message above, the criteria of being local-test or test are easily spelled out: A is local-test iff for each object a of A, the projection A[a]\times \Omega_A\to A[a] realises to a weak equivalence, where \Omega_A is the subobject classifier of A^ and A[a] is the presheaf represented by a. A is test iff A is local-test and the nerve of A is weakly contractible. With best regards, Clemens Berger. From rrosebru@mta.ca Thu Mar 2 11:07:28 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Mar 2006 11:07:28 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FEpKS-0003e4-Ja for categories-list@mta.ca; Thu, 02 Mar 2006 11:02:56 -0400 Date: Thu, 2 Mar 2006 10:50:26 -0400 (AST) From: Bob Rosebrugh To: categories Subject: categories: Graphical Database for Category Theory (GDCT 3.0) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 6 Dear Colleagues, This is to announce a new release of a Java application for storage and display of finitely presented categories and functors among them. GDCT is a tool for the creation, editing, display and storage of finitely presented categories. Categories can be opened and saved from local files as well as loaded from a user-specified server. Open categories can be displayed graphically, and the graphical display can be manipulated (in 3 dimensions) and stored. Several tools for testing properties of objects and arrows can be applied. Functors between stored categories can also be created and stored. Open functors can be displayed as diagrams and with a domain/codomain display. This version of GDCT contains some enhancements: - Moved to a desktop-style interface, where multiple categories and functors can be opened in their own respective windows - Implemented pullback, pushout, create product, create sum, and partial order category tools. A built-in help system describes how to use GDCT. The application is available for download at http://mathcs.mta.ca/research/rosebrugh/gdct/ and is linked from http://www.mta.ca/~rrosebru/ Instructions for setting up GDCT are are on the Web pages: - users may download a zipped archive providing the application; - source code is available. Graphical Database for Category Theory was developed by Jeremy Bradbury, Ian Rutherford, M. Graves, J. Tweedle and R. Rosebrugh with support from NSERC Canada. We are interested in any response from users, and would also appreciate receiving any reports of difficulties in function or usability. Bob Rosebrugh From rrosebru@mta.ca Thu Mar 2 22:32:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Mar 2006 22:32:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FF00O-0006Sh-51 for categories-list@mta.ca; Thu, 02 Mar 2006 22:26:56 -0400 Date: Thu, 2 Mar 2006 13:32:32 -0500 (EST) From: F W Lawvere X-Sender: wlawvere@joxer.acsu.buffalo.edu Reply-To: wlawvere@acsu.buffalo.edu To: categories@mta.ca Subject: categories: Re: Undirected graph citation Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 7 As Clemens Berger reminds us, the category of small categories is a reflective subcategory of simplicial sets, with a reflector that preserves finite products. But as I mentioned, there is a similar "advantage" for the Boolean algebra classifier (=presheaves on non-empty finite cardinals, or "symmetric" simplicial sets): The category of small groupoids is reflective in this topos, with the reflector preserving finite products. Thus the Poincare' groupoid of a simplicial complex is directly available. (The simplicial complexes are merely the objects generated weakly by their points, a relation which defines a cartesian closed reflective subcategory of any topos.) It is not clear how one is to measure the loss or gain of combinatorial information in composing the various singular and realization functors between these different models. Is there such a measure? Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Fri Mar 3 14:53:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 14:53:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFFHE-0003uD-4w for categories-list@mta.ca; Fri, 03 Mar 2006 14:45:20 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: Re: Undirected graph citation Date: Fri, 3 Mar 2006 10:04:57 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 Undirected versus directed Going along with the last messages of C Berger and FW Lawvere, I would like to list the following parallel notions, undirected versus directed. Of course, it is not a question of saying which is better, but only of separating them to make things clearer. --- Undirected: - symmetric simplicial sets (sss) - simplicial complexes (classical) = sets with distinguished subsets = sss where each simplex is determined by its vertices - undirected graphs - groupoids (fundamental groupoids) - abelian groups (homology groups) - spaces - classical metric spaces - undirected algebraic topology --- Directed: - simplicial sets - "directed simplicial complexes" (not classical) = sets with distinguished words = simplicial sets where each simplex is determined by (the family of) its vertices - directed graphs - categories (fundamental categories) - preordered abelian groups ("directed homology groups") - "directed spaces" (preordered, locally preordered, etc.) - generalised metric spaces (Lawvere) - "directed algebraic topology" --- Spaces are plainly an undirected structure. Note that their singular simplicial set already has a natural symmetric structure (by "permuting vertices" on tetrahedra); there is no need of symmetrising it and loosing information. Classical algebraic topology is mostly undirected (since spaces, groupoids, abelian groups are so), but it has also used directed structures, like simplicial sets, for undirected purposes: simulating spaces and computing undirected algebraic structures, like groupoids and homology groups. The study of "directed algebraic topology" is quite recent. (There are some papers on that in my web page, from which one can see the literature; present applications are concerned with concurrency and rewriting. But the general aim should be modeling non-reversible phenomena.) Finally, I would like to point out - once more - that the term "simplicial complex" is highly confusing: this notion (as Bill recalls) is a simplified version of a symmetric simplicial set, while the corresponding simplified version of a simplicial set is a "set with distinguished words" (the reflexive cartesian closed subcategory of "objects determined by their vertices", in the presheaf topos of simplicial sets). But I have noticed that people can get nervous about terminology, and it might be better to forget about this last point. Marco Grandis From rrosebru@mta.ca Fri Mar 3 14:53:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 14:53:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFFK5-0004Ai-Ps for categories-list@mta.ca; Fri, 03 Mar 2006 14:48:17 -0400 Message-ID: <004201c63eec$282ab7d0$0b00000a@C3> From: "George Janelidze" To: References: Subject: categories: Re: Undirected graph citation Date: Fri, 3 Mar 2006 19:59:01 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 9 I am not sure if I really understand what is the target of this discussion, but I would like to make some comments to Bill's messages: The Poincare' groupoid is (up to an equivalence) nothing but the largest Galois groupoid, and it is directly available as soon as one has what I call Galois structure in my several papers, if we assume that every object of the ground category has a universal covering. This is certainly the case for every locally connected topos with coproducts and enough projectives. Therefore this is certainly the case for every presheaf topos. Therefore what Bill means by "directly available" should be not "available without going through geometric realization" but just "can be calculated as the result of reflection" (probably this is exactly what Bill had in mind). Moreover, it was Grothendieck's observation that Galois/fundamental groupoids are to be defined as quotients of certain equivalence relations - in fact kernel pairs, and this observation was used by many authors in topos theory and elsewhere; my own observation (1984) then was that one can make Galois theory purely categorical by using not "quotients" but "images under a left adjoint" (the first prototype for me was actually not Grothendieck's but Andy Magid's "componentially locally strongly separable" Galois theory of commutative rings). What I am trying to conclude is that the Galois/fundamental groupoids actually arise not from anything simplicial but from abstract category theory: it is just a result of a game with adjoint functors between categories with pullbacks. In another message Bill says: "A similar lacuna of explicitness occurs in many papers on Galois theory where pregroupoids are an intermediate step ; the description of the pregroupoid concept is really just a presentation of the monoid of endomaps of the 4-element set..." Assuming that everyone understands that this is not about classical Galois theory (I don't think somebody like J.-P. Serre ever mentions pregroupoids) and not about what Anders Kock calls pregroupoids, let me again return to the categorical Galois theory: If p : E ---> B is an "extension" in a category C, R its kernel pair, and F : C ---> X the left adjoint involved in a given Galois theory, then one wants to define the Galois groupoid Gal(E,p) as F(R) = the image of R under F (I usually write I instead of F, but in an email message this does not look good...). But if our extension p : E ---> B is not normal, then, since F usually does not preserve pullbacks, F(R) is not a groupoid - it is a weaker structure, the "equational part" of groupoid structure. This weaker structure is still good enough to define its internal actions in X and these internal actions classify covering objects over B split by (E,p). Hence this weaker structure needs a name and I called it "pregroupoid" (I did not know that this term was already overused for almost the same and for unrelated concepts). I cannot speak for everyone, but for my own purposes there are actually several possible candidates for the notion of pregroupoid and half of them can certainly be defined as monoid actions for a specific monoid, like the one Bill mentions. However, in each case we deal with a "very small" category actions and it is a triviality to observe that that category can be replaced with a monoid. Essentially, what you need is to check that the terminal object (in your category of pregroupoids) has either no proper subobjects or only one such, which must be initial. In this observation - due to Max Kelly, about the categories monadic over powers of Sets being monadic over Sets, one usually says "strictly initial"; but we can omit "strictly" here since it is about a topos. George Janelidze ----- Original Message ----- From: "F W Lawvere" To: Sent: Thursday, March 02, 2006 8:32 PM Subject: categories: Re: Undirected graph citation > > As Clemens Berger reminds us, the category of small categories > is a reflective subcategory of simplicial sets, with a reflector that > preserves finite products. But as I mentioned, there is a similar > "advantage" for the Boolean algebra classifier (=presheaves on non-empty > finite cardinals, or "symmetric" simplicial sets): > The category of small groupoids is reflective in this topos, with the > reflector preserving finite products. Thus the Poincare' groupoid of a > simplicial complex is directly available. (The simplicial complexes are > merely the objects generated weakly by their points, a relation which > defines a cartesian closed reflective subcategory of any topos.) > > It is not clear how one is to measure the loss or gain of combinatorial > information in composing the various singular and realization functors > between these different models. Is there such a measure? > > > Bill Lawvere > > ************************************************************ > F. William Lawvere > Mathematics Department, State University of New York > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > Tel. 716-645-6284 > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > ************************************************************ > > > > > > > From rrosebru@mta.ca Fri Mar 3 14:53:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 14:53:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFFGi-0003rN-GT for categories-list@mta.ca; Fri, 03 Mar 2006 14:44:48 -0400 MIME-Version: 1.0 From: "Dr. Cyrus F Nourani" To: categories@mta.ca Date: Thu, 02 Mar 2006 13:16:45 -0500 Subject: categories: Re: Undirected graphs Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-1" Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 10 This is very interesting. The enclosed papers were in part=20 indicating that models can be designed with undirected graphs on Hasse diagrams where disenfrachised computing can create=20 models in pieces. Further insights would be great. Cyrus Functors Computing Hasse Diagram Models February 17, 1997, MiniConferece, Maine, April 1997. Functorial Hasse Models=20 SLK 2002, Eurpean Sumer Logic Colloquium, Munster, Germany, July 2002 wwwmath.uni-muenster.de/LC2002/presentedbytitle.html > ----- Original Message ----- > From: "Vaughan Pratt" > To: categories@mta.ca > Subject: categories: Re: Undirected graphs citation > Date: Wed, 01 Mar 2006 12:58:14 -0800 >=20 >=20 > Marco Grandis wrote: > > It is the 2-truncation of "symmetric simplicial sets" as presheaves > > on finite cardinals, cf (*). >=20 > Marco, thanks for that, this is really nice. It hadn't occurred to me > to extend undirected graphs to higher dimensions but ... of course! >=20 > While "symmetric" is technically correct terminology here (and indeed > graph theorists often define undirected graphs as the symmetric case of > directed graphs), "undirected" conveys the appropriate intuition that > the edges and higher-dimensional cells have no specific orientation. > Whereas the automorphism group of a directed n-cell is the trivial > group, that of an undirected n-cell is S_N where N=3Dn+1, i.e. undirected > n-cells are permitted to "flop around" in all N! possible ways. > Moreover the group as a whole behaves like a single cell with regard to > identification: if one of the N! copies of an undirected edge is > identified with a copy of another undirected edge, all copies are > identified bijectively, i.e. the two undirected cells are identified. >=20 > So without taking issue with Marco's terminology "symmetric" here, since > it is correct and natural, I would nevertheless like to suggest that in > the context of simplicial complexes, and with ordinary graphs as a > precedent, that "undirected" be considered an acceptable synonym for > "symmetric". >=20 > But that connection leads to another that hadn't previously occurred to > me (though again this is unlikely to be news to at least some). This is > the question of an appropriate language for the signature of simplicial > complexes in general. >=20 > Each operation can be named with a lambda-calculus term of the form > \xyz.xyzzy, that is, a string of (distinct of course) variables followed > by another string of the same variables with repetitions or omissions > allowed. Dually to undirected simplicial complexes being a special case > of (directed) simplicial complexes, the language for the latter is the > special case of that for the former in which the body of the lambda term > preserves the order of the formal parameter list; the smallest term thus > disallowed is \xy.yx. >=20 > In particular s and t (source and target) arise as respectively \xy.x > and \xy.y: given an edge, bind x and y to its source and target > respectively and return the designated vertex. Similarly \x.xx denotes > the distinguished self-loop at a given vertex x (these being reflexive > graphs since we allow contraction). The lambda terms with N=3Dn+1 > parameters have as domain the set of n-cells. >=20 > The one operation that undirected graphs have that is absent in the > general directed case is \xy.yx, which names the other member of the > group of automorphic copies of an undirected edge. These two copies > always travel together (literally as a group), justifying the intuition > that the group of both of them constitutes a single edge (or n-cell). > For general n these copies of a given cell are named by the linear > lambda terms, those with exactly one occurrence of each formal > parameter. Any given cell of a graph attaches to the rest of the graph > at various points around that cell, but graph homomorphisms cannot > disturb those points of attachment or incidence, though it can certainly > map the cell to any of the N! isomorphic copies of itself. >=20 > It should be pointed out that "undirected graph" as a "special case" of > "directed graph" has its syntactic rather than semantic meaning here, in > the sense that UGraph (undirected graphs) does not embed in DGraph > (directed graphs), at least not in the expected way. Consider a graph > with two vertices x,y, two edges from x to y, and two edges from y to x. > If a graph homomorphism identifies the two edges from x to y, it need > not identify the other two edges in DGraph, but it does need to identify > them in UGraph. >=20 > Unless I've overlooked some subtlety, 2-UGraph does however embed in the > expected way in 2-DGraph, where 2 =3D {0,1} (=3D V in enriched parlance) = are > the possible cardinalities of "homsets", i.e. at most one edge in each > direction. This is because the implicit pairing in 2-DGraph perfectly > mimics the explicit pairing in 2-UGraph. This would explain why graph > theorists, who usually work in 2-DGraph, encounter no ambiguity of the > Set-UGraph < Set-DGraph kind when they define an undirected graph as > simply a symmetric graph, one with no one-way streets. >=20 > Vaughan Pratt >=20 > Marco Grandis wrote: >=20 > > Vaughan Pratt asked about: > > > >> undirected graphs ... as presheaves on the full subcategory 1 and > >> 2 of Set? > > > > > > > > It is the 2-truncation of "symmetric simplicial sets" as presheaves > > on finite cardinals, cf (*). > > > > Curiously, symmetric simplicial sets have been rarely considered. > > Even if simplicial complexes (well-known!) are a symmetric notion and > > have a natural embedding in symmetric simplicial sets. While > > simplicial sets are a directed notion, used as an undirected one in > > classical Algebraic Topology. > > > > (*) M. Grandis, Finite sets and symmetric simplicial sets, Theory > > Appl. Categ. 8 (2001), No. 8, 244-252. > > > > > > Marco Grandis > > > --=20 _______________________________________________ Search for businesses by name, location, or phone number. -Lycos Yellow Pa= ges http://r.lycos.com/r/yp_emailfooter/http://yellowpages.lycos.com/default.as= p?SRC=3Dlycos10 --_----------=_114132340527320-- From rrosebru@mta.ca Fri Mar 3 14:53:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 14:53:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFFHn-0003x2-DK for categories-list@mta.ca; Fri, 03 Mar 2006 14:45:55 -0400 Date: Fri, 3 Mar 2006 12:41:29 +0000 (GMT) From: Paul B Levy To: categories@mta.ca Subject: categories: Re: intuitionism and disjunction In-Reply-To: <200603011517.k21FHYEH009161@fb04209.mathematik.tu-darmstadt.de> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 On Wed, 1 Mar 2006, Thomas Streicher wrote: > W.r.t. to the discussion on intuitionism and disjunction I would like to > remark that this relation depends heavily on whether the ambient setting is > predicative or not. In an impredicative setting like toposes (or the Calculus > of Constructions) one may define disjunction and existential quantification > `a la Russell-Prawitz by quantification over \Omega (that's what (some) > logicians call impredicative, i.e. the possibility to quantify over > propositions and thus also predicates). In the topos situation, we are really concerned with intuitionistic provability, not with proof equality. A model of intuitionistic provability is a bi-Heyting algebra. I agree that, in the provability setting, disjunction can be encoded in terms of impredicative universal quantification, However, when we are concerned with proof-equality, this encoding doesn't work as a translation of equational theories. I mean that the eta-law for the (encoded) sum types cannot be proved in the beta-eta-theory for polymorphism. (If we incorporate Plotkin-Abadi logic into the target theory into, then the encoding works, if I recall correctly, but the target theory isn't equational any more.) > However in a first order setting this is not possible. E.g. if one considers > Heyting arithmetic with ist usual axioms and as logic the > \neg,\wedge,->,\forall fragment of first order logic then all formulas are > provably double negation closed. This shows that disjunction and existence > are not definable from the rest via first order logic. Agreed. > So far the logical side. What Paul Levy had in mind I didn't have CBPV in mind; that is a calculus for effects (such as divergence). My point was just that intuitionistic logic corresponds to "pure" type theory: no divergence or other effects. Therefore a bi-ccc is required to model it. Categories that are suitable for modelling effects, such as the category of domains, won't do for modelling intuitionistic proofs, Admittedly, if we follow John's suggestion of weakening beta-eta laws into morphisms a la Neil Ghani, then we won't need a bi-ccc any more. All I am saying is that, *if* we require the eta law for function types, then we should also require it for sum types. Paul From rrosebru@mta.ca Fri Mar 3 17:00:13 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 17:00:13 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFHLE-0002Ed-31 for categories-list@mta.ca; Fri, 03 Mar 2006 16:57:36 -0400 Date: Fri, 3 Mar 2006 15:20:49 -0500 (EST) From: Robert Seely To: Categories List Subject: categories: Re: intuitionism and disjunction In-Reply-To: Message-ID: References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 12 On Fri, 3 Mar 2006, Paul B Levy wrote: > Admittedly, if we follow John's suggestion of weakening beta-eta laws into > morphisms a la Neil Ghani, then we won't need a bi-ccc any more. All I am > saying is that, *if* we require the eta law for function types, then we > should also require it for sum types. A couple of remarks about this. First the trivial one: "bi-ccc" is a misleading term (I've used it myself, with the same confusion induced in others), when all you mean is a ccc with coproducts. More importantly, when you look at the structure of proofs in intuitionist logic, you find that sum and product are not really dual - though I have to admit this is very much a matter of presentation. The problem is an old one: Zucker (in the early 70's I think) pointed out that normalization steps are not the same as cut elimination steps, particularly for disjunction. I looked at this in my 1977 thesis, using natural deduction (in the Gentzen-Prawitz formulation) for the logic (this has the advantage over the sequent calculus of giving a category without need for any equivalence relation on derivations). Taking eta and beta (one an expansion, the other a reduction) as 2-cells, I looked at the 2-categorical structure of the connectives (and quantifiers, though really forall and exists are very similar respectively to conjunction and disjunction, not surprisingly). It's easy to see that conjunction is a weak adjoint without any further structure, but disjunction is NOT - unless you add some permuting conversions (equivalences): in fact exactly the permuting rewrites Prawitz had already identified earlier. And function types are a mess in this setting, unless you make the conjunction a real product (again by introducing equivalences - this time just beta and eta). You can find a sketch of all this in two extended abstracts I published: one in the Durham 1977 Category Meeting ("Applications of sheaves ...", SLNM 753), and the other in LICS 1987. (Both are available on my webpage.) (BTW - this was all done long before Ghani looked at the matter ... Barry Jay also subsequently looked at this. But the higher dimensional structure of intuitionist logic is still largely un-studied, and merits serious attention.) -= rags =- -- From rrosebru@mta.ca Sun Mar 5 10:10:25 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 05 Mar 2006 10:10:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFto2-0002W8-AI for categories-list@mta.ca; Sun, 05 Mar 2006 10:01:54 -0400 Message-ID: <003801c6404f$70d2e020$9ea24e51@brown1> From: "Ronald Brown" To: Subject: categories: Alexander Grothendieck on `speculation' Date: Sun, 5 Mar 2006 12:21:31 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 13 I am in the process of editing my correspondence 1982-1991 with AG which has been latexed through George Malsiniotis for appearing as part of an Appendix to a published edition of `Pursuing Stacks'. I came across the following extract which seemed to me to contain points of general interest about mathematical methodology and sociology, so I give this below, to invite comments. Ronnie Brown www.bangor.ac.uk/r.brown ---------------------------------------------------------------------------- from a letter dated 14 June, 1983, Your idea of writing a ``frantically speculative" article on groupoids seems to me a very good one. It is the kind of thing which has traditionally been lacking in mathematics since the very beginnings, I feel, which is one big drawback in comparison to all other sciences, as far as I know. Of course, no creative mathematician can afford not to ``speculate", namely to do more or less daring guesswork as an indispensable source of inspiration. The trouble is that, in obedience to a stern tradition, almost nothing of this appears in writing, and preciously little even in oral communication. The point is that the disrepute of ``speculation" or ``dream" is such, that even as a strictly private (not to say secret!) activity, it has a tendency to vegetate - much like the desire and drive of love and sex, in too repressive an environment. Despite the ``repression", in the one or two years before I unexpectedly was led to withdraw from the mathematical milieu and to stop publishing, it was more or less clear to me that, besides going on pushing ahead with foundational work in SGA and EGA, I was going to write a wholly science-fiction kind [of] book on ``motives'', which was then the most fascinating and mysterious mathematical being I had come to meet so far. As my interests and my emphasis have somewhat shifted since, I doubt I am ever going to write this book - still less anyone else is going to, presumably. But whatever I am going to write in mathematics, I believe a major part of it will be ``speculation" or ``fiction", going hand in hand with painstaking, down-to-earth work to get hold of the right kind of notions and structures, to work out comprehensive pictures of still misty landscapes. The notes I am writing up lately are in this spirit, but in this case the landscape isn't so remote really, and the feeling is rather that, as for the specific program I have been out for is concerned, getting everything straight and clear shouldn't mean more than a few years work at most for someone who really feels like doing it, maybe less. But of course surprises are bound to turn up on one's way, and while starting with a few threads in hand, after a while they may have multiplied and become such a bunch that you cannot possibly grasp them all, let alone follow. From rrosebru@mta.ca Sun Mar 5 10:10:25 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 05 Mar 2006 10:10:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFtnM-0002Ub-8N for categories-list@mta.ca; Sun, 05 Mar 2006 10:01:12 -0400 Date: Sat, 4 Mar 2006 20:21:20 -0500 (EST) From: F W Lawvere X-Sender: wlawvere@callisto.acsu.buffalo.edu Reply-To: wlawvere@acsu.buffalo.edu To: categories@mta.ca Subject: categories: Re: Undirected graph citation In-Reply-To: <004201c63eec$282ab7d0$0b00000a@C3> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 14 Dear George, Concerning undirected graphs, the Boolean algebra classifier, and the intermediate sub-topos that suffices for groupoids: The special feature of these toposes I wanted to emphasize is not that some of them can be generated by monoids, but rather (whether one splits idempotents or not) that the site of operators is itself a full subcategory of the category of sets. This is a small part of the point that Vaughn wanted to make, I believe. Having the direct visualization of this system of operators available as merely maps between certain small sets is a useful auxiliary to formal presentations of the d,s kind. As you mention, a basic way in which such presheaves can arise is by applying a non-exact functor F to a group; the fact that the exponents on the group are just these ordinary sets explains why we obtain an object in this sort of topos (which can serve as a presentation of another group, if desired). As noted in my unpublished (but widely distributed) paper on toposes generated by codiscrete objects, the Yoneda embedding in these cases produces of course a full sub-category of a topos, one which looks exactly like (a piece of) the category of sets; of course this is not the discrete inclusion, but its dialectical opposite, the codiscrete one. Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Fri, 3 Mar 2006, George Janelidze wrote: > I am not sure if I really understand what is the target of this discussion, > but I would like to make some comments to Bill's messages: > > The Poincare' groupoid is (up to an equivalence) nothing but the largest > Galois groupoid, and it is directly available as soon as one has what I call > Galois structure in my several papers, if we assume that every object of the > ground category has a universal covering. This is certainly the case for > every locally connected topos with coproducts and enough projectives. > Therefore this is certainly the case for every presheaf topos. Therefore > what Bill means by "directly available" should be not "available without > going through geometric realization" but just "can be calculated as the > result of reflection" (probably this is exactly what Bill had in mind). > > Moreover, it was Grothendieck's observation that Galois/fundamental > groupoids are to be defined as quotients of certain equivalence relations - > in fact kernel pairs, and this observation was used by many authors in topos > theory and elsewhere; my own observation (1984) then was that one can make > Galois theory purely categorical by using not "quotients" but "images under > a left adjoint" (the first prototype for me was actually not Grothendieck's > but Andy Magid's "componentially locally strongly separable" Galois theory > of commutative rings). What I am trying to conclude is that the > Galois/fundamental groupoids actually arise not from anything simplicial but > from abstract category theory: it is just a result of a game with adjoint > functors between categories with pullbacks. > > In another message Bill says: "A similar lacuna of explicitness occurs in > many papers on Galois theory where pregroupoids are an intermediate step ; > the description of the pregroupoid concept is really just a presentation > of the monoid of endomaps of the 4-element set..." Assuming that everyone > understands that this is not about classical Galois theory (I don't think > somebody like J.-P. Serre ever mentions pregroupoids) and not about what > Anders Kock calls pregroupoids, let me again return to the categorical > Galois theory: > > If p : E ---> B is an "extension" in a category C, R its kernel pair, and F > : C ---> X the left adjoint involved in a given Galois theory, then one > wants to define the Galois groupoid Gal(E,p) as F(R) = the image of R under > F (I usually write I instead of F, but in an email message this does not > look good...). But if our extension p : E ---> B is not normal, then, since > F usually does not preserve pullbacks, F(R) is not a groupoid - it is a > weaker structure, the "equational part" of groupoid structure. This weaker > structure is still good enough to define its internal actions in X and these > internal actions classify covering objects over B split by (E,p). Hence this > weaker structure needs a name and I called it "pregroupoid" (I did not know > that this term was already overused for almost the same and for unrelated > concepts). I cannot speak for everyone, but for my own purposes there are > actually several possible candidates for the notion of pregroupoid and half > of them can certainly be defined as monoid actions for a specific monoid, > like the one Bill mentions. However, in each case we deal with a "very > small" category actions and it is a triviality to observe that that category > can be replaced with a monoid. Essentially, what you need is to check that > the terminal object (in your category of pregroupoids) has either no proper > subobjects or only one such, which must be initial. In this observation - > due to Max Kelly, about the categories monadic over powers of Sets being > monadic over Sets, one usually says "strictly initial"; but we can omit > "strictly" here since it is about a topos. > > George Janelidze > > ----- Original Message ----- > From: "F W Lawvere" > To: > Sent: Thursday, March 02, 2006 8:32 PM > Subject: categories: Re: Undirected graph citation > > > > > > As Clemens Berger reminds us, the category of small categories > > is a reflective subcategory of simplicial sets, with a reflector that > > preserves finite products. But as I mentioned, there is a similar > > "advantage" for the Boolean algebra classifier (=presheaves on non-empty > > finite cardinals, or "symmetric" simplicial sets): > > The category of small groupoids is reflective in this topos, with the > > reflector preserving finite products. Thus the Poincare' groupoid of a > > simplicial complex is directly available. (The simplicial complexes are > > merely the objects generated weakly by their points, a relation which > > defines a cartesian closed reflective subcategory of any topos.) > > > > It is not clear how one is to measure the loss or gain of combinatorial > > information in composing the various singular and realization functors > > between these different models. Is there such a measure? > > > > > > Bill Lawvere > > > > ************************************************************ > > F. William Lawvere > > Mathematics Department, State University of New York > > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > > Tel. 716-645-6284 > > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > > ************************************************************ > > > > > > > > > > > > > > > > > > > From rrosebru@mta.ca Mon Mar 6 09:27:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 09:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGFgU-00030s-Up for categories-list@mta.ca; Mon, 06 Mar 2006 09:23:34 -0400 Message-ID: <005d01c64089$1b437220$0b00000a@C3> From: "George Janelidze" To: References: Subject: categories: Re: Undirected graph citation Date: Sun, 5 Mar 2006 21:15:01 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 15 Dear Bill, Indeed, there were no monoids in Vaughan's original message of February 28, but since you have mentioned them in your message of March 1, and since you were talking there about "...lacuna of explicitness ... in many papers on Galois theory...", I simply wanted to say that: I do not see any relevance of these kinds of presentations in Galois theory (apart from the fact the internal pre-whatever-s in a category X form an X-valued presheaf category). On the other hand Galois theory is not the end of the World, and I think the beauty and importance of those your ideas is clear to everyone who saw them. Putting myself in risk of making my message boring, I would like to make one more remark concerning your last message and Galois theory: You say: "...applying a non-exact functor F to a group..." - true and fine, but I have actually mentioned F(R) for R being not a group, but another extreme case of a groupoid, namely an equivalence relation. What seems to be most amazing is, that, because F preserves not-all-but-some pullbacks, there are beautiful examples where R is an equivalence relation and F(R) is a group; in simple words, F creates a group out of nothing! The classical example, as you know, is: if R is the kernel pair of a universal covering map E ---> B of a "good" connected topological space B, and F is the functor sending ("good") topological spaces to the sets of their connected components, then F(R) is the fundamental group of B. The same thing is true in other Galois theories of course. George ----- Original Message ----- From: "F W Lawvere" To: Sent: Sunday, March 05, 2006 3:21 AM Subject: categories: Re: Undirected graph citation > > Dear George, > > Concerning undirected graphs, the Boolean algebra classifier, and > the intermediate sub-topos that suffices for groupoids: > > The special feature of these toposes I wanted to emphasize is not that > some of them can be generated by monoids, but rather (whether one splits > idempotents or not) that the site of operators is itself a full > subcategory of the category of sets. This is a small part of the point > that Vaughn wanted to make, I believe. Having the direct visualization of > this system of operators available as merely maps between certain small > sets is a useful auxiliary to formal presentations of the d,s kind. > As you mention, a basic way in which such presheaves can arise is by > applying a non-exact functor F to a group; the fact that the exponents on > the group are just these ordinary sets explains why we obtain an object > in this sort of topos (which can serve as a presentation of another group, > if desired). > > As noted in my unpublished (but widely distributed) paper on toposes > generated by codiscrete objects, the Yoneda embedding in these cases > produces of course a full sub-category of a topos, one which looks > exactly like (a piece of) the category of sets; of course this is not > the discrete inclusion, but its dialectical opposite, the codiscrete one. > > Bill > ************************************************************ > F. William Lawvere > Mathematics Department, State University of New York > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > Tel. 716-645-6284 > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > ************************************************************ > > > > On Fri, 3 Mar 2006, George Janelidze wrote: > > > I am not sure if I really understand what is the target of this discussion, > > but I would like to make some comments to Bill's messages: > > > > The Poincare' groupoid is (up to an equivalence) nothing but the largest > > Galois groupoid, and it is directly available as soon as one has what I call > > Galois structure in my several papers, if we assume that every object of the > > ground category has a universal covering. This is certainly the case for > > every locally connected topos with coproducts and enough projectives. > > Therefore this is certainly the case for every presheaf topos. Therefore > > what Bill means by "directly available" should be not "available without > > going through geometric realization" but just "can be calculated as the > > result of reflection" (probably this is exactly what Bill had in mind). > > > > Moreover, it was Grothendieck's observation that Galois/fundamental > > groupoids are to be defined as quotients of certain equivalence relations - > > in fact kernel pairs, and this observation was used by many authors in topos > > theory and elsewhere; my own observation (1984) then was that one can make > > Galois theory purely categorical by using not "quotients" but "images under > > a left adjoint" (the first prototype for me was actually not Grothendieck's > > but Andy Magid's "componentially locally strongly separable" Galois theory > > of commutative rings). What I am trying to conclude is that the > > Galois/fundamental groupoids actually arise not from anything simplicial but > > from abstract category theory: it is just a result of a game with adjoint > > functors between categories with pullbacks. > > > > In another message Bill says: "A similar lacuna of explicitness occurs in > > many papers on Galois theory where pregroupoids are an intermediate step ; > > the description of the pregroupoid concept is really just a presentation > > of the monoid of endomaps of the 4-element set..." Assuming that everyone > > understands that this is not about classical Galois theory (I don't think > > somebody like J.-P. Serre ever mentions pregroupoids) and not about what > > Anders Kock calls pregroupoids, let me again return to the categorical > > Galois theory: > > > > If p : E ---> B is an "extension" in a category C, R its kernel pair, and F > > : C ---> X the left adjoint involved in a given Galois theory, then one > > wants to define the Galois groupoid Gal(E,p) as F(R) = the image of R under > > F (I usually write I instead of F, but in an email message this does not > > look good...). But if our extension p : E ---> B is not normal, then, since > > F usually does not preserve pullbacks, F(R) is not a groupoid - it is a > > weaker structure, the "equational part" of groupoid structure. This weaker > > structure is still good enough to define its internal actions in X and these > > internal actions classify covering objects over B split by (E,p). Hence this > > weaker structure needs a name and I called it "pregroupoid" (I did not know > > that this term was already overused for almost the same and for unrelated > > concepts). I cannot speak for everyone, but for my own purposes there are > > actually several possible candidates for the notion of pregroupoid and half > > of them can certainly be defined as monoid actions for a specific monoid, > > like the one Bill mentions. However, in each case we deal with a "very > > small" category actions and it is a triviality to observe that that category > > can be replaced with a monoid. Essentially, what you need is to check that > > the terminal object (in your category of pregroupoids) has either no proper > > subobjects or only one such, which must be initial. In this observation - > > due to Max Kelly, about the categories monadic over powers of Sets being > > monadic over Sets, one usually says "strictly initial"; but we can omit > > "strictly" here since it is about a topos. > > > > George Janelidze > > > > ----- Original Message ----- > > From: "F W Lawvere" > > To: > > Sent: Thursday, March 02, 2006 8:32 PM > > Subject: categories: Re: Undirected graph citation > > > > > > > > > > As Clemens Berger reminds us, the category of small categories > > > is a reflective subcategory of simplicial sets, with a reflector that > > > preserves finite products. But as I mentioned, there is a similar > > > "advantage" for the Boolean algebra classifier (=presheaves on non-empty > > > finite cardinals, or "symmetric" simplicial sets): > > > The category of small groupoids is reflective in this topos, with the > > > reflector preserving finite products. Thus the Poincare' groupoid of a > > > simplicial complex is directly available. (The simplicial complexes are > > > merely the objects generated weakly by their points, a relation which > > > defines a cartesian closed reflective subcategory of any topos.) > > > > > > It is not clear how one is to measure the loss or gain of combinatorial > > > information in composing the various singular and realization functors > > > between these different models. Is there such a measure? > > > > > > > > > Bill Lawvere > > > > > > ************************************************************ > > > F. William Lawvere > > > Mathematics Department, State University of New York > > > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > > > Tel. 716-645-6284 > > > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > > > ************************************************************ > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > From rrosebru@mta.ca Mon Mar 6 09:27:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 09:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGFfX-0002wa-0u for categories-list@mta.ca; Mon, 06 Mar 2006 09:22:35 -0400 Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <003801c6404f$70d2e020$9ea24e51@brown1> References: <003801c6404f$70d2e020$9ea24e51@brown1> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Krzysztof Worytkiewicz Subject: categories: Alexander Grothendieck on `speculation' Date: Sun, 5 Mar 2006 10:38:42 -0500 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 16 Today's motto sadly appears to be: "the presentable individuals are precisely the discrete ones, what an exemplary society!!" (Gabriel- Ulmer 1971) Cheers Krzysztof -- my government will categorically deny the incident ever occurred From rrosebru@mta.ca Mon Mar 6 09:28:46 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 09:28:46 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGFkq-0003NO-Uz for categories-list@mta.ca; Mon, 06 Mar 2006 09:28:04 -0400 From: Thomas Streicher Message-Id: <200603061028.k26ASVcH032704@fb04305.mathematik.tu-darmstadt.de> Subject: categories: Re: intuitionism and disjunction In-Reply-To: To: categories@mta.ca Date: Mon, 6 Mar 2006 11:28:31 +0100 (CET) X-Mailer: ELM [version 2.4ME+ PL100 (25)] MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 I just want to remark that the second order encoding of disjunction `a la Russell-Prawitz is not bound to work only for provability. E.g. in PER(N) it holds that (\forall X) (A->X) -> (B->X) -> X is isomorphic to A + B. (A related result for "polymorphic" natural numbers was proved by Peter Freyd and extension of this result to polymorphic encodings free algebras was obtained a bit later by Hyland, Robinson and Rosolini). If your data type A lives in PER(N) then the 2nd order order encoding of existential quantification (\exists x:A) P(x) \equiv (\forall Q) ((\forall x:A) P(x) -> Q) -> Q is isomorphic to (\Sigma x:A) P(x) and thus we can project out the witness. In the extended calculus of constructions XCC where Prop (i.e. PER(N)) closed under small sums one can syntactically prove (\exists x:A) P(x) \equiv (\Sigma x:A) P(x) and thus one can extract witnesses from proves of (\exists x:A) P(x). Admittedly, this equivalence is not an isomorphism in general but it is in PER(N) and a lot of other relizability models. (A tricky counterexample for the general case was proided by Ivar Rummelhoff a cople of years ago!) Thus in an impredicative setting the positive operations turn out as 2nd order definable from -> and \forall. The main defect of Domains (with bottom) as a model for proofs is that every type = domain contains an element, namely bottom. But weak coproducts are definitely available. If one integrates dynamics = rewriting = computation into the model by considering 2-categories things might turn out as different. However, there are no indications (as far as I know) that this 2-categorical setting makes sums definable from lambda calculus. Thomas PS Peter Freyd has remarked that the early constructivists (implicitly) considered as an earmark of constructivity that (E) |= \exists x:A. P(x) entails that |= P([a/x] for some a : 1 -> A I am not going to contradict that BUT it definitely fails already for boolean toposes: consider the topos Psh(G) for G a nontrivial group and A the representable presheaf then it holds in Psh(G) that \exists x:A. x=x although A hasn't any global section. Of course, Peter Freyd's glueing argument shows that in the free topos (E) holds. This certainly is a most beautiful argument. Nevertheless, I think it doesn't tell that (E) is necessarily an earmark of constructivity since the free models (i.e. formulas modulo provability) are not necessarily the intended models. Moreover, (E) holds in realizability models (as long as type A is \neg\neg-separated, i.e. an assembly). From rrosebru@mta.ca Mon Mar 6 16:13:35 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 16:13:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGLyI-0003JR-Qm for categories-list@mta.ca; Mon, 06 Mar 2006 16:06:22 -0400 From: "Petr Ivankov" To: Subject: categories: Category Theory software Date: Mon, 6 Mar 2006 18:36:21 +0300 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="koi8-r";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 18 Dear colleagues. I am developing category theory software. It includes objects, morphisms, diagrams, functors, direct and inverse limits. Software math categories include vector spaces, modules over fuclidean rings, finite sets, finitely generated commutative algebras. The software works with algebraic fields, complex and real field, Galois fields. The software supports following functors: Tensor products, Hom, Ext, Tor. Additional software feature is Homology calculations. The software requires .NET 2.0 You can download sofware from http://sourceforge.net/projects/categorytheory . Recent instructions you can download from http://prdownloads.sourceforge.net/categorytheory/CategoryTheory-doc-0.9.zip?download If you wish improvement of the sofware vizit its forum http://sourceforge.net/forum/?group_id=160444 Yours sincerely Petr. From rrosebru@mta.ca Mon Mar 6 16:13:36 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 16:13:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGLzI-0003Oy-LZ for categories-list@mta.ca; Mon, 06 Mar 2006 16:07:24 -0400 Message-ID: <440C7F19.7080705@math.upenn.edu> Date: Mon, 06 Mar 2006 13:27:37 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Alexander Grothendieck on `speculation' References: <003801c6404f$70d2e020$9ea24e51@brown1> In-Reply-To: <003801c6404f$70d2e020$9ea24e51@brown1> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 19 This suggests two possibiloities: for the brave, start your own blog for speculations for the timid, same input but into a file only you can access until late in life and famous you can show how you had the ideas all along jim Ronald Brown wrote: > I am in the process of editing my correspondence 1982-1991 with AG which has > been latexed through George Malsiniotis for appearing as part of an Appendix > to a published edition of `Pursuing Stacks'. I came across the following > extract which seemed to me to contain points of general interest about > mathematical methodology and sociology, so I give this below, to invite > comments. > > Ronnie Brown > www.bangor.ac.uk/r.brown > From rrosebru@mta.ca Mon Mar 6 19:11:44 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 19:11:44 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGOoc-00064o-HU for categories-list@mta.ca; Mon, 06 Mar 2006 19:08:34 -0400 To: categories@mta.ca Subject: categories: Re: Undirected graph citation Date: Mon, 06 Mar 2006 15:08:47 -0500 From: wlawvere@buffalo.edu Message-ID: <1141675727.440c96cf4fe0c@mail2.buffalo.edu> References: <005d01c64089$1b437220$0b00000a@C3> In-Reply-To: <005d01c64089$1b437220$0b00000a@C3> X-Mailer: University at Buffalo WebMail Cyrusoft SilkyMail v1.1.11 MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 20 Dear George By a presentation in mathematics I mean generators and relations for an algebraic structure of a certain kind. Occasionally we are fortunate to have also another more direct description of the same algebra, which it is useful to make explicit; a well known example of the usefulness of making explicit such a conceptual (as opposed to syntactical) description is the pair of definitions for the algebra of operators that defines the notion of simplicial set. In your 2001 book with Borceux, Definition 7.2.1 involves five generators and five relations. Sometimes this is augmented by symmetry. What is actually being presented ? a certain full finite subcategory of the category of finite sets. Why should diagrams of this shape occur so often and be transported by functors even when they do not satisfy any exactness ? That is especially evident in the case of the Amitsur complex on page 264: the family of powers of a given object is a functor of the exponents, which are sets from that little category. That groupoids form a subcategory of the topos permits to take images, in the topos, of maps between groupoids; surprisingly, that can be useful. I prefer to consider one more finite set, so that "associativity" is a structure even when it is not an exact property (and analogously in the case of categories vs truncated simplicial sets - the question is how truncated). Then to be a groupoid is just a pullback-preservation condition. Bill Quoting George Janelidze : > Dear Bill, > > Indeed, there were no monoids in Vaughan's original message of > February 28, > but since you have mentioned them in your message of March 1, and > since you > were talking there about "...lacuna of explicitness ... in many > papers on > Galois theory...", I simply wanted to say that: > > I do not see any relevance of these kinds of presentations in Galois > theory > (apart from the fact the internal pre-whatever-s in a category X form > an > X-valued presheaf category). > > On the other hand Galois theory is not the end of the World, and I > think the > beauty and importance of those your ideas is clear to everyone who > saw them. > > Putting myself in risk of making my message boring, I would like to > make one > more remark concerning your last message and Galois theory: > > You say: "...applying a non-exact functor F to a group..." - true and > fine, > but I have actually mentioned F(R) for R being not a group, but > another > extreme case of a groupoid, namely an equivalence relation. What > seems to be > most amazing is, that, because F preserves not-all-but-some > pullbacks, there > are beautiful examples where R is an equivalence relation and F(R) is > a > group; in simple words, F creates a group out of nothing! The > classical > example, as you know, is: if R is the kernel pair of a universal > covering > map E ---> B of a "good" connected topological space B, and F is the > functor > sending ("good") topological spaces to the sets of their connected > components, then F(R) is the fundamental group of B. The same thing > is true > in other Galois theories of course. > > George > From rrosebru@mta.ca Tue Mar 7 10:02:45 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Mar 2006 10:02:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGcgL-0007Ar-PL for categories-list@mta.ca; Tue, 07 Mar 2006 09:56:57 -0400 Message-ID: <001501c64183$14d23700$0b00000a@C3> From: "George Janelidze" To: References: <005d01c64089$1b437220$0b00000a@C3> <1141675727.440c96cf4fe0c@mail2.buffalo.edu> Subject: categories: Re: Undirected graph citation Date: Tue, 7 Mar 2006 03:04:24 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 Dear Bill, > By a presentation in mathematics I mean generators and relations for an > algebraic structure of a certain kind. Occasionally we are fortunate to > have also another more direct description of the same algebra, which it is > useful to make explicit; a well known example of the usefulness of making > explicit such a conceptual (as opposed to syntactical) description is the > pair of definitions for the algebra of operators that defines the notion > of simplicial set. By a presentation in mathematics I mean exactly the same thing, I agree with every word you said in this paragraph, and I know that what we call today Lawvere theories was your beautiful discovery along this lines to thoughts. > In your 2001 book with Borceux, Definition 7.2.1 involves five generators > and five relations. Sometimes this is augmented by symmetry. > > What is actually being presented ? a certain full finite subcategory of > the category of finite sets. I am sorry, what is "presented" in Definition 7.2.1 can of course be considered as a subcategory of the category of finite sets, but certainly not full (because, say, there are no arrows from C_1 to C_2). I suppose you have noticed this, and so you are suggesting to modify the Definition 7.2.1 - since in "all" examples in fact there are more arrows. Well, one could do so, but there is also a good reason not to do so: those five generators and five relations is exactly the minimum needed to define internal actions (I have actually first used this definition in my CT90 paper "Precategories and Galois theory" and many other people used similar definitions for other purposes, probably long before). Having said this, I again agree with every word of the rest of your message. Can you accept the fact I agree with it and at the same time I do like Definition 7.2.1? Note that if we go one step down in dimension, there will be reflexive graphs whose "theory" is a full subcategory of sets and just graphs whose "theory" is not. Are you telling me that this is a good reason to forget the notion of graph, and use only reflexive graphs? George ----- Original Message ----- From: To: Sent: Monday, March 06, 2006 10:08 PM Subject: categories: Re: Undirected graph citation > > Dear George > > By a presentation in mathematics I mean generators and relations for an > algebraic structure of a certain kind. Occasionally we are fortunate to > have also another more direct description of the same algebra, which it is > useful to make explicit; a well known example of the usefulness of making > explicit such a conceptual (as opposed to syntactical) description is the > pair of definitions for the algebra of operators that defines the notion > of simplicial set. > > In your 2001 book with Borceux, Definition 7.2.1 involves five generators > and five relations. Sometimes this is augmented by symmetry. > > What is actually being presented ? a certain full finite subcategory of > the category of finite sets. Why should diagrams of this shape occur so > often and be transported by functors even when they do not satisfy any > exactness ? That is especially evident in the case of the Amitsur complex > on page 264: the family of powers of a given object is a functor of the > exponents, which are sets from that little category. > > That groupoids form a subcategory of the topos permits to take images, in > the topos, of maps between groupoids; surprisingly, that can be useful. > > I prefer to consider one more finite set, so that "associativity" is a > structure even when it is not an exact property (and analogously in the > case of categories vs truncated simplicial sets - the question is how > truncated). Then to be a groupoid is just a pullback-preservation > condition. > > Bill > From rrosebru@mta.ca Tue Mar 7 10:02:45 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Mar 2006 10:02:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGci1-0007J5-NC for categories-list@mta.ca; Tue, 07 Mar 2006 09:58:41 -0400 Message-ID: <440D0F71.8080405@cs.stanford.edu> Date: Mon, 06 Mar 2006 20:43:29 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu Organization: Stanford University User-Agent: Mozilla Thunderbird 0.9 (Windows/20041103) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Undirected graph citation References: <005d01c64089$1b437220$0b00000a@C3> In-Reply-To: <005d01c64089$1b437220$0b00000a@C3> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 22 George Janelidze wrote: > > Indeed, there were no monoids in Vaughan's original message of February 28, My take on monoids vs. initial segments of Delta, FinSet, etc. as sites for a category of presheaves is that it is like Hasse diagrams vs. posets, or axioms vs. theories. The former should be understood only as a convenient representation of its idempotent completion, just as a Hasse diagram of a poset is a convenient representation of its reflexive transitive closure, or an axiom system a convenient representation of a theory. In the case of reflexive undirected graphs as a presheaf category, the monoid Set(2,2) works as a site but is not idempotent closed (the two constant functions don't split). In the category of sets with one or two elements however, the terminator splits the constant functions, as it does in any category with a terminator if one defines "constant morphism" as an idempotent split by the terminator. The benefit of idempotent closed sites is that equivalent presheaf categories must then have equivalent sites, as I learned from Jiri Adamek's post on this list the other week asking for the earliest reference to that fact. I subsequently learned the proof from Borceux Vol I (Theorem 6.5.11, where idempotent completion is called by its synonym Cauchy completion). My question was about the theory, for which Bill pointed out a nice axiomatization. As it turns out, the earliest reference answering my original question may well be this very list! At the risk of embarrassing Marco Grandis (to whom I therefore apologize in advance), the 1999 monoid-on-graphs thread at http://www.mta.ca/~cat-dist/catlist/1999/monoid-on-graphs includes two posts by Marco, the first asserting that FinSet could be substituted for Delta in the definition of reflexive graphs as presheaves on the truncation of that site, the second recanting a day later and pointing out the impact of the twist:2->2 in creating what he called at the time involutive reflexive graphs. Marco subsequently wrote about symmetric simplicial complexes as the higher-dimensional generalization of the impact of the twist. So far no one has mentioned an earlier explicit reference than this March 1999 one in response to my question. Bill mentioned Sets for Mathematics, but that was 2003. I did however receive two private responses from a La Jolla 1965 participant who first protested that surely presheaves on the site I asked about were *directed* graphs, but then with the same one-day pause as Marco pointed out the role of the twist in binding together the two directions of an undirected edge. (I assume this is history repeating itself and not the email counterpart of a standup routine the experts do from time to time for our edutainment. In standup, timing is as important as content.) My own excuse for not noticing Marco's 1999 posts, or for that matter Francois Lamarche's citation question sparking that whole thread, is that I was in Hanover that week exhibiting at CeBIT what Guinness Records 2000 subsequently listed as the world's smallest webserver (p.162, nestled interestingly when the book is closed). Bad timing on my part, that was a useful thread. It's impressive what can come out of a simple request for citations on this list. Vaughan From rrosebru@mta.ca Wed Mar 8 23:11:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Mar 2006 23:11:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHBPd-0001VD-Ns for categories-list@mta.ca; Wed, 08 Mar 2006 23:02:01 -0400 Subject: categories: Glasgow PSSL: Second announcement From: Tom Leinster To: categories@mta.ca Content-Type: text/plain Date: Wed, 08 Mar 2006 15:42:23 +0000 Message-Id: <1141832543.735.23.camel@tl-linux.maths.gla.ac.uk> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 23 Dear All, You are warmly welcomed to the 83rd Peripatetic Seminar on Sheaves and Logic, to be held in Glasgow, Scotland, on 6-7 May. Our invited speaker is Jon Woolf, who will give us an introduction to derived categories. To register, request accommodation, and find out more, visit http://www.maths.gla.ac.uk/~tl/pssl/ Best wishes, Tom Leinster Richard Steiner From rrosebru@mta.ca Wed Mar 8 23:11:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Mar 2006 23:11:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHBRw-0001eq-Bs for categories-list@mta.ca; Wed, 08 Mar 2006 23:04:24 -0400 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-1" MIME-Version: 1.0 From: "Dr. Cyrus F Nourani" To: categories@mta.ca Date: Wed, 08 Mar 2006 15:22:23 -0500 Subject: categories: Re: Undirected graph citation Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 24 Hmmm, a paper entitled Funcotrial Generic Filters was written July 2005, abstract to ASL, where you can observe ejecting on=20 initial segments towards models. What it might do on preshaeves was sent to a conference a month ago. Like I had told the list there=20 were papers I published over several years ago on functors computing models on Hasse diagrams.=20=20 I'm not in a position to escalate and have to keep you on a holding as to what it was doing on sheaves. On the surface it appears as if we are living in parallel worlds getting a message through.=20 Cyrus > ----- Original Message ----- > From: "Vaughan Pratt" > To: categories@mta.ca > Subject: categories: Re: Undirected graph citation > Date: Mon, 06 Mar 2006 20:43:29 -0800 >=20 >=20 > George Janelidze wrote: > > > > Indeed, there were no monoids in Vaughan's original message of February= 28, >=20 > My take on monoids vs. initial segments of Delta, FinSet, etc. as sites > for a category of presheaves is that it is like Hasse diagrams vs. > posets, or axioms vs. theories. The former should be understood only as > a convenient representation of its idempotent completion, just as a > Hasse diagram of a poset is a convenient representation of its reflexive > transitive closure, or an axiom system a convenient representation of a etc, etc... From rrosebru@mta.ca Wed Mar 8 23:27:28 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Mar 2006 23:27:28 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHBmk-0002uW-MM for categories-list@mta.ca; Wed, 08 Mar 2006 23:25:54 -0400 Subject: categories: Undirected graphs To: categories@mta.ca Date: Wed, 8 Mar 2006 11:07:20 -0400 (AST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> From: Robert Pare Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 25 [note from moderator: resent due to faulty From: ] I've been following the recent posts on undirected graphs with interest. But I have a question. I think it's being said that undirected graphs are the same as directed graphs with involution. (Presheaves on the full subcategory of SET determined by 1 and 2, or just 2.) Which is nice but what about loops? The involution might fix a loop or not. So wouldn't we be getting undirected graphs with two kinds of loops, whole loops and semiloops? What am I missing? Bob From rrosebru@mta.ca Fri Mar 10 05:51:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 05:51:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHeCq-0001n2-TF for categories-list@mta.ca; Fri, 10 Mar 2006 05:46:44 -0400 Date: Thu, 9 Mar 2006 17:38:04 +0000 (GMT) From: Chris Wensley To: categories@mta.ca Subject: categories: Re: Undirected graphs Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 26 Dear Bob > I've been following the recent posts on undirected graphs > with interest. But I have a question. I think it's being said > that undirected graphs are the same as directed graphs with > involution. (Presheaves on the full subcategory of SET determined > by 1 or 2, or just 2.) Which is nice but what about loops? > The involution might fix a loop or not. So wouldn't we be > getting undirected graphs with two kinds of loops, whole loops > and semiloops? What am I missing? There has been some 'work in progress' at Bangor on this very question for the past 8 years. This is intended to present a categorical approach to graph theory to workers in combinatorics, and is not intended for category theorists. The current draft, 06.04, is available at http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/06/ (follow the link 06.04 to 06_04.pdf). We do indeed discuss two types of loop, which we call 'loops' and 'bands'. Ronnie is away today, but may well add his own comment tomorrow. Best wishes, Chris Wensley From rrosebru@mta.ca Fri Mar 10 05:51:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 05:51:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHeEi-0001px-1B for categories-list@mta.ca; Fri, 10 Mar 2006 05:48:40 -0400 In-Reply-To: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> References: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <95202C2D-742C-4D18-8863-23AD165EB912@dima.unige.it> Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: Re: Undirected graphs Date: Fri, 10 Mar 2006 09:07:03 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 27 Dear Bob, Involutive graphs are what you are saying, of course. If I had to choose, I would take this as my favoured notion of "undirected graph", because it is a presheaf topos on a very simple site. A graph theorist would probably say that an "undirected graph" is what you are hinting at, which amounts to taking the involutive graphs where all loops are fixed by the involution (or the ones where no loop is fixed, except the trivial ones?). Then, he might want to forget about trivial loops, and allow vertices with no loops. Being in a category list, another reason of "preferring" the first notion might be: - a category has an underlying graph, - an involutive category has an underlying involutive graph, - involutive categories where all endomorphisms are fixed by the involution are rather unnatural; not to mention the ones where no endomorphism is fixed except the identities. Of course, there might be reasons in favour of the other choices, or of considering different choices at a time. Life is complicated and mathematics too. Even working in category theory, I think we should avoid being too "categorical"... Best regards Marco On 8 Mar 2006, at 16:07, cat-dist@mta.ca wrote: > I've been following the recent posts on undirected graphs > with interest. But I have a question. I think it's being said > that undirected graphs are the same as directed graphs with > involution. (Presheaves on the full subcategory of SET determined > by 1 and 2, or just 2.) Which is nice but what about loops? > The involution might fix a loop or not. So wouldn't we be > getting undirected graphs with two kinds of loops, whole loops > and semiloops? What am I missing? > > Bob > > > From rrosebru@mta.ca Fri Mar 10 05:51:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 05:51:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHeBZ-0001lV-UZ for categories-list@mta.ca; Fri, 10 Mar 2006 05:45:25 -0400 Date: Thu, 9 Mar 2006 09:05:13 -0500 (EST) From: F W Lawvere X-Sender: wlawvere@hercules.acsu.buffalo.edu Reply-To: wlawvere@acsu.buffalo.edu To: categories@mta.ca Subject: categories: Re: Undirected graphs In-Reply-To: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 28 Dear all, Yes, there are two kinds of loops in the topos of right actions of the four-element monoid A, where A consists of endomaps of the two-element set. Consider for example the concrete structure of the truth-value object in that topos, which is forced to contain a truth-value called "foray". Rather than as "semi"loops, my colleagues and I usually think of them as one-lane in the sense that some other edges are really two lanes, related by the involution operator in the site. My old paper "Qualitative distinctions..." tried to make the point that there are several precise toposes all deserving the rough name of "graph" or "network" and that each of these precise toposes may have a role to play. For example, in any given topos, for any given object L, the category of objects over L, or "L-labelled graphs" (which in practice may serve as a category of networks) is another topos of "graphs". In my experience it is important to consider the whole topos in order to get good exactness properties but, moreover, because the truth-value object and other specific objects which may seem rather far from an initial prejudice about what one wants the objects to mean, nonetheless turn out in a systematic theory to play a key role in representing concepts directly related to the original particular subject matter. A simple example is the representability of gender and moitie in the topos of kinship systems. This example is treated briefly in Conceptual Mathematics and in more detail, (again actually involving several related toposes rather than a single choice) in "Kinship and mathematical categories". Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Wed, 8 Mar 2006 cat-dist@mta.ca wrote: > I've been following the recent posts on undirected graphs > with interest. But I have a question. I think it's being said > that undirected graphs are the same as directed graphs with > involution. (Presheaves on the full subcategory of SET determined > by 1 and 2, or just 2.) Which is nice but what about loops? > The involution might fix a loop or not. So wouldn't we be > getting undirected graphs with two kinds of loops, whole loops > and semiloops? What am I missing? > > Bob > > > > From rrosebru@mta.ca Fri Mar 10 05:51:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 05:51:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHe9b-0001jC-Ek for categories-list@mta.ca; Fri, 10 Mar 2006 05:43:23 -0400 Subject: categories: Re: Undirected graphs From: Sebastiano Vigna To: categories@mta.ca In-Reply-To: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> References: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> Content-Type: text/plain Date: Thu, 09 Mar 2006 07:56:00 +0100 Message-Id: <1141887360.12171.6.camel@localhost.localdomain> Mime-Version: 1.0 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 29 On Wed, 2006-03-08 at 11:07 -0400, Robert Pare wrote: > by 1 and 2, or just 2.) Which is nice but what about loops? > The involution might fix a loop or not. So wouldn't we be > getting undirected graphs with two kinds of loops, whole loops > and semiloops? What am I missing? Yes, you'll get two kind of loops. This explains why in topological graph theory books sometimes you'll find a remark like "we will count loops once" or "we will count loops twice" (in the first case, sometimes loops are depicted as segments going out of vertices with a dashed ending). The problem is that the standard representation for undirected graphs (subsets of unordered pairs) fails to distinguish between the two kind of loops. The presheaf representation makes this distinction clear. In most cases you can forget about this problem, but when studying covering the difference is huge: a loop fixed by the involution is covered by an edge, whereas a pair of loops exchanged by the involution are covered by a line. We discussed this issue at some length in our paper "Fibrations of graphs" (Discrete Math., 2002). Ciao, seba From rrosebru@mta.ca Fri Mar 10 18:25:05 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 18:25:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHpxx-0003j9-LR for categories-list@mta.ca; Fri, 10 Mar 2006 18:20:09 -0400 Subject: categories: CT2006 - Mac Lane, Eilenberg session and deadline To: categories@mta.ca (Categories List) Date: Fri, 10 Mar 2006 09:33:30 -0400 (AST) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20060310133330.8ED6010854@sigma.mathstat.dal.ca> From: selinger@mathstat.dal.ca (Peter Selinger) Status: RO X-Status: X-Keywords: X-UID: 30 Dear Category Theorists: This is a reminder, for those who wish to contribute to the special session on Mac Lane and Eilenberg at CT 2006, to send their abstracts by March 15. The special session will be chaired by Bill Lawvere. Due to popular demand, abstracts for the general sessions are also still being accepted until March 15. We would also like to remind people who are attending the conference to make their reservations with White Point as soon as possible as White Point is planning to release a part of our room block early next week. - The organizers, CT 2006 http://www.mathstat.dal.ca/~selinger/ct2006/ From rrosebru@mta.ca Fri Mar 10 18:28:22 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 10 Mar 2006 18:28:22 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FHq4Y-0004aA-1G for categories-list@mta.ca; Fri, 10 Mar 2006 18:26:58 -0400 Message-ID: <4411B4C5.2040702@cs.stanford.edu> Date: Fri, 10 Mar 2006 09:17:57 -0800 From: Vaughan Pratt X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Undirected graphs References: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> <1141887360.12171.6.camel@localhost.localdomain> In-Reply-To: <1141887360.12171.6.camel@localhost.localdomain> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 31 Sebastiano Vigna wrote on 3/8/06: > The problem is that the standard representation for undirected > graphs (subsets of unordered pairs) fails to distinguish between the two > kind of loops. The presheaf representation makes this distinction > clear. I wouldn't call this a "failure" of the set-of-unordered-pairs notion of undirected graph, which should be understood in 2-enriched (2 = 0->1) terms rather than Set-enriched. In my March 1 post I distinguished the "set-enriched" or presheaf graph categories Set-DGraph and Set-UGraph from the "2-enriched" graph categories 2-DGraph and 2-UGraph ("homsets" of edges are either 0 or 1, i.e. empty or singleton), pointing out that 2-UGraph was a full subcategory of 2-DGraph but Set-UGraph was not a full subcategory of 2-DGraph, via the same mechanism that creates two kinds of loops in Set-UGraph. There is only one kind of loop in 2-UGraph, which category theory is faithful to when this kind of undirected graph is properly described in categorical language. One description would be as the full subcategory of Set-UGraph (= Set^M^op for M the monoid Set(2,2)) induced by the evident functor Nonempty:Set->2 collapsing nonempty homsets to singletons; it would be nice if 2-UGraph arose more naturally, e.g. as some sort of 2-enriched presheaf category, though I don't see how. Vaughan Pratt From rrosebru@mta.ca Sun Mar 12 18:48:00 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 12 Mar 2006 18:48:00 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIZAV-0004Pf-HV for categories-list@mta.ca; Sun, 12 Mar 2006 18:36:07 -0400 Message-ID: <44139F28.7070608@cs.stanford.edu> Date: Sat, 11 Mar 2006 20:10:16 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu Organization: Stanford University User-Agent: Mozilla Thunderbird 0.9 (Windows/20041103) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Undirected graphs References: <20060308150720.9D5D07375B@chase.mathstat.dal.ca> <1141887360.12171.6.camel@localhost.localdomain> <4411B4C5.2040702@cs.stanford.edu> In-Reply-To: <4411B4C5.2040702@cs.stanford.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 32 My definition of 2-UGraph (undirected graphs with at most one edge from any given vertex to another) as "the full subcategory of Set-UGraph (= Set^M^op for M the monoid Set(2,2)) induced by the evident functor Nonempty:Set->2 collapsing nonempty homsets to singletons" was an attempt to say "2-UGraph is a retract of Set-UGraph" with both too few words and too many. 2-UGraph is the full subcategory of Set-UGraph consisting of graphs with at most one edge per "homset", and at the same time the quotient of Set-UGraph arising from identifying all members of each homset of each graph. I.e. a retract. This is something of an eye-opener for me as I have for decades thought of the undirected graph (of the one-edge-per-homset kind) as the algebraically impoverished cousin of the directed graph. I am tickled pink to find it arising as a retract of a presheaf category, and moreover without either of the two quirks that have been pointed out for the more general undirected graphs allowing multiple edges per homset (Set-UGraph has two types of distinguished loop, and does not embed in Set-DGraph). Vaughan Pratt From rrosebru@mta.ca Mon Mar 13 01:05:27 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Mar 2006 01:05:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIfAn-0001eG-Vn for categories-list@mta.ca; Mon, 13 Mar 2006 01:00:50 -0400 Date: Sun, 12 Mar 2006 19:51:36 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: Re: Undirected graphs In-Reply-To: <44139F28.7070608@cs.stanford.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 33 Dear Vaughan, I hope you will also like the following remark: Inside any presheaf or other topos there is a canonical subcategory which is an adjoint retract with the adjoint preserving finite products (but not equalizers). This category is therefore cartesian closed in the obvious way, but its special property is that any two maps from X to Y are distinguished by a map from 1 to X. It is never a topos and its exactness properties are rather bad, but it does recapture classical restrictions in many cases. For example, within the Boolean algebra classifier (=presheaves on the category of non-empty finite sets) this canonical subcategory is the category of all classical simplicial complexes, a "higher dimensional" version of your remark about undirected graphs. Let me take this opportunity to make another remark about undirected graphs. They are treated on pages 176 - 180 in the book by Bob Rosebrugh and me where in particular the two kinds of loops are clearly pointed out. But I like Steve Schanuel's proposal of a way of picturing these objects: there is a "geometric realization" functor from undirected graphs to topological spaces which preserves all colimits ("gluing") (but not finite limits), namely the Kan extension of the covariant inclusion of the monoid into topological spaces which interprets the involution as 1-t on the unit interval. Since the one-lane "loop" is the coequalizer of two maps from I to I in the graph topos, the same coequalizer statement remains true for the geometric realizations. Therefore, the resulting picture of the one-lane loop is as a cul-de-sac, with one starting point, a parameterization which goes until t=1/2, then returns to the starting point, without encountering any other points. For example, the truth value "foray" can be pictured this way. This flattened loop picture not only has the foregoing rigorous justification but should make visualizing the objects less nerve-wracking. Of course, the two-lane loops are now pictured simply as loops, parameterized in two canonical ways by the unit interval. Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Sat, 11 Mar 2006, Vaughan Pratt wrote: > My definition of 2-UGraph (undirected graphs with at most one edge from > any given vertex to another) as "the full subcategory of Set-UGraph (= > Set^M^op for M the monoid Set(2,2)) induced by the evident functor > Nonempty:Set->2 collapsing nonempty homsets to singletons" was an > attempt to say "2-UGraph is a retract of Set-UGraph" with both too few > words and too many. > > 2-UGraph is the full subcategory of Set-UGraph consisting of graphs with > at most one edge per "homset", and at the same time the quotient of > Set-UGraph arising from identifying all members of each homset of each > graph. I.e. a retract. > > This is something of an eye-opener for me as I have for decades thought > of the undirected graph (of the one-edge-per-homset kind) as the > algebraically impoverished cousin of the directed graph. I am tickled > pink to find it arising as a retract of a presheaf category, and > moreover without either of the two quirks that have been pointed out for > the more general undirected graphs allowing multiple edges per homset > (Set-UGraph has two types of distinguished loop, and does not embed in > Set-DGraph). > > Vaughan Pratt From rrosebru@mta.ca Mon Mar 13 18:53:32 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Mar 2006 18:53:32 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIvtR-00016m-Dd for categories-list@mta.ca; Mon, 13 Mar 2006 18:52:01 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <2DE2CF1F-BF75-4606-990C-19C860157CF4@dima.unige.it> Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: An autonomous category Date: Mon, 13 Mar 2006 14:45:25 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 34 The Lawvere category of extended positive real numbers has also an autonomous structure, with a multiplicative tensor product (instead of the original additive one). Has this been considered somewhere? To be more explicit: The well-known article of Lawvere on "Metric spaces..." (Rend. Milano 1974, republished in TAC Reprints n. 1) introduced the category of extended positive real numbers, from 0 to oo (infinity included), with arrows x \geq y, equipped with a strict symmetric monoidal closed structure: the tensor product is the sum, the internal hom is truncated difference (with oo - oo = 0). Now, the same category can be equipped with a multiplicative tensor product, x.y. Provided we define 0.oo = oo (so that tensoring by any element preserves the initial object oo), this is again a strict symmetric monoidal closed structure, with hom(y, z) = z/y. Now, the 'undetermined forms' 0/0 and oo/oo are defined to be 0. The new multiplicative structure is even *-autonomous, with involution x* = 1/x (and 'nearly' compact). (Note that this choice of values of the undetermined forms comes from privileging the direction x \geq y, which is necessary if we want to view metric spaces, normed categories etc. as enriched categories). Marco Grandis From rrosebru@mta.ca Mon Mar 13 18:54:59 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Mar 2006 18:54:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIvuv-0001Dk-Ml for categories-list@mta.ca; Mon, 13 Mar 2006 18:53:33 -0400 Date: Mon, 13 Mar 2006 15:17:37 +0100 From: Jiri Rosicky To: categories@mta.ca Subject: categories: symmetric simplicial sets Message-ID: <20060313141737.GA24776@queen.math.muni.cz> Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-2 Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 One advantage of symmetric simplicial sets is that their model category structure is determined, cf. (*), by taking cofibrations as monomorphisms. It means that the classical homotopy category Ho (of CW-complexes) can be obtained just from the category S of symmetric simplicial sets (without any additional data). Simplicial sets do not have this property (it was also observed by J. H. Smith). In (*), we have used this property of S to give an elementary proof of an important result of D. Dugger saying that Ho is a free homotopy theory over a point. (*) J. Rosicky and W. Tholen, Left-determined model categories and universal homotopy theories, TAMS 355 (2003), 3611-3623. From rrosebru@mta.ca Mon Mar 13 18:50:29 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Mar 2006 18:50:29 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIviW-0000Ji-JW for categories-list@mta.ca; Mon, 13 Mar 2006 18:40:44 -0400 Reply-To: marta.bunge@mcgill.ca From: "Marta Bunge" To: categories@mta.ca Subject: categories: cracks and pots Date: Sun, 12 Mar 2006 17:29:42 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 36 Hi, I just came across the following pages http://motls.blogspot.com/2004/11/category-theory-and-physics.html http://motls.blogspot.com/2004/11/this-week-208-analysis.html written by Lubos Motl, a physicist (string theorist). Some of you may find these articles interesting and probably revealing. Are we category theorists as a whole going to quietly accept getting discredited by a minority of us presumably applying category theory to string theory? It is surely not too late to react and point out that this is not what (all of) category theory is about. Please give a thought about what we, as a community, can urgently do to repair this damaging impression. Unless we are prepared to wait until things change by themselves within our lifetime. Hopefully disturbing your weekend, Cordially, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************ From rrosebru@mta.ca Mon Mar 13 18:56:05 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Mar 2006 18:56:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FIvw6-0001Iq-8m for categories-list@mta.ca; Mon, 13 Mar 2006 18:54:46 -0400 From: Andrej Bauer To: categories Date: Mon, 13 Mar 2006 16:01:17 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: 7bit Content-Disposition: inline Message-Id: <200603131601.18902.Andrej.Bauer@andrej.com> Subject: categories: When are all monos regular? Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 37 This might be an embarrassingly easy question, but I always get confused about it. When are all monos in an algebraic category regular (or more generally, when are all monos in a regular category regular)? What are some sensible sufficient or necessary conditions? For example, all monos in the category of groups are regular. How about the category of lattices, or lattices with a top element? Andrej Bauer From rrosebru@mta.ca Tue Mar 14 18:44:58 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 18:44:58 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIAB-00027A-Ts for categories-list@mta.ca; Tue, 14 Mar 2006 18:38:47 -0400 Date: Mon, 13 Mar 2006 23:02:52 -0500 (EST) From: Robert Seely To: Categories List Subject: categories: Categorical Logic & Quantum Computation: ASL special session Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 38 There will be a special session on Categorical Logic & Quantum Computation (organized by Michael Makkai, Prakash Panangaden, and Robert Seely) at the forthcoming ASL annual meeting at UQAM, 17 - 21 May 2006. The speakers in this session will include Richard Blute Robin Cockett Sergey Slavnov Bob Coecke Dusko Pavlovic Peter Selinger Claudio Hermida Nicola Gambino In addition, Peter Selinger will give a tutorial on Quantum Information Theory, as part of the main program. Titles and abstracts may be found on the (unofficial!) webpage http://www.math.mcgill.ca/rags/seminar/CLQC.html More (official) information about the meeting in general may be found at the meeting website http://asl2006.uqam.ca/ If you are interested we hope to see you there. -= rags =- -- From rrosebru@mta.ca Tue Mar 14 18:44:58 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 18:44:58 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIDE-0002Lc-9T for categories-list@mta.ca; Tue, 14 Mar 2006 18:41:56 -0400 In-Reply-To: References: Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: David Yetter Subject: categories: Re: cracks and pots Date: Tue, 14 Mar 2006 01:08:49 -0500 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 39 Dear Marta, My reaction to the blog posts you cite is that this is a sting theorist holding his breath and refusing to learn category theory. My guess is that Motl wouldn't want to learn the heavily categorical formulations of mirror symmetry that Yan Soibelman uses, even though they are motivated by string theory. Basically categorical ideas aren't part of the standard bag of tricks physicists use (even though they often give much more elegant, concise, and insightful formulations of some of those tricks), and the proverb about 'old dogs' and 'new tricks' applies to physicists as well. His attack on Baez is fairly standard stuff: in the mode of "string theory is the theory of nature, so we don't want to think about alternatives like loop quantum gravity." It is a polemical defense of a scientific theory that hasn't produced a testable prediction in the 40 plus years since its inception, and worse than that, unless one adds bells and whistles to fix it (in the manner of 'gaseous Vulcan' or Ptolemaic epicycles), predicts the existence of a massless scalar field *not observed in nature*. It really has nothing at all to say about category theory, which is after all a mathematical theory which stands irrespective of its extra-mathematical applications. Categorical ideas are absolutely central to several competitors to string theory: the Barrett-Crane model of quantum gravity (and to a lesser extent 'loop quantum gravity' with which the BC model is often conflated) and Connes' recovery of the Standard Model from non-commutative geometry (a part of mathematics which has obliged reluctant mathematicians to think about categorical ideas deeper than they originally were comfortable with). There is nothing cracked or crackpot about either. It is simply a fact we have to live with that our subject has found legitimate uses in physics, but uses which are unpopular with the dominant school of physics in the North America. If (I suspect when) the string theory emperor turns out to have no clothes, category theory will suddenly become de rigeur in physics. (As it should, since categorical expressions of physical ideas are the logical conclusion of 20th century physics drive to express everything in coordinate-free terms.) Best Thoughts, David Yetter On 12 Mar 2006, at 17:29, Marta Bunge wrote: > Hi, > > I just came across the following pages > > http://motls.blogspot.com/2004/11/category-theory-and-physics.html > http://motls.blogspot.com/2004/11/this-week-208-analysis.html > > written by Lubos Motl, a physicist (string theorist). Some of you may > find > these articles interesting and probably revealing. > > Are we category theorists as a whole going to quietly accept getting > discredited by a minority of us presumably applying category theory to > string theory? It is surely not too late to react and point out that > this is > not what (all of) category theory is about. Please give a thought > about what > we, as a community, can urgently do to repair this damaging impression. > Unless we are prepared to wait until things change by themselves > within our > lifetime. > > > Hopefully disturbing your weekend, > Cordially, > Marta > > > > ************************************************ > Marta Bunge > Professor Emerita > Dept of Mathematics and Statistics > McGill University > 805 Sherbrooke St. West > Montreal, QC, Canada H3A 2K6 > Office: (514) 398-3810 > Home: (514) 935-3618 > marta.bunge@mcgill.ca > http://www.math.mcgill.ca/bunge/ > ************************************************ > > From rrosebru@mta.ca Tue Mar 14 18:52:22 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 18:52:22 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIJQ-0002mj-Ot for categories-list@mta.ca; Tue, 14 Mar 2006 18:48:20 -0400 From: "Walter Tholen" Message-Id: <1060314101419.ZM224374@pascal.math.yorku.ca> Date: Tue, 14 Mar 2006 10:14:19 -0500 References: <200603131601.18902.Andrej.Bauer@andrej.com> X-Mailer: Z-Mail (4.0.1 13Jan97) To: categories Subject: categories: Re: When are all monos regular? MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 40 To say that monos are regular (in a variety or, more generally, in a general category satisfying some very minor hypotheses) amounts to the so-called Intersection Property of Amalgamations: for any two algebras A, B with a common subalgebra C, if there are monomorphisms f : A --> D, g: B --> D that coincide on C, then one can choose f, g with the additional property that C is their pullback. References: C.M. Ringel: JPAA 2 (1972) 341-42 W. Tholen: Algebra Univ. 14 (1982) 391-397 E.W. Kiss, L. Marki, P. Prohle, W. Tholen: Studia Sci. Math. Hungaricum 18 (1983) 79-141. The last paper contains a large table of specific categories, including lattices (the affirmative answer is attributed to Gratzer in this case), plus an extensive list to the literature. For some categories, the question whether monos are regular can get quite involved, for example in the category of compact (Hausdorff) groups, for which an affirmative answer was provided by Poguntke (Math. Z. 130 (1973) 107-117). The property in question obviously implies "epimorphisms are surjective", but examples witnessing failure of the converse statement are harder to find: see again the four-author paper. Hope this helps. Walter Tholen. On Mar 13, 4:01pm, Andrej Bauer wrote: > Subject: categories: When are all monos regular? > This might be an embarrassingly easy question, but I always get confused about > it. When are all monos in an algebraic category regular (or more generally, > when are all monos in a regular category regular)? What are some sensible > sufficient or necessary conditions? > > For example, all monos in the category of groups are regular. How about the > category of lattices, or lattices with a top element? > > Andrej Bauer From rrosebru@mta.ca Tue Mar 14 18:52:23 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 18:52:23 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIGe-0002Zp-9q for categories-list@mta.ca; Tue, 14 Mar 2006 18:45:28 -0400 Subject: categories: Re: cracks and pots From: Eduardo Dubuc To: categories@mta.ca Date: Tue, 14 Mar 2006 11:55:07 -0300 (ART) In-Reply-To: from "Marta Bunge" at Mar 12, 2006 05:29:42 PM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 41 I congratulate Marta for her posting, I just read Motls's http://motls.blogspot.com/2004/11/category-theory-and-physics.html and find it very revealing as Marta said, and more than that, I find it a very clear exposition (by way of philosophy and by way of examples) about what is good science and mathematicas and about what it is not. Marta is right about that it concerns all of us category theoricist. I will like to see here a debate about Motls's writing quoted above. Just about this writing, NOT ABOUT Motls himself or other things he may have done or represent !! I do not feel capable to say something because in particular am ignorant of physics, but many of you are not. I think in Bill for example. So long Eduardo Dubuc From rrosebru@mta.ca Tue Mar 14 18:53:00 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 18:53:00 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIMQ-00031K-GC for categories-list@mta.ca; Tue, 14 Mar 2006 18:51:26 -0400 Message-ID: <4416E9D7.4030704@cs.stmarys.ca> Date: Tue, 14 Mar 2006 12:05:43 -0400 From: "Robert J. MacG. Dawson" User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.1 required=5.0 tests=FORGED_RCVD_HELO autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 Marta Bunge wrote: > Hi, > > I just came across the following pages > > http://motls.blogspot.com/2004/11/category-theory-and-physics.html > http://motls.blogspot.com/2004/11/this-week-208-analysis.html > > written by Lubos Motl, a physicist (string theorist). Some of you may find > these articles interesting and probably revealing. > > Are we category theorists as a whole going to quietly accept getting > discredited by a minority of us presumably applying category theory to > string theory? It is surely not too late to react and point out that > this is not what (all of) category theory is about. I don't see that we have any more need to do this than (for instance) algebraic topologist, group theorists, or differential geometers have when somebody floats a perhaps-too-conjectural theory using those branches of mathematics. Heck, physicists have managed to come up with what are now generally seen as dubious theories using nothing more than elementary arithmetic (Dirac's Big Numbers hypothesis, say.) Do the number theorists have to protest this? Big problems in physics have tended to be solved only after a lot of attempts that look pretty strange in retrospect (think of some of the early models of the atom!) But correct theories (or at least theories that represent a major improvement in understanding and prediction) can also look pretty strange; think how general relativity, or even special relativity, must have looked in their day. I seem to recall that the periodic table was originally considered at least as dubious as Bode's Law - and if they had been able to measure molecular masses more accurately in Mendeleev's day, they would have seen that the main idea was actually _wrong_, and its acceptance would probably have had to await the technology to separate individual isotopes, which do have (reasonably) predictable masses. Quaternions were fashionable in Victorian days to represent motions in space, dropped out of fashion when people decided that the restriction of their applicability to three-dimensional space was parochial, and dropped back in again when people realized that in fact a three-plus-one-dimensional spacetime had some rather special properties. Mathematics, like the phone service, is a "common carrier". We develop it; we use it; but we have neither the right nor the obligation to police how others apply it (unless they get the mathematics itself wrong?). Moreover, given the historical difficulty of recognizing good physical theories ahead of time, it would be impossible to do so wisely even if we had the right. I do not see how anybody can possibly discredit category theory by applying it to string theory, even inappropriately, any more than "The da Vinci Code" discredits classical geometry and number theory. -Robert Dawson From rrosebru@mta.ca Tue Mar 14 19:00:54 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 19:00:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJITr-0003dN-H3 for categories-list@mta.ca; Tue, 14 Mar 2006 18:59:07 -0400 Reply-To: marta.bunge@mcgill.ca In-Reply-To: <4416E9D7.4030704@cs.stmarys.ca> From: "Marta Bunge" To: categories@mta.ca Subject: categories: Re: cracks and pots Date: Tue, 14 Mar 2006 11:30:21 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 43 Dear Robert, I agree with most of what you say, and I was not suggesting that we police how categorists choose to apply their field. Nothing further from my mind. > Mathematics, like the phone service, is a "common carrier". We develop it; >we use it; but we have neither the right nor the obligation to police how >others apply it (unless they get the mathematics itself wrong?). Moreover, >given the historical difficulty of recognizing good physical theories ahead >of time, it would be impossible to do so wisely even if we had the right. But organizers of meetings in category-related subjects can certainly direct attention to certain trends in category theory, thereby promoting certain areas over others, and this they can easily do by their choice of invited speakers of (series of) lectures. They may have neither the right nor the obligation to do so, but they certainly have the power to do so. If this happens consistently, then the outcome is predictable. Students (and their advisors) might flock to certain areas of research just because they are fashionable and can thus get funding that otherwise will not be easily obtained. This may lead to narrow developments of any subject that they approach with this objective in mind, and that is dangerous for the future of category theory (of mathematics, in general). That is my main concern. My posting tried to call attention to what I think is a sad state of affairs in category theory, when it need not be. Best wishes, Marta From rrosebru@mta.ca Tue Mar 14 19:01:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 19:01:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIVg-0003mJ-T6 for categories-list@mta.ca; Tue, 14 Mar 2006 19:01:00 -0400 Message-ID: <4416F8A2.9080300@cs.stmarys.ca> Date: Tue, 14 Mar 2006 13:08:50 -0400 From: "Robert J. MacG. Dawson" User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 44 Marta Bunge wrote; This [inviting researchers in fashionable applied areas to speak at category theory meetings] may lead to narrow > developments of any subject that they approach with this objective in > mind, and that is dangerous for the future of category theory (of > mathematics, in general). That is my main concern. My posting tried to > call attention to what I think is a sad state of affairs in category > theory, when it need not be. It is not clear to me that the majority of theoretical physicists agree with the negative view of categorical string theory held by the cited blog writers; and in the absence of a consensus among the physicists, I for one (with an undergradate degree and some graduate courses in physics) do not feel qualified to take sides; if anything, errors should be on the side of trying out too many ideas, not too few. I have this image of differential geometers saying to each other, a century ago, "Don't you think somebody ought to tell that Einstein to stop trying to use differential geometry to explain gravity, before our whole field gets a bad name?" Of course, the pioneering knot theorists probably thought that Lord Kelvin ought to stop trying to explain atomic nuclei as knotted loops of ether, too. But I think Einstein did differential geometry more good than Kelvin did harm to knot theory. A mathematical technique powerful enough to show that a physical theory does *not* work has shown its own value. What has sometimes gone on, at least for a while, is that very abstract physical theories have continued to be studied after it had become obvious that their predictions were wildly at variance with observation, or that they would never make any predictions. Even then I don't think the reputation of the mathematical theory being abused suffers, though that of the neighboring theoretical physicists may. I don't think this is the case with string theory yet, though I could be wrong. Cheers, Robert Dawson From rrosebru@mta.ca Tue Mar 14 19:03:04 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 19:03:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIWr-0003u8-D2 for categories-list@mta.ca; Tue, 14 Mar 2006 19:02:13 -0400 Reply-To: marta.bunge@mcgill.ca In-Reply-To: <4416F8A2.9080300@cs.stmarys.ca> From: "Marta Bunge" To: categories@mta.ca Subject: categories: Re: cracks and pots Date: Tue, 14 Mar 2006 12:48:33 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 45 Robert Dawson wrote: > It is not clear to me that the majority of theoretical physicists agree >with the negative view of categorical string theory held by the cited blog >writers; and in the absence of a consensus among the physicists, I for one >(with an undergradate degree and some graduate courses in physics) do not >feel qualified to take sides; if anything, errors should be on the side of >trying out too many ideas, not too few. > I was trying to elicit an open response from those who *do* know about the value (or lack of it) of categorical string theory. In particular, I would like to have an answer to this question. Why is it that anything which even remotedly claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles? Best, Marta From rrosebru@mta.ca Tue Mar 14 19:03:56 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 19:03:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIXu-00042Y-6h for categories-list@mta.ca; Tue, 14 Mar 2006 19:03:18 -0400 Message-Id: <200603141956.k2EJu9625544@math-cl-n03.ucr.edu> To: categories@mta.ca Subject: categories: Re: cracks and pots Date: Tue, 14 Mar 2006 11:56:09 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 46 Hi - > I just came across the following pages > > http://motls.blogspot.com/2004/11/category-theory-and-physics.html > http://motls.blogspot.com/2004/11/this-week-208-analysis.html > > written by Lubos Motl, a physicist (string theorist). Some of you may find > these articles interesting and probably revealing. > > Are we category theorists as a whole going to quietly accept getting > discredited by a minority of us presumably applying category theory to > string theory? I can't tell if you're kidding. I'll assume you're not. There's nothing wrong with applying category theory to string theory. The papers by Michael Douglas and Paul Aspinwall cited above by Motl are some nice examples of using derived categories to study D-branes. Further examples: the Moore-Seiberg relations turn out to be little more than the definition of a balanced monoidal category, and the Segal-Moore axioms for open-closed topological strings are nicely captured using category theory here: http://arxiv.org/abs/math.AT/0510664 There were a lot of nice talks on the borderline between category theory and string theory at the Streetfest. Perhaps more to the point, Lubos Motl is famous for his heated rhetoric. He doesn't like me, or anyone else who criticizes string theory. The articles you mention above are mainly reactions to my This Week's Finds. He's actually being very gentle - for him. He even says "the role of category theory can therefore be described as a `progressive direction' within string theory". I'm sure you'll all be pleased to know that. :-) > It is surely not too late to react and point out that this is > not what (all of) category theory is about. I would urge everyone not to react - at least, not until they are well aware of what a discussion with him is like. See his blog and his comments on Peter Woit's blog if you don't understand what I mean. For example: http://pitofbabel.org/blog/?p=51 > Please give a thought about what > we, as a community, can urgently do to repair this damaging impression. Since Motl's personality is well known, any damage will be minimal. I think we should relax and take it easy. Best, jb From rrosebru@mta.ca Tue Mar 14 19:08:55 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Mar 2006 19:08:55 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJIcY-0004SK-Gz for categories-list@mta.ca; Tue, 14 Mar 2006 19:08:06 -0400 Message-ID: <00cf01c647b8$cc302660$e462893e@brown1> From: "Ronald Brown" To: References: <003801c6404f$70d2e020$9ea24e51@brown1> <440C7F19.7080705@math.upenn.edu> Subject: categories: Re: Alexander Grothendieck on `speculation' Date: Tue, 14 Mar 2006 22:43:17 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 47 Jim Stasheff writes: > This suggests two possibilities: > > for the brave, start your own blog for speculations > for the timid, same input but into a file only you can access > until late in life and famous you can show how you had the ideas all > along The situation is more complicated in that what could be classed as speculation may get published as theorem and proof. For example, in algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago by Eldon Dyer in relation to results on local fibration implies global fibration (for paracompact spaces) where he and Eilenberg felt Dold's paper on this contained the first complete proof. I have been unable to complete the proof in Spanier's book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn't it continuous?') Eldon speculated (!) that perhaps 50% of published algebraic topology was seriously wrong! van Kampen's original 1935 `proof' of what is called his theorem is incomprehensible today, and maybe was then also. Efforts to give full details of a major result, i.e. to give a proof, are sometimes derided. Of course credit should be given to the originator of the major steps towards a proof. Grothendieck's efforts to develop structures and language which would reduce proofs to a sequence of tautologies are notable here. Colin McLarty's excellent article on `The rising sea: Grothendieck on simplicity and generality ' is relevant. Some scientists snear at the mathematical notion of rigour and of proof. On the other hand many are attracted to math because it can give explanations of why something is true. But `explanations' need a higher level of structural language than for what might be called proofs. I can't resist mentioning that one student questionaire on my first year analysis wrote `Professor Brown puts in too many proofs.' So I determined to rectify the situation, and next year there were no theorems, and no proofs. However there were lots of statements labelled `FACT' followed by several paragraphs labelled `EXPLANATION'. This did modify the course because something labelled `explanation' ought really to explain something! I leave you all to puzzle this out! In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used. Surely much early algebraic topology is speculative, in that the language has not yet been developed to express concepts with rigour so that a clear proof can be written down. It would be a curious ahistorical assumption that there is not at this date another future level of concepts which require a similar speculative approach to reach towards them. Ronnie Brown From rrosebru@mta.ca Wed Mar 15 19:18:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfF0-0003XW-6G for categories-list@mta.ca; Wed, 15 Mar 2006 19:17:18 -0400 Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <200603141956.k2EJu9625544@math-cl-n03.ucr.edu> References: <200603141956.k2EJu9625544@math-cl-n03.ucr.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <3618C00E-DF36-4BD9-9BC3-A51AB6D1D422@mac.com> Content-Transfer-Encoding: 7bit From: Krzysztof Worytkiewicz Date: Wed, 15 Mar 2006 12:26:53 -0500 To: categories@mta.ca Subject: categories: Re: cracks and pots Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 48 The blog in question is indeed more than dubious. Besides the "scientific" manicheism (group good, monoid bad...), what to think about ranking countries according to a "civilization index"? The blogger also claims he was mastering differential geometry and particle physics at age of 15, so he obviously was too busy and missed the provocative phase. Not a reason however to try to catch it up as an "adult". Cheers Krzysztof -- my government will categorically deny the incident ever occurred From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJf8G-00032v-7q for categories-list@mta.ca; Wed, 15 Mar 2006 19:10:20 -0400 X-MimeOLE: Produced By Microsoft Exchange V6.5.7226.0 Content-class: urn:content-classes:message MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Subject: categories: RE: An autonomous category Date: Wed, 15 Mar 2006 11:58:36 +1100 Message-ID: <039A7CE5BC8F554C81732F2505D511720290EF4B@BONHAM.AD.UWS.EDU.AU> From: "Stephen Lack" To: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 49 Dear Marco, This has been considered by Brian Day. He spoke about it in a talk *-autonomous convolution=20 in the Australian Category Seminar on 5 March 1999, You can also transform this via the log/exponential functions to an=20 additive tensor product on the extended (positive and negative) reals. Regards, Steve Lack. -----Original Message----- From: cat-dist@mta.ca on behalf of Marco Grandis Sent: Tue 14/03/2006 12:45 AM To: categories@mta.ca Subject: categories: An autonomous category =20 The Lawvere category of extended positive real numbers has also an autonomous structure, with a multiplicative tensor product (instead of the original additive one). Has this been considered somewhere? To be more explicit: The well-known article of Lawvere on "Metric spaces..." (Rend. Milano 1974, republished in TAC Reprints n. 1) introduced the category of extended positive real numbers, from 0 to oo (infinity included), with arrows x \geq y, equipped with a strict symmetric monoidal closed structure: the tensor product is the sum, the internal hom is truncated difference (with oo - oo =3D 0). Now, the same category can be equipped with a multiplicative tensor product, x.y. Provided we define 0.oo =3D oo (so that tensoring by any element preserves the initial object oo), this is again a strict symmetric monoidal closed structure, with hom(y, z) =3D z/y. Now, the 'undetermined forms' 0/0 and oo/oo are defined to be 0. The new multiplicative structure is even *-autonomous, with involution x* =3D 1/x (and 'nearly' compact). (Note that this choice of values of the undetermined forms comes from privileging the direction x \geq y, which is necessary if we want to view metric spaces, normed categories etc. as enriched categories). Marco Grandis From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJf5l-0002tR-RE for categories-list@mta.ca; Wed, 15 Mar 2006 19:07:45 -0400 Date: Tue, 14 Mar 2006 18:18:56 -0500 (EST) From: Robert Seely To: categories@mta.ca Subject: categories: Re: cracks and pots In-Reply-To: Message-ID: References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 50 I just posted a notice of a special session on Categorical Logic and Quantum Computation in the upcoming ASL meeting at UQAM - although the preparation for that session goes back many months, and it was always my intention to post a schedule for the session here, I was reminded to do so upon reading Marta's message; in a sense I see it as a partial reply (even if the applications to physics are not those of the postings Marta quoted). I think the mathematics of that session will be of a high standard - we hope many of you will attend to judge for yourselves! -= rags =- -- From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJf6e-0002x6-Ic for categories-list@mta.ca; Wed, 15 Mar 2006 19:08:40 -0400 X-Authentication-Warning: thue.Stanford.EDU: dominic owned process doing -bs Date: Tue, 14 Mar 2006 15:26:31 -0800 (PST) From: Dominic Hughes To: categories@mta.ca Subject: categories: Re: cracks and pots In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 51 Roger Penrose, page 960 of "The Road to Reality - A Complete Guide to the Laws of the Universe": Another idea that may someday find a significant role to play in physical theory is *category* theory and its generalisation to n-category theory. [...] It would not altogether surprise me to find these notions playing some significant role in superseding conventional spacetime notions in the physics of the 21st century. Dominic http://boole.stanford.edu/~dominic From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfDt-0003TD-OG for categories-list@mta.ca; Wed, 15 Mar 2006 19:16:09 -0400 From: "RFC Walters" To: Subject: categories: Re: cracks and pots Date: Wed, 15 Mar 2006 14:35:38 +0100 Message-ID: <001101c64835$5a2a8380$f0a2cec1@WaltersDeskTop> MIME-Version: 1.0 Content-Type: text/plain;charset="US-ASCII" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 52 I also would like to support the remarks of Marta with which I am in full agreement. The category theory community seems happy to accept uncritically, and give centre-stage to, any interest shown by an external field. In this context one should certainly look the gift horse in the mouth. Bob Walters From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfA1-000393-Eo for categories-list@mta.ca; Wed, 15 Mar 2006 19:12:09 -0400 From: Gaucher Philippe To: categories@mta.ca Subject: categories: Re: Alexander Grothendieck on `speculation' Date: Wed, 15 Mar 2006 10:31:40 +0100 User-Agent: KMail/1.8.2 References: <003801c6404f$70d2e020$9ea24e51@brown1> <440C7F19.7080705@math.upenn.edu> <00cf01c647b8$cc302660$e462893e@brown1> In-Reply-To: <00cf01c647b8$cc302660$e462893e@brown1> MIME-Version: 1.0 Content-Disposition: inline Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Message-Id: <200603151031.40471.gaucher@pps.jussieu.fr> Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 53 Le Mardi 14 Mars 2006 23:43, vous avez =E9crit=A0: > Jim Stasheff writes: > > This suggests two possibilities: > > > > for the brave, start your own blog for speculations > > for the timid, same input but into a file only you can access > > until late in life and famous you can show how you had the ideas all > > along > > The situation is more complicated in that what could be classed as > speculation may get published as theorem and proof. For example, in > algebraic topology, sometimes proofs of continuity are omitted as if this > was an exercise for the reader, yet the formulation of why the maps are > continuous (if they are necessarily so) may contain a key aspect of what > should be a complete proof. This difficulty was pointed out to me years a= go > by Eldon Dyer in relation to results on local fibration implies global > fibration (for paracompact spaces) where he and Eilenberg felt Dold's pap= er > on this > contained the first complete proof. I have been unable to complete the > proof in Spanier's book, even the second edition. (I sent a correction to > Spanier as the key function in the first edition was not well defined, > after Spanier had replied `Isn't it continuous?') Eldon speculated (!) > that perhaps 50% of published algebraic topology was seriously wrong! My guess is that most of the algebraic topologists assume that the map=20 they are constructing is automatically continuous since the proof will work= =20 for example for simplicial sets (in which there is no continuity to check).= =20 And this argument is wrong : because the category of general topological=20 spaces is not cartesian closed while the category of simplicial sets is=20 cartesian closed. And most of these proofs of continuity become possible on= ly=20 by working in a more convenient category of topological spaces. pg. From rrosebru@mta.ca Wed Mar 15 19:18:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfCK-0003LO-EL for categories-list@mta.ca; Wed, 15 Mar 2006 19:14:32 -0400 Reply-To: marta.bunge@mcgill.ca In-Reply-To: <200603141956.k2EJu9625544@math-cl-n03.ucr.edu> From: "Marta Bunge" To: categories@mta.ca Subject: categories: Re: cracks and pots Date: Wed, 15 Mar 2006 07:23:28 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 54 Hi, I am relieved to learn (from the postings by David Yetter and John Baez) that Motl's blog on the issue of categories and string theory is based on 1) (Yetter) Motl's reluctance, as is the case with many string theorists, to refuse to learn category theory, and 2) (Baez) Motl's personal dislike of John Baez and of many other people, so that since Motl's personality is well-known, any damage will be minimal. I have also been reminded that 1) (Yetter) categorical ideas are central to several competitors of string theory, and that there is nothing cracked or crackpotish about them, and 2) (Baez) there is some serious work in the borderline of category theory and string theory as exemplified by several speakers at the StreetFest. I thank David and John for taking the trouble to respond in detail to what may have seem as a "provocation" on my part (well, perhaps it was...). But these informative responses do not address my main concern, which is one that others (publicly, as Eduardo Dubuc, but several others privately) have expressed to me following my posting. I was aiming at the fact that there is a certain trend within category theory (when did it start?) to consistely give center stage to anything that claims to have connections with physics (in particular string theory). Is this because (it is believed that) the state of category theory is now so poor (as "evidenced" by the lack of grants) that they (the organizers of meetings) want to repair this image at any cost? Also, by so doing, are we not becomeing vulnerable? Are we not pushing students to work on a certain area on the grounds that it is fashionable and likely to be funded, even if those students may lack the motivation and sound background knowledge? I feel that this is dangerous for category theory (and mathematics in general), as it may lead (is leading?) to narrow developments of any subject that is approached with these objectives in mind. I did point these concerns of mine already, in response to the posting by Robert MacDawson, whom I also thank for giving me the opportunity to make clearer what my real concerns are. On the subject of what constitutes good mathematics, Ronnie Brown has pointed out to me a beautiful expose (with Tim Porter) which you can find in www.bangor.ac.uk/r.brown/publar.html I urge you to read it. I end with a quote from the end of David Yetter's posting in reply to mine. "If (I suspect when) the string theory emperor turns out to have no clothes, category theory will suddenly become de rigeur in physics". I share his optimism. Most cordially, Marta Bunge ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************ >From: "John Baez" >To: categories@mta.ca >Subject: categories: Re: cracks and pots >Date: Tue, 14 Mar 2006 11:56:09 -0800 (PST) > >Hi - > > > I just came across the following pages > > > > http://motls.blogspot.com/2004/11/category-theory-and-physics.html > > http://motls.blogspot.com/2004/11/this-week-208-analysis.html > > > > written by Lubos Motl, a physicist (string theorist). Some of you may >find > > these articles interesting and probably revealing. > > > > Are we category theorists as a whole going to quietly accept getting > > discredited by a minority of us presumably applying category theory to > > string theory? > >I can't tell if you're kidding. I'll assume you're not. > >There's nothing wrong with applying category theory to string theory. >The papers by Michael Douglas and Paul Aspinwall cited above by Motl >are some nice examples of using derived categories to study D-branes. > >Further examples: the Moore-Seiberg relations turn out to be little >more than the definition of a balanced monoidal category, and the >Segal-Moore axioms for open-closed topological strings are nicely >captured using category theory here: > >http://arxiv.org/abs/math.AT/0510664 > >There were a lot of nice talks on the borderline between category >theory and string theory at the Streetfest. > >Perhaps more to the point, Lubos Motl is famous for his heated >rhetoric. He doesn't like me, or anyone else who criticizes >string theory. The articles you mention above are mainly reactions >to my This Week's Finds. > >He's actually being very gentle - for him. He even says "the >role of category theory can therefore be described as a `progressive >direction' within string theory". > >I'm sure you'll all be pleased to know that. :-) > > > It is surely not too late to react and point out that this is > > not what (all of) category theory is about. > >I would urge everyone not to react - at least, not until they are >well aware of what a discussion with him is like. See his blog >and his comments on Peter Woit's blog if you don't understand what >I mean. For example: > >http://pitofbabel.org/blog/?p=51 > > > Please give a thought about what > > we, as a community, can urgently do to repair this damaging impression. > >Since Motl's personality is well known, any damage will be minimal. >I think we should relax and take it easy. > >Best, >jb > > > > > > > > From rrosebru@mta.ca Wed Mar 15 19:18:50 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:18:50 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfB3-0003F0-Kx for categories-list@mta.ca; Wed, 15 Mar 2006 19:13:13 -0400 Message-ID: <44180179.2080105@mcs.le.ac.uk> Date: Wed, 15 Mar 2006 11:58:49 +0000 From: Alexander Kurz MIME-Version: 1.0 To: categories@mta.ca Subject: categories: MGS 2006 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 55 ************************************************************* Midlands Graduate School 2006 in the Foundations of Computing and MGS Workshop 2006 ************************************************************* 2nd Call for Participation The Midlands Graduate School is taking place 8 - 12 April 2006 at the University of Leicester, UK. A timetable is now available at our updated website http://www.cs.le.ac.uk/~mgs2006 The School finishes on Wednesday at lunch time. Afterwards a small workshop will be held to give PhD students the opportunity to present their own work. If you are interested send a title and abstract to mgs2006 at mcs.le.ac.uk. The School provides an intensive course of lectures on the Foundations of Computing. It is very well established, having run annually for the past six years, and has always proved a popular and successful event. This year we have Luke Ong, Oxford University and Thomas Streicher, Darmstadt University as guest lecturers. The lectures are aimed at graduate students, typically in their first or second year of study for a PhD. However, the school is open to anyone who is interested in learning more about mathematical computing foundations, and we especially invite participants from UK universities and from sites participating in the APPSEM working group. Foundational courses: R Crole Leicester Operational Semantics P Levy Birmingham Typed Lambda Calculus D Pattinson Leicester Category Theory Advanced courses: T Altenkirch Nottingham Quantum Programming M Escardo Birmingham Operational Domain Theory & Topology H Nilsson Nottingham Advanced Functional Programming L Ong Oxford Game Semantics T Streicher Darmstadt Constructive Logic E Tuosto Leicester Concurrency and Mobility We still have a small number grants for students resident in the UK, while APPSEM funds can be used to support students from APPSEM affiliated sites. For further details and registration please visit http://www.cs.le.ac.uk/~mgs2006 From rrosebru@mta.ca Wed Mar 15 19:19:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:19:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfGB-0003d8-Uz for categories-list@mta.ca; Wed, 15 Mar 2006 19:18:31 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: preprint: Categories, norms and weights Date: Wed, 15 Mar 2006 19:18:16 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 56 The following preprint is available: Marco Grandis Categories, norms and weights Dip. Mat. Univ. Genova, Preprint 538 (2006), 14 p. http://www.dima.unige.it/~grandis/wCat.pdf http://www.dima.unige.it/~grandis/wCat.ps Abstract. The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous and a CL-algebra (linked with classical linear logic). Normed sets, with a norm in R, inherit thus two symmetric monoidal closed structures, and categories enriched on one of them have a 'subadditive' or 'submultiplicative' norm, respectively. Typically, the first case occurs when the norm expresses a cost, the second with Lipschitz norms. This paper is a preparation for a sequel, devoted to 'weighted algebraic topology', an enrichment of directed algebraic topology. The structure of R, and its extension to the complex projective line, might be a first step in abstracting a notion of algebra of weights, linked with physical measures. From rrosebru@mta.ca Sat Mar 4 10:38:44 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 04 Mar 2006 10:38:44 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFXqP-0003bg-1E for categories-list@mta.ca; Sat, 04 Mar 2006 10:34:53 -0400 Message-ID: <004701c63f0e$499dacc0$239d4c51@brown1> From: "Ronald Brown" To: Subject: categories: Now available: Topology and groupoids, by Ronald Brown Date: Fri, 3 Mar 2006 22:02:29 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: A X-Keywords: X-UID: 57 This is a retitled, revised, refined, expanded edition of my topology book previously published in 1968 and 1988, and now available at the amazing price of $23.99. To order, please use the following link: http://www.booksurge.com/product.php3?bookID=GPUB05314-00001&affiliateID=A001356 Their Global Publishing System should allow more local printing and delivery. 538 pages. List of chapters: Some topology on the real line Topological spaces Connected spaces, compact spaces Identification spaces and cell complexes Projective and other spaces The fundamental groupoid Some combinatorial groupoid theory Cofibrations Computation of the fundamental groupoid Covering spaces, covering groupoids Orbit spaces, orbit groupoids Conclusion It combines a geometric and a categorical approach, where the latter allows clear analogies between areas of mathematics, such as topology and algebra. This is seen very clearly in the chapters on covering spaces and orbit spaces. Here are some examples of topics covered not available in other texts at this level: the initial topology on joins of spaces; the convenient category of k-spaces in the non Hausdorff case; a gluing theorem for homotopy equivalences; the Phragmen-Brouwer property, proved using the groupoid van Kampen theorem, and applied to the Jordan Curve theorem; the equivalence between covering maps of spaces and covering morphisms of groupoids; the fundamental groupoid of an orbit space by a discontinuous action of a group as the orbit groupoid of the fundamental groupoid. The book is graced by a cover by John Robinson and Ben Dickens. Ronnie Brown www.bangor.ac.uk/r.brown www.popmath.org.uk From rrosebru@mta.ca Wed Mar 15 19:36:39 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Mar 2006 19:36:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJfWD-0004zN-RS for categories-list@mta.ca; Wed, 15 Mar 2006 19:35:05 -0400 From: Eduardo Dubuc Date: Wed, 15 Mar 2006 18:00:07 -0300 (ART) To: categories@mta.ca Subject: categories: Re: cracks and pots MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 58 Hi: I will put quotations from different postings or Molt's writing in between two "**" Well, I can see the classical reaction of some groups when one of its members points out that something is really wrong with the group. Marta will suffer all kind of "polite" (nothing of the sort of the Benabou-Taylor confrontation) attacks, but not for this less devious or sanguine. Typically she will be taken out of context, or get answers to questions she never had asked, or be treated ironically or in disbelief (**I can't tell if you're kidding. I'll assume you're not **) There are two principal points here: 1. The real value of some contributions of category theory to physics. 2. The lot of rubbish written using category theory and which is fashionable because it claims to have applications to physics. Marta was forced to explicit some of the questions we can clearly see in between lines in her original posting: ** I was trying to elicit an open response from those who *do* know about the value (or lack of it) of categorical string theory. In particular, I would like to have an answer to this question. Why is it that anything which even remotely claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles? Best, Marta ** I will like to see a clear answer to this question. Or a clear refutation proving that it is not the case. Notice that the existence of point 2. above is perfectly consistent with the existence of really valuable contributions of category theory to string theory, which is one of the points treated by Motl. ** There's nothing wrong with applying category theory to string theory. The papers by Michael Douglas and Paul Aspinwall cited above by Motl are some nice examples of using derived categories to study D-branes.** This make us think that they may be some valuable contributions, but this possibility is also left open by Motl himself. Quoting myself: ** I will like to see here a debate about Motls's writing quoted above. Just about this writing, NOT ABOUT Motls himself or other things he may have done or represent !! ** No luck, just discredit Motl, not refute his sayings: ** Perhaps more to the point, Lubos Motl is famous for his heated rhetoric. He doesn't like me, or anyone else who criticizes string theory. The articles you mention above are mainly reactions to my This Week's Finds. ** ** My reaction to the blog posts you cite is that this is a sting theorist holding his breath and refusing to learn category theory. My guess is that Motl wouldn't want to learn the heavily categorical formulations of mirror symmetry that Yan Soibelman uses, even though they are motivated by string theory.** The following is better in answering Motl: ** Categorical ideas are absolutely central to several competitors to string theory: the Barrett-Crane model of quantum gravity (and to a lesser extent 'loop quantum gravity' with which the BC model is often conflated) and Connes' recovery of the Standard Model from non-commutative geometry (a part of mathematics which has obliged reluctant mathematicians to think about categorical ideas deeper than they originally were comfortable with). There is nothing cracked or crackpot about either. ** I am unable to judge, but it seems to me this gives category theory strong support But does not go against what Motl says concerning category theory. Neither against Marta's warning that category theory is being discredited by many (she says a minority) category theory people. Motl writes: ** I've asked the same elementary questions to many people who've been trying to explain me derived categories - some of them with some success, most of them with no success whatsoever: Are these notions and statements of category theory something that you can prove - or at least check in many situations - to be valid for string theory as we know it, or is it just an unproven conjecture that derived categories describe D-branes? ** Can somebody give a an answer ? He also writes: ** I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts". ** I would say that any serious scientist or mathematician would feel the same way !!, and also that this seems to be a common practice in many papers that claim applications of category theory to physics. ** I have this image of differential geometers saying to each other, a century ago, "Don't you think somebody ought to tell that Einstein to stop trying to use differential geometry to explain gravity, before our whole field gets a bad name?" ** Well, Einstein was not "trying to", he was using it, and presented this use as an accomplished fact. Also, you forgot to mention that he flunk a high-school exam or something of the sort proving by this very fact that a lot of people were stupid, just as they are those which have doubts about the real value of some applications of category theory to physics ! ** I do not see how anybody can possibly discredit category theory by applying it to string theory, even inappropriately, any more than "The da Vinci Code" discredits classical geometry and number theory. ** ** Since Motl's personality is well known, any damage will be minimal. I think we should relax and take it easy.** Well, rubbish category theory always discredits the whole of category theory, specially given the fact that it is not yet a prestigious and established subject (think in SGA4 and the introduction of SGA41/2) It will be nice to relax and take it easy. Will all of us do so ? I hope we will read in this cat-list some valuable considerations about Motl's questions and doubts, and about Marta's courageous warnings. e.d. From rrosebru@mta.ca Thu Mar 16 05:29:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 05:29:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJoia-0005na-9f for categories-list@mta.ca; Thu, 16 Mar 2006 05:24:28 -0400 Message-ID: <4418C532.20706@math.upenn.edu> Date: Wed, 15 Mar 2006 20:53:54 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To categories@mta.ca Subject: categories: Re: cracks and pots References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 59 Mostly well said, David I would only modify/deform ;-) what you say by doubting therte are that many physicists who are anti-cat theory (not pro-) but watch out once some leader of the school adopts it the school will follow - much faster than if they were mathematicians jim David Yetter wrote: > Dear Marta, > > My reaction to the blog posts you cite is that this is a sting theorist > holding > his breath and refusing to learn category theory. My guess is that Motl > wouldn't > want to learn the heavily categorical formulations of mirror symmetry > that Yan > Soibelman uses, even though they are motivated by string theory. > Basically > categorical ideas aren't part of the standard bag of tricks physicists > use (even > though they often give much more elegant, concise, and insightful > formulations of some of those tricks), and the proverb about 'old dogs' > and > 'new tricks' applies to physicists as well. > > His attack on Baez is fairly standard stuff: in the mode of "string > theory > is the theory of nature, so we don't want to think about alternatives > like > loop quantum gravity." It is a polemical defense of a scientific > theory that > hasn't produced a testable prediction in the 40 plus years since its > inception, > and worse than that, unless one adds bells and whistles to fix it (in > the manner > of 'gaseous Vulcan' or Ptolemaic epicycles), predicts the existence > of a massless scalar field *not observed in nature*. It really has > nothing at > all to say about category theory, which is after all a mathematical > theory > which stands irrespective of its extra-mathematical applications. > > Categorical ideas are absolutely central to several competitors to > string theory: > the Barrett-Crane model of quantum gravity (and to a lesser > extent 'loop quantum gravity' with which the BC model is often > conflated) > and Connes' recovery of the Standard Model from non-commutative geometry > (a part of mathematics which has obliged reluctant mathematicians to > think about > categorical ideas deeper than they originally were comfortable with). > There is nothing > cracked or crackpot about either. > > It is simply a fact we have to live with that our subject has found > legitimate uses > in physics, but uses which are unpopular with the dominant school of > physics in > the North America. If (I suspect when) the string theory emperor turns > out > to have no clothes, category theory will suddenly become de rigeur in > physics. (As it should, since categorical expressions of physical > ideas are the logical conclusion of 20th century physics drive to > express > everything in coordinate-free terms.) > > Best Thoughts, > David Yetter > > > > > > > > > > On 12 Mar 2006, at 17:29, Marta Bunge wrote: > >> Hi, >> >> I just came across the following pages >> >> http://motls.blogspot.com/2004/11/category-theory-and-physics.html >> http://motls.blogspot.com/2004/11/this-week-208-analysis.html >> >> written by Lubos Motl, a physicist (string theorist). Some of you may >> find >> these articles interesting and probably revealing. >> >> Are we category theorists as a whole going to quietly accept getting >> discredited by a minority of us presumably applying category theory to >> string theory? It is surely not too late to react and point out that >> this is >> not what (all of) category theory is about. Please give a thought >> about what >> we, as a community, can urgently do to repair this damaging impression. >> Unless we are prepared to wait until things change by themselves >> within our >> lifetime. >> >> >> Hopefully disturbing your weekend, >> Cordially, >> Marta >> >> >> >> ************************************************ >> Marta Bunge >> Professor Emerita >> Dept of Mathematics and Statistics >> McGill University >> 805 Sherbrooke St. West >> Montreal, QC, Canada H3A 2K6 >> Office: (514) 398-3810 >> Home: (514) 935-3618 >> marta.bunge@mcgill.ca >> http://www.math.mcgill.ca/bunge/ >> ************************************************ >> >> >> >> From rrosebru@mta.ca Thu Mar 16 05:29:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 05:29:10 -0400 Message-ID: <4418C655.4020109@math.upenn.edu> Date: Wed, 15 Mar 2006 20:58:45 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To categories@mta.ca Subject: categories: Re: cracks and pots In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 60 Marta Bunge wrote: > > I was trying to elicit an open response from those who *do* know about the > value (or lack of it) of categorical string theory. In particular, I would > like to have an answer to this question. Why is it that anything which even > remotedly claims to have applications to physics (particularly string > theory) is > given (what I view as) uncritical support in our circles? > > Best, > Marta > It's not so much the applications that seduce some of us but rather the *new* structures the physicists suggest that turn out to have neat mathemaical, e.g. categorical, aspects. e.g quantum groups jim From rrosebru@mta.ca Thu Mar 16 05:29:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 05:29:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJolS-0005rH-Jo for categories-list@mta.ca; Thu, 16 Mar 2006 05:27:26 -0400 Message-ID: <4418C85D.10506@math.upenn.edu> Date: Wed, 15 Mar 2006 21:07:25 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To categories@mta.ca Subject: categories: Re: cracks and pots In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 61 > I was aiming at the fact that > there is > a certain trend within category theory (when did it start?) to consistely > give center stage to anything that claims to have connections with physics > (in particular string theory). Is this because (it is believed that) the > state of category theory is now so poor (as "evidenced" by the lack of > grants) that they (the organizers of meetings) want to repair this image at > any cost? Also, by so doing, are we not becomeing vulnerable? Are we not > pushing students to work on a certain area on the grounds that it is > fashionable and likely to be funded, even if those students may lack the > motivation and sound background knowledge? I feel that this is dangerous > for > category theory (and mathematics in general), as it may lead (is leading?) > to narrow developments of any subject that is approached with these > objectives in mind. I did point these concerns of mine already, in response > to the posting by Robert MacDawson, whom I also thank for giving me the > opportunity to make clearer what my real concerns are. > Consider instead what happened in algebraic topology in the last century (or in invariant theory of polynomial forms in the previous one): classic internal problems e.g. homotopy groups of spheres ground on and on while the enthusiasm and excitement of `application' motivated problems died with a lack of such problems (I have in mind vector fields on spheres and allsorts of diff geom motivations). > On the subject of what constitutes good mathematics, Ronnie Brown has > pointed out to me a beautiful expose (with Tim Porter) which you can > find in > www.bangor.ac.uk/r.brown/publar.html > I urge you to read it. Exactly - if it's good math, it's not tainted by being invented by physicists. jim > > I end with a quote from the end of David Yetter's posting in reply to mine. > "If (I suspect when) the string theory emperor turns out to have no > clothes, > category theory will suddenly become de rigeur in physics". I share his > optimism. > > > Most cordially, > Marta Bunge > > > > > ************************************************ > Marta Bunge > Professor Emerita > Dept of Mathematics and Statistics > McGill University > 805 Sherbrooke St. West > Montreal, QC, Canada H3A 2K6 > Office: (514) 398-3810 > Home: (514) 935-3618 > marta.bunge@mcgill.ca > http://www.math.mcgill.ca/bunge/ > ************************************************ > > > > >> From: "John Baez" >> To: categories@mta.ca >> Subject: categories: Re: cracks and pots >> Date: Tue, 14 Mar 2006 11:56:09 -0800 (PST) >> >> Hi - >> >> > I just came across the following pages >> > >> > http://motls.blogspot.com/2004/11/category-theory-and-physics.html >> > http://motls.blogspot.com/2004/11/this-week-208-analysis.html >> > >> > written by Lubos Motl, a physicist (string theorist). Some of you may >> find >> > these articles interesting and probably revealing. >> > >> > Are we category theorists as a whole going to quietly accept getting >> > discredited by a minority of us presumably applying category theory to >> > string theory? >> >> I can't tell if you're kidding. I'll assume you're not. >> >> There's nothing wrong with applying category theory to string theory. >> The papers by Michael Douglas and Paul Aspinwall cited above by Motl >> are some nice examples of using derived categories to study D-branes. >> >> Further examples: the Moore-Seiberg relations turn out to be little >> more than the definition of a balanced monoidal category, and the >> Segal-Moore axioms for open-closed topological strings are nicely >> captured using category theory here: >> >> http://arxiv.org/abs/math.AT/0510664 >> >> There were a lot of nice talks on the borderline between category >> theory and string theory at the Streetfest. >> >> Perhaps more to the point, Lubos Motl is famous for his heated >> rhetoric. He doesn't like me, or anyone else who criticizes >> string theory. The articles you mention above are mainly reactions >> to my This Week's Finds. >> >> He's actually being very gentle - for him. He even says "the >> role of category theory can therefore be described as a `progressive >> direction' within string theory". >> >> I'm sure you'll all be pleased to know that. :-) >> >> > It is surely not too late to react and point out that this is >> > not what (all of) category theory is about. >> >> I would urge everyone not to react - at least, not until they are >> well aware of what a discussion with him is like. See his blog >> and his comments on Peter Woit's blog if you don't understand what >> I mean. For example: >> >> http://pitofbabel.org/blog/?p=51 >> >> > Please give a thought about what >> > we, as a community, can urgently do to repair this damaging impression. >> >> Since Motl's personality is well known, any damage will be minimal. >> I think we should relax and take it easy. >> >> Best, >> jb >> >> >> >> >> >> >> >> > > > From rrosebru@mta.ca Thu Mar 16 05:30:36 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 05:30:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FJoml-0005tF-Eo for categories-list@mta.ca; Thu, 16 Mar 2006 05:28:47 -0400 Message-ID: <4418C8AA.7050201@math.upenn.edu> Date: Wed, 15 Mar 2006 21:08:42 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To categories@mta.ca Subject: categories: Re: cracks and pots <3618C00E-DF36-4BD9-9BC3-A51AB6D1D422@mac.com> In-Reply-To: <3618C00E-DF36-4BD9-9BC3-A51AB6D1D422@mac.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 62 For that remember (if any are as old as I) matrices good, groups bad the gruppenpest jim Krzysztof Worytkiewicz wrote: > The blog in question is indeed more than dubious. Besides the > "scientific" manicheism (group good, monoid bad...), what to think > about ranking countries according to a "civilization index"? The > blogger also claims he was mastering differential geometry and > particle physics at age of 15, so he obviously was too busy and > missed the provocative phase. Not a reason however to try to catch it > up as an "adult". > > Cheers > > Krzysztof > > -- my government will categorically deny the incident ever occurred > > > > > From rrosebru@mta.ca Thu Mar 16 20:28:57 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:28:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2lF-0007M5-Vd for categories-list@mta.ca; Thu, 16 Mar 2006 20:24:10 -0400 Mime-Version: 1.0 (Apple Message framework v623) To: "categories@mta. ca" From: Marquis Jean-Pierre Subject: categories: Cracks and pots and the gruppenpest Date: Thu, 16 Mar 2006 09:32:19 -0500 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 63 With respect to the gruppenpest (and perhaps the actual situation concerning groupoids and categories), here is a quote from John Slater, who was the head of the MIT Physics departement and a leading American physicist: "It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their "Gruppenpest": the pest of the group theory.... The authors of the "Gruppenpest" wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practical consequences appreared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest. I had what I can only describe as a feeling of outrage at the turn which the subject had taken... As soon as this (Slater's) paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as "Slater has slain the '"Gruppenpest"'. I believe that no other piece of work I have done was so universally popular". I take this quote from Sternberg's book "Group Theory and Physics" who has taken it from Slater's autobiography. Maybe it has something to do with MIT ; -). Best, Jean-Pierre Marquis > Date: 15 mars 2006 21:08:42 GMT-05:00 > To categories@mta.ca > Subject: categories: Re: cracks and pots > > For that remember (if any are as old as I) > matrices good, groups bad > > the gruppenpest > > jim > > > Krzysztof Worytkiewicz wrote: >> The blog in question is indeed more than dubious. Besides the >> "scientific" manicheism (group good, monoid bad...), what to think >> about ranking countries according to a "civilization index"? The >> blogger also claims he was mastering differential geometry and >> particle physics at age of 15, so he obviously was too busy and >> missed the provocative phase. Not a reason however to try to catch it >> up as an "adult". >> >> Cheers >> >> Krzysztof >> From rrosebru@mta.ca Thu Mar 16 20:28:57 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:28:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2e7-0006x2-AE for categories-list@mta.ca; Thu, 16 Mar 2006 20:16:47 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: dusko Subject: categories: Re: cracks and pots To: categories@mta.ca Date: Thu, 16 Mar 2006 04:05:46 -0800 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 64 On Mar 15, 2006, at 5:35 AM, RFC Walters wrote: > The category theory community seems happy to accept uncritically, and > give centre-stage to, any interest shown by an external field. In this > context one should certainly look the gift horse in the mouth. i think this is a very nice metaphor. but i am not sure that being critical about science is as easy as looking in horses mouth. already hilbert was largely wrong when he tried to prescribe a shape of a science. and nowadays it is a much harder task. everyone sees just a very small fragment. research advances by evolution, not by intelligent design. the division between pure and applied mathematics is not as simple as it used to be. 20 years ago, if you wanted to work on something that would never ever degrade into applications, then algebraic geometry probably seemed like a good bet. nowadays, at each moment, millions of transactions on the internet are secured using elliptic and hyperelliptic curves; the structure of their picard groups is discussed in standardisation bodies. if a bank protects its customers from phishing by identity-based keys, they are using weil or tate pairing... so the purest math has become the most applied; the most spiritual the most concrete. the other way around, these applications put a babylonian library on everyone's desk. what was picard group again? google for it. biology research is based on large public databases. physics is documented (driven?) by blogs. even category theory is discussed online. so i think it is great that people get nasty, or personal about category theory. the landscape of babylon: "the dog barks while the caravan goes by." just my 2p, -- dusko From rrosebru@mta.ca Thu Mar 16 20:28:57 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:28:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2c8-0006r8-8z for categories-list@mta.ca; Thu, 16 Mar 2006 20:14:44 -0400 Date: Thu, 16 Mar 2006 19:48:25 -0500 From: Bob Rosebrugh To: Subject: categories: From moderator: reposting Stasheff/interruption Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 65 Due to an editing error on my part some of you will have received four posts from Jim Stasheff with no subject line, and in some cases (as Nimish Shah pointed out to Jim, who also alerted me) spam filters will have prevented delivery. With apologies to Jim, these messages are repeated below (except for some of the material repeated from earlier posts). Since I will be away from email access there will be an interruption in postings to this list until Wednesday starting a little more than 24 hours from now. Several new items in the current thread will be posted shortly, and responses to them arriving in the next day will be posted, but I would encourage you not to submit further items for posting until after Wednesday. best wishes, Bob Rosebrugh ----------------------------------------------------------------------------- Date: Wed, 15 Mar 2006 20:53:54 -0500 From: jim stasheff Subject: categories: Re: cracks and pots Mostly well said, David I would only modify/deform ;-) what you say by doubting therte are that many physicists who are anti-cat theory (not pro-) but watch out once some leader of the school adopts it the school will follow - much faster than if they were mathematicians jim David Yetter wrote: > Dear Marta, > > My reaction to the blog posts you cite is that this is a sting theorist > holding > his breath and refusing to learn category theory. My guess is that Motl > wouldn't > want to learn the heavily categorical formulations of mirror symmetry > that Yan > Soibelman uses, even though they are motivated by string theory. > Basically > categorical ideas aren't part of the standard bag of tricks physicists > use (even > though they often give much more elegant, concise, and insightful > formulations of some of those tricks), and the proverb about 'old dogs' > and > 'new tricks' applies to physicists as well. ... Date: Wed, 15 Mar 2006 20:58:45 -0500 From: jim stasheff Subject: categories: Re: cracks and pots Marta Bunge wrote: > > I was trying to elicit an open response from those who *do* know about the > value (or lack of it) of categorical string theory. In particular, I would > like to have an answer to this question. Why is it that anything which even > remotedly claims to have applications to physics (particularly string > theory) is > given (what I view as) uncritical support in our circles? > > Best, > Marta > It's not so much the applications that seduce some of us but rather the *new* structures the physicists suggest that turn out to have neat mathemaical, e.g. categorical, aspects. e.g quantum groups jim Date: Wed, 15 Mar 2006 21:07:25 -0500 From: jim stasheff Subject: categories: Re: cracks and pots > I was aiming at the fact that > there is > a certain trend within category theory (when did it start?) to consistely > give center stage to anything that claims to have connections with physics > (in particular string theory). Is this because (it is believed that) the > state of category theory is now so poor (as "evidenced" by the lack of > grants) that they (the organizers of meetings) want to repair this image at > any cost? Also, by so doing, are we not becomeing vulnerable? Are we not > pushing students to work on a certain area on the grounds that it is > fashionable and likely to be funded, even if those students may lack the > motivation and sound background knowledge? I feel that this is dangerous > for > category theory (and mathematics in general), as it may lead (is leading?) > to narrow developments of any subject that is approached with these > objectives in mind. I did point these concerns of mine already, in response > to the posting by Robert MacDawson, whom I also thank for giving me the > opportunity to make clearer what my real concerns are. > Consider instead what happened in algebraic topology in the last century (or in invariant theory of polynomial forms in the previous one): classic internal problems e.g. homotopy groups of spheres ground on and on while the enthusiasm and excitement of `application' motivated problems died with a lack of such problems (I have in mind vector fields on spheres and allsorts of diff geom motivations). > On the subject of what constitutes good mathematics, Ronnie Brown has > pointed out to me a beautiful expose (with Tim Porter) which you can > find in > www.bangor.ac.uk/r.brown/publar.html > I urge you to read it. Exactly - if it's good math, it's not tainted by being invented by physicists. jim Date: Wed, 15 Mar 2006 21:08:42 -0500 From: jim stasheff Subject: categories: Re: cracks and pots For that remember (if any are as old as I) matrices good, groups bad the gruppenpest jim Krzysztof Worytkiewicz wrote: > The blog in question is indeed more than dubious. Besides the > "scientific" manicheism (group good, monoid bad...), what to think > about ranking countries according to a "civilization index"? The > blogger also claims he was mastering differential geometry and > particle physics at age of 15, so he obviously was too busy and > missed the provocative phase. Not a reason however to try to catch it > up as an "adult". > > Cheers > > Krzysztof From rrosebru@mta.ca Thu Mar 16 20:28:57 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:28:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2oW-0007XA-9p for categories-list@mta.ca; Thu, 16 Mar 2006 20:27:32 -0400 To: categories@mta.ca Subject: categories: Re: cracks and pots Mime-Version: 1.0 Content-Type: text/plain; format=flowed From: "V. Schmitt" Date: Thu, 16 Mar 2006 09:51:00 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 66 Dear Marta, My english is so, so. I am french. But this is to give briefly my opinion (I agree with you more or less). I know a little of category and mathematics in general. I love the category theory developped in the 70's and I would have appreciated some category meetings at the time. But i am too young. Category theory like any good mathematics will never die - but may "our" category community will. Of course the problem is the way research is sponsored. Leading researchers are not so much good mathematicians but good salesmen. Category theory is just not very trendy at the minute and to get the money one needs to do theoretical physics (there had been also Computer Science at some point - that was poor is not it?). There were a couple of Fields medals and a new train called TQFT that everybody just jumps in to get funded. Now as a *community* what shall *we* do? First the question regards mainly the established people in the community (not me!). 1/ One can try to sell category theory in a better way. This is a bit like tomato sauce that you can put everywhere. And try to make new friends - inviting them to give talks... - from different disciplines. 2/ We may claim loudly that cat theory is real mathematics and really try hard to do good mathematics. There are certainly good mathematicians definitely willing to use cat theory. I saw many coming to category theory to develop their own maths (- this happens for instance in France with Berger who will never claim that he is a "categorician". Though he is completely in it!) My feeling is the attitude 1/ pushed to the extreme may be very damaging. These talks about category everywhere and for everything are just poor and sound really stupid. They do not serve the cause. 2/ Will be the rebirth of category - I bet! Sorry for the message written in haste and the poor english. Good e-mails from you on the list! best regards, Vincent. From rrosebru@mta.ca Thu Mar 16 20:30:03 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:30:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2pl-0007br-9E for categories-list@mta.ca; Thu, 16 Mar 2006 20:28:49 -0400 Message-ID: <44197C2E.2090300@cs.stmarys.ca> Date: Thu, 16 Mar 2006 10:54:38 -0400 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 67 Eduardo wrote: > Well, Einstein was not "trying to", he was using it, and presented this > use as an accomplished fact. He didn't just wake up one morning with the whole thing in finished form. Moreover, it was some time before it was experimentally verified; some details, such as the presence or absence of a cosmological constant, took some time to settle; some predictions (black holes, Big Bang) were not generally accepted for some time; and even now it is known *not* to be a good description of the universe at a very small scale. > Also, you forgot to mention that he flunk a high-school exam or something > of the sort proving by this very fact that a lot of people were stupid, > just as they are those which have doubts about the real value of some > applications of category theory to physics ! I did not "forget" to, it never occurred to me to do so, for two good reasons. Firstly, I don't see the relevance. Are you suggesting that (1) Einstein must have been stupid to flunk an exam, or that (2) his teacher and N-1 unspecified others were stupid because (2a) an exam was set that Einstein could flunk, or (2b) Einstein having flunked the exam, they did not recognize his future genius & change the grade? None of these conclusions seem justified to me... as my records at Dalhousie and Cambridge will show, people can flunk exams on bad days; I don't *think* I'm stupid, and I know the instructors who set the exams were not. But, secondly and more to the point, recent research suggests that the story of Einstein's failing grades is apocryphal. What seems to have happened is that his school changed over from a grading scheme with 1 high and 6 low to one with 6 high and 1 low, and a surviving report card had been misinterpreted. See for instance: http://www.abc.net.au/science/k2/moments/s1115185.htm -Robert From rrosebru@mta.ca Thu Mar 16 20:31:22 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:31:22 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2qt-0007hL-DV for categories-list@mta.ca; Thu, 16 Mar 2006 20:29:59 -0400 To: categories@mta.ca Subject: categories: pots and kettles From: Paul Taylor Date: Thu, 16 Mar 2006 15:56:55 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 68 I wonder what Eduardo Dubuc means by "the Benabou-Taylor confrontation". Could he possibly be referring to the Christmas flame-war to which Eduardo was himself a major contributor, but which I attempted to settle with him privately and amicably? I suggest that he and Marta Bunga should go for a long walk to calm down. I'm not pointing any fingers at Eduardo or Marta, but those who were active in category theory in the 1960s and 70s would be well advised to think twice before accusing those who came later of bringing category theory into disrepute. I wonder when the phrases "generalised abstract nonsense" and "empty set theory" came into circulation? I seem to remember hearing them when I was an undergraduate. I wonder which generation it was that got prestigeous jobs in mathematics departments at a time when (at least in Britain) they were being given away in corn flakes packets? Which generation is it that struggles to get jobs in computer science departments? Which subject was it, according to the participants of a parallel conference in Vancouver in 2004, whose followers were "easily recognisable because they're all so old"? Which generation was it that alienated other mathematicians by making outrageous claims about the foundations of mathematics that it never backed up with theorems? Which generation actually got its hands dirty and proved the theorems that relate category theory to other foundational disciplines? I have no idea what John Baez's standing is amongst physicists. Therefore Marta is quite correct in warning me that it would be foolish for me to base public claims of the value of category theory on its applications to physics. John is nevertheless very welcome in our community, in my humble opinion, BECAUSE he brings in new ideas from physics, BECAUSE he exports some of our ideas in return, and because he is good at presenting ideas of either kind. Also, I don't need to be specific about them to say that the applications to physics, computation, topology, geometry, analysis, ... are the reasons why I consider that category theory is a Good Thing. In my experience, category theory as a tool usually points me in the right direction, whereas set theory usually points me in the wrong one. (To give credit where it is due, Jamie Gabbay's work on alpha-equivalence is a counterexample.) Category theory has been successful on many many occasions - starting in algebraic topology - in removing the clutter of ad hoc definitions that have no conceptual anchor. For example, I was once told that "a matrix is an array of numbers". I am currently looking into Interval Analysis, in which all of the textbooks begin with a chapter of miscellaneous notation based on defining an interval as a pair [a,b] or as the subset of classical real numbers that lie between them - "analysis with double vision", as I call it. (To give credit again where it is due, the "Interval Newton" algorithm is a genuine insight into solving equations - it does not just do what you might imagine it does.) So I urge Marta, Eduardo and everybody else to stop trying to "police" category theory, and instead celebrate its achievements in many areas of mathematics. Paul Taylor www.cs.man.ac.uk/~pt - new web pages and new stuff. From rrosebru@mta.ca Thu Mar 16 20:32:11 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:32:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2rr-0007mu-Nu for categories-list@mta.ca; Thu, 16 Mar 2006 20:30:59 -0400 Subject: categories: 4th workshop on Quantum Programming Languages To: categories@mta.ca (Categories List) Date: Thu, 16 Mar 2006 12:42:40 -0400 (AST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 69 [Category theory is used heavily in the semantics of programming languages, and it used in interesting ways in the semantics of quantum programming languages. Therefore, this announcement might be of interest to the categories community. -PS] CALL FOR PAPERS 4th International Workshop on Quantum Programming Languages (QPL2006) July 17-19, 2006, Oxford http://www.mathstat.dal.ca/~selinger/qpl2006/ * * * The goal of this workshop is to bring together researchers working on mathematical foundations and programming languages for quantum computing. In the last few years, there has been a growing interest in logical tools, languages, and semantical methods for analyzing quantum computation. These foundational approaches complement the more mainstream research in quantum computation which emphasizes algorithms and complexity theory. Possible topics include the design and semantics of quantum programming languages, new paradigms for quantum programming, specification of quantum algorithms, higher-order quantum computation, quantum data types, reversible computation, axiomatic approaches to quantum computation, abstract models for quantum computation, properties of quantum computing resources and primitives, concurrent and distributed quantum computation, compilation of quantum programs, semantical methods in quantum information theory, and categorical models for quantum computation. Previous workshops in this series were held in Ottawa (2003), Turku (2004), and Chicago (2005). This year's workshop will be held in Oxford, as part of the week-long event "Cats, Kets and Cloisters", July 17-23, 2006, which will include four workshops on related topics (See http://se10.comlab.ox.ac.uk:8080/FOCS/CKCinOXFORD_en.html). TUTORIALS: The first day of the workshop, July 17, will consist of tutorials, followed by two days of contributed research talks. SUBMISSION PROCEDURE: Prospective speakers should submit a detailed abstract (or extended abstract) of 5-12 pages. Submissions of works in progress are encouraged, but must be more substantial than a research proposal. Submissions must provide sufficient detail to allow the program committee to assess the merits of the work. Submissions should be in Postscript or PDF format, and should be sent to selinger@mathstat.dal.ca by May 10 (please put "workshop submission" in the subject line). Receipt of all submissions will be acknowledged by return email. IMPORTANT DATES/DEADLINES: Submissions: May 10, 2006 Notification of acceptance: May 31, 2006 Corrected papers: June 14, 2006 Workshop: July 17-19, 2006 PROGRAM COMMITTEE: Samson Abramsky (Oxford) Bob Coecke (Oxford) Simon Gay (Glasgow) Philippe Jorrand (Grenoble) Prakash Panangaden (McGill) Peter Selinger (Dalhousie) CONTACT INFORMATION: Organizer: Peter Selinger Dalhousie University, Halifax, Canada Email: selinger@mathstat.dal.ca Local organizer: Bob Coecke Oxford Computing Laboratory Email: Bob.Coecke@comlab.ox.ac.uk (revised Mar 16, 2006) From rrosebru@mta.ca Thu Mar 16 20:34:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:34:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2uE-0000AN-TQ for categories-list@mta.ca; Thu, 16 Mar 2006 20:33:26 -0400 Subject: categories: Re: cracks and pots From: Eduardo Dubuc To: categories@mta.ca Date: Thu, 16 Mar 2006 14:29:12 -0300 (ART) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 70 Well Robert, 1) > Well, Einstein was not "trying to"; he was using it, and presented this > use as an accomplished fact. I just wanted to put in evidence the following fallacy that you are pushing forward: To attack the use of category theory (by some people) in string theory is at the same level that it would have been to attack the use (by Einstein) of differential geometry in general relativity. General relativity was born with differential geometry; it has no meaning without differential geometry. String theory was already there when a category theory approach began. It is not the same thing. Putting everything in the same bag is a well-known strategy to confuse an issue. Also, to have a poor opinion of many papers on applications of category theory to physics is one thing, to say that category theory has no future in physics is a completely different one. Nobody (including Motl's writing I am discussing (*)) has said the latter!! Quoting now from David Yetter: ** If (I suspect when) the string theory emperor turns out to have no clothes, Category theory will suddenly become de rigeur in physics". ** I start to believe that independently from what it finally happens with string theory; it is possible, even with the emperor well dressed, that category theory will with time become the rigeur in physics. 2) > Also, you forgot to mention that he flunk a high-school exam or > something of the sort proving by this very fact that a lot of people > were stupid, just as they are those which have doubts about the real > value of some applications of category theory to physics ! I can only say that I am sorry about your reaction to this. It was just an irony, and I thought this was evident. e.d. (*) Motl said (if I remember correctly) something of the sort that he thinks that to reach some goals of string theory certain category theory approach will not be helpful. From rrosebru@mta.ca Thu Mar 16 20:36:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:36:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2vX-0000Eg-JI for categories-list@mta.ca; Thu, 16 Mar 2006 20:34:47 -0400 Date: Thu, 16 Mar 2006 14:41:35 -0400 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 71 Eduardo Dubuc wrote: > Well Robert, > > 1) > > >>Well, Einstein was not "trying to"; he was using it, and presented this >>use as an accomplished fact. ... > General relativity was born with differential geometry; it has no meaning > without differential geometry. String theory was already there when a > category theory approach began. Sorry, Eduardo! That's a little oversimplified. See, for instance, section 17.7 of Misner, Thorne, and Wheeler's "Gravitation", among other references. General relativity (though of course not in its modern form) goes back to Einstein's formulation of the equivalence principle in 1907 (only two years after special relativity), and the prediction of the gravitational red shift. In 1911 Einstein also predicted the bending of light by massive bodies; this too is intrinsically part of GR. But it was only in 1912 that he realized that Euclidean geometry awas not compatible with this, and (encouraged by Grossmann and Levi-Civita) started looking at differential geometry as a way to handle non-Euclidean spacetime. Einstein and Grossmann's 1913 attempt at a general relativity theory was wrong; it did not transform correctly. Some time after this, Planck specifically warned him that the differential geometry approach would not work and would not be believed if it did. In November 1915 Einstein submitted two papers. The first of these explained some observations such as the precession of the perihelion of Mercury, but in other ways made wildly nonphysical predictions (essentially ignoring many of the effects of mass -though this "linearized theory" does have some uses as an approximation) He corrected this soon with a second paper in which he finally got it right. Sort of. In 1917 Einstein introduced a cosmological constant into his field equations to account for the "fact" that the universe wasn't expanding. In the 1920's he took it out again when it turned out that the universe *was* expanding. Now astronomers think there ought to be one, but with a value very different from what Einstein originally put forward. So GR got by without differential geometry for five years; and it was another decade or so before it was a mature theory with enough of the bugs out to do what was expected of it. And, as you know, there are still scales, almost a century later, on which its predictions are unsatisfactory. -Robert From rrosebru@mta.ca Thu Mar 16 20:37:13 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Mar 2006 20:37:13 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FK2wW-0000Jy-Qe for categories-list@mta.ca; Thu, 16 Mar 2006 20:35:48 -0400 Message-Id: <200603162047.k2GKlrX03245@math-cl-n03.ucr.edu> Subject: categories: cracks and pots To: categories@mta.ca (categories) Date: Thu, 16 Mar 2006 12:47:53 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 72 Dear Marta - You write: > I am relieved to learn (from the postings by David Yetter and John Baez) > that Motl's blog on the issue of categories and string theory is based on 1) > (Yetter) Motl's reluctance, as is the case with many string theorists, to > refuse to learn category theory, and 2) (Baez) Motl's personal dislike of > John Baez and of many other people, so that since Motl's personality is > well-known, any damage will be minimal. Good! > I thank David and John for taking the trouble to respond in detail to what > may have seem as a "provocation" on my part (well, perhaps it was...). By the way, I should explain why I thought you might be kidding in your original post. I had never heard anyone before suggest that category theory could be discredited by applications to string theory. It completely surprised me. I'm used to the opposite complaint: that category theory is discredited by its *lack* of applications. Of course, this always comes from people who 1) haven't taken the time to learn of its applications, 2) don't know enough category theory to appreciate its *intrinsic* interest. But it's good to hear your real concern: > But these informative responses do not address my main concern, which is one > that others (publicly, as Eduardo Dubuc, but several others privately) have > expressed to me following my posting. I was aiming at the fact that there is > a certain trend within category theory (when did it start?) to consistently > give center stage to anything that claims to have connections with physics > (in particular string theory). Is this because (it is believed that) the > state of category theory is now so poor (as "evidenced" by the lack of > grants) that they (the organizers of meetings) want to repair this image at > any cost? Since I began as a mathematical physicist and got interested in n-categories for their applications to topological quantum field theory, only later falling in love with category theory per se, I'm the wrong one to answer this question. I don't even know if it's true that applications to physics are given center stage, much less when this started, or why. I know a bit more about how people in differential geometry and differential topology got excited about work with links to physics. This trend probably started around the time of the Atiyah-Singer index theorem, which uses characteristic classes to compute the Euler characteristics of certain chain complexes built using differential operators. At the time this result was proved (1962-1965), it seemed an audacious blend of analysis and topology. That's one reason it caught people's interest. Another reason people liked the index theorem so much was that it turned out to be related to "anomalies" in quantum field theory, a phenomenon discovered by Adler, Bell and Jackiw around 1969. These nasty "anomalies" are actually a very practical issue in particle physics: they're related to the lifetime of the pion, and you can rule out field theories that have certain kinds of anomalies. I guess the relation between the index theorem and anomalies only became clear in the late 70's. I guess people were shocked and excited when it turned out that such sophisticated topology had practical applications to physics. Most topologists didn't know any quantum field theory, and most quantum field theorists didn't know that much topology. So, a kind of mutual fascination developed: both sides began learning about each other. People gave lots of proofs of the index theorem that illustrated very different ways of looking at it. The first proof had used a lot of K-theory and cobordism theory; later proofs used more facts about the heat equation, but by the time I was in grad school (1982-86) Quillen was giving lectures in which he tried to find a proof that only used multivariable calculus and "super" reasoning - i.e., lots of Z/2-graded linear algebra. This was when supersymmetry was just hitting the shores of mathematics, and Witten was starting to work his wonders. Anyway, index theory is just one of the first of many developments where ideas from physics met ideas from branches of math that seemed to have nothing to do with physics. In the heyday of Bourbaki, I guess pure mathematics seemed very removed from physics. It's fun to read what Dieudonne says about mathematical physics in his "Panorama of Pure Mathematics". By now, the situation has completely reversed in many fields, starting with differential geometry and topology, but then moving on to certain areas of algebra, and algebraic geometry, and now category theory, especially higher category theory.... This process has caused friction at every stage. Physicists don't always enjoy the intrusion of more mathematics into their various fields! Mathematicians don't always enjoy the intrusion of more physics - or the fast-paced, exploratory, sometimes sloppy cognitive style of physicists. You may recall Jaffe and Quinn's worries about the impact of physics on mathematics: http://www.arxiv.org/abs/math.HO/9307227 and how Atiyah in reply called for mathematicians to adopt the more "buccaneering" style of physics: http://www.ams.org/bull/pre-1996-data/199430-2/199430-2TOC.html which led Mac Lane to respond with the ballad of Captain Kidd: http://www.math.nsc.ru/LBRT/g2/english/ssk/proof_is_necessary.pdf The interesting big question is: how has this increased interaction both helped and hurt mathematics and physics? Clearly there are benefits. But does math become too "trendy" by chasing after links with the latest ideas of string theory? Does physics lose sight of its real purpose by focusing too much on mathematical elegance? There are lots of issues here. I've gone on too long already to want to tackle them now. But I think it's fair to say that that mathematics has benefited more than physics. One reason is that theories of physics do not need to be correct - i.e., apply to this particular universe of ours - to be mathematically interesting. Indeed, the funny thing about string theory is that while leading to an abundant harvest of rigorous mathematical results, it has not yet correctly predicted a single result from a single experiment, even after more than 20 years of work on the part of many smart people. This is part of a more general malaise in the theoretical side of fundamental physics, which various people have been commenting on recently: http://www.math.columbia.edu/~woit/wordpress/?p=307 http://www.nyas.org/publications/UpdateUnbound.asp?UpdateID=41 http://math.ucr.edu/home/baez/where_we_stand/ So, it's possible that string theory will eventually fall out of fashion. This could change the current dynamic between math and physics. A lot will depend on the results from the LHC particle accelerator, due to start operation in 2007. It may get evidence for string theory; it may not. Anyway, I'm sure these comments won't put your worries to rest! They're not really meant to. I just think it's good to see the issue of "category theory and string theory" as part of a much bigger and more complicated mess. :-) Best, jb From rrosebru@mta.ca Fri Mar 17 23:15:16 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:15:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRm2-0003Hl-Qn for categories-list@mta.ca; Fri, 17 Mar 2006 23:06:38 -0400 Reply-To: marta.bunge@mcgill.ca From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: cracks and pots Date: Fri, 17 Mar 2006 03:06:32 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 73 Dear John, Thanks for your candid and informative letter. I feel that few people who so far have responded to me (or to others in the discussion that I started) really understand what my concerns are. >Anyway, I'm sure these comments won't put your worries to rest! >They're not really meant to. I just think it's good to see the >issue of "category theory and string theory" as part of a much >bigger and more complicated mess. :-) > Your comments were very interesting and of course they will not put my worries to rest. I am grateful, though, for your taking this as part of a larger issue, on which I could expand more, but will not, since all I wanted was to raise awareness, not to preach (or even less to police). Best thoughts, Marta -------------------------------------------------------------------------------------------------------- >From: "John Baez" >To: categories@mta.ca (categories) >Subject: categories: cracks and pots >Date: Thu, 16 Mar 2006 12:47:53 -0800 (PST) > >Dear Marta - > >You write: > > > I am relieved to learn (from the postings by David Yetter and John Baez) > > that Motl's blog on the issue of categories and string theory is based >on 1) > > (Yetter) Motl's reluctance, as is the case with many string theorists, >to > > refuse to learn category theory, and 2) (Baez) Motl's personal dislike >of > > John Baez and of many other people, so that since Motl's personality is > > well-known, any damage will be minimal. > >Good! ... [Further repetition deleted by moderator.] From rrosebru@mta.ca Fri Mar 17 23:15:16 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:15:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRf8-00036i-Ep for categories-list@mta.ca; Fri, 17 Mar 2006 22:59:30 -0400 Date: Thu, 16 Mar 2006 17:52:03 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 74 Category theory, and for that matter modern (as opposed to elementary) algebra, is to mathematics as mathematics is to physics, and for that matter to computer science. Whereas mathematics organizes reasoning about the phenomena studied by physicists and computer scientists, algebra and category theory perform a similar function for mathematics. In any setting organization is desirable, and arguably necessary on occasion. But the use of algebra and category theory to organize physics and computer science is a double whammy here. One should therefore be doubly sympathetic of those physicists and computer scientists who want to know what substantive contribution is being made to their subject and can't evaluate the answers because they are one if not two levels removed from the necessary abstractions. Vaughan Pratt From rrosebru@mta.ca Fri Mar 17 23:15:17 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:15:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRhm-0003Bj-Fg for categories-list@mta.ca; Fri, 17 Mar 2006 23:02:14 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: pots and kettles Date: Fri, 17 Mar 2006 02:36:49 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 75 Dear Paul, >I suggest that he and Marta Bunga should go for a long walk to calm down. First of all, my name is Marta Bunge (not Bunga), in case you never saw it written. Secondly, I do not understand why Eduardo and I are put together in this, when we wrote independently to the categolries list -- is it because we are both Argentinians? (The Malvinas-Falklands war is over and both of our governments profitted from it; no need to keep the antagonism open.) Thirdly, I cannot go for a long walk in this weather (very cold in Montreal, in case you did not know), although your advice is probably good (my doctors would agree with you). Finally, I think that it is you who needs to take it easy. I hope that from now on this discussion will be civilized and about ideas, not people -- or else I will be really sorry that I started it! Best wishes, Marta From rrosebru@mta.ca Fri Mar 17 23:15:26 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:15:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRrI-0003RH-15 for categories-list@mta.ca; Fri, 17 Mar 2006 23:12:04 -0400 Reply-To: marta.bunge@mcgill.ca From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: cracks and pots Date: Fri, 17 Mar 2006 03:49:47 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 76 Dear Vincent, I am glad that you posted your reply to me. You raise questions that many of us have and that really relate to what I was trying to convey. I hope that your letter is widely read. I will only comment on one aspect of it. >Of course the problem is the way research is sponsored. >Leading researchers are not so much good mathematicians >but good salesmen. Category theory is just not very trendy at the >minute and to get the money one needs to do theoretical physics >(there had been also Computer Science at some point - that was poor >is not it?). >There were a couple of Fields medals and a new train called >TQFT that everybody just jumps in to get funded. I see nothing wrong in seeking funding for serious and well-motivated research. Young people have to eat too! What I worry about (this I did not say before) is that this craze for funding may drive researchers to accept *any* kind of funding, thinking naively that there are no strings (no pun intended) attached. When I was young, I once rejected NATO funding and, since there was no other source of funding for me at the moment, I had to go back to Argentina for two years and thus interrupt my graduate studies at Penn. Nowadays, it is the turn of organizations such as the Templeton Foundation (seeking to conciliate science with religion) which offer "graciously" to fund (and lavishly so) many projects in philosophy, physics and mathematics. Examples of Templeton funding are increasingly found: take the Perimeter Insitute (String Theory), the Godel Centennary Symposium in Vienna (Logic and Foundations), the workshop organized by A. Connes at the Sir Isaac Newton Insititue in Cambridge (Non Commutative Algebra), and others that are mentioned in Nature, for instance. This is all for public consumption -- just look at their web sites. Some of us find this really scary. That is why I do not put the getting of grants as a priority-- good science and good mathematics should always be the main priority. But then, you will ask, how do we feed graduate students, postdocs and unemployed category theorists? I do not know, but certainly not by resorting to dubious sources of funding. Not that it has happened yet! Forgive my "using" your comment to give way to a deep source of worry, certainly not unrelated to what I have been saying since the beginning of this discussion. As for >2/ Will be the rebirth of category - I bet! This is partly what I was asking -- are we (CT) in such a poor state that we need to be reborn in another guise? Maybe so, but I am just too immersed in my own (certainly not main stream) work to really judge. You are not the only one to suggest that we need an uplift. That may be so, but is it the reason for thinking it merely that there are no grants coming our way these days -- except when we (say that we) work in matters of interest to physics? It seems that I have only questions to ask -- not solutions to give. I apologize for that. Best wishes, Marta >From: "V. Schmitt" >To: categories@mta.ca >Subject: categories: Re: cracks and pots >Date: Thu, 16 Mar 2006 09:51:00 +0000 > >Dear Marta, >My english is so, so. I am french. >But this is to give briefly my opinion (I agree with you >more or less). > >I know a little of category and mathematics in general. >I love the category theory developped in the 70's and >I would have appreciated some category meetings at the >time. But i am too young. > >Category theory like any good mathematics will never >die - but may "our" category community will. > >Of course the problem is the way research is sponsored. >Leading researchers are not so much good mathematicians >but good salesmen. Category theory is just not very trendy at the >minute and to get the money one needs to do theoretical physics >(there had been also Computer Science at some point - that was poor >is not it?). >There were a couple of Fields medals and a new train called >TQFT that everybody just jumps in to get funded. > >Now as a *community* what shall *we* do? > >First the question regards mainly the established >people in the community (not me!). >1/ One can try to sell category theory in a better way. >This is a bit like tomato sauce that you can put everywhere. >And try to make new friends - inviting them to give talks... - >from different disciplines. >2/ We may claim loudly that cat theory is real mathematics >and really try hard to do good mathematics. There are >certainly good mathematicians definitely willing to use >cat theory. I saw many coming to category theory to develop >their own maths (- this happens for instance in France with Berger who >will never claim that he is a "categorician". Though he is completely >in it!) > >My feeling is the attitude 1/ pushed to the extreme may be >very damaging. These talks about category everywhere and >for everything are just poor and sound really stupid. >They do not serve the cause. > >2/ Will be the rebirth of category - I bet! > >Sorry for the message written in haste >and the poor english. Good e-mails from you >on the list! > >best regards, >Vincent. > > > > From rrosebru@mta.ca Fri Mar 17 23:17:54 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:17:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRtb-0003VR-5u for categories-list@mta.ca; Fri, 17 Mar 2006 23:14:27 -0400 Message-ID: <004101c649a6$51d28df0$0b00000a@C3> From: "George Janelidze" To: Subject: categories: Re: cracks and pots Date: Fri, 17 Mar 2006 11:36:46 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 77 I join Bob in saying that I fully agree with Marta, and I fully agree with Bob's second sentence. However, I have a problem with "look the gift horse in the mouth", since the horses we get are so often headless... I would also like to make just one comment to Paul's message (although I disagree with most of it; sorry!). Paul says: "Which generation was it that alienated other mathematicians by making outrageous claims about the foundations of mathematics that it never backed up with theorems? Which generation actually got its hands dirty and proved the theorems that relate category theory to other foundational disciplines?" Well, our colleagues active in the 1960s and 70s invented elementary toposes, for example, and proved many theorems about them. Those theorems did not convince set-theorists to forget sets, but are they convinced now? On the other hand those theorems were very beautiful, along with many others from several areas of category theory; I would describe 1960s and 70s as Golden Age of category theory. I am not saying of course that nothing important was discovered after 70s, but I see problems, and growing chaos, often created by ambitiously presented pseudo-relations with "other foundational disciplines". Moreover, talking about "relations": According to the classical work of Sammy and Saunders, the first "relation" was with algebraic topology. As we all know, there are various (co)homology/homotopy functors from topological spaces to groups, or to more complicated algebraic (or coalgebraic, Hopf, etc.) structures. There are also simplicial sets and other combinatorial intermediate players, and the relationship between geometric and combinatorial objects goes back to Euler (if not to Plato...). As we know from 1960s, the universal property of Yoneda embedding yields various adjoint functors, including those between simplicial sets and topological spaces - and this is why combinatorial objects are there! And what do recent algebraic topology text books do instead of explaining this? They are still talking about gluing cells instead. I think if we really care about relations between category theory and "other foundational disciplines", we should begin by explaining that category theory is not just a language allowing one to call homology a functor, but that category theory has beautiful constructions and results (some already from 1940s and 50s!) making enormous simplifications/applications/illuminations in neighbour areas of pure mathematics, such as abstract algebra, geometry, and logic. George Janelidze ----- Original Message ----- From: "RFC Walters" To: Sent: Wednesday, March 15, 2006 3:35 PM Subject: categories: Re: cracks and pots > I also would like to support the remarks of Marta with which I am in > full agreement. > The category theory community seems happy to accept uncritically, and > give centre-stage to, any interest shown by an external field. In this > context one should certainly look the gift horse in the mouth. > > Bob Walters From rrosebru@mta.ca Fri Mar 17 23:20:11 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:20:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRvu-0003bf-43 for categories-list@mta.ca; Fri, 17 Mar 2006 23:16:50 -0400 Date: Fri, 17 Mar 2006 09:25:09 -0500 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 78 Robert J. MacG. Dawson wrote: And, as you know, there are still scales, almost a century later, on which its predictions are unsatisfactory. For us ignorant of these, please explicate. thanks jim > Eduardo Dubuc wrote: >> Well Robert, >> >> 1) >> >> >>> Well, Einstein was not "trying to"; he was using it, and presented this >>> use as an accomplished fact. > > ... > >> General relativity was born with differential geometry; it has no meaning >> without differential geometry. String theory was already there when a >> category theory approach began. > > Sorry, Eduardo! That's a little oversimplified. See, for instance, > section 17.7 of Misner, Thorne, and Wheeler's "Gravitation", among other > references. > > General relativity (though of course not in its modern form) goes back > to Einstein's formulation of the equivalence principle in 1907 (only two > years after special relativity), and the prediction of the gravitational > red shift. In 1911 Einstein also predicted the bending of light by > massive bodies; this too is intrinsically part of GR. > > But it was only in 1912 that he realized that Euclidean geometry awas > not compatible with this, and (encouraged by Grossmann and Levi-Civita) > started looking at differential geometry as a way to handle > non-Euclidean spacetime. Einstein and Grossmann's 1913 attempt at a > general relativity theory was wrong; it did not transform correctly. > Some time after this, Planck specifically warned him that the > differential geometry approach would not work and would not be believed > if it did. > > In November 1915 Einstein submitted two papers. The first of these > explained some observations such as the precession of the perihelion of > Mercury, but in other ways made wildly nonphysical predictions > (essentially ignoring many of the effects of mass -though this > "linearized theory" does have some uses as an approximation) He > corrected this soon with a second paper in which he finally got it > right. Sort of. > > In 1917 Einstein introduced a cosmological constant into his field > equations to account for the "fact" that the universe wasn't expanding. > In the 1920's he took it out again when it turned out that the universe > *was* expanding. Now astronomers think there ought to be one, but with > a value very different from what Einstein originally put forward. > > So GR got by without differential geometry for five years; and it was > another decade or so before it was a mature theory with enough of the > bugs out to do what was expected of it. And, as you know, there are > still scales, almost a century later, on which its predictions are > unsatisfactory. > > -Robert > > > > From rrosebru@mta.ca Fri Mar 17 23:23:18 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:23:18 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKRyl-0003lJ-Uk for categories-list@mta.ca; Fri, 17 Mar 2006 23:19:47 -0400 Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Krzysztof Worytkiewicz Subject: categories: Re: cracks and pots Date: Fri, 17 Mar 2006 11:24:23 -0500 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 79 Vincent, you sound like this Beatles song, you know, in the White Album... Fully agree with you on the essentials of point 2 (you know that). However, uncritically referring to vociferations out of a some hate blog only because the blogger is labeled "string theorist" is not unlike point 1, at least in my modest opinion. Among the problems with the way research is sponsored there is this particularly modern one: the commitment to the short-term. It is quite similar to what happens in other sectors of the globalised society (of "high civilisation index" as L.Motl would presumably say -:( ) and leads to a growing disbalance in the allocation of resources. Cats are a very fine tool to organise concepts and proofs. Surprisingly enough, most mathematicians are quite reluctant or openly hostile. On the high-end, cat theory is crucial when it comes down to unify seemingly disparate areas of maths (which is unlikely a goal for itself) and this kind of work is quite clearly long-term. My 2 p: cat theory needs to be demystified in first place rather than to be sold. In particular, I think that the (still somehow ongoing) debate if it is a better foundation for maths or not is absolutely pointless. > Category theory is just not very trendy at the > minute and to get the money one needs to do theoretical physics > (there had been also Computer Science at some point - that was poor > is not it?). Now we have the best of both worlds: quantum computing :-)) Cheers Krzysztof -- my government will categorically deny the incident ever occurred From rrosebru@mta.ca Fri Mar 17 23:25:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:25:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKS0l-0003p0-0S for categories-list@mta.ca; Fri, 17 Mar 2006 23:21:51 -0400 Subject: categories: Re: cracks and pots From: Eduardo Dubuc To: categories@mta.ca Date: Fri, 17 Mar 2006 14:26:38 -0300 (ART) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 80 Well Robert, you are right in every particular fact and detail about Einstein and about relativity, there is not question about this. But there is not question either that is not Einstein or relativity that concern us here. I am right about the fact that introducing Einstein and differential geometry into our present discussion on the interaction of string theory and category theory was an infantile attempt to attack Motl's views. Worst than that, it introduces a distraction to Marta's principal issues. I can not be on the side of Motl, neither on the side of Baez, since I am ignorant about string theory. I point out that the Motl-Baez interaction gives rise to important issues that concern what it is good and what is bad mathematics or physics. In particular, what is good and what is bad category theory. Also, the meaning and profitable consequences of being in a fashionable subject. These issues were clearly exposed in Marta's postings, and we should be ready to talk about them publicly. In March 16 we got very good and pertinent postings that make us see other angles, think and learn: Baez's, Paul Taylor's, Dusko's and Vincent Shmitt's postings. Silences are also meaningful. e.d. From rrosebru@mta.ca Fri Mar 17 23:27:15 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:27:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKS2p-0003t0-Be for categories-list@mta.ca; Fri, 17 Mar 2006 23:23:59 -0400 Date: Fri, 17 Mar 2006 14:29:16 -0400 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 81 jim stasheff wrote: > > > Robert J. MacG. Dawson wrote: > > And, as you know, there are > still scales, almost a century later, on which its predictions are > unsatisfactory. > > For us ignorant of these, please explicate. (1) At the very small scale, nobody has really managed to unify quantum mechanics (which is as thoroughly tested on its home turf as relativity is on its own) with general relativity. QM works astonishingly well on the atomic scale, GR works astonishingly well on the astronomical scale, but there is a big gap, "in which we live", in which neither is particularly evident and classical Newtonian mechanics works pretty well for most purposes. (2) At very large scales there is some question as to whether additional forces, not predicted by general relativity, are needed to explain some cosmological observations. This is more speculative, but a lot of physicists seem to think *something* needs to be done. -Robert From rrosebru@mta.ca Fri Mar 17 23:31:59 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Mar 2006 23:31:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FKS72-00042x-8h for categories-list@mta.ca; Fri, 17 Mar 2006 23:28:20 -0400 Message-ID: <441B080B.7050102@math.upenn.edu> Date: Fri, 17 Mar 2006 14:03:39 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: pots and kettles Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 82 "generalised abstract nonsense" I'm pretty sure it was current while I was a grad student at princeton 1956-58 it wasn't necessarily perjorative category theory certainly applies marvelously to knot theory but Yetter should wade in on that aspect jim Paul Taylor wrote: > > I wonder when the phrases "generalised abstract nonsense" and "empty set > theory" came into circulation? I seem to remember hearing them when I was > an undergraduate. > From rrosebru@mta.ca Tue Mar 21 23:29:59 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 21 Mar 2006 23:29:59 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FLtrk-0007VM-93 for categories-list@mta.ca; Tue, 21 Mar 2006 23:18:32 -0400 Date: Mon, 20 Mar 2006 09:55:48 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Jon Beck Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 83 I just got a note from Jon's daughter that Jon died suddenly on March 11. She did not go into detail as to the circumstances. Nadine's email address is pearbeck@aol.com if anyone would like to write her. She is soliciting personal reminiscences either to be read at the memorial service she has planned or privately to her, whichever you choose. From rrosebru@mta.ca Wed Mar 22 22:58:03 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 22:58:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMFzh-0004fV-Gm for categories-list@mta.ca; Wed, 22 Mar 2006 22:56:13 -0400 Message-ID: <002301c64ac9$bdc5b310$4610a8c0@sanet.ge> From: "Mamuka Jibladze" To: References: Subject: categories: Re: cracks and pots Date: Sun, 19 Mar 2006 00:22:48 +0400 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=response Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 84 > Category theory, and for that matter modern (as opposed to elementary) > algebra, is to mathematics as mathematics is to physics, and for that > matter to computer science. Whereas mathematics organizes reasoning > about the phenomena studied by physicists and computer scientists, > algebra and category theory perform a similar function for mathematics. > > In any setting organization is desirable, and arguably necessary on > occasion. But the use of algebra and category theory to organize > physics and computer science is a double whammy here. One should > therefore be doubly sympathetic of those physicists and computer > scientists who want to know what substantive contribution is being made > to their subject and can't evaluate the answers because they are one if > not two levels removed from the necessary abstractions. > > Vaughan Pratt It just occurred to me that to justify such viewpoint we might have to look at the point in time when mathematics began to become abstracted out from natural sciences to see whether category theory is already in the same position with respect to the rest of mathematics. Although I certainly do not know enough history of science, I will still dare to speculate that the situation now is completely different from what it was then. I believe mathematics as a substantial part of the body of scientific knowledge did exist and evolve long long before it began to be considered as some separate entity which can be used to organize the rest - in fact many people still think of mathematics as just another science on completely equal footing with, say, biology or physics. Whereas birth and development of category theory has been, I think, much more deliberate, abrupt and discontinuous in comparison. If this is so, one possible conclusion might be that probably category theorists simply want too much too soon. Maybe they should be more patient and let their discipline become stronger within the body of mathematics before forcibly declaring it a new organizing force outside the rest of mathematics. This is as if a child would be forced to care for its parents shortly after being born. Mamuka Jibladze From rrosebru@mta.ca Wed Mar 22 22:58:03 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 22:58:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMFwM-0004Se-La for categories-list@mta.ca; Wed, 22 Mar 2006 22:52:46 -0400 Date: Sat, 18 Mar 2006 10:19:53 -0500 (EST) From: James Stasheff To: categories@mta.ca Subject: categories: RE: cracks and pots MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 85 Dear all, Picking up on the funding issue. I've served on the NSF Advisory panel, though many years ago, so I'm familiar with some of the issues with federal funding. The worst of it from my experience is that it takes universities off the hook as to supporting research directly, as opposed to being only a channel for external funds. Worse yet, tenure and promotion decisions are increasingly based on external support (at least in the US) thus the university doesn't trust its own faculty. Also involved is a bureaucratic aspect: it's more efficient to process a large grant with multiple researchers. In the ``good ole days'', math in the US has essentially NO post-docs. Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds On Fri, 17 Mar 2006, Marta Bunge wrote: > Dear Vincent, > > I am glad that you posted your reply to me. You raise questions that many of > us have and that really relate to what I was trying to convey. I hope that > your letter is widely read. I will only comment on one aspect of it. > > >Of course the problem is the way research is sponsored. > >Leading researchers are not so much good mathematicians > >but good salesmen. Category theory is just not very trendy at the > >minute and to get the money one needs to do theoretical physics > >(there had been also Computer Science at some point - that was poor > >is not it?). > >There were a couple of Fields medals and a new train called > >TQFT that everybody just jumps in to get funded. > > I see nothing wrong in seeking funding for serious and well-motivated > research. Young people have to eat too! What I worry about (this I did not > say before) is that this craze for funding may drive researchers to accept > *any* kind of funding, thinking naively that there are no strings (no pun > intended) attached. When I was young, I once rejected NATO funding and, > since there was no other source of funding for me at the moment, I had to > go back to Argentina for two years and thus interrupt my graduate studies > at Penn. Nowadays, it is the turn of organizations such as the Templeton > Foundation (seeking to conciliate science with religion) which offer > "graciously" to fund (and lavishly so) many projects in philosophy, > physics and mathematics. > > Examples of Templeton funding are increasingly found: take the Perimeter > Insitute (String Theory), the Godel Centennary Symposium in Vienna (Logic > and Foundations), the workshop organized by A. Connes at the Sir Isaac > Newton Insititue in Cambridge (Non Commutative Algebra), and others that > are mentioned in Nature, for instance. This is all for public consumption > -- just look at their web sites. Some of us find this really scary. That > is why I do not put the getting of grants as a priority-- good science and > good mathematics should always be the main priority. > > But then, you will ask, how do we feed graduate students, postdocs and > unemployed category theorists? I do not know, but certainly not by resorting > to dubious sources of funding. Not that it has happened yet! Forgive my > "using" your comment to give way to a deep source of worry, certainly not > unrelated to what I have been saying since the beginning of this discussion. > > > As for > > >2/ Will be the rebirth of category - I bet! > > > This is partly what I was asking -- are we (CT) in such a poor state that we > need to be reborn in another guise? Maybe so, but I am just too immersed in > my own (certainly not main stream) work to really judge. You are not the > only one to suggest that we need an uplift. That may be so, but is it the > reason for thinking it merely that there are no grants coming our way these > days -- except when we (say that we) work in matters of interest to > physics? > > It seems that I have only questions to ask -- not solutions to give. I > apologize for that. > > > Best wishes, > Marta > > >From: "V. Schmitt" > >To: categories@mta.ca > >Subject: categories: Re: cracks and pots > >Date: Thu, 16 Mar 2006 09:51:00 +0000 > > > >Dear Marta, > >My english is so, so. I am french. > >But this is to give briefly my opinion (I agree with you > >more or less). > > > >I know a little of category and mathematics in general. > >I love the category theory developped in the 70's and > >I would have appreciated some category meetings at the > >time. But i am too young. > > > >Category theory like any good mathematics will never > >die - but may "our" category community will. > > > >Of course the problem is the way research is sponsored. > >Leading researchers are not so much good mathematicians > >but good salesmen. Category theory is just not very trendy at the > >minute and to get the money one needs to do theoretical physics > >(there had been also Computer Science at some point - that was poor > >is not it?). > >There were a couple of Fields medals and a new train called > >TQFT that everybody just jumps in to get funded. > > > >Now as a *community* what shall *we* do? > > > >First the question regards mainly the established > >people in the community (not me!). > >1/ One can try to sell category theory in a better way. > >This is a bit like tomato sauce that you can put everywhere. > >And try to make new friends - inviting them to give talks... - > >from different disciplines. > >2/ We may claim loudly that cat theory is real mathematics > >and really try hard to do good mathematics. There are > >certainly good mathematicians definitely willing to use > >cat theory. I saw many coming to category theory to develop > >their own maths (- this happens for instance in France with Berger who > >will never claim that he is a "categorician". Though he is completely > >in it!) > > > >My feeling is the attitude 1/ pushed to the extreme may be > >very damaging. These talks about category everywhere and > >for everything are just poor and sound really stupid. > >They do not serve the cause. > > > >2/ Will be the rebirth of category - I bet! > > > >Sorry for the message written in haste > >and the poor english. Good e-mails from you > >on the list! > > > >best regards, > >Vincent. > > > > > > > > > > > > From rrosebru@mta.ca Wed Mar 22 22:58:03 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 22:58:03 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMFxq-0004Ya-0S for categories-list@mta.ca; Wed, 22 Mar 2006 22:54:18 -0400 Date: Sat, 18 Mar 2006 10:21:24 -0500 (EST) From: James Stasheff To: categories@mta.ca Subject: categories: Re: cracks and pots In-Reply-To: Message-ID: References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 86 I think knot theory is particularly helpful here but I'll let Yetter and Freyd reply further. Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds On Thu, 16 Mar 2006, Vaughan Pratt wrote: > Category theory, and for that matter modern (as opposed to elementary) > algebra, is to mathematics as mathematics is to physics, and for that > matter to computer science. Whereas mathematics organizes reasoning > about the phenomena studied by physicists and computer scientists, > algebra and category theory perform a similar function for mathematics. > > In any setting organization is desirable, and arguably necessary on > occasion. But the use of algebra and category theory to organize > physics and computer science is a double whammy here. One should > therefore be doubly sympathetic of those physicists and computer > scientists who want to know what substantive contribution is being made > to their subject and can't evaluate the answers because they are one if > not two levels removed from the necessary abstractions. > > Vaughan Pratt > > From rrosebru@mta.ca Wed Mar 22 23:00:45 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 23:00:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMG3K-0004vk-I5 for categories-list@mta.ca; Wed, 22 Mar 2006 22:59:58 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <9CC67FE4-AAAD-4ABD-90BB-06D5BA30CAF2@cs.bham.ac.uk> To: Content-Transfer-Encoding: 7bit From: Steve Vickers Subject: categories: Re: cracks and pots Date: Sun, 19 Mar 2006 18:25:08 +0000 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 87 On 17 Mar 2006, at 09:36, George Janelidze wrote: > ... I think if we really care about > relations between category theory and "other foundational > disciplines", we > should begin by explaining that category theory is not just a language > allowing one to call homology a functor, but that category theory has > beautiful constructions and results (some already from 1940s and 50s!) > making enormous simplifications/applications/illuminations in > neighbour > areas of pure mathematics, such as abstract algebra, geometry, and > logic. Dear George, I think the straight answer is that it is genuinely difficult. Even for elementary applications it is not easy. Try asking non- categorical topologists how they explain the product topology to students. Many will say, "This definition may look odd, but it turns out to work best." Others will produce various ad hoc justifications, such as "It's the definition that makes Tychonoff's theorem true." (Though that may be at least historically correct.) You point out that the product topology is the unique one such that projections are continuous and tupling preserves continuity, but they still don't see that as anything special. But with regard to certain advanced applications, there are pictures in the minds of the category theorists that do not translate at all easily to paper. Even the master expositors find it hard. I'm thinking for example of the idea of topos as generalized space. I have been working seriously with toposes (usually as generalized spaces) for about 15 years now and in some respects my understanding of them is quite deep. Yet there is still a huge gap in my understanding when it comes to their applications in algebraic geometry, Galois theory and algebraic topology, the kind of fields that gave rise to toposes in the first place. Somehow when I read the accounts I see a mass of machinery but no clear intuitions for what it is doing. This surprises me. A characteristic strength of category theory is that it is particularly good at explaining the underlying meaning of constructions, with its notion of universal properties, and with some beautiful tricks of categorical logic. So is it possible to explain, or illuminate, those particular categorical applications to someone like me? (Perhaps the challenge has already been met, and I've just missed the right book; and of course I eagerly await vol. 3 of the Elephant.) Here's a sample question where my categorical understanding falls short of the applications. If A is an Abelian group, then the space ^A of A-torsors is also a group (modulo canonical isomorphisms - the equational laws of group theory do not necessarily hold up to equality). The identity element is the regular representation of A on itself, group multiplication is "tensor product" of torsors, and inverses are got by inverting the A- action. It follows that if X is any space, then the collection of continuous maps from X to ^A is also a group, and this construction is self- evidently contravariant in X. Obviously it takes topos theory to formalize this, but already we can paint a picture. For example, suppose A is the cyclic group C_2 of order 2 and X is the circle. Then there are (classically) two isomorphism classes of maps T: X -> ^A, essentially because in going once right round the circle the variable torsor T(x) can come back either just how it started or with an automorphism swapping its two elements. The corresponding group is C_2. This looks like some kind of cohomology, so is it already part of the standard theory? I've never managed to follow all the machinery through. All the best, Steve Vickers From rrosebru@mta.ca Wed Mar 22 23:04:42 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 23:04:42 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMG73-0005Bl-K3 for categories-list@mta.ca; Wed, 22 Mar 2006 23:03:49 -0400 Message-ID: <441F0B7E.1000207@gc.cuny.edu> Date: Mon, 20 Mar 2006 15:07:26 -0500 From: Sergei Artemov User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.6) Gecko/20040113 X-Accept-Language: en-us, ru MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Logical Foundations of Computer Science'07. First Call for Papers Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 88 ******************************************************************************************** SYMPOSIUM ON LOGICAL FOUNDATIONS OF COMPUTER SCIENCE (LFCS'07) Call for papers New York City, June 4 - 7, 2007 URL: www.cs.gc.cuny.edu/lfcs07 Email: lfcs07@gmail.com * Purpose. The LFCS series provides an outlet for the fast-growing body of work in the logical foundations of computer science, e.g., areas of fundamental theoretical logic related to computer science. * Theme. Constructive mathematics and type theory; logical foundations of programming; logical aspects of computational complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justification; nonmonotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; system design logics; other logics in computer science. * All submissions must be done electronically (15 pages, according to LNCS standards) to lfcs07@gmail.com. * Submission Deadline. December 18, 2006. * Steering Committee. Anil Nerode (Cornell, General Chair); Stephen Cook (Toronto); Dirk van Dalen (Utrecht); Yuri Matiyasevich (St.Petersburg); John McCarthy (Stanford); J. Alan Robinson (Syracuse); Gerald Sacks (Harvard); Dana Scott (Carnegie-Mellon). * Program Committee. Samson Abramsky (Oxford); Sergei Artemov (New York City, PC Chair); Matthias Baaz (Vienna); Lev Beklemishev (Moscow); Andreas Blass (Ann Arbor); Lenore Blum (CMU); Samuel Buss (San Diego); Thierry Coquand (Go"teborg); Ruy de Queiroz (Recife, Brazil); Denis Hirschfeldt (Chicago); Bakhadyr Khoussainov (Auckland); Yves Lafont (Marseille); Joachim Lambek (McGill); Daniel Leivant (Indiana); Victor Marek (Kentucky); Anil Nerode (Cornell, General LFCS Chair); Philip Scott (Ottawa); Anatol Slissenko (Paris); Alex Simpson (Edinburgh); V.S. Subrahmanian (Maryland); Michael Rathjen (Columbus); Alasdair Urquhart (Toronto). From rrosebru@mta.ca Wed Mar 22 23:07:05 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 23:07:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMG8y-0005JZ-6d for categories-list@mta.ca; Wed, 22 Mar 2006 23:05:48 -0400 Message-ID: <441FB94F.7050607@cs.ru.nl> Date: Tue, 21 Mar 2006 09:29:03 +0100 From: Herman Geuvers User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051010) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Logic Colloquium 2006 (Nijmegen NL) Call for Participation/Contributed talks Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 89 [The usual apologies for multiple copies apply: we are sending the announcement both to personal e-mail addresses and to mailing lists. Please distribute this e-mail to all interested people.] ------------- CALL for PARTICIPATION ------------------- | | | LOGIC COLLOQUIUM 2006 | | (ASL European Summer Meeting) | | July 27 -- August 2, 2006 | | | | Institute for Computing and Information Science | | Radboud University Nijmegen (The Netherlands) | | | ---------------------------------------------------------- The European summer meeting of the ASL in the year 2006 will be held in Nijmegen, the Netherlands: http://www.cs.ru.nl/lc2006/ Plenary invited speakers: ======================== Samson Abramsky (Oxford) Marat Arslanov (Kazan) Harvey Friedman (Ohio) Martin Goldstern (Vienna) Ehud Hrushovski (Jerusalem) Jochen Koenigsmann (Freiburg) Andy Lewis (Leeds) Antonio Montalban (Chicago) Erik Palmgren (Uppsala) Wolfram Pohlers (Muenster) Ernest Schimmerling (Pittsburgh) John Steel (Berkeley) William Tait (Chicago) Frank Wagner (Lyon) Tutorials by: ============ Rodney Downey (Wellington) Ieke Moerdijk (Utrecht) Boban Velickovic (Paris) Plenary Discussion: ================== On the occasion of the 100th birthday of the great logician Kurt Goedel, there will be a plenary discussion on "Goedel's Legacy", discussing his influence on set theory, proof theory and philosophical logic. Special sessions: ================ * Computability Theory Speakers: Noam Greenberg, Bjorn Kjos-Hanssen, Peter Hertling, Joe Miller, Jan Reimann, Frank Stephan * Computer Science Logic Speakers: Ulrich Berger, Venanzio Capretta, Martin Escardo John Harrison, Martin Hofmann, Andy Pitts * Model Theory Speakers: Raf Cluckers, Clifton Ealy, Piotr Kowalski, Assaf Hasson, Sonia L'Innocente, Tim Mellor * Proof Theory and Type Theory Speakers: Klaus Aehlig, Andrey Bovykin, Nicola Gambino, Joost Joosten, Thomas Studer, Henry Towsner * Set Theory Speakers: Natasha Dobrinen, John Krueger, Paul Larson, Jordi Lopez-Abad, Christian Rosendal, Martin Zeman REGISTRATION ============ You can now register by submitting the online registration form (or you can send it by fax). You can find the registration form and all other information related to the conference at: http://www.cs.ru.nl/lc2006/ The Organizing Committee From rrosebru@mta.ca Thu Mar 23 23:15:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:15:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMcjO-00020u-Vw for categories-list@mta.ca; Thu, 23 Mar 2006 23:12:55 -0400 Date: Thu, 23 Mar 2006 10:56:34 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: Jon Beck Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 90 Dear Friends, It is happening much too often during this past year. Discussions, whose continuation has been too long delayed, are forever ended by the sad passing of a colleague. Particularly poignant for me is the loss of my friend and collaborator from Varenna and the ETH in 1966 and in the Zurich Triples book (SLNM No. 80). Beyond his famous and far-reaching results on tripleability, intensive discussions with Jon led to some of the points raised in my paper in that book. The word "doctrine" itself is entirely due to him and signifies something which is like a theory, except appropriate to be interpreted in the category of categories, rather than, for example, in the category of sets; of course, an important example of a doctrine is a 2-monad, and among 2-monads there are key examples whose category of "algebras" is actually a category of theories in the set-interpretable sense. Among such "theories of theories", there is a special kind whose study I proposed in that paper. This kind has come to be known as "Kock-Zoeberlein" doctrine in honor of those who first worked out some of the basic properties and ramifications, but the recognition of its probable importance had emerged from those discussions with Jon. In those days Jon was insistent on mathematical clarity and did much to encourage precision in discussions and in the formulation of mathematical results. We lovingly remember him from those youthful days. Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Thu Mar 23 23:15:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:15:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMce8-0001kK-06 for categories-list@mta.ca; Thu, 23 Mar 2006 23:07:28 -0400 Date: Thu, 23 Mar 2006 12:17:05 +0000 From: Robin Houston To: categories@mta.ca Subject: categories: Products in a compact closed category Message-ID: <20060323121705.GA26650@rpc142.cs.man.ac.uk> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 91 Dear categorists, I recently proved that products (or coproducts) in a compact closed category are necessarily biproducts, and I'm wondering whether this is a known theorem. I can't find any reference to it in the literature, but the proof is not hugely complicated and it would not surprise me to learn that someone noticed it before this week! (More precisely, I can prove that given a monoidal category that has (finite) sums and products, if the tensor distributes over the sums on one side and the products on the other -- e.g. for every object A, -*A preserves sums and A*- preserves products -- then the products and coproducts are both really biproducts.) Any references or recollections will be much appreciated. Yours, Robin From rrosebru@mta.ca Thu Mar 23 23:15:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:15:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMcg1-0001ph-1s for categories-list@mta.ca; Thu, 23 Mar 2006 23:09:25 -0400 Date: Thu, 23 Mar 2006 09:06:08 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: George Mackey, 1916-2006 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 92 Dear friends, Together with many others, I am deeply saddened by the death of George Mackey. For far too long, I had been delaying the trip to see him and continue our discussions which had started in the "Weyl'sche Kammer" at the ETH in Zurich and continued in the physics center in Trieste. Long before I met him, his insights into mathematics and into quantum mechanics had been informing my own thinking. It was the study of his book on quantum mechanics in 1967 which led directly to the joint course by Saunders Mac Lane and me at the University of Chicago. But his relation to category theory goes back much further than that, as Saunders and Sammy had explained to me. George Mackey's Ph.D. thesis displayed remarkable thinking of a categorical nature, even before categories had been defined. Specifically, the fact that the category of Banach spaces and continuous linear maps is fully embedded into a category of pairings of abstract vector spaces, together with the definition and use of "Mackey convergence" of a sequence in a "bornological" vector space were discovered there and have played a basic role in some form in nearly every book on functional analysis since. What is perhaps unfortunately not clarified in nearly every book on functional analysis, is that these concepts are intensely categorical in character and that further enlightenment would result if they were so clarified. And who, despite initial skepticism, permitted the first paper giving an exposition of the theory of categories to see the light of day in the Transactions of the AMS in 1945? None other than the referee, George Whitelaw Mackey. Sincerely, F. William Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Thu Mar 23 23:17:05 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:17:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMcmj-0002Di-0v for categories-list@mta.ca; Thu, 23 Mar 2006 23:16:21 -0400 Subject: categories: re: cracks and pots From: Eduardo Dubuc To: categories@mta.ca Date: Thu, 23 Mar 2006 13:50:45 -0300 (ART) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 93 Hi To follow are the contents of two postings that Bob (always vigilant, ja!) thought best to concatenate in only one. On spite of Robert's erudition and his knowledgeable discourse, I still think Einstein using differential geometry to develop general relativity is not at the same level that John Baez using category theory to develop and/or understand string theory. His arguments are valid in a court of law, but do not convince me. I imagine John himself is probably the first to laugh at such a comparison. But this is not the issue of my present posting. He touches also some pertinent points that go more to the core of the "cracks_and_pots" debate. (In between ** are Robert words) What Motl says certainly does not make people using category theory in string theory laugh. Applications of category theory to string (or to other physical theories competing with string theory ?, see Yetter's posting, it is all very confusing !!) may be valuable or may not. I (and a lot of us) can not tel. ** In which case demands that they ($) be read out of the meeting are premature. ($) papers that claim applications to physics ** This is a difficult question. Marta was saying (and Bob Walters and others agree) that when a paper was claiming applications to physics it was easily accepted without knowledgeable and close examination, and that there were a lot of them. Probably a lot of them should be read out, but not by policy against (as it was erroneously interpreted in these postings). Serious refereeing is a healthy practice that should not be equaled with censorship. **Remember - in mathematics it's a matter of "In God We Trust, everybody else must provide a proof."** This is not so much so. Speculations in math are very difficult. If not well founded they are vacuous. Only great mathematicians can do them (example close to us, Grothendieck), the rest of us must provide a proof. **If the math itself meets mathematical standards of rigor, its application to physics need surely only meet the standards appropriate to that subject.** The math itself must also meet standards of quality, not only of rigor. Besides that, "standards appropriate to that subject" does not mean "free for anything". Motl writes: "I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts." Clearly he is saying that these standards are not being fulfilled (in his opinion of course) by claimed applications of math to physics. Motl may be wrong or he may be right, what we have not seen yet in these postings is a convincing or clear answer to the questions he arises. I would say, not even an answer at all. These questions triggered Marta's original posting, which in turn was arising other (not exactly the same) questions. I do not agree necessarily with Marta's implicit views, what I support is her courage to point out that they are serious problems in the category theory community (for example, quality of the publications, abuse of fashionable topics to get grants, invited speakers in CT meetings). Best wishes e.d. From rrosebru@mta.ca Thu Mar 23 23:20:36 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:20:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMcps-0002Ro-WE for categories-list@mta.ca; Thu, 23 Mar 2006 23:19:37 -0400 Date: Thu, 23 Mar 2006 13:09:07 -0400 From: "Robert J. MacG. Dawson" X-Accept-Language: en-us, en MIME-Version: 1.0 To: cat group Subject: categories: One-stop version of my postings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 94 Eduardo having preferred (wisely) not to try to edit down both sides of a discussion with a lot of quoting, I will follow his lead and summarize one side of it with context quotes from the other side. Eduardo Dubuc wrote (in part) > I am right about the fact that introducing Einstein and differential > geometry into our present discussion on the interaction of string theory > and category theory was an infantile attempt to attack Motl's views. Worst > than that, it introduces a distraction to Marta's principal issues. I responded: Sorry, Eduardo. I refuse to write out a hundred times "I will not attempt to attack the views of Lubos Motl". As for introducing distractions, Marta suggested (Mar. 12) that we should as a community not "quietly accept getting discredited by a minority of us presumably applying category theory to string theory", and that we should "react and point out that this is not what (all of) category theory is about" and "give a thought about what we, as a community, can urgently do to repair this damaging impression." On Mar. 14, it was suggested that meeting organizers might want to take this into account while choosing invited speakers. Now, I do accept that there are such things as bad mathematics and bad physics. However, this call for collective action against an entire field of research seems uncomfortably close to an organized boycott, an extreme enough breach of tradition that only an emergency - if that - could justify it. And, in such circumstances, the question of whether the emergency really exists is certainly relevant, and not (_pace_ George W. Bush) a presumptuous distraction. My response (Mar. 14) was twofold. I suggested that the general credibility of category theory does not depend on the credibility of its applications to physics. But I also pointed out, which I think is even more relevant, that physicists are quite used to tentative theories being studied for many years before they are accepted or rejected, and that there is no loss of credibility among physicists for those involved in this process. Neils Bohr is not remembered as "that crackpot who thought the atom was like a solar system". Note that only credibility among physicists is at stake here. Mathematicians are unlikely to base a negative judgement of _any_ branch of mathematics on what physicists may be doing with it, and in particular know perfectly well that this is not "what all of category theory is about". The rest of the world will not have a clue what all of category theory is about, and will never know the debate took place. We expect mathematics to be correct by the time it sees print; internal consistency is a comparatively easy goal. Physics, despite using almost as many equations, has a real and sometimes uncooperative universe to contend with, and cannot afford the luxury of progressing only through unassailable, permanently-established truths. Mathematics was once in a similar position: we recognize the genius of Euclid's axiomatic system even while realizing that (by our standards) it was fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his axioms assert that any pair of circles whatsoever do so.) Ramanujan is perhaps alone among modern mathematicians in having been permitted, by general consent, something like the latitude to err freely that the physics community customarily grants itself - and his was a very special case. Had he lived to see the greater part of his work in print, he too would have proved, retracted, or labelled as conjecture many of the speculative claims that ended up as his legacy. I mentioned Einstein in passing, assuming that readers would have some idea of the initial skepticism with which relativity was viewed, and the length of time it took before that skepticism ceased to be justified. (Actual hostility was a separate, later, phenomenon, more related to antisemitism and Stalinism than to physics.) If I assumed too much, and should have explained more, I apologize. Eduardo then responded (in part) > Einstein using differential geometry to develop general relativity is not at the same level that John Baez using category theory to develop and/or understand string theory. I imagine John himself is probably the first to laugh at such a comparison. and I responded to that (in part) If category theory happens to be the approach that works, and significant progress is made in fundamental physics as a result, it will be very much on the same level as relativity. If it isn't, it will be, from most physicists' point of view, an honorable attempt on the same level as many other theories such as the steady-state universe, tachyons (Remember tachyons? No? _They_ remember _you_...), magnetic monopoles, or the luminiferous ether, that were seriously considered in the past by respectable physicists. Until the results are in, it is, I think, not unreasonable to make comparisons with Einstein's early work on relativity - at a time when Einstein could not get results consistent with observation and many of his talented contemporaries had serious doubts about whether he was using the right approach. (Not whether he was a fool or a charlatan, just whether he was using the right approach.) Or, equally, with Einstein's later work on quantum mechanics, when history seems to have shown that he backed the wrong horse - but it was not so clear at the time. -Robert Dawson From rrosebru@mta.ca Thu Mar 23 23:26:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:26:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMcvW-0002lZ-89 for categories-list@mta.ca; Thu, 23 Mar 2006 23:25:26 -0400 Message-ID: <2cc0d36c0603231145x5ed9e737pf44482eec90d56e4@mail.gmail.com> Date: Thu, 23 Mar 2006 20:45:38 +0100 From: "Peter Arndt" To: categories@mta.ca Subject: categories: Re: cracks and pots MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 95 Dear category theorists, I would like to support Krzysztof Worytkiewicz's remark that "cat theory needs to be demystified in first place rather than to be sold" from a different side: I have recently come across several publications and research projects of philosophers who have become over-enthusiastic with category theory. In certain circles category theory seems to have gained a nimbus of an all-encompassing theory of everything, be it part of mathematics or not, see for example http://lists.debian.org/debian-devel/2000/10/msg02048.html for an expression of such opinions or http://ru.philosophy.kiev.ua/rodin/Endurance.htm for a crude offspring of them. Such exaggerated propaganda is very likely to cause railings like the one of Lubos Motl. Has anyone observed the same phenomenon or does it only exist among the people I have to do with? All the best, Peter From rrosebru@mta.ca Sat Mar 25 00:47:18 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Mar 2006 00:47:18 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FN0aO-0004v0-5h for categories-list@mta.ca; Sat, 25 Mar 2006 00:41:12 -0400 Date: Fri, 24 Mar 2006 09:10:16 -0500 (EST) From: Michael Barr To: categories@mta.ca Subject: categories: Re: George Mackey, 1916-2006 In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 96 I had not heard that Mackey had died and am also saddened, although I never met him. Bill does not mention that Mackey's thesis, I think it was published in the same volume of the Transactions as "General Theory of natural equivalances", was the direct source of the Chu construction. As Bill mentioned the category of pairs embeds the category of Banach spaces (and continuous linear maps). It is in fact equivalent to the category of what are now called Mackey spaces, which are characterized as the locally convex topological vector spaces that have the finest possible topology for their set of continuous linear functionals. I once wrote to Mackey asking him if he had had any intention of considering a space of linear functionals as a _replacement_ for a topology or merely an adjunct to it. Unfortunately, he did not reply, even to a snail mail "letter" (as we used to call them). As it happens it was just yesterday that I sent off the followin abstract of the talk I will give in the category session of the Canadian Math. Soc. meeting in Calgary in early June: A standard theorem says that any locally convex topological vector space has a finer topology, its \emph{Mackey topology} with the same set of continuous linear functionals and that is the finest possible topology with that property. If $E$ and $F$ are two such spaces topologize the space $\mbox{Hom(E,F)}$ of continuous linear transformations $E\to F$ with the weak topology induced by the algebraic tensor product $E\otimes F'$ and then let $[E,F]$ denote the associated Mackey topology. Let $F^*$ denote the dual $F'$ topologized by the Mackey topology on the weak dual and let $E\otimes F=[E,F^*]^*$ (whose underlying vector space is the algebraic tensor product). Then for any Mackey spaces $E$, $F$, and $G$, \begin{enumerate} \item $[E\otimes F,G]\cong [E,[F,G]]$ \item $E\cong E^*{}^*$ \item $[E,F]\cong (E\otimes F^*)^*$ \end{enumerate} which is summarized by saying that the category of Mackey spaces and continuous linear transformations is $*$-autonomous. This category is equivalent to the category of weakly topologized locally convex topological vector spaces (which have the coarsest possible topology for their set of continuous linear functionals) which is therefore also $*$-autonomous. They are also equivalent to the chu category of vector spaces (which will be explained). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% On Thu, 23 Mar 2006, F W Lawvere wrote: > > Dear friends, > > Together with many others, I am deeply saddened by the death of > George Mackey. For far too long, I had been delaying the trip to see him > and continue our discussions which had started in the "Weyl'sche Kammer" > at the ETH in Zurich and continued in the physics center in Trieste. Long > before I met him, his insights into mathematics and into quantum mechanics > had been informing my own thinking. It was the study of his book on > quantum mechanics in 1967 which led directly to the joint course by > Saunders Mac Lane and me at the University of Chicago. But his relation to > category theory goes back much further than that, as Saunders and Sammy > had explained to me. > > George Mackey's Ph.D. thesis displayed remarkable thinking of a > categorical nature, even before categories had been defined. Specifically, > the fact that the category of Banach spaces and continuous linear maps is > fully embedded into a category of pairings of abstract vector spaces, > together with the definition and use of "Mackey convergence" of a sequence > in a "bornological" vector space were discovered there and have played a > basic role in some form in nearly every book on functional analysis since. > What is perhaps unfortunately not clarified in nearly every book on > functional analysis, is that these concepts are intensely categorical in > character and that further enlightenment would result if they were so > clarified. > > And who, despite initial skepticism, permitted the first paper giving > an exposition of the theory of categories to see the light of day in the > Transactions of the AMS in 1945? None other than the referee, George > Whitelaw Mackey. > > Sincerely, > F. William Lawvere > > > ************************************************************ > F. William Lawvere > Mathematics Department, State University of New York > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > Tel. 716-645-6284 > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > ************************************************************ > > > > > From rrosebru@mta.ca Sat Mar 25 00:47:18 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Mar 2006 00:47:18 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FN0c0-0004y4-9F for categories-list@mta.ca; Sat, 25 Mar 2006 00:42:52 -0400 Date: Fri, 24 Mar 2006 11:13:08 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: Re: George Mackey, 1916-2006 In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 97 Dear Mike, Looking into the MathSciNet, I see that: Mackey's 1942 thesis was published in abbreviated form in the Proceedings of the National Academy of Sciences, vol. 29 (1943). The Math Reviews reviewer seems to clearly understand that part of idea was to replace open sets with linear functionals. The extended publication in the Transactions was in vol. 57 (1945) whereas the publication of Eilenberg and Mac Lane's famous paper was in vol. 58. In the same volume 58 there is a paper by Clifford Truesdell. Best, Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Fri, 24 Mar 2006, Michael Barr wrote: > I had not heard that Mackey had died and am also saddened, although I > never met him. > > Bill does not mention that Mackey's thesis, I think it was published in > the same volume of the Transactions as "General Theory of natural > equivalances", was the direct source of the Chu construction. As Bill > mentioned the category of pairs embeds the category of Banach spaces > (and continuous linear maps). It is in fact equivalent to the category of > what are now called Mackey spaces, which are characterized as the locally > convex topological vector spaces that have the finest possible topology > for their set of continuous linear functionals. > > I once wrote to Mackey asking him if he had had any intention of > considering a space of linear functionals as a _replacement_ for a > topology or merely an adjunct to it. Unfortunately, he did not reply, > even to a snail mail "letter" (as we used to call them). > > As it happens it was just yesterday that I sent off the followin abstract > of the talk I will give in the category session of the Canadian Math. Soc. > meeting in Calgary in early June: > > > A standard theorem says that any locally convex topological vector space > has a finer topology, its \emph{Mackey topology} with the same set of > continuous linear functionals and that is the finest possible topology > with that property. If $E$ and $F$ are two such spaces topologize the > space $\mbox{Hom(E,F)}$ of continuous linear transformations $E\to F$ > with the weak topology induced by the algebraic tensor product $E\otimes > F'$ and then let $[E,F]$ denote the associated Mackey topology. Let > $F^*$ denote the dual $F'$ topologized by the Mackey topology on the > weak dual and let $E\otimes F=[E,F^*]^*$ (whose underlying vector space > is the algebraic tensor product). Then for any Mackey spaces $E$, $F$, > and $G$, > \begin{enumerate} > \item $[E\otimes F,G]\cong [E,[F,G]]$ > \item $E\cong E^*{}^*$ > \item $[E,F]\cong (E\otimes F^*)^*$ > \end{enumerate} > which is summarized by saying that the category of Mackey spaces and > continuous linear transformations is $*$-autonomous. > > This category is equivalent to the category of weakly topologized > locally convex topological vector spaces (which have the coarsest > possible topology for their set of continuous linear functionals) which > is therefore also $*$-autonomous. They are also equivalent to the chu > category of vector spaces (which will be explained). > > %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% > > On Thu, 23 Mar 2006, F W Lawvere wrote: > > > > > Dear friends, > > > > Together with many others, I am deeply saddened by the death of > > George Mackey. For far too long, I had been delaying the trip to see him > > and continue our discussions which had started in the "Weyl'sche Kammer" > > at the ETH in Zurich and continued in the physics center in Trieste. Long > > before I met him, his insights into mathematics and into quantum mechanics > > had been informing my own thinking. It was the study of his book on > > quantum mechanics in 1967 which led directly to the joint course by > > Saunders Mac Lane and me at the University of Chicago. But his relation to > > category theory goes back much further than that, as Saunders and Sammy > > had explained to me. > > > > George Mackey's Ph.D. thesis displayed remarkable thinking of a > > categorical nature, even before categories had been defined. Specifically, > > the fact that the category of Banach spaces and continuous linear maps is > > fully embedded into a category of pairings of abstract vector spaces, > > together with the definition and use of "Mackey convergence" of a sequence > > in a "bornological" vector space were discovered there and have played a > > basic role in some form in nearly every book on functional analysis since. > > What is perhaps unfortunately not clarified in nearly every book on > > functional analysis, is that these concepts are intensely categorical in > > character and that further enlightenment would result if they were so > > clarified. > > > > And who, despite initial skepticism, permitted the first paper giving > > an exposition of the theory of categories to see the light of day in the > > Transactions of the AMS in 1945? None other than the referee, George > > Whitelaw Mackey. > > > > Sincerely, > > F. William Lawvere > > > > > > ************************************************************ > > F. William Lawvere > > Mathematics Department, State University of New York > > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > > Tel. 716-645-6284 > > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > > ************************************************************ > > > > > > > > > > > > > From rrosebru@mta.ca Sat Mar 25 00:47:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Mar 2006 00:47:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FN0Z6-0004sf-5W for categories-list@mta.ca; Sat, 25 Mar 2006 00:39:52 -0400 Date: Fri, 24 Mar 2006 00:08:15 -0800 From: Vaughan Pratt X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories list Subject: categories: Progressive or linear or ... monoids? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 98 1. Is the quasivariety of monoids generated by the groups and the free monoids finitely based? That is, is there a finite set of universal Horn formulas entailing the common universal Horn theory of groups and free monoids? In other words, what do groups and free monoids have in common, besides being monoids? Apart from the (equational) axioms for monoids, the only members of that theory I can think of are xy=x -> y=1 and yx=x -> y=1. 2. How different is the abelian case? More or fewer axioms? Vaughan Pratt From rrosebru@mta.ca Sat Mar 25 00:47:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Mar 2006 00:47:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FN0XW-0004pQ-0x for categories-list@mta.ca; Sat, 25 Mar 2006 00:38:14 -0400 Date: Thu, 23 Mar 2006 20:31:47 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu Organization: Stanford University User-Agent: Mozilla Thunderbird 0.9 (Windows/20041103) X-Accept-Language: en-us, en MIME-Version: 1.0 To: cat group Subject: categories: Re: One-stop version of my postings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 99 Robert J. MacG. Dawson wrote: > ...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his > axioms assert that any pair of circles whatsoever do so.) In axiomatic mathematics, everything that is not forbidden is permitted. Circles can climb trees and drape themselves over branches in Dali's axiomatization of geometry, not because he says they can but because he is not the strict disciplinarian that Euclid is. Euclid insists that his circles shape up or else. This creates many problems for the circles but few problems for mathematicians as their managers. Dali runs a looser ship, which lets the circles lead a less structured life while creating more problems for mathematicians. This is a win-win situation: the circles end up with fewer neuroses while the mathematicians thrive, problems being their lifeblood. Thank god for Lobachevsky and the others we can't remember because Tom Lehrer didn't. Vaughan Pratt From rrosebru@mta.ca Sat Mar 25 00:54:37 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Mar 2006 00:54:37 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FN0mF-0005In-H1 for categories-list@mta.ca; Sat, 25 Mar 2006 00:53:27 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: re: cracks and pots Date: Fri, 24 Mar 2006 11:24:25 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 100 Hi, I thought that my intention in raising the issues that I did in my original posting of March 12 were clear enough. Now it seems that they were not, to some. 1. I find ridiculous the suggestion put forward by Robert Dawson (March 23) that my presumed "call for collective action against an entire field of research seems uncomfortably close to an organized boycott, an extreme breach of tradition that only an emergency -if that - could justify it". The invention of an alleged "boycott" plot seems aimed at dismissing the questions that I (and other concerned mathematicians who joined the discussion) have raised. Anybody who, like Robert Dawson, resorts to such inventions appears to be panicking in that he is trying to divert attention from, rather than help, a healthy discussion. 2. Eduardo Dubuc writes: "I do not agree necessarily with Marta's implicit views". There is nothing implicit in my views. Just take a second look at my various postings of March 14, 15, and 17 in reply to some people. If Eduardo refers to my bringing in the Templeton Foundation into the discussion, then I would like to add some comments, partly expanding (and correcting) my reply to Vincent Schmitt (March 17). I can back up my contentions in reference to the the Goedel Centenary Symposium in Vienna http://www.logic.at/goedel2006/ and the workshop organized by A. Connes at the Sir Isaac Newton Institue in Cambridge (Non Commutative Algebra) http://www.newton.cam.ac.uk/programmes/NCG/ncgw02 I should, however, make more precise my reference to the Perimeter Institute for Mathematical Physicts. What sems clear is that one of its most prominent long-term researchers is at the same time one of the prominent particpants in Templeton funding and activties, for instance the Foundational Questions Institute. I quote from the last issue of Nature http://www.fqxi.org/about.html "Phycists to confront those big questions. Time travel, multiple universes and extraterrestrial intelligence seem more the purview of Star Trek scriptwriters than of serious researchers. (...) The FQI was set up last October with a grant from the Templeton Foundation, which promotes research at the boundary of religion and science. With US$8 million in seed money, the FQI will fund dozens of researcher's part-time work on these questions. (...) "I am very happy to see that a project has started to address these needs" says Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, who is also on the FQI's scientific advisory board. -- Geoff Brumfield. Nature. 2 March 2006." I stated incorrectly that the Pi is devoted to String Theory, when it seems, judging from the work of Lee Smolin, that Pi rather promotes Loop Quantum Gravity, a competitor to String Theory. By the way, an article by Lee Smolin entitled "Atoms of Space and Time" on LQG has been issued already three times (with minor variations) in Scientific American (200, 2004, 2006), so many of you must have seen it. 3. I have never suggested that "an entire field of research" should be suspect of constituting bad mathematics. If by this entire field of research it is meant n-categories, theta-categories, operads, topological quantum theories, and so on, there is, as in any other field, good and bad mathematics. Perhaps I should bring to your attention my comments to the organizers of the StreetFest, requested by them of all participants, and posted in their website as http://streetfest.maths.mq.edu.au/feedback?lastname=Bunge&firstname=Marta I stand by this, and only hope that my remarks in the "cracks and pots" postings have not been misinterpreted by the people mentioned in my comment above, and by others, like Ieke Moerdijk, not mentioned in it since they were not there. 4. I also think that a problem persists in the emphasis given to the "you do not want to know" general message in Baez postings, not because of them intrinsically, or of himself, but of the use others (for what purposes, I do not know) are making of this general trend. One instance of this trend (although in a different casting) is the following http://www.math.uchicago.edu/~eugenia/morality/ of a lecture that Eugenia Cheng gave in Cambridge last year. With best wishes for (and absolute faith in) category theory, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************ >From: Eduardo Dubuc >To: categories@mta.ca >Subject: categories: re: cracks and pots >Date: Thu, 23 Mar 2006 13:50:45 -0300 (ART) > >Hi > >To follow are the contents of two postings that Bob (always vigilant, ja!) >thought best to concatenate in only one. > >On spite of Robert's erudition and his knowledgeable discourse, I still >think Einstein using differential geometry to develop general relativity >is not at the same level that John Baez using category theory to develop >and/or understand string theory. His arguments are valid in a court of >law, but do not convince me. I imagine John himself is probably the first >to laugh at such a comparison. > >But this is not the issue of my present posting. He touches also some >pertinent points that go more to the core of the "cracks_and_pots" debate. > >(In between ** are Robert words) > >What Motl says certainly does not make people using category theory in >string theory laugh. Applications of category theory to string (or to >other physical theories competing with string theory ?, see Yetter's >posting, it is all very confusing !!) may be valuable or may not. I (and a >lot of us) can not tel. > >** In which case demands that they ($) be read out of the meeting are >premature. >($) papers that claim applications to physics ** > >This is a difficult question. > >Marta was saying (and Bob Walters and others agree) that when a paper was >claiming applications to physics it was easily accepted without >knowledgeable and close examination, and that there were a lot of them. > >Probably a lot of them should be read out, but not by policy against (as >it was erroneously interpreted in these postings). Serious refereeing is a >healthy practice that should not be equaled with censorship. > >**Remember - in mathematics it's a matter of "In God We Trust, >everybody else must provide a proof."** > >This is not so much so. Speculations in math are very difficult. If not >well founded they are vacuous. Only great mathematicians can do them >(example close to us, Grothendieck), the rest of us must provide a proof. > >**If the math itself meets mathematical standards of rigor, its >application to physics need surely only meet the standards appropriate to >that subject.** > >The math itself must also meet standards of quality, not only of rigor. >Besides that, "standards appropriate to that subject" does not mean "free >for anything". Motl writes: > >"I always feel very uneasy if the mathematically oriented people present >their conjectures about physics, quantum gravity, or string theory as some >sort of "obvious facts." > >Clearly he is saying that these standards are not being fulfilled (in his >opinion of course) by claimed applications of math to physics. > >Motl may be wrong or he may be right, what we have not seen yet in these >postings is a convincing or clear answer to the questions he arises. I >would say, not even an answer at all. > >These questions triggered Marta's original posting, which in turn was >arising other (not exactly the same) questions. I do not agree necessarily >with Marta's implicit views, what I support is her courage to point out >that they are serious problems in the category theory community (for >example, quality of the publications, abuse of fashionable topics to get >grants, invited speakers in CT meetings). > >Best wishes e.d. > > > > > > > > From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRzO-0004vh-Qw for categories-list@mta.ca; Sun, 26 Mar 2006 05:56:50 -0400 From: "George Janelidze" To: "categories list" References: Subject: categories: Re: Progressive or linear or ... monoids? Date: Sat, 25 Mar 2006 20:50:39 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 101 Dear Vaughan, 1. Instead of universal Horn formulas I shall talk about quasiidentities, which is the same thing in your context. Recall: A quasiidentity is a fir= st order formula of the form (P1&...&Pn)=3D>Q, where P1,...,Pn,Q are atomic formulas (i.e. "equations of terms"). A class of universal algebras is sa= id to be a quasivariety if it can be axiomatized with quasiidentities. Quasivarieties can be characterized as classes of algebras closed under subalgebras, products, and filtered colimits (the only reason for filtere= d colimits is that we want n above to be finite; products and filtered colimits can be replaced here with filtered products). 2. Since the free monoid on a set X can be considered as a submonoid of t= he free group on the same X, the quasivariety of monoids generated by groups contain all free monoids. In other words, every quasiidentity that holds = for all groups, will also hold for all free monoids. That is, asking your question, forget free monoids! 3. The quasivariety of monoids generated by groups obviously consists exactly of all those monoids that can be embedded into groups. Hence one should probably see Malcev, A. =DCber die Einbettung von assoziativen Systemen in Gruppen. (Russian) Rec. Math. [Mat. Sbornik] N.S. 6 (48), (1939). 331--336. MR0002= 152 (2,7d) Malcev, A. =DCber die Einbettung von assoziativen Systemen in Gruppen. II= . (Russian) Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940). 251--264. MR00028= 95 (2,128b) 4. I do not have these papers here in Cape Town. But I seem to remember t= hat Mal'tsev had an infinite list of quasi-identities which describes the quasi-variety of monoids that can be embedded into groups. Furthermore it seems (I am not sure) that he actually introduced the term "quasiidentity= " exactly for this purpose. The first non-trivial one, which he called Condition Z, was (as =3D bt) & (cs =3D dt) & (au =3D bv) =3D> (cu =3D dv); this condition obviously holds in every group and therefore in every submonoid of a group, but he constructed a semigroup with cancellation in which it does not hold. (Here the difference between monoids and semigrou= ps is irrelevant here). I do not know much did Mal'tsev know about universal constructions; once Mac Lane told me that Zariski knew them... Knowing th= em, one would of course begin by looking at the universal group with a homomorphism from a given monoid into it, and taking the list of quasiidentities from the description of the congruence involved in the construction of that group. 5. Your idea of xy =3D x =3D> y =3D 1 and yx =3D x =3D> y =3D 1 is too ba= d (sorry!), because it is even weaker than the usual cancellation xy =3D xz =3D> y =3D z and yx =3D zx =3D> y =3D z. For, take the quotient of the free monoid on {x,y} by identifying xx =3D = xy (and =3D yx =3D yy if you like); it satisfies your implications but not t= he cancellation. 6. In the abelian case (obviously) it is just one quasiidentity, namely t= he cancellation. That is, the quasivariety of abelian monoids that can be embedded into groups is determined by the axiom x + y =3D x + z =3D> y =3D= z. George Janelidze ----- Original Message ----- From: "Vaughan Pratt" To: "categories list" Sent: Friday, March 24, 2006 10:08 AM Subject: categories: Progressive or linear or ... monoids? > 1. Is the quasivariety of monoids generated by the groups and the free > monoids finitely based? > > That is, is there a finite set of universal Horn formulas entailing the > common universal Horn theory of groups and free monoids? > > In other words, what do groups and free monoids have in common, besides > being monoids? > > Apart from the (equational) axioms for monoids, the only members of tha= t > theory I can think of are xy=3Dx -> y=3D1 and yx=3Dx -> y=3D1. > > 2. How different is the abelian case? More or fewer axioms? > > Vaughan Pratt > > > > From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRuQ-0004pn-F4 for categories-list@mta.ca; Sun, 26 Mar 2006 05:51:42 -0400 Date: Sat, 25 Mar 2006 08:27:04 -0500 (EST) From: Peter Freyd Message-Id: <200603251327.k2PDR4fp029927@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Re: Progressive or linear or ... monoids? Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 102 Vaughan asks: Is the quasivariety of monoids generated by the groups and the free monoids finitely based? The free monoids seem to be red herrings. The question is equivalent to Is the quasivariety of monoids generated by the groups finitely based? (since every free monoid is a submonoid of a group.) If the answer is known I suspect it would be well-known as a theorem about which monoids can be embedded in groups. I note that a 46-year- old paper by S.I.Adyan is still cited. It establishes just the special case of cancelation monoids given by single defining relations. Vaughn goes on to ask How different is the abelian case? More or fewer axioms? This case is easy. Again, the free commutative monoids are free herrings. Just add to the equational theory of commutative monoids the cancelation principle: xy = xz -> y = z. (Every such monoid appears in the quasivariety since its reflector into the subcat of abelian groups is an embedding.) From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRwr-0004sP-47 for categories-list@mta.ca; Sun, 26 Mar 2006 05:54:13 -0400 To: categories list Subject: categories: Re: Progressive or linear or ... monoids? Date: Sat, 25 Mar 2006 08:46:14 -0500 From: wlawvere@buffalo.edu Message-ID: <1143294374.442549a6154a9@mail2.buffalo.edu> References: In-Reply-To: X-Mailer: University at Buffalo WebMail Cyrusoft SilkyMail v1.1.11 MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit X-Originating-IP: 128.205.248.196 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 103 Dear Vaughan A related question, originating with the problem of the quality of the truth value space in M- sets, led me to the discovery of an equational class that some years later, Peter Freyd rediscovered from an entirely different point of view. What is the Structure of the union of the "variety" groups with the variety of commutative monoids ? The motivation was the common feature of the Heyting algebra of right ideals of a monoid in either class, and the partial answer is in my "Taking categories seriously", reprinted in TAC. Not only do we have to think about the meaning of "union" and about the analogy with closed subschemes (probably the source of the term "variety") but also about the fact that the inclusion of groups in monoids is more "open" than closed and is certainly not a (sub) variety even though both categories are algebraic. Again analogously, the Structure 2-functor, adjoint to Semantics does not carry full inclusions to surjective interpretations, Thus in particular in my example like others, the algebraic theory that results has more operations, not only more equations. It may be true in your case as well. Bill Quoting Vaughan Pratt > 1. Is the quasivariety of monoids generated by the groups and the > free > monoids finitely based? > > That is, is there a finite set of universal Horn formulas entailing > the > common universal Horn theory of groups and free monoids? > > In other words, what do groups and free monoids have in common, > besides > being monoids? > > Apart from the (equational) axioms for monoids, the only members of > that > theory I can think of are xy=x -> y=1 and yx=x -> y=1. > > 2. How different is the abelian case? More or fewer axioms? > > Vaughan Pratt > > > > From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNS0H-0004x6-VP for categories-list@mta.ca; Sun, 26 Mar 2006 05:57:45 -0400 Date: Sat, 25 Mar 2006 21:32:47 -0000 (GMT) Subject: categories: Monads on finite categories From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-15 Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 104 Dear All, 1. Are there significant or interesting examples of monads on finite categories? I want to look beyond monads on posets, a.k.a. closure operators. (Since a finite category with binary sums or products is necessarily a poset, some of the usual examples of monads reduce to this case.) I can only think of one class of examples (described below), and I don't know if it's particularly significant. 2. Any monad on a finite category is idempotent. Is this widely known? Thanks. Tom * * * The class of examples: let A be a finite Cauchy-complete category. Let M be the 2-element monoid consisting of the identity and an idempotent, so that [M, A] is the category of idempotents in A. Then the diagonal functor A ---> [M, A] has adjoints on both sides. The induced monad on A is trivial, but that on [M, A] is not. (It sends an idempotent e to 1_a, where a is the object through which e splits.) From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRyI-0004uK-8O for categories-list@mta.ca; Sun, 26 Mar 2006 05:55:42 -0400 Date: Sat, 25 Mar 2006 16:58:11 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories list Subject: categories: Re: Progressive or linear or ... monoids? In-Reply-To: Message-ID: References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 105 The Horn formulae quoted by Vaughan can of course be generalized to the cancellation laws: xy = xz => y = z (and similarly on the other side). In fact the free monoids aren't needed: every free monoid embeds as a submonoid of a (free) group, and so satisfies all Horn formulae true in groups. Thus Vaughan is asking for the universal Horn theory of those monoids which are embeddable in groups. I'm fairly sure that the answer to this is known, but I can't find a reference for it. In the commutative case, it's easy to see that the cancellation law is all you need (the proof is essentially the same as the proof that every integral domain embeds in a field), but I vaguely remember that in the non-commutative case you need something more than the cancellation laws. Generalizing in an obvious way: can one characterize those categories which admit faithful functors to groupoids by a finite collection of Horn formulae? Peter Johnstone ---------------- On Fri, 24 Mar 2006, Vaughan Pratt wrote: > 1. Is the quasivariety of monoids generated by the groups and the free > monoids finitely based? > > That is, is there a finite set of universal Horn formulas entailing the > common universal Horn theory of groups and free monoids? > > In other words, what do groups and free monoids have in common, besides > being monoids? > > Apart from the (equational) axioms for monoids, the only members of that > theory I can think of are xy=x -> y=1 and yx=x -> y=1. > > 2. How different is the abelian case? More or fewer axioms? > > Vaughan Pratt > > > From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNS1F-0004xr-PE for categories-list@mta.ca; Sun, 26 Mar 2006 05:58:45 -0400 Date: Sat, 25 Mar 2006 19:25:43 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu MIME-Version: 1.0 To: categories list Subject: categories: Re: Progressive or linear or ... monoids? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 106 Thanks to Peter, Peter, and George for the uniformly equivalent answers to my question (in striking contrast to the decided lack of uniformity of the responses to other recent topics on this list). (Bill's response was less an answer than a pointer to related work with "commutative" in place of "free" in my question, interesting but for different reasons.) I should have noticed that free monoids were a red herring, and I have no idea why I restricted to z=1 in xy=xz->y=z. The crux of the obstacle to axiomatizing the nonabelian case is clearly brought out by Malcev's Condition Z cited by George Janelidze as > > (as = bt) & (cs = dt) & (au = bv) => (cu = dv); > If you could just shuffle the terms around, Z along with the larger "quasiidentities" would all collapse under cancellation, as everyone pointed out. At first glance one might think to organize Condition Z with proof nets of some kind, something along the lines of au = bv | | as = bt .| | cs = dt ------- cu = dv and so on for longer such sorites along with (independent?) reflections of each side. (The stray dot is against mailers that delete leading spaces, apologies to those not receiving this in a fixed-width font.) However googling for relevant material led to a 194-page St. Andrew's thesis submitted last July on "Presentations for subsemigroups of groups" by Alan Cain, available at http://www-history.mcs.st-and.ac.uk/~alanc/maths/publications/c_phdthesis/c_phdthesisonesided.ps The preface indicates how one can do better via J.-C. Spehner's notion of a Malcev presentation. (Anyone know if this is the same J.-C. Spehner as does computational geometry at Labo MAGE?) > This thesis is largely concerned with subsemigroups of groups, and > from its pages a reader may discover much of their character and a > little of their history. The title is perhaps a little restrictive: > the body of the thesis approaches subsemigroups of groups from three > directions: `ordinary' semigroup presentations, Malcev presentations, > and automatic structures. > > Malcev presentations are semigroup presentations of a special type for > group-embeddable semigroups, introduced by Spehner (1977). Informally, > whilst an `ordinary' semigroup presentation defines a semigroup by > means of generators and defining relations, a Malcev presentation > defines a semigroup using generators, defining relations, and a > rule of group-embeddability. This rule of group-embeddability is > worth an infinite number of defining relations, in the sense that > a semigroup may admit a finite Malcev presentation but no finite > ordinary presentation. During the three decades since Spehner's > definition, little research was carried out in the area. This > thesis should convince the reader that this neglect has been unfair. > In preparation for the main body of the thesis, Chapter 1 formally > defines Malcev presentations and establishes their basic properties. Spehner's basic idea, on the semantic side, is that if it is known by all parties (writers and readers) that the intended congruence on a semigroup S makes S/~ embeddable in a group, then one can uniquely determine ~ with less of it than without that knowledge. The crucial fact here is that the family of all congruences ~ on a semigroup S for which S/~ embeds in a group is closed under arbitrary (including empty) intersection (consider the product of the witnessing embedding groups). Hence any binary relation R on a semigroup S has a unique minimal extension to such a congruence, namely the intersection of all such congruences containing R. A Malcev presentation of a desired such congruence is any binary relation R that generates it in this way. So in particular if au = bv, as = bt, and cs = dt are all in R, then the conclusion cu = dv of Condition Z is in this "Malcev closure" of R. On the syntactic side, the appropriate term rewriting system incorporating this knowledge of group-embeddability augments the vocabulary with left and right inverses of generators, allowing this conclusion to be obtained equationally, without resorting to quasiidentities, as cu = csSu = dtSu = dBbtSu = dBasSu = dBau = dBbv = dv where I've written S for the right inverse of s and B for the left inverse of b for want of a third flavor of s and b in ASCII. After developing the basis for, and variants of, all this, the thesis then dives into automatic semigroups, a subject with a rich literature and some very deep results that I have so far been completely unable to motivate myself to participate in (starting with Michael Arbib's lectures on this general genre at Sydney University in the Australian winter of 1967, give or take a winter, and then more lectures in the same vein in 1969 by John Rhodes at Berkeley). Maybe some day I'll see the point of it all, though so far I've had more success eventually seeing the point of Max Kelly's 1965 lectures on category theory, that only took two decades (slow learner). I'm fine with algebraic logic (which was my route back to category theory in 1983), algebraic automata theory seems an altogether different kettle of fish -- the deployment of weapons of math destruction in an unwinnable war on the combinatorial complexity of automata and semigroups. Again, many thanks for the answers and pointers! Vaughan Pratt From rrosebru@mta.ca Sun Mar 26 06:03:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Mar 2006 06:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNRt4-0004o8-UG for categories-list@mta.ca; Sun, 26 Mar 2006 05:50:18 -0400 Mime-Version: 1.0 (Apple Message framework v623) Content-Type: multipart/alternative; boundary=Apple-Mail-1--406493223 Message-Id: <40e3ffe6ed730ffe4d5f9efe5975c939@math.ksu.edu> From: David Yetter Subject: categories: Re: cracks and pots Date: Fri, 24 Mar 2006 22:22:27 -0500 To: Categories Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=WINDOWS-1252; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 107 Fellow categorists, Jim Stasheff has been appealing to me to comment on the role of=20 category theory in knot theory in the context of the =91cracks and pots=92= =20 thread. In that regard, let me begin with the story I tell in the introduction=20= to my book: At a Joint Summer Research conference a number of years back, Moishe=20 Flato at some point offered the usual dismissal of category theory--=91it=20= is a mere language=92. Kolya Reshetikhin and I undertook that evening = to=20 disabuse him of the notion by explaining Shum=92s coherence theorem (the=20= one-object version, being =91the braided monoidal category with = two-sided=20 duals compatible with the braiding is monoidally equivalent to the=20 category of framed tangles=92). This is a remarkable theorem--a=20 structure absolutely natural from the internal structure of category=20 theory is essentially identical to the key geometric sturctures in 3-=20 and 4-manifold topology, framed tangles being simply =91relative = versions=20 of=92 the framed links on which the Kirby calculi for 3- and 4-manifolds=20= depend. It is one of several theorems relating category theory, and=20 with it a great deal of algebra, to geometric topology, all of which=20 had a =93who ordered that?=94 feel about them. It is also the only = basis=20 on which the connection between knot theory and quantum groups can be=20 explained: the category of representations of a quantum group has the=20= algebraic structure for which framed tangles are a free model! I retired before the point had sunk in, leaving Kolya to continue the=20 discussion. The next morning as I sat in the back of the main lecture=20= hall, Flato came in, tapped me on the shoulder, and with a thumbs up,=20 said =93Hey! Viva les categories!. . .these new ones, the braided=20 monoidal ones.=94 Now, Shum=92s theorem is merely the first of several, all of which give=20= one the =93who ordered that?=94 impression, at least once on starts=20 thinking of TQFT=92s. The other two that come to mind require a bit of=20= set up to state fully, but I will spare you all now: they are Abrams=20 theorem that a 2-dimensional TQFT is equivalent to a Frobenius algebra,=20= and a theorem due to myself and Crane, and Kerler, that in a certain=20 category of cobordisms between surfaces with boundary, the handle (a=20 torus with a hole cut in it) has the structure of a Hopf algebra (CY,=20 K) which is self-dual (CY) and admits a right integral (K), and that=20 every surface with a circle boundary is a Yetter-Drinfel=92d module over=20= the handle. (For an ordinary finite dimensional Hopf algebra,=20 YD-modules are modules over the Drinfel=92d double, but they exist more=20= generally, for infinite dimensional Hopf algebras or Hopf algebra=20 objects in arbitrary monoidal categories, where Drinfel=92d doubles = don=92t=20 exist.) All of these are part and parcel of a different face of category theory=20= than one saw in the old days: category theory as algebra, rather than=20= category theory as foundations. Flato=92s dismissal was directed at category theory as foundations. It=20= is easily ignored if one is interested in foundations of mathematics,=20 since most mathematicians really don=92t care about foundations. For=20 example, many mathematicians pay lip service to the attitude =91set=20 theory is =91the=92 foundation of mathematics=92, but then turn around = and=20 talk about =91the real numbers=92. Which =91real numbers=92? Dedekind = cuts? =20 equivalence classes of Cauchy sequences? a complete Archimedian field=20 constructed from surreal numbers? Now we, as categorists, know the=20 question is silly: one doesn=92t bother asking which of a family of=20 isomorphic structures one means, because they are isomorphic. It is in=20= the practical sense of describing the basic structure of what=20 mathematicians actually do, that category theory is a superior=20 =91foundation=92 to set theory. (Was there ever a time when the epsilon=20= tree defining an element of a smooth manifold ever mattered to anyone?)=20= The dim view of category theory in many mathematical circles is surely=20= due to mathematicians=92 boredom with foundations--and attitude which=20 might be summed up as =93Set theory was bad enough. Why open up all=20 those questions again? Just let me do my geometry, algebra, or=20 whatever.=94 I reading this thread, I wonder how much of the concern about public=20 perceptions of category theory is really concern that =91categories as=20= algebra=92 has become the public face of category theory, concern on the=20= part of those who are fond of =91categories as foundations=92. =91Categories as foundations=92 served the subject poorly in relations = to=20 most mathematicians, but well in relation to computer science: only=20 categorists were willing to take up the challenge of polymorphic type=20 theory. If you thought =91set theory is *the* foundation=92, you bashed=20= your head against Russell=92s paradox and were no help the the folk in=20= CS. We (really those of *you* who took up the challenge) were the only=20= mathematicians who had any hope of being helpful. On the other hand, those of us who set our sights on =91core = mathematics=92=20 have been better served by =91categories as algebra=92: the = applications=20 to knot theory (and geometric topology more generally), homotopy=20 theory, deformation theory, and physics all flow from this =91face=92 of=20= category theory. Even if those of us whose love is =91categories as foundations=92 can be = a=20 little uneasy with the other face of the subject getting applied to=20 physics and drawing fire from outside mathematics, those of us whose=20 love is =91categories as algebra=92 can be uneasy about applications of = the=20 other face to philosophy (as pointed to in Peter Arndt=92s last post),=20= which are sure to be vilified (philosophers and humanists always vilify=20= their rivals, as I am learning from my daughter who is studying=20 philosophy). =91Categories as algebra=92 at least got a =91Viva!=92 = from one=20 of the fathers of deformation quantization. Best Thoughts, David Yetter On 23 Mar 2006, at 14:45, Peter Arndt wrote: > Dear category theorists, > I would like to support Krzysztof Worytkiewicz's remark that "cat=20 > theory > needs to be demystified in first place rather than to be sold" from a > different side: I have recently come across several publications and > research projects of philosophers who have become over-enthusiastic=20 > with > category theory. In certain circles category theory seems to have=20 > gained a > nimbus of an all-encompassing theory of everything, be it part of > mathematics or not, see for example > http://lists.debian.org/debian-devel/2000/10/msg02048.html for an=20 > expression > of such opinions or http://ru.philosophy.kiev.ua/rodin/Endurance.htm=20= > for a > crude offspring of them. Such exaggerated propaganda is very likely to=20= > cause > railings like the one of Lubos Motl. Has anyone observed the same=20 > phenomenon > or does it only exist among the people I have to do with? > > All the best, > > Peter > --Apple-Mail-1--406493223 Content-Transfer-Encoding: quoted-printable Content-Type: text/enriched; charset=WINDOWS-1252 TimesFellow categorists, Jim Stasheff has been appealing to me to comment on the role of category theory in knot theory in the context of the =91cracks and pots=92= thread. In that regard, let me begin with the story I tell in the introduction to my book: =20 At a Joint Summer Research conference a number of years back, Moishe Flato at some point offered the usual dismissal of category theory--=91it is a mere language=92. Kolya Reshetikhin and I undertook that evening to disabuse him of the notion by explaining Shum=92s coherence theorem (the one-object version, being =91the braided monoidal category with two-sided duals compatible with the braiding is monoidally equivalent to the category of framed tangles=92). This is a remarkable theorem--a structure absolutely natural from the internal structure of category theory is essentially identical to the key geometric sturctures in 3- and 4-manifold topology, framed tangles being simply =91relative versions of=92 the framed links on which the Kirby calculi for 3- and 4-manifolds depend. It is one of several theorems relating category theory, and with it a great deal of algebra, to geometric topology, all of which had a =93who ordered that?=94= feel about them. It is also the only basis on which the connection between knot theory and quantum groups can be explained: the category of representations of a quantum group has the algebraic structure for which framed tangles are a free model! I retired before the point had sunk in, leaving Kolya to continue the discussion. The next morning as I sat in the back of the main lecture hall, Flato came in, tapped me on the shoulder, and with a thumbs up, said =93Hey! Viva les categories!. . .these new ones, the braided monoidal ones.=94 Now, Shum=92s theorem is merely the first of several, all of which give one the =93who ordered that?=94 impression, at least once on starts thinking of TQFT=92s. The other two that come to mind require a bit of set up to state fully, but I will spare you all now: they are Abrams theorem that a 2-dimensional TQFT is equivalent to a Frobenius algebra, and a theorem due to myself and Crane, and Kerler, that in a certain category of cobordisms between surfaces with boundary, the handle (a torus with a hole cut in it) has the structure of a Hopf algebra (CY, K) which is self-dual (CY) and admits a right integral (K), and that every surface with a circle boundary is a Yetter-Drinfel=92d module over the handle. (For an ordinary finite dimensional Hopf algebra, YD-modules are modules over the Drinfel=92d double, but they exist more generally, for infinite dimensional Hopf algebras or Hopf algebra objects in arbitrary monoidal categories, where Drinfel=92d doubles don=92t exist.) All of these are part and parcel of a different face of category theory than one saw in the old days: category theory as algebra, rather than category theory as foundations. Flato=92s dismissal was directed at category theory as foundations. It is easily ignored if one is interested in foundations of mathematics, since most mathematicians really don=92t care about foundations. For example, many mathematicians pay lip service to the attitude =91set theory is =91the=92 foundation of mathematics=92, but then turn around = and talk about =91the real numbers=92. Which =91real numbers=92? Dedekind = cuts?=20 equivalence classes of Cauchy sequences? a complete Archimedian field constructed from surreal numbers? Now we, as categorists, know the question is silly: one doesn=92t bother asking which of a family of isomorphic structures one means, because they are isomorphic. It is in the practical sense of describing the basic structure of what mathematicians actually do, that category theory is a superior =91foundation=92 to set theory. (Was there ever a time when the epsilon tree defining an element of a smooth manifold ever mattered to anyone?) The dim view of category theory in many mathematical circles is surely due to mathematicians=92 boredom with foundations--and attitude which might be summed up as =93Set theory was bad enough. Why open up all those questions again? Just let me do my geometry, algebra, or whatever.=94=20 I reading this thread, I wonder how much of the concern about public perceptions of category theory is really concern that =91categories as algebra=92 has become the public face of category theory, concern on the part of those who are fond of =91categories as foundations=92. =20 =91Categories as foundations=92 served the subject poorly in relations = to most mathematicians, but well in relation to computer science: only categorists were willing to take up the challenge of polymorphic type theory. If you thought =91set theory is *the* foundation=92, you bashed your head against Russell=92s paradox and were no help the the folk in CS. We (really those of *you* who took up the challenge) were the only mathematicians who had any hope of being helpful. On the other hand, those of us who set our sights on =91core mathematics=92 have been better served by =91categories as algebra=92: = the applications to knot theory (and geometric topology more generally), homotopy theory, deformation theory, and physics all flow from this =91face=92 of category theory. =20 Even if those of us whose love is =91categories as foundations=92 can be = a little uneasy with the other face of the subject getting applied to physics and drawing fire from outside mathematics, those of us whose love is =91categories as algebra=92 can be uneasy about applications of the other face to philosophy (as pointed to in Peter Arndt=92s last post), which are sure to be vilified (philosophers and humanists always vilify their rivals, as I am learning from my daughter who is studying philosophy). =91Categories as algebra=92 at least got a = =91Viva!=92 from one of the fathers of deformation quantization. Best Thoughts, David Yetter On 23 Mar 2006, at 14:45, Peter Arndt wrote: Dear category theorists, I would like to support Krzysztof Worytkiewicz's remark that "cat theory needs to be demystified in first place rather than to be sold" from a different side: I have recently come across several publications and research projects of philosophers who have become over-enthusiastic with category theory. In certain circles category theory seems to have gained a nimbus of an all-encompassing theory of everything, be it part of mathematics or not, see for example http://lists.debian.org/debian-devel/2000/10/msg02048.html for an expression of such opinions or http://ru.philosophy.kiev.ua/rodin/Endurance.htm for a crude offspring of them. Such exaggerated propaganda is very likely to cause railings like the one of Lubos Motl. Has anyone observed the same phenomenon or does it only exist among the people I have to do with? All the best, Peter = --Apple-Mail-1--406493223-- From rrosebru@mta.ca Mon Mar 27 03:07:01 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:07:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNljm-0005mw-JQ for categories-list@mta.ca; Mon, 27 Mar 2006 03:02:02 -0400 Date: Sun, 26 Mar 2006 14:25:16 +0100 From: "Urs Schreiber" To: categories@mta.ca Subject: categories: re: cracks and pots In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 108 Dear Category Theorists, I have begun trying to compile a list with information (mainly links to reviews and other literature) on applications of categories in mathematical physics and string theory. (It is not finished yet, though.) See http://golem.ph.utexas.edu/string/archives/000775.html . Best regards, Urs Schreiber From rrosebru@mta.ca Mon Mar 27 03:07:23 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:07:23 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNloG-0005rL-Ke for categories-list@mta.ca; Mon, 27 Mar 2006 03:06:40 -0400 Date: Sun, 26 Mar 2006 14:37:25 +0100 From: "V. Schmitt" X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots Content-Type: text/plain; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 109 David Yetter wrote: > Fellow categorists, > > Jim Stasheff has been appealing to me to comment on the role of > category theory in knot theory in the context of the =91cracks and pots= =92 > thread. > [lengthy quotations omitted...] Hi David, then, i again, to precise my thoughts. Knot theory is trivially a good thing. That category has to do with it does not surprise anybody reading this thread. You can relax... Personnaly, and as a matter of taste, i would not put for instance polymorphic types is the same bag. But... ok, say. Now that theoretical physics, computer science, phylo., a mix of those, or whatever? , is used to justify poor "categorical" work is, in my view, an existing problem. More or less everyone is conscious of it (come on!...) but so far that has not been publically debated. I am happy that it happens now. So I am sorry not share the enthusiastic mood that everything is good in maths and I wish that our colleagues "categorists" take categories... humm... seriously. Again, i should not be the one who says that. Best, Vincent. PS: since you averted your book - can we get a good price? From rrosebru@mta.ca Mon Mar 27 03:23:12 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:23:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNm38-0006E2-5D for categories-list@mta.ca; Mon, 27 Mar 2006 03:22:02 -0400 Date: Sun, 26 Mar 2006 16:43:31 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: WHY ARE WE CONCERNED? I Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 110 WHY ARE WE CONCERNED? I When Saunders Mac Lane penned his hard-hitting 1997 Synthese article, he was defending mathematics from an attack many of us hoped would just go away. But Saunders was aware of the seriousness of the threat, which indeed is still here with greater determination. Although the title of that article was "Despite physicists, proof is essential in mathematics", he was not opposing physics, nor even that immediate handful who, assuming the mantle of "mathematical physicists", gave themselves license to insult generations of scrupulously serious physicists and to demand that mathematics adopt a culture that considers conjecture as nearly-established truth. In essence it was an attack on science itself, as the highest form of knowing, that Saunders was opposing. The increased determination of that attack is expressed in two ways. To equip and organize the attack, finance capital has set up several institutions, some of which rather openly proclaim their goal of submitting science to the service of medieval obscurantism. Others say that they support mathematical research, but encourage a barrage of "popular" writings to shock and awe the public into continuing in the belief that they will never understand mathematics and hence never be able to actively participate in science. The contempt for Mac Lane's fight, recently expressed in articles supposedly memorializing him, takes the form of the claim that category theory itself is a "cool" instrument for deepening obscurantism. Not only Harvard's "When is one thing equal to another thing?" and the Cambridge "morality" muddle, but also a 2003 article aimed at teachers of undergraduates, quite explicitly support that claim. In the MAA Monthly, a Clay Fellow states as fact that category theory "is mathematics with the substance removed". Mastering the technique of disinformation whereby the readers are first told that now finally they will be informed, the article suggests that some raising of the level of understanding of the relationship between space and intensively variable quantity is going to be achieved. Then the author short-circuits any such understanding via the simplifying assumption that omits the distinction between covariant and contravariant functors as "unwieldy". As final display of the mastery of expositional technique, the categorical object which has, for nearly twenty pages, been heralded as simple, is revealed in the final pages in the most complicated and unexplained form possible. (Totally passed over is the issue that had led Grothendieck to the considerations allegedly being treated: not only the category of affine schemes, but also the category of all its presheaves, where the author implicitly wants us to work, fails to have the geometrically correct colimits needed to define projective space.) Another level of attack was launched when Cornell University was given very large sums of money to develop methods of teaching geometry without mentioning any geometrical concepts. No proof of the desirability of such a draconian excising of content needed to be given, beyond some phrases from the Dalai Lama. "Dumbing down" is an attack not only on school children and on undergraduates, but also one taking measured aim at colleagues in adjacent fields and at the general public. The general public is thirsty for genuinely informational articles to replace the science fiction gruel served constantly by journals like the Scientific American and the New York Times "Science" section. Those journals have never published anything resembling a mathematical proof and hence have rarely actually explained any scientific subject in a usable way. Nor have they even undertaken any program to raise the level of knowledge of calculus or linear algebra among their readers in a way which would make such explanations feasible. Instead, they provide games and amusements to divert the mathematically-interested public. In January of 2005 the Notices of the AMS announced that they had for a full ten years been strictly following a certain editorial policy. There had been a widespread demand for expository articles. To that demand, the response was a new definition of "expository": all precise definitions of mathematical concepts must be eliminated. Authors of expository articles were forced to compromise their presentation, or to withdraw their paper. Mathematicians, who were for several years becoming aware that these new expository articles are absolutely useless for developing a mathematical thought, were shocked to learn that a conscious policy had forced that situation. A peculiar sort of anti-authoritarianism seems to be the only justification offered for degrading the role of definition, theorem, and proof; certainly, serious expositors have never considered that the use of those three pillars of geometrical enlightenment excludes explanations and examples. Others have urged, however, that those instruments be eliminated even from lectures at meetings and from professional papers. That threat is part of the background for the concern expressed in the many messages to the categories list over the past weeks. Deeply concerned mathematicians ask me "How can we know?". Indeed, how can we know whether it is worthwhile to attend a certain meeting or a certain talk, and how can a scientific committee know whether a proposed talk is scientifically viable? If the "you don't want to know" culture of no proofs, no definitions, is accepted, we will truly have no way of knowing, and will be pressured to fall back on unsupported faith. ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Mon Mar 27 03:25:51 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:25:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNm67-0006Ix-8h for categories-list@mta.ca; Mon, 27 Mar 2006 03:25:07 -0400 From: "Ronnie Brown" To: Subject: categories: Re: George Mackey, 1916-2006 Date: Sun, 26 Mar 2006 22:48:29 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 111 I met George Mackey in April 1967 at the British Mathematical Colloquium in Swansea, where I gave an invited talk on the groupoid van Kampen theorem, and I overheard some people in the common room saying they were not completely convinced. You win some, you lose some! But Mackey came up to me at tea and said: `That was very interesting. I have been using groupoids for years. My name is Mackey.' He then told me of his work in ergodic theory using `virtual groups'. It occurred to me that if the groupoid idea can be arrived at from two quite different directions, then there couild be more in the groupoid idea than met the eye. It became clear that he used the action groupoid of a group action, and this convinced me that I should add to my planned book a chapter on covering spaces and covering groupoids. For those who are unaware of the idea of a virtual group, Mackey's idea was that since a transitive action of a group corresponded to a conjugacy class of subgroups, then an ergodic action (i.e. one where the orbits are of measure 0 or 1) should correspond to an analogous concept. His exposition went through various phases, including a cocycle formulation, and eventually involved the measured groupoid corresponding to an ergodic action. This work, and that of his students, such as Arlan Ramsay, has been a foundation, as I understand it, for much work on the C^*-algebras of measured groupoids. We met a few more times, and he was always most friendly and genuine. When I went to Bangor, Tony Seda came there from Warwick in order to help his mother who was ill and lived in Llandudno. His MSc project had been in measure theory. So we agreed he should look at Mackey's work. In the end he developed Haar measure in this area, and when I told Mackey he said *his* student was doing the same! Tony's excellent papers in this area have perhaps not been as well noticed as they should, so Tony in the end moved into theoretical computer science. So my conclusion is that George Mackey was a great pioneer in structural ideas in mathematics, with a broad range of interests, and a really nice guy. Ronnie Brown From rrosebru@mta.ca Mon Mar 27 03:26:42 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:26:42 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNm72-0006KL-Sf for categories-list@mta.ca; Mon, 27 Mar 2006 03:26:04 -0400 Subject: categories: Geocal 06 To: categories@mta.ca (categories) Date: Sun, 26 Mar 2006 13:50:19 -0800 (PST) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 112 Here's some stuff about Geocal 06 and n-categories in proof theory. Best, jb Also available at http://math.ucr.edu/home/baez/week227.html March 12, 2006 This Week's Finds in Mathematical Physics (Week 227) John Baez [stuff about physics deleted] I had a great time in Marseille. The area around there is great for mathematicians. Algebraists can visit the beautiful nearby city of Aix - pronounced "x". Logicians will enjoy the dry, dusty island of If - pronounced "eef", just like a French logician would say it. And everyone will enjoy the medieval hill town of Les Baux, which looks like something out of Escher. Actually the Chateau D'If, on the island of the same name, is where Edmond Dantes was imprisoned in Alexander Dumas' novel "The Count of Monte Cristo". It's in this formidable fortress that the wise old Abbe Faria tells Dantes the location of the treasure that later made him rich. I guess everyone except me read this story as a kid - I'm just reading it now. But how many of you remember that Faria spent his time in prison studying the works of Aristotle? There's a great scene where Dantes asks Faria where he learned so much about logic, and Faria replies: "If - and only If!" That Dumas guy sure was a joker. Luckily I didn't need to be locked up on a deserted island to learn some logic in Marseille. There were lots of great talks on this topic at the conference I attended: 7) Geometry of Computation 2006 (Geocal06), http://iml.univ-mrs.fr/geocal06/ For example, Yves Lafont gave a category-theoretic approach to Boolean logic gates which explains their relation to Feynman diagrams: 8) Yves Lafont, Towards an algebraic theory of Boolean circuits, Journal of Pure and Applied Algebra 184 (2003), 257-310. Also available at http://iml.univ-mrs.fr/~lafont/publications.html and together with Yves Guiraud, Francois Metayer and Albert Burroni, he gave a detailed introduction to the homology of n-categories and its application to rewrite rules. The idea is to study any sort of algebraic gadget (like a group) by creating an n-category where the objects are "expressions" for elements in the gadget, the morphisms are "ways of rewriting expressions" by applying the rules at hand, the 2-morphisms are "ways of passing from one way of rewriting expressions to another" by applying certain "meta-rules", and so on. Then one can use ideas from algebraic topology to study this n-category and prove stuff about the original gadget! To understand how this actually works, it's best to start with Craig Squier's work on the word problem for monoids. I explained this pretty carefully back in "week70" when I first heard Lafont lecture on this topic - it made a big impression on me! You can read more here: 9) Yves Lafont and A. Proute, Church-Rosser property and homology of monoids, Mathematical Structures in Computer Science, Cambridge U. Press, 1991, pp. 297-326. Also available at http://iml.univ-mrs.fr/~lafont/publications.html 10) Yves Lafont, A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier), Journal of Pure and Applied Algebra 98 (1995), 229-244. Also available at http://iml.univ-mrs.fr/~lafont/publications.html Then you can go on to the higher-dimensional stuff: 11) Albert Burroni, Higher dimensional word problem with application to equational logic, Theor. Comput. Sci. 115 (1993), 43-62. Also available at http://www.math.jussieu.fr/~burroni/ 12) Yves Guiraud, The three dimensions of proofs, Annals of Pure and Applied Logic (in press). Also available at http://iml.univ-mrs.fr/%7Eguiraud/recherche/cos1.pdf 13) Francois Metayer, Resolutions by polygraphs, Theory and Applications of Categories 11 (2003), 148-184. Available online at http://www.tac.mta.ca/tac/volumes/11/7/11-07abs.html I was also lucky to get some personal tutoring from folks including Laurent Regnier, Peter Selinger and especially Phil Scott. Ever since "week40", I've been trying to understand something called "linear logic", which was invented by Jean-Yves Girard, who teaches in Marseille. Thanks to all this tutoring, I think I finally get it! To get a taste of what Phil Scott told me, you should read this: 14) Philip J. Scott, Some aspects of categories in computer science, Handbook of Algebra, Vol. 2, ed. M. Hazewinkel, Elsevier, New York, 2000. Available as http://www.site.uottawa.ca/~phil/papers/ Right now, I'm only up to explaining a microscopic portion of this stuff. But since the typical reader of This Week's Finds may know more about physics than logic, maybe that's good. In fact, I'll use this as an excuse to simplify everything tremendously, leaving out all sorts of details that a real logician would want. Logic can be divided into two parts: SYNTAX and SEMANTICS. Roughly speaking, syntax concerns the symbols you scribble on the page, while semantics concerns what these symbols mean. A bit more precisely, imagine some kind of logical system where you write down some theory - like the axioms for a group, say - and use it to prove theorems. In the realm of syntax, we focus on the form our theory is allowed to have, and how we can deduce new sentences from old ones. So, one of the key concepts is that of a PROOF. The details will vary depending on the kind of logical system we're studying. In the realm of semantics, we are interested in gadgets that actually satisfy the axioms in our theory - for example, actual groups, if we're thinking about the theory of groups. Such a gadget is called a MODEL of the theory. Again, the details vary immensely. In the realm of syntax, we say a list of axioms X "implies" a sentence P if we can prove P from X using some deduction rules, and we write this as X |- P In the realm of semantics, we say a list of axioms X "entails" a sentence P if every model of X is also a model of P, and we write this as X |= P Syntax and semantics are "dual" in a certain sense - a sense that can be made very precise if one fixes a specific class of logical systems. This duality is akin to the usual relation between vector spaces and their duals, or more generally groups and their categories of representations. The idea is that given a theory T you can figure out its models, which form a category Mod(T) - and conversely, given the category of models Mod(T), perhaps with a little extra information, you can reconstruct T. A little extra information? Well, in some cases a model of T will be a *set* with some extra structure - for example, if T is the theory of groups, a model of T will be a group, which is a set equipped with some operations. So, in these cases there's a functor U: Mod(T) -> Set assigning each model its underlying set. And, you can easily reconstruct T from Mod(T) *together* with this functor. This idea was worked out by Lawvere for a class of logical systems called algebraic theories, which I discussed in "week200". But, the same idea goes by the name of "Tannaka-Krein duality" in a different context: a Hopf algebra H has a category of comodules Rep(H), which comes equipped with a functor U: Rep(H) -> Vect assigning each comodule its underlying vector space. And, you can reconstruct H from Rep(H) together with this functor. The proof is even very similar to Lawvere's proof for algebraic theories! I gave a bunch of talks in Marseille about algebraic theories, some related logical systems called PROPs and PROs, and their relation to quantum theory, especially Feynman diagrams: 14) John Baez, Universal algebra and diagrammatic reasoning, available as http://math.ucr.edu/home/baez/universal/ I came mighty close to explaining how to compute the cohomology of an algebraic theory... and you can read more about that here: 15) Mauka Jibladze and Teimuraz Pirashvili, Cohomology of algebraic theories, J. Algebra 137 (1991) 253-296. Mauka Jibladze and Teimuraz Pirashvili, Quillen cohomology and Baues-Wirsching cohomology of algebraic theories, Max-Planck-Institut fuer Mathematik, preprint series 86 (2005). But alas, I didn't get around to talking about the duality between syntax and semantics. For that Lawvere's original thesis is a good place to go: 16) F. William Lawvere, Functorial Semantics of Algebraic Theories, Ph.D. thesis, Columbia University, 1963. Also available at http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html Anyway, the stuff Phil Scott told me about was mainly over on the syntax side. Here categories show up in another way. Oversimplifying as usual, the idea is to create a category where an object P is a *sentence* - or maybe a list of sentences - and a morphism f: P -> Q is a *proof* of Q from P - or maybe an equivalence class of proofs. We can compose proofs in a more or less obvious way, so with any luck this gives a category! And, different kinds of logical system give us different kinds of categories. Quite famously, intuitionistic logic gives cartesian closed categories. The "multiplicative fragment" of linear logic gives *-autonomous categories. And, full-fledged linear logic gives us certain fancier kinds of categories. If you want to learn about these examples, read the handbook article by Phil Scott mentioned above. One thing that intrigues me is the equivalence relation we need to get a category whose morphisms are equivalence classes of proofs. In Gentzen's "natural deduction" approach to logic, there are various deduction rules. Here's one: P |- Q P |- Q' ------------------ P |- Q & Q' This says that if P implies Q and it also implies Q', then it implies Q & Q'. Here's another: P |- Q => R ------------ P and Q |- R And here's a very important one, called the "cut rule": P |- Q Q |- R ----------------- P |- R If P implies Q and Q implies R, then P implies R! There are a bunch more... and to get the game rolling we need to start with this: P |- P In this setup, a proof f: P -> Q looks vaguely like this: f-crud f-crud f-crud f-crud f-crud f-crud ------------- P |- Q The stuff I'm calling "f-crud" is a bunch of steps which use the deduction rules to get to P |- Q. Suppose we also we also have a proof g: Q -> R There's a way to stick f and g together to get a proof fg: P -> R This proof consists of setting the proofs f and g side by side and then using the cut rule to finish the job. So, fg looks like this: f-crud g-crud f-crud g-crud f-crud g-crud f-crud g-crud -------- -------- P |- Q Q |- R --------------------- P |- R Now let's see if composition is associative. Suppose we also have a proof h: R -> S We can form proofs (fg)h: P -> S and f(gh): P -> S Are they equal? No! The first one looks like this: f-crud g-crud f-crud g-crud f-crud g-crud h-crud f-crud g-crud h-crud -------- -------- h-crud P |- Q Q |- R h-crud --------------------- ----------- P |- R R |- S ---------------------------- P |- S while the second one looks like this: g-crud h-crud g-crud h-crud f-crud g-crud h-crud f-crud g-crud h-crud f-crud -------- -------- f-crud Q |- R R |- S --------- --------------------- P |- Q Q |- S ---------------------------- P |- S So, they're not quite equal! This is one reason we need an equivalence relation on proofs to get a category. Both proofs resemble trees, but the first looks more like this: \ / / \/ / \ / | while the second looks more like this: \ \ / \ \/ \ / | So, we need an equivalence relation that identifies these proofs if we want composition to be associative! This sort of idea, including this "tree" business, is very familiar from homotopy theory, where we need a similar equivalence relation if we want composition of paths to be associative. But in homotopy theory, people have learned that it's often better NOT to impose an equivalence relation on paths! Instead, it's better to form a *weak 2-category* of paths, where there's a 2-morphism going from this sort of composite: \ / / \/ / \ / | to this one: \ \ / \ \/ \ / | This is called the "associator". In our logic context, we can think of the associator as a way to transform one proof into another. The associator should satisfy an equation called the "pentagon identity", which I explained back in "week144". However, it will only do this if we let 2-morphisms be *equivalence classes* of proof transformations. So, there's a kind of infinite regress here. To deal with this, it would be best to work with a "weak omega-category" with sentences (or sequences of sentences) as objects, proofs as morphisms, proof transformations as 2-morphisms, transformations of proof transformations as 3-morphisms,... and so on. With this, we would never need any equivalence relations: we keep track of all transformations explicitly. This is almost beyond what mathematicians are capable of at present, but it's clearly a good thing to strive toward. So far, it seems Seely has gone the furthest in this direction. In his thesis, way back in 1977, he studied what one might call "weak cartesian closed 2-categories" arising from proof theory. You can read an account of this work here: 17) R.A.G. Seely, Weak adjointness in proof theory, in Proc. Durham Conf. on Applications of Sheaves, Springer Lecture Notes in Mathematics 753, Springer, Berlin, 1979, pp. 697-701. Also available at http://www.math.mcgill.ca/rags/WkAdj/adj.pdf R.A.G. Seely, Modeling computations: a 2-categorical framework, in Proc. Symposium on Logic in Computer Science 1987, Computer Society of the IEEE, pp. 65-71. Also available at http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf Can we go all the way and cook up some sort of omega-category of proofs? Interestingly, while the logicians at Geocal06 were talking about n-categories and the geometry of proofs, the mathematician Vladimir Voevodsky was giving some talks at Stanford about something that sounds pretty similar: 18) Vladimir Voevodsky, lectures on homotopy lambda calculus, notice at http://math.stanford.edu/distinguished_voevodsky.htm Voevodsky has thought hard about n-categories, and he won the Fields medal for his applications of homotopy theory to algebraic geometry. The typed lambda calculus is another way of thinking about intuitionistic logic - or in other words, cartesian closed categories of proofs. The "homotopy lambda calculus" should thus be something similar, but where we keep track of transformations between proofs, transformations between transformations between proofs... and so on ad infinitum. But that's just my guess! Is this what Voevodsky is talking about??? I haven't managed to get anyone to tell me. Maybe I'll email him and ask. There were a lot of other cool talks at Geocal06, like Girard's talk on applications of von Neumann algebras (especially the hyperfinite type II_1 factor!) in logic, and Peter Selinger's talk on the category of completely positive maps, diagrammatic methods for dealing with these maps, and their applications to quantum logic: 19) Peter Selinger, Dagger compact closed categories and completely positive maps, available at http://www.mscs.dal.ca/~selinger/papers.html But, I want to finish writing this and go out and have some waffles for my Sunday brunch. So, I'll stop here! ----------------------------------------------------------------------- Addendum: An anonymous correspondent had this to say about the "homotopy lambda calculus": Several years ago, Kontsevich explained to me an idea he had about "homotopy proof theory" (or model theory, or logic, ...). As soon as I saw Voevodsky's abstract it reminded me of what Kontsevich said; perhaps it's a well-known idea in the Russian-Fields-medallist club. Somewhere I have notes from what Kontsevich said, but as far as I remember it went roughly like this. In certain set-ups (such as Martin-Lof type theory) every statement carries a proof of itself. Of course, a statement may have many proofs. If we imagine that all the statements are of the form "A = B", then what we're saying is that every equals sign carries with it a *reason* for equality, or proof of equality. If I remember rightly, Kontsevich's idea was to do a topological analogue, so that every term (like A and B) is assigned a point in some fixed space, and equalities of terms induce paths between points. There was more, pushing the idea further, but I forget what. ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html From rrosebru@mta.ca Mon Mar 27 03:29:09 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Mar 2006 03:29:09 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FNm9E-0006OJ-Ab for categories-list@mta.ca; Mon, 27 Mar 2006 03:28:20 -0400 Date: Sun, 26 Mar 2006 17:04:38 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: Foundations? Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 113 Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Tue Mar 28 19:01:39 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Mar 2006 19:01:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FON7f-00067M-En for categories-list@mta.ca; Tue, 28 Mar 2006 18:57:11 -0400 Date: Mon, 27 Mar 2006 12:05:28 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu MIME-Version: 1.0 To: cat group Subject: categories: The Yoneda axiom (was: One-stop version of my postings) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 114 Robert J. MacG. Dawson wrote: > Vaughan Pratt wrote: >> In axiomatic mathematics, everything that is not forbidden is permitted. > > > Yes, in a sense... but theorems along the lines of "there exists a > model of X such that Y" were a long, long way in the future Reuben Hersh, cited in Eugenia Cheng's fascinating paper "Mathematics, Morally," put it succinctly in 1991. "There is an amazingly high consensus in mathematics as to what is 'correct' or 'accepted'." It's really too bad we can't go back and show those old-timers what their stuff has evolved into. Most of it would presumably evoke the ancient counterpart of "Wow," but that's boringly predictable. What would really be interesting would be the less predictable reactions of the form "I knew that but didn't write it down because...". For example the basic concept of a model must surely have occurred to many over the millennia but seemed pointless formulating as a mathematical concept in its own right, perhaps for want of the concept of any alternative model to that one, perhaps for want of a suitable framework, perhaps because everyone else seemed to be taking it for granted as part of the ambient logic so why should I rock that boat? One can see this in the late Robert Floyd's seminal 1967 paper "Assigning meanings to programs" on the verification of imperative programs, where he clearly has some model of his axioms in mind in his completeness proof for his axiomatization of assignment x:=e (the dual of Hoare's subsequent axiomatization), yet it does not occur to him to say what the model is. In my 1976 paper "Semantical considerations on Floyd-Hoare logic" I made explicit the model Floyd apparently had in mind in his proof by defining it to be a Kripke structure (of the kind introduced by Kripke in "Semantical considerations on modal logic"), showed that Floyd's informal completeness argument worked just fine when formalized for that model (so Floyd was morally correct), and extended it to a complete Floyd-style axiomatization of array assignment (whose completeness is complicated by the possibility of aliasing as in a[a[i]]:=a[a[j]]). (The paper then went on to define modal dynamic logic, which I put at the end as merely a syntactic sidebar of marginal interest; unexpectedly it caught people's fancy and appears to have settled down as a staple of linguistic theory.) I believe this was the first application of model theoretic methods to the verification of programs containing assignment statements, the kind of program forming the mainstream of programming languages. (Functional programming in contrast was designed from the outset to support semantics, whence its popularity with logicians.) It certainly was the first to use Kripke structures for that purpose, which today are the standard model for imperative programming. The paper cites and treats a number of references to prior applications of modal logic to programming, none of which however offered a semantics, seemingly having such a semantics only in the back of their mind as with Floyd. This neglect of semantics continues even today: most programmers, along with their managers, remain oblivious to the important backroom role of semantics in the verification of their programs, to the extent of being unaware even of its existence. A basic reason for not writing down what one is thinking is lack of awareness of one's thought processes. Another is want of a suitable framework for formulating those thoughts in a way other than as what later mathematicians would judge as handwaving. Yet another is the difficulty, even today, of separating mathematics from religion, defined as matters of faith. The irrationality of the diagonal of the unit square was at one point unthinkable, giving us no way of knowing for how many centuries it was a dark secret of certain mathematical cults bordering on heresy or worse. Today's mathematical religions have shifted their attention away from existence, where nowadays everything goes (though Robinson-style infinitesimals are a tiny pill that some find hard to swallow), instead focusing on relevance, utility, and other subjective criteria for acceptance. Foundations reared its head in the 19th C, forcing mathematicians to hold up their intuitions to the bright lights and microscopes of logic. This process is somewhat akin to fashion designers being asked to appraise the apparel of their emperor. Once a consensus has built, even if artificially as necessary for underdressed emperors, it becomes very hard to stick to your guns. This phenomenon makes it impossible to tell, post-Cantor, whether those intuitions were more set theoretical or category theoretical in nature. Answers to that must be sought in pre-Cantorian mathematics. My personal belief is that when the pathways of the human brain are finally unravelled and understood, they will be shown to be based on Yoneda, not as a lemma however but as an axiom postulating a certain dense embedding. In the millions of years of evolution of primate thinking, no productive mathematical mechanism has a higher probability of being stumbled on than mathematics founded on the Yoneda axiom. I know of no better explanation of how human thought could have evolved to its present form than evolution finding and exploiting the Yoneda principle, that (for example) the unique 2-element semigroup whose Cayley table has distinct constant columns can generate every graph and graph homomorphism in a simple, uniform, and economical way. The likelihood of evolution stumbling on a serviceable fragment of Zermelo's axioms other than by our conscious thought seems considerably lower. If evolution discovered both, chances are it found Yoneda first. Yoneda needs so much less neural machinery than any of its competitors. Yet even today Google has not stumbled on the Yoneda axiom (as a phrase). I put that down to the mechanism being such a fundamental part of how we think that we have no more mental access to it than we have physical access to quarks. Proof that our brains incorporate the Yoneda axiom will have to await more advanced technology, until then that model of the evolution of human thought will remain a conceit of the cracks and pots promoting it. Vaughan Pratt From rrosebru@mta.ca Tue Mar 28 19:01:40 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Mar 2006 19:01:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FON21-0005iw-Dy for categories-list@mta.ca; Tue, 28 Mar 2006 18:51:21 -0400 From: "Reinhard Boerger" Organization: FernUniversitaet To: categories@mta.ca Date: Mon, 27 Mar 2006 10:06:02 +0200 MIME-Version: 1.0 Subject: categories: Re: Monads on finite categories Message-ID: <4427B908.13265.48C862@reinhard.boerger.fernuni-hagen.de> Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Content-description: Mail message body X-prewhitelist: your reply will pass through without greylisting Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 115 Hello, Tom Leinster wrote: > 1. Are there significant or interesting examples of monads on finite > categories? I want to look beyond monads on posets, a.k.a. closure > operators. (Since a finite category with binary sums or products > is necessarily a poset, some of the usual examples of monads reduce > to this case.) I can only think of one class of examples > (described below), and I don't know if it's particularly > significant. Idempotent monads correspond to full reflective subcategories; so the only examples are induced by full reflective subcategories of finite categories. Greetings Reinhard From rrosebru@mta.ca Tue Mar 28 19:01:40 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Mar 2006 19:01:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FON4n-0005tu-HV for categories-list@mta.ca; Tue, 28 Mar 2006 18:54:13 -0400 Date: Mon, 27 Mar 2006 09:56:58 -0400 From: "Robert J. MacG. Dawson" User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To:cat group Subject: categories: Re: One-stop version of my postings References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 116 Vaughan Pratt wrote: > Robert J. MacG. Dawson wrote: > >> ...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his >> axioms assert that any pair of circles whatsoever do so.) > > > In axiomatic mathematics, everything that is not forbidden is permitted. Yes, in a sense... but theorems along the lines of "there exists a model of X such that Y" were a long, long way in the future -Robert From rrosebru@mta.ca Tue Mar 28 19:01:40 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Mar 2006 19:01:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FON6K-00061Y-3g for categories-list@mta.ca; Tue, 28 Mar 2006 18:55:48 -0400 Subject: categories: Re: cracks and pots To: categories@mta.ca Date: Mon, 27 Mar 2006 10:28:57 -0400 (AST) In-Reply-To: from "Marta Bunge" at Mar 14, 2006 12:48:33 PM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 117 I just returned from a vacation and caught up with this thread, so please bear with me as I back up to the central question posed by Marta Bunge. She suggested that > anything which even remotedly claims to have applications to physics > (particularly string theory) is given (what I view as) uncritical > support in our circles. Is there any evidence to support this claim? I.e., actual examples where such research was disproportionally supported that was uncritical and perhaps unwarranted? There have been several posts seemingly agreeing that this is the case, but none have given concrete evidence. I feel that it is necessary to establish that such practices indeed exist, before discussing what, if anything, needs to be done about it. Can one rule out another possibility, namely that such research is supported because it is original, timely, and interesting? -- Peter Marta Bunge wrote: > > Robert Dawson wrote: > > > It is not clear to me that the majority of theoretical physicists agree > >with the negative view of categorical string theory held by the cited blog > >writers; and in the absence of a consensus among the physicists, I for one > >(with an undergradate degree and some graduate courses in physics) do not > >feel qualified to take sides; if anything, errors should be on the side of > >trying out too many ideas, not too few. > > > > I was trying to elicit an open response from those who *do* know about the > value (or lack of it) of categorical string theory. In particular, I would > like to have an answer to this question. Why is it that anything which even > remotedly claims to have applications to physics (particularly string > theory) is given (what I view as) uncritical support in our circles? > > Best, > Marta > > > > From rrosebru@mta.ca Wed Mar 29 07:35:26 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 07:35:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOYsA-00053A-WA for categories-list@mta.ca; Wed, 29 Mar 2006 07:29:59 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Message-Id: To: Categories From: dusko Subject: categories: Re: cracks and pots Date: Tue, 28 Mar 2006 00:01:38 -0800 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=WINDOWS-1252; delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 118 i think david yetter's analysis of the dichotomy "categories as =20 foundations" vs "categories as algebra" was spot on --- with respect =20= to people and the community. indeed, one could split most of our =20 papers into one category or the other. but at the end of the day, i think, we'll all agree that the source =20 of the unreasonable effectiveness of categorical algebra is its =20 foundational content (although there is probably a lot of it that we =20 dont understand yet); and the other way around. eg, if you look at =20 grothendieck's work, he started working in algebra, and ended up =20 developing foundational structures, because he needed them. and a lot =20= on the "algebra" side now is built upon them. ok, then for a while it =20= was thought that he exaggerated with foundations, and that a more =20 direct approach "could have been in better taste" (to cite =20 eilenberg). but maby the fermat theorem would have a more useful =20 proof if it was developed in grothendieck style. and nowadays, there =20 is a lot of foundational content in tannaka duality etc, in TQFT in =20 general, but we only see hints of it at the moment (and i for one =20 just see the reflections of these hints in other people's eyes). i am of course saying things very clear and familiar to many people =20 on this list, but maby they are worth saying nevertheless. it might =20 be good if the links between "categories as algebras" and "categories =20= as foundations" would not boil down just to the greatest of the =20 category theorists, leaving the rest of us in two camps. -- dusko From rrosebru@mta.ca Wed Mar 29 07:35:26 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 07:35:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOYuN-00056s-15 for categories-list@mta.ca; Wed, 29 Mar 2006 07:32:15 -0400 Message-ID: <44295E5F.4050504@math.upenn.edu> Date: Tue, 28 Mar 2006 11:03:43 -0500 From: jim stasheff MIME-Version: 1.0 To: Categories List Subject: categories: how science (math) should work Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 119 The latter part of http://www.oxfordtoday.ox.ac.uk/2005-06/v18n2/01.shtml has some very relevant comments response? jim From rrosebru@mta.ca Wed Mar 29 07:36:19 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 07:36:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOYxc-0005F9-1f for categories-list@mta.ca; Wed, 29 Mar 2006 07:35:36 -0400 Message-ID: <4429A1E1.8080907@math.upenn.edu> Date: Tue, 28 Mar 2006 15:51:45 -0500 From: jim stasheff User-Agent: Thunderbird 1.5 (Windows/20051201) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 120 I beg to differ - a little F W Lawvere wrote: > WHY ARE WE CONCERNED? I > > "Dumbing down" is an attack not only on school children and on > undergraduates, but also one taking measured aim at colleagues in adjacent > fields and at the general public. The general public is thirsty for > genuinely informational articles to replace the science fiction gruel > served constantly by journals like the Scientific American and the New > York Times "Science" section. so far so good Those journals have never published anything > resembling a mathematical proof why should they? and hence have rarely actually explained > any scientific subject in a usable way. a math proof is hardly necessary to explain a scientific subject in a usable way. now for a mathematical subject a math proof is sometimes but not always necessary > In January of 2005 the Notices of the AMS announced that they had > for a full ten years been strictly following a certain editorial policy. > There had been a widespread demand for expository articles. To that > demand, the response was a new definition of "expository": all precise > definitions of mathematical concepts must be eliminated. Authors of > expository articles were forced to compromise their presentation, or to > withdraw their paper. Not all of us and notice you are talking about the NOTICES not the Bulletin Mathematicians, who were for several years > becoming aware that these new expository articles are absolutely useless > for developing a mathematical thought, developing a mathematical thought, depends what you mean by that developing in the sense of enough to be active in the field - of course not developing a sense of what the thought of the experts are so that one might want to learn more or NOT or might see relevance to ones own disparate research - they work fine were shocked to learn that a > conscious policy had forced that situation. > A peculiar sort of anti-authoritarianism seems to be the only > justification offered for degrading the role of definition, theorem, and > proof; certainly, serious expositors have never considered that the use of > those three pillars of geometrical enlightenment excludes explanations and > examples. Others have urged, however, that those instruments be > eliminated even from lectures at meetings and from professional papers. Examples ? I certinaly have not seen such In fact as an editor and referee and all the referees I've used have never tolerated such elimination. in fact, due to cross fertilization, even some physics papers now have defintions > That threat is part of the background for the concern expressed in > the many messages to the categories list over the past weeks. Deeply > concerned mathematicians ask me "How can we know?". Indeed, how can we > know whether it is worthwhile to attend a certain meeting or a certain > talk, and how can a scientific committee know whether a proposed talk is > scientifically viable? If the "you don't want to know" culture of no > proofs, no definitions, is accepted, we will truly have no way of knowing, > and will be pressured to fall back on unsupported faith. > Me thinks thou doth protest too much or you've run into some alternate universe I'm unfamiliar with ;-D jim From rrosebru@mta.ca Wed Mar 29 07:38:41 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 07:38:41 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOYzv-0005L2-1u for categories-list@mta.ca; Wed, 29 Mar 2006 07:37:59 -0400 Date: Tue, 28 Mar 2006 21:30:59 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: WHY ARE WE CONCERNED? II Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 121 WHY ARE WE CONCERNED? II Misconceptions The question is not whether mathematics should be applied. Most of us agree that it should. The concern is rather that our subject is sometimes being used as a mystifying smoke screen to protect pseudo-applications against the scrutiny of the general public and of the scientific colleagues in adjacent disciplines. We need to ensure that applications themselves be maximally effective, not clouded by misunderstanding. Some of the most important applications of our unifying efforts as categorists have been to the teaching of algebraic topology teaching of algebraic geometry teaching of logic and set theory teaching of differential geometry These subjects all arose from the efforts to clarify and apply calculus; thus some of us have applied category theory to the teaching of calculus. But it seems that we have not taught category theory itself well enough. Several recent writings reveal that basic misunderstandings about category theory are still prevalent, even among people who use it. Some of these concern the myth that category theory is the "insubstantial part" of mathematics and that it heralds an era when precise axioms are no longer needed. (Other myths revolve around the false belief that there are "size problems" if one tries to do category theory in a way harmonious with the standard practice of professional set theorists; see next posting.) The first of these misunderstandings is connected with taking seriously the jest "sets without elements". The traditions of algebraic geometry and of category theory are completely compatible about elements, as I now show. Contrary to Fregean rigidity, in mathematics we never use "properties" that are defined on the universe of "everything". There is the "universe of discourse" principle which is very important: for example, any given group, (or any given topological space, etc.) acts as a universe of discourse. As these examples suggest, a universe of discourse typically carries a structure which permits interesting properties and constructions on it. As the examples also show, there are typically many objects of a given mathematical category and also many categories, so transformation is an essential part of the content. As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries. (His conjecture that the continuum hypothesis holds in that realm is probably true. [Bulletin of Symbolic Logic 9 (2003) 213-223].) Dedekind, Hausdorff, and most of 20th century mathematics followed the paradigm whereby structures have two aspects, a theory and an interpretation of it in such a featureless background. Because the background thus contributes minimal distortion to the assumptions of the theory, the completeness theorems of first-order logic, the Nullstellensatz, and related results are available. The more geometric background categories which receive models are also viewed as structures (of an opposite kind) in abstract sets, for example the classifying topos for local rings as a background for algebraic groups. Such is "set theory" in the practice of mathematics; it is part of the essence from which organization emerges. By contrast, the "set theory" studied by 20th century set theorists has a different aim and architecture. The aim is "justification" of mathematics, and the architecture is that of the cumulative hierarchy. The alleged need for justification arose in connection with the re-naming of Cantor's theorem as "Russell's paradox"; Cantor's theorem had shown that the system proposed by Frege was inconsistent, but there were those who dreamed nonetheless of restoring that rigidity. There was a bitter controversy between Cantor and Frege, and Zermelo swore allegiance to Frege [Cantor G.: Abhandlungen mathematischen und philosophischen Inhalts, 1966, page 441, remarks of Zermelo on Cantor's 1884 review of Frege]. Von Neumann based himself on Zermelo and made explicit the cumulative hierarchy, which Bernays and Goedel used and which many subsequent set theorists presumed was the only architecture to be studied. The justificational aspect stems from the supposed construction of the hierarchy by a bizarre parody of ordinary iteration, parameterized by infinite ordinal numbers (Cantor's third discovery), entities which from the point of view of ordinary mathematics are even more in need of justification than the analysis that supposedly needed it. (Indeed, in attempting to describe what these alleged infinite ordinals are and do, people often resort to stories about gods and demons.) Little or no progress has been made on this "justification" problem in a century, but work with the hierarchy has produced some knowledge about the possibilities for categories of sets. By adopting a standard definition of map and discarding the mock iteration (with its concomitant complicated structure), each model of the cumulative hierarchy yields a category of abstract nearly featureless sets; most of the usual set-theoretical issues depend only on the mere category: measurable cardinals, Goedel-constructibility, the continuum hypothesis, etc. Having thus briefly understood the two visions which are called set theory (1) a category of Cantorian featureless sets which serves as the background recipient for the structures of algebra, geometry and analysis; (2) the cumulative hierarchy with its rigid Fregean structure aiming to justify mathematics, it is not surprising that the precise nature of the elementhood relations appropriate to each are quite different. While the Fregean image involves rigid inclusion and elementhood relations imagined to be given once and for all for mathematics as a whole, the usual mathematical practice instead considers inclusion and membership relations for subsets of a given universe of discourse (such as R^3). Thanks to Grothendieck's Tohoku observation, these mathematical local belonging relations are well globalized within the notion of category, whose primitives are domain, codomain, identity, and composition. [The notion of category is a simple first-order theory of a semi-algebraic kind. It has myriads of interpretations, some in "classes", some "locally small" etc., but such undefined restrictions on interpretations have nothing to do with the notion of category per se. Many properties are best expressed within the first-order theory itself.] Composition is a kind of non-commutative multiplication, hence there are two kinds of division problems. In any category, given any two morphisms a and b we can ask whether there exists a morphism p such that a = bp; if so, we may say that a belongs to b. This forces a and b to have as codomain the same object, which serves as their common universe of discourse. (The dual relation, f determines g, defined by "there exists m with mf = g", is probably equally important in mathematics.) There are two special cases of this belonging relation which are of special interest. First we say that b is a part (or subset in the case of a category of sets) of its codomain, if for all a belonging to b, the proof p of that belonging is unique; this is immediately seen to be equivalent to the usual notion of monomorphism. Then, if a and b are parts of the same object, we say a included in b iff a belongs to b. Any arbitrary morphism x with codomain X may be considered an element of X in the sense of Volterra (also known as a figure in X); we say that x is a member of b iff x belongs to b. Then clearly a is included in b iff for all x, if x is a member of a, then x is a member of b. The usual relationship between these two relations is thus maintained. Because in category theory the domain relation is as important as the codomain relation, we can be more precise about elements: very often it is appropriate to consider a special property of objects, and restrict the term element (or figure) to elements whose domain has that property, that is, to figures whose shape has the property. For example, in algebraic geometry the connected separable objects are appropriate domains for the figures known as "points"; in the algebraically closed case it suffices to consider elements with domain a terminal object 1 as points. On the other hand, frequently it is of interest to choose a small class of figure shapes which generates in the sense of Grothendieck, i.e. so that the above equivalence between inclusion and universal implication of memberships holds even when the figures x are restricted to those of the prescribed shapes. A basic property of categories of Cantorian sets is that this holds with x restricted to those with terminal domain 1. In algebraic geometry, the figures whose domains have trivial cohomology are adequate. Note that if f is a morphism from A to B and if x is an element of A, then fx is an element of B of the same shape (of course in general figures are singular in that they distort their shape, for example, fx may be more singular than the figure x). Properties of x in A may be quite different from the properies of fx in B. The mysterious distinction between x and singleton(x) in the hierarchical Frege architecture takes quite a different form in the categorical architecture where there is a natural transformation from the identity functor to the covariant power set functor; this natural transformation can be called singleton: singleton(x) is simply x considered as a special element of PX, rather than of the original X. Professors may not consider the possibility of learning from undergraduate text books, and some may feel bored that I have once again repeated the above basic definitions and observations. But if these basics were widely understood among algebraic geometers, perhaps misconceptions like "category theory is the insubstantial part of mathematics" would not have arisen. (As we know from experience, all of the substance of mathematics can be fully expressed in categories.) Perhaps the general term "A-points" for arbitrary rings A was confusing. "Spec(A)-shaped figures" is a more accurate rendering of Volterra's "elements"; that could be abbreviated to "A-figures", but points are in some sense special among figures. On the other hand, we often vary the background category, so that alternative terminology might involve passing from a category E to E/spec(A), and restricting the notion of "point" in any category to mean figure of terminal shape; then the A-figures become, on pulling back to the new category, literally "moving points". Whatever the particular chosen terminology, the important conclusion is to actively eliminate the mythology that spaces in categories have no elements, because as we see, this mythology obscures the simplicity of certain matters and thus provides a bogus basis for insulating one field of mathematics from another. [The belonging relation is just the poset collapse of the categories E/X, whose actual maps serve as incidence relations, especially between figures in X. Thus every category E supports a certain geometrical imagery wherein all maps are geometrically continuous, in that they map figures to figures without tearing the incidence relations. Precise axioms about E are a key to further progress because they explicitly sum up and guide our experience with the objects and maps in E.] ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Wed Mar 29 23:30:16 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:30:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOnmj-0007M2-Sd for categories-list@mta.ca; Wed, 29 Mar 2006 23:25:21 -0400 To: Categories List Subject: categories: Re: how science (math) should work Date: Wed, 29 Mar 2006 08:40:27 -0500 From: wlawvere@buffalo.edu Message-ID: <1143639627.442a8e4b5fc43@mail2.buffalo.edu> MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 122 Thanks, Jim, for pointing this out. The concern expressed by Sir Edwin Southern is in the minds of many. For example Pierre Cartier has asked "where are the youth? " Indeed, if they submit to a culture that emphasizes carrer above content, how can our children become serious scientists? Bill Quoting jim stasheff : > The latter part of > > http://www.oxfordtoday.ox.ac.uk/2005-06/v18n2/01.shtml > > has some very relevant comments > > response? > > jim > > > > From rrosebru@mta.ca Wed Mar 29 23:30:16 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:30:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOniL-00077T-4t for categories-list@mta.ca; Wed, 29 Mar 2006 23:20:49 -0400 Date: Wed, 29 Mar 2006 13:57:05 +0100 From: Alex Simpson To: categories@mta.ca Subject: categories: Re: cracks and pots References: In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain;charset=UTF-8; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 123 Quoting dusko : > but at the end of the day, i think, we'll all agree that the source > of the unreasonable effectiveness of categorical algebra is its > foundational content (although there is probably a lot of it that we > dont understand yet); and the other way around. eg, if you look at > grothendieck's work, he started working in algebra, and ended up > developing foundational structures, because he needed them. and a lot > on the "algebra" side now is built upon them. ok, then for a while > it was thought that he exaggerated with foundations, and that a more > direct approach "could have been in better taste" (to cite > eilenberg). but maby the fermat theorem would have a more useful > proof if it was developed in grothendieck style. and nowadays, there > is a lot of foundational content in tannaka duality etc, in TQFT in > general ... TQFT!? It seems dusko has finally discovered the shift key on his keyboard. Alex -- Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://homepages.inf.ed.ac.uk/als Fax: +44 (0)131 667 7209 From rrosebru@mta.ca Wed Mar 29 23:34:15 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:34:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOnuS-0007lO-OY for categories-list@mta.ca; Wed, 29 Mar 2006 23:33:20 -0400 Date: Wed, 29 Mar 2006 08:54:48 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Appreciation of Jon Beck Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 124 It would seem appropriate for me to write something on Jon's mathematics. His career can be divided into three parts. The first was on (co)homology theories, (co)triples, and resolutions and is really the only one I understand. The second was on things related to H-spaces and there were only two published papers [LNM #88, 139--153 and LNM #196, 54--62]. One thing I know he claimed (and never got credit for) was that an A_\infty space is homotopic (or maybe weakly homotopic) to a topological monoid. Since I never really understood this work, I will leave it to others to comment on. The third was his attempt to apply homotopy theory to understand the operations of a finite calculator. This was evidently an outgrowth of the second, since the operations of a finite calculator are not associative, but perhaps more like an A_\infty space. Since that work was not perfected, I will say no more about it. Jon's thesis has many things in it, but here are the parts I remember best. One of the most ingenious was the idea of a Beck module. He observed that if *C* is a category and C an object, then the category Ab(*C*/C), the abelian group objects in the category of objects over C is defined to be the category of C-modules. Here is what that amounts to in specific cases: Groups, left C-modules; rings, C-bimodules; commutative rings, left C-modules; Lie algebras, left C-modules. It is apparent that in each case (including that of commutative algebras whose cohomology theory appeared only in 1962 and Jon did not know of until later), these Beck modules are exactly the modules used as coefficients in the cohomology theories. It also turns out that if C' --> C is an object over C, and M is a C-module, then Hom(C',M) in the category *C*/C can be identified in all those cases as the group Der(C',M) of derivations. I think that this brilliant observation alone would have deserved a PhD, but it was only one chapter (of five) in Jon's thesis. Given a cotriples \G on the category *C*, define a resolution of an object C as the simplicial object ---> ---> ... .. G^3C .. G^2C ===> GC ---> ---> (This is an awful medium to do this in. You also have to imagine degeneracies.) and the cohomology H^.(C,M) as the cohomology groups of the chain complex 0 ---> Der(GC,M) ---> Der(G^2C,M) ---> Der(G^3C,M) ---> ... Jon proved that H^0(C,M) = Der(C,M) and that H^1(C,M) could be interpretated as the group of "singular extensions" (he had to find the appropriate definitions of that too) of C with kernel M. These definitions are thus, in low dimensions, closely related to the classical cohomology groups modulo the usual shift in dimensions. He also defined what you might call the inhomogenous chain complex that used the triple in the underlying category to define the same groups. Along the way, he also proved the PTT, the theorem that characterizes the category of Eilenberg-Moore algebras for a triple. Although all this was available in a draft dated 1964, the final version was completed only in 1967. Fortunately, the thesis is now available as a TAC reprint. Until that reprint, the only source were various n-generation photocopies that were being passed hand to hand. I spent the two years 1962-64 at Columbia, but I don't recall that I had much contact with Jon until the spring term of 1964. And even then, Sammy ordered me to keep away from Jon because he wanted him to write up his thesis. But we talked about showing that the higher dimensional groups he had defined were the same as the higher dimensional groups that had been defined originally. We had no handle on the question. This discussion continued through the fall of 1964 when we had both moved to Urbana. The two resolutions looked so different that we just could not see any way to procede. In mid-December, we both went east for the vacation, Jon to NY and me to Philadelphia and eventually also in NY. One day, I ran into Jon absolutely accidentally at the Times Square subway station, if you can imagine, and Jon asked me if I had ever heard of acyclic models. I hadn't. Jon had spoken to Harry Appelgate, another Eilenberg student, who was writing a thesis on a categorical version of acyclic models. When we got back to Urbana, we looked at acyclic models a la Appelgate. As I recall it, Appelgate's version used a Yoneda extension to induce a triple on a functor category. Since we were starting with a triple, it turned out to be much simpler for us and we quickly (well, fairly quickly) carried out the required verifications. We both spoke on this at the first midwest category meeting in Chicago in April of 1965 and Jon spoke on it at the La Jolla meeting and it appeared in the La Jolla prodeedings [Springer, 336-343]. The final collaboration that Jon and I had was that of using a simplicial resolution, not derived from a cotriple, to define cohomology and derived functors. This was a long paper in LNM #80 (informally called the "Zurich Triples Book") [345--435]. I think we did this after (but I cannot recall) seeing Michel Andre's "step-by-step" resolution for the particular case of commutative algebras. One more notable thing he did in that period was discover distributive laws between triples, since they do not naturally compose [LNM #80, 119-140]. This concept was crucial to my analysis of Shukla cohomology, which required the original distributive law between multiplication and addition in rings, moved up to the level of triples. After that our interests diverged and I leave it to others to comment on his later work. Let me just make a comment on the word "triple". Although Jon never thought it was a good term he thought that "monad" was at least as bad and didn't approve of the idea of replacing one poor term by another. I went along as a gesture of solidarity, although I don't have any deep feeling on the subject (although my title TTT is really nice). In the end, of course, Mac Lane prevailed. I once asked Sammy why he gave the idea such a poor name. Especially given the care that he and Steenrod had taken to naming "exact". His answer was that the concept seemed to have no importance, so he and John Moore didn't spend any time on it! Michael From rrosebru@mta.ca Wed Mar 29 23:36:35 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:36:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOnwn-00005s-N9 for categories-list@mta.ca; Wed, 29 Mar 2006 23:35:45 -0400 Mime-Version: 1.0 (Apple Message framework v623) Message-Id: From: David Yetter Subject: categories: Re: cracks and pots Date: Wed, 29 Mar 2006 09:02:27 -0500 To: Categories Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=WINDOWS-1252;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 125 I used the word 'faces' to describe the two aspects of category theory.=20= I see no actual separation in content, only a difference in emphasis=20= (esp. as regards applications) and public presentation. Even as Saunders, late in his life, gave lectures entitled 'All=20 Mathematics Belongs Together', so all category theory belongs together. D. Y. On 28 Mar 2006, at 03:01, dusko wrote: > i think david yetter's analysis of the dichotomy "categories as=20 > foundations" vs "categories as algebra" was spot on ---=A0 with = respect=20 > to people and the community. indeed, one could split most of our=20 > papers into one category or the other.=A0 > > but at the end of the day, i think, we'll all agree that the source of=20= > the unreasonable effectiveness of categorical algebra is its=20 > foundational content (although there is probably a lot of it that we=20= > dont understand yet); and the other way around.=A0eg,=A0if you look at=20= > grothendieck's work, he started working in algebra, and ended up=20 > developing foundational structures, because he needed them. and a lot=20= [lengthy further quotation omitted ...] From rrosebru@mta.ca Wed Mar 29 23:48:21 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:48:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOo7t-0000kb-4v for categories-list@mta.ca; Wed, 29 Mar 2006 23:47:13 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: dusko Subject: categories: Re: cracks and pots Date: Wed, 29 Mar 2006 11:23:06 -0800 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 126 when i said > eg, if you look at grothendieck's work, he started working in > algebra, and ended up developing foundational structures, because > he needed them. i meant that he ended up working on toposes, fibrations, and descent (as foundational structures). i did not mean that he observed the grothendieck universes (which are perhaps foundational, but not much of a structure), as my hasty formulation had suggested to some people. sorry about the confusion (and about taking bandwidth to correct it), -- dusko From rrosebru@mta.ca Wed Mar 29 23:49:21 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOo9I-0000oP-O4 for categories-list@mta.ca; Wed, 29 Mar 2006 23:48:40 -0400 Date: Wed, 29 Mar 2006 12:10:40 -0800 From: Vaughan Pratt X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I References: <4429A1E1.8080907@math.upenn.edu> In-Reply-To: <4429A1E1.8080907@math.upenn.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 127 jim stasheff wrote: > now for a mathematical subject a math proof is sometimes but not always > necessary Absolutely. I would add publication date as a factor here. As an example, a few decades ago an elementary exposition of the Fundamental Theorem of Algebra would not be expected to include an elementary proof since the extant proofs were either lengthy arguments or nonelementary appeals to the minimum modulus principle, properties of holomorphic functions such as Liouville's theorem, or other results the reader would be unlikely to be on top of. The dominant belief was that the only short proofs were nonelementary ones. But for an audience aware only that z^i for any nonnegative integer i maps circles at the origin to i-fold circles of radius r^i at the origin, an entirely elementary notion, an expositor today would be morally obligated to include a full proof since there is hardly anything left to explain. The polynomial a_d z^d + ... + a_0 maps little circles to the neighborhood of a_0 and big circles to a loop tending to a very big d-fold circle of radius a_d r^d, whence the smoothly growing image, under the polynomial, of a smoothly growing circle is obliged to cross the origin at some stage. Still a topological argument, but now an entirely elementary one. Except, that is, for the theorem that a loop wound d times around the hole in the punctured plane cannot be continuously retracted to a point, which was tacitly smuggled in there. But that statement is less intimidating than anything based on holomorphic functions. This slick proof seems only to have emerged in the past couple of decades. It is an interesting commentary on mathematics that it took this long for people to come up with an argument "for the rest of us." Maybe some people "knew" it all along, but in that case they were keeping pretty quiet about it. Vaughan Pratt From rrosebru@mta.ca Wed Mar 29 23:51:21 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:51:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOoBE-0000wB-4v for categories-list@mta.ca; Wed, 29 Mar 2006 23:50:40 -0400 Date: Wed, 29 Mar 2006 14:17:57 -0700 From: Robin Cockett X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Reminder: CMS Summer Meeting in Calgary 2006 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 128 A gentle reminder: CMS 2006 meeting and other (categorical) events in Calgary : The summer Canadian Math Society meeting is in Calgary this year (June 3rd - June 5th) Steve Awodey is a plenary speaker for the meeting and there is a Category Theory session. (Note: Abstracts for speakers are due April 10th) IN ADDITION there are various other events of categorical interest occurring in Calgary round those dates: (i) June 2nd: Categories and Semigroup Workshop (University of Calgary) (ii) June 7th-9th: Foundational methods in Computer Science (Kananaskis Field Station, Alberta) LOCAL ARRANGEMENTS: If you wish to participate in any of these events you should let either myself (robin@cpsc.ucalgary.ca) or Pieter Hofstra (hofstrap@cpsc.ucalgary.ca) know so that we can keep you informed of local organizational details. WEBSITES: Please check the CMS site for registration details (http://www.cms.math.ca/Events/summer06). For the remaining events please check (http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_06/CatMeetings.html) Robin Cockett From rrosebru@mta.ca Wed Mar 29 23:54:31 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:54:31 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOoE8-00017O-MD for categories-list@mta.ca; Wed, 29 Mar 2006 23:53:40 -0400 From: "Reinhard Boerger" Organization: FernUniversitaet To: categories@mta.ca Date: Wed, 29 Mar 2006 15:22:00 +0200 MIME-Version: 1.0 Subject: categories: Re: WHY ARE WE CONCERNED? I Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Content-description: Mail message body Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 129 Hello, just a few remarks. Jim Stasheff wrote: > F W Lawvere wrote: > > WHY ARE WE CONCERNED? I > > > > "Dumbing down" is an attack not only on school children and on > > undergraduates, but also one taking measured aim at colleagues in > > adjacent fields and at the general public. The general public is > > thirsty for genuinely informational articles to replace the science > > fiction gruel served constantly by journals like the Scientific > > American and the New York Times "Science" section. > > so far so good > > Those journals have never published anything > > resembling a mathematical proof > > why should they? Because otherwise the readers do not learn what mathematics is about. > a math proof is hardly necessary to explain a scientific subject in a > usable way. > > now for a mathematical subject a math proof is sometimes but not > always necessary That depends on what you mean by explaining a subject. Of course, many people know what a prime is, and if a journal reports that some larger (Mersenne) prime has been found, or if the journal contains some nice pictures of fractals, they may either admire this or ask "so what?" In any case they do not see what a mathematical result is. I met several people with an academic education in another field. When I told them that I am a mathematician, some of them replied: "I always liked maths - except proofs." If this misconception is so wide-spread among educated people - at least in Germany, Canada and the United States - I think it is more important that these people see easy proofs of mathematical results (e.g. Euclid's proof for the existence of infinitely many primes) than that they see mysterious mathematical statements, which they don't understand. Mathematics is thinking rather than computation, and if one does not know what a proof is, one does not know what mathematics is. So for which subject do you think that a proof is not necessary? Greetings Reinhard From rrosebru@mta.ca Wed Mar 29 23:56:21 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:56:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOoG1-0001HC-Jk for categories-list@mta.ca; Wed, 29 Mar 2006 23:55:37 -0400 Date: Wed, 29 Mar 2006 10:42:52 -0500 (EST) From: James Stasheff To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 130 I deliberately overstated the case presenting accessible proofs should of course be done and certainly equations should not be eschewed pace Penrose but there's more to math than proofs cf. Reinhard's own reference to *thinking* i.e. *before* a formal proof is worked out if we could convey even that Jim Stasheff jds@math.upenn.edu On Wed, 29 Mar 2006, Reinhard Boerger wrote: > Hello, > > just a few remarks. > Jim Stasheff wrote: > > > F W Lawvere wrote: > > > WHY ARE WE CONCERNED? I > > > > > > "Dumbing down" is an attack not only on school children and on > > > undergraduates, but also one taking measured aim at colleagues in > > > adjacent fields and at the general public. The general public is > > > thirsty for genuinely informational articles to replace the science > > > fiction gruel served constantly by journals like the Scientific > > > American and the New York Times "Science" section. > > > > so far so good > > > > Those journals have never published anything > > > resembling a mathematical proof > > > > why should they? > > Because otherwise the readers do not learn what mathematics is about. > > > a math proof is hardly necessary to explain a scientific subject in a > > usable way. > > > > now for a mathematical subject a math proof is sometimes but not > > always necessary > > That depends on what you mean by explaining a subject. Of course, many people > know what a prime is, and if a journal reports that some larger (Mersenne) prime has > been found, or if the journal contains some nice pictures of fractals, they may either > admire this or ask "so what?" In any case they do not see what a mathematical result > is. I met several people with an academic education in another field. When I told > them that I am a mathematician, some of them replied: "I always liked maths - except > proofs." If this misconception is so wide-spread among educated people - at least in > Germany, Canada and the United States - I think it is more important that these > people see easy proofs of mathematical results (e.g. Euclid's proof for the existence > of infinitely many primes) than that they see mysterious mathematical statements, > which they don't understand. Mathematics is thinking rather than computation, and if > one does not know what a proof is, one does not know what mathematics is. So for > which subject do you think that a proof is not necessary? > > > Greetings > Reinhard > From rrosebru@mta.ca Thu Mar 30 18:52:09 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:52:09 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP5uo-0006Jy-O6 for categories-list@mta.ca; Thu, 30 Mar 2006 18:46:54 -0400 Subject: categories: Re: WHY ARE WE CONCERNED? I From: Graham White To: Categories@mta.ca Content-Type: text/plain Date: Thu, 30 Mar 2006 09:13:52 +0100 Message-Id: <1143706432.8570.16.camel@localhost.localdomain> Mime-Version: 1.0 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 131 > jim stasheff wrote: > > > now for a mathematical subject a math proof is sometimes but not always > > necessary > There's a saying about Lefschetz that he "never wrote a valid proof, and never made a false conjecture". Now it's not an attitude that want to encourage, but if you have great mathematicians who are like that (and Lefschetz was not just a good mathematician, but a great mathematician, without whom a good deal of modern algebraic geometry would be unimaginable), then this ought to tell us something. What it tells us is, of course, not easy to formulate: it's an example that causes severe problems for almost every philosophy of mathematics that I know. But it ought to stop us saying things of the form "if we don't do category theory in such and such a way, then it won't be mathematics at all". (Of course we'll all keep saying this, because we all have a secret fear that, if we aren't really careful about what we do, the grown up mathematicians will kick sand in our face, but that's a psychological problem and not a mathematical problem.) -- Dr. Graham White Lecturer Department of Computer Science Queen Mary, University of London Mile End Road London E1 4NS http://www.dcs.qmul.ac.uk/~graham (+44)(020)7882 5242 From rrosebru@mta.ca Thu Mar 30 18:52:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:52:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP5wD-0006P9-Im for categories-list@mta.ca; Thu, 30 Mar 2006 18:48:21 -0400 Date: Thu, 30 Mar 2006 10:03:57 +0100 (BST) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 132 On Wed, 29 Mar 2006, Vaughan Pratt wrote: > a few decades ago an elementary exposition of the Fundamental > Theorem of Algebra would not be expected to include an elementary proof > since the extant proofs were either lengthy arguments or nonelementary > appeals to the minimum modulus principle, properties of holomorphic > functions such as Liouville's theorem, or other results the reader would > be unlikely to be on top of. The dominant belief was that the only > short proofs were nonelementary ones. > > But for an audience aware only that z^i for any nonnegative integer i > maps circles at the origin to i-fold circles of radius r^i at the > origin, an entirely elementary notion, an expositor today would be > morally obligated to include a full proof since there is hardly anything > left to explain. The polynomial a_d z^d + ... + a_0 maps little circles > to the neighborhood of a_0 and big circles to a loop tending to a very > big d-fold circle of radius a_d r^d, whence the smoothly growing image, > under the polynomial, of a smoothly growing circle is obliged to cross > the origin at some stage. Still a topological argument, but now an > entirely elementary one. > > Except, that is, for the theorem that a loop wound d times around the > hole in the punctured plane cannot be continuously retracted to a point, > which was tacitly smuggled in there. But that statement is less > intimidating than anything based on holomorphic functions. > > This slick proof seems only to have emerged in the past couple of > decades. Has it? It seems to me no more than (an explicity homotopy-theoretic formulation of) the (implicitly homotopy-theoretic) proof via Rouch\'e's Theorem, which I was taught as an undergraduate (and which I've taught to undergraduates on many occasions since then). Peter Johnstone From rrosebru@mta.ca Thu Mar 30 18:52:10 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:52:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP5yY-0006ZF-DP for categories-list@mta.ca; Thu, 30 Mar 2006 18:50:46 -0400 From: Nikita Danilov MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <17451.46095.476795.214410@gargle.gargle.HOWL> Date: Thu, 30 Mar 2006 14:33:51 +0400 To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 133 Vaughan Pratt writes: [...] > > This slick proof seems only to have emerged in the past couple of > decades. It is an interesting commentary on mathematics that it took > this long for people to come up with an argument "for the rest of us." > Maybe some people "knew" it all along, but in that case they were > keeping pretty quiet about it. This proof was used by Pontryagin in the April 1982 issue of "Kvant" magazine (targeting school-children!): http://kvant.mccme.ru/1982/04/osnovnaya_teorema_algebry.htm (in Russian, but pictures should be enough). > > Vaughan Pratt Nikita. From rrosebru@mta.ca Thu Mar 30 18:52:39 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:52:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP5zg-0006dI-Hf for categories-list@mta.ca; Thu, 30 Mar 2006 18:51:56 -0400 Subject: categories: Re: WHY ARE WE CONCERNED? I Date: Thu, 30 Mar 2006 10:08:44 -0400 (AST) To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20060330140845.3397A10854@sigma.mathstat.dal.ca> From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 134 > F W Lawvere wrote: > > WHY ARE WE CONCERNED? I > > > > "Dumbing down" is an attack not only on school children and on > > undergraduates, but also one taking measured aim at colleagues in > > adjacent fields and at the general public. The general public is > > thirsty for genuinely informational articles to replace the > > science fiction gruel served constantly by journals like the > > Scientific American and the New York Times "Science" section. > > Those journals have never published anything resembling a > > mathematical proof and hence have rarely actually explained any > > scientific subject in a usable way. jim stasheff wrote: > > a math proof is hardly necessary to explain a scientific subject in a > usable way. > > now for a mathematical subject a math proof is sometimes but not always > necessary I agree with Bill that the prevailing style of expository writing, especially in newspapers, is often of poor quality. It would be nice if such articles more often gave a glimpse into the nature of research, rather than serving, as Bill puts it, entertainment. However, I disagree on the role of proofs in expository writing. Clearly, proofs are central in mathematics. But to say that mathematics is only about proofs is a bit like saying that dentistry is only about clinical research. Of course, the research is important, and most of us who have root canals are very glad that it is being done. However, I would like to believe that mathematics is ultimately about solving problems that *matter*, and the reason they matter often has nothing to do with their proofs. I am of course not advocating replacing proofs by conjecture. I am only speaking of expository writing, where I believe it is often more important to explain the results than their proofs. And sometimes, it can even be justified to give an "approximate" proof, i.e., a proof idea, or even an "approximate" definition, if it is stated clearly that there has been some simplification. The poor state of mathematical exposition is not confined to articles about mathematics. The following quote, from an ordinary new article in yesterday's Times, send my logic-circuits spinning: French lawmakers, for example, gave preliminary support this month to a measure that would require the company to open the iPod to play music purchased from any online music service; currently, songs purchased from iTunes can be played only on iPods. New York Times, 2006/03/29, "Apple vs. Apple in Dispute Over Trademark" This is of course not a logical contradiction; but I would be very surprised if it is what the writer really meant to say. Sadly, most readers probably won't know the difference one way or the other. -- Peter From rrosebru@mta.ca Thu Mar 30 18:53:48 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:53:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP60n-0006jW-Iu for categories-list@mta.ca; Thu, 30 Mar 2006 18:53:05 -0400 Date: Thu, 30 Mar 2006 09:27:30 -0500 From: jim stasheff MIME-Version: 1.0 To: Categories List Subject: categories: May and Beck Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 135 I asked Peter May where he had commented on the relevance of Jon's work to his own: -------- Original Message -------- Subject: We'll talk (This refers to being together at the Mac Lane memorial fest - where others of you will be also) Date: Thu, 30 Mar 2006 08:11:07 -0600 From: Peter May To: jds@math.upenn.edu No time to write, though. The paper was ``On H-spaces and infinite loop spaces''. The work of his I most liked concerned distributivity laws and monads, which I only learned about after rediscovering it for myself while doing multiplicative infinite loop space theory. See you next week, Peter From rrosebru@mta.ca Thu Mar 30 18:58:14 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:58:14 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP64n-00071h-Ol for categories-list@mta.ca; Thu, 30 Mar 2006 18:57:13 -0400 Date: Thu, 30 Mar 2006 09:10:53 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 136 In response to Peter Johnstone (and those who responded privately), my point about the Fundamental Theorem of Algebra was not that this particular proof (based on the limiting behaviors of small and large circles) was not known to anyone, but that it had not emerged, instead being effectively sat on by those in the know, even if not intentionally. At this risk of sounding like an Abu Ghraib interrogator, "who knew?" My claim is that no extant proof at all, that or any other, was considered fit for an elementary exposition more than a couple of decades ago. If that estimate is right, the 1982 Pontrjagin article cited by Nikita Danilov would be one of the earliest popular expositions based on the circles argument, assuming the section containing Fig. 6 is the relevant one (my Russian is even rustier than my algebra). I'd be very interested in seeing an earlier popular account that didn't claim that every proof necessarily either was long or depended on out-of-scope material. As a case in point, just now I checked a relatively recent Brittanica article on algebra (1987 ed.), which states flatly (p.260a) that "No elementary algebraic proof of [the FTAlg] exists, and the result is not proved here." (Not even "is known" but "exists"; an expository article should not assume that the reader knows the jargon meaning of this term as "exists in the literature".) The authors taking responsibility for this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter Hilton, and Paul Cohn. They go into detail to show that z^n = a has n roots, starting with the geometry of addition and multiplication in the Argand diagram, so it's not as if their exposition was at too elementary a level to talk in terms of mapping circles, or that "algebraic" ruled out simple geometric arguments. I submit their nonexistence claim as prima facie evidence for my claim that the very few who knew this argument weren't even letting the likes of Birkhoff, Hall, etc. in on it, let alone "the rest of us." The general message in the literature prior to the 1980's seemed to be, if Gauss couldn't find a simple proof in half a dozen tries, there isn't one. If you don't possess the necessary higher maths or the stamina for an intricate argument, we can't help you with that result, ask us about solvability of z^n = a. Good for Pontrjagin for promoting FTAlg to school children! Vaughan Pratt From rrosebru@mta.ca Thu Mar 30 19:00:39 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 19:00:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP675-0007Bq-JN for categories-list@mta.ca; Thu, 30 Mar 2006 18:59:35 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: WHY ARE WE CONCERNED? I Date: Thu, 30 Mar 2006 14:28:54 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 137 Dear Bill, Congratulations on your posting, particularly in what refers to Mac Lane, which is very revealing. > When Saunders Mac Lane penned his hard-hitting 1997 Synthese >article, he was defending mathematics from an attack many of us hoped >would just go away. But Saunders was aware of the seriousness of the >threat, which indeed is still here with greater determination. >Although the title of that article was "Despite physicists, proof is >essential in mathematics", he was not opposing physics, nor even that >immediate handful who, assuming the mantle of "mathematical physicists", >gave themselves license to insult generations of scrupulously serious >physicists and to demand that mathematics adopt a culture that considers >conjecture as nearly-established truth. In essence it was an attack on >science itself, as the highest form of knowing, that Saunders was >opposing. In case there may be somebody not acquainted with MacLane's excellent article, here is a link to it: http://www.math.nsc.ru/LBRT/g2/english/ssk/proof_is_necessary.pdf > The contempt for Mac Lane's fight, recently expressed in articles >supposedly memorializing him, takes the form of the claim that category >theory itself is a "cool" instrument for deepening obscurantism. Not only >Harvard's "When is one thing equal to another thing?" and the Cambridge >"morality" muddle, but also a 2003 article aimed at teachers of >undergraduates, quite explicitly support that claim. I suppose that you cannot (or do not want to) be more explicit. I do not know (for the most part) which articles you are referring to. Best wishes, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************ From rrosebru@mta.ca Thu Mar 30 20:53:06 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 20:53:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP7pq-0006wZ-Iw for categories-list@mta.ca; Thu, 30 Mar 2006 20:49:54 -0400 Subject: categories: cracks and pots 93 From: Eduardo Dubuc To: cat-dist@mta.ca Date: Thu, 30 Mar 2006 15:31:17 -0300 (ART) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: QUOTED-PRINTABLE Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 138 Hi, The 93 is because I have by now 92 msages in my cracks and pots file. I apologize for the length of this posting. It is intended to be a (may= be biased) partial account of the debate, and some comments. Well, by now the "cracks and pots" debate is establishing itself as, in= my opinion, an interesting and worth-wile event. Congratulations Marta != ! We are learning about: a) Understand (for many of us) better what is mathematics, and what is physics, what is rigor and what is buccaneering, and also what is bullshit. =20 b) "Something is rotten in the state of category theory community"=20 Pay attention that The Bard does not say "category theory", but he says "category theory community" I start from who has made the more refreshing, humorous, down to earth, honest and intelligent contributions to this debate: **Vicent Schmitt: that theoretical physics, computer science, phylo., a mix of those, or whatever? , is used to justify poor "categorical" work is, in my view, an existing problem. More or less everyone is conscious= of it (come on!...) but so far that has not been publicly debated.** Yes Vincent!!, you point right to what it is at the center (or very nea= r it) the problem raised in Marta=D5s original "cracks and pots" posting!= =2E And the "(come on!...)", beautiful !. Now, talking about rigor, conjectures and proofs: **Maclane : If a result has not yet been given valid proof, it isn't ye= t mathematics. This however does not deny the many preliminary stages of insight, experiment, speculation or conjecture, which can lead to mathematics. It states simply that a conjectured result is not yet a theorem ** It is relevant to compare this with Motl's distinction between physics = and mathematics: **Motl: In physics, we propose different conjectures about the real wor= ld, and it is important that we're not guaranteed that these conjectures wi= ll be true. String theory itself is not just a conjecture, but rather a seemingly consistent mathematical framework. Once we accept string theo= ry as an objectively existing mathematical structure, a structure that we treat as a part of "generalized physics" - which is of course what all string theorists are doing every day - we can ask a lot of questions ab= out its properties.** He does distinguish between "physics as conjecture" and mathematics wi= th applications to physics. He call this mathematics "generalized physics= " But "conjecture" to be acceptable is not unrigourous neither buccaneeri= ng. he says: **Motl: the statements about string theory are just conjectures, and th= ey need to be proved or supported by evidence, otherwise they're irrelevan= t and "wrong", in the physical sense.** He also says: ** Motl: I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or str= ing theory as some sort of "obvious facts". He is clearly saying that those "mathematically oriented people" are lacking rigor.=20 Many postings in this debate confound mathematical rigor with formalism= , and push forward the idea that a formal and logically correct statement has automatically rigor. Even if it is foolish: **V. Pratt: In axiomatic mathematics, everything that is not forbidden = is permitted. ** **R. Dawson: If the math itself meets mathematical standards of rigor, = its application to physics need surely only meet the standards appropriate = to that subject.** It seems to me that he is equating here "mathematical standards of rigo= r" with "logically correct", and "the standards appropriate to that subje= ct" (in this case, physics) with " buccaneering " Nothing more wrong!! . In both cases, failing to convey what it should = be considered "rigor in mathematics" and "rigor in physics" But again Saunders and Lubos: **MacLane: real proof is not simply a formalized document, but a sequen= ce of ideas and insights** ** Motl: the primary physical motivation is to locate the right ideas a= nd equations that describe the real world. Category theory has been used b= y many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor.** He however seems to be pushing forward the same misconception of "rigor= ": **Motl: It may be nice to be rigorous, but it's always more important t= o be correct: if the specific kind of rigor leads us to stupid conclusion= s in physics, we should avoid it.** =46rom the original Marta's "cracks and pots" **M.Bunge: Are we category theorists as a whole going to quietly accep= t getting discredited by a minority of us presumably applying category theory to string theory?** **J. Baez: I had never heard anyone before suggest that category theory could be discredited by applications to string theory. It completely surprised me. I'm used to the opposite complaint: that category theory= is discredited by its *lack* of applications.** Here it is a clear and rigorous answer: (1) **W. Lawvere: The question is not whether mathematics should be applied. Most of us agree that it should. The concern is rather that ou= r subject is sometimes being used as a mystifying smoke screen to protect pseudo-applications against the scrutiny of the general public and of t= he scientific colleagues in adjacent disciplines. We need to ensure that applications themselves be maximally effective, not clouded by misunderstanding.** Now, an example of superficial conclusions: ** J. Baez: Indeed, the funny thing about string theory is that while leading to an abundant harvest of rigorous mathematical results, it has not yet correctly predicted a single result from a single experiment, even after more than 20 years of work on the part of many smart people.= ** There is nothing funny about this. Lubos say: ** Motl: One of the fascinating features of string theory is that its objects and investigations, even though they've been partially disconnected from the daily exchanges with the experimentalists, remain= ed extremely physical in character. All of the objects that we deal with a= re analogous to some objects in well-known working physical theories, to s= ay the least.** Bill has made a serious, well fundamented and non-bullshit contribution= to "crack and pots" (he utilizes a different heading:" WHY ARE WE CONCERNE= D?") In contrast to many passages of some contributors that it will be tires= ome to reproduce here, and where one founds an overwhelming proliferation o= f =20 highly technical, sophisticated, difficult and impressively sounding wo= rds=20 such that it becomes impossible to see what they are saying, unless you are an expert, in which case you may find out that it is only superfici= al thinking (I am thinking specially in certain parts of Davis Yetter's postings). ** W. Lawvere: Professors may not consider the possibility of learning from undergraduate text books, and some may feel bored that I have once again repeated the above basic definitions and observations.** If you have some real thoughts, you do not need impressive jargon. See what an original and deep insight: ** W. Lawvere: As quantity includes zero, so structure includes the cas= e of no structure, which Cantor considered one of his most profound and exciting discoveries** Superficial thinking (which could be malicious, but very often is simp= ly stupid) has manifested itself in these postings by pushing forward the idea that there are two different kinds of category theory: =20 "Categories as Foundation" and "Categories as Algebra", the first implicitly (but not explicitly said) the "bad one", and the second the "good" one.=20 ** D. Yetter: All of these are part and parcel of a different face of category theory than one saw in the old days: category theory as algebr= a, rather than category theory as foundations.** We have an excellent analysis of this fallacy in Bill's postings, which should be read carefully and slowly. =20 I imagine now to add something that Lawvere himself pointed out a long time ago: The laws of logic are a particular instance of the categorica= l concept of adjoint functors, a concept that grew out of mathematical experience. =20 There is any way some explanation to Yetter's prejudice against=20 "categories as foundation". Often very poor category theory has been justified by people writing on foundations. Bill's quote (1) above also applies to this and related use of category theory in theoretical compu= ter science. =20 Somebody else that does not need either noisy language sees better: ** Dusko: I am of course saying things very clear and familiar to many people on this list, but maybe they are worth saying nevertheless.** ** Dusko: but at the end of the day, I think, we'll all agree that the source of the unreasonable effectiveness of categorical algebra is its foundational content **=20 Then, he passes to consider Grothendiek's ("the greatest of the category theorists") work on Topos theory as work on foundations, whic= h agrees with the analysis of foundations made by Lawvere. I can not restrain myself to quote the following magnificent piece of meaningless hallucinogenic discourse: **V. Pratt: In the millions of years of evolution of primate thinking, = no productive mathematical mechanism has a higher probability of being stumbled on than mathematics founded on the Yoneda axiom. I know of no better explanation of how human thought could have evolved to its prese= nt form than evolution finding and exploiting the Yoneda principle** Now, some serious business:=20 In recent years J.Baez and his followers have been occupying more and m= ore space in the categorical community (this fact is at the starting point = of the present debate).=20 I think this is so because they have some interesting category theory t= o show, but they are occupying more space than their mathematics deserves= =20 because they bring a refreshing air to a community until now dominated = by an old guard that has not shown signs of necessary evolution, and that = has not being able to attract very good and talented young mathematicians t= o the community. There is now not other exiting body of developments with= in the community. The old guard is being pushed out (prone or supine ?), b= ut, alas, not by better mathematicians. =20 Category theory is in good shape (in particular pushed forward by the Russian school), and it is now passing over the category community. I have lost the information now, but recently it was in Europe an importa= nt congress that it had two subjects: one was a prestigious subject (that = I do not remember now), the other was category theory. Not a single name (including Baez group) that we see in the category theory community meetings was there. Best wishes to all e.d. From rrosebru@mta.ca Thu Mar 30 20:53:28 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 20:53:28 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP7sf-00076E-3F for categories-list@mta.ca; Thu, 30 Mar 2006 20:52:49 -0400 From: Colin McLarty To: Categories@mta.ca Date: Thu, 30 Mar 2006 18:44:19 -0500 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: WHY ARE WE CONCERNED? I Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 139 > There's a saying about Lefschetz that he "never wrote a valid > proof, and never made a false conjecture". Now it's not an attitude > that want to encourage, but if you have great mathematicians who > are like that (and Lefschetz was not just a good mathematician, but > a great mathematician, without whom a good deal of modern algebraic > geometry would be unimaginable), then this ought to tell us something. This, and much else about Lefschetz has to tell us a lot. As to proof, Lefschetz also never published a theorem without a purported proof, and he often came to feel very strongly that his proofs were not good enough. He wrote two long books on topology in the attempt to repair the bad proofs in his influential booklet on cohomology in algebraic topology, L'Analysis situs et la Topologie Algebrique. It was so important to him that he enlisted many others. Notably for us, he asked Eilenberg and Mac Lane to contribute an appendix to his 1942 TOPOLOGY. This was their first published collaboration "On homology groups of infinite complexes and compacta" and pursued the questions that quickly led to category theory. Lefschetz had encouraged work on solving specific problems just over the edge of what well-understood foundations for homology could handle. Apparently he believed such solutions would lead to significantly deeper understanding. He had encouraged Steenrod to work on p-adic solenoids because existing methods did not seem adequate to it. But whatever his motive, he was determined to see rigorous solutions to quite specific problems. Colin From rrosebru@mta.ca Thu Mar 30 20:56:13 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 20:56:13 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP7vG-0007HG-LZ for categories-list@mta.ca; Thu, 30 Mar 2006 20:55:30 -0400 Date: Thu, 30 Mar 2006 19:43:55 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Fundamental theorem Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 140 The proof that Vaughan outlined is found in almost exactly the same form in Courant & Robbins, published around 60 years ago. Here is a proof I learned in grad school, based on three facts, one analytic and two algebraic. The analytic fact, which is an irreducible minimum, given that the reals cannot be defined algebraically (except as a real closed field of continuum transcendence degree, which misses the point) is the intermediate value theorem. In other words, order completeness. From this follows the fact that every odd order polynomial has a (real) root and that every real number--and with a bit of manipulation, every complex number--has a complex square root. The algebraic facts are the existence of a splitting field and the theorem on elementary symmetric functions, neither of which is quite trivial, nor very deep. The way you do it is by proving that every real polynomial of degree n = 2^k*m with m odd has a complex root, by induction on k. The case k = 0 is quite trivial, of course. So suppose that f is a real polynomial of even degree n and, in some splitting field has roots r_1,...,r_n. For each integer s form the polynomial f_s = \prod{i Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPSsL-0004VM-SJ for categories-list@mta.ca; Fri, 31 Mar 2006 19:17:53 -0400 Subject: categories: re: fundamental theorem of algebra Date: Thu, 30 Mar 2006 20:01:48 -0800 (PST) From: "John Baez" To: categories@mta.ca (categories) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 141 Dear Vaughan - You write: > As a case in point, just now I checked a relatively recent Brittanica > article on algebra (1987 ed.), which states flatly (p.260a) that "No > elementary algebraic proof of [the FTAlg] exists, and the result is not > proved here." (Not even "is known" but "exists"; an expository article > should not assume that the reader knows the jargon meaning of this term > as "exists in the literature".) The authors taking responsibility for > this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter > Hilton, and Paul Cohn. They go into detail to show that z^n = a has n > roots, starting with the geometry of addition and multiplication in the > Argand diagram, so it's not as if their exposition was at too elementary > a level to talk in terms of mapping circles, or that "algebraic" ruled > out simple geometric arguments. > > I submit their nonexistence claim as prima facie evidence for my claim > that the very few who knew this argument weren't even letting the likes > of Birkhoff, Hall, etc. in on it, let alone "the rest of us." I really doubt those authors were unaware of the topological proof of the fundamental theorem of calculus in 1987. After all, it's exercise H.5 in chapter 1 of Spanier's "Algebraic Topology", copyright 1966. This book used to be the canonical textbook on algebraic topology, and Peter Hilton is a darn good algebraic topologist. I think I learned this topological proof sometime in grad school, around 1986. So, I don't think it was any sort of secret by then. I don't know what counts as an "elementary algebraic proof", but people often say that there is no "purely algebraic proof" of the fundamental theorem of calculus. After all, this theorem is about the complex numbers, which are often defined in terms of the real numbers, which are often defined as a topological completion of the rational numbers. I hope this is what the Encyclopedia article was trying to say. There are some so-called "algebraic proofs": http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra that use a bare minimum of topology. These proofs tend to have a purely algebraic core, namely "if odd-degree polynomials and the polynomial x^2 + 1 have roots in some field, this field is algebraically closed". But, they use the intermediate value theorem for continuous functions f: [0,1] -> R to show that C meets these conditions. So, I wouldn't call them "purely algebraic". It's sort of ironic that the so-called "fundamental theorem of algebra" doesn't have a purely algebraic proof. Gauss is famous for having given a proof of the fundamental theorem of algebra in his dissertation back in 1799. On the St. Andrews math history website they write: Gauss's proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof. They don't say how the proof went. So, I decided to find out! I was hoping I could irritate you by showing that it was just the topological proof you claim is so new. There's a discussion of it here: Hans Willi Siegberg Some Historical Remarks Concerning Degree Theory, American Mathematical Monthly, 88 (1981), 125-139. (Available on JSTOR, or via Google Scholar.) As the title hints, Gauss' proof uses ideas closely related to the winding number. Unfortunately, it's slightly different than the proof you like. The idea is to take a polynomial of degree n, say P: C -> C break it into real and imaginary parts P = U + iV, see where they vanish: S = {z: U(z) = 0} T = {z: V(z) = 0} and show that the intersection of S and T is nonempty. Gauss argues that far from the origin, S and T are smooth curves. Because the leading term of the polynomial dominates the rest, each of these curves intersects any sufficiently large circle transversely at n points. If we go around the circle these intersection points alternate: first a point in S, then one in T, then one in S, and so on. Moreover, the curves I'm talking about can't just disappear as we follow them into the disk, since they separate the region where U (resp. V) is positive from the region where it's negative. They may become singular, or intersect, but they can't just end! "So", S and T must intersect somewhere. This is true, but it takes more topology to prove it rigorously than was available to Gauss. Gauss knew his proof wasn't completely rigorous, so he invented some other arguments. The "winding number" idea you like is lurking in Gauss' third proof, which he wrote up in 1816 - but he only gave this winding number proof explicitly in 1840. According to Siegberg, Indeed, in a lecture "Theorie der imaginaeren Groessen (1840), Gauss mentioned [see Fraenkel, 1922] that his third proof of the fundamental theorem of the algebra originated from his first one, and he gave the function-theoretic argument that the winding number W(P|S, 0) equals n [the degree of the polynomial], whereas the winding number of any map F: (B,S) -> (R^2, R^2 - {0}) vanishes if there is no zero of F in B [see Fraenkel, 1922]. However, this argument cannot be found explicitly in [Gauss, 1816]. So, I guess that except perhaps for Gauss, nobody knew the proof you're talking about until 1840. Best, jb From rrosebru@mta.ca Fri Mar 31 19:24:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPStW-0004Yt-FB for categories-list@mta.ca; Fri, 31 Mar 2006 19:19:06 -0400 Date: Thu, 30 Mar 2006 23:20:26 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories Subject: categories: Re: fundamental theorem of algebra Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 142 John Baez wrote: > I really doubt those authors were unaware of the topological proof > of the fundamental theorem of calculus in 1987. After all, it's Right, both my claim and its premises needed a fair bit of tuning (as with my recent question about the quasivariety "groups+free monoids" -- this is a good list to get corrective feedback from). (But a neat piece of historical research there, John.) The issue seems to be coming down to Mike Barr's question, which if I can paraphrase it without changing its intent, was, what is the proper status of an appeal to the very plausible in a proof? My suggestion in my last message to Peter Freyd was that the prover should point out the gap, its cause (lack of a simple proof), and its plausibility notwithstanding. This suggestion raises more questions than it answers. 1. Is a proof with a gap more acceptable for expository purposes when the bridgability of the gap is more plausible? (The case in point being an extreme example.) 2. How is plausibility to be judged? By consensus, or are there objective criteria? 3. It is certainly not necessary to prove A before B merely because B depends on A; indeed one common-sense practice when proving a two-lemma proof is to get the easier lemma out of the way first, even if it depends on the harder one. Is it kosher to truncate such a proof after the first lemma (or in this case the final result), call it an exposition, and point to the literature for the second lemma? Regarding 3, the authors of the Britannica article seemed not to think so, but perhaps this just reflects Garrett Birkhoff's attitude that "I don't consider this algebra, but this doesn't mean that algebraists can't use it" cited by Michael Artin when proving FTAlg in his 1991 book "Algebra". Who on this list considers the fundamental theorem of algebra "not algebra"? These questions are probably more appropriate for a philosophy of mathematics list than this one. What makes FTAlg such an interesting case study for those with something at stake in such questions is that the tensions here are so extreme. The final result (FTAlg) is not at all obvious, whereas the lemma it rests on, whether it be that |P(z)| attains its minimum, or that circles around a hole don't retract, or the intermediate value theorem, or the existence of a root for a real polynomial of odd degree, seems self-evident. Yet the one that is hard to see is easy to prove, while the one that is easy to see is hard to prove. If seeing is believing, what is proof? In the real world, when something is easy to see it is up to the opposition to demonstrate that it is nonetheless false. How did mathematics evolve to play by a different rule book? Vaughan Pratt From rrosebru@mta.ca Fri Mar 31 19:24:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPSqc-0004P1-Nl for categories-list@mta.ca; Fri, 31 Mar 2006 19:16:06 -0400 Date: Thu, 30 Mar 2006 18:46:13 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: cracks and pots 93 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 143 Eduardo is quite right to take me task for tailing off into utter incomprehensibility at the end of one of my longer postings. There's an argument to be made there, but that was neither the place nor the way to make it. One should not wait till the end to stop. For anyone wondering what on earth I had in mind, try googling for dipolar theories, which is what my CT'04 talk morphed into. Feedback welcome. Apropos of googling for "Yoneda axiom," Steve Lack offered the useful hint "Yoneda structures". Vaughan Pratt From rrosebru@mta.ca Fri Mar 31 19:24:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPSpK-0004LL-JF for categories-list@mta.ca; Fri, 31 Mar 2006 19:14:46 -0400 Message-ID: <442C9295.4000109@cs.stanford.edu> Date: Thu, 30 Mar 2006 18:23:17 -0800 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Fundamental theorem Content-Type: text/plain; charset=3Dwindows-1252; format=3Dflowed Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 144 The illustrator of Hal Abelson's 1970 "Calculus of Elementary Functions", Ellis Cooper, pointed me at it as an earlier reference for a proof in print (or out of print in this case) of the growing-circles argument for FTAlg than Pontrjagin=92s 1982 article. But then Mike Barr just now doubled that with Courant and Robbins! (So how far back *does* that very nice argument go?) Mike concluded with exactly the right question, which is what I wish I'd asked in the first instance: =93what does it mean to give an intuitively appealing non-proof and call it a proof?" Apropos of both my original assertion and that question, Peter Freyd and I went back and forth a bit, and he suggested I forward the ensuing correspondence to the list, following. My answer to Mike's question I think would be in my fourth last sentence, "By all means tell the audience..." Vaughan Pratt =3D=3D=3D=3D=3D PJF: Vaughan, surely you're not saying that the standard homotopy argument for the FToA is new, are you" It's the first one I ever heard -- and that was over 50 years ago. Peter =3D=3D=3D=3D=3D VRP: Hi, Peter, As usual you're in the vanguard in these things. But then why didn't you or whoever told you the proof pass it on to one of Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter Hilton (who *surely* should have known, yes?), or Paul Cohn? They're the ones taking responsibility for the algebra article for the Brittanica (1987 ed.). On p.260a they say "No elementary algebraic proof of [the FTAlg] exists, and the result is not proved here." I was being told the same thing in the 1960s - if Gauss couldn't find a simple proof in half a dozen tries, there isn't one. The basic message was, if you don't possess the necessary higher maths or the stamina for an intricate argument, we can't help you with that result, ask us about solvability of z^n =3D a. My cohort may well have run into you in the Edgeworth David building then, you would appear to have missed the opportunity to disabuse the rest of us of this notion. If you know of an elementary exposition of any proof of FTAlg appearing in the 1970's or earlier I'd love to see it. Vaughan =3D=3D=3D=3D=3D PJF: The proof was shown to me over 50 years ago as the standard argument of why one should believe the FToA .It's an easily understandable proof. But it is not elementary. To complete the proof there's a lot of work to be done. Just try proving -- from scratch -- that the circle is not contractible (if it were, there's no way you could use it to prove the FToA). There is a pretty elementary way that starts with the proof that for any continuous self-map on the unit circle C --> C there's a continuous R --> R such that h R --> R g | | g v v C --> C f where g(x) =3D exp(xi2\pi). (A proof of this doesn't require a lot of background but it's a bit tedious.) Then the "winding number' of f is the constant value (after you prove it's constant) of h(x+2\pi) - h(x)..And one can then construct a tedious proof that the winding number is a homotopy invariant. But even with this there's still a lot of work to do. There's another easy proof of the FToA (but also not elementary) that rests on the fact that a polynomial mapping on the complex plane is open and proper. Peter Do you want to forward our correspondence (when done) to the cat net? =3D=3D=3D=3D=3D VRP: No problem, just let me know when. My position is that this is a perfect example of a nonelementary proof that *is* fit for general consumption, in contrast to one that can't be grasped without first understanding the definition of some nonelementary concept such as holomorphic function [better example: local compactness =96vp]. Any child knows intuitively that a rubber band wrapped one or more times around a pencil can only be removed from the pencil by pulling it off one end or the other or by magic. By all means tell the audience that we're not going to prove that bit because a proper proof is surprisingly hard for such an intuitively obvious result. But otherwise I would say that, at least morally, this is an elementary proof. If "the rest of us" have been deprived of this argument all this time (up to the 1980s or whenever) on the ground that this property of loops is not an elementary one, I for one am very sorry not to have been shown the argument long ago, in place of the ones I *was* shown. It's a good thing sailors and scouts aren't taught by mathematicians or they wouldn't be allowed to study knots until they were officers or Eagle scouts. Vaughan =3D=3D=3D=3D=3D PJF: Damned good point. From rrosebru@mta.ca Fri Mar 31 19:24:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPSum-0004dj-TB for categories-list@mta.ca; Fri, 31 Mar 2006 19:20:24 -0400 From: "Peter McBURNEY" To: Subject: categories: Re: WHY ARE WE CONCERNED? Date: Fri, 31 Mar 2006 10:18:01 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed;charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 145 Peter Selinger wrote: > > "French lawmakers, for example, gave preliminary support this month to > a measure that would require the company to open the iPod to play > music purchased from any online music service; currently, songs > purchased from iTunes can be played only on iPods." > > New York Times, 2006/03/29, "Apple vs. Apple in Dispute Over Trademark" > > This is of course not a logical contradiction; but I would be very > surprised if it is what the writer really meant to say. Sadly, most > readers probably won't know the difference one way or the other. > I am not sure that mathematicians should cast stones here, given the decades it took for the ambiguities of First-Order Logic to be recognized, and then rectified with Independence-Friendly Logic. -- Peter McBurney University of Liverpool, UK. From rrosebru@mta.ca Fri Mar 31 19:31:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:31:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPT4N-0005Ge-LK for categories-list@mta.ca; Fri, 31 Mar 2006 19:30:19 -0400 Date: Fri, 31 Mar 2006 09:15:28 -0500 (EST) From: F W Lawvere To: categories@mta.ca Subject: categories: WHY ...CONCERNED? III MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 146 WHY ARE WE CONCERNED? III The second main misconception about category theory Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. Since such perceptions hold back the learning of category theory, and hence facilitate its misuse as a mystifying shield, they are among our concerns. We need to deal with the size preoccupation head on. Experience has shown that we cannot build up or construct mathematical concepts from nothing. On the contrary, centuries of experience become concentrated in concepts such as "there must be a group of all rotations" and we then place ourselves conceptually within that creation; we state succinctly the properties which that creation as a structure seems to have, and then develop rigorously the consequences of those properties taken as axioms. The notion of category arose in that way, and in turn serves as a powerful instrument for guiding further such developments. Placing ourselves conceptually within the metacategory of categories, we routinely make use of the leap which idealizes the category of all finite sets as an object. The question is, what more? Of course we make use of the experience of those who have labored to justify mathematics, and it is fortunate that ultimately our results are compatible with theirs. (Mac Lane's use of the term metacategory is not mysterious; it simply refers to the universe of discourse of any model, in the special context where the elements of such a model are themselves called categories and functors. In the spirit of algebra, we do not concentrate on the cumulative hierarchy which might have been used to present the metacategory, but rather on the mathematical category itself.) The supposed size problems of category theory are often concentrated in functor category formation. For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness). That existence statement is compatible with standard set theory, although it is often presumed to be incompatible. In the original 1945 exposition of category theory, it was the Goedel-Bernays account of the cumulative hierarchy (see posting II) that was cited as probably relevant (in case the problem of justifying category theory should come up). As a result, category theorists have been worried about supposed "illegitimacies" that might arise from violating the Goedel-Bernays rules (which in essence stemmed from von Neumann). These rules expressed an expediency which was a very effective trick at the time, identifying two kinds of membership relation and truncating the content at a plausible level. The Goedel-Bernays theory is well known to have the same logical strength as the Zermelo-Fraenkel system. An important advantage is that the greater expressive power of Goedel-Bernays permits it to be finitely axiomatizable, whereas Zermelo-Fraenkel is not; the greater expressive power concerns an element V of any model in which all small sets of the model can be embedded (just as another smaller element captures all finite sets). But the greater expressive power still allows mutual relative consistency: To every model of Goedel-Bernays, a model of Zermelo-Fraenkel can be constructed in a fairly straightforward manner: just take the small elements; in the converse direction there are two procedures (left and right adjoint?): given a model of Zermelo-Fraenkel, one can take all definable subsets of it, or just all subsets, and in either case a model of Goedel-Bernays apparently results. Because these mutual interpretations are hypothetical, relatively weak assumptions are required on the background category of sets taken as the recipient of models. In fact, with only slightly stronger assumptions on the background category one can construct, for any model of Zermelo-Fraenkel, a model of what set theorists use daily as BG+, which contains as elements not only V but W = V^V, V^W etc. Our practice is consistent with the minimal assumptions of professional set theorists: For any model of BG+ the presented metacategory of categories is both cartesian closed (in the usual elementary sense) and also has an object S of small sets. (Those facts strongly augment well-known properties, such as the existence of the first four finite ordinals and their adequacy in the metacategory relative to the sub-metacategory of discrete categories; of course these same ordinals also co-represent one of the "2-category" structures on the metacategory). The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it. Of course functor categories may no longer enjoy the same enrichment, just as functor categories starting from finite sets may not have finite hom-sets; but that is no reason to avoid considering them, and functionals on them, etc. when such considerations serve mathematics. It is of special interest to note that the restrictive "law" (under which categorists have been chafing) was already repealed forty years ago by Goedel and Bernays themselves. In their correspondence of 1963, it appears that they had been informed that a student of Eilenberg was working on a project to base set theory and mathematics on category theory; their immediate response was that mathematics will have to consider finite types over the class of small sets. (The relative consistency was presumably obvious to them.) Even though most set-theorists have themselves maintained clarity on the distinction, the identification of two kinds of membership in a formalized theory may have fostered in the minds of others a confusion between smallness (of a class or set) and existence as an element of the (meta)universe. Certainly, the specific meaning of smallness needs to be clarified (although for some purposes it can be taken as a parameter). There is a way of specifying smallness that is directly related to fundamental space/quantity dualities (rather than to imagined "building up" by stronger and stronger closure properties). Just as Dedekind finite sets X are characterized by the condition that a natural map X --->Hom(Q^X, Q) is an isomorphism, so indications from the study of rings of continuous functions and other branches of analysis strongly suggest that all small sets X should satisfy the same sort of isomorphism, with the truth-value space Q being replaced by the real line (in both cases, Hom refers to the binary algebraic operations on the object Q). There is the possibility to assume that conversely all sets X satisfying that isomorphism are small i.e. that, like the Dedekind-finite sets, they belong to a single uniquely-determined category S. That possibility in itself would imply no commitment concerning the existence or non-existence of super-huge objects in the metacategory "beyond" S, S^S, etc. Such an axiom would be somewhat stronger than ZF, but much weaker than the standard discussions of contemporary set theorists. ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ From rrosebru@mta.ca Fri Mar 31 19:32:29 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:32:29 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPT5p-0005Mu-8Q for categories-list@mta.ca; Fri, 31 Mar 2006 19:31:49 -0400 Date: Fri, 31 Mar 2006 09:30:58 -0500 From: jim stasheff MIME-Version: 1.0 To: Categories@mta.ca Subject: categories: Re: WHY ARE WE CONCERNED? I Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 147 Proofs may be of ultimate importance but a lot can be accomplished at the penulitmate level or even sooner jim Colin McLarty wrote: >> There's a saying about Lefschetz that he "never wrote a valid >> proof, and never made a false conjecture". Now it's not an attitude >> that want to encourage, but if you have great mathematicians who >> are like that (and Lefschetz was not just a good mathematician, but >> a great mathematician, without whom a good deal of modern algebraic >> geometry would be unimaginable), then this ought to tell us something. > > > This, and much else about Lefschetz has to tell us a lot. As to proof, > Lefschetz also never published a theorem without a purported proof, and > he often came to feel very strongly that his proofs were not good > enough. He wrote two long books on topology in the attempt to repair > the bad proofs in his influential booklet on cohomology in algebraic > topology, L'Analysis situs et la Topologie Algebrique. It was so > important to him that he enlisted many others. Notably for us, he > asked Eilenberg and Mac Lane to contribute an appendix to his 1942 > TOPOLOGY. This was their first published collaboration "On homology > groups of infinite complexes and compacta" and pursued the questions > that quickly led to category theory. > > Lefschetz had encouraged work on solving specific problems just over > the edge of what well-understood foundations for homology could > handle. Apparently he believed such solutions would lead to > significantly deeper understanding. He had encouraged Steenrod to work > on p-adic solenoids because existing methods did not seem adequate to > it. But whatever his motive, he was determined to see rigorous > solutions to quite specific problems. > > Colin > > From rrosebru@mta.ca Fri Mar 31 19:33:36 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:33:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPT6q-0005Sd-DW for categories-list@mta.ca; Fri, 31 Mar 2006 19:32:52 -0400 Subject: categories: re: fundamental theorem of algebra To: categories@mta.ca (categories) Date: Fri, 31 Mar 2006 11:39:16 -0800 (PST) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 148 A couple of mistakes. I wrote: >I really doubt those authors were unaware of the topological proof >of the fundamental theorem of calculus in 1987. I meant "fundamental theorem of algebra". >Gauss argues that far from the origin, S and T are smooth curves. >Because the leading term of the polynomial dominates the rest, >each of these curves intersects any sufficiently large circle >transversely at n points. Should be: any sufficiently large circle centered at the origin. Best, jb From rrosebru@mta.ca Fri Mar 31 19:35:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:35:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPT8P-0005c0-U5 for categories-list@mta.ca; Fri, 31 Mar 2006 19:34:29 -0400 Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Message-Id: <622e5ac70de8ee0b69105fc9f9868f91@math.ksu.edu> Content-Transfer-Encoding: quoted-printable To: categories@mta.ca From: David Yetter Subject: categories: Re: cracks and pots 93 Date: Fri, 31 Mar 2006 10:48:52 -0500 To: Eduardo Dubuc X-Mailer: Apple Mail (2.623) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 149 Eduardo, I think it is you who are suffering from superficial thinking, or at=20 least superficial reading. Admittedly there was a deliberate superficiality in my topic--I=20 describe 'faces' of category theory, aspects presented on the surface, to which those approaching=20 the subject react, even as people in social contexts react to the face of those they meet.=20= But the entire discussion has been about reactions to the public face=20 of category theory, and about what and who should be that face. You evidently did not read my post carefully enough. It is not I, but=20= the mathematical community as a whole that has a prejudice against=20 'categories as foundations'--and indeed against foundations, which most=20= mathematicians try devoutly to ignore as my discussion of the attitude=20= toward constructions of the real numbers illustrated. Both category =20 theory and categorists have suffered as a result. In the 1980's, when=20= I fell in love with category theory, in part because it did address big=20= foundational issues, this prejudice resulted in the marginalization and=20= ghettoization of category theory within mathematics. I began my post with the story of Moishe Flato's dismissal of category=20= theory as 'a mere language' and his repentance from that view. I chose this because it=20 was the most cheerful story I could tell to illustrate the prejudice against foundations, and=20= category theory as such, and probably one most had not heard. Category theory is breaking out of its ghetto not by finding=20 foundational applications in computer science--excellent though those are, both for=20= the intellectual life of our community and job prospects for categorists--and certainly=20= not by asserting its foundational role in mathematics, by showing its=20 face as algebra to mathematics as a whole. Your last paragraph suggests, perhaps categorists are not. The attitude evinced by your reply to my post--dismissing the=20 mathematical content of my remarks as "highly technical, sophisticated, difficult and impressively=20= sounding words" (doesn't all mathematics sound that way until one=20 masters the relevant concepts and definitions?), and adopting a 'blame=20= the messenger' attitude to my report of anti-foundational prejudice among mathematicians--suggests that you are content to remain in the ghetto, and want to keep the rest of us there=20= with you. Peevishly yours, D. Yetter On 30 Mar 2006, at 13:31, Eduardo Dubuc wrote: > Hi, > > The 93 is because I have by now 92 msages in my cracks and pots file. > > I apologize for the length of this posting. It is intended to be a=20 > (may be > biased) partial account of the debate, and some comments. > > > Well, by now the "cracks and pots" debate is establishing itself as,=20= > in my > opinion, an interesting and worth-wile event. Congratulations Marta=20= > !! > > We are learning about: > > a) Understand (for many of us) better what is mathematics, and what = is > physics, what is rigor and what is buccaneering, and also what is > bullshit. > > b) "Something is rotten in the state of category theory community" > > Pay attention that The Bard does not say "category theory", but he = says > "category theory community" > > I start from who has made the more refreshing, humorous, down to > earth, honest and intelligent contributions to this debate: > > **Vicent Schmitt: that theoretical physics, computer science, phylo., = a > mix of those, or whatever? , is used to justify poor "categorical" = work > is, in my view, an existing problem. More or less everyone is=20 > conscious of > it (come on!...) but so far that has not been publicly debated.** > > Yes Vincent!!, you point right to what it is at the center (or very=20 > near > it) the problem raised in Marta=D5s original "cracks and pots" = posting!.=20 > And > the "(come on!...)", beautiful !. > > Now, talking about rigor, conjectures and proofs: > > **Maclane : If a result has not yet been given valid proof, it isn't=20= > yet > mathematics. This however does not deny the many preliminary stages of > insight, experiment, speculation or conjecture, which can lead to > mathematics. It states simply that a conjectured result is not yet a > theorem ** > > It is relevant to compare this with Motl's distinction between physics=20= > and > mathematics: > > **Motl: In physics, we propose different conjectures about the real=20 > world, > and it is important that we're not guaranteed that these conjectures=20= > will > be true. > > String theory itself is not just a conjecture, but rather a > seemingly consistent mathematical framework. Once we accept string=20 > theory > as an objectively existing mathematical structure, a structure that we > treat as a part of "generalized physics" - which is of course what all > string theorists are doing every day - we can ask a lot of questions=20= > about > its properties.** > > He does distinguish between "physics as conjecture" and mathematics=20= > with > applications to physics. He call this mathematics "generalized=20 > physics" > > But "conjecture" to be acceptable is not unrigourous neither=20 > buccaneering. > he says: > > **Motl: the statements about string theory are just conjectures, and=20= > they > need to be proved or supported by evidence, otherwise they're=20 > irrelevant > and "wrong", in the physical sense.** > > He also says: > > ** Motl: I always feel very uneasy if the mathematically oriented > people present their conjectures about physics, quantum gravity, or=20 > string > theory as some sort of "obvious facts". > > He is clearly saying that those "mathematically oriented people" are > lacking rigor. > > Many postings in this debate confound mathematical rigor with=20 > formalism, > and push forward the idea that a formal and logically correct = statement > has automatically rigor. Even if it is foolish: > > **V. Pratt: In axiomatic mathematics, everything that is not forbidden=20= > is > permitted. ** > > **R. Dawson: If the math itself meets mathematical standards of rigor,=20= > its > application to physics need surely only meet the standards appropriate=20= > to > that subject.** > > It seems to me that he is equating here "mathematical standards of=20 > rigor" > with "logically correct", and "the standards appropriate to that=20 > subject" > (in this case, physics) with " buccaneering " > > Nothing more wrong!! . In both cases, failing to convey what it should=20= > be > considered "rigor in mathematics" and "rigor in physics" > > But again Saunders and Lubos: > > **MacLane: real proof is not simply a formalized document, but a=20 > sequence > of ideas and insights** > > ** Motl: the primary physical motivation is to locate the right ideas=20= > and > equations that describe the real world. Category theory has been used=20= > by > many to achieve completely wrong physical conclusions - for example, = by > considering the "pompously foolish" quantization functor.** > > He however seems to be pushing forward the same misconception of=20 > "rigor": > > **Motl: It may be nice to be rigorous, but it's always more important=20= > to > be correct: if the specific kind of rigor leads us to stupid=20 > conclusions > in physics, we should avoid it.** > > =46rom the original Marta's "cracks and pots" > > **M.Bunge: Are we category theorists as a whole going to quietly=20 > accept > getting discredited by a minority of us presumably applying category > theory to string theory?** > > **J. Baez: I had never heard anyone before suggest that category = theory > could be discredited by applications to string theory. It completely > surprised me. I'm used to the opposite complaint: that category=20 > theory is > discredited by its *lack* of applications.** > > Here it is a clear and rigorous answer: > > (1) **W. Lawvere: The question is not whether mathematics should be > applied. Most of us agree that it should. The concern is rather that=20= > our > subject is sometimes being used as a mystifying smoke screen to = protect > pseudo-applications against the scrutiny of the general public and of=20= > the > scientific colleagues in adjacent disciplines. We need to ensure that > applications themselves be maximally effective, not clouded by > misunderstanding.** > > Now, an example of superficial conclusions: > > ** J. Baez: Indeed, the funny thing about string theory is that while > leading to an abundant harvest of rigorous mathematical results, it = has > not yet correctly predicted a single result from a single experiment, > even after more than 20 years of work on the part of many smart=20 > people.** > > There is nothing funny about this. Lubos say: > > ** Motl: One of the fascinating features of string theory is that its > objects and investigations, even though they've been partially > disconnected from the daily exchanges with the experimentalists,=20 > remained > extremely physical in character. All of the objects that we deal with=20= > are > analogous to some objects in well-known working physical theories, to=20= > say > the least.** > > Bill has made a serious, well fundamented and non-bullshit=20 > contribution to > "crack and pots" (he utilizes a different heading:" WHY ARE WE=20 > CONCERNED?") > > In contrast to many passages of some contributors that it will be=20 > tiresome > to reproduce here, and where one founds an overwhelming proliferation=20= > of > > highly technical, sophisticated, difficult and impressively sounding=20= > words > > such that it becomes impossible to see what they are saying, unless = you > are an expert, in which case you may find out that it is only=20 > superficial > thinking (I am thinking specially in certain parts of Davis Yetter's > postings). > > ** W. Lawvere: Professors may not consider the possibility of learning > from undergraduate text books, and some may feel bored that I have = once > again repeated the above basic definitions and observations.** > > If you have some real thoughts, you do not need impressive jargon. > > See what an original and deep insight: > > ** W. Lawvere: As quantity includes zero, so structure includes the=20 > case > of no structure, which Cantor considered one of his most profound and > exciting discoveries** > > Superficial thinking (which could be malicious, but very often is=20 > simply > stupid) has manifested itself in these postings by pushing forward = the > idea that there are two different kinds of category theory: > > "Categories as Foundation" and "Categories as Algebra", the first > implicitly (but not explicitly said) the "bad one", and the second the > "good" one. > > ** D. Yetter: All of these are part and parcel of a different face of > category theory than one saw in the old days: category theory as=20 > algebra, > rather than category theory as foundations.** > > We have an excellent analysis of this fallacy in Bill's postings, = which > should be read carefully and slowly. > > I imagine now to add something that Lawvere himself pointed out a long > time ago: The laws of logic are a particular instance of the=20 > categorical > concept of adjoint functors, a concept that grew out of mathematical > experience. > > There is any way some explanation to Yetter's prejudice against > "categories as foundation". Often very poor category theory has been > justified by people writing on foundations. Bill's quote (1) above = also > applies to this and related use of category theory in theoretical=20 > computer > science. > > Somebody else that does not need either noisy language sees better: > > ** Dusko: I am of course saying things very clear and familiar to many > people on this list, but maybe they are worth saying nevertheless.** > > ** Dusko: but at the end of the day, I think, we'll all agree that the > source of the unreasonable effectiveness of categorical algebra is its > foundational content ** > > Then, he passes to consider Grothendiek's ("the greatest of the > category theorists") work on Topos theory as work on foundations,=20 > which > agrees with the analysis of foundations made by Lawvere. > > I can not restrain myself to quote the following magnificent piece of > meaningless hallucinogenic discourse: > > **V. Pratt: In the millions of years of evolution of primate thinking,=20= > no > productive mathematical mechanism has a higher probability of being > stumbled on than mathematics founded on the Yoneda axiom. I know of no > better explanation of how human thought could have evolved to its=20 > present > form than evolution finding and exploiting the Yoneda principle** > > Now, some serious business: > > In recent years J.Baez and his followers have been occupying more and=20= > more > space in the categorical community (this fact is at the starting point=20= > of > the present debate). > > I think this is so because they have some interesting category theory=20= > to > show, but they are occupying more space than their mathematics = deserves > because they bring a refreshing air to a community until now dominated=20= > by > an old guard that has not shown signs of necessary evolution, and that=20= > has > not being able to attract very good and talented young mathematicians=20= > to > the community. There is now not other exiting body of developments=20 > within > the community. The old guard is being pushed out (prone or supine ?),=20= > but, > alas, not by better mathematicians. > > Category theory is in good shape (in particular pushed forward by the > Russian school), and it is now passing over the category community. I > have lost the information now, but recently it was in Europe an=20 > important > congress that it had two subjects: one was a prestigious subject (that=20= > I > do not remember now), the other was category theory. Not a single name > (including Baez group) that we see in the category theory community > meetings was there. > > Best wishes to all e.d. > > > > > > > > > From rrosebru@mta.ca Fri Mar 31 19:37:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:37:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPTAi-0005kI-U3 for categories-list@mta.ca; Fri, 31 Mar 2006 19:36:53 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: cracks and pots 93 Date: Fri, 31 Mar 2006 17:17:26 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=3Dflowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 150 Dear Eduardo, Your message contains many important observations with which I very much agree, and a few others that need discussing. I may comment on it some time soon, publicly or privately. Right now, my purpose for replying to you is a different and more pressing one. I noticed that in your "cracks and pots 93" you implicitly refer to a lette= r which I had sent to categories in reply to Peter Selinger, and which I had also sent privately to various people, including you. The letter in questio= n never appeared!. It contained an attachment (to MacLane's article) which, according to Bob Rosebrugh, was difficult to include, hence the delay of it= s posting, and ultimately my replacing it by a brief message in response to Bill Lawvere, in which I included the URL, as suggested by Bob. It seems imperative now that I post the original message, without the attachment. Thus, portions of your letter may make more sense. It is below this message= =2E By the way, by my own count, there are only 75 messages posted in the "thread", but 183 more that were written privately to me in connection with it. Maybe you included some private messages in your count? Either way, thi= s means a *lot* of messages. Best, Marta ---------------------------------------------------------------------------= --------------------------------------------------------- Dear Peter, You were lucky to have been away on vacation, but perhaps quickly reading (how else) the mass of postings in the "cracks and pots" has caused you intellectual indigestion. Your reaction is therefore quite understandable. For your sake (and that of others in similar situations), I will sum up wha= t caused my postings, and be more explicit concerning >Is there any evidence to support this claim? I.e., actual examples where >such research was disproportionally supported that was uncritical and >perhaps unwarranted? There have been several posts seemingly agreeing that >this is the case, but none have given concrete evidence. 1. On March 13, I shared with you all a disturbing posting in Motl's blog, criticizing category theory in its applications to physics, and more particularly, John Baez. My concern was based on the possibility that any o= f this criticism might be justified because I could not failed to notice how John Baez had become more or less a prominent figure (as speaker/member of the scientific committee) in recent(ly announced) meetings in CT. Explicitly, I was thinking of Firenze, Ramifications of CT, Nov 13-19, 2003, Sydney, StreetFest, July 11-16, 2005, Union College, UC Mathematical Conference, December 3-4, 2005, Chicago, MacLane Memorial Conference (Unni Nambondiri Lectures), April 7,10,11, 2006, Halifax (near), CT'06, June 25-July 1, 2006. 2. On March 14, and in response to some, I asked more explicitly what caused organizers of meetings to bring to center stage one aspect of CT over others, particulaly one which seemed to me not to be in good standing after Motl's postings. Was it because it is indeed the case that CT is in disrepute, and if so its reputation needs to be restored, this being the best way to do it? Was it because it is funding for CT (notoriously lacking in the USA) that may be more easily secured that way? I wanted to know myself, but also possibly alert organizers of meetings to reflect on this issues, since their power and responsibility is indeed enormous in promotin= g a certain kind of research over another. 3. >From the many responses that I got (some public, and many more privately),= I picked on one (March 17) to add some information that I had just come acros= s by reading Nature (on our coffee table, along with a dozen or so scientific journals), in an article which connected Lee Smolin of the Perimeter Institute with the Templeton Foundation, the latter a promoter of anything they can in the borderline of science and religion. In the Scientific American articles by Lee Smolin on Loop Quantum Gravity and the discretenes= s of the universe, a paper John Baez is quoted among the few references give= n at the end of the article. This, in turn, led me to research the Templeton Foundation itself, and with some help from a fellow categorist who seemed t= o know a lot about it, I easily located references to Templeton funding to th= e Goedel Centenary Symposium in Vienna, and to the A.Connes workshop on NCA a= t the Sir Isaac Newton Institute in Cambridge. I was, however, relieved not t= o find any direct connection between Templeton and Category Theory. Still, I meant to warn those unaware of this easy source of funding (with strings attached). In a subsequengt posting (March 27) I gave explicit references t= o these claims in response to some queries. 4. In short, I do not think that I can be blamed for not being explicit enough in matters that I could be explicit about. I still do not have all the answers to my questions. As I mentioned on March 27, I was mistaken in thinking of John Baez as a promoter of string theory when, in fact, he promotes a competitor thery, LQG. But the general question of categorical applications to physics remained. Why are they promoted now? As you, Peter, kindly offer as a possible explanation, >Can one rule out another possibility, namely that such research is >supported because it is original, timely, and interesting? No, of course not -- one cannot rule it out. Here, I am ignorant of physics so I cannot answer this question (David Yetter has supported the view that they are original, timely and interesting, and has contrasted "algebraic" t= o "foundational" aspects of CT). But even if the answer were "yes", I would welcome responses to the question which still remains unaswered (except that most of us surely have a formed opinion) -- is CT in such a poor state that it needs revamping? Sould we not wait a few years until several original and interesting (maybe not timely) contributions to CT in connection with other fields of mathematics are appreciated and incorporate= d into the mainstream? What do we gain by pushing those under the rag? To imply, perhaps, that we ourselves do not value them? These, I believe, are crucial and timely questions, and I do not regret unwilingly having brought them up 5. I take this opportunity to thank Bill Lawvere for his first posting "Why ar= e we concerned? I", in which the lucid article by Saunders MacLane (Synthese, 1997) is recalled in connection with the discussions that arose in the "cracks and pots" so-called-thread (why "thread"?). I am sure that most of you have read it, but just in case you have not, I attach it here it in pdf form. This is very timely in view of the upcoming MacLane Memorial Conference in Chicago. Peter, I hope that I have answered your questions. I can't speak for the others who have contributed to this "thread". Unlike what has been suggested, what I originated on March 13 was far from a "complot". It was a genuine concern of mine and I see now, by many of the responses, that it is also a concern of others. On the other hand, getting personally attacked (for the wrong reasons, to boot) is a necessary price that I have to pay an= d it does not concern me as much. Yours, Marta ---------------------------------------------------------------------------= ------------------- >From: Eduardo Dubuc >To: cat-dist@mta.ca >Subject: categories: cracks and pots 93 >Date: Thu, 30 Mar 2006 15:31:17 -0300 (ART) > >Hi, > >The 93 is because I have by now 92 msages in my cracks and pots file. > >I apologize for the length of this posting. It is intended to be a (may be >biased) partial account of the debate, and some comments. > > >Well, by now the "cracks and pots" debate is establishing itself as, in my >opinion, an interesting and worth-wile event. Congratulations Marta !! > >We are learning about: > >a) Understand (for many of us) better what is mathematics, and what is >physics, what is rigor and what is buccaneering, and also what is >bullshit. > >b) "Something is rotten in the state of category theory community" > >Pay attention that The Bard does not say "category theory", but he says >"category theory community" > >I start from who has made the more refreshing, humorous, down to >earth, honest and intelligent contributions to this debate: > >**Vicent Schmitt: that theoretical physics, computer science, phylo., a >mix of those, or whatever? , is used to justify poor "categorical" work >is, in my view, an existing problem. More or less everyone is conscious of >it (come on!...) but so far that has not been publicly debated.** > >Yes Vincent!!, you point right to what it is at the center (or very near >it) the problem raised in Marta=D5s original "cracks and pots" posting!. A= nd >the "(come on!...)", beautiful !. > >Now, talking about rigor, conjectures and proofs: > >**Maclane : If a result has not yet been given valid proof, it isn't yet >mathematics. This however does not deny the many preliminary stages of >insight, experiment, speculation or conjecture, which can lead to >mathematics. It states simply that a conjectured result is not yet a >theorem ** > >It is relevant to compare this with Motl's distinction between physics and >mathematics: > >**Motl: In physics, we propose different conjectures about the real world, >and it is important that we're not guaranteed that these conjectures will >be true. > >String theory itself is not just a conjecture, but rather a >seemingly consistent mathematical framework. Once we accept string theory >as an objectively existing mathematical structure, a structure that we >treat as a part of "generalized physics" - which is of course what all >string theorists are doing every day - we can ask a lot of questions about >its properties.** > >He does distinguish between "physics as conjecture" and mathematics with >applications to physics. He call this mathematics "generalized physics" > >But "conjecture" to be acceptable is not unrigourous neither buccaneering. >he says: > >**Motl: the statements about string theory are just conjectures, and they >need to be proved or supported by evidence, otherwise they're irrelevant >and "wrong", in the physical sense.** > >He also says: > >** Motl: I always feel very uneasy if the mathematically oriented >people present their conjectures about physics, quantum gravity, or string >theory as some sort of "obvious facts". > >He is clearly saying that those "mathematically oriented people" are >lacking rigor. > >Many postings in this debate confound mathematical rigor with formalism, >and push forward the idea that a formal and logically correct statement >has automatically rigor. Even if it is foolish: > >**V. Pratt: In axiomatic mathematics, everything that is not forbidden is >permitted. ** > >**R. Dawson: If the math itself meets mathematical standards of rigor, its >application to physics need surely only meet the standards appropriate to >that subject.** > >It seems to me that he is equating here "mathematical standards of rigor" >with "logically correct", and "the standards appropriate to that subject" >(in this case, physics) with " buccaneering " > >Nothing more wrong!! . In both cases, failing to convey what it should be >considered "rigor in mathematics" and "rigor in physics" > >But again Saunders and Lubos: > >**MacLane: real proof is not simply a formalized document, but a sequence >of ideas and insights** > >** Motl: the primary physical motivation is to locate the right ideas and >equations that describe the real world. Category theory has been used by >many to achieve completely wrong physical conclusions - for example, by >considering the "pompously foolish" quantization functor.** > >He however seems to be pushing forward the same misconception of "rigor": > >**Motl: It may be nice to be rigorous, but it's always more important to >be correct: if the specific kind of rigor leads us to stupid conclusions >in physics, we should avoid it.** > >From the original Marta's "cracks and pots" > >**M.Bunge: Are we category theorists as a whole going to quietly accept >getting discredited by a minority of us presumably applying category >theory to string theory?** > >**J. Baez: I had never heard anyone before suggest that category theory >could be discredited by applications to string theory. It completely >surprised me. I'm used to the opposite complaint: that category theory is >discredited by its *lack* of applications.** > >Here it is a clear and rigorous answer: > >(1) **W. Lawvere: The question is not whether mathematics should be >applied. Most of us agree that it should. The concern is rather that our >subject is sometimes being used as a mystifying smoke screen to protect >pseudo-applications against the scrutiny of the general public and of the >scientific colleagues in adjacent disciplines. We need to ensure that >applications themselves be maximally effective, not clouded by >misunderstanding.** > >Now, an example of superficial conclusions: > >** J. Baez: Indeed, the funny thing about string theory is that while >leading to an abundant harvest of rigorous mathematical results, it has >not yet correctly predicted a single result from a single experiment, >even after more than 20 years of work on the part of many smart people.** > >There is nothing funny about this. Lubos say: > >** Motl: One of the fascinating features of string theory is that its >objects and investigations, even though they've been partially >disconnected from the daily exchanges with the experimentalists, remained >extremely physical in character. All of the objects that we deal with are >analogous to some objects in well-known working physical theories, to say >the least.** > >Bill has made a serious, well fundamented and non-bullshit contribution to >"crack and pots" (he utilizes a different heading:" WHY ARE WE CONCERNED?"= ) > >In contrast to many passages of some contributors that it will be tiresome >to reproduce here, and where one founds an overwhelming proliferation of > >highly technical, sophisticated, difficult and impressively sounding words > >such that it becomes impossible to see what they are saying, unless you >are an expert, in which case you may find out that it is only superficial >thinking (I am thinking specially in certain parts of Davis Yetter's >postings). > >** W. Lawvere: Professors may not consider the possibility of learning >from undergraduate text books, and some may feel bored that I have once >again repeated the above basic definitions and observations.** > >If you have some real thoughts, you do not need impressive jargon. > >See what an original and deep insight: > >** W. Lawvere: As quantity includes zero, so structure includes the case >of no structure, which Cantor considered one of his most profound and >exciting discoveries** > >Superficial thinking (which could be malicious, but very often is simply >stupid) has manifested itself in these postings by pushing forward the >idea that there are two different kinds of category theory: > >"Categories as Foundation" and "Categories as Algebra", the first >implicitly (but not explicitly said) the "bad one", and the second the >"good" one. > >** D. Yetter: All of these are part and parcel of a different face of >category theory than one saw in the old days: category theory as algebra, >rather than category theory as foundations.** > >We have an excellent analysis of this fallacy in Bill's postings, which >should be read carefully and slowly. > >I imagine now to add something that Lawvere himself pointed out a long >time ago: The laws of logic are a particular instance of the categorical >concept of adjoint functors, a concept that grew out of mathematical >experience. > >There is any way some explanation to Yetter's prejudice against >"categories as foundation". Often very poor category theory has been >justified by people writing on foundations. Bill's quote (1) above also >applies to this and related use of category theory in theoretical computer >science. > >Somebody else that does not need either noisy language sees better: > >** Dusko: I am of course saying things very clear and familiar to many >people on this list, but maybe they are worth saying nevertheless.** > >** Dusko: but at the end of the day, I think, we'll all agree that the >source of the unreasonable effectiveness of categorical algebra is its >foundational content ** > >Then, he passes to consider Grothendiek's ("the greatest of the >category theorists") work on Topos theory as work on foundations, which >agrees with the analysis of foundations made by Lawvere. > >I can not restrain myself to quote the following magnificent piece of >meaningless hallucinogenic discourse: > >**V. Pratt: In the millions of years of evolution of primate thinking, no >productive mathematical mechanism has a higher probability of being >stumbled on than mathematics founded on the Yoneda axiom. I know of no >better explanation of how human thought could have evolved to its present >form than evolution finding and exploiting the Yoneda principle** > >Now, some serious business: > >In recent years J.Baez and his followers have been occupying more and more >space in the categorical community (this fact is at the starting point of >the present debate). > >I think this is so because they have some interesting category theory to >show, but they are occupying more space than their mathematics deserves >because they bring a refreshing air to a community until now dominated by >an old guard that has not shown signs of necessary evolution, and that has >not being able to attract very good and talented young mathematicians to >the community. There is now not other exiting body of developments within >the community. The old guard is being pushed out (prone or supine ?), but, >alas, not by better mathematicians. > >Category theory is in good shape (in particular pushed forward by the >Russian school), and it is now passing over the category community. I >have lost the information now, but recently it was in Europe an important >congress that it had two subjects: one was a prestigious subject (that I >do not remember now), the other was category theory. Not a single name >(including Baez group) that we see in the category theory community >meetings was there. > >Best wishes to all e.d. > > > > > > > > > > From rrosebru@mta.ca Fri Mar 31 19:43:27 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:43:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPTFl-0006C9-Qt for categories-list@mta.ca; Fri, 31 Mar 2006 19:42:05 -0400 Subject: categories: Re: cracks and pots 93 From: Eduardo Dubuc Date: Fri, 31 Mar 2006 13:56:55 -0300 (ART) To: categories@mta.ca X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 151 David, your message below is clear and positive. It does clarify and help to understand the issues in this debate. I think my own message was worth if only to trigger your reply. My message touch sensitive places and I hope it will motivate more enlightening replies as it was yours. Concerning your last remark "suggests that you are content to remain in the ghetto, and want to keep the rest of us there with you". I do not see the logic by which you think this statment follows from my message. I do no want to remain in the ghetto, and much less want to keep anybody else in it. I am happy to see that you also acknowledge in public that such a ghetto exists. Some will be able to escape, and some others not. Probably with time the ghetto will disolve in nothingness. I do not know what "Peevishly" means, but please !! do not explain it to me. Some day I will look in a dictionary. yours Eduardo Dubuc > Eduardo, > > I think it is you who are suffering from superficial thinking, or at > least superficial reading. > [... quotation omitted] > > Peevishly yours, > D. Yetter >