From rrosebru@mta.ca Tue Nov 1 16:04:49 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 01 Nov 2005 16:04:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EX2DA-0000RF-CQ for categories-list@mta.ca; Tue, 01 Nov 2005 15:54:24 -0400 X-ME-UUID: 20051101125025832.CB288C400082@mwinf3216.me.freeserve.com Message-ID: <013401c5dee2$816ff360$cee74c51@brown1> From: "Ronald Brown" To: "categories" Subject: categories: weak multiple categories Date: Tue, 1 Nov 2005 12:47:23 -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: X-Keywords: X-UID: 1 Just been away so it was interesting to see the correspondence on all this, in the double case. It is useful first to have some examples of weak double categories. As usual, these can come from homotopy theory. Let A,B be subspaces of a space X. R= Let R_2(X:A,B) be the space of maps I^2 \to X which map the edges in direction 1 to A and the edges in direction 2 to B (so A,B had better have a non empty intersection). Then we get partially defined compositions in both directions, but no identities, no associativity. You can do better by considering `Moore rectangles'. There are also involutions in each directions, giving -_1, -_2. It seems hard to get a homotopy double groupoid out of this, unless one of A,B is contained in the other. However Loday showed that the fundamental group of R at the constant map does inherit the other 2 structures, giving a cat^2-group (a double groupoid in groups). It is a nice exercise to generalise the above to an n-ad (n-subspaces). Curiously, the interchange law is exactly valid in R_2. I expect that this law is one of those to go, for certain applications, such as multiple holonomy, and rewriting. The first could be important for gravity theory (that would be nice!). There is a way of handling the first (?) obstruction to the interchange law, by considering cubical multiple categories with connection as defined in 116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118. The monoidal closed structure allows for the notion of `cubical algebra'. i.e. with a monoid structure w.r.t. tensor product. C \otimes C \to C. This includes a whiskering operation of C_0 on the left and right of C_1, and also a map say b: C_1 \times C_1 \to C_2, which somehow measures the failure of the interchange law for the each of the two binary operations which can be defined on C_1 using whiskering. Such a cubical algebra should (?) be related to Sjoerd Crans' tesis, but the cubical format could be more convenient: I have always found that format more useful in several ways than the globular approach, as multiple compositions are easy to understand, but the relation between the two, when it exists, is important. Analogues of the map b occur in braided crossed modules, 2-crossed modules, automorphisms of crossed modules, .... In the paper 123. (with I.ICEN), `Towards a 2-dimensional notion of holonomy', Advances in Math, 178 (2003) 141-175. we tried to get towards a double groupoid associated with 2-holonomy, but this now seems naive. We would like a map such as the above b to be involved with some physical phenomena! Properties of the globular version may not yet have been written down! But groupoid/crossed module analogues are in 59. (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism structures for crossed modules'', {\em Proc. London Math. Soc.} (3) 59 (1989) 51-73. Ronnie Brown www.bangor.ac.uk/r.brown From rrosebru@mta.ca Tue Nov 1 20:10:12 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 01 Nov 2005 20:10:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EX68N-0001gd-Hx for categories-list@mta.ca; Tue, 01 Nov 2005 20:05:43 -0400 Date: Tue, 1 Nov 2005 14:37:32 -0500 (EST) From: Peter Freyd Message-Id: <200511011937.jA1JbWQd001371@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Free abelian stuff Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 Given an additive category *C*, its finite co-completion, R(*C*), may be constructed as the full subcategory of finitely presented objects (see footnote for a definition of f.p.) in the category of abelian- group-valued presheaves on *C*. The finite completion, L(*C*) may therefore be constructed as (R(*C*`))` (using ` to denote here the dual category). Remarkably: R(L(*C*)) = L(R(*C*)) Moreover: It is the free abelian category generated by *C*. To be precise, it is an abelian category and every additive functor from *C* to an abelian category extends to an exact functor on R(L(*C*)), uniquely up to natural equivalence. (If the target of the additive functor is not abelian then there are extensions, but not necessarily any that preserve both finite limits and co-limits.) If one starts with a one-object additive category -- that is, a ring -- then its finite co-completion, the category of f.p. presheaves, is, of course, just the category of f.p. modules (as usually understood). So the free abelian category generated by a ring, R, may be constructed as the full category of f.p. covariant abelian-group- valued functors on the category of f.p. R-modules. If R is commutative (or if it has an anti-involution) then the freeness means this category of f.p. functors is self-dual. Moreover: It is a *-autonomous.category. There are a lot of functors in this category. Besides being closed under the abelian operations it is closed under composition. (Any group-valued functor lifts canonically to an R-module-valued functor, hence composition is defined.) The dual of Hom(A,--) is A * -- (using * for tensor product) and the dual of Ext^n (A,--) is Tor_n (A,--). The dual of a composition ST is the composition of their duals (do not reverse the order of composition). I found all this when trying recently to simplify the proofs for the symmetric construction of free abelian categories (Murray Abelian, Categories Over Additive Ones. J. Pure Appl. Algebra 3 (1973), 103--117). The point of departure was the recognition that free abelian categories always have enough projectives (and, dually, enough injectives). Since at least the world's first category theory conference it has been known that an abelian category with enough projectives is the finite co-completion of its full subcat of projectives. (Representations in abelian categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95--120 Springer, New York) Herein is the outline of this alternate approach. I find it hard to believe that it is new. We will assume that *C* is already additive in the stronger sense, that is, it has finite direct sums. For a first-order construction of R(*C*) start with the category of maps in *C* (that is, the category of functors from the ordinal 2 to *C* ) and then reduce by a notion of "homotopy": a map from A' --> A to B' --> B, a A'--> A f'| | f B'--> B b (all vertical arrows point down) is null-homotopic if a map A --> B' can be inserted making the _lower_ of the two triangles commute. The functor *C* --> R(*C*) sends A to O --> A. A cokernel of the displayed f-map is constructible as: B' ---> B | | A + B'--> B where each map is given by a matrix in which each entry is either 1 or a single letter (b,f), if such fits, else 0. Given a functor, T, to a finitely co-complete category, define T(A'--> A) by choosing a cokernel of T(A') --> T(A). It is easy to verify that T preserves cokernels. LEMMA: If *C* is finitely complete, then R(C) is abelian and *C* --> R(*C*) preserves finite limits. BECAUSE: An additive category is abelian if each map can be factored as a cokernel (of something) followed by a kernel. The f-map displayed map above factors as: a A'--> A | | 1 P --> A | | f B'--> B b with the lower square is a pullback diagram. It is a good finger- exercise to see that the upper square is a cokernel of the pullback square: P'--> P | | A'--> A (which being monic is therefore the kernel of the given f-map) and the lower square is a kernel of the already mentioned cokernel: B' ---> B | | A + B'--> B Finally then, given a functor, T, from a finitely complete *C* to an abelian category it is easy to verify that if T preserves finite limits then the right-exact extension of T to R(*C*) is not just right exact but exact. When *C* is finitely complete one may verify that every projective in R(*C*) is isomorphic to an object in the image of *C* --> R(*C*) Because that functor preserves kernels we see easily that R(*C*) has global dimension at most two. One may prove that if a free abelian category has global dimension at most one then it has global dimension zero. Hence the only global dimensions that can occur are 0 and 2. (Every abelian category of global dimension zero is a free abelian category -- of itself. All additive functors therefrom preserve finite limits and co-limits.) The dual of the free abelian category generated by *C* is necessarily isomorphic to the free abelian category generated by the dual of *C*. Hence (still using ` to denote duals) we have (R(L(*C*`)))` = R(L(*C*)). But (R(L(*C*`))` = L(L(*C*`)`) = L(R(*C*)). The *-autonomous structures must wait. Footnote: By a FINITELY PRESENTED object in a co-complete category is meant an object whose corresponding covariant representable functor preserves filtered co-limits. If *C* has finite direct sums then a presheaf (all presheaves will be understood to be valued in the category of abelian groups) is f.p. iff it appears as the cokernel of a map between representables. Postscript: Where is the abelianess of the target category being used? If it is abelian then -- using the notation above -- one can prove that T(P') --> T(P) --> Cok(Ta) --> Cok(Tb) is exact. The extension of T is defined so that it preserves the cokernel of T(P') --> T(P). One invokes abelianess of the target to know that it therefore preserves the kernel of Cok(Ta) --> Cok(Tb). From rrosebru@mta.ca Tue Nov 1 20:10:13 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 01 Nov 2005 20:10:13 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EX69N-0001jr-GE for categories-list@mta.ca; Tue, 01 Nov 2005 20:06:45 -0400 Message-Id: <200511012143.jA1LhbK12967@math-cl-n03.ucr.edu> Subject: categories: 2007 Fields program: postdoctoral positions To: categories@mta.ca (categories) Date: Tue, 1 Nov 2005 13:43:37 -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: 3 Some of you may be interested in this.... .................................................................... Postdoctoral positions: Fields Institute program "Geometric Applications of Homotopy Theory", January-June, 2007. Postdoctoral positions will be available in connection with this research program. *The application deadline is December 15, 2005.* The program will develop new applications of homotopy theory in algebraic geometry, number theory and mathematical physics, and will include subprograms on the homotopy theory of schemes, stacks in geometry and topology, and higher categories and their applications. Further information about the program and the postdoc application procedure is available at the program web site: http://www.fields.utoronto.ca/programs/scientific/06-07/homotopy/ Rick Jardine, Lead Organizer. From rrosebru@mta.ca Wed Nov 2 11:58:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 02 Nov 2005 11:58:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXKwY-0004Mg-EG for categories-list@mta.ca; Wed, 02 Nov 2005 11:54:30 -0400 Message-ID: <2cc0d36c0511012253p1a15630ay1c0eb35f724905db@mail.gmail.com> Date: Wed, 2 Nov 2005 03:53:43 -0300 From: Peter Arndt To: categories@mta.ca Subject: categories: Two topos questions 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: 4 Hi, category theorists, 1. In a message to the categories list from 15. jan.1997 (that message can be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere talks about "the ... internal topos ... which parametrizes the decidable K-finites". Does anyone know what exactly is that internal topos? Is there some morphism that can be seen as the indexed family of decidable K-finites (just like the generic cardinal "is" the indexed family of finite cardinals and can be used to construct the full internal subcategory of finite cardinals)? 2. An object Y of a topos is said to have locally a property P if there is an object Z with global support such that Z*(Y) has the property P. For the topos of sheaves on a T1-space X (and a property P stable under pullback along subterminals), I convinced myself that this implies the existence of = a covering of X, such that P holds on the restriction of Y to each open set o= f the covering. Can this also be proved for schemes or other classes of topological spaces, maybe with additional conditions on P? Thanks a lot! Peter From rrosebru@mta.ca Wed Nov 2 20:29:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 02 Nov 2005 20:29:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXSqG-0005yD-Ve for categories-list@mta.ca; Wed, 02 Nov 2005 20:20:33 -0400 Date: Wed, 2 Nov 2005 21:22:46 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Two topos questions In-Reply-To: <2cc0d36c0511012253p1a15630ay1c0eb35f724905db@mail.gmail.com> 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: 5 On Wed, 2 Nov 2005, Peter Arndt wrote: > Hi, category theorists, > 1. In a message to the categories list from 15. jan.1997 (that message can > be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere > talks about "the ... internal topos ... which parametrizes the decidable > K-finites". Does anyone know what exactly is that internal topos? Is there > some morphism that can be seen as the indexed family of decidable K-finites > (just like the generic cardinal "is" the indexed family of finite cardinals > and can be used to construct the full internal subcategory of finite > cardinals)? I can't remember exactly what Bill was talking about in that posting. However, there is no hope of `parametrizing' decidable K-finite objects by an internal category, unless the ambient topos has a natural number object (cf. the remarks on pp. 1058-9 of "Sketches of an Elephant"), and if it does the decidable K-finites are exactly the objects locally isomorphic to finite cardinals. So I suspect that he was referring to the internal category of finite cardinals. > 2. An object Y of a topos is said to have locally a property P if there is > an object Z with global support such that Z*(Y) has the property P. For the > topos of sheaves on a T1-space X (and a property P stable under pullback > along subterminals), I convinced myself that this implies the existence of a > covering of X, such that P holds on the restriction of Y to each open set of > the covering. Can this also be proved for schemes or other classes of > topological spaces, maybe with additional conditions on P? Yes, of course -- this is exactly the geometric intuition behind this use of "locally". One needs to assume that P is stable under arbitrary pullback (which will certainly be the case if it's expressible in the internal language of a topos). Then, in any topos generated by subterminals (in particular, in any topos of sheaves on a space), every cover Z -->> 1 is dominated by one of the form \coprod_i U_i -->> 1, where the U_i are a family of subterminals covering 1 in the classical sense. So P holds locally for Y iff it holds for the restriction of Y to each member of some cover in the classical sense. Peter Johnstone From rrosebru@mta.ca Fri Nov 4 07:25:42 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 04 Nov 2005 07:25:42 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXzYh-0001fW-9U for categories-list@mta.ca; Fri, 04 Nov 2005 07:16:35 -0400 To: categories@mta.ca Subject: categories: Re: Two topos questions Date: Thu, 03 Nov 2005 11:18:38 -0500 From: wlawvere@buffalo.edu Content-Type: text/plain Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 What I was talking about 15 Jan 1997 was (not hoping for an axiom of infinity without assuming one, but) the fact that most of the mathematical uses of the rig N of natural numbers do not work in a topos, if one interprets that rig to mean the one characterized by Dedekind recursion. 1. starting with characteristic functions of subobjects, then adding and multiplying them for various combinatorial calculations 2. applying the least number principle 3. measuring the fiber dimension of a bundle of linear spaces all require the inf-completion of N, also known as the semicontinuous natural numbers. It contains the truth-value object omega and is contained in the semicontinuous reals (themselves indispensible for norming internal Banach spaces, and constructible simply as one-sided Dedekind cuts). Yet another way to picture these objects in the case of a Grothendieck topos E is to consider the sheaf of germs of continuous maps from E to the appropriate locale : the order topology (not the discrete one) on N, the order topology (not the interval topology) on nonnegative reals. Is any more known now as opposed to 9 years ago about the mathematical applications of finiteness to variable and cohesive sets ? The fact that K-finiteness is appropriate for some applications and that its theory resembles the classical theory for constant discrete sets should not distract us from the achievements of geometers in using coherence, Notherianness,etc., nor from the fact that our "logic" should serve to partly guide the learning of also those developments of thought. From rrosebru@mta.ca Fri Nov 4 07:25:42 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 04 Nov 2005 07:25:42 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXzak-0001hU-MI for categories-list@mta.ca; Fri, 04 Nov 2005 07:18:42 -0400 Date: Thu, 3 Nov 2005 21:19:49 +0100 From: Hans-Wolfgang Loidl To: appsem05-announce@tcs.ifi.lmu.de Subject: categories: CfP: Applied Semantics, Special Issue of J of TCS 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: 7 [ -- apologies for multiple copies of this CfP -- HWL ] CALL FOR PAPERS Special Issue on APPLIED SEMANTICS of the Journal of Theoretical Computer Science http://lionel.tcs.ifi.lmu.de/APPSEM05/journal_call.php We invite the submission of full papers on the topic of Applied Sema= ntics, as described below, for publication in a special issue of the Journal of T= heoretical Computer Science (TCS). Papers should be revised versions of those papers= submitted to and presented at the APPSEM05 Workshop, Frauenchiemsee, Germany, Septem= ber 12-15. However, we will consider submissions of papers not presented there, pro= vided they fall into the scope of the call and clearly work out a novel contributi= on to the field that wasn't mature enough to be presented at the aforementioned works= hop. Programming languages are the basic tools with which all applications of= computers are built. It is important, therefore, that they should be well designe= d and well implemented. Achieving these goals requires both a good theoretical unders= tanding of programming language designs, and practical skills in the development of hi= gh quality compilers. This special issue will cover all these areas and will focus on = the formal basis for programming languages. The general areas covered by this special issue are as follows: 1. Program structuring: object-oriented programming, modules, 2. Proof assistants, functional programming, and dependent types, 3. Program analysis, generation, and configuration, 4. Specification and verification methods, 5. Types and type inference in programming, 6. Games, sequentiality, and abstract machines, 7. Semantic methods for distributed computing, 8. Resource models and web data, 9. Continuous phenomena in Computer Science. 10. Industrial applications. This list is non-exclusive, but contributions that do not clearly fall into= one of these topics should carefully work out their relationship. We particularly invite industrial contributions covering the areas above. = This could mean development of a novel language, a novel compiler, program analysis= tools, or indeed, just a semantic model for a new kind of application. Again, this l= ist is not exclusive and we welcome papers on any kind of industrial work which is i= nformed by the science of programming languages, clearly states the problem being = solved and elaborates on the main techniques of the above research areas being used to= solve it. Programme committee: . Gavin Bierman, Microsoft Research . Olivier Danvy, University of Aarhus . Peter Dybjer, Chalmers University of Technology . Martin Hofmann, Ludwig-Maximilians-Universit=E4t M=FCnchen (Chair) . Neil Jones, University of Copenhagen . Hans Wolfgang Loidl, Ludwig-Maximilians-Universit=E4t M=FCnchen . Peter O'Hearn, Queen Mary College, University of London . Uday Reddy, University of Birmingham . Didier Remy, INRIA Rocquencourt . Ian Stark, University of Edinburgh . Thomas Streicher, Technische Universit=E4t Darmstadt . Peter Thiemann, Universit=E4t Freiburg Format of submission (see links below):=20 Papers should be formatted according to Elsevier's elsart document style= , used for articles in the Journal of Theoretical Computer Science. Submission = should be electronically in .pdf format via the APPSEM Workshop page. Papers should = have 20-25 pages, including appendices. Papers exceeding the upper bound may be reject= ed without refereeing. Important dates: . Paper submission: 8.1.2006 . Notification: 27.2.2006 . Camera-ready copy: 27.3.2006 Links: . APPSEM05 page and paper submissions: http://lionel.tcs.ifi.lmu.de/APPSEM0= 5/ . TCS page: http://www.elsevier.com/wps/find/journaldescription.cws_home/50= 5625/description . Document style (elsart.cls): http://authors.elsevier.com/latex From rrosebru@mta.ca Sat Nov 5 16:03:16 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 05 Nov 2005 16:03:16 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EYU2Z-0005Jo-CZ for categories-list@mta.ca; Sat, 05 Nov 2005 15:49:27 -0400 Message-ID: <6468629.1131158050664.JavaMail.teamon@b104.teamon.com> Date: Fri, 4 Nov 2005 18:34:10 -0800 (PST) From: flinton@flinton.mail.wesleyan.edu To: Categories@mta.ca Subject: categories: Seeking Osvaldo Acuna-Ortega Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 Greetings. I'm hoping to make Osvaldo Acuna-Ortega aware, either directly or through the intervention of some kind reader, that in 5-6 days or so I'll be in Costa Rica for about a week, including San Jose. -- Fred From rrosebru@mta.ca Sun Nov 6 14:31:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 06 Nov 2005 14:31:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EYpD2-0005t5-En for categories-list@mta.ca; Sun, 06 Nov 2005 14:25:40 -0400 Message-ID: <436B526E.8040806@inf.u-szeged.hu> Date: Fri, 04 Nov 2005 13:22:06 +0100 From: Zoltan Esik 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: Ackermann Award 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: 9 ------------------------------------------------------------------- AA06 ------------------------------------------------------------------- Ackermann Award 2006 --------------------- Nominations are solicited for the Ackermann Award 2006. The EACSL Outstanding Dissertation Award for Logic in Computer Science (The Ackermann Award) will be presented to the recipients at the annual conference of the EACSL (CSL'06). The jury is entitled to give more than one award per year. The first Ackermann Award was presented at CSL'05. The 2005 recipients were Mikolaj Bojanczyk Konstantin Korovin Nathan Segerlind Eligible for the 2006 Ackermann Award are PhD dissertations in topics specified by the EACSL and LICS conferences, which were formally accepted as PhD theses at a university or equivalent institution between 1.1.2004 and 31.12. 2005. ------------------------------------------ The deadline for submission is 31.1.2006. ------------------------------------------ Submission details are available at www.dimi.uniud.it/~eacsl/award.html www.cs.technion.ac.il/eacsl The award consists of * a diploma, * an invitation to present the thesis at the CSL conference, * the publication of the abstract of the thesis and the laudatio in the CSL proceedings, * travel support to attend the conference. The jury consists of seven members: * The president of EACSL, J. Makowsky (Haifa); * The vice-president of EACSL, D. Niwinski (Warsaw); * One member of the LICS organizing committee, S. Abramnsky (Oxford); * B. Courcelle (Bordeaux); * E. Graedel (Aachen); * M. Hyland (Cambridge); * A. Razborov (Moscow and Princeton). From rrosebru@mta.ca Sun Nov 6 14:31:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 06 Nov 2005 14:31:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EYpDy-0005vZ-Or for categories-list@mta.ca; Sun, 06 Nov 2005 14:26:38 -0400 Date: Fri, 04 Nov 2005 14:31:32 +0100 From: Computer Science Logic '06 Conference User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: CSL'06 CALL FOR PAPERS 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: 10 ********************************************************************** * CSL'06 * * Annual Conference of the European Association for * * Computer Science Logic * * September 25 -- 29, 2006, Szeged, Hungary * * http://www.inf.u-szeged.hu/~csl06/ * * PRELIMINARY CALL FOR PAPERS * ********************************************************************** Computer Science Logic (CSL) is the annual conference of the European Association for Computer Science Logic (EACSL). The conference is intended for computer scientists whose research activities involve logic, as well as for logicians working on issues significant for computer science. CSL'06, the 15th annual EACSL conference will be organized by the Institute of Informatics, University of Szeged. Suggested topics of interest include: automated deduction and interactive theorem proving, constructive mathematics and type theory, equational logic and term rewriting, automata and formal logics, modal and temporal logic, model checking, logical aspects of computational complexity, finite model theory, computational proof theory, logic programming and constraints, lambda calculus and combinatory logic, categorical logic and topological semantics, domain theory, database theory, specification, extraction and transformation of programs, logical foundations of programming paradigms, verification of security protocols, linear logic, higher-order logic, nonmonotonic reasoning, logics and type systems for biology. Programme Committee: Invited speakers: Krzysztof Apt(Amsterdam/Singapore) Martin Escardo (Birgmingham) Matthias Baaz (Vienna) Paul-Andre Mellies (Paris) Michael Benedikt (Chicago) Luke Ong (Oxford) Pierre-Louis Curien (Paris) Luc Segoufin (Orsay) Rocco De Nicola (Florence) Miroslaw Truszczynski(Lexington,KY) Zoltan Esik (Szeged, chair) Dov Gabbay (London) Fabio Gadducci (Pisa) Organizing Committee: Neil Immerman (Amherst) Michael Kaminski (Haifa) Bakhadyr Khoussainov (Auckland) Zoltan Esik (Szeged, co-chair) Ulrich Kohlenbach (Darmstadt) Zsolt Gazdag (Szeged) Marius Minea (Timisoara) Eva Gombas (Szeged, co-chair) Damian Niwinski (Warsaw) Szabolcs Ivan (Szeged) R. Ramanujam (Chennai) Zsolt Kakuk (Szeged) Philip Scott (Ottawa) Zoltan L. Nemeth (Szeged) Philippe Schnoebelen (Cachan) Sandor Vagvolgyi Alex Simpson (Edinburgh) (Szeged, workshop-chair) It is anticipated that the proceedings will be published in the LNCS series. Each paper accepted by the Programme Committee must be presented at the conference by one of the authors, and final copy prepared according to Springer's guidelines. Submitted papers must be in Springer's LNCS style and of no more than 15 pages, presenting work not previously published. They must not be submitted concurrently to another conference with refereed proceedings. The PC chair should be informed of closely related work submitted to a conference or journal by 1 April, 2006. Papers authored or coauthored by members of the Programme Committee are not allowed. Submitted papers must be in English and provide sufficient detail to allow the Programme Committee to assess the merits of the paper. Full proofs may appear in a technical appendix which will be read at the reviewer's discretion. The title page must contain: title and author(s), physical and e-mail addresses, identification of the corresponding author, an abstract of no more than 200 words, and a list of keywords. The submission deadline is in two stages. Titles and abstracts must be submitted by 24 April, 2006 and full papers by 1 May, 2006. Notifications of acceptance will be sent by 12 June, 2006, and final versions are due 3 July, 2006. A submission server will be available from 1 April, 2006. The Ackermann Award for 2006 will be presented to the recipients at CSL'06. Important Dates: Submission - title & abstract: 24 April, 2006 - full paper: 1 May, 2006 Notification: 12 June, 2006 Final papers: 3 July, 2006 Conference address: CSL'06 c/o Prof. Zoltan Esik Institute of Informatics, University of Szeged H-6701, Szeged, P.O.B. 652, Hungary web site: http://www.inf.u-szeged.hu/~csl06/ e-mail: csl06@inf.u-szeged.hu phone: +36-62-544-289 or +36-62-544-205 fax: +36-62-544-895 or +36-62-546-397 ********************************************************************** ********************************************************************** From rrosebru@mta.ca Tue Nov 8 20:26:43 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 08 Nov 2005 20:26:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EZdg7-0002DV-Gb for categories-list@mta.ca; Tue, 08 Nov 2005 20:19:03 -0400 Subject: categories: Temporary job in Glasgow From: Tom Leinster To: categories@mta.ca Date: Tue, 08 Nov 2005 14:30:29 +0000 Mime-Version: 1.0 X-Mailer: Evolution 2.2.1 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 11 There's a six-month lectureship in Glasgow being advertised. It's for Jan-June next year, and the closing date is soon (18 Nov). See http://www.jobs.ac.uk/jobfiles/PX792.html We'll be continuing to hire new permanent staff over the next few years, so if you're interested, this may be a good way of selling yourself. Tom -- Tom Leinster From rrosebru@mta.ca Tue Nov 8 20:26:44 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 08 Nov 2005 20:26:44 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EZdfW-0002Ba-UY for categories-list@mta.ca; Tue, 08 Nov 2005 20:18:26 -0400 Date: Tue, 8 Nov 2005 13:47:36 +0100 From: Ralf Treinen To: categories@mta.ca Subject: categories: RTA'06: 1st Call for Papers Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 ************************************ * * * RTA'06 FIRST CALL FOR PAPERS * * * ************************************ http://rta06.csl.sri.com/ Seattle, WA, USA August 12-14, 2006 Affiliated workshops Aug 11 & 15 The 17th International Conference on Rewriting Techniques and Applications (RTA'06) is organized as part of the Federated Logic Conference (FLoC), collocated with CAV, ICLP, IJCAR, LICS, SAT, and several affiliated workshops. IMPORTANT DATES: Feb 15, 2006: Deadline for electronic submission of title and abstract Feb 22, 2006: Deadline for electronic submission of papers May 01, 2006: Notification of acceptance of papers Jun 01, 2006: Deadline for final versions of accepted papers RTA is the major forum for the presentation of research on all aspects of rewriting. Typical areas of interest include (but are not limited to): * Applications: case studies; rule-based (functional and logic) programming; symbolic and algebraic computation; theorem proving; system synthesis and verification; proof checking; reasoning about programming languages and logics; * Foundations: matching and unification; narrowing; completion techniques; strategies; constraint solving; explicit substitutions; tree automata; termination; * Frameworks: string, term, graph, and proof rewriting; lambda-calculus and higher-order rewriting; proof nets; constrained rewriting/deduction; categorical and infinitary rewriting; * Implementation: compilation techniques; parallel execution; rewrite tools; termination checking; * Semantics: equational logic; rewriting logic. The following workshops are affiliated with RTA'06: * HOR'06: 3rd International Workshop on Higher-Order Rewriting * RULE'06: 7th International Workshop on Rule-Based Programming * UNIF'06: 20th International Workshop on Unification * WG1.6: Annual meeting of the IFIP Working Group 1.6 on Term Rewriting. * WRS'06: 6th International Workshop on Reduction Strategies in Rewriting and Programming * WST'06: 8th International Workshop on Termination Please refer to the RTA'06 web site for further information on the workshops. INVITED SPEAKERS: Randy Bryant will be the joint plenary speaker of LICS, RTA and SAT. More RTA invited speakers will be announced later. BEST PAPER AWARDS AND TRAVEL GRANTS: An award is given to the best paper or papers as decided by the program committee. A limited number of travel grants may be available for students who are (co-)authors of RTA-papers. To apply for grants, students should send an e-mail to the PC chair together with their submission. RTA'06 PROGRAM COMMITTEE CHAIR: * Frank Pfenning, Carnegie Mellon University RTA'06 PROGRAM COMMITTEE: * Zena Ariola, University of Oregon * Franz Baader, Technical University Dresden * Gilles Dowek, Ecole Polytechnique and INRIA * Guillem Godoy, Technical University of Catalonia * Deepak Kapur, University of New Mexico * Delia Kesner, University Paris 7 * Denis Lugiez, University of Provence * Claude Marche, University Paris-Sud * Jose Meseguer, University of Illinois at Urbana-Champaign * Frank Pfenning, Carnegie Mellon University (Chair) * Ashish Tiwari, SRI International * Yoshihito Toyama, Tohoku University * Eelco Visser, Utrecht University * Hans Zantema, Eindhoven University of Technology RTA'06 CONFERENCE CHAIR: * Ashish Tiwari, SRI International RTA'06 SUBMISSIONS: Submissions must be original and not submitted for publication elsewhere. Submission categories include regular research papers and system descriptions. Problem sets and submissions describing interesting applications of rewriting techniques are also welcome. The page limit for submissions is 15 pages in Springer Verlag LNCS style (10 pages for system descriptions). Please consult http://rta06.csl.sri.com/ for further instructions. LOCATION, TRAVEL, ACCOMMODATION, AND REGISTRATION: RTA'06 will be part of the 2006 Federated Logic Conference (FLoC 2006) which will be held August 10-22, 2006, at the Seattle Sheraton Hotel and Towers, in Seattle, Washington state, USA. Further information will be made available at the FLoC 2006 home page http://research.microsoft.com/floc06/index.htm. From rrosebru@mta.ca Tue Nov 8 20:26:44 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 08 Nov 2005 20:26:44 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EZde8-00027U-FN for categories-list@mta.ca; Tue, 08 Nov 2005 20:17:00 -0400 Date: Mon, 07 Nov 2005 10:14:35 +0100 From: Jonathan Scott To: categories@mta.ca Subject: categories: job: post-doctoral positions at EPFL Content-Type: text/plain; charset=3DISO-8859-1; format=3Dflowed Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 13 The Institute of Geometry, Algebra and Topology (IGAT) of the Ecole Polytechnique F=E9d=E9rale de Lausanne (EPFL) invites applications for full-time postdoctoral positions from 1 October 2006 through 30 September 2007, with possibility of extension for a second year. In addition to research, duties include teaching within the framework of the Mathematics Section of the EPFL. Candidates must have completed their PhD within the last four years and have shown promise of excellence in research in geometry, algebra or topology. Applications, including curriculum vitae, publication list, research plan, statement of teaching experience, and two references, must be submitted by 15 January 2005 to Prof. Kathryn Hess, EPFL SB-IGAT, B=E2timent BCH, CH-1015 Lausanne, Switzerland. For further information concerning the IGAT, see http://igat.epfl.ch/ igat/. ------------------------------------------------------------------------ ---------------------- L'Institut de g=E9om=E9trie, alg=E8bre et topologie (IGAT) de l'Ecole Polytechnique F=E9d=E9rale de Lausanne (EPFL) sollicite des candidatures pour un poste =E0 plein temps de collaborateur scientifique du 1er octobre 2006 au 30 septembre 2007, avec possibilit=E9 d'extension. La charge principale du titulaire de ce poste sera l'enseignement d'un compl=E9ment acad=E9mique de math=E9matiques pour les =E9tudiants de la Ha= ute Ecole P=E9dagogique, futurs enseignants au niveau primaire et secondaire. Tout candidat =E0 ce poste doit =EAtre un chercheur actif, ayant un doctorat en math=E9matiques, de pr=E9f=E9rence en alg=E8bre, g=E9om=E9trie = ou topologie. Par ailleurs, une ma=EEtrise parfaite de la langue fran=E7aise est essentielle. Les candidatures doivent comprendre: -- curriculum vitae; -- description de la vision de l'enseignement du candidat; -- plan de recherche; -- liste de publications; et -- trois r=E9f=E9rences, dont l'une concernant l'enseignement. Toute candidature =E0 ce poste doit =EAtre soumise au plus tard le 15 janvier 2006 =E0 la Prof. Kathryn Hess, EPFL SB-IGAT, B=E2timent BCH, CH-1015 Lausanne, Suisse. Pour plus d'informations concernant l'IGAT, veuillez visiter la page http://igat.epfl.ch/igat. From rrosebru@mta.ca Sat Nov 12 11:39:36 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 12 Nov 2005 11:39:36 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EaxJz-0007Cs-Rk for categories-list@mta.ca; Sat, 12 Nov 2005 11:29:39 -0400 Message-ID: <011d01c5e77f$0ef00140$d78a4c51@brown1> From: "Ronald Brown" To: categories Subject: categories: Question on rewriting and program specification Date: Sat, 12 Nov 2005 11:48:53 -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: 14 I would be grateful for advice and background on the following question or issues, on which I do not know the computer science background. I have read that rewriting for monoid presentations is relevant to program specification. Now at Bangor we have been involved with computing induced actions of monoids (and categories) (with Anne Heyworth), `Using rewriting systems to compute left Kan extensions and induced actions of categories', J. Symbolic Computation 29 (2000) 5-31. where there is defined the notion of presentation for the data of a Kan extension (= induced action). -------------------------------------- Question: is it reasonable to say that rewriting for such a presentation is relevant to program specification in which information is also given on the input to the program? ----------------------------------------- The intuitive idea is that a program may be very complex, even undecidable, or inconsistent, but if the input is trivial, or simple, then the output may be decidable, and simple. Perhaps there are commercial examples of this, where most users give simple inputs? Recall that we get induced actions for monoids when given a morphism u: M --> N of monoids, an action of M on X, and so get an action of N on a set u_*(X). A presentation for this involves a presentation for N, and generators A for M, and the values of these generators in terms of the presentation of N. If these generators act trivially on X, and u(A) generates N, then the induced action is on X again, with trivial action of N. A recent application of these ideas of induced actions of categories is for computing double cosets (math.CO/0508391 with Neil Ghani, Anne Heyworth, Chris Wensley, JSC to appear). Ronnie Brown www.bangor.ac.uk/r.brown www.popmath.org.uk From rrosebru@mta.ca Sun Nov 13 09:25:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 13 Nov 2005 09:25:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EbHms-0001bK-Ej for categories-list@mta.ca; Sun, 13 Nov 2005 09:20:50 -0400 Date: Sat, 12 Nov 2005 22:23:13 -0500 From: Jacques Carette Organization: McMaster University To: categories Subject: categories: Re: Question on rewriting and program specification References: <011d01c5e77f$0ef00140$d78a4c51@brown1> In-Reply-To: <011d01c5e77f$0ef00140$d78a4c51@brown1> 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: 15 There is one part of your questions that I can comment on: Ronald Brown wrote: >The intuitive idea is that a program may be very complex, even undecidable, >or inconsistent, but if the input is trivial, or simple, then the output may >be decidable, and simple. Perhaps there are commercial examples of this, >where most users give simple inputs? > > Consider a Computer Algebra System (ie like Maple or Mathematica). Almost everything interesting it does in the area of Analysis is formally undecidable. This is because zero-equivalence for any interesting subset of the (constructive) reals is undecidable, and almost all CAS algorithms to do analysis (integration, solving equations, limits, simplification, etc) requires many zero-equivalence tests, amongst many undecidable problems. Nevertheless, these systems are very powerful, and frequently return interesting answers to even fairly complex user input. This is because most users give ``structured'' input, for which semi-decision procedures seem to be quite adequate. Of course, if you happen to know exactly which semi-decision procedures are being used, you can fool them and get the system to either go into an infinite loop or give a wrong answer. But that requires a huge amount of knowledge to do so. The average user would be unable to manufacture such examples. It is very important to note that the distinction is between ``structured'' input and ``generic'' input, rather than between simple and complex. In other words, what seems to matter most is the Kolmogorov Complexity (or in the specification case, the logical succinctness) of an input. [See http://www.cas.mcmaster.ca/~carette/publications/simplification.pdf for one approach to this]. There are similar examples in the automated theorem proving world, where in particular PVS and IMPS can frequently prove theorems which are not a priori known to be in a decidable subclass. Here again, a combination of semi-decision procedures driven by intelligent heuristics seems to be highly effective. Jacques From rrosebru@mta.ca Mon Nov 14 16:39:04 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 14 Nov 2005 16:39:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ebkwq-00021x-DC for categories-list@mta.ca; Mon, 14 Nov 2005 16:29:04 -0400 Date: Mon, 14 Nov 2005 12:11:59 +0000 From: cie06@swansea.ac.uk To: categories@mta.ca Subject: categories: CiE06: 2nd Call for Papers Message-ID: <43787F0F.mailO0B1IJU5M@swansea.ac.uk> User-Agent: nail 11.4 8/29/04 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: 16 [Apologies for the inevitable multiple copies of this announcement] CiE 2006 Computability in Europe 2006 : Logical Approaches to Computational Barriers 30 June - 5 July 2006 Swansea University http://www.cs.swansea.ac.uk/cie06/ 2nd CALL FOR PAPERS Deadline: JANUARY 6, 2006 CiE 2006 is the second of a new conference series on Computability Theory and related topics which started in Amsterdam in 2005. CiE 2006 will focus on (but not be limited to) logical approaches to computational barriers: - practical and feasible barriers, e.g., centred around the P vs. NP problem; - computable barriers connected to models of computers and programming languages; - hypercomputable barriers related to physical systems. Tutorials: Samuel R. Buss (San Diego, CA) Julia Kempe (Paris) Invited Speakers: Jan Bergstra (Amsterdam) Luca Cardelli (Microsoft Cambridge) Martin Davis (New York, NY) John W Dawson (York, PA) (Special Address on Kurt Goedel) Jan Krajicek (Prague) Elvira Mayordomo Camara (Zaragoza) Istvan Nemeti (Budapest) Helmut Schwichtenberg (Munich) Andreas Weiermann (Utrecht) Special Sessions: Proofs and Computation Computable Analysis Challenges in Complexity Foundations of Programming Mathematical Models of Computers and Hypercomputers Goedel Centenary: His Legacy for Computability The Programme Committee cordially invites all researchers (European and non-European) in the area of Computability Theory to submit their papers (in PDF-format, max 10 pages) for presentation at CiE 2006. We particularly invite papers that build bridges between different parts of the research community. Since women are underrepresented in mathematics and computer science, we emphatically encourage submissions by female authors. The proceedings will be published within Springer's LNCS series. To submit a paper and for more information on the submission process, go to our web site http://www.cs.swansea.ac.uk/cie06/ Important dates: Submission Deadline: January 6th, 2006. Notification of Authors: March 3rd, 2006. Deadline for Final Version: March 24th, 2006. (Please note that the dates have changed.) Programme Committee: Samson Abramsky (Oxford) Klaus Ambos-Spies (Heidelberg) Arnold Beckmann (Swansea, co-chair) Ulrich Berger (Swansea) Olivier Bournez (Nancy) Barry Cooper (Leeds) Laura Crosilla (Firenze) Costas Dimitracopoulos (Athens) Abbas Edalat (London) Fernando Ferreira (Lisbon) Ricard Gavalda (Barcelona) Giuseppe Longo (Paris) Benedikt Loewe (Amsterdam) Yuri Matiyasevich (St.Petersburg) Dag Normann (Oslo) Giovanni Sambin (Padova) Uwe Schoening (Ulm) Andrea Sorbi (Siena) Ivan Soskov (Sofia) Leen Torenvliet (Amsterdam) John Tucker (Swansea, co-chair) Peter van Emde Boas (Amsterdam) Klaus Weihrauch (Hagen) Sponsors: Financial support: British Logic Colloquium (BLC) Engineering and Physical Sciences Research Council (EPSRC) Kurt Goedel Society (KGS) London Mathematical Society (LMS) Welsh Development Agency (WDA) Other sponsors: Association for Symbolic Logic (ASL) European Association for Theoretical Computer Science (EATCS) British Computer Society (BCS) IT Wales Grants: Some UK student grants, funded by the EPSRC, are available. A limited number of UK student grants and former Soviet Union grants, funded by the LMS, is available. Registered students, who are members of the ASL, may also apply for ASL travel funds. For more information on the conference series, please check the CiE conference series http://www.illc.uva.nl/CiE/ and our web page http://www.cs.swansea.ac.uk/cie06/. From rrosebru@mta.ca Mon Nov 14 16:39:04 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 14 Nov 2005 16:39:04 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EbkwK-0001yH-8K for categories-list@mta.ca; Mon, 14 Nov 2005 16:28:32 -0400 Message-ID: <43778201.5060703@andrej.com> Date: Sun, 13 Nov 2005 19:12:17 +0100 From: Andrej Bauer To: categories Subject: categories: Re: Question on rewriting and program specification Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 Jacques Carette wrote: > Of course, if you happen to know exactly > which semi-decision procedures are being used, you can fool them and get > the system to either go into an infinite loop or give a wrong answer. > But that requires a huge amount of knowledge to do so. The average user > would be unable to manufacture such examples. I have a rather unfortunate experience with "average users" who get wrong answers from CAS's, namely our undergraduate math majors. In their first-year analysis course they learn how to compute limits. Invariably, they are given some limits which Mathematica gets wrong, e.g.: Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]), x -> 0] The answer is 0, _except_ when the parameter a equals e, in which case the answer is 6. Yes, this is a nasty limit pulled out of a hat, but it is precisely the sort of thing we test our students on. It is rather disappointing that Mathematica falls into exactly the same sort of trap as the average student. Another example is the use of l'Hospital rule, which is used by every CAS. There is a side condition which is not checked by them, which makes them give wrong answers. (The side condition is very nasty to check, namely, whether the zero of a derivative is isolated.) The situation is even worse when engineers and physicsts use CAS. They trust them blindly (I suspect). One day they're going to build a nuclear power plant based on a faulty limit computed by Mathematica or Maple. Andrej Bauer From rrosebru@mta.ca Wed Nov 16 17:11:01 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 16 Nov 2005 17:11:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcURb-0005JO-DU for categories-list@mta.ca; Wed, 16 Nov 2005 17:03:51 -0400 Date: Mon, 14 Nov 2005 17:15:59 -0500 From: Jacques Carette MIME-Version: 1.0 To: categories Subject: categories: Computer Algebra weaknesses References: <43778201.5060703@andrej.com> In-Reply-To: <43778201.5060703@andrej.com> 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: 18 I should have been more precise when I spoke of "decision procedures" in the context of a CAS. There are many semi-decision procedures for non-parametric problems. For parametric problems, these procedures break down. The issue is almost always the same: a theorem is applied without properly checking side-conditions. Frequently, the issue is one of zero recognition in the presence of parameters, but there can be more complex situations, such as the one Andrej points out. CASes are typically very good at getting good answers at non-parametric problems, and my comments were aimed mostly at that class. Computer Algebra implementers (and I used to be one) are typically very afraid of combinatorial explosions, as a lot of computer algebra people are ``algorithms'' people. Their mathematics background was often such that they prefered a ``generic'' answer quickly than a thorough answer that was a) a mess, and b) slow to get. This is, in part, why I am now in academia, as I want to ``fix'' that. Andrej Bauer wrote: >I have a rather unfortunate experience with "average users" who get >wrong answers from CAS's, namely our undergraduate math majors. In their >first-year analysis course they learn how to compute limits. Invariably, >they are given some limits which Mathematica gets wrong, e.g.: > >Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]), > x -> 0] > >The answer is 0, _except_ when the parameter a equals e, in which case >the answer is 6. Yes, this is a nasty limit pulled out of a hat, but it >is precisely the sort of thing we test our students on. It is rather >disappointing that Mathematica falls into exactly the same sort of trap >as the average student. > > This is exactly the kind of parameter specialization problem where CAS designers have ``chosen'' to ignore and return a generic answer. This has been documented since at least 1991 in a nice paper in the Bulletin of the AMS. >Another example is the use of l'Hospital rule, which is used by every >CAS. There is a side condition which is not checked by them, which makes >them give wrong answers. (The side condition is very nasty to check, >namely, whether the zero of a derivative is isolated.) > > I don't know about Mathematica, but Maple does NOT use l'Hospital's rule. Almost all limits are computed using one-sided asymptotic series expansions, as l'Hospital's rule is just not very suitable to large-scale computations. If interested, I can provide various published references to the algorithms involved. Of course, one still needs to check that the leading term is non-zero. For the limit above, observe (in Maple): > assume(a::real, x::real); > f := ((1 + 4*x^2)^(1/4) - (1 + 5*x^2)^(1/5))/(a^(-x^2/2) - cos(x)): > normal(series(f, x=0,5)); 1 2 4 - ---------- x~ + O(x~ ) ln(a~) - 1 which clearly indicates that a=exp(1) is a problem point. That the limit is computed anyways is an instance of choosing the ``generic'' answer [whatever that means]. >The situation is even worse when engineers and physicsts use CAS. They >trust them blindly (I suspect). One day they're going to build a nuclear >power plant based on a faulty limit computed by Mathematica or Maple. > > Engineers and physicists don't use CAS - they use Matlab. The errors you get there are both worse and better: worse because numerical algorithms are so much more prone to giving (silent) nonsense, and better because Matlab cannot phrase any problems which are parametric! In Ontario (where I teach to Engineers), an Engineer who used either a CAS or Matlab in a computation for their safety critical system and did not check the correctness of the result, would be fully liable for any ensuing problems on their design. Thus, theoretically, Professional Engineers are supposed to given full justifications for the answers of the tools they use. In practice, they trust the tools blindly a bit too often. Again in Ontario, I have some knowledge of the process used for safety-critical software in nuclear power plants [I even have some grant money associated to that]. Tools like Matlab or Maple would not be allowed to give ``final'' answers, in any step of the process. But this is getting just a bit off-topic for the categories mailing-list... Jacques From rrosebru@mta.ca Wed Nov 16 17:11:01 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 16 Nov 2005 17:11:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcUPt-0005A3-EG for categories-list@mta.ca; Wed, 16 Nov 2005 17:02:05 -0400 Message-Id: <200511142119.jAELJaD28664@math-cl-n03.ucr.edu> Subject: categories: Schreier theory To: categories@mta.ca (categories) Date: Mon, 14 Nov 2005 13:19:36 -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: 19 Dear Categorists - This issue of my column has an introduction to nonabelian cohomology that some of you might enjoy. You'll have to skip past some other stuff to get to it. Best, jb ........................................................................ Also available as http://math.ucr.edu/home/baez/week223.html November 14, 2005 This Week's Finds in Mathematical Physics - Week 223 John Baez This week I'd like to talk about two aspects of higher gauge theory: p-form electromagnetism and nonabelian cohomology. Lurking behind both of these is the mathematics of n-categories, but I'll do my best to hide that until the end, to build up the suspense. But first, some cool pictures. Astronomy is booming these days, and it's a great way to see beautiful complexity emerging from simple laws in this wonderful universe of ours. So, I'd like the freedom to occasionally start This Week's Finds with some pictures from the skies. Think of it as an appetizer before the main course. Sometimes I'll explicitly relate these pictures to math and physics; other times not. Here's Saturn's moon Hyperion, photographed up close by the Cassini probe: 1) Cassini-Huyghens Mission, Hyperion: Odd World, http://saturn.jpl.nasa.gov/multimedia/images/image-details.cfm?imageID=1762 It seems to be a huge pile of rubble loosely held together by gravity and heavily cratered by meteor bombardments. Hyperion is interesting because it's the only known moon that tumbles chaotically on a short time scale, thanks to its eccentric shape and gravitational interactions with Saturn and Titan. This leads to some interesting math. We can think of Hyperion's angular momentum vector as a point on a sphere. If we started out knowing this point lay inside some small disk, time evolution would warp this disk into an ever more complicated region as time passed. This region would always have the same area, thanks to the wonders of symplectic geometry. But it would sprout ever more complicated tendrils, with its perimeter growing by a factor of e about every 100 days or so! That's chaos for you. Indeed, only quantum mechanics would stop the intricacy from growing forever, by blurring it out. After about 37 years, the area of a typical tendril would equal Planck's constant. At this point, classical mechanics would no longer be accurate. You'd really need to describe Hyperion's spin state using quantum theory: for example, a holomorphic section of some line bundle on the sphere. Well... at least you would if it weren't for decoherence caused by the interaction of Hyperion with its environment, for example solar radiation! For an explanation of how this changes the story, try: 2) Michael Berry, Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?), in Quantum Mechanics: Scientific Perspectives on Divine Action, CTNS Publications, Vatican Observatory, 2001. Also available at http://www.phy.bris.ac.uk/people/berry_mv/the_papers/berry337.pdf Here's another great picture: 3) The Hubble Heritage Project, Cat's Eye Nebula - NGC 6543, http://heritage.stsci.edu/2004/27/index.html This is a star about the size of the Sun, nearing the end of its life, emitting pulses of gas and dust. Astronomers call such a thing a "planetary nebula", though it has nothing to do with planets. It's in our galaxy, about 3000 light years from us. When it's done shedding its outer layers, all that's left of this star will be a white dwarf. Our own Sun will become a planetary nebula in about 6.9 billion years, after two separate stages of being a red giant - one as it runs out hydrogen, and one as it runs out of helium. When the helium is all gone, the Sun will start to pulsate every 100,000 years, ejecting more and more mass in each pulse, eventually throwing off all but the hot inner core made of heavier elements. The astronomer Bruce Balick has written eloquently on what this will mean for the Earth: Here on Earth, we'll feel the wind of the ejected gases sweeping past, slowly at first (a mere 5 miles per second!), and then picking up speed as the spasms continue - eventually to reach 1000 miles per second!! The remnant Sun will rise as a dot of intense light, no larger than Venus, more brilliant than 100 present Suns, and an intensely hot blue-white color hotter than any welder's torch. Light from the fiendish blue "pinprick" will braise the Earth and tear apart its surface molecules and atoms. A new but very thin "atmosphere" of free electrons will form as the Earth's surface turns to dust. So, don't keep procrastinating - enjoy life now! For other pictures of planetary nebulae, try Balick's webpage: 4) Bruce Balick, Hubble Space Telescope images of planetary nebulae, http://www.astro.washington.edu/balick/WFPC2/index.html For a timeline of the universe, including the future life of our Sun, try: 5) John Baez, A brief history of the universe, http://math.ucr.edu/home/baez/timeline.html Now... on to p-form electromagnetism! In ordinary electromagnetism, the secret star of the show turns out to be not the electromagnetic field but the "vector potential", A. At least locally, we can think of this as a 1-form. A 1-form is just a gadget that you can integrate along a path and get a number. In the case of the vector potential, this number describes the change in phase that a charged particle acquires as it moves along this path. The 1-form A gives rise to a 2-form F called the "electromagnetic field". A 2-form is a gadget you can integrate over a surface and get a number. Here's how we get F from A. Suppose we move a charged particle around a loop that's the boundary of some surface. Then the integral of F over this surface is defined to be the integral of A around the loop! We summarize this by saying that F is the "exterior derivative" of A, and writing F = dA. F is called the electromagnetic field because... that's what it is! It contains both the electric and magnetic fields in a single neat package. In 4d spacetime, the magnetic field describes the change in a phase of a charged particle that loops around a surface in the xy, yz or zx planes. The electric field describes the change in phase of a charged particle that loops around a surface in the xt, yt or zt planes. If you don't know this stuff, you're missing some of the best fun life has to offer. For an easy introduction with lots of gorgeous pictures, see: 6) Derek Wise, Electricity, magnetism and hypercubes, available at http://math.ucr.edu/~derek/talks/050916bw.pdf The idea of p-form electromagnetism is to replace point particles by strings or higher-dimensional membranes. To see how this goes, it's enough to look at 2-form electromagnetism. In 2-form electromagnetism, the star of the show is a 2-form, A. As already mentioned, a 2-form is a gadget you can integrate over a surface and get a number. In 2-form electromagnetism, this number describes the change in phase that a charged string acquires as it moves along, tracing out a surface in spacetime. The 2-form A gives rise to a 3-form, F. A 3-form is a gadget you can integrate over a 3-dimensional region and get a number. Suppose we move a charged string and let it trace out a surface that's the boundary of some 3-dimensional region. Then the integral of F over this region is defined to be the integral of A over the surface! Again we write this as: F = dA. So, we're just adding one to the dimensions of things. This makes it easy to keep on going. In fact, for any integer p, we can write down a generalization of Maxwell's equations. It goes like this. We start with a p-form A. We define a (p+1)-form F = dA This automatically implies some of Maxwell's equations: dF = 0 but the nontrivial Maxwell equations say that *d*F = J where * is the Hodge star operator and J is a p-form called the "current", which is produced by charged matter. What does this mean, physically? The idea is that we have charged matter consisting of (p-1)-dimensional membranes. These trace out p-dimensional surfaces in spacetime as time passes. The current J is a p-form that's concentrated on these surfaces. The current affects the A field in a manner governed by Maxwell's equations. Conversely, the A field affects the motion of the membranes. Classically, we just integrate the A field over the surface traced out by a membrane and add the result to the *action* for the membrane. In the path integral approach to quantum mechanics, this number gives a change in phase, as already mentioned. Maxwell's equations and their p-form generalization make sense when spacetime is any Lorentzian manifold. However, to get a theory where initial data determine a unique global solution, we want our spacetime to be "globally hyperbolic", which means that it has a "Cauchy surface": roughly, a spacelike surface that any sufficiently long timelike curve hits precisely once. To get a good *quantum* theory of p-form electromagnetism with a Hilbert space of states on which time evolution acts as unitary operators, we need more: our spacetime should be "stationary", meaning that it has time translation symmetry. Otherwise there's no way to define energy and the vacuum state - which is defined to be the least-energy state. My student Miguel Carrion-Alvarez tackled an important special case in his thesis, namely "static" globally hyperbolic spacetimes: 7) Miguel Carrion-Alvarez, Loop quantization versus Fock quantization of p-form electromagnetism on static spacetimes, available as math-ph/0412032. There's a lot of interesting analysis involved, especially when space (the Cauchy surface) is noncompact. When it's compact, we can use "Hodge's theorem" to relate its deRham cohomology to its topology, and this turns out to be crucial for understanding p-form electromagnetism - especially issues like the p-form Bohm-Aharonov effect. When it's noncompact we need something called "twisted L^2 cohomology" instead, and Miguel proved a generalization of Hodge's theorem for this. With the analysis under control, Miguel was able to set up a very beautiful approach to "loop quantum electromagnetism" and its p-form generalization. Here the idea is to write Maxwell's equations in terms of the integrals of A around all possible loops in space - or more generally, over all p-dimensional surfaces. People interested in loop quantum gravity should like this. As you can guess, either from seeing all the "d" operators or seeing all the buzzwords I'm throwing around, p-form electromagnetism is really just cohomology incarnated as physics! My student Derek Wise made this very precise for a version of the theory where spacetime is *discrete* - so-called "lattice p-form electromagnetism": 8) Derek Wise, Lattice p-form electromagnetism and chain field theory, available as gr-qc/0510033. Version with better graphics and related material at http://math.ucr.edu/~derek/pform/index.html In this paper, he shows lattice p-form electromagnetism is a "chain field theory": something like a topological quantum field theory, but where what matters is not spacetime itself so much as the cochain complex of differential forms *on* spacetime, equipped with just enough extra geometrical structure to write down the p-form version of Maxwell's equations. Both Miguel's thesis and Derek's papers are great if you want to learn lots of math and physics. I seem to attract students who enjoy explaining things. Speaking of which.... Next I want to explain some stuff Danny Stevenson told me at a mall in the little town of Cabazon while we were recovering from a hike in the desert followed by pancakes at the Wheel Inn - a roadside restaurant famous for its enormous statues of dinosaurs. Danny works on gerbes, stacks, and higher gauge theory. Last year we wrote a paper with Alissa Crans and Urs Schreiber constructing 2-groups (categorified groups) from the math of string theory - more precisely, from central extensions of loop groups. Since then I've been spending a lot of time writing a paper with Urs on higher gauge theory, where we set up a theory of parallel transport along surfaces. 2-form electromagnetism is the simplest case of this theory. Meanwhile, Danny has been thinking about connections on 2-vector bundles and their relation to the cohomology of Lie 2-algebras. This has led him to generalize Schreier theory in some interesting ways. So, let me tell you about Schreier theory! Schreier theory is a way to classify short exact sequences of groups. I'll say what I mean by that in a minute... but what makes Schreier theory special is that avoids some simplifying assumptions you might have seen if you've studied short exact sequences before. Normally people water down their short exact sequences by assuming some of the groups in question are *abelian*. This lets them use "cohomology theory" to do the classification. See "week210" for a nice book that takes this approach. This standard approach is great - I'm not knocking it - but Schreier theory is more general: it's really a branch of "nonabelian cohomology theory". It's not all that hard to explain, either. So, I'll explain it and then talk about various simplifying assumptions people make. The goal of Schreier theory is to classify short exact sequences of groups: 1 -> F -> E -> B -> 1 for a given choice of F and B. "Exact" means that the arrows stand for homomorphisms and the image of each arrow is the kernel of the next. Here this just means that F is a normal subgroup of E and B is the quotient group E/F. Such a short exact sequence is also called an "extension of B by F", since E is bigger than B and contains F. The simplest choice is to let E be the direct sum of F and B. Usually there are other more interesting extensions as well. To classify these, the trick is to use the analogy between group theory and topology. As I explained in "week213", you can think of a group as a watered-down version of a connected space with a chosen point. The reason is that given such a space, we get a group consisting of homotopy classes of loops based at the chosen point. This is called the "fundamental group" of our space. There's a lot more information in our space than this group. But pretty much anything you can do for groups, you can do for such spaces. It's usually harder, but it's completely analogous! In particular, classifying short exact sequences is a lot like classifying "fibrations": 1 -> F -> E -> B -> 1 where now the letters stand for connected spaces with a chosen point, and the arrows stand for continuous maps. If you're a physicist or geometer you may prefer fiber bundles to "fibrations" - but luckily, they're so similar we can ignore the difference in a vague discussion like this. The idea is basically just that E maps onto B, and sitting over each point of B we have a copy of F. We call B the "base space", E the "total space" and F the "fiber". If we want to classify such fibrations we can consider carrying the fiber F around a loop in B and see how it twists around. For example, if all our spaces are smooth manifolds, we can pick a connection on the total space E and see what parallel transport around a loop in the base space B does to points in the fiber F. This gives a kind of homomorphism Omega(B) -> Aut(F) sending loops in B to invertible maps from F to itself. And, the cool thing is: this homomorphism lets us classify the fibration! Here I say "kind of homomorphism" since Omega(B), the space of loops in B based at the chosen point, is only "kind of" a topological group: the group laws only hold up to homotopy. But let's not worry about this technicality - especially since I'm being vague about all sorts of other equally important issues! The reason I can get away with not worrying about these issues is that I'm trying to explain a very robust powerful principle - one that can easily survive a dose of vagueness that would kill a lesser idea. Namely, if B is a connected space with a chosen basepoint, FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. This could be called "the basic principle of Galois theory", for reasons explained in "week213". There I explained the special case where the fiber is discrete. Then our fibration called a "covering space", and the basic principle of Galois theory boils down to this: COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F. Okay. Now let's use the same principle to classify extensions of a group B by a group F: 1 -> F -> E -> B -> 1 The group B here acts like "loops in the base". But what acts like "automorphisms of the fiber"? You might guess it's the group of automorphisms of F. But, it's actually the *2-group* of automorphisms of F! A 2-group is a categorified version of a group where all the usual group laws hold up to natural isomorphism. They play a role in higher gauge theory like that of groups in ordinary gauge theory. In higher gauge theory, parallel transport along a path is described by an *object* in a 2-group, while parallel transport along a path-of-paths is described by a *morphism*. In 2-form electromagnetism we use a very simple "abelian" 2-group, which has one object and either the real line or the circle as morphism. But there are other more interesting "nonabelian" examples. If you want to learn more about 2-form electromagnetism from this perspective, try "week210". For 2-groups in general, try this paper: 9) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 12 (2004), 423-491. Available online at http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html or as math.QA/0307200. Anyway: it turns out that any group F gives a 2-group AUT(F) where the objects are automorphisms of F and the morphisms are "conjugations" - elements of F acting to conjugate one automorphism and yield another. And, extensions 1 -> F -> E -> B -> 1 are classified by homomorphisms B -> AUT(F) where we think of B as a 2-group with only identity morphisms. More precisely: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) It's fun to work out the details, but it's probably not a good use of our time together grinding through them here. So, I'll just sketch how it works. Starting with our extension i p 1 --> F --> E --> B --> 1 we pick a "section" s E <-- B meaning a function with p(s(b)) = b for all b in B. We can find a section because p is onto. However, the section usually *isn't* a homomorphism. Given the section s, we get a function alpha: B -> Aut(F) where Aut(F) is the group of automorphisms of F. Here's how: alpha(b) f = s(b) f s(b)^{-1} However, usually alpha *isn't* a homomorphism. So far this seems a bit sad: functions between groups want to be homomorphisms. But, we can measure how much s fails to be a homomorphism using the function g: B^2 -> F given by g(b,b') = s(bb') s(b')^{-1} s(b)^{-1} Note that g = 1 iff s is a homomorphism. We can then use this function g to save alpha. The sad thing about alpha is that it's not a homomorphism... but the good thing is, it's a homomorphism up to conjugation by g! In other words: alpha(bb') f = g(b,b') [alpha(b) alpha(b') f] g(b,b')^{-1} Taken together, alpha and g satisfy some equations ("cocycle conditions") which say precisely that they form a homomorphism from B to the 2-group AUT(F). Conversely, any such homomorphism gives an extension of B by F. In fact, isomorphism classes of extensions of B by F correspond in a 1-1 way with isomorphism classes of homorphisms from B to AUT(F). So, we've classified these extensions! In fact, something even better is true! It's evil to "decategorify" by taking isomorphism classes as we did in the previous paragraph. To avoid this, we can form a groupoid whose objects are extensions of B by F, and a groupoid whose objects are homomorphisms B -> AUT(F). I'm pretty sure that if you form these groupoids in the obvious way, they're equivalent. And that's what this slogan really means: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) Next, let me say how Schreier theory reduces to more familiar ideas in two special cases. People have thought a lot about the special case where F is abelian and lies in the center of E. These are called "central extensions". This is just the case where alpha = 1. The set of isomorphism classes of central extensions is called H^2(B,F) - the "second cohomology" of B with coefficients in F. People have also thought about "abelian extensions". That's an even more special case where all three groups are abelian. The set of isomorphism classes of such extensions is called Ext(B,F). Since we don't make any simplifying assumptions like this in Schreier theory, it's part of a subject called "nonabelian cohomology". It was actually worked out by Dedecker in the 1960's, based on much earlier work by Schreier: 10) O. Schreier, Ueber die Erweiterung von Gruppen I, Monatschefte fur Mathematik and Physick 34 (1926), 165-180. Ueber die Erweiterung von Gruppen II, Abh. Math. Sem. Hamburg 4 (1926), 321-346. 11) P. Dedecker, Les foncteuers Ext_Pi, H^2_Pi and H^2_Pi non abeliens, C. R. Acad. Sci. Paris 258 (1964), 4891-4895. More recently, Schreier theory was pushed one step up the categorical ladder by Larry Breen. As far as I can tell, he essentially classified the extensions of a 2-group B by a 2-group F in terms of homomorphisms B -> AUT(F), where AUT(F) is the *3-group* of automorphisms of F: 12) Lawrence Breen, Theorie de Schreier superieure, Ann. Sci. Ecole Norm. Sup. 25 (1992), 465-514. Also available at http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1992_4_25_5 We can keep pushing Schreier theory upwards like this, but we can also expand it "sideways" by replacing groups with groupoids. You should have been annoyed by how I kept assuming my topological spaces were connected and equipped with a specified point. I did this to make them analogous to groups. For example, it's only spaces like this for which the fundamental group is sufficiently powerful to classify covering spaces. For more general spaces, we should use the fundamental *groupoid* instead of the fundamental group. And, we can set up a Schreier theory for extensions of groupoids: 13) V. Blanco, M. Bullejos and E. Faro, Categorical non abelian cohomology, and the Schreier theory of groupoids, available as math.CT/0410202. In fact, these authors note that Grothendieck did something similar back in 1971: he classified *all* groupoids fibered over a groupoid B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of groupoids! The point here is that Gpd contains AUT(F) for any fixed groupoid F: 14) Alexander Grothendieck, Categories fibrees et descente (SGA I), Lecture Notes in Mathematics 224, Springer, Berlin, 1971. Having extended the idea "sideways" like this, one can then continue marching "upwards". I don't know how much work has been done on this, but the slogan should be something like this: n-GROUPOIDS FIBERED OVER AN n-GROUPOID B ARE "THE SAME" AS WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpd Grothendieck also studied this kind of thing with categories replacing groupoids, so there should also be an n-category version, I think... but it's more delicate to define "fibrations" for categories than for groupoids, so I'm a bit scared to state a slogan suitable for n-categories. However, I'm not scared to go from n-groupoids to omega-groupoids, which are basically the same as spaces. In terms of spaces, the slogan goes like this: SPACES FIBERED OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE SPACE OF ALL SPACES This is how James Dolan taught it to me. Most mortals are scared of "the space of all spaces" - both for fear of Russell's paradox, and because we really need a *space* of all spaces, not just a mere set of them. To avoid these terrors, you can water down Jim's slogan by choosing a specific space F to be the fiber: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) where AUT(F) is the topological group of homotopy self-equivalences of F. The fearsome "space of all spaces" is then the disjoint union of the classifying spaces of all these topological groups AUT(F). It's too large to be a space unless you pass to a larger universe of sets, but otherwise it's perfectly fine. Grothendieck invented the notion of a "Grothendieck universe" for precisely this purpose: 14) Wikipedia, Grothendieck universe, http://en.wikipedia.org/wiki/Grothendieck_universe ----------------------------------------------------------------------- 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/twf.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 Wed Nov 16 17:11:01 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 16 Nov 2005 17:11:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcUTe-0005Sk-Jo for categories-list@mta.ca; Wed, 16 Nov 2005 17:05:58 -0400 Message-ID: <2e0980f30511151533h6eb4840bpec6c1a4ef86dbaba@mail.gmail.com> Date: Tue, 15 Nov 2005 17:33:22 -0600 From: Michael Mislove To: categories@mta.ca Subject: categories: First Announcement and Call for Papers for MFPS 22 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: 20 Dear Colleagues, Below is the First Announcement and Call for Papers for MFPS 22, which will take place at the University of Genoa, IT from Wednesday, May 24 through Saturday, May 27, 2006. Best regards, Mike Mislove Professor Michael Mislove Phone: +1 504 862-3441 Department of Mathematics FAX: +1 504 865-5063 Tulane University URL: http://www.math.tulane.edu/~mwm New Orleans, LA 70118 USA First Announcement and Call for Papers MFPS XXII Twenty-second Conference on the Mathematical Foundations of Programming Semantics University of Genoa Genova Italy May 24 - May 27, 2006 Partially Supported by US Office of Naval Research The Twenty-second Conference on the Mathematical Foundations of Programming Semantics will take place at the University of Genoa, Italy from Wednesday, May 24 through Saturday, May 27, 2006. The invited speakers for MFPS XXII are Marcelo Fiore, Cambridge Eugenio Moggi, Genova Prakash Panangaden, McGill Davide Sangiorgi, Bologna Peter Selinger, Dalhousie Steve Zdancewic, Penn In addition to the invited addresses, there will be a Special Session on Security organized by Catherine Meadows. Other special sessions also are planned, and details will be announced as they are available. There also will be a Tutorial Day on May 23 devoted to Separation Logic. Details about this event also will be forthcoming later. The remainder of the program will be composed of papers selected by the Program Committee from submissions received in response to this Call for Papers. The Program Committee is being chaired by Stephen Brookes (CMU) and Michael Mislove (Tulane). It also includes: o Mariangiola Dezani-Ciancaglini (Torino) o Martin Escardo (Birmingham) o Joshua Guttman (Mitre) o Cedric Fournet (Microsoft) o Radha Jagadeesan (DePaul) o Achim Jung (Birmingham) o Peeter Laud (Tartu) o Catherine Meadows (NRL) o Catuscia Palamidessi (INRIA) o Prakash Panangaden (McGill) o Roberto Segala (Vernoa) o Phil Scott (Ottawa) o Simona Ronchi della Rocha (Torino) o Alex Simpson (Edinburgh) Submissions should consist of original work that has not been published elsewhere. Submissions should be no longer than 12 pages, and they should be in the form of either PostScript or pdf files that can be printed on a standard printer. They can be made using the link that will be available on the MFPS 22 Home Page http://www.math.tulane.edu/~mfps/mfps22.htm - submissions will open in early January. Submissions must be received by midnight, Pacific Standard Time on Friday, February 15, 2005. Authors will be notified of acceptance by March 25, 2005. The MFPS conferences are devoted to those areas of mathematics, logic and computer science which are related to the semantics of programming languages. The series particularly has stressed providing a forum where both mathematicians and computer scientists can meet and exchange ideas about problems of common interest. We also encourage participation by researchers in neighboring areas, since we strive to maintain breadth in the scope of the series. The Organizing Committee for MFPS consists of Stephen Brookes (CMU), Achim Jung (Birmingham), Catherine Meadows (NRL), Michael Mislove (Tulane) and Prakash Panangaden (McGill). The local arrangements for MFPS XXI are being overseen by Giuseppe Rosolini (Genova). In addition to supporting the conference overall, the support we anticipate from the Office of Naval Research makes funds available to help offset expenses of graduate students. Women and minorities also are encouraged to inquire about possible support to attend the meeting. Participation Information Information about MFPS XXII can be found at the URL http://www.math.tulane.edu/~mfps/mfps22.htm Registration information will be available at this site shortly after the New Year. If you have problems accessing the link above, then send email to mfps@math.tulane.edu. From rrosebru@mta.ca Thu Nov 17 10:34:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Nov 2005 10:34:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcknT-0005pW-OQ for categories-list@mta.ca; Thu, 17 Nov 2005 10:31:31 -0400 Date: Wed, 16 Nov 2005 22:26:16 -0600 (CST) From: Peter May To: categories@mta.ca Subject: categories: Ancient history Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 21 In his posting today, John Baez advertised the slogan: FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. He hedged it with a ``dose of vagueness'', but in fact I proved a completely precise and general version of exactly this result in ``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. Using Moore loops on B, LB, one has a topological monoid, and one also has the topological monoid Aut(F) of homotopy equivalences of $F$. A ``transport'' is a homomorphism of topological monoids from LB to Aut(F). Allowing F to vary by a homotopy equivalence, one can define an equivalence relation on transports such that the equivalence classes are in natural bijective correspondence with the equivalence classes of `fibrations over the base space B with fiber F'. One can generalize the context by allowing fibers in some nice category and prove the same result. See opus cit, Theorem 14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking about categorification, but I would imagine that the methods categorify. Some questions from more recent work (in progress in fact): 1. In work on (equivariant, stable) parametrized homotopy theory, Johann Sigurdsson and I need and develop duality in ``symmetric bicategories B'', which are not to be confused with the reasonably standard symmetric monoidal bicategories. Rather there must be a prescribed involution on the bicategory B, a pseudo-equivalence t between B and its opposite bicategory (not completely general: we find it helpful to require tt = id on 0-cells). For example, the standard bicategory whose 0-cells are rings, whose 1-cells R >--> S are (S,R)-bimodules, and whose 2-cells are maps of bimodules is symmetric; t takes R to its opposite ring and takes an (S,R)-bimodule to the same Abelian group regarded as a (tR,tS)-bimodule. Is there a pre-existing theory of such bicategories and their duality theory, analogous to duality theory in symmetric monoidal categories? 2. The example in 1 is additionally a symmetric monoidal bicategory under the tensor product over Z, and there is an analogous bicategory starting with a commutative ground ring replacing Z. These assemble nicely into a tricategory of commutative rings, algebras, bimodules, and maps of bimodules. Moreover, the bicategory in 1 is actually part of a pseudo double category with maps of algebras as vertical 1-cells. Promoting this to the tricategory just mentioned, one has maps of commutative rings as vertical 1-cells and maps of algebras as vertical 2-cells. I don't know a name for the resulting notion, something like a pseudo triple category. Here again, what is most important is duality theory. Has anybody studied such structures? There are derived versions of the cited example, and such structures also appear naturally in our work on parametrized homotopy theory. Thankfully, we do not (yet) seem to need tetracategories! From rrosebru@mta.ca Thu Nov 17 10:34:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Nov 2005 10:34:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ecklk-0005hL-Jm for categories-list@mta.ca; Thu, 17 Nov 2005 10:29:44 -0400 Message-ID: Date: Wed, 16 Nov 2005 21:42:43 -0600 From: Michael Shulman To: categories Subject: categories: distributors 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: 22 Hi, Can anyone tell me the motivation for the use of the term "distributors" as a synonym for profunctors/bimodules? Is there something being distributed? Thanks, Mike From rrosebru@mta.ca Thu Nov 17 16:29:49 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Nov 2005 16:29:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcqMR-0007C7-6t for categories-list@mta.ca; Thu, 17 Nov 2005 16:27:59 -0400 Message-ID: <437C9CD2.8040603@math.upenn.edu> Date: Thu, 17 Nov 2005 10:08:02 -0500 From: jim stasheff User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Ancient history 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: 23 Even more ancient: Parallel transport in fibre spaces," Bol. Soc. Mat. Mexicana (1968), 68-86. Unfortunately that's a hard paper to get a hold of somewhat related is Associated fibre spaces," Michigan Math. Journal 15 (1968), 457-470. and at the survey level H-spaces and classifying spaces, I-IV", AMS Proc. Symp. Pure Math. 22 (1971), 247-272. Of course, as you might expect, I describe things in terms of A_\infty-morphisms from the space of loops into Aut(F) of homotopy equivalences of $F$ Now that some of us are comfortable with A_\infty-cats, categborification should proceed perhaps with some technical details. jim Peter May wrote: >In his posting today, John Baez advertised the slogan: > > FIBRATIONS OVER THE BASE SPACE B WITH FIBER F > ARE "THE SAME" AS > HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. > >He hedged it with a ``dose of vagueness'', but in fact I proved a >completely precise and general version of exactly this result in >``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. Using >Moore loops on B, LB, one has a topological monoid, and one also has the >topological monoid Aut(F) of homotopy equivalences of $F$. A ``transport'' >is a homomorphism of topological monoids from LB to Aut(F). Allowing F to >vary by a homotopy equivalence, one can define an equivalence relation on >transports such that the equivalence classes are in natural bijective >correspondence with the equivalence classes of `fibrations over the base >space B with fiber F'. One can generalize the context by allowing fibers >in some nice category and prove the same result. See opus cit, Theorem >14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking >about categorification, but I would imagine that the methods categorify. > > From rrosebru@mta.ca Thu Nov 17 16:29:49 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Nov 2005 16:29:49 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcqLE-00076Z-JN for categories-list@mta.ca; Thu, 17 Nov 2005 16:26:44 -0400 Message-ID: <437C9478.7010604@ll319dg.fsnet.co.uk> Date: Thu, 17 Nov 2005 14:32:24 +0000 From: Ronald Brown User-Agent: Mozilla Thunderbird 0.7.2 (Windows/20040707) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Schreier theory 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: 24 John Baez gave an interesting account of nonabelian cohomology and extension theory. Here are a few more references with which I have been involved, all using crossed complexes and free crossed resolutions: 1) (with P.J. HIGGINS), ``Crossed complexes and non-abelian extensions'', {\em Category theory proceedings, Gummersbach}, 1981, Lecture Notes in Math. 962 (ed. K.H. Kamps et al, Springer, Berlin, 1982), pp. 39-50. This generalises classical Schreier theory to extensions of groupoids. 2) (with O. MUCUK), ``Covering groups of non-connected topological groups revisited'', {\em Math. Proc. Camb. Phil. Soc}, 115 (1994) 97-110. This relates the theory of covering topological groups of non connected topological groups to the classical theory of extensions and obstructions to a Q-kernel with an invariant in H^3. It uses the properties of the internal hom in crossed complexes CRS(F,C) , and exact sequences derived from a fibration C \to D and the induced fibration on CRS(F, -). 3) (with T. PORTER), ``On the Schreier theory of non-abelian extensions: generalisations and computations''. {\em Proceedings Royal Irish Academy} 96A (1996) 213-227. This establises a generalisation of the Schreier theory in two ways (but only for groups). One is using coefficients in a crossed module, following Dedecker's key ideas, as in the references in John's account. Second it shows how to compute with such extensions A \to E \to G in terms of presentations of the group G. This involves identities among relations for the presentation, as shown originally by Turing in Turing, A. 1938. The extensions of a group. {\em Compositio Mathematica.} {\bf 5 } 357-367 The advantage of this method is that one can actually do sums, even when the group G may be infinite. The example given by us is G= trefoil group with two generators x,y and relation x^3=y^2. This presentation has no identities among relations, and so the calculation is especially simple. Equivalence of extensions is described in terms of homotopies of morphisms of crossed complexes, and this relates the ideas to other forms of homological or homotopical algebra. An advantage of this approach is the ability to calculate some small free crossed resolutions of some groups: this is one reason for using crossed complexes. Note that a convenient monoidal closed structure on the category of crossed complexes has been explicitly written down, and this allows convenient calculation and representations of homotopies, using the `unit interval' groupid, as a crossed complex. One of the problems I have with the globular approach is the difficulty of writing down homotopies, and higher homotopies. For example, Ilhan Icen and I found it difficult to rewrite in terms of group-groupoids the well known Whitehead theory of automorphisms of crossed modules, explained for the crossed modules of groupoids case in (with \.I. \.I\c cen ), `Homotopies and automorphisms of crossed modules of groupoids', Applied Categorical Structures, 11 (2003) 185-206. It looks as if it would be better expressed in terms of automorphisms of 2-groupoids: good marks to anyone who writes it down in that way! One knows such homotopies of globular infinity groupoids exist because globular infinity-groupoids are equivalent to crossed complexes (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and crossed complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981) 371-386. (This paper contains an early definition of a (strict, globular) infinity category.) This raises a question: what is the crossed complex associated to the free globular groupoid on one generator of dimension n? I have a round-about sketch proof, using also cubical theory, and a van Kampen theorem, that it *is* the fundamental crossed complex of the n-globe. Does anyone have a purely algebraic proof? The idea of singular nonabelian cohomology of a space X with coefficients in a crossed complex C is given in (with P.J.HIGGINS), ``The classifying space of a crossed complex'', {\em Math. Proc. Camb. Phil. Soc.} 110 (1991) 95-120. This cohomology is give by [\Pi SX, C], homotopy classes of maps from the fundamental crossed complex of the singular complex of X, to C. There is also a Cech version (current work with Jim Glazebrook and Tim Porter). An interesting problem is to classify extensions of crossed complexes! There is an interesting account of extensions of principal bundles and transitive Lie groupoids by Androulidakis, developing work of Mackenzie, at math.DG/0402007 (not using crossed complexes). Ronnie Brown www.bangor.ac.uk/r.brown Ronnie Brown From rrosebru@mta.ca Fri Nov 18 08:37:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 18 Nov 2005 08:37:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ed5OK-0006uf-P4 for categories-list@mta.ca; Fri, 18 Nov 2005 08:30:56 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <3A0F3A28-F822-452A-801E-267BBD3ED685@math.mq.edu.au> Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: Not getting it wrong Date: Fri, 18 Nov 2005 11:20:25 +1100 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 25 I took the liberty of circulating to our Math Dept Andrej Bauer's message which included the example >> >> Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]), >> x -> 0] In retaliation my colleague Alf van der Poorten commented as below. --Ross ------------------------------------------------------------------------ - Dear Ross, I've long said that "mathematics ain't about getting it right but about not getting it wrong". Our students of course think that we teach them methods/algorithms/recipes for getting it right --- indeed, we often give them partial reward just for seeming to have started to attempt to use some reasonable way that might possibly have led them to a correct answer. In that context looking it up or having a machine look it up (or asking/copying from a wise friend) is a perfectly fine recipe for maybe getting it right. The trick is verifying that it is indeed very likely right (or, occasionally, actually \emph{is} right) and much better yet, noticing that it is wrong or that there are grounds for suspicion warranting their consulting a dinkum mathematician to get the good oil. By the way, the particular limit cited didn't tempt me towards error at all: Almost automatically I scribbled $(1+4x^2)^{1/4}=1+x^2+\ldots \,$, and so on, realising a line or so later that I had better warm up those $\ldots\,$ to $-\frac32x^4+\ldots\,$, and so on. I hadn't realised that I was checking whether some zero is isolated and am sorry to learn that I might have been. Ideally our students will learn from us to see when they've surely got it wrong and, more subtly, when they might well possibly have got it wrong ("Yup, it does indeed work for $a=1$ but, oops! Why is there an unspecified $a$ at all? I'd better phone a mathematician to ask --- hoping she does something sensible such as reminding me of Taylor expansions rather than waffling about isolated zeros."). Unfortunately, as always, distinguishing right from wrong requires quite a bit of knowledge (both of what's right and of what isn't) that students are often slow in acquiring. ------------------------ Baseball and Quantum Physics Umpire: "Some is balls and some is strikes, but until I calls 'em they ain't nothing." Centre for Number Theory Research 1 Bimbil Place, Killara NSW 2071 +61 2 9416 6026 From rrosebru@mta.ca Fri Nov 18 17:02:56 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 18 Nov 2005 17:02:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EdDIe-0000ml-SF for categories-list@mta.ca; Fri, 18 Nov 2005 16:57:36 -0400 Date: Fri, 18 Nov 2005 08:21:39 -0500 From: jim stasheff User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Schreier theory References: <437C9478.7010604@ll319dg.fsnet.co.uk> In-Reply-To: <437C9478.7010604@ll319dg.fsnet.co.uk> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Scanned-By: MIMEDefang 2.53 on 128.91.55.26 X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 26 Ronald Brown wrote: > > There is an interesting account of extensions of principal bundles and > transitive Lie groupoids by Androulidakis, developing work of Mackenzie, > at math.DG/0402007 (not using crossed complexes). > > Ronnie Brown > www.bangor.ac.uk/r.brown > > Ronnie Brown > This brings to mind an interesting (but insufficiently well known) treatment by I htink) Dennis Johnson of extensions A \to E \to G of topological groups which are also principal bundles the classification of such extensions fits into an exact sequence X --> Y --> Z where Z classifies the bundle - forgetting the group structure and X is the appropriate H^2 classifying split topolgocical group extensions i.e. A x G as bundles jim > > > > > > > > > From rrosebru@mta.ca Sat Nov 19 09:06:37 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 19 Nov 2005 09:06:37 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EdSIH-0004ON-1s for categories-list@mta.ca; Sat, 19 Nov 2005 08:58:13 -0400 Message-ID: <437DEB4E.1080105@andrej.com> Date: Fri, 18 Nov 2005 15:55:10 +0100 From: Andrej Bauer User-Agent: Debian Thunderbird 1.0.2 (X11/20050331) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories Subject: categories: Re: Not getting it wrong References: <3A0F3A28-F822-452A-801E-267BBD3ED685@math.mq.edu.au> In-Reply-To: <3A0F3A28-F822-452A-801E-267BBD3ED685@math.mq.edu.au> Content-Type: text/plain; charset=ISO-8859-2 Content-Transfer-Encoding: 7bit X-SA-Exim-Connect-IP: 127.0.0.1 X-SA-Exim-Mail-From: Andrej.Bauer@andrej.com X-SA-Exim-Scanned: No (on haka.fmf.uni-lj.si); SAEximRunCond expanded to false X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 27 Ross Street wrote: > I took the liberty of circulating to our Math Dept Andrej Bauer's > message which included the example > >>> >>> Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]), >>> x -> 0] > > In retaliation my colleague Alf van der Poorten commented as below. > --Ross I don't quite understand who is retaliating for what in the posted comment. Even though this is not strictly speaking category theory, I will take the further liberty to tell you the whole story. The above limit cropped up when I told my students in "Computer Science I" to bring examples of problems they do in Analysis and Algebra. The idea was to solve them using Mathematica. Someone bought the above limit, except that instead of the parameter 'a' he had the constant 'e' (base of natural logarithm). Now, as it happens, in Mathematica this constant is written 'E' instead of 'e'. The student of course typed in 'e' which Mathematica took as a "free parameter" and gave the "generic" answer 0, while the correct answer was 6. It took a while to explain the whole mess. I suspect the only thing the students learned was that (a) the limit is stupid, (b) Mathematica is stupid and (c) the teacher is a pedantic fanatic. By the way, to see what is going on in the above limit, draw a family of functions for various values of 'a' approaching and passing through a=e. All will be clear as you see two "waves" passing through each other. I guess I am trying to point out that current Computer Alegbra Systems are very tricky to use _correctly_. In Mathematica's defense it should be said that it also found a couple of errors where the Analsyis teacher did not (one involved cancelling "a common factor of b" between b^2 and sqrt(b^2) for a real parameter b). Andrej From rrosebru@mta.ca Sat Nov 19 09:06:37 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 19 Nov 2005 09:06:37 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EdSNG-0004Uv-MF for categories-list@mta.ca; Sat, 19 Nov 2005 09:03:22 -0400 In-Reply-To: References: From: Marco Grandis Subject: categories: pseudo triple categories Date: Fri, 18 Nov 2005 19:59:20 +0100 To: categories@mta.ca Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 28 In reply to P. May's message. I do not know of any work on "pseudo triple category". On the other hand, the pseudo double category Rng of rings, =20 homomorphisms and bimodules is briefly considered in our first work on pseudo double categories, =20 subsection 5.3 M. Grandis - R. Par=E9, Limits in double categories, Cah. Topol. =20 G=E9om. Diff. Cat=E9g. 40 (1999), 162-220 as a substructure of the pseudo double category of Ab-categories, Ab-=20= functors and Ab-profunctors. As an interesting construction, in this pseudo double category: - the "tabulator" of a bimodule u: R -+-> S (u is a left-R, right-=20= S bimodule) (i.e., its double limit) can be constructed as a ring of =20= triangular 2x2 matrices, with "matrix product" r x r' x' rr' rx'+xs' . =3D 0 s 0 s' 0 ss' (with r. r' in R, s, s' in S and x, x' in u). Tabulators are crucial for double limits, since all of them can be =20 constructed from double products, double equalisers and tabulators. [ In a bicategory, the tabulator (of the vertical identity of A) is =20 the cotensor 2*A, and the previous result amounts to the =20 construction of weighted limits, in R.H. Street, Limits indexed by category valued 2-functors, J. Pure =20= Appl. Algebra 8 (1976), 149-181. ] Best regards Marco Grandis On 17 Nov 2005, at 05:26, Peter May wrote: > > In his posting today, John Baez advertised the slogan: > > FIBRATIONS OVER THE BASE SPACE B WITH FIBER F > ARE "THE SAME" AS > HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. > > He hedged it with a ``dose of vagueness'', but in fact I proved a > completely precise and general version of exactly this result in > ``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. =20 > Using > Moore loops on B, LB, one has a topological monoid, and one also =20 > has the > topological monoid Aut(F) of homotopy equivalences of $F$. A =20 > ``transport'' > is a homomorphism of topological monoids from LB to Aut(F). =20 > Allowing F to > vary by a homotopy equivalence, one can define an equivalence =20 > relation on > transports such that the equivalence classes are in natural bijective > correspondence with the equivalence classes of `fibrations over the =20= > base > space B with fiber F'. One can generalize the context by allowing =20 > fibers > in some nice category and prove the same result. See opus cit, =20 > Theorem > 14.2, page 83. That was over 30 years ago, so naturally I wasn't =20 > thinking > about categorification, but I would imagine that the methods =20 > categorify. > > > Some questions from more recent work (in progress in fact): > > 1. In work on (equivariant, stable) parametrized homotopy theory, > Johann Sigurdsson and I need and develop duality in ``symmetric > bicategories B'', which are not to be confused with the reasonably > standard symmetric monoidal bicategories. Rather there must be a > prescribed involution on the bicategory B, a pseudo-equivalence t > between B and its opposite bicategory (not completely general: > we find it helpful to require tt =3D id on 0-cells). For example, the > standard bicategory whose 0-cells are rings, whose 1-cells R >--> S =20= > are > (S,R)-bimodules, and whose 2-cells are maps of bimodules is symmetric; > t takes R to its opposite ring and takes an (S,R)-bimodule to the same > Abelian group regarded as a (tR,tS)-bimodule. Is there a pre-existing > theory of such bicategories and their duality theory, analogous to > duality theory in symmetric monoidal categories? > > 2. The example in 1 is additionally a symmetric monoidal bicategory > under the tensor product over Z, and there is an analogous bicategory > starting with a commutative ground ring replacing Z. These assemble > nicely into a tricategory of commutative rings, algebras, bimodules, > and maps of bimodules. Moreover, the bicategory in 1 is actually part > of a pseudo double category with maps of algebras as vertical 1-cells. > Promoting this to the tricategory just mentioned, one has maps of > commutative rings as vertical 1-cells and maps of algebras as > vertical 2-cells. I don't know a name for the resulting notion, > something like a pseudo triple category. Here again, what is most > important is duality theory. Has anybody studied such structures? > There are derived versions of the cited example, and such structures > also appear naturally in our work on parametrized homotopy theory. > Thankfully, we do not (yet) seem to need tetracategories! > From rrosebru@mta.ca Sat Nov 19 09:06:37 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 19 Nov 2005 09:06:37 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EdSOF-0004X1-AV for categories-list@mta.ca; Sat, 19 Nov 2005 09:04:23 -0400 Message-ID: Date: Fri, 18 Nov 2005 13:31:13 -0600 From: Michael Shulman To: categories Subject: categories: Re: distributors 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 Status: O X-Status: X-Keywords: X-UID: 29 Thanks to everyone who replied to my email. The consensus from people who claim to know (although there were some plausible guesses) is that the name is by analogy with "distributions" from functional analysis, which are "generalized functions." Thus changing "ion" to "or" we get that "distributors" are "generalized functors." On 11/16/05, Michael Shulman wrote: > Hi, > > Can anyone tell me the motivation for the use of the term > "distributors" as a synonym for profunctors/bimodules? Is there > something being distributed? > > Thanks, > Mike > > > From rrosebru@mta.ca Sun Nov 20 21:09:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 20 Nov 2005 21:09:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ee03v-0004vw-SA for categories-list@mta.ca; Sun, 20 Nov 2005 21:01:39 -0400 Message-Id: <200511202248.jAKMmEc05975@math-cl-n03.ucr.edu> Subject: categories: Re: fibrations as ... To: categories@mta.ca (categories) Date: Sun, 20 Nov 2005 14:48:13 -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: 30 Jim Stasheff wrote: > John and anyone else who cares to weigh in, > here are some comments from the purely topological > or rather homotopy theory side: > > For both bundles and fibrations (e.g. over a paracompact base), your > last slogan is the oldest: > > FIBRATIONS WITH FIBER F OVER THE SPACE B > ARE "THE SAME" AS > MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) > > `the same as' referring to homotopy classes. It's certainly old, but I mentioned another that may be older: COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F. although one usually sees this special case (which I didn't bother to mention): CONNECTED COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS TRANSITIVE ACTIONS OF THE FUNDAMENTAL GROUP OF B ON F which is usually disguised as follows: CONNECTED COVERING SPACES OVER B ARE "THE SAME" AS SUBGROUPS OF THE FUNDAMENTAL GROUP OF B Anyway, I wasn't trying to present things in historical order. I was trying present them roughly in order of increasing "dimension", starting with extensions of groups, then going up to 2-groups, then expanding out to groupoids, then going up to n-groupoids, and finally omega-groupoids... which are the same as homotopy types! And here, as usual, the n-category theorists meet up with the topologists - and find that the topologists have already done everything there is to do with omega-groupoids ... but usually by thinking of them of them as *spaces*, rather than omega-groupoids! It's sort of like climbing a mountain, surmounting steep cliffs with the help of ropes and other equipment, and then finding a Holiday Inn on top and realizing there was a 4-lane highway going up the other side. So, as usual, the main point of calling homotopy types "omega-groupoids" instead of "spaces" is not to reinvent topology, but to see how ideas from topology generalize to n-category theory. Think of spaces as omega-groupoids but use those as a springboard for omega-categories - or at least n-categories, perhaps just for low values of n if one is feeling tired. In the case at hand, the omega-groupoidal slogan: FIBRATIONS OF OMEGA-GROUPOIDS WITH FIBER F AND BASE B ARE "THE SAME" AS WEAK OMEGA-FUNCTORS FROM B TO AUT(F) is just a reformulation of: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) but it suggests a grandiose generalization: FIBRATIONS OF OMEGA-CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK OMEGA-FUNCTORS FROM B^{op} TO THE OMEGA-CATEGORY OF OMEGA-CATEGORIES! I guess we can thank Grothendieck for making precise and proving a version of this with omega replaced by n = 1: FIBRATIONS OF CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK 2-FUNCTORS FROM B^{op} TO THE 2-CATEGORY OF CATEGORIES. More recently people have been thinking about the n = 2 case, especially Claudio Hermida: 22) Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Applied Categorical Structures, 12 (2004), 427-459. Also available at http://maggie.cs.queensu.ca/chermida/papers/2-descent.pdf He states something that hints at this: FIBRATIONS OF 2-CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK 3-FUNCTORS FROM B^{op} TO THE WEAK 3-CATEGORY OF 2-CATEGORIES. where I'm using B^{op} to mean B with the directions of both 1-morphisms and 2-morphisms reversed. (Hermida follows tradition and calls this B^{coop} - "op" for reversing 1-morphisms and "co" for reversing 2-morphisms. But, it looks like we'll be needing to reverse all kinds of morphisms in n-category case, so we'll need a short name for that.) Best, jb From rrosebru@mta.ca Sun Nov 20 21:09:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 20 Nov 2005 21:09:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ee02z-0004s8-Bg for categories-list@mta.ca; Sun, 20 Nov 2005 21:00:41 -0400 Message-ID: <438085A9.3090403@math.upenn.edu> Date: Sun, 20 Nov 2005 09:18:17 -0500 From: jim stasheff User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories List Subject: categories: fibrations as ... Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Scanned-By: MIMEDefang 2.53 on 128.91.55.26 X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 31 John and anyone else who cares to weigh in, here are some comments from the purely topological or rather homtopy theory side: For both bundles and fibrations (e.g. over a paracompact base), your last slogan is the oldest: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) `the same as' referring to homotopy classes For bundles, Aut(F) now means the structure group of the bundle This standard approach is great - I'm not knocking it - but Schreier theory is more general: it's really a branch of "nonabelian cohomology theory". It's not all that hard to explain, either. So, I'll explain it and then talk about various simplifying assumptions people make. The goal of Schreier theory is to classify short exact sequences of groups: 1 -> F -> E -> B -> 1 for a given choice of F and B. "Exact" means that the arrows stand for homomorphisms and the image of each arrow is the kernel of the next. Here this just means that F is a normal subgroup of E and B is the quotient group E/F. Such a short exact sequence is also called an "extension of B by F", since E is bigger than B and contains F. The simplest choice is to let E be the direct sum of F and B. Usually there are other more interesting extensions as well. To classify these, the trick is to use the analogy between group theory and topology. As I explained in "week213", you can think of a group as a watered-down version of a connected space with a chosen point. The reason is that given such a space, we get a group consisting of homotopy classes of loops based at the chosen point. This is called the "fundamental group" of our space. There's a lot more information in our space than this group. But pretty much anything you can do for groups, you can do for such spaces. It's usually harder, but it's completely analogous! In particular, classifying short exact sequences is a lot like classifying "fibrations": 1 -> F -> E -> B -> 1 where now the letters stand for connected spaces with a chosen point, and the arrows stand for continuous maps. If you're a physicist or geometer you may prefer fiber bundles to "fibrations" - but luckily, they're so similar we can ignore the difference in a vague discussion like this. The idea is basically just that E maps onto B, and sitting over each point of B we have a copy of F. We call B the "base space", E the "total space" and F the "fiber". If we want to classify such fibrations we can consider carrying the fiber F around a loop in B and see how it twists around. For example, if all our spaces are smooth manifolds, we can pick a connection on the total space E and see what parallel transport around a loop in the base space B does to points in the fiber F. This gives a kind of homomorphism Omega(B) -> Aut(F) sending loops in B to invertible maps from F to itself. And, the cool thing is: this homomorphism lets us classify the fibration! Here I say "kind of homomorphism" since Omega(B), the space of loops in B based at the chosen point, is only "kind of" a topological group: the group laws only hold up to homotopy. As Peter May points out, you can use the strict monoid of Moore loops instead of Poincare's, but kind of homomorphism" still does not mean strict but rather A-\infty map even though both domain and range are strictly associative So intrepreted, the equivalence with B --> Aut F is clear. If you wnat a strict homomorphism, try using a classical connection - at least in the smooth case. But let's not worry about this technicality - especially since I'm being vague about all sorts of other equally important issues! The reason I can get away with not worrying about these issues is that I'm trying to explain a very robust powerful principle - one that can easily survive a dose of vagueness that would kill a lesser idea. Namely, if B is a connected space with a chosen basepoint, FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. This could be called "the basic principle of Galois theory", for reasons explained in "week213". There I explained the special case where the fiber is discrete. Then our fibration called a "covering space", and the basic principle of Galois theory boils down to this: COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F. Okay. Now let's use the same principle to classify extensions of a group B by a group F: 1 -> F -> E -> B -> 1 The group B here acts like "loops in the base". But what acts like "automorphisms of the fiber"? You might guess it's the group of automorphisms of F. But, it's actually the *2-group* of automorphisms of F! A 2-group is a categorified version of a group where all the usual group laws hold up to natural isomorphism. They play a role in higher gauge theory like that of groups in ordinary gauge theory. In higher gauge theory, parallel transport along a path is described by an *object* in a 2-group, while parallel transport along a path-of-paths is described by a *morphism*. In 2-form electromagnetism we use a very simple "abelian" 2-group, which has one object and either the real line or the circle as morphism. But there are other more interesting "nonabelian" examples. If you want to learn more about 2-form electromagnetism from this perspective, try "week210". For 2-groups in general, try this paper: 9) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 12 (2004), 423-491. Available online at http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html or as math.QA/0307200. Anyway: it turns out that any group F gives a 2-group AUT(F) where the objects are automorphisms of F and the morphisms are "conjugations" - and conjugations correspond to homtopies at the classifying space level elements of F acting to conjugate one automorphism and yield another. And, extensions 1 -> F -> E -> B -> 1 are classified by homomorphisms B -> AUT(F) where we think of B as a 2-group with only identity morphisms. More precisely: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) It's fun to work out the details, but it's probably not a good use of our time together grinding through them here. So, I'll just sketch how it works. Starting with our extension i p 1 --> F --> E --> B --> 1 we pick a "section" s E <-- B meaning a function with p(s(b)) = b for all b in B. We can find a section because p is onto. However, the section usually *isn't* a homomorphism. Given the section s, we get a function alpha: B -> Aut(F) where Aut(F) is the group of automorphisms of F. Here's how: alpha(b) f = s(b) f s(b)^{-1} However, usually alpha *isn't* a homomorphism. So far this seems a bit sad: functions between groups want to be homomorphisms. But, we can measure how much s fails to be a homomorphism using the function g: B2 -> F given by g(b,b') = s(bb') s(b')^{-1} s(b)^{-1} Note that g = 1 iff s is a homomorphism. We can then use this function g to save alpha. The sad thing about alpha is that it's not a homomorphism... but the good thing is, it's a homomorphism up to conjugation by g! In other words: alpha(bb') f = g(b,b') [alpha(b) alpha(b') f] g(b,b')^{-1} Taken together, alpha and g satisfy some equations ("cocycle conditions") which say precisely that they form a homomorphism from B to the 2-group AUT(F). Conversely, any such homomorphism gives an extension of B by F. In fact, isomorphism classes of extensions of B by F correspond in a 1-1 way with isomorphism classes of homorphisms from B to AUT(F). So, we've classified these extensions! In fact, something even better is true! It's evil to "decategorify" by taking isomorphism classes as we did in the previous paragraph. To avoid this, we can form a groupoid whose objects are extensions of B by F, and a groupoid whose objects are homomorphisms B -> AUT(F). I'm pretty sure that if you form these groupoids in the obvious way, they're equivalent. And that's what this slogan really means: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) Ah, your 2-group must be hiding my homotopies - I'd love to see that Next, let me say how Schreier theory reduces to more familiar ideas in two special cases. People have thought a lot about the special case where F is abelian and lies in the center of E. These are called "central extensions". This is just the case where alpha = 1. The set of isomorphism classes of central extensions is called H2(B,F) - the "second cohomology" of B with coefficients in F. People have also thought about "abelian extensions". That's an even more special case where all three groups are abelian. The set of isomorphism classes of such extensions is called Ext(B,F). Since we don't make any simplifying assumptions like this in Schreier theory, it's part of a subject called "nonabelian cohomology". It was actually worked out by Dedecker in the 1960's, based on much earlier work by Schreier: 10) O. Schreier, Ueber die Erweiterung von Gruppen I, Monatschefte fur Mathematik and Physick 34 (1926), 165-180. Ueber die Erweiterung von Gruppen II, Abh. Math. Sem. Hamburg 4 (1926), 321-346. 11) P. Dedecker, Les foncteuers Ext_Pi, H2_Pi and H2_Pi non abeliens, C. R. Acad. Sci. Paris 258 (1964), 4891-4895. More recently, Schreier theory was pushed one step up the categorical ladder by Larry Breen. As far as I can tell, he essentially classified the extensions of a 2-group B by a 2-group F in terms of homomorphisms B -> AUT(F), where AUT(F) is the *3-group* of automorphisms of F: 12) Lawrence Breen, Theorie de Schreier superieure, Ann. Sci. Ecole Norm. Sup. 25 (1992), 465-514. Also available at http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1992_4_25_5 We can keep pushing Schreier theory upwards like this, but we can also expand it "sideways" by replacing groups with groupoids. You should have been annoyed by how I kept assuming my topological spaces were connected and equipped with a specified point. I did this to make them analogous to groups. For example, it's only spaces like this for which the fundamental group is sufficiently powerful to classify covering spaces. For more general spaces, we should use the fundamental *groupoid* instead of the fundamental group. And, we can set up a Schreier theory for extensions of groupoids: 13) V. Blanco, M. Bullejos and E. Faro, Categorical non abelian cohomology, and the Schreier theory of groupoids, available as math.CT/0410202. In fact, these authors note that Grothendieck did something similar back in 1971: he classified *all* groupoids fibered over a groupoid B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of groupoids! The point here is that Gpd contains AUT(F) for any fixed groupoid F: 14) Alexander Grothendieck, Categories fibrees et descente (SGA I), Lecture Notes in Mathematics 224, Springer, Berlin, 1971. Having extended the idea "sideways" like this, one can then continue marching "upwards". I don't know how much work has been done on this, but the slogan should be something like this: n-GROUPOIDS FIBERED OVER AN n-GROUPOID B ARE "THE SAME" AS WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpd aha, weak or lax as much above will likely be also Grothendieck also studied this kind of thing with categories replacing groupoids, so there should also be an n-category version, I think... but it's more delicate to define "fibrations" for categories than for groupoids, so I'm a bit scared to state a slogan suitable for n-categories. However, I'm not scared to go from n-groupoids to omega-groupoids, which are basically the same as spaces. In terms of spaces, the slogan goes like this: SPACES FIBERED OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE SPACE OF ALL SPACES This is how James Dolan taught it to me. Most mortals are scared of "the space of all spaces" - both for fear of Russell's paradox, and because we really need a *space* of all spaces, not just a mere set of them. To avoid these terrors, you can water down Jim's slogan by choosing a specific space F to be the fiber: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) where AUT(F) is the topological group of homotopy self-equivalences of F. The fearsome "space of all spaces" is then the disjoint union of the classifying spaces of all these topological groups AUT(F). It's too large to be a space unless you pass to a larger universe of sets, but otherwise it's perfectly fine. Grothendieck invented the notion of a "Grothendieck universe" for precisely this purpose: 14) Wikipedia, Grothendieck universe, http://en.wikipedia.org/wiki/Grothendieck_universe ----------------------------------------------------------------------- From rrosebru@mta.ca Mon Nov 21 16:54:02 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 21 Nov 2005 16:54:02 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EeIa4-0005VH-Jf for categories-list@mta.ca; Mon, 21 Nov 2005 16:48:04 -0400 Subject: categories: PSSL in Glasgow: preliminary announcement From: Tom Leinster To: categories@mta.ca Content-Type: text/plain Date: Mon, 21 Nov 2005 11:25:46 +0000 Message-Id: <1132572346.13208.23.camel@tl-linux.maths.gla.ac.uk> Mime-Version: 1.0 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 32 Dear All, The 83rd Peripatetic Seminar on Sheaves and Logic will be held on the weekend of 6-7 May 2006 in Glasgow, Scotland. Further information will be circulated closer to the time. For those unfamiliar with PSSLs, they are a long-running series of meetings, usually held on a weekend at a university somewhere in Europe. They are fairly informal, e.g. people sometimes present work in progress. The name is a (charming) historical relic: talks cover all aspects of category theory, not just sheaves and logic. Tom -- Tom Leinster From rrosebru@mta.ca Mon Nov 21 16:54:02 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 21 Nov 2005 16:54:02 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EeIcy-0005k2-Co for categories-list@mta.ca; Mon, 21 Nov 2005 16:51:04 -0400 Date: Mon, 21 Nov 2005 11:51:14 -0500 (EST) From: Peter Freyd Message-Id: <200511211651.jALGpEKx024980@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: *-Autonomous Abelian Functor Categories Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 33 Let R be a commutative ring. {Footnote: If, instead, R is a ring with anti-involution (such as a group-ring) all that is below appears to still hold.} Let R-fpmod be the category of finitely presented R-modules and let *F* be the full category of finitely presented covariant abelian-group-valued functors on R-fpmod. A functor is in *F* iff it appears in an exact sequence H^A --> H^B --> T --> O where the first two objects are representable functors. It is an easy exercise to see that *F* is an exact subcategory of the ambient functor category, hence an abelian category. For any f.p.module A we will use A@ to denote the functor that carries B to A@B (using, here, @ for $\otimes$). If we choose a finite-rank free resolution F --> G --> A --> O we obtain exact F@ --> G@ --> A@ --> O. Since F@ = H^F, G@ = H^G and *F* is closed under cokernel formation, we know that A@ is in *F*. (I'll continue to denote isomorphisms here with equality signs.) Let I denote the forgetful functor from R-fpmod to the category of abelian groups. It has two other notations, to wit, H^R and R@. I pointed out in my last note that *F* is the free abelian category generated by R. (That is, given any abelian category with an object and an action of R thereon there is an exact functor from *F* that carries I to the given object, unique up to natural equivalence.) But I'm not going to use that fact here. (I did use it to learn all this stuff, though.). NOTATION: Every group-valued functor from a category of R-modules, commutative R, can be canonically lifted to a module-valued functor. Given two such functors, S and T, we follow the CS tradition of denoting their composition, "first apply S then T" as S;T (hence (S;T)(A) = T(S(A)). For fixed S the functor S;T is exact in the second variable. (That property, together with S:I = S, characterizes this bifunctor; we don't actually need to define it in terms of composition.) We will need: (H^A);(H^B) = H^{A@B} and (A@);(B@) = (A@B)@ DEFINE: [S,T] is the functor such that sends A to the group of natural transformations Hom((H^A;S),T) Note: [S,T] carries right exact sequences in the first variable to left exact sequences, and preserves left-exactness in the second variable. We will need the formula [H^A,T] = (A@);T verifiable by evaluating on an arbitrary B: [H^A,T](B) = Hom((H^B;H^A),T) = Hom(H^{A@B},T) = T(A@B) = ((A@);T)(B) A couple of things implicit in the definition of [S,T] should be explicated. First: LEMMA: If S and T are in *F* then so is S;T. BECAUSE: If T is in *F* let H^A --> H^B --> T --> 0 be exact. We obtain an exact sequence S;H^A --> S;H^B --> S;T --> O Since *F* is closed under cokernel formation it suffices to show that S;H^A is in *F* for any f.p. R-module A. Let F and G be finite-rank free modules and F --> G --> A --> O exact. Then so is O --> S;H^A --> S;H^G --> S;H^F Since *F* is closed under kernel formation it suffices to show that S;H^F is in *F* for any finite-rank free module F. But S;H^F is just a finite direct sum of copies of S and *F* is, of course, closed under direct sums. [] Second explication: LEMMA: If S and T are in *F* then so is [S,T]. BECAUSE: If S is in *F* let H^A --> H^B --> S --> 0 be exact. We obtain an exact sequence O --> [S,T] --> [H^B,T] --> [H^A,T] which we know can be rewritten as O --> [S,T] --> (B@);T --> (A@);T Since *F* is closed under kernel formation and composition we're done. [] DEFINE a *-FUNCTOR, S* as [S,I]. LEMMA: The duality functor is exact BECAUSE: If S --> T --> U is exact then H^A;S --> H^A;T --> H^A;U is exact for every A. We will obtain, therefore, the exactness of U*(A) --> S*(A) --> T*(A) if we know: LEMMA: The object I is injective in *F*. BECAUSE: Being a representable functor, I is, of course, projective in the ambient functor category, hence in *F*. Let O | I | H^A --> H^B --> T --> O be exact (all vertical arrows point down). We seek a retraction for I --> T. Since I is projective we can choose a map I --> H^B to yield a commutative triangle. The fact that I --> T is monic says that O ----> I | | H^A --> H^B is a pullback. It is, therefore, the Yoneda-image of a pushout square: B --> A | | R --> O Let O --> K --> B --> A. It is an exercise in abelian categories that K --> B --> R is epi. (The dual exercise is easy in the category of abelian groups -- which, of course, suffices.) We may choose a retraction R --> K. The map it induces, H^B --> H^K --> I, is such that H^A --> H^B --> H^K --> I is a zero-map and we obtain a factorization H^B --> I = H^B --> T --> I. The map T --> I is what we seek. [] {Footnote: The object I need not be injective in the ambient functor category. In the case R = Z let T be the functor that assigns to a group its torsion subgroup. If I were injective then any endo- transformation on T would extend to an endo-transformation on I. But there are uncountably many endo-transformations on T and only countably many on I (the ring of endo-transformations on T is the "n-adic" completion of the integers -- it may be constructed as the product of all the p-adic completions.)} THEOREM: The *-functor on is a duality on *F*. WE NEED TO SHOW that S** = S. First note that I* = I, hence, of course, I** = I. Second, for any finite-rank free module, F, (H^F)** = H^F, just because the *-functor is additive. Third, for any f.p. module A let F and G be finite rank free modules and F --> G --> A --> O exact. Then O --> H^A --> H^G --> H^F is exact and is carried to exact O --> (H^A)** --> H^G --> H^F, hence (H^A)** = H^A. Finally, for any T in *F*, choose H^A --> H^B --> T --> O exact. The **-functor carries it to exact H^A --> H^B --> T** --> O Hence T** = T. [] Note that (H^A)* = [H^A,I] = (A@);I = A@. Hence (A@)* = H^A. SLOGAN: H^A and A@ are dual. It follows that A@ is injective in *F*. (It's a good exercise to find a direct proof.) And, of course, every object has an injective resolution (if H^A --> H^B --> T*--> O is exact then so is O --> T --> B@ --> A@). Note that an f.p, functor is projective iff it is representable and, dually, it is injective iff it is of the form A@. {Footnote: The category of f.p. abelian-group-valued sheaves on R-fpmod, that is, the category of f.p. abelian-group-valued _contravariant_ functors on R-fpmod need not have injective resolutions. In the case that R = Z any f.p. functor will carry Z to an f.p. abelian group. If E is injective in the category of f.p functors then for any n > 0 the map n:Z --> Z induces monic H_Z --> H_Z hence epic (H_Z,E) --> (H_Z,E). The latter is just multiplication by n on E(Z), hence E(Z) is a divisible group. But the only divisible finitely generated abelian group is O, thus H_Z can not have an injective extension.} DEFINE S@T = [S,T*]*. It follows immediately that this bi-functor on *F* is right-exact in both variables and that it is a commutative with I as unit. Note: H^A @ H^B = H^{A@B} (which together with the right-exactness characterizes @). THEOREM *F* is *-autonomous. BECAUSE: The only thing left to prove is Hom(S@T,U) = Hom(S,[T,U]) Both sides of this equation carry right-exact sequences in the first two positions to left-exact sequences in the category of abelian groups. It therefore suffices to verify the isomorphism when S = H^A and T = H^B: Hom(H^A @ H^B, U) = Hom(H^{A@B},U) = U(A@B) Hom(H^A,[H^B,U]) = Hom(H^A,[H^B,U]) = ]H^B,U](A) = U(A@B) [] COR: [S,T] = [T*,S*] LEMMA: For S and T in *F*, (S;T)* = S*;T*. BECAUSE: Both (S;T)* and S*;T* are exact in T. Because every object in *F* is obtainable using finite limits and co-limits starting with the object I, it therefore suffices to establish that (S;I)* = S*;I* and that's immediate. [] COR: (H^A)@S = [H^A,S*]* = ((A@);S*)* = H^A;S An interesting adjointness arises: PROPOSITION: For S and T in *F* Hom( H^A;S , T ) = Hom( S , A@;T ) BECAUSE: H^A;S = S@H^A and A@;T = [H^A,T]. [] {Footnote: When the compositions are reversed we obtain the routine adjointness: Hom( S;A@ , T ) = Hom( S , T;H^A ).} COR: [S,T](A) = Hom(S, A@;T) S*(A) = Hom(S,A@). {Footnote: The last equation is the one I used at Ottawa to compute S*. If all one knows is that there is a duality on *F* such that (H^A)* = A@ then we may infer S*(A) = Hom(H^A,S*) = Hom(S**,(H^A)*) = Hom(S,A@).} Recall that a coherent ring is one such that all finitely generated ideals are finitely presented. It follows that f.p. modules are closed under finite limits (hence form an exact subcategory). {Footnote: Clearly Noetherian implies coherent. For a quick separating example take the group ring of the rationals (usually called "polynomials with rational exponents"). All finitely generated ideals are principal. The important non-Noetherian examples arise in algebraic geometry.} LEMMA: For f.p. modules over a coherent ring Ext^n(A,-) and Tor_n(A,-) are dual. BECAUSE: Choose a free resolution ...--> F_{n+1} --> F_n -->....--> F_2 --> F_1 --> A --> 0 The Ext^n(A,--) functors are obtainable as the homology of the sequence: 0 --> H^A --> H^{F_1} --> H^{F_2} --> ... The Tor_n(A,--) functors are obtainable as the homology of the sequence: ... --> (F_2)@ --> (F_1)@ --> A@ --> O. These complexes are dual. [] It has not escaped my notice that this *-autonomous structure suggests a possible alternate approach to classical homological algebra. Just one example: a "connecting homomorphism" between half-exact f.p. functors S and T may be identified as an exact sequence of the form O --> S --> E --> P --> T --> O where E is injective and P is projective and this says that the duality works well with the notion of satellites and derived functors. From rrosebru@mta.ca Wed Nov 23 16:36:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 23 Nov 2005 16:36:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ef1Df-0000le-Jr for categories-list@mta.ca; Wed, 23 Nov 2005 16:27:55 -0400 X-Authentication-Warning: client122.comlab: lo owned process doing -bs Date: Tue, 22 Nov 2005 12:37:33 +0000 (GMT) From: Luke Ong To: categories@mta.ca Subject: categories: job: Postdoc at Oxford 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: 34 Samson Abramsky, Tom Melham, Oege de Moor and myself have recently founded the "Centre for Metacomputation" at Oxford. We are looking for a 4-year senior postdoc to help coordinate the activities of the centre: http://web.comlab.ox.ac.uk/oucl/jobs/platform Topics of interest include types for quotation or reflection, termination analysis, compositional model checking of (higher-order) programs, and games semantics for aspect-orientation. Applications from readers of this list would be very welcome! Please bring this opportunity to the attention of anyone who might be interested; naturally I'd be delighted to discuss the particulars on an informal basis. Many thanks, Luke Ong From rrosebru@mta.ca Wed Nov 23 16:36:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 23 Nov 2005 16:36:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ef1GN-00010g-DE for categories-list@mta.ca; Wed, 23 Nov 2005 16:30:43 -0400 Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <77132699-0BA8-4186-894D-A1114F3C597B@maths.adelaide.edu.au> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories List From: David Roberts Subject: categories: Equivalence relations Date: Wed, 23 Nov 2005 17:01:03 +1030 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 Dear all, Considering the well known fact that an equivalence relation R on a set S gives a groupoid S_R with object set S, and the quotient of S by R is pi_0(S_R), has anyone done any work on "equivalence relations" on categories? Taking the skeleton of a cat is the prototypical example, but what I had in mind was a more "relative" construction. Given a groupoid enriched in categories, taking a sort of Pi_1 would give us a groupoid mod "equivalent morphisms". There is a smell of relative homotopy about, and I don't know enough in that area. I realise there are a couple of levels to this game, as evidenced by Kapranov and Voevodsky in their paper on 2-cats and the Zamolodchikov tetrahedron equations - do we take a "skeleton" at one or more dimensions? Any pointers appreciated ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts@maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts www.trf.org.au "Go ye into all the world, and preach the gospel to every creature." - Mark 16:15 CRICOS Provider Number 00123M ----------------------------------------------------------- This email message is intended only for the addressee(s) and contains information that may be confidential and/or copyright. If you are not the intended recipient please notify the sender by reply email and immediately delete this email. Use, disclosure or reproduction of this email by anyone other than the intended recipient(s) is strictly prohibited. No representation is made that this email or any attachments are free of viruses. Virus scanning is recommended and is the responsibility of the recipient. From rrosebru@mta.ca Wed Nov 23 16:36:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 23 Nov 2005 16:36:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ef1Fx-0000yT-Gb for categories-list@mta.ca; Wed, 23 Nov 2005 16:30:17 -0400 Message-ID: <438366BB.20108@math.upenn.edu> Date: Tue, 22 Nov 2005 13:43:07 -0500 From: jim stasheff User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories List Subject: categories: [Fwd: Schreier theory discussion] 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: 36 -------- Original Message -------- Subject: Schreier theory discussion Date: Tue, 22 Nov 2005 18:01:42 +0100 (CET) From: Johannes Huebschmann To: ronnie@ll319dg.fsnet.co.uk, baez@math.ucr.edu, jds@math.upenn.edu, derek@math.ucr.edu CC: Johannes Huebschmann Dear Friends and Colleagues A few addenda to the Schreier theory discussion etc. which you might find interesting. (I am reacting to a message I received via J. Stasheff.) 1) As pointed out by R. Brwon, an approach to non-abelian cohomology may be phrased in terms of crossed modules. There is a notion more general than crossed modules, that of "crossed pair" which I introduced in the paper Group extensions, crossed pairs, and an eight term exact sequence, J. reine angew. Math. 321 (1981), 150--172. Crossed pairs may be used, for example, to explore group extensions, in particular, to examine differentials in the spectral sequence of a group extension. I worked this out in the paper Automorphisms of group extensions and differentials in the Lyndon--Hochschild--Serre spectral sequence, J. of Algebra 72 (1981), 296--334. I discovered later that the idea of crossed pair was in the literature before, under the name "pseudo module" in the 50's. Crossed pairs arise under other circumstances as well, for example in the Galois theory of Azumaya algebras. I have known this all my scientific life but never found the time to write it up properly. 2) I have worked out a rigorous approach to lattice gauge theory in the paper Extended moduli spaces, the Kan construction, and lattice gauge theory Topology 38 (1999), 555--596. In this paper, I discretize a space of based gauge equivalence classes of connections in terms of a combinatorial structure, and the resulting object is a COSIMPLICIAL space. The geometric realization of this space is G-equivariantly weakly homotopy equivalent to the space of based gauge equivalence classes of connections, where G refers to the structure group of the corresponding principal bundle. On such a space, for example, path integrals are well defined. As an illustration I worked out a calculation of the Chern-Simons invariant for lens spaces. The calculation involves identities among relations, a notion which arises in the structure theory of crossed modules. Best regards Johannes HUEBSCHMANN Johannes Professeur de Mathematiques USTL, UFR de Mathematiques UMR 8524 Laboratoire Paul Painleve F-59 655 Villeneuve d'Ascq Cedex France http://math.univ-lille1.fr/~huebschm TEL. (33) 3 20 43 41 97 (33) 3 20 43 42 33 (secretariat) (33) 3 20 43 48 50 (secretariat) Fax (33) 3 20 43 43 02 e-mail Johannes.Huebschmann@math.univ-lille1.fr --------------030708090300040304040700 Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit

-------- Original Message --------
Subject: Schreier theory discussion
Date: Tue, 22 Nov 2005 18:01:42 +0100 (CET)
From: Johannes Huebschmann <Johannes.Huebschmann@math.univ-lille1.fr>
To: ronnie@ll319dg.fsnet.co.uk, baez@math.ucr.edu, jds@math.upenn.edu, derek@math.ucr.edu
CC: Johannes Huebschmann <Johannes.Huebschmann@math.univ-lille1.fr> 


Dear Friends and Colleagues

A few addenda to the Schreier theory discussion etc. which you might find
interesting. (I am reacting to a message I received via J. Stasheff.)

1) As pointed out by R. Brwon, an approach to non-abelian cohomology
may be phrased in terms of crossed modules.
There is a notion more general than crossed modules, that of
"crossed pair" which I introduced in the paper

Group extensions, crossed pairs, and an eight term exact sequence,
J. reine angew. Math. 321 (1981), 150--172.

Crossed pairs may be used, for example, to explore group extensions, in
particular, to examine differentials in the spectral sequence of a group extension.
I worked this out in the paper

Automorphisms of group extensions and differentials in the
Lyndon--Hochschild--Serre spectral sequence,
J. of  Algebra 72 (1981), 296--334.

I discovered later that the idea of crossed pair was in the literature
before, under the name "pseudo module" in the 50's.


Crossed pairs arise under other circumstances as well,
for example in the Galois theory of Azumaya algebras.
I have known this all my scientific life but never found the time to write
it up properly.


2) I have worked out a rigorous approach to lattice gauge theory
in the paper

Extended moduli spaces, the Kan construction, and lattice gauge theory
Topology 38 (1999), 555--596.

In this paper, I discretize a space of based gauge equivalence classes of
connections in terms of a combinatorial structure, and the resulting
object is a COSIMPLICIAL space. The geometric realization of this space
is G-equivariantly weakly homotopy equivalent to the space of based gauge
equivalence classes of connections, where G refers to the structure group
of the corresponding principal bundle.

On such a space, for example, path integrals are well defined.
As an illustration I worked out a calculation of the Chern-Simons
invariant for lens spaces. The calculation involves identities among
relations, a notion which arises in the structure theory of crossed
modules.


Best regards

Johannes


HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

TEL. (33) 3 20 43 41 97
     (33) 3 20 43 42 33 (secretariat)
     (33) 3 20 43 48 50 (secretariat)
Fax  (33) 3 20 43 43 02

e-mail Johannes.Huebschmann@math.univ-lille1.fr
--------------030708090300040304040700-- From rrosebru@mta.ca Thu Nov 24 15:59:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 24 Nov 2005 15:59:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EfNCY-0007Gl-Qs for categories-list@mta.ca; Thu, 24 Nov 2005 15:56:14 -0400 From: Topos8@aol.com Message-ID: <281.4f1b3b.30b73c65@aol.com> Date: Thu, 24 Nov 2005 10:55:17 EST Subject: categories: Semigroups with many objects To: categories@mta.ca 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: 37 Is there an accepted terminology for semigroups with many objects, i.e. gadgets that satisfy the all the axioms satisfied by categories excepting those which refer to identities ? Thanks Carl Futia From rrosebru@mta.ca Thu Nov 24 15:59:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 24 Nov 2005 15:59:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EfNBI-00079N-ND for categories-list@mta.ca; Thu, 24 Nov 2005 15:54:56 -0400 Date: Wed, 23 Nov 2005 13:44:18 -0800 From: Toby Bartels To: Categories List Subject: categories: Internal anafunctors. Message-ID: <20051123214418.GB10753@math-rs-n04.ucr.edu> 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: 38 Has anybody worked out a theory of internal anafunctors? On the one hand, there is a notion of internal category, that is a category internal to some other category, such as (were these the first?) Ehresmann's differential categories. There are also (I assume that Ehresmann discussed these too) internal functors between these internal categories. In particular, one component of an internal functor from X to Y is a morphism from the object of objects of X to the object of objects of Y, just as one component of an ordinary functor from C to D is a function from the set of objects of C to the set of objects of D. On the other hand, Marco Makkai has argued that, if you don't believe in the axiom of choice (either because you disbelieve or wish to be agnostic), then you should use anafunctors (possibly always saturated) instead of functors in general category theory. In particular, an anafunctor from C to D does *not* necessarily include a function from the set of objects of C to the set of objects of D (although such a function does follow from the axiom of choice). Now, even if you believe in the axiom of choice, still there are many topoi in which choice does not hold. Yet Makkai's theory of anafunctors (being constructive) can be expressed in the internal language of a topos, so there is automatically a theory of internal anfunctor between internal categories in an arbitrary topos. (Arguably, this should be regarded as the right way to internalise category theory into a topos, or more generally to treat models of constructive category theory.) My question, then, is whether anybody has worked this out in arbitrary categories, or at least more generally than in topoi (for example, in an arbitrary site). In particular, has anybody worked out differentiable anafunctors between differentiable categories (internal to the category of differentiable spaces)? I am pretty sure that I know how to do this, and it will be used in my PhD dissertation. But I would prefer to give the proper credit, and even replace as many proofs as possible with citations to others' papers! ^_^ -- Toby Bartels From rrosebru@mta.ca Thu Nov 24 15:59:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 24 Nov 2005 15:59:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EfNBn-0007Cq-7r for categories-list@mta.ca; Thu, 24 Nov 2005 15:55:27 -0400 Message-ID: <438596F8.6080101@math.ist.utl.pt> Date: Thu, 24 Nov 2005 10:33:28 +0000 From: Amilcar Sernadas 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: Lisbon positions in quantum computation and information Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 39 (we apologize for multiple postings) The Center for Logic and Computation http://clc.math.ist.utl.pt/ of the Department of Mathematics of Instituto Superior T=E9cnico, Lisbon, Portugal, invites applications for visiting scientists and postdoctoral positions. There are no teaching duties associated with these positions. The visiting scientist positions are for three to six months (grant of around two thousand Euro per month). Applicants should have a strong research record in topics relevant to the QuantLog project. For details see: http://clc.math.ist.utl.pt/quantlog.html The postdoctoral positions are for one year (grant of around fifteen hundred Euro per month), with the possibility of renewal upon mutual agreement. Applicants should have a recent PhD and show high research potential in the areas of quantum computation, information and logic. Messages of intent should be sent by December 9, 2005 to Am=EDlcar Sernadas in order to get detailed information about the formal application procedure and deadline. The selection process will take place in early January, 2006. Positions can start in February, 2006. --=20 ++++++++++++++++++++++++++++++++++++++++++++++++++ Amilcar Sernadas Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL tel: 351-21-8417150 fax: 351-21-8417598 e-mail: acs@math.ist.utl.pt www: http://slc.math.ist.utl.pt/acs.html ++++++++++++++++++++++++++++++++++++++++++++++++++ From rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Effq0-0001kd-Vk for categories-list@mta.ca; Fri, 25 Nov 2005 11:50:13 -0400 Date: Fri, 25 Nov 2005 13:41:37 +0000 (GMT) From: Kirill Mackenzie X-X-Sender: pm1kchm@makar.shef.ac.uk To: categories@mta.ca Subject: categories: Equivalence relations 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: 40 Here are a couple of pointers for double groupoids (strict, but with arbitrary side structures) : 1. Given a double groupoid S, say over groupoids H and V with common base M, the set of orbits for S over V is a groupoid over the set of orbits for H over M. This is set out in \3 of @ARTICLE{, author = {K.~C.~H. Mackenzie}, title = {Double {L}ie algebroids and second-order geometry, {I}}, journal = {Adv. Math.}, volume = 94, number = 2, pages = {180--239}, year = 1992, } It gives an elegant way of passing from an affinoid to the corresponding `butterfly diagram' or `Morita equivalence'. 2. General quotients for a groupoid G can be formulated in terms of congruences, which in turn are sub double groupoids of the double groupoid structure on G\times G. This (done with Philip Higgins) is in \S2.4 of @book {MR2157566, AUTHOR = {Mackenzie, Kirill C. H.}, TITLE = {General theory of {L}ie groupoids and {L}ie algebroids}, SERIES = {London Mathematical Society Lecture Note Series}, VOLUME = {213}, PUBLISHER = {Cambridge University Press}, ADDRESS = {Cambridge}, YEAR = {2005}, PAGES = {xxxviii+501}, ISBN = {978-0-521-49928-3; 0-521-49928-3}, MRCLASS = {58H05 (53D17)}, MRNUMBER = {MR2157566}, } Kirill ---------- Forwarded message ---------- Date: Wed, 23 Nov 2005 17:01:03 +1030 From: David Roberts To: Categories List Subject: categories: Equivalence relations Dear all, Considering the well known fact that an equivalence relation R on a set S gives a groupoid S_R with object set S, and the quotient of S by R is pi_0(S_R), has anyone done any work on "equivalence relations" on categories? Taking the skeleton of a cat is the prototypical example, but what I had in mind was a more "relative" construction. Given a groupoid enriched in categories, taking a sort of Pi_1 would give us a groupoid mod "equivalent morphisms". There is a smell of relative homotopy about, and I don't know enough in that area. I realise there are a couple of levels to this game, as evidenced by Kapranov and Voevodsky in their paper on 2-cats and the Zamolodchikov tetrahedron equations - do we take a "skeleton" at one or more dimensions? Any pointers appreciated ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts@maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts www.trf.org.au From rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EffpI-0001gc-12 for categories-list@mta.ca; Fri, 25 Nov 2005 11:49:28 -0400 Date: Thu, 24 Nov 2005 23:49:22 +0000 From: Vincent Schmitt User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: semi-categories? 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: 41 I think that Isar Stubbe 's thesis treated those gadgets and he called them "semi-categories". From rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Effou-0001e3-7Z for categories-list@mta.ca; Fri, 25 Nov 2005 11:49:04 -0400 Message-ID: <32830.219.175.16.113.1132890984.squirrel@secure.octopus.com.au> In-Reply-To: <281.4f1b3b.30b73c65@aol.com> References: <281.4f1b3b.30b73c65@aol.com> Date: Fri, 25 Nov 2005 14:56:24 +1100 (EST) Subject: categories: Re: Semigroups with many objects From: duraid@octopus.com.au To: categories@mta.ca User-Agent: SquirrelMail/1.4.4 MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 X-Priority: 3 (Normal) Importance: Normal Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 > Is there an accepted terminology for semigroups with many objects, i.e. > gadgets that satisfy the all the axioms satisfied by categories excepti= ng > those > which refer to identities ? Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in bicategories" (TAC vol 3, no 8 (1997)). Duraid From rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Effo7-0001af-6B for categories-list@mta.ca; Fri, 25 Nov 2005 11:48:15 -0400 From: Topos8@aol.com Message-ID: <1e8.47042c41.30b7e030@aol.com> Date: Thu, 24 Nov 2005 22:34:08 EST Subject: categories: Re: Semigroups with many objects To: categories@mta.ca 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: 43 In a message dated 11/24/2005 2:20:54 PM Central Standard Time, P.B.Levy@cs.bham.ac.uk writes: Dear Carl, > Is there an accepted terminology for semigroups with many objects, i.e. > gadgets that satisfy the all the axioms satisfied by categories > excepting those which refer to identities ? Do you have any examples of such things? I'd be interested to know. Paul The principal examples I know are all related to the category of Moore paths in a topological space X . It turns out to be convenient not to have any degenerate paths when making certain constructions, especially when working with the higher dimensional versions of these gadgets. This can be arranged as follows. Consider the set of non-identity paths of the Moore category. Define the domain (respectively, the codomain) of a path f to be the path of UNIT length that is constantly f ( 0 ) (resp., constantly f ( 1 ) ). Two paths f , g, can be composed if codomain g = domain f and the composite is then the concatenation ( f followed by g ). This is a semigroup with many objects, i.e. a directed graph with an associative composition law. The multiplication is stricly associative but there are no identities or even any idempotents. I don't like the term "semigroupoids" because it evokes (for me) the notion of invertibility which I want to avoid. I know Anders Kock has suggested "fair categories" to describe category-like objects in which there are identities unique only "up to homotopy", but semigroups with many objects don't have any identities at all. The term "near category" occurred to me but I seem to recall this being used to describe something else and I can't put my finger on that reference. Of course, when the day comes that "higher dimensional algebra" is just "algebra" maybe semigroups with many objects will just be called semigroups ( and semigroups with one object called a proper semigroups ?), groupoids will be called groups (and groups with one object called proper groups ?) and categories will be called monoids (and monoids with one object called proper monoids?). Carl From rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EffqW-0001mr-To for categories-list@mta.ca; Fri, 25 Nov 2005 11:50:45 -0400 Date: Fri, 25 Nov 2005 16:19:30 +0100 (CET) From: Jaap van Oosten Reply-To: Jaap van Oosten Subject: categories: Master Class in Mathematical Logic, 2006-7 To: categories@mta.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: cZH4eqwjI4XbSSrUydNuKA== Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 44 [Apolgies for multiple postings] Dear Colleagues, please bring the following to the attention of your students. 2006-2007 MASTER CLASS IN MATHEMATICAL LOGIC In the academic year 2006-2007 a year-long program of courses in Mathematical Logic is organized by MRI (a cooperation of Dutch Universities). The program is intended for advanced undergraduate and beginning graduate students, and aims to provide them with a solid preparation for a possible Ph.D. studentship in the area. There are possibilities for fellowships for students. Students interested in fellowships should apply before January 15, 2006. Details can be found at http://www.math.uu.nl/people/jvoosten/mclogic In particular, a brochure and a poster (in pdf format) can be downloaded there; one also finds a list of the courses that will be given. Jaap van Oosten From rrosebru@mta.ca Fri Nov 25 17:10:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 17:10:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EfkmY-0005KM-4R for categories-list@mta.ca; Fri, 25 Nov 2005 17:06:58 -0400 Date: Fri, 25 Nov 2005 16:36:59 +0000 (GMT) From: Kirill Mackenzie To: categories@mta.ca Subject: categories: Re: Schreier theory In-Reply-To: <437C9478.7010604@ll319dg.fsnet.co.uk> Message-ID: References: <437C9478.7010604@ll319dg.fsnet.co.uk> 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: 45 On Thu, 17 Nov 2005, Ronald Brown wrote: > > An interesting problem is to classify extensions of crossed complexes! > > There is an interesting account of extensions of principal bundles and > transitive Lie groupoids by Androulidakis, developing work of Mackenzie, > at math.DG/0402007 (not using crossed complexes). > Apropos the classification of extensions of principal bundles (equivalently, of locally trivial Lie groupoids) : The 1989 paper on extensions of principal bundles which Iakovos Androulidakis developed, @article { , AUTHOR = {Mackenzie, Kirill}, TITLE = {Classification of principal bundles and {L}ie groupoids with prescribed gauge group bundle}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {58}, YEAR = {1989}, NUMBER = {2}, PAGES = {181--208}, ISSN = {0022-4049}, CODEN = {JPAAA2}, MRCLASS = {58H05 (20L05 22E99 55R15 58A99)}, } actually set out to demonstrate that non-abelian cohomology is _not_ needed for the classification. The standard classification of principal bundles P(M,G) with M, G given is by nonabelian H^1. My point was that it is at least equally interesting to classify possible P(M,G) with not just G prescribed, but the bundle of Lie groups associated to P through the inner automorphism action. This is variously called the gauge group bundle or (I think misleadingly) the adjoint group bundle. Given M and a bundle of Lie groups B on M, there is an obstruction class to the existence of a principal bundle P(M,G) with gauge group bundle B. If such a P exists then all possible such P are classified in the usual way by abelian cohomology. This approach extends in principle to general extensions of principal bundles. This approach arose because in the corresponding problem on the infinitesimal level, it is certainly more natural to classify transitive Lie algebroids with prescribed adjoint bundle. (The adjoint bundle of a transitive Lie algebroid is the kernel of the anchor map. For the Atiyah sequence of a principal bundle it is the bundle associated to P through the adjoint representation.) Kirill ============================================= Kirill C H Mackenzie Department of Pure Mathematics University of Sheffield Sheffield S3 7RH United Kingdom http://www.shef.ac.uk/~pm1kchm/ ============================================= From rrosebru@mta.ca Fri Nov 25 17:10:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 17:10:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EfknJ-0005Nk-U4 for categories-list@mta.ca; Fri, 25 Nov 2005 17:07:45 -0400 Mime-Version: 1.0 (Apple Message framework v734) To: categories@mta.ca Message-Id: <58287EFA-C101-43F5-8DA2-1DD8391D2593@mac.com> Content-Type: multipart/alternative; boundary=Apple-Mail-5-10634711 From: Krzysztof Worytkiewicz Subject: categories: re: semi-categories? Date: Fri, 25 Nov 2005 11:37:37 -0500 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 46 In addition to I.Stubbe's, there are works by Schroeder, Herrlich-=20 Schroeder and Borceux et al. Further examples of (topologically enriched) semicategories are =20 ghizmos called =A8Flows=A8, promoted by Ph.Gaucher (this author crafts =20= everything from scratch, without any mention of semicategories and =20 enrichment). From rrosebru@mta.ca Fri Nov 25 17:10:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 17:10:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Efkoi-0005Sj-9A for categories-list@mta.ca; Fri, 25 Nov 2005 17:09:12 -0400 Date: Fri, 25 Nov 2005 17:51:45 +0000 From: Miles Gould To: categories@mta.ca Subject: categories: Re: Semigroups with many objects Message-ID: <20051125175145.A544@assyrian.org.uk> References: <1e8.47042c41.30b7e030@aol.com> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.2.5i In-Reply-To: <1e8.47042c41.30b7e030@aol.com>; from Topos8@aol.com on Thu, Nov 24, 2005 at 10:34:08PM -0500 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 47 On Thu, Nov 24, 2005 at 10:34:08PM -0500, Topos8@aol.com wrote: > I don't like the term "semigroupoids" because it evokes (for me) the notion > of invertibility which I want to avoid. Google's not a perfect metric for popularity, but it returns about 350 hits for "semigroupoid", about 10 for "fair category", and none for "near category" (the one hit it returns is spurious). Wikipedia has an entry for "semigroupoid" (with the definition you're thinking of) and nothing on any of the others. Looking at MathSciNet, we find 180 hits for "semigroupoid", none for "fair category", and one for "near category". It's worse than that, though, because that paper uses "near category" to mean something different, namely a category-like object with identities but without associativity! All this suggests to me that "semigroupoid" is the standard term, and certainly it's the only one I've ever heard before. I don't think you need to worry about implied invertibility: if you know what both a groupoid and a semigroup are, the term "semigroupoid" strongly suggests a multi-object structure with associatively-composable arrows, but not necessarily with identities. At least, it suggests that to me :-) Hope that helps, Miles -- If you want to see your plays performed the way you wrote them, become President. -- Vaclav Havel From rrosebru@mta.ca Fri Nov 25 17:10:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 17:10:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Efkp5-0005Uq-VF for categories-list@mta.ca; Fri, 25 Nov 2005 17:09:36 -0400 Date: Fri, 25 Nov 2005 12:26:13 -0800 From: Toby Bartels To: Categories List Subject: categories: Re: Internal anafunctors. Message-ID: <20051125202613.GA25555@math-cl-n03.ucr.edu> References: <20051123214418.GB10753@math-rs-n04.ucr.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: <20051123214418.GB10753@math-rs-n04.ucr.edu> User-Agent: Mutt/1.4i Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 48 I wrote in small part: >Marco Makkai has argued that you should use anafunctors. Several people have pointed out to me that I mean Michael Makkai here (not to be confused with Marco Mackaay). ^^^^^^^ Since Makkai may read this list (at least, he's posted in the past), I hope he's not offended! I apologise. --Toby From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXMU-0000Gf-Ii for categories-list@mta.ca; Sun, 27 Nov 2005 20:59:18 -0400 From: J=FCrgen Koslowski Message-Id: <200511272338.jARNc2O14251@lxt5.iti.cs.tu-bs.de> Subject: categories: Re: Semigroups with many objects (fwd) To: categories@mta.ca (categories list) Date: Mon, 28 Nov 2005 00:38:02 +0100 (CET) X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=3Diso-8859-1 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 49 [I answered this on Friday, but haven't seen my reply so far.] =3D20 > > Is there an accepted terminology for semigroups with many objects, i.= =3D e. > > gadgets that satisfy the all the axioms satisfied by categories excep= =3D ting > > those > > which refer to identities ? >=3D20 > Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in > bicategories" (TAC vol 3, no 8 (1997)). Well, not quite. In a taxonomy the identity axiom is not simply removed, but replaced by a weaker requirement. Essentially this says that every morphism factors. In the corresponding semigroups, which I would call "interpolative", this can also be formulated as follows: the multiplication * : S x S ---> S is a coequalizer of S x * and * x S . (This formulation can be lifted to taxonomies as well.) It would be interesting to know, whether the gadget that prompted this question is a taxonomy in this sense. [By the way, the term "taxonomy" resulted from a misunderstanding on my part of a remark by Robert Pare and Richard Wood. It has since been abbreviated to "taxon".] The notion of "category without units" also shows up in Azumaya=3D20 theory as studied by Francis Borceux and others. -- J=3DFCrgen --=3D20 Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR) From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXKX-0000Ap-6r for categories-list@mta.ca; Sun, 27 Nov 2005 20:57:17 -0400 From: Topos8@aol.com Message-ID: <284.505656.30b9d225@aol.com> Date: Sat, 26 Nov 2005 09:58:45 EST Subject: categories: correction to "semigroupoid algebras" remark To: categories@mta.ca 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: 50 In my comment on Miles' post which surveyed current terminology I said: "The algebra closure in the weak operator topology then defines the "semigroupoid" C star algbera. Of course this algebra does contain one idempotent for each object, but this is a consequence of taking the algebra- closure of the set of patial isometries defined by the arrows. " The last sentence is incorrect. I have to stop associating "... oid" with invertibility. Carl From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXLF-0000Co-7a for categories-list@mta.ca; Sun, 27 Nov 2005 20:58:01 -0400 Date: Sun, 27 Nov 2005 13:40:19 -0500 (EST) From: Susan Niefield To: categories@mta.ca Subject: categories: Union College Conference 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 UNION COLLEGE MATHEMATICS CONFERENCE Saturday and Sunday Schenectady December 3-4, 2005 New York This is the second announcement for the twelfth Union College Mathematics Conference featuring: John Baez (University of California, Riverside) David Cox (Amherst College) Jesper Grodal (University of Chicago) In addition to these plenary speakers, there will be shorter contributed talks in parallel sessions in algebraic topology, category theory, and commutative algebra. If you plan to attend and have not yet registered, please contact one of the organizers as soon as possible so that we can plan accordingly. The meeting will begin with a reception 8:00 to 10:00 PM on Friday, December 2 in Bailey Hall 204, and end at 3:30 on Sunday afternoon. For a detailed schedule and other information see our Web page at . CONFERENCE ORGANIZERS Category Theory Susan Niefield niefiels@union.edu Algebraic Topology Brenda Johnson johnsonb@union.edu Kathryn Lesh leshk@union.edu Commutative Algebra Pedro Teixeira teixeirp@union.edu David Vella vellad@union.edu We hope to see you next week! From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXHD-0007mh-02 for categories-list@mta.ca; Sun, 27 Nov 2005 20:53:51 -0400 Subject: categories: CT 2006 announcement To: categories@mta.ca (Categories List) Date: Fri, 25 Nov 2005 19:33:55 -0400 (AST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20051125233355.9BF5973736@chase.mathstat.dal.ca> From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 52 International Category Theory Conference CT 2006 June 25 - July 1, 2006 White Point, Nova Scotia http://www.mathstat.dal.ca/~selinger/ct2006/ * * * The International Category Theory Conference (CT) covers all areas of pure and applied category theory. Topics of interest include, but are not limited to: higher dimensional categories, categorical logic, applications of categories in algebra, topology, combinatorics, and other areas of mathematics, applications of category theory to computer science, physics and other mathematical sciences. Previous meetings in this series were held in Vancouver (2004), Como (2000), Coimbra (1999), Vancouver (1997), Halifax (1995), Tours (1994), Isle of Thorns (1992), Montreal (1991), and Como (1990). CT 2006 will be an active research-oriented conference, in something of an "Oberwolfach style". The seaside setting will offer more opportunities for informal discussion and collaboration in the evenings than might be the case in an urban setting. There will also be many opportunities for recreational activities. All those interested in category theory and its applications are welcome. LOCATION: The CT 2006 conference will be held at the White Point Beach Resort (http://www.whitepoint.com/). White Point is a seaside resort in Nova Scotia (Canada), about 100 minutes' drive from Halifax. The conference venue offers a choice of accommodation, both in comfortable finished cabins and in "hotel-style" rooms. All meals are included in the price of the rooms. (There will be a moderate registration fee to cover other conference costs.) For more details on accommodations and reservations, as well as details of how to get there, please see the conference web page. TO GIVE A TALK: Prospective speakers should submit an abstract of up to one page. Abstracts should be sufficiently detailed to allow the scientific committee to assess the merits of the work. Submissions should be in plain text, Postscript, or PDF format, and must be sent to ct06@mathstat.dal.ca by February 27. Receipt of all submissions will be acknowledged by return email. Authors will be notified by March 30 of the acceptance of their talks. BOOKING YOUR ROOM We would like to stress that, due to the secluded rural location of the conference, there are no convenient alternative accommodations. White Point Beach Resort has reserved a large block of rooms and cottages for us, but cannot hold them indefinitely; if numbers are below expectation by late January, they may have to reduce the number of rooms reserved. While all the rooms at White Point are very nice, there are various different styles; booking early will assure you the best choice of accommodation. (Should unforeseen circumstances make it impossible for you to attend, you can cancel for a $5 processing fee.) IMPORTANT DEADLINES: Feb 27, 2006: submission of abstracts Mar 1, 2006: early registration Mar 30, 2006: notification of authors Apr 20, 2006: registration SCIENTIFIC COMMITTEE: Jiri Adamek John Baez Michael Barr Eugenia Cheng Maria Manuel Clementino Marcelo Fiore Peter Freyd Jonathon Funk Peter Johnstone Steve Lack Susan Niefield Phil Scott Ross Street Walter Tholen (chair) Enrico Vitale ORGANIZERS: Robert Dawson (rdawson@cs.stmarys.ca) Dorette Pronk (pronk@mathstat.dal.ca) Peter Selinger (selinger@mathstat.dal.ca) CONFERENCE EMAIL AND WEBSITE: ct06@mathstat.dal.ca http://www.mathstat.dal.ca/~selinger/ct2006/ From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXIS-00004d-Vh for categories-list@mta.ca; Sun, 27 Nov 2005 20:55:09 -0400 Date: Sat, 26 Nov 2005 07:50:40 -0500 (EST) From: Peter Freyd Message-Id: <200511261250.jAQCoetu005155@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: *-Autonomous Functor Categories, revision Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 53 Mike Barr has pointed out that the proof in my last posting of LEMMA: The object I = H^R is injective in *F*. doesn't work. (It was actually the fourth proof I had come up with. I wondered why it was so much simpler). So here's one that does work (and is just about as simple). Let O | H^R | H^A --> H^B --> T --> O be exact (all vertical arrows point down). We seek a retraction for H^R --> T. Since H^R is projective (as is any representable) we may choose a map H^R --> H^B to yield a commutative triangle. The full subcategory of representables is closed under finite limits, so let H^C --> H^R | | H^A --> H^B be a pullback in *F* and let B --> A | | R --> C be the corresponding pushout in the category of f.p R-modules. The map from H^C to T is the zero map and we use the hypothesis that H^R --> T is monic to infer that H^C --> H^R, hence R --> C, are zero maps. Let O --> K --> B --> A be exact. It is an exercise in abelian categories that R --> C = 0 implies K --> B --> R is epi. Now (finally using the projectivity of R) choose a retraction R --> K. The map H^A --> H^B --> H^K --> H^R is of course, a zero map and we may factor H^B --> H^K --> H^R as H^B --> T --> H^R. The map T --> H^R is easily checked to be the retraction we seek. From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXJr-00008X-Ps for categories-list@mta.ca; Sun, 27 Nov 2005 20:56:35 -0400 From: Topos8@aol.com Message-ID: <2ba.97b7ad.30b9cfe3@aol.com> Date: Sat, 26 Nov 2005 09:49:07 EST Subject: categories: Re: Semigroups with many objects To: categories@mta.ca 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: 54 Well, it does seem that "semigroupoids" is the preferred terminology. Searching on this term through the math xxx archive pulls up a number of papers which study various kinds of C star algebras built on top of semigroupoids. The idea is to use the objects of the semigroupoid to index a basis in a (separable) Hilbert space and to use the arrows to define partial isometries of this Hilbert space in the obvious way. The algebra closure in the weak operator topology then defines the "semigroupoid" C star algbera. Of course this algebra does contain one idempotent for each object, but this is a consequence of taking the algebra- closure of the set of patial isometries defined by the arrows. Carl Futia From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXG0-0007jM-HX for categories-list@mta.ca; Sun, 27 Nov 2005 20:52:36 -0400 Date: Fri, 25 Nov 2005 22:24:37 +0100 From: Joachim Kock Subject: categories: Re: Semigroups with many objects In-reply-to: <281.4f1b3b.30b73c65@aol.com> X-Sender: kock@127.0.0.1:55110 To: categories@mta.ca Message-id: MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-transfer-encoding: QUOTED-PRINTABLE References: <281.4f1b3b.30b73c65@aol.com> X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 55 >Is there an accepted terminology for semigroups with many objects, i= .e. >gadgets that satisfy the all the axioms satisfied by categories exce= pting those which refer to identities ? Perhaps 'semi-category' is the most widely used term.=20 The word 'taxonomy' has also been used (Par=E9, Wood, Ageron), but Koslowski has used that word for something a bit more complicated ('interpolads in SPAN'). On the other hand, Schroeder has used the word 'semi-category' for the 'multiplicative graphs' of Ehresmann (some structure where composition of arrows is not always defined even if their source and target match). (Curiously, in a preliminary version of the paper by Moens, Berni-Canani, and Borceux, 'On regular presheaves and regular semi-categories', the term 'multiplicative graph' was used for 'semi-category' -- the final version uses 'semi-category'.) I would also like to advogate 'semi-monoid' instead of 'semi-group', and 'semi-monoidal category' for 'monoidal category without unit'. It seems to be too late at this point to convince operadists to say 'semi-operad' for operads without unit. In the same spirit I find it convenient to use 'semi-simplicial set' for presheaves on Delta-mono, but I am told that this is confusing, since apparently 'semi-simplicial set' meant something else fifty years ago... Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock Departament de Matem=E0tiques -- Universitat Aut=F2noma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 25 34 Fax: +34 93 581 27 90 ---------------------------------------------------------------- From rrosebru@mta.ca Sun Nov 27 21:02:15 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:02:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXHz-000033-W1 for categories-list@mta.ca; Sun, 27 Nov 2005 20:54:40 -0400 From: Philippe Gaucher Reply-To: gaucher@pps.jussieu.fr To: categories@mta.ca Subject: Re: categories: Re: Semigroups with many objects Date: Sat, 26 Nov 2005 05:30:06 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="windows-1252" Content-Disposition: inline Message-Id: <200511260530.06987.gaucher@pps.jussieu.fr> Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 56 Le vendredi 25 Novembre 2005 04:56, duraid@octopus.com.au a =E9crit : > > Is there an accepted terminology for semigroups with many objects, i.= e. > > gadgets that satisfy the all the axioms satisfied by categories excep= ting > > those > > which refer to identities ? > > Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in > bicategories" (TAC vol 3, no 8 (1997)). > > Duraid Dear all, I call a "small semigroup with many objects enriched over the model categ= ory=20 of compactly generated topological spaces" a "flow" in my work (these obj= ects=20 are interesting for me only if they are enriched over very particular mod= el=20 categories satisfying particular properties). The terminology comes from = the=20 fact that I use them to study the time flow of a higher dimensional autom= aton=20 (up to directed homotopy).=20 For "taxonomy", I would be very curious to know the origin of the termino= logy.=20 What does it mean exactly ? In the paper q-alg/9608025 "Flexible sheaves", Carlos Simpson calls a "(n= ot=20 necessarily small) semigroup with many objects enriched over the category= of=20 topological spaces" a continuous semicategory.=20 I had also seen the word "precategory" but I cannot remember where. Bewar= e of=20 the fact that the word precategory is also used for categories *with=20 identities* such that the composition law is partially defined : that is = the=20 fact that the codomain of F is equal to the domain of G is not sufficient= for=20 GoF to exist. Once again, I cannot remember where I read this word. The o= nly=20 thing I remember is that that was a computer-scientific work. The word "non-unital category" is also used sometime in mathematical pape= rs.=20 pg. From rrosebru@mta.ca Sun Nov 27 21:03:12 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 27 Nov 2005 21:03:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EgXPb-0000RT-Pu for categories-list@mta.ca; Sun, 27 Nov 2005 21:02:31 -0400 Date: Sat, 26 Nov 2005 18:42:05 +0100 From: Andree Ehresmann To: TAC Subject: categories: Quasi-categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 57 In answer to Carl Futia Charles Ehresmann has called such "gadgets" quasi-categories in "Introduction to the theory of structured categories" (Kansas 1966) reprinted in "Charles Ehresmann: Oeuvres completes et commentees" Part III-2. Sincerely Andree C. Ehresmann From rrosebru@mta.ca Mon Nov 28 12:08:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 28 Nov 2005 12:08:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EglTx-0001wR-35 for categories-list@mta.ca; Mon, 28 Nov 2005 12:03:57 -0400 Date: Mon, 28 Nov 2005 07:26:29 -0500 (EST) From: Peter Freyd Message-Id: <200511281226.jASCQTx3009541@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: re: semigroups with many objects Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 58 Jacobson coined "rng" for a ring without identity and Bill returned the favor by proposing "rig" for a ring without negation (at least Bill's proposal can be pronounced). Alas, "catgory" is the only approximation for the instant case -- the only one, that is, if you refuse to count "ctegory" (category without automorphisms). Seriously though, "semi-category" is the one proposal not needing explanation. May I suggest that its very obviousness is why it was avoided. From rrosebru@mta.ca Mon Nov 28 12:08:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 28 Nov 2005 12:08:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EglSo-0001qW-LH for categories-list@mta.ca; Mon, 28 Nov 2005 12:02:46 -0400 Message-ID: <438A7BCE.9070609@cs.stanford.edu> Date: Sun, 27 Nov 2005 19:38:54 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: *-Autonomous Abelian Functor Categories References: <200511211651.jALGpEKx024980@saul.cis.upenn.edu> In-Reply-To: <200511211651.jALGpEKx024980@saul.cis.upenn.edu> 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: 59 Peter Freyd wrote on 11/21/2005: > ... > NOTATION: Every group-valued functor from a category of R-modules, > commutative R, can be canonically lifted to a module-valued functor. > Given two such functors, S and T, we follow the CS tradition of > denoting their composition, "first apply S then T" as S;T (hence > (S;T)(A) = T(S(A)). Actually it is an RA (Relation Algebra) tradition dating back to the 19th century. In my LICS'92 evening history talk, "Origins of the Calculus of Binary Relations", http://boole.stanford.edu/pub/ocbr.pdf, I attributed it as follows. > But this view of composition/concatenation as a form of conjunction > predates even Peirce and would appear to be due to De Morgan in 1860 > [DeM]. The following footnote appears exactly one-third of the > way through De Morgan's ``On the Syllogism IV'' (p.221 in Heath's > anthology ``On the Syllogism'' [DeM66]). Here De Morgan argues > that, allowing for the obvious differences, composition L;M of > relations L and M resembles conjunction XY of ``terms'' > (predicates) X and Y. Indeed he notates composition LM the > better to suggest conjunction---the L;M notation which is now in > almost universal use, and is in (fortuitous?) agreement with Algol 60 > and dynamic logic [Pr76], was introduced later by Peirce. I still don't know whether RA played any role in the adoption of ; by Algol 60. However Algol 60 used ; not as a statement terminator (as in C or Java) but as an associative infix operator between statements, suggestive of RA influence. (Perhaps so as not to overly inconvenience those who tended to think of semicolon as a terminator anyway, the empty string was permitted as a statement, inadvertently complicating the task of generating the language Algol 60 with an unambiguous context-free grammar.) Vaughan Pratt From rrosebru@mta.ca Mon Nov 28 12:08:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 28 Nov 2005 12:08:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EglTO-0001tR-Rz for categories-list@mta.ca; Mon, 28 Nov 2005 12:03:22 -0400 From: Philippe Gaucher Reply-To: gaucher@pps.jussieu.fr To: categories@mta.ca Subject: Re: categories: Quasi-categories Date: Mon, 28 Nov 2005 08:06:08 +0100 User-Agent: KMail/1.7.1 References: In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset="windows-1252" Content-Disposition: inline Message-Id: <200511280806.09112.gaucher@pps.jussieu.fr> Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 60 Le samedi 26 Novembre 2005 18:42, vous avez =E9crit : > In answer to Carl Futia > > Charles Ehresmann has called such "gadgets" quasi-categories in > "Introduction to the theory of structured categories" (Kansas 1966) > reprinted in "Charles Ehresmann: Oeuvres completes et commentees" Part > III-2. > > Sincerely > Andree C. Ehresmann Dear All, One told me very recently that Joyal is writting a book about=20 "quasi-categories". But with a different meaning. A quasi-category is a=20 simplicial set satisfying a condition a little bit weaker than the Kan=20 condition. Morally speaking, two composable arrows have several possible=20 compositions, up to homotopy. I dont know whether Joyal reads this=20 mailing-list ?=20 pg. From rrosebru@mta.ca Wed Nov 30 15:04:50 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:50 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX7D-0001wy-8i for categories-list@mta.ca; Wed, 30 Nov 2005 14:55:39 -0400 Message-Id: <200511290711.jAT7BeT11834@math-cl-n03.ucr.edu> Subject: categories: higher gauge theory To: categories@mta.ca (categories) Date: Mon, 28 Nov 2005 23:11:40 -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: 61 Dear Categorists - Here's a new paper: http://math.ucr.edu/home/baez/higher.pdf http://math.ucr.edu/home/baez/higher.ps John Baez and Urs Schreiber Higher Gauge Theory Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing's theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing's framework, this requires that the `fake curvature' must vanish. In this paper we summarize the main results of our theory without proofs. Fans of Lie groupoids may enjoy the "smooth 2-groupoid" used in this paper. Best, jb From rrosebru@mta.ca Wed Nov 30 15:04:50 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:50 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX83-00022B-VB for categories-list@mta.ca; Wed, 30 Nov 2005 14:56:31 -0400 Message-Id: X-Mailer: Eudora F6.0.2 Date: Tue, 29 Nov 2005 14:55:58 +0100 To: categories@mta.ca From: Francis Borceux Subject: categories: semi-categories Content-Type: text/plain; charset="iso-8859-1" ; format="flowed" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 62 There has been a long discussion on the list=20 about "categories without identities", whatever=20 you decide to call them. And the attention has=20 been brought to axioms which could -- in this=20 more general context -- replace the identity=20 axiom. I would like to focus on a very striking categorical aspect of this problem. A (right) module M on a ring R with unit must satisfy the axiom m1=3Dm =2E.. but what about the case when R does not have a unit ? Simply dropping the axiom m1=3Dm leaves you with=20 the unpleasant situation where you have two=20 different notions of module, in the case where R=20 has a unit. Therefore people working in linear algebra have considered the axiom the scalar multiplication M@R ---> M is an isomorphism (@=3Dtensor product sign) which is equivalent to the axiom m1=3Dm, when the=20 ring has a unit ... but makes perfect sense when=20 the ring does not have a unit. Such modules are=20 generally called "Taylor regular". A ring R with unit is simply a one-objet additive=20 category and a right module M on R is simply an=20 additive presheaf M ---> Ab (=3Dthe category of=20 abelian groups). A ring without unit is thus a "one-object=20 additive category without identity", again=20 whatever you decide to call this. But what is the analogue of the axiom M@R ---> M is an isomorphism when R is now an arbitrary small (enriched)=20 "category without identities" and M is an=20 arbitrary (enriched) presheaf on it ? All of us know that to define a (co)limit, we do=20 not need at all to start with an indexing=20 category: an arbitrary graph with arbitrary=20 commutativity conditions works perfectly well. In=20 particular, a "category without identities" is=20 all right. And the same holds in the enriched=20 case, with (co)limits replaced by "weighted=20 (co)limits". Now every presheaf on a small category is=20 canonically a colimit of representable ones ...=20 but this result depends heavily on the existence=20 of identities ! When you work with a presheaf M=20 on a "category R without identities", you still=20 have a canonical morphism canonical colimit of representables ---> R and you can call M "Taylor regular" when this is=20 an isomorphism. Again in the enriched case,=20 "colimit" means "weighted colimit". This=20 recaptures exactly the case of "Taylor regular=20 modules", when working with Ab-enriched=20 categories. A sensible axiom to put on a "category R without=20 identities" is the fact that the representable=20 functors are "Taylor regular". (We should=20 certainly call this something else than "Taylor=20 regular", but let me keep this terminology in=20 this message.) And when R is a "Taylor regular category without identities", the constructi= on presheaf on R |---> corresponding canonical colimit of representables yields a reflection for the inclusion of Taylor=20 regular presheaves in all presheaves. A very striking property is the existence of a=20 further (necessarily full and faithful) left=20 adjoint to this reflection. This second inclusion=20 provides in fact an equivalence with the full=20 subcategory of those presheaves which satisfy the=20 Yoneda isomorphism. This yields thus a nice example of what Bill=20 Lawvere calls the "unity of opposites": the two=20 inclusions identify the category of Taylor=20 regular presheaves with * on one side, those presheaves which are colimits of representables; * on the other side, those presheaves which satisfy the Yoneda lemma. This underlines the pertinence of these "Taylor=20 regular categories without identities". To my knowledge, the best treatment of these=20 questions is to be found in various papers by=20 Marie-Anne Moens and by Isar Stubbe, in=20 particular in the "Cahiers" and in "TAC". And very interesting examples occur in functional=20 analysis (the identity on a Hilbert space is a=20 compact operator ... if and only if the space is=20 finite dimensional) and also in the theory of=20 quantales. =46rancis Borceux -- =46rancis BORCEUX D=E9partement de Math=E9matique Universit=E9 Catholique de Louvain 2 chemin du Cyclotron 1348 Louvain-la-Neuve (Belgique) t=E9l. +32(0)10473170, fax. +32(0)10472530 borceux@math.ucl.ac.be From rrosebru@mta.ca Wed Nov 30 15:04:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX6m-0001tS-NW for categories-list@mta.ca; Wed, 30 Nov 2005 14:55:12 -0400 Date: Mon, 28 Nov 2005 14:07:38 -0500 (EST) From: Peter Freyd Message-Id: <200511281907.jASJ7cEW026208@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Categories Anonymous X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 63 While checking Jacobson's responsibility for "rng" Google directed me to www.math.binghamton.edu/alex/Yale_Math_Skit.html where we learn a most appropriate pronunciation for the word. And it reminds those of us old enough just what we used to be up against. Script of skit performed at the Yale Math Department Christmas Party in 1974 by the second year graduate students: Chico Miraglia, Craig Huneke, Paul Blanchard, Bob Bix, Richard Foote, Alex Feingold, Monica Baratieri, Dan Coro, John Elton (Piano!), John M(?). * * * Jacobson Interview Johnny: Now, please welcome our next guest, the current President of Categories Anonymous, Mr. Nathan Jacobson. (He enters) Mr. Jacobson, just what does Categories Anonymous do? Jacobson: Johnny, our organization was founded to lead the fight against Categories. This dreaded disease can strike any mathematician, regardless of age, sex, sponsoring institution, or expected area of concentration. Categories is now the Number One Killer of Mathematics. Johnny, did you know that Categories has destroyed more mathematics than co- and homology combined? Johnny: No. Jacobson: Did you know that algebraic geometry used to make some sense? Johnny: I didn't know that. Jacobson: Now all our rings have units. Where did we go rng? I'm sorry, Johnny, but when something this terrible strikes so close to home -- Johnny: I understand. But tell me, is there a cure for categories? Jacobson: Not yet. But if diagnosed early enough, categories can be controlled; the patient is forced to factor integers into primes and multiply matrices out until his insanity passes. Johnny: Can a person examine himself for categories? (Ed McMath looks himself over) Jacobson: I'm glad you asked me that. Remember that categories can strike anyone, any time, any place. So learn the Early Warning Signs for Categories: 1. Have you stopped caring whether you understand what you're doing? 2. Do you like your theorems general but vacuous? 3. Do you consider yourself a social arrow-chaser? 4. Do you think of everything as a universal object? * * * From rrosebru@mta.ca Wed Nov 30 15:04:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX7a-0001yx-Fr for categories-list@mta.ca; Wed, 30 Nov 2005 14:56:02 -0400 From: Thomas Streicher Message-Id: <200511290914.jAT9EpI3003111@fb04209.mathematik.tu-darmstadt.de> Subject: categories: further references on semi-categories To: categories@mta.ca Date: Tue, 29 Nov 2005 10:14:51 +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: 64 In order to provide a categorical semantics of typed $\lambda$-calculus without $\eta$-rule Susumu Hayashi used semicategories in MR0841025 (87i:18005) Hayashi, Susumu Adjunction of semifunctors: categorical structures in nonextensional lambda calculus. Theoret. Comput. Sci. 41 (1985), no. 1, 95--104. This was later taken up by R.Hoofman in his Thesis on "Nonstable Models of Linear Logic" (see the first 7 items when you type in "Hoofman, R*" in Math.Reviews). However, in my opinion for the purpose of modelling $\lambda\beta$-calculus it is more natural to use the following kind of structures: (small) categories \C with finite products such that y(Y)^{y(X)} (taken in Psh(\C)) is a retract of some y(E). Thomas Streicher From rrosebru@mta.ca Wed Nov 30 15:04:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhX9K-00027w-LN for categories-list@mta.ca; Wed, 30 Nov 2005 14:57:50 -0400 Date: Tue, 29 Nov 2005 17:51:04 +0100 Subject: categories: Fwd: Mathematica and CAS Mime-Version: 1.0 (Apple Message framework v543) To: Categories From: jean benabou Message-Id: <546F06B4-60F8-11DA-9B52-000393B90F2C@wanadoo.fr> Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 65 As I know nothing about Computer Algebra Systems, but was surprised by=20= examples given by Andrej Bauer on the Category list of limits that=20 Mathematica gets wrong, I forwarded his mail to a computer scientist=20 friend of mine, specialist of Mathematca, asking for her opinion. She wrote the following answer, where she disagrees with Bauer, and=20 asked me to forward it to the Category list as she would like to know=20 the reaction of some "experts" to her statements. To tell the truth, so=20= do I. Cordially to all, Jean Benabou D=E9but du message r=E9exp=E9di=E9 : > De: Jacqueline Zizi > Date: Ven 25 nov 2005 11:12:03 Europe/Paris > =C0: Jean B=E9nabou > Objet: Mathematica and CAS > > Thanks, Jean, for forwarding me some exchanges about Mathematica. > > Please find below my opinion. If you think that it might bring some=20 > light, please feel free to forward it to the discussion list =20 > "categories". > > > A) Andrej Bauer points out very interesting questions, but I think he=20= > is wrong: > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > > The interesting questions are: > 1) Symbolic systems and students fall in the same trap; > 2) People trust blindly computer results obtained via computers (not=20= > only using CAS); > 3) Utilization of scientific results by politic or technical people=20 > without checking the results could be very dangerous; > > But I have the impression that Andrej himself falls in the trap. And=20= > especially when he says that : > > " I guess I am trying to point out that current Computer Alegbra=20 > Systems are very tricky to use_correctly" > > Indeed CAS are very complex systems built over several thousands of=20 > functions, called primitive. CAS are NOT only tools and moreover they=20= > are NOT global closed tools. They live like a science. Improving all=20= > the time. Growing all the time. > > Each of the functions of a CAS has it's own rules of application. And=20= > exactly as in Mathematics you can't use a theorem if some of the=20 > hypothesis are not satisfied, you can't use properly a function in a=20= > CAS if you are not inside the limits of application of this primitive.=20= > The rules for application of the primitives are clearly given in=20 > Mathematica, in the "Help" menu. For example, I put a screen shot for=20= > the primitive "Limit" as EXAMPLE 1, at this address: > > http://homepage.mac.com/jacquelinezizi/CategoriesQA/ > > As you can see in this screen shot it is written : "Limit by default=20= > makes no explicit assumptions about symbolic functions" . That clearly=20= > means that you can't hope any discussion about the symbol "a" of the=20= > Andrej's limit. > > Nevertheless the solution of this limit can be easily done and=20 > discussed using Mathematica as you can see in the EXAMPLE 2, at the=20= > same address. > > But this does not mean at all that Mathematica is not able to deal=20 > with parameters, as we are going to see. > > > > B) Jacques Carette said: > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > "Engineers and physicists don't use CAS - they use Matlab. The errors=20= > you get there are both worse and better: worse because numerical=20 > algorithms are so much more prone to giving (silent) nonsense, and=20 > better because Matlab cannot phrase any problems which are parametric!=20= > " > > I agree with that. For example, in all numerical systems, the Integers=20= > are limited, depending on the machine you work on. That is not the=20 > case in CAS were Integers are as large as you want, like in=20 > Mathematics. This is important as it produces sometimes hidden=20 > mistakes in embedded computations, that lead to a wrong result. > > Now, I must say that I don't agree with what Jacques Carette says in=20= > the following paragraph about people developing CAS: > > "This is exactly the kind of parameter specialization problem where=20 > CAS designers have "chosen" to ignore and return a generic answer.=20 > This has been documented since at least 1991 in a nice paper in the=20 > Bulletin of the AMS" > > For example, Mathematica has been giving results, for quite a few=20 > years now, using "Assumptions" for some primitives. I give an example=20= > for the primitive " Integral" in EXAMPLE 3 at the same address. You=20= > can see, on this example, that Mathematica deals quite well with=20 > parameters, both for questions AND answers. Better than I can do... > > > Conclusion > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > Happily there are more and more people working hard and well in CAS=20 > theory! The problems that they cannot solve, just as in Mathematics,=20= > are infinite. But as Mathematics, Mathematica can already solve, to=20 > day, quite a lot... This has very little to do with specific numerical=20= > tools programmed for specific aims. > > > Jacqueline Zizi > From rrosebru@mta.ca Wed Nov 30 15:04:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhXAD-0002DK-1x for categories-list@mta.ca; Wed, 30 Nov 2005 14:58:45 -0400 Message-ID: <438DCA67.4020801@tzi.de> Date: Wed, 30 Nov 2005 16:51:03 +0100 From: Lutz Schroeder User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050715) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories Subject: categories: Re: Semigroups with many objects References: <200511260530.06987.gaucher@pps.jussieu.fr> In-Reply-To: <200511260530.06987.gaucher@pps.jussieu.fr> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 66 Dear all, > I had also seen the word "precategory" but I cannot remember where. Beware of > the fact that the word precategory is also used for categories *with > identities* such that the composition law is partially defined : that is the > fact that the codomain of F is equal to the domain of G is not sufficient for > GoF to exist. Once again, I cannot remember where I read this word. The only > thing I remember is that that was a computer-scientific work. That would have been my paper with Paulo Mateus "Universal aspects of probabilistic automata" in MSCS (and also "Monads on composition graphs" in APCS). We do indeed use the word "precategory" for strucures with identities, and with a partially defined composition law satisfying the identity laws (strongly) and the associative law in the sense that f(gh)=(fg)h holds strongly (or Kleene) provided that both gh and fg are defined. Moreover, as pointed out in a previous message, I have used the word "semicategory" for similar structures, but with a stronger associative law, requiring that f(gh)=(fg)h are both defined whenever fg and gh are defined (or slight variations of this). Ehresmann used the term "multiplicative graph" (and also sometimes "neocategory", I believe) for structures satisfying the identity law, with no associativity imposed at all. -- Lutz -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode ----------------------------------------------------------------------------- From rrosebru@mta.ca Wed Nov 30 15:04:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 30 Nov 2005 15:04:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhXBH-0002Kz-U3 for categories-list@mta.ca; Wed, 30 Nov 2005 14:59:51 -0400 X-Authentication-Warning: u00.math.uiuc.edu: ralphw owned process doing -bs Date: Wed, 30 Nov 2005 10:50:41 -0600 (CST) From: Ralph Leonard Wojtowicz To: cc: Subject: categories: job: non-academic position 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: 67 Metron, Inc. is a small consulting firm with headquarters in the Washington, D.C. area and offices in San Diego. The Systems and Analysis Division seeks to hire an analyst with a background in category theory and an interest in applications to engineering or computer science. Please see http://www.metsci.com for information about the company and current projects. A list of the East Cost technical staff is included below. Analysts are encouraged to pursue their own research and funding directions and may contribute to a number of on-going projects at Metron. Salaries and benefits are excellent. Interested individuals should complete the on-line form: Go to http://www.metsci.com, click on "Careers", "openings", "Analyst(Virginia)", then "Apply online". For "Source", indicate "-Other-", then enter "category theory". Please send three recommendation letters and any other supporting material to Recruiting (attn: category theory) Metron, Inc. 11911 Freedom Drive Suite 800 Reston, VA 20190-5602 Since most of our current work is for Department of Defense clients, applicants must be U.S. citizens. Review of applications will begin immediately. Questions directed to the contact listed below are welcome. Sincerely, Ralph Wojtowicz Metron, Inc. 11911 Freedom Drive Suite 800 Reston, VA 20190-5602 e-mail: wojtowicz@metsci.com Phone: 703-787-8700 x437 Fax: 703-787-3518 East Coast Metron technical staff: Analysis L. Corwin (Chairman of the Board) Ph.D., Statistics, Princeton University Lawrence D. Stone (President, CEO) Ph.D., Mathematics, Purdue University Thomas L. Mifflin (VP Federal Division) Ph.D., Mathematics, George Washington University Tom A. Stefanick (Vice President) Ph.D., Mechanical Engineering, George Washington University Gregory A. Godfrey (Senior Engineer) Ph.D., Statistics & Operations Research, Princeton University Mark R. Anderson Ph.D., Aeronautics and Astronautics, Purdue University Stephen L. Anderson Ph.D., Mathematics, Brown University Terence J. Bazow Ph.D., Physics, Catholic University Christopher M. Boner Ph.D., Mathematics, University of Virginia Christopher Carlson B.S., Physics, James Madison University Dennis P. Carroll Ph.D., Chemistry, John Hopkins University Tofer A. Chagnon B.S., Mathematics, East Tennessee State University Gordon W. Clark Ph.D., Mathematics, University of Texas at Austin James P. Ferry Ph.D., Applied Math, Brown University Thomas E. Giddings Ph.D., Mechanical Engineering, Rensselaer Polytechnic University Joshua Hughes Ph.D., Computational Analysis and Modeling, Louisiana Tech Christopher P. Husband Ph.D., Computational & Applied Mathematics, Rice University Summer M. Husband Ph.D., Computational & Applied Mathematics, Rice University Thomas M.Kratzke Ph.D., Mathematics, University of Illinois at Urbana-Champaign Ray Jakobovits Ph.D., Operations Research, Cornell University Mary L. Kohl Ph.D., Physics, University of Colorado Stanley D. Kuo M.S., Applied Physics, Harvard University Darren Lo Ph.D., Mathematics, University of Wisconsin Bradley S. Moskowitz Ph.D., Mathematics, University of California, Los Angeles James P. Ochoa Ph.D., Mathematics, University of North Texas Bryan R. Osborn M.S., Applied Mathematics, University of Maryland Jeffrey M. Roach Ph.D., Mathematics, University of Virginia Joseph J. Shirron Ph.D., Applied Mathematics, University of Maryland Anna Domnich Tirat-Gefen M.S., Electrical Engineering, University of Michigan Roy L. Streit Ph.D., Mathematics, University of Rhode Island Vinh-Thy Minh Tran Ph.D., Mathematics, Northwestern University Didier H. Vergamini Ph.D., Mathematics/Computer Science, Ecole Des Mines de Paris/INRIA Harold H. Wadleigh Ph.D., Mathematics, University of California, Los Angeles Ralph L Wojtowicz Ph.D., Mathematics, University of Illinois at Urbana Software Analysts: John R. Cunningham B.S., Computer Science, James Madison University Robert Foster M.S., Systems Engineering, George Mason University Paul B. Goger B.S., Computer Science & Mathematics, College of William & Mary David S. Grogan B.S., Computer Science, University of Virginia Michael A. Hogye B.S., Computer Science, University of Virginia Alexis J. Humphreys M.S., Mathematics, Rutgers University Christine M. R. Judd M.C.S., Computer Science, University of Virginia James A. Kilgore, Jr. M.S., Computer Science, George Mason University Aren G. Knutsen B.S., Applied Mathematics, James Madison University Tuan A. Tran M.S., Computer Science, George Mason University Geoffrey T.Ullman B.S., Computer Science & Mathematics, Rose-Hulman David A. Vanderson; B.S., Computer Science, University of Virginia From rrosebru@mta.ca Thu Dec 1 16:03:20 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Dec 2005 16:03:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhuYS-0006mv-70 for categories-list@mta.ca; Thu, 01 Dec 2005 15:57:20 -0400 To: categories@mta.ca Subject: categories: Re: semi-categories Date: Wed, 30 Nov 2005 17:24:35 -0500 From: wlawvere@buffalo.edu Message-ID: <1133389475.438e26a388e62@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: 128.205.249.221 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 68 Perhaps it has not been sufficiently emphasized that semi-categories and the like are not really "generalizations" of categories (though formally they may appear so). Actually they present possibly-useful SPECIAL classes of categories. That is because we represent one ultimately in an actual large category (such as sets or abelian groups) and those representations are indeed representations of a certain ordinary (V-) category, namely the one freely generated by the given semicategory. The forgetful 2-functor has a left adjoint, just as does the one from categories to directed graphs etc. To be a value of such a left adjoint means that the large category of representations may have special properties, for example it may unite by a bicontinuous quotient p a pair of subcategories i, j whose domains are identical but where i, j are themselves opposite in that they are the respective adjoints to the same p. This is the kind of UIAO that Francis refers to. Is there a convincing example showing that it can be useful mathematically to treat operator ideals (such as compact, nuclear, etc) as semicategories? I always believed that Jacobson invented rngs because algebraic practice (not the dreaded categorists) had convinced him to grudgingly conclude that after all ideals in rings are ideals but not subrings, whereas the opposite view is not a convenience but a confusion which denies ideals their dignity. Bill Lawvere Quoting Francis Borceux : > > There has been a long discussion on the list > about "categories without identities", whatever > you decide to call them. And the attention has > been brought to axioms which could -- in this > more general context -- replace the identity > axiom. > > I would like to focus on a very striking categorical aspect of this > problem. > > A (right) module M on a ring R with unit must satisfy the axiom > m1=3Dm > ... but what about the case when R does not have a unit ? > > Simply dropping the axiom m1=3Dm leaves you with > the unpleasant situation where you have two > different notions of module, in the case where R > has a unit. > > Therefore people working in linear algebra have considered the axiom > > the scalar multiplication M@R ---> M is an isomorphism > (@=3Dtensor product sign) > > which is equivalent to the axiom m1=3Dm, when the > ring has a unit ... but makes perfect sense when > the ring does not have a unit. Such modules are > generally called "Taylor regular". > > A ring R with unit is simply a one-objet additive > category and a right module M on R is simply an > additive presheaf M ---> Ab (=3Dthe category of > abelian groups). > > A ring without unit is thus a "one-object > additive category without identity", again > whatever you decide to call this. > > But what is the analogue of the axiom > > M@R ---> M is an isomorphism > > when R is now an arbitrary small (enriched) > "category without identities" and M is an > arbitrary (enriched) presheaf on it ? > > All of us know that to define a (co)limit, we do > not need at all to start with an indexing > category: an arbitrary graph with arbitrary > commutativity conditions works perfectly well. In > particular, a "category without identities" is > all right. And the same holds in the enriched > case, with (co)limits replaced by "weighted > (co)limits". > > Now every presheaf on a small category is > canonically a colimit of representable ones ... > but this result depends heavily on the existence > of identities ! When you work with a presheaf M > on a "category R without identities", you still > have a canonical morphism > > canonical colimit of representables ---> R > > and you can call M "Taylor regular" when this is > an isomorphism. Again in the enriched case, > "colimit" means "weighted colimit". This > recaptures exactly the case of "Taylor regular > modules", when working with Ab-enriched > categories. > > A sensible axiom to put on a "category R without > identities" is the fact that the representable > functors are "Taylor regular". (We should > certainly call this something else than "Taylor > regular", but let me keep this terminology in > this message.) > > And when R is a "Taylor regular category without identities", the > constructi> on > > presheaf on R |---> corresponding canonical colimit of > representables > > yields a reflection for the inclusion of Taylor > regular presheaves in all presheaves. > > A very striking property is the existence of a > further (necessarily full and faithful) left > adjoint to this reflection. This second inclusion > provides in fact an equivalence with the full > subcategory of those presheaves which satisfy the > Yoneda isomorphism. > > This yields thus a nice example of what Bill > Lawvere calls the "unity of opposites": the two > inclusions identify the category of Taylor > regular presheaves with > * on one side, those presheaves which are colimits of > representables; > * on the other side, those presheaves which satisfy the Yoneda > lemma. > This underlines the pertinence of these "Taylor > regular categories without identities". > > To my knowledge, the best treatment of these > questions is to be found in various papers by > Marie-Anne Moens and by Isar Stubbe, in > particular in the "Cahiers" and in "TAC". > > And very interesting examples occur in functional > analysis (the identity on a Hilbert space is a > compact operator ... if and only if the space is > finite dimensional) and also in the theory of > quantales. > > Francis Borceux > > -- > Francis BORCEUX > D=E9partement de Math=E9matique > Universit=E9 Catholique de Louvain > 2 chemin du Cyclotron > 1348 Louvain-la-Neuve (Belgique) > t=E9l. +32(0)10473170, fax. +32(0)10472530 > borceux@math.ucl.ac.be > > > > From rrosebru@mta.ca Thu Dec 1 16:03:20 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Dec 2005 16:03:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhuVV-0006V4-4i for categories-list@mta.ca; Thu, 01 Dec 2005 15:54:17 -0400 Date: Wed, 30 Nov 2005 13:43:32 -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: Categories Anonymous References: <200511281907.jASJ7cEW026208@saul.cis.upenn.edu> In-Reply-To: <200511281907.jASJ7cEW026208@saul.cis.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: RO X-Status: X-Keywords: X-UID: 69 Peter Freyd wrote: > While checking Jacobson's responsibility for "rng" Google directed > me to www.math.binghamton.edu/alex/Yale_Math_Skit.html where we > learn a most appropriate pronunciation for the word. Semiliterate: able to tell a right module from a rng module. Vaughan Pratt From rrosebru@mta.ca Thu Dec 1 16:03:20 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Dec 2005 16:03:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EhuZG-0006rY-0o for categories-list@mta.ca; Thu, 01 Dec 2005 15:58:10 -0400 X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Wed, 30 Nov 2005 20:48:52 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Name for a concept 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: 70 Is there a standard name for a square A ----> B | | | | | | v v C ----> D in which the canonical map A ---> B x_D C is epic? I had always called it a weak pullback, but Peter Freyd claims that that phrase is reserved for the case that it satisfies the existence, but not necessarily the uniqueness of the definition of pullback. In fact, he claims it means that Hom(E,-) converts it to the kind of square I am talking about. What is interesting is that in an abelian category, it satisfies this condition iff it satisfies the dual condition iff the evident sequence A ---> B x C ---> D is exact. Putting a zero at the left end characterizes a genuine pullback and at the other end a pushout. Michael