From rrosebru@mta.ca Fri Dec 1 15:15:17 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB1IaLO15944 for categories-list; Fri, 1 Dec 2000 14:36:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 1 Dec 2000 14:06:12 GMT Message-Id: <9087.200012011406@craro.dcs.ed.ac.uk> To: categories@mta.ca Subject: categories: CFP: Workshop on Implicit Computational Complexity From: "Martin Hofmann" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 1 Third international workshop on Implicit Computational Complexity-2001 (ICC'01) ------------------------------------------------------------------------------- (affiliated with PADO and MFPS) The Implicit Complexity Workshop (ICC'01) will be held on Sunday, 20 May 2001 in Aarhus as part of the joint PADO/MFPS 2000 conferences. Topics of Interest: ------------------- The synergy between Logic, Computational Complexity and Programming Language Theory has gained importance and vigour in recent years, cutting across areas such as Proof Theory, Computation Theory, Applicative Programming, and Philosophical Logic. Several machine-independent approaches to computational complexity have been developed, which characterize complexity classes by conceptual measures borrowed primarily from mathematical logic. Collectively these approaches might be dubbed IMPLICIT COMPUTATIONAL COMPLEXITY. Practically, implicit computational complexity provides a framework for a streamlined incorporation of computational complexity into areas such as formal methods in software development, programming language theory. In addition to research reports on theoretical advances in implicit computational complexity, practical contributions bridging the gap between Computational Complexity and Programming Language Theory are therefore of particular interest. Previous Workshops ------------------ have been held in Indianapolis 1994, Baltimore 1998, Trento 1999, Santa Barbara 2000. Programme committee ------------------- Samson Abramsky (University of Oxford, UK) Martin Hofmann (University of Edinburgh, UK) (Chair) Bruce Kapron (University of Victoria, Canada) Harry Mairson (Brandeis University) Jean-Yves Marion (Loria, Nancy, France) So/ren Riis (Queen Mary and Westfield College, London) Helmut Schwichtenberg (University of Munich, Germany) Steering committee ------------------ Daniel Leivant (University of Indiana at Bloomington) Jean-Yves Marion (Loria, Nancy, France) Submission procedure -------------------- E-mail your contribution as a PostScript file to the programme chair (mxh@dcs.ed.ac.uk) to bereceived by 16 March 2001. Alternatively, you can send 5 hardcopies by air mail to the program chair. Authors with restricted copying facilities may also send a single hardcopy. Important dates --------------- 16 March 2001 Submission deadline 10 April 2001 Notification of authors of accepted papers 20 May 2001 Workshop Workshop WWW-page ----------------- http://www.dcs.ed.ac.uk/home/mxh/ICC01.html Contact information ------------------- Martin Hofmann Division of Informatics University of Edinburgh JCMB, KB Mayfield Road Edinburgh EH9 3JZ UK tel : (44) 131 650 5187 fax : (44) 131 667 7209 mxh@dcs.ed.ac.uk From rrosebru@mta.ca Sat Dec 2 10:59:09 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2ERxD24485 for categories-list; Sat, 2 Dec 2000 10:27:59 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 1 Dec 2000 17:19:59 -0500 (EST) From: Michael MAKKAI To: categories@mta.ca Subject: categories: Re: Categories ridiculously abstract In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 In "Towards a categorical foundation of mathematics" (Logic Colloquium '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic no.11, 1998; pp.153-190) and in subsequent work, I am proposing an approach to a foundation whose universe consists of the weak n-categories and whatever things are needed to connect them. This is done on the basis of a general point of view concerning the role of identity of mathematical objects. Readers of said paper who have followed developments on weak higher dimensional categories will note that much has been done since towards fleshing out the program. Michael Makkai On Thu, 30 Nov 2000, Tom Leinster wrote: > > Michael Barr wrote: > > > > And here is a question: are categories more abstract or less abstract than > > sets? > > A higher-dimensional category theorist's answer: > "Neither - a set is merely a 0-category, and a category a 1-category." > > There's a more serious thought behind this. Sometimes I've wondered, in a > vague way, whether the much-discussed hierarchy > > 0-categories (sets) form a (1-)category, > (1-)categories form a 2-category, > ... > > has a role to play in foundations. After all, set-theorists seek to found > mathematics on the theory of 0-categories; category-theorists sometimes talk > about founding mathematics on the theory of 1-categories and providing a > (Lawverian) axiomatization of the 1-category of 0-categories; you might ask > "what next"? Axiomatize the 2-category of (1-)categories? Or the > (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek > and head in clouds, that general n-categories provide a more natural > foundation than either 0-categories or 1-categories? > > > Tom > From rrosebru@mta.ca Sat Dec 2 11:04:59 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2EQfZ26225 for categories-list; Sat, 2 Dec 2000 10:26:41 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <200012012113.WAA10773@irmast2.u-strasbg.fr> Date: Fri, 1 Dec 2000 22:13:06 +0100 (MET) From: Philippe Gaucher Reply-To: Philippe Gaucher Subject: categories: localization : more precise question To: categories@mta.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: NPPm/u4YBwp+GuR5icyu9Q== X-Mailer: dtmail 1.3.0 CDE Version 1.3 SunOS 5.7 sun4u sparc Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 3 Re-bonjour, Thank you for your answers. My question was very general. So here is the example. I am going to define the category C and the collection of morphisms S, with respect to what I would like to localize. The object of C are the oriented graph. Such an object X is a topological space obtained by choosing a discrete set X^0 and by attaching 1-dimensional cells *with orientations*. It is a 1-dimensional CW-complex with oriented arrows. The morphisms of C are the continuous maps f from X to Y satisfying this conditions : 1) f(X^0)\subset Y^0 2) f is orientation-preserving 3) f is non-contracting in the sense that a 1-cell is never contracted to one point. Remark I : in C, an arrow x--> is not isomorphic to a point. Remark II : an arrow a-->b can be mapped on the loop a-->a with one oriented arrow from a to a. A morphism f of C is in S if and only if f induces an homeomorphism on the underlying topological spaces. Now here is an example of f\in S which is not invertible : a--->b mapped on a-->x-->b This morphism has no inverse in C because the image of x must be equal to a or b by 1) and therefore one of the arrows would be contracted by 2), which contredicts 3). I would like to know if C[S^{-1}] exists or no (in the same universe). The irresistible conjecture is of course that C[S^{-1}] is equivalent to the category whose objects are that of C and whose morphisms from A to B are the subset of C^0(A,B) (the set of continuous maps from A to B) containing all composites of the form g_1.f_1^{-1}.g_2...g_n.f_n^{-1}.g_{n+1} where g_1,...,g_{n+1}, are morphisms of C and f_1,...,f_n morphisms in S. The Ore condition is not satisfied by S because of this example. The Ore condition says that for any s:A-->B in S, and any f:X-->B, there exists t:Y-->X in S and g:Y-->A such that s.g=f.t. Now the counterexample : A is a--->b, B is a-->x-->b with s as above ; X is a-->x with the inclusion f from X in B. Then necessarily Y=X and t=Id. And s.g(x)=b and f.t(x)=x. Thanks in advance. pg. From rrosebru@mta.ca Sat Dec 2 11:07:17 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2EOTw26183 for categories-list; Sat, 2 Dec 2000 10:24:29 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Sat, 02 Dec 2000 09:34:22 -0400 From: "Robert J. MacG. Dawson" Subject: categories: Re: Categories ridiculously abstract To: categories@mta.ca Message-id: <3A28FA5E.9640F981@stmarys.ca> MIME-version: 1.0 X-Mailer: Mozilla 4.72 [en] (WinNT; I) Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7bit X-Accept-Language: en References: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 4 Tom Leinster wrote: > > Michael Barr wrote: > > > > And here is a question: are categories more abstract or less abstract than > > sets? > > A higher-dimensional category theorist's answer: > "Neither - a set is merely a 0-category, and a category a 1-category." > > There's a more serious thought behind this. Sometimes I've wondered, in a > vague way, whether the much-discussed hierarchy > > 0-categories (sets) form a (1-)category, > (1-)categories form a 2-category, > ... > > has a role to play in foundations. After all, set-theorists seek to found > mathematics on the theory of 0-categories; category-theorists sometimes talk > about founding mathematics on the theory of 1-categories and providing a > (Lawverian) axiomatization of the 1-category of 0-categories; you might ask > "what next"? Axiomatize the 2-category of (1-)categories? Or the > (n+1)-category of n-categories? Surely we should start with the set of (-1)-categories? -Robert Dawson From rrosebru@mta.ca Sat Dec 2 16:50:25 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB2K5Aa12517 for categories-list; Sat, 2 Dec 2000 16:05:10 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f To: categories@mta.ca Subject: categories: Re: localization : more precise question References: <200012012113.WAA10773@irmast2.u-strasbg.fr> From: Dan Christensen Date: 02 Dec 2000 13:05:01 -0500 In-Reply-To: <200012012113.WAA10773@irmast2.u-strasbg.fr> Message-ID: <87elzq7l82.fsf@julian.uwo.ca> Lines: 68 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.7 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 Philippe Gaucher writes: > The object of C are the oriented graph. Such an object X is a > topological space obtained by choosing a discrete set X^0 and by > attaching 1-dimensional cells *with orientations*. It is a > 1-dimensional CW-complex with oriented arrows. > > The morphisms of C are the continuous maps f from X to Y satisfying > this conditions : > > 1) f(X^0)\subset Y^0 > 2) f is orientation-preserving > 3) f is non-contracting in the sense that a 1-cell is never contracted > to one point. > > A morphism f of C is in S if and only if f induces an homeomorphism on > the underlying topological spaces. > > I would like to know if C[S^{-1}] exists or no (in the same universe). > > The irresistible conjecture is of course that C[S^{-1}] is equivalent > to the category whose objects are that of C and whose morphisms from A > to B are the subset of C^0(A,B) (the set of continuous maps from A to > B) containing all composites of the form > g_1.f_1^{-1}.g_2...g_n.f_n^{-1}.g_{n+1} where g_1,...,g_{n+1}, are > morphisms of C and f_1,...,f_n morphisms in S. Call the category you describe D. There is an obvious functor C --> D which inverts the morphisms of S. So there is an induced functor C[S^{-1}] --> D, which is the identity on objects and is clearly full, since the morphisms from A to B in C[S^{-1}] can be described as the *formal* composites g_1.f_1^{-1}.....f_n^{-1}.g_{n+1} (modulo certain relations), where g_1,...,g_{n+1}, are morphisms of C and f_1,...,f_n morphisms in S. The question is whether the functor C[S^{-1}] --> D is faithful. I suspect that this is true in general, but can only prove it if you restrict yourself to CW-complexes with a finite number of cells. If you do this, then I believe that the reverse Ore condition holds: given s A ---> B | v C with s in S, there exists s A ---> B | | v v C ---> D t with t in S. (B is just A with a finite number of vertices added; just add the images of those points in C as new vertices to get D.) With this, it isn't hard to see that the functor is faithful. For infinite CW-complexes, this Ore condition doesn't hold, but I still suspect that the functor is faithful. In part it depends upon what you mean by "orientation preserving". Does this mean "having a 'positive' derivative at all times"? Or 'non-negative'? Or can the map go forwards and backwards as long as overall it has degree one? Dan From rrosebru@mta.ca Sun Dec 3 10:42:04 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB3E9eI18068 for categories-list; Sun, 3 Dec 2000 10:09:40 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <200012022333.AAA11616@irmast2.u-strasbg.fr> Date: Sun, 3 Dec 2000 00:33:28 +0100 (MET) From: Philippe Gaucher Reply-To: Philippe Gaucher Subject: categories: Re: localization : more precise question To: categories@mta.ca, jdc@julian.uwo.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: j4IgY+mMIPsZwfCwzuww1w== X-Mailer: dtmail 1.3.0 CDE Version 1.3 SunOS 5.7 sun4u sparc Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 6 >The question is whether the functor C[S^{-1}] --> D is faithful. > >I suspect that this is true in general, but can only prove it if >you restrict yourself to CW-complexes with a finite number of cells. I believe that you are wrong somewhere. The explanation is in post-scriptum (borrowed from a question in sci.math.research which is not yet posted by now). Or maybe I am wrong in the reasonning ? >For infinite CW-complexes, this Ore condition doesn't hold, but I >still suspect that the functor is faithful. In part it depends upon >what you mean by "orientation preserving". Does this mean "having a >'positive' derivative at all times"? Or 'non-negative'? Or can the >map go forwards and backwards as long as overall it has degree one? I meant 'non-negative'. Maybe the definition of the category still needs to be debugged. I don't know. (The motivation of this question was to encode the notion of 1-dimensional HDA up to dihomotopy for those who know the subject in a "true" category such that isomorphism classes represent 1-dimensional HDA up to dihomotopy). "having a 'positive' derivative at all times" would be also sufficient I think. Cheers. pg. PS : The natural conjecture is that C[S^{-1}] is equivalent to the category D whose objects are that of C and whose morphisms from A to B are the subset of C^0(A,B) (the set of continuous maps from A to B) containing all composites of the form g_1.f_1^{-1}.g_2...g_n.f_n^{-1}.g_{n+1} where g_1,...,g_{n+1} are morphisms of C and f_1,...,f_n morphisms in S. If U is a universe containing all sets, let V be a universe with U\in V. The categorical construction of C[S^{-1}] (let us call it "C[S^{-1}]") gives a V-small category. "C[S^{-1}]"(A,B) is generally not a set. To see that, take an object like g_1.f_1^{-1}.g_2 with g_1 and g_2 not invertible in C (this is a reduced form which cannot be simplified in "C[S^{-1}]"). Then replace f_1^{-1} by f_1^{-1} \sqcup Id_{codom(f_1^{-1})} \sqcup Id_{codom(f_1^{-1})} \sqcup... and g_1 by g_1 \sqcup g_1 \sqcup g_1 ... Then "C[S^{-1}]"(dom(g_2),codom(g_1)) has as many elements as the sets of U-small cardinals. Therefore "C[S^{-1}]"(dom(g_1),codom(g_2)) is not a set. The relation between "C[S^{-1}]" and D is as follows. There is a canonical V-small map g : "C[S^{-1}]"(A,B) --> Sets(A,B) and D(A,B) is the quotient of the V-small set "C[S^{-1}]"(A,B) by the V-small equivalence relation "x equivalent to y iff g(x)=g(y)". The above element of "C[S^{-1}]"(dom(g_2),codom(g_1)) are all of them identified by this equivalence relation : it is the reason why the homset from dom(g_2) to codom(g_1) becomes a set. The obvious functor from C-->D does invert the morphisms of S. But one has to prove that for any functor C-->E inverting the morphisms of S, C-->E factorizes through C-->D by a unique functor from D-->E. Such functor C-->E factorizes through "C[S^{-1}]" but for proving the factorization through D, one has to prove that E is a sort of concrete category (a category with a faithful functor to Sets). Of course there is no reason for E to be concrete but because of the functor F:C-->E, Im(F) is not too far from a concrete category. C is a concrete category, constructed with oriented graphs. I never heard about a general way of constructing localizations of concrete categories. Does it exist ? From rrosebru@mta.ca Mon Dec 4 11:55:59 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB4FKIE08739 for categories-list; Mon, 4 Dec 2000 11:20:18 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <200012040530.VAA06421@coraki.Stanford.EDU> To: categories@mta.ca Subject: categories: Re: Categories ridiculously abstract Date: Sun, 03 Dec 2000 21:30:39 -0800 From: Vaughan Pratt Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 7 1 ab.stract \ab-'strakt, 'ab-,\ adj (15c) [ML abstractus, fr. L, pp. of abstrahere to draw away, fr. abs-, ab- + trahere to draw -- more at DRAW] 1a: disassociated from any specific instance 1b: difficult to understand: ABSTRUSE 1c: IDEAL 1d: insufficiently factual: FORMAL 2: expressing a quality apart from an object 3a: dealing with a subject in its abstract aspects: THEORETICAL 3b: IMPERSONAL, DETACHED 4: having only intrinsic form with little or no attempt at pictorial representation or narrative content -- ab.stract.ly \ab-'strak-(t)l, 'ab-,\ adv -- ab.stract.ness \ab-'strak(t)-ns, 'ab-,\ n 1a: Sets and categories as mathematical abstractions are equally disassociated from specific instances. 1b: For almost every interesting known theorem of category theory there is a harder interesting known theorem of set theory, and vice versa. It is plausible that the exceptions from set theory outnumber those from category theory, but it is equally plausible that a majority of mathematical literates judge category theory harder than set theory. No clear winner here. 1c: Sets and categories are both ideal entities. 1d: Set theory and category theory are equally factual, and equally formal. 2: In this sense set theory and category theory are both abstract while sets and categories are objects and so not abstract. 3a: Set theory and category theory deal equally with the abstract aspects of their respective subjects. 3b: The FOM mailing list tends to get worked up much more often and rather more heatedly about the set-vs-category debate than does the categories mailing list. 4. Categories lend themselves better to diagrams than do sets. Conclusions (organized by dictionary meaning of "abstract"): 1 to 3a: No difference. 3b: Category theorists are more abstract than set theorists. 4: Sets are more abstract than categories. -- Vaughan Pratt O res ridicula! immensa stultitia. --Chorus of Old Men, Catulli Carmina From rrosebru@mta.ca Mon Dec 4 11:56:00 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB4FJo330401 for categories-list; Mon, 4 Dec 2000 11:19:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f To: categories@mta.ca Subject: categories: Re: localization : more precise question References: <200012022333.AAA11616@irmast2.u-strasbg.fr> From: Dan Christensen Date: 03 Dec 2000 17:39:20 -0500 In-Reply-To: <200012022333.AAA11616@irmast2.u-strasbg.fr> Message-ID: <878zpx5duv.fsf@julian.uwo.ca> Lines: 52 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.7 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: 8 Philippe Gaucher writes: > Dan Christensen wrote: > > >I suspect that this is true in general, but can only prove it if > >you restrict yourself to CW-complexes with a finite number of cells. > > I believe that you are wrong somewhere. ... > If U is a universe containing all sets, let V be a universe with U\in > V. The categorical construction of C[S^{-1}] (let us call it > "C[S^{-1}]") gives a V-small category. "C[S^{-1}]"(A,B) is generally > not a set. To see that, take an object like g_1.f_1^{-1}.g_2 with g_1 > and g_2 not invertible in C (this is a reduced form which cannot be > simplified in "C[S^{-1}]"). Then replace f_1^{-1} by > > f_1^{-1} \sqcup Id_{codom(f_1^{-1})} \sqcup Id_{codom(f_1^{-1})} > \sqcup... > > and g_1 by > > g_1 \sqcup g_1 \sqcup g_1 ... > > Then "C[S^{-1}]"(dom(g_2),codom(g_1)) has as many elements as the sets > of U-small cardinals. Therefore "C[S^{-1}]"(dom(g_1),codom(g_2)) is > not a set. The maps you've described all represent the same map in C[S^{-1}]. For example, the diagram g_1 f_1 g_2 A ---> C <---------- D ------------> B | | | | |1 | | |1 v v v v g_1 f_1 v 1 g_2 v g_2 A ---> C v D <------ D v D --------> B shows that the top map is equal to the bottom map in C[S^{-1}]. So this doesn't seems to be a counterexample to my guess that C[S^{-1}] exists in general. (By the way, my argument for the case when the CW complexes are finite, while possibly useful as an idea for how to approach the general case, is more complicated than is necessary, since with this assumption C is equivalent to a small category, and so any localization of C exists.) Dan From rrosebru@mta.ca Mon Dec 4 11:56:02 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB4FJOh32551 for categories-list; Mon, 4 Dec 2000 11:19:24 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <200012031644.RAA12007@irmast2.u-strasbg.fr> Date: Sun, 3 Dec 2000 17:44:54 +0100 (MET) From: Philippe Gaucher Reply-To: Philippe Gaucher Subject: categories: Re: localization : more precise question To: categories@mta.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: vouqimf63rZfbrHamQEPeQ== X-Mailer: dtmail 1.3.0 CDE Version 1.3 SunOS 5.7 sun4u sparc Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 9 >I meant 'non-negative'. Maybe the definition of the category still needs >to be debugged. I don't know. (The motivation of this question was to encode >the notion of 1-dimensional HDA up to dihomotopy for those who know the >subject in a "true" category such that isomorphism classes represent >1-dimensional HDA up to dihomotopy). "having a 'positive' derivative at >all times" would be also sufficient I think. I would like to add : I meant 'non-negative' locally. Because one needs that the morphism from an arrow a-->b to a loop a-->a exists. The exact definition is : morphism of local po-spaces (see "Algebraic topology and concurrency", by Fajstrup, Goubault & Rau{\ss}en ; preprint R-99-2008, Aalborg University). pg. From rrosebru@mta.ca Mon Dec 4 20:04:16 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB4NaI921280 for categories-list; Mon, 4 Dec 2000 19:36:18 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: baez@newmath.UCR.EDU Message-Id: <200012042041.eB4KfaM14920@math-cl-n06.ucr.edu> Subject: categories: (-1)-categories and (-2)-categories To: categories@mta.ca Date: Mon, 4 Dec 2000 12:41:36 -0800 (PST) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 Robert Dawson wrote: Surely we should start with the set of (-1)-categories? Actually we should start at least a little bit before that, with (-2)-categories. Let me explain.... Once upon a time I showed what I called the "periodic table" to Chris Isham, a physicist who works on quantum gravity. It starts like this: k-tuply monoidal n-categories n = 0 n = 1 n = 2 k = 0 sets categories 2-categories k = 1 monoids monoidal monoidal categories 2-categories k = 2 commutative braided braided monoids monoidal monoidal categories 2-categories k = 3 " " symmetric sylleptic monoidal monoidal categories 2-categories k = 4 " " " " symmetric monoidal 2-categories k = 5 " " " " " " and it extends infinitely in both directions. The basic idea is that a "k-tuply monoidal n-category" is a weak (n+k)-category with only one j-morphism for j < k. There's a lot of evidence from homotopy theory and elsewhere that each column of this table must "stabilize" when k reaches n + 2. Of course, this observation needs to be made more precise before it can become a theorem, or even a conjecture, so James Dolan and I called it the "stabilization hypothesis". Carlos Simpson found one way to make it precise and prove it: On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories, math.CT/9810058. but everyone who has a definition should take a whack at it! Anyway, when I showed this pattern to Chris Isham, I was very proud of it, so I was annoyed when he instantly found fault with it. He said: what about the (-2)-categories, (-1)-categories, and monoidal (-1)-categories? You've drawn this big triangle, but it's missing the upper left-hand corner! I told him I'd have to think about that. After that, I kept trying to guess what (-2)-categories, (-1)-categories and monoidal (-1)-categories should be. Clearly a monoidal (-1)-category should be a set with just one element. But what about the other two? Later, when explaining the concepts of "property", "structure" and "stuff" to Toby Bartels, James Dolan figured out what (-1)-categories are. Toby then helped him figure out what (-2)-categories were, too. Actually, I should be a bit careful here: they really figured out what (-1)-groupoids and (-2)-groupoids are. However, I believe that these coincide with (-1)-categories and (-2)-categories. I have a lot to do today, and this article is already getting too long, so I'll stop here and leave these as a puzzle for all of you. It's sort of fun! I should however mention this: after James and I came to understand this stuff, someone pointed out an error in our definition of n-categories, and we were very perturbed until we realized it could be fixed by changing just one number in our existing definition - which would have been obvious from the start if we'd understood about (-1)-categories. John Baez From rrosebru@mta.ca Tue Dec 5 20:50:10 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB5Nuop22119 for categories-list; Tue, 5 Dec 2000 19:56:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Reals as final coalgebra (exercise) To: categories@mta.ca Date: Tue, 5 Dec 2000 19:55:56 +0000 (GMT) Cc: T.Leinster@dpmms.cam.ac.uk (Tom Leinster) X-Mailer: ELM [version 2.5 PL3] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: Tom Leinster Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 Fans of the characterization of the real interval as a final coalgebra might be interested to see the exercise I set the Part III (first year graduate) students in Cambridge taking this year's Category Theory course. It's a bit too cumbersome to reproduce in plain text, but can be found as the last question on sheet 3 at http://www.dpmms.cam.ac.uk/~leinster/categories and consists of a guided proof of the fact that the closed real interval is terminal in a certain category of coalgebras. I tried to make the exercise as quick and easy as possible, which is why there is no mention of the wedge product, nor orderings, nor digital expansions. I don't know whether this (particularly the omission of the wedge) is at some cost to understanding. If anyone has any suggestions as to how the exercise might be made more transparent then I'd be interested to hear; I'm aware that there are plenty of people on this list who have a deeper understanding of this result than I do. Flagged "highly optional" as it was, no-one as far as I'm aware has actually tried *doing* this question... Tom From rrosebru@mta.ca Wed Dec 6 16:21:21 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB6Jos225630 for categories-list; Wed, 6 Dec 2000 15:50:54 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 6 Dec 2000 16:56:07 +0100 To: categories@mta.ca From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Can we ignore smallness? Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 12 Dear categorists, in the last week there were some messages about categories of fractions and the smallness of their hom-sets, set forth by a question of Ph. Gaucher (Subject: category of fraction and set-theoretic problem; 30 Nov). I was puzzled by this sentence, in M. Barr's reply (30 Nov): > ... "But first, I might ask why it matters. Gabriel-Zisman ignores the >question and I think they are right to. Every category is small in >another universe." ... The reason why I think it matters should be clear from this example. U is a universe and Set is the category of U-small sets. Set has U-small hom-sets and is U-complete (has all limits based on U-small categories); it is not U-small. Of course it is V-small for every universe V to which U belongs; but then, it is not V-complete. The relevant fact, here, should be: - to have U-small hom sets and U-small limits for the SAME universe, i.e., a balance between a property (small hom-sets) which automatically extends to larger universes and another (small completeness) which automatically extends the other way, to smaller ones. Similar balances arise, less trivially, in categories of fractions. I think that the interest of proving they have small hom-sets (when possible) is related to other properties of such categories, holding for the same universe but not in larger ones. Thus: HoTop (the homotopy category of U-small topological spaces) has U-small hom-sets and U-small products. (It lacks equalisers; but it has weak equalisers, whence U-small weak limits.) [HoTop is the category of fractions of Top with respect to homotopy equivalences. One proves that it has U-small hom sets by realising it as the quotient of Top modulo the homotopy congruence. U-small products (as well as U-small sums) are inherited from Top, because they are "2-products" there, i.e. satisfy the universal property also for homotopies. Weak equalisers are provided by homotopy equalisers in Top.] With best regards Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ From rrosebru@mta.ca Wed Dec 6 16:21:22 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB6Jqmx27369 for categories-list; Wed, 6 Dec 2000 15:52:48 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <003501c05fb9$51f038a0$560318c4@Macs200002> From: "DR Mawanda" To: References: Subject: categories: Re: Categories ridiculously abstract Date: Wed, 6 Dec 2000 21:18:24 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2919.6700 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2919.6700 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 13 My understanding of the relation between category theory and set theory is that category theory is a formal theory built on abstract concepts (objects and morphisms). The way of defining category theory need a metalanguage which is closed to the logic of set theory language (a particular case of what is called boolean logic). There is a sort of dichothomy between logic behind the two theories. This dichothomy come from our limitation of talking about category theory. We use already two-valued logic (true and false) which we cannot avoid if we need to talk about identity of objects and morphisms. Now a kind of Godel's arguments about natural numbers (If N is consistent, then there is no proof of its consistency by method formalizable within the theory ) is what is going on. This doesn't stop the category theory 'game'. When you give birth to a child you will never know in advance if the child will be an honest person or a criminal. Category theory have generated many structures which can help us to understand why many mathematicians have work differently to describe a same mathematical concept in different ways. As an example we know, from category theory, that Cauchy and Dedekind were defining real numbers from rational numbers but the two definitions are not saying the same thing. ----- Original Message ----- From: "Michael MAKKAI" To: Sent: Saturday, December 02, 2000 12:19 AM Subject: categories: Re: Categories ridiculously abstract > In "Towards a categorical foundation of mathematics" (Logic Colloquium > '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic > no.11, 1998; pp.153-190) and in subsequent work, I am proposing an > approach to a foundation whose universe consists of the weak n-categories > and whatever things are needed to connect them. This is done on the basis > of a general point of view concerning the role of identity of mathematical > objects. Readers of said paper who have followed developments on weak > higher dimensional categories will note that much has been done since > towards fleshing out the program. > > Michael Makkai > > > On Thu, 30 Nov 2000, Tom Leinster wrote: > > > > > Michael Barr wrote: > > > > > > And here is a question: are categories more abstract or less abstract than > > > sets? > > > > A higher-dimensional category theorist's answer: > > "Neither - a set is merely a 0-category, and a category a 1-category." > > > > There's a more serious thought behind this. Sometimes I've wondered, in a > > vague way, whether the much-discussed hierarchy > > > > 0-categories (sets) form a (1-)category, > > (1-)categories form a 2-category, > > ... > > > > has a role to play in foundations. After all, set-theorists seek to found > > mathematics on the theory of 0-categories; category-theorists sometimes talk > > about founding mathematics on the theory of 1-categories and providing a > > (Lawverian) axiomatization of the 1-category of 0-categories; you might ask > > "what next"? Axiomatize the 2-category of (1-)categories? Or the > > (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek > > and head in clouds, that general n-categories provide a more natural > > foundation than either 0-categories or 1-categories? > > > > > > Tom > > > From rrosebru@mta.ca Thu Dec 7 12:08:24 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB7FPow03123 for categories-list; Thu, 7 Dec 2000 11:25:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f To: categories@mta.ca Subject: categories: Re: Can we ignore smallness? References: From: Dan Christensen Date: 06 Dec 2000 16:49:35 -0500 In-Reply-To: Message-ID: <873dg15ifk.fsf@jdc.math.uwo.ca> Lines: 34 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.7 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: 14 I agree with what Marco Grandis wrote, suggesting that sometimes it is important to know that the hom sets in a category are small, and want to just supplement what he said with some examples from topology. In topology, one often wants to use a generalized homology or cohomology theory E to compute something, and it can be useful to "localize" a space with respect to this (co)homology theory. The localization X --> L_E X can be characterized as the terminal map from X which induces an isomorphism under E. The existence of such localizations for all X is equivalent to the category Top[(E-isomorphisms)^{-1}] having small hom sets, and so knowing that the latter is true means that one has an important tool for practical computations. The paper by Bousfield Bousfield, A. K. The localization of spaces with respect to homology. Topology 14 (1975), 133--150. is considered quite important because it showed that for any generalized homology theory E, localizations exist, and these localizations now play a central role in homotopy theory. Bousfield proved the existence by showing that the category of fractions above has small hom sets. And he did that by showing that there is a model structure on the category Top with the E-isomorphisms as the weak equivalences. Note that it is still an open question as to whether *co*homological localizations exist for every cohomology theory E! Casacuberta, Scevenels and Jeff Smith have recently shown that they exist if you assume Vopenka's principle, but if anyone can prove this in general or show it is independent of ZFC, that would be considered very interesting. Dan From rrosebru@mta.ca Thu Dec 7 22:41:22 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB829dH26229 for categories-list; Thu, 7 Dec 2000 22:09:39 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Thesis: `Operads in higher-dimensional category theory' To: categories@mta.ca Date: Fri, 8 Dec 2000 01:38:50 +0000 (GMT) X-Mailer: ELM [version 2.5 PL3] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: Tom Leinster Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 15 My PhD thesis, `Operads in higher-dimensional category theory', is now (approved and) available electronically as math.CT/0011106 - that is, at http://arXiv.org/abs/math.CT/0011106 The summary follows. Tom Operads in Higher-Dimensional Category Theory Tom Leinster The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak omega-category. This theory of operads and multicategories has various other applications too: for instance, to the opetopic approach to n-categories expounded by Baez, Dolan and others, and to the theory of enrichment of higher-dimensional categorical structures. We sketch some of these further developments, without exploring them in full. We start with a look at bicategories (Chapter 1). Having reviewed the basics of the classical definition, we define `unbiased bicategories', in which n-fold composites of 1-cells are specified for all natural n (rather than the usual nullary and binary presentation). We go on to show that the theories of (classical) bicategories and of unbiased bicategories are equivalent, in a strong sense. The heart of this work is the theory of generalized operads and multicategories. More exactly, given a monad T on a category E, satisfying simple conditions, there is a theory of T-operads and T-multicategories. In Chapter 2 we set up the basic concepts of the theory, including the important definition of an algebra for a T-multicategory. In Chapter 3 we cover an assortment of further operadic topics, some of which are used in later parts of the thesis, and some of which pertain to the applications mentioned in the first paragraph. Chapter 4 is a (proposed) definition of weak omega-category, a modification of that given by Batanin (Adv Math 136 (1998), 39-103). Having given the definition formally, we take a long look at why it is a *reasonable* definition. We then explore weak n-categories (for finite n), and show that weak 2-categories are exactly unbiased bicategories. The four appendices take care of various details which would have been distracting in the main text. Appendix A contains the proof that unbiased bicategories are essentially the same as classical bicategories. Appendix B describes how to form the free T-multicategory on a given T-graph. In Appendix C we discuss various facts about strict omega-categories, including a proof that the category they form is monadic over an appropriate category of graphs. Finally, Appendix D is a proof of the existence of an initial object in a certain category, as required in Chapter 4. From rrosebru@mta.ca Tue Dec 12 11:14:41 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBCE3AR11700 for categories-list; Tue, 12 Dec 2000 10:03:10 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.3.32.20001211112530.006b28c8@mat.uc.pt> X-Sender: cita2001@mat.uc.pt X-Mailer: QUALCOMM Windows Eudora Light Version 3.0.3 (32) Date: Mon, 11 Dec 2000 11:25:30 +0000 To: categories@mta.ca From: Congresso Ibero-Americano de Topologia e suas Aplicacoes Subject: categories: First Announcement 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: 16 Please circulate to all interested colleagues. ===================================================== FIRST ANNOUNCEMENT ===================================================== IV Iberoamerican Conference on Topology and its Applications Department of Mathematics University of Coimbra Coimbra, Portugal April 18-21, 2001 The Department of Mathematics of the University of Coimbra is organizing the "IV Congresso Ibero-Americano de Topologia e suas Aplicacoes" to be held at the University of Coimbra, from 18th to 21st of April, 2001. The conference arrival day is Tuesday April 17. The scientific programme will begin Wednesday morning April 18 and will finish before lunch on Saturday April 21. The conference will consist of 50mn plenary lectures delivered by invited speakers and 25mn contributed talks. It will open with a special invited lecture by Prof. Bernhard Banaschewski, on the occasion of his 75th birthday. Invited Speakers ================ Bernhard Banaschewski (McMaster Univ., Canada) Marcel Erne' (Univ. Hannover, Germany ) Manuel Sanchis (Univ. Jaume I, Spain) Manuela Sobral (Univ. Coimbra, Portugal) Franklin D. Tall (Univ. Toronto, Canada) Artur H. Tomita (Univ. Sao Paulo, Brazil) F. Javier Trigos-Arrieta (California State Univ., USA) Jorge Vielma (Univ. Los Andes, Venezuela) Scientific Committee ==================== Ofelia Alas (Univ. Sao Paulo, Brazil) Maria Manuel Clementino (Univ. de Coimbra, Portugal) Salvador Garcia-Ferreira (UNAM, Mexico) Valentin Gregori (Univ. Politecnica de Valencia, Spain) Salvador Hernandez (Univ. Jaume I, Spain) Horst Herrlich (Univ. Bremen, Germany) Hans Peter Kuenzi (Univ. Cape Town, South Africa) Robert Lowen (Univ. Antwerp, Belgium) Stephen Watson (York Univ., Canada) Organizing Committee ==================== Maria Manuel Clementino (Univ. Coimbra) Jorge Picado (Univ. Coimbra) Lurdes Sousa (Inst. Politecnico de Viseu) Maria Joao Ferreira (Univ. Coimbra) Goncalo Gutierres (Univ. Coimbra) Dirk Hofmann (Univ. Coimbra) Deadlines ========= Registration and payment of fees: January 31, 2001 Reservation of accommodation: January 31, 2001 Abstract submission: February 28, 2001 Conference Homepage =================== The Registration and Accommodation Form and the Submission of Abstracts, as well as up-to-date information, are available at our web page http://www.mat.uc.pt/~cita2001/ Conference Addresses ==================== E-mail address: cita2001@mat.uc.pt Postal address: CITA 2001 (c/o Prof. Maria Manuel Clementino) Departamento de Matematica Universidade de Coimbra 3001-454 Coimbra Portugal From rrosebru@mta.ca Tue Dec 12 17:03:39 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBCKBgr06947 for categories-list; Tue, 12 Dec 2000 16:11:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3a35cdd73a39f901@amyris.wanadoo.fr> (added by amyris.wanadoo.fr) X-Mailer: Microsoft Internet Mail & News for Macintosh - 3.0 Date: Tue, 12 Dec 2000 08:19:37 +0000 Subject: categories: Terminology From: Jean Benabou To: Category list Mime-version: 1.0 Content-type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id eBC73rt15252 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 17 I am confronted with problems of "contradictory terminology" which I would like to solve and, since english is not my language, I need some suggestions. Let F: Y-----> X be a functor such that for every object x of X the comma category (x,F) is connected.Such functors, although they are not defined in all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory. The "cofinal" name comes obviously from the vocabulary of ordered sets which are special cases, but in category theory "co" is now associated with dual notions. The "initial" name is even less satisfactory, because: (i) If Y=1, F is identified with an object x of X and F is "initial" iff x is a terminal object of X ! (ii) More generally, if Y has a terminal object t then F is "initial" iff F(t) is terminal ! (iii) Even more generally yet, without assuming the existence of terminal objects in Y or X : Let X^ and Y^ be the categories of presheaves on X and Y, and F! :X^-----> Y^ the canonical extension of F to these categories.If T is the terminal object of Y^ one can easily show that F has the previous property iff F!(T) is terminal in X^.(Which by the way, gives the nicest proof of the stability under composition of such functors) I propose to call these functors either "terminal" or better "final" but I would like to know if this would not conflict with previous terminology. Thanks for your help. From rrosebru@mta.ca Wed Dec 13 09:21:10 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBDChOd26649 for categories-list; Wed, 13 Dec 2000 08:43:24 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from zent.mta.ca (zent.mta.ca [138.73.101.4]) by mailserv.mta.ca (8.11.1/8.11.1) with SMTP id eBD1Hot05503 for ; Tue, 12 Dec 2000 21:17:50 -0400 (AST) X-Received: FROM siv.maths.usyd.edu.au BY zent.mta.ca ; Tue Dec 12 21:19:09 2000 -0400 X-Received: from smap@localhost by siv.maths.usyd.edu.au (8.8.8/5.7 to SMTP) id MAA192177 for cat-dist@mta.ca; Wed, 13 Dec 2000 12:17:32 +1100 (EST) X-Received: from milan.maths.usyd.edu.au (stevel(.pmstaff;2406.2002)@milan.maths.usyd.edu.au) [129.78.69.163] by siv.maths.usyd.edu.au via smtpdoor V12.3 id 192306 for cat-dist@mta.ca; Wed, 13 Dec 2000 12:17:32 +1100 X-Received: from stevel@localhost by milan.maths.usyd.edu.au (8.8.8/5.5 to SMTP) id MAA21581; Wed, 13 Dec 2000 12:17:32 +1100 (EST) From: Steve Lack MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Message-ID: <14902.52780.44914.922801@milan.maths.usyd.edu.au> Date: Wed, 13 Dec 2000 12:17:32 +1100 (EST) To: categories@mta.ca Subject: categories: Re: Terminology X-Mailer: VM 6.71 under 21.1 (patch 7) "Biscayne" XEmacs Lucid Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id eBD1Hot05503 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 18 Jean Benabou writes: > I am confronted with problems of "contradictory terminology" which I would > like to solve and, since english is not my language, I need some > suggestions. > Let F: Y-----> X be a functor such that for every object x of X the comma > category (x,F) is connected.Such functors, although they are not defined in > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's > handbook (Vol.1-§2.11-p.69) but none of these terms is satisfactory. Mac Lane calls such functors ``final'' in Categories for the Working Mathematician. I do too. Steve Lack. From rrosebru@mta.ca Wed Dec 13 09:21:14 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBDCg4q27816 for categories-list; Wed, 13 Dec 2000 08:42:04 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Re: Terminology To: categories@mta.ca Date: Wed, 13 Dec 2000 11:10:32 +0000 (GMT) In-Reply-To: <3a35cdd73a39f901@amyris.wanadoo.fr> (added by amyris.wanadoo.fr) from "Jean Benabou" at Dec 12, 2000 08:19:37 AM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: "Dr. P.T. Johnstone" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 19 > I am confronted with problems of "contradictory terminology" which I would > like to solve and, since english is not my language, I need some > suggestions. > Let F: Y-----> X be a functor such that for every object x of X the comma > category (x,F) is connected.Such functors, although they are not defined in > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's > handbook (Vol.1-'2.11-p.69) but none of these terms is satisfactory. > The "cofinal" name comes obviously from the vocabulary of ordered sets > which are special cases, but in category theory "co" is now associated with > dual notions. There was some discussion of this point on the categories mailing list a year or two back. I think there was general consensus that the "co" in "cofinal" was redundant, and that such functors should simply be called "final". This is the term used in Mac Lane's book (section IX 3, p.217) -- I believe Mac Lane was the first to shorten "cofinal" to "final". For some reason, Borceux chose to use the opposite convention regarding "initial" and "final" in his book (although, in Exercise 2.17.8 on page 94, he seems to have reverted to the same convention as Mac Lane). Peter Johnstone From rrosebru@mta.ca Wed Dec 13 09:59:39 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBDDIqS01714 for categories-list; Wed, 13 Dec 2000 09:18:52 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 13 Dec 2000 13:09:53 GMT From: Paul Taylor Message-Id: <200012131309.NAA26562@koi-pc.dcs.qmw.ac.uk> To: categories@mta.ca Subject: categories: final functors Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 20 As Steve Lack and Peter Johnstone said, the functors to which Jean Benabou referred, > Let F: Y-----> X be a functor such that for every object x of X the comma > category (x,F) is connected.Such functors, although they are not defined in > all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's > handbook (Vol.1-'2.11-p.69) but none of these terms is satisfactory. are called "final" in Saunders Mac Lane's famous book. As Peter also said, there was a lengthy discussion on "categories" in July 1998 on this topic, which you can look up in the archive at ftp://tac.mta.ca/pub/categories/1998/98-7 The footnote that I wrote on p389 of "Practical Foundations" http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s73.html summing up this discussion reads > The prefix ``co-'' in the original word cofinal carried the usual > Latin--English meaning of ``together,'' rather than the meaning of > dualisation inherited from (co)homology (and maybe trigonometry > before that). > Although final functors are the analogue, not the dual, of cofinal > monotone functions, the prefix was dropped in [Mac Lane, p. 213] as it > was considered inappropriate. > I feel that it was unnecessary to introduce this confusion, as > Proposition 3.2.10 associates them with \emph{co}limits (but cf > Exercise 3.32). > Even so, the definitions are not the same: any surjective function > between discrete posets is cofinal and they give rise to the same > joins, but to different coproducts. This difference is attributable > to the hidden existential quantifier mentioned in the footnote on page 129. Paul From rrosebru@mta.ca Thu Dec 14 10:10:17 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEDT3j10837 for categories-list; Thu, 14 Dec 2000 09:29:03 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 14 Dec 2000 17:17:41 +1100 (EST) From: maxk@maths.usyd.edu.au (Max Kelly) Message-Id: <200012140617.RAA02673@milan.maths.usyd.edu.au> To: categories@mta.ca Subject: categories: Re: Terminology Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 In response to Jean Benabou's question about the terminology for what some call "cofinal" functors, may I refer him to Section 4.5 of my book "Basic Concepts of Enriched Category Theory", where such notions are considered in considerable generality? In so far as we deal with functors - meaning "V-functors" in the context of V-enriched category theory - the terms I used, which are those common here at Sydney, are "final functor" and "initial functor". These notions, however, make sense only when V is cartesian closed; for a more general symmetric monoidal closed V, what is said to be initial is a pair (K,x) where K is a V-functor A --> C and x is a V-natural transformation H --> FK, where H: A --> V and F: C --> V are V-functors with codomain V, and thus are "weights" for weighted limits. The 2-cell x expresses F as the left Kan extension of H along K if and only if, for every V-functor T: C --> B of domain C, the canonical comparison functor (induced by K and x) between the weighted limits, of the form (K,x)* : {F,T} ----> {H,TK}, is invertible (either side existing if the other does); the book contains a third equivalent form making sense whether the limits exist or not. When these equivalent properties hold, the pair (K,x) is said to be INITIAL. The point is that, in this case, the F-weighted limit of any T can be calculated as the H-weighted limit of TK. When V is cartesian closed, we have for each V-category C the V-functor C ---> V constant at the object 1, limits weighted by which are the CONICAL limits, which when V = Set are the classical limits. For such a V we can consider the special case of the situation considered above, where each of H and F is the functor constant at the object 1, and where x is the unique 2-cell between H and FK; we call the functor K "initial" when this pair (K,x) is so; equivalently when the canonical lim T ---> lim TK is invertible for every T (for which one side exists -- or better put in terms of cones), or equivalently again when colim C(K-,c) == 1 for each object c of C. When V = Set, this is just to say that each comma-category K/c is connected. When the category C is filtered, a fully-faithful K: A --> C is final (dual to initial) precisely when each c/K is non-empty. The book goes on to discuss the Street-Walters factorization of any (ordinary) functor into an initial one followedby a discrete op-fibration. The above being so, it seems that Jean's good taste has led him to suggest the very same nomenclature that recommended itself to us at Sydney. I should have been happier, though, if he had recalled the treatment I gave lovingly those many years ago. There are many other expositions in the book that I am equally happy with, and which I am sure Jean would enjoy. By the way, someone spoke recently on this bulletin board of the book's being out of print and hard to get; I've been meaning to find the time to reply to that, and discuss what might be done. The copyright has reverted to me; but the text does not exist in electronic form - it was written before TEX existed, and prepared on an IBM typewriter by an excellent secretary with nine balls. I suppose I could have some copies - one or more hundreds - printed from the old master, after correcting the observed typos. But the photocopying and binding and the postage would cost a bit. I'ld be happy to receive suggestions, especially from such colleagues as would like to get hold of a copy. By the way, I sent out preprint copies to about 100 colleagues back in 1980 or 1981; if any of those are still around, I point out that they contain the full text. So too do those copies which appeared in the Hagen Seminarberichte series. Once again, I look forward to any comments, either in favour of or against making further copies. Max Kelly. From rrosebru@mta.ca Thu Dec 14 10:11:11 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBECix628138 for categories-list; Thu, 14 Dec 2000 08:44:59 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 13 Dec 2000 20:19:28 -0800 (PST) From: Bill Rowan Message-Id: <200012140419.eBE4JSF19580@transbay.net> To: categories@mta.ca Subject: categories: Query about Ab[C] Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 22 Hi all, Ab[C] is just my notation for the category of abelian group objects in the category C. I was wondering if there is a simple characterization of those categories C for which Ab[C] is abelian. Bill Rowan From rrosebru@mta.ca Thu Dec 14 16:18:41 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJNpG12236 for categories-list; Thu, 14 Dec 2000 15:23:51 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 14 Dec 2000 09:44:57 -0500 (EST) From: Peter Freyd Message-Id: <200012141444.eBEEiv005794@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Re: Query about Ab[C] Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 23 Bill Rowan asks if there is a simple characterization of those categories C for which Ab[C] is abelian. I doubt if there can be a useful necessary and sufficient condition. A sufficient condition can be found on page 91 of Cats and Alligators, to wit, that the category be effective regular (where "effective" means that every equivalence relation is effective,i.e. it appears as a pullback of a map against itself). Note that the conclusion (abelian) is self-dual but the condition (effective regular) is not. I'm pessimistic about a useful necessary and sufficient condition because of the following: Let C be a category with cartesian squares (needed to define abelian-group-object) such that Ab[C] is abelian. Let C' be a full subcategory closed under cartesian squaring that contains the image of the forgetful functor from Ab[C] back to C. Then Ab[C'] = Ab[C]. An example of the sort of pathological categories to be found among such C' is the category of all groups in which the commutator subgroup is a product of a finite number of simple groups each of which was described prior to 30 June 1973. From rrosebru@mta.ca Thu Dec 14 16:19:01 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJP2T03622 for categories-list; Thu, 14 Dec 2000 15:25:02 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: rags owned process doing -bs Date: Thu, 14 Dec 2000 10:05:22 -0500 (EST) From: "Robert A.G. Seely" To: categories@mta.ca Subject: categories: Re: Max's LMS book (out-of-print) In-Reply-To: <200012140617.RAA02673@milan.maths.usyd.edu.au> 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: 24 > I suppose I could have some copies - one or more hundreds - printed > from the old master, after correcting the observed typos. But the > photocopying and binding and the postage would cost a bit. I'ld be > happy to receive suggestions, especially from such colleagues as would > like to get hold of a copy. ... > I look forward to any comments, either in favour of or against making > further copies. Why not get it retyped in TeX (making minor changes - but don't attempt a serious revision, which would just take time with little reward towards the project in mind)? - there are bound to be secretaries who could do that for a reasonable fee. This would only be worth it if you can get a "subscription" system to offset that fee of course. I do have a copy myself - picked up in a second hand shop in Cambridge during one of the category meetings there - I think it was in the late 80's. That copy has been duplicated several times in recent years; I think at least one of its offspring has made it back to Sydney! So, although I may not buy another copy, I would encourage any efforts to keep this book in circulation. Mike Barr may have some advice - both the books he and Charles wrote are now available cheaply, one on the web, the other via the CRM (at U de Montreal). -= rags =- ("secretaries with 9 balls" - only in Sydney ... :-) ) ================== R.A.G. Seely From rrosebru@mta.ca Thu Dec 14 16:19:01 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJRRa11515 for categories-list; Thu, 14 Dec 2000 15:27:27 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Thu, 14 Dec 2000 12:39:57 -0500 (EST) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: categories@mta.ca Subject: categories: Re: Query about Ab[C] In-Reply-To: <200012140419.eBE4JSF19580@transbay.net> 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: 25 I suspect that it is too much to hope for a characterization. But it is sufficient that either C or C^op be exact. For C to be exact, it is required that it have finite limits, coequalizers of equivalence relations, that regular epis be stable under pullback and that every equivalence relation is the kernel pair of its coequalizer. On Wed, 13 Dec 2000, Bill Rowan wrote: > > Hi all, > > Ab[C] is just my notation for the category of abelian group objects in > the category C. I was wondering if there is a simple characterization of > those categories C for which Ab[C] is abelian. > > Bill Rowan > From rrosebru@mta.ca Thu Dec 14 16:20:59 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJQgx06464 for categories-list; Thu, 14 Dec 2000 15:26:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Re: Query about Ab[C] To: categories@mta.ca Date: Thu, 14 Dec 2000 16:25:39 +0000 (GMT) In-Reply-To: <200012140419.eBE4JSF19580@transbay.net> from "Bill Rowan" at Dec 13, 2000 08:19:28 PM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: "Dr. P.T. Johnstone" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 26 > > > Hi all, > > Ab[C] is just my notation for the category of abelian group objects in > the category C. I was wondering if there is a simple characterization of > those categories C for which Ab[C] is abelian. > > Bill Rowan You can't hope to characterize them: knowing properties of Ab[C] can't tell you everything about C. For example, if C has a strict terminal object, then Ab[C] is abelian (because it's degenerate), but that gives you no information about what else C might contain. If you're looking for a sufficient condition on C, a canonical one is "Barr-exact" (= effective regular, in Freyd's terminology): Ab[C] inherits Barr-exactness from C, and abelian is equivalent to Barr-exact plus additive. Conversely, every abelian category A is isomorphic to Ab[C] for a suitable Barr-exact C, namely C = A. Peter Johnstone From rrosebru@mta.ca Thu Dec 14 19:29:18 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEMCDL23236 for categories-list; Thu, 14 Dec 2000 18:12:13 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: baez@newmath.UCR.EDU Message-Id: <200012142107.eBEL7WQ24612@math-cl-n03.ucr.edu> Subject: categories: (-1)-categories and (-2)-categories To: categories@mta.ca (categories) Date: Thu, 14 Dec 2000 13:07:32 -0800 (PST) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 27 Frank Atanassow emailed me saying that he's "dying to know" the answer to my puzzle about (-1)-categories and (-2)-categories. That's good! So far the silence has been deafening, and I can't tell if it indicates bewilderment, lack of interest, or disgust with an imprecisely posed question. I guess I'll give away the answer. As I said, the trick is to figure out what (-1)-groupoids and (-2)-groupoids are, and then cross our fingers and hope that the answer is the same for (-1)-categories and (-2)-categories. We start with the basic principle that people use when trying to cook up definitions of "weak n-groupoid": weak n-groupoids should be the same as nice topological spaces of homotopy dimension n. Here "nice" means something like a CW complex, and "the same" must be taken in a suitably n-categorical/homotopical spirit. But what does "homotopy dimension n" mean? Well, the usual definition is that a space has homotopy dimension n if all its homotopy groups above the nth are trivial. But if we carefully unpack this, we'll see it's an interesting condition even when n is -1 or -2. Here goes: A topological space X has homotopy dimension n if given k > n, any continuous map from the k-sphere to X extends to a continuous map from the (k+1)-disc to X. So: X has homotopy dimension 0 if its arc-components have vanishing homotopy groups. If X is nice, this means it's a disjoint union of contractible spaces, so it's the same as a *set*. That's good: 0-groupoids should be sets. (See? I'm using "the same" in a suitably homotopical spirit! A disjoint union of nice contractible spaces is homotopy equivalent to a set with the discrete topology.) Next: X has homotopy dimension -1 if it has homotopy dimension 0 and also any continuous map from the 0-sphere to X extends to a continuous map from the 1-disc to X. In other words, X is either empty, or it consists of a single arc-component with vanishing homotopy groups. If X is nice, this means it's the same as a *set with cardinality 0 or 1*. So we declare that a (-1)-groupoid is a set with cardinality 0 or 1. Next: X has homotopy dimension -2 if it has homotopy dimension -1 and also any continuous map from the (-1)-sphere to X extends to a continuous map from the 0-disc to X. The 0-disc is a single point, and its boundary the (-1)-sphere is the empty set. So X must be a *nonempty* space with homotopy dimension -1. So X consists of a single arc-component with vanishing homotopy groups. If X is nice, this means it's the same as a *set with cardinality 1*. So we declare that a (-2)-groupoid is a set with cardinality 1. Crossing our fingers, we therefore guess: A (-1)-category is a set with cardinality 0 or 1. A (-2)-category is a set with cardinality 1. This may seem silly, but it's not! There is a nice relation between all this business and the notion of "n-stuff". But I'm getting worn out, so instead of explaining that, I'll just quote some articles that James Dolan and Toby Bartels wrote on sci.physics.research when they were first figuring out this "n-stuff" stuff. Frank also asked if the error in our definition of n-categories appears in HDA3. Yes! We will correct it in HDA5, which will be about Feynman diagrams and "n-stuff". As I said, it's easy to fix: just one number is wrong. John Baez ......................................................................... From: "james dolan" Subject: Re: Just Categories now Date: 16 Nov 1998 00:00:00 GMT Message-ID: <72pusp$3o5$1@pravda.ucr.edu> Organization: Department of Mathematics, University of California, Riverside Newsgroups: sci.physics.research X-Newsposter: Pnews 4.0-test50 (13 Dec 96) toby bartels writes: -john baez wrote: ->I will leave it to James Dolan to explain the technical ->distinction between "extra properties", "extra structure", ->and "extra stuff" - there is a nice category-theoretic way ->of making this precise. - -Ooh, let me guess! - -Given a functor U: C -> D, interpret U as a forgetful functor. -Then C is D with extra *structure* if U is surjective on the -objects and, given a pair of objects, injective on the -morphisms between them; and C is D with extra *properties* if -U is injective on the morphisms (meaning injective on the -objects and on the morphisms between a given pair); Otherwise, -I guess C is just D with extra *stuff* if, given a pair of -objects, U is injective on the morphisms between them. here's my classification: given groupoids c,d and a functor u:c->d, the objects of c can be thought of via the forgetful functor u as objects of d with an extra _property_ iff u is full and faithful, as objects of d with extra _structure_ iff u is faithful, and as objects of d with extra _stuff_ regardless. (some category-theoretic jargon: 1. a "groupoid" is a category where all the morphisms are invertible. it may very well be interesting to generalize the subject matter of this discussion to the case where c and d are not necessarily groupoids, but to keep things simple for now i won't do that in this post. 2. a functor u:c->d is "full" iff for any pair c1,c2 of objects in c, the map from the hom-set hom(c1,c2) to the hom-set hom(u(c1),u(c2)) given by u is surjective. 3. a functor u:c->d is "faithful" iff for any pair c1,c2 of objects in c, the map from the hom-set hom(c1,c2) to the hom-set hom(u(c1),u(c2)) given by u is injective.) one reason i don't (as i think toby was suggesting) require the forgetful functor u to be surjective on (isomorphism classes of) objects in order for the objects of c to qualify as objects of d with extra "structure" is as follows: consider for example the case where c is the category of rings, d is the category of groups, and u is the functor assigning to each ring its underlying additive group. clearly the objects of c are objects of d with extra "structure" in the intuitive sense that i'm trying to capture; we can say that "a ring is defined to be a group (henceforward referred to as "the underlying additive group of the ring") equipped with an extra multiplication operation on it satisfying certain equational laws...", and although this may sound like the equational laws only constrain the ring structure on the additive group, they in fact also implicitly constrain the additive group itself: it's easy to show that even if you don't explicitly require the additive group of a ring to be commutative, it will automatically be forced to be commutative by the other clauses in the usual definition of "ring" (left and right distributivity plus multiplicative unit laws, in combination with the group axioms for addition, should do it, i think). thus this example is supposed to demonstrate the fact that as soon as you generally allow yourself to invent a new "type of structure that an object of d can be equipped with" by starting with an arbitrary existing such type of structure and constraining the structures to satisfy some property, it's awkward to prevent an arbitrary "property that can be predicated of an object of d" from being considered as a "type of structure that an object of d can be equipped with" by being looked at as a constraint on [the degenerate "type of structure that an object of d can be equipped with" given by "no extra structure at all"]. thus you should probably broaden your concept of "type of structure that an object of d can be equipped with" to include "property that can be predicated of an object of d" as a special case. for similar reasons you should probably broaden your concept of "type of stuff that an object of d can be equipped with" to include "type of structure that an object of d can be equipped with" as a special case, if it isn't that broad already. -For example, the forgetful functor Groups -> Sets shows tha -groups are sets with extra structure, while the forgetful -functor Abelian Groups -> Groups shows that Abelian groups are -groups with extra properties. i agree with those examples (at least if i interpret them in accordance with my self-imposed restriction to consider only the case where all of the morphisms in c and d are invertible). -Or you can turn around and use -the free functor Sets -> Groups and say that sets are groups -with extra properties (to wit, the property of being free). i disagree with that example, for reasons that hopefully are clear from my explanations above. thus i would _not_ say that a set is a group with the extra property of being free; rather i'd say that a set is a group with the extra _structure_ of being equipped with a favored "basis" of mutually free mutual generators. -OTOH, the Abelianization functor Groups -> Abelian groups is -surjective on the objects (and on the morphisms for that -matter), but groups are not Abelian groups with extra -structure, because the functor isn't injective on the -morphisms between a given pair. i think i agree with this, but it sounds like you're using my rules here rather than the rules i thought you tried to spell out in your post. another example of an object equipped with extra "stuff" would be a set equipped with another _set_; that is, take c to be the category of ordered pairs of sets, d to be the category of sets, and u to be the "projection" functor assigning to an ordered pair (x,y) of sets its first coordinate x. i hope this example helps to show why i consider the terminology "stuff" reasonably descriptive of the intuition involved. another example (maybe or maybe not causing some additional (?) number of people to see this post as having some relevance to physics) of an object equipped with extra "stuff" rather than merely with extra "structure" is a manifold equipped with an unfortunately so-called "spin structure". the point is that if we define the concept of "morphism between spin manifolds" in what seems to me to be the most advantageous way, then taking c to be the category of spin manifolds, d the category of manifolds, and u the hopefully obvious forgetful functor assigning to a spin manifold its underlying ordinary manifold, u is not faithful. thus a "spin structure" is not merely "structure"; it's "stuff". so what is this extra "stuff"?? you can think of it as "spin frames" if you want to. (a "spin frame" is what a spin manifold has two of where an ordinary manifold has only one.) or you can think of it as "spinors"; morphisms between spin manifolds have an extra discrete degree of freedom to flip the sign of spinors even after their action on ordinary manifold points has been completely nailed down. a deeper understanding of how the classification offered here arises involves the relationship between groupoid theory and homotopy theory, as follows: for any integer n greater than or equal to -1, a space x is defined to be of "homotopy dimension n" iff for any integer j strictly greater than n, every continuous map from the j-dimensional sphere s^j to x is homotopic to a constant map. using this terminology, every space of homotopy dimension n is also of homotopy dimension m for any integer m greater than n. a crucial fact is that the world of spaces of homotopy dimension 1 is secretly isomorphic in a very strong way to the world of groupoids; there's an amazingly perfect "dictionary" linking concepts from the world of spaces of homotopy dimension 1 to their secret equivalents in the world of groupoids. the groupoid corresponding to a space x of homotopy dimension 1 is called the "fundamental groupoid" of x, and the space of homotopy dimension 1 corresponding to a groupoid g is called the "classifying space" of g. inside the world of spaces of homotopy dimension 1 are of course the sub-world of spaces of homotopy dimension 0, and the sub-sub-world of spaces of homotopy dimension -1. the secret equivalent inside the world of groupoids of the sub-world of spaces of homotopy dimension 0 is the sub-world of so-called "discrete groupoids", and the secret equivalent of the sub-sub-world of spaces of homotopy dimension -1 is the sub-sub-world of just those special discrete groupoids which have either just one object and one morphism, or no objects and morphisms at all. the discrete groupoids are also known as "sets", or (exploiting the [homotopy dimension 1]/groupoids dictionary) "groupoids of homotopy dimension 0". the special discrete groupoids corresponding to the spaces of homotopy dimension -1 are called "truth values", or "groupoids of homotopy dimension -1". the groupoid with just one object and one morphism is called "true" (aka "the terminal groupoid" aka "yes" aka "in") while the empty groupoid is called "false" (aka "the initial groupoid" aka "no" aka "out"). given a pair c,d of groupoids and a functor u:c->d and an object d1 in d, we can construct a new groupoid called "the homotopy fiber of u over d1". roughly, the homotopy fiber of u over d1 is the groupoid of "objects of c equipped with designated isomorphisms from their images under u to d1"; the morphisms in the homotopy fiber are required to preserve the designated isomorphisms. as you might guess from the name "homotopy fiber", the groupoid-theoretic concept of "homotopy fiber" has a very direct equivalent in homotopy theory. we can now re-state the definitions of "property", "structure", and "stuff" in terms of homotopy dimension of homotopy fibers, as follows: given groupoids c,d and a functor u:c->d, the objects of c can be thought of via the forgetful functor u as objects of d with an extra _property_ iff the homotopy fibers of u are all of homotopy dimension -1, as objects of d with extra _structure_ iff the homotopy fibers of u are all of homotopy dimension 0, and, and as objects of d with extra _stuff_ iff the homotopy fibers of u are all of homotopy dimension 1. hopefully this makes the intuition behind the concepts a bit clearer. a "property" is something which, if you possess it at all, then you have no choice in _how_ to possess it, you just do. a "structure" is something which if you possess it then possessing it involves picking a particular structure in a way analogous to picking an element of a set. "stuff" is something which if you possess it then possessing it amounts to picking some particular stuff in a way analogous to picking an object of a groupoid. of course as with most concepts of groupoid theory, the concepts discussed here should be generalized to the case of "higher-dimensional groupoid theory" which corresponds to the homotopy theory of spaces with arbitrary homotopy dimension in the same way that ordinary groupoid theory corresponds to the homotopy theory of spaces of homotopy dimension 1. thus the stunted progression property, structure, stuff becomes a genuine open-ended progression: property, structure, stuff, eka-stuff, eka-eka-stuff, ... . thus given arbitrary spaces c,d and a continuous map u:c->d, we should say that "the objects of the fundamental infinity-groupoid of c can be thought of via the forgetful infinity-functor induced by u as objects of the fundamental infinity-groupoid of d equipped with extra eka^n-stuff" iff all of the homotopy fibers of u are of homotopy-dimension n+1. From: james dolan Subject: Re: Just Categories now Date: 05 Jan 1999 00:00:00 GMT Message-ID: <76rulr$h7d$1@pravda.ucr.edu> References: <726r22$qr$1@news-1.news.gte.net> <3647A029.6FB4@easyon.com> <72agsd$ook$1@pravda.ucr.edu> <72b3pd$709@gap.cco.caltech.edu> Organization: Department of Mathematics, University of California, Riverside Newsgroups: sci.physics.research X-Newsposter: Pnews 4.0-test50 (13 Dec 96) toby bartels writes: -james dolan wrote: - ->given groupoids c,d and a functor u:c->d, the objects of c can ->be thought of via the forgetful functor u as objects of d with ->an extra _property_ iff u is full and faithful, as objects of d ->with extra _structure_ iff u is faithful, and as objects of d ->with extra _stuff_ regardless. - ->A "groupoid" is a category where all the morphisms are ->invertible. it may very well be interesting to generalize the ->subject matter of this discussion to the case where c and d are ->not necessarily groupoids, but to keep things simple for now i ->won't do that in this post. - -You seem to agree with John Baez's classification, -but he doesn't feel the need to limit to groupoids; -perhaps a word on how you think that complicates things? it complicates things in the obvious way: a single concept in groupoid theory (for example the concept of "faithful functor between groupoids") may bifurcate into non-equivalent concepts in category theory (for example the concepts of "faithful functor between categories" and "functor between categories which is faithful on isomorphisms"); the necessity of worrying about the distinctions between such non-equivalent concepts is eliminated by discussing only the groupoid case. but presumably you're also asking why it is that in this tradeoff between simplicity and generality i chose simplicity, so i'll try to say something about that too. -Or is it just that groupoids are needed for the deep homotopy connection? that's part of my motivation by now, but i think my original motivation had less to do with the "dictionary" that relates groupoid theory to a special part of homotopy theory than with a different but in its own way equally powerful "dictionary" relating groupoid theory to a special kind of predicate logic. in the world of predicate logic there's an obvious sense in which adding extra "properties" to the models of a theory means adding new axioms to the theory, adding extra "structure" to the models means adding new predicate symbols (possibly supplemented by new axioms) to the theory, and adding extra "stuff" to the models means adding new "types" (possibly supplemented by new predicate symbols and axioms) to the theory. this property/structure/stuff distinction in predicate logic matches perfectly the property/structure/stuff distinction in groupoid theory if groupoids are interpreted as a certain sort of logical theories in a certain way. the more i think about this the more it seems that there should be some nice big picture that links together the predicate logic aspects of the situation with the homotopy theory aspects of the situation, but if so it's a bit too big for me to fully grasp yet so i won't try to say any more about it at the moment. i will say though that if someone would show how to generalize the correspondence between groupoids and logical theories of a certain sort to a correspondence between categories and logical theories of some more general sort, then i might be willing to agree that there is some obvious way of extending the property/structure/stuff classification of groupoid theory to apply to category theory as well. i have a vague suspicion that in fact this has already been done and that the logical theories corresponding to categories differ from the logical theories corresponding to groupoids more or less precisely in being "intuitionistic" rather than "classical", but i'm not at all clear on the details of how this works if it's even correct. ->a deeper understanding of how the classification offered here ->arises involves the relationship between groupoid theory and ->homotopy theory, as follows: - ->for any integer n greater than or equal to -1, a space x is ->defined to be of "homotopy dimension n" iff for any integer ->j strictly greater than n, every continuous map from the ->j-dimensional sphere s^j to x is homotopic to a constant map. - -You can even generalize this to n = -2, noting that s^{-1} is the empty set. yes, very much so, though i don't think i thought about this until afterwards. -Of course, no map from s^{-1} to any space can ever be homotopic to a -constant, yet there is always some map from s^{-1} to any space (the -empty map), so no space has homotopy dimension -2, which must be why -nobody talks about it. hmm. first of all, i think i should revise my definition of homotopy dimension to eliminate the idea of "homotopic to a constant map", because people seem to disagree on the meaning of "constant map" when the domain is empty. (some people think that constantness of maps is the property of factoring through the one-point set, others think it's the _structure_ of being equipped with a specific factorization through the one-point set, and toby apparently thinks it's the property of having the one-point set as image.) here's the revised version: for any integer n greater than or equal to -2, a space x is defined to be of "homotopy dimension n" iff for every continuous map m from the [n+1]-dimensional sphere s^[n+1] to x, the space of extensions of m to the [n+2]-dimensional disk d^[n+2] is contractible. (here the sphere s^[-1] is defined to be empty, the disk d^[j+1] is defined to be the mapping cylinder of the map s^j->1, and the sphere s^[j+1] is defined to be the pushout d^[j+1] +_[s^j] d^[j+1]. "contractible" means equivalent to the 1-point space.) hopefully with this revised definition it's still true that being of homotopy dimension n implies being of homotopy dimension n+1. the spaces of homotopy dimension -2 are the contractible spaces, and the spaces of homotopy dimension n for higher n are hopefully just as before. the spaces of homotopy dimension n taken from a sufficiently "convenient" category s of spaces form a cartesian closed category s_n, and the spaces of homotopy dimension n+1 in s are the spaces equivalent to classifying spaces of groupoids enriched over s_n. the class of continuous maps with all homotopy fibers of homotopy dimension -2 is the class of all homotopy equivalences. in the world of groupoids this corresponds to the class of all functors that are "invertible up to natural isomorphism". thus eka^[-3]-stuff is _vacuous_ properties; that is, given groupoids c,d and a functor f:c->d with f invertible up to natural isomorphism, objects of c can be thought of as objects of d equipped with a _vacuous_ property. (as throughout this discussion, we are interested in everything only "up to natural isomorphism" or "up to homotopy" in groupoid theory or in homotopy theory theory respectively.) notice that the class of all maps with all homotopy fibers of homotopy dimension n is closed under composition because the homotopy fibers of a composite map fg are themselves the total spaces of fibrations with base spaces which are homotopy fibers of g and fibers which are homotopy fibers of f, and because the class of spaces of homotopy dimension n is closed under the process of forming a new space as the total space of a fibration with its base and all its fibers in the class. finally, if there's anything such as "spaces of homotopy dimension -3", i don't want to hear about it. From: jdolan@galaxy.ucr.edu (james dolan) Subject: Re: Just Categories now Date: 15 Jan 1999 00:00:00 GMT Message-ID: <77k9fr$va6$1@pravda.ucr.edu> References: <3647A029.6FB4@easyon.com> <72b3pd$709@gap.cco.caltech.edu> <76rulr$h7d$1@pravda.ucr.edu> <77h2qd$ohj@gap.cco.caltech.edu> Organization: none Newsgroups: sci.physics.research toby bartels wrote: -james dolan wrote: - ->Toby Bartels wrote: - ->>Or is it just that groupoids are needed for the deep homotopy ->>connection? - ->that's part of my motivation by now, but i think my original ->motivation had less to do with the "dictionary" that relates groupoid ->theory to a special part of homotopy theory than with a different but ->in its own way equally powerful "dictionary" relating groupoid theory ->to a special kind of predicate logic. in the world of predicate logic ->there's an obvious sense in which adding extra "properties" to the ->models of a theory means adding new axioms to the theory, adding extra ->"structure" to the models means adding new predicate symbols (possibly ->supplemented by new axioms) to the theory, and adding extra "stuff" to ->the models means adding new "types" (possibly supplemented by new ->predicate symbols and axioms) to the theory. this ->property/structure/stuff distinction in predicate logic matches ->perfectly the property/structure/stuff distinction in groupoid theory ->if groupoids are interpreted as a certain sort of logical theories in ->a certain way. - -OK, I tried to think about this, but I don't really know where to -start. Give me a clue: what famous groupoid corresponds to what I've -been taught to regard as the basic predicate calculus: ordinary logic -with forall, forsome, and equality? the correspondence is between individual groupoids and individual _theories_ of a particular form of predicate logic. the particular form of predicate logic involved is pretty much just "the basic" form, with the allowed syntactic constructions including: 1. the usual finitary boolean connectives obeying the usual finitary boolean equational laws 2. the universal quantifier "for all" (and therefore also the existential quantifier "for some" via the equivalence between "for some x, p(x)" and "not (for all x, (not p(x)))") 3. the built-in binary predicate "equality" with it's usual built-in reflexivity, symmetry, transitivity, and substitutability properties plus one more construction going beyond what's ordinarily considered "the basic": 4. the restriction in #1 above against the _infinitary_ boolean connectives (such as n-fold conjunction for an arbitrary infinite cardinality n) is lifted. given a theory t expressed in this kind of logic, we obtain the groupoid of models of t. when all the i's are dotted and the t's crossed in the right way, this process of passing from the theory t to the groupoid of models of t becomes a "bi-equivalence from the bi-category of theories to the bi-category of groupoids". for example, let t be the theory presented by giving no predicate symbols, plus the one axiom "there are exactly seven things". (of course this axiom can be expressed using the allowed syntatic constructions.) the groupoid of models of t is the groupoid of seven-element sets. this groupoid has just one isomorphism class because the theory t is "categorical" (in a sense of the word "categorical" having not much relationship to category theory!). that's not a complete exposition of the situation, rather just a clue of the sort i hope you wanted. i will mention further though that to develop the full correspondence between theories and groupoids, the theories should be allowed to be "multi-typed". if only "single-typed" theories are considered then the most straightforward correspondence is not with "abstract" groupoids but rather with "concrete" groupoids, a "concrete groupoid" being a groupoid equipped with a faithful functor to the groupoid of sets. it might be a good idea to develop the correspondence between single-typed theories and concrete groupoids before developing the full correspondence between multi-typed theories and abstract groupoids. one of the basic lemmas you should try to understand is as follows: let x be a set. let c be the collection of all pairs (s,p) with s a (possibly infinite) set and p an s-ary relation on x. let d be the hyper-collection of all sub-collections of c that are closed under all of the operations on relations alluded to in #1-#4 above. then d is in canonical bijection with the set of subgroups of the group of permutations of x (taking "permutation" to mean "auto-bijection"). (in the above lemma, among the operations that should count as "alluded to" is the operation of replacing an s-ary relation by the obvious corresponding t-ary relation given a bijection from s to t, even though this operation was perhaps _not_ very explicitly alluded to.) From rrosebru@mta.ca Fri Dec 15 09:39:21 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBFD30R11188 for categories-list; Fri, 15 Dec 2000 09:03:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: From: S.J.Vickers@open.ac.uk To: categories@mta.ca Subject: categories: RE: Max's LMS book (out-of-print) Date: Fri, 15 Dec 2000 09:11:23 -0000 MIME-Version: 1.0 X-Mailer: Internet Mail Service (5.5.2650.21) Content-Type: text/plain; charset="iso-8859-1" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 28 > Why not get it retyped in TeX .... This would only > be worth it if you can get a "subscription" system to offset that fee > of course. ... > R.A.G. Seely I was thinking of suggesting the same myself - I'd be very interested in subscribing. All the better if, assuming Max is willing, you can find subscribers public spirited enough to allow free electronic availability once it's TeXed. (Perhaps subscribers get a bound copy signed by the author?) What would it cost? Can OCR scanning help the initial input for such texts? Steve Vickers. From rrosebru@mta.ca Fri Dec 15 11:20:50 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBFEcgf26628 for categories-list; Fri, 15 Dec 2000 10:38:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <4.1.20001215091112.013f67e0@mail.oberlin.net> X-Sender: cwells@mail.oberlin.net X-Mailer: QUALCOMM Windows Eudora Pro Version 4.1 Date: Fri, 15 Dec 2000 09:23:03 -0800 To: categories@mta.ca From: Charles Wells Subject: categories: Scanning mathematical text 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: 29 Steve Vickers raised the possibility of scanning mathematical text as a start toward getting it into TeX form. I thought I would report on my experience with getting Triples, Toposes and Theories onto the web. Some dozens of pages were missing from the original TeX computer file. I retyped some of them and I scanned some of them and then turned the special symbols into TeX notation when the OCR program said it didn't understand. Scanning works fine and is faster than retyping (for me) if the mathematical symbolism is not too dense. Most of the part I had to redo WAS too dense, and scanning those pages took me about the same length of time as retyping them. Michael Barr had retyped the diagrams so they didn't slow me down. If I typed maybe 25% faster than I do scanning would not be worth it except for things like the introduction and historical notes. If you do want to scan, get good software. I use Omnipage Pro 10.0 which you can get from http://www.caere.com/products/omnipage/pro/ for $500, but if you bought a scanner with a free version of Omnipage you can upgrade for $100. Do not try to use the free version, it is worth what it costs. Academic pricing might be lower. There are probably other good scanning programs out there but I don't know anything about them. Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA. email: charles@freude.com. home phone: 440 774 1926. professional website: http://www.cwru.edu/artsci/math/wells/home.html personal website: http://www.oberlin.net/~cwells/index.html NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm From rrosebru@mta.ca Fri Dec 15 11:39:44 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBFF8ZY07213 for categories-list; Fri, 15 Dec 2000 11:08:35 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 15 Dec 2000 14:54:47 GMT From: Paul Taylor Message-Id: <200012151454.OAA11901@koi-pc.dcs.qmw.ac.uk> To: categories@mta.ca Subject: categories: Re: Scanning mathematical text Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 30 As Charles Wells said (embedded in his message), scanning would not be worth it Even translating MicroSoft Word into LaTeX (which I helped other people to do with Steve Vickers' book "Reasonned Programming") is vastly more trouble than retyping in LaTeX directly. (Except perhaps for plain text, in which case you might as well have used Word to save it in plain text, or, better, not used Word in the first place.) In fact I have a recollection of making a resolution that (not only would I not do such a traslation but that) I would never again help someone with a LaTeX document that had resulted from such a translation. Honestly. When you're pulled all your hair out with the frustration of trying to do all the necessary incidental corrections, remember what I said. Paul From rrosebru@mta.ca Fri Dec 15 16:42:20 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBFK0jf15981 for categories-list; Fri, 15 Dec 2000 16:00:45 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: baez@newmath.UCR.EDU Message-Id: <200012151908.eBFJ8fP14853@math-cl-n06.ucr.edu> Subject: categories: (-1)-categories and (-2)-categories To: categories@mta.ca (categories) Date: Fri, 15 Dec 2000 11:08:41 -0800 (PST) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 31 Steve Vickers asked by email whether in constructive mathematics a (-1)-category should be allowed to be an arbitrary "subsingleton", i.e. a set with at most one element. I think that yes, these should be allowed - but maybe other things too! Indeed, a space has homotopy dimension -1 iff 1) given 2 points in the space there exists a path joining them, and 2) given 2 paths joining them, there exists a path of paths joining *them* and 3) given 2 paths of paths joining *them*, there exists a path of paths of paths joining THEM, and so on ad infinitum. As a nonconstructivist, I would say that either the space is empty in which case clause 1) is vacuous, or it's nonempty in which case we go on and read the resulting infinite list of clauses. But a constructive mathematician would have to proceed differently here, not being allowed to use the excluded middle. Do the remaining clauses provide any extra challenges for the constructivist? Then there's the bit where, having gotten a space that's either empty or contains one arc-component with vanishing homotopy groups, I conclude that if it's *nice* (e.g. a CW complex) it's homotopy equivalent to a space that's either empty or one point. I don't know how this reasoning (which uses the Whitehead theorem) gets affected by constructivism. Steve's idea sounds interesting, for this reason. If you plow through the detailed exchange between James Dolan and Toby Bartels, you'll see that (-1)-categories secretly represent TRUTH VALUES. In any approach to math where "truth values" are more interesting than merely 0 or 1, (-1)-categories will be correspondingly more interesting than merely sets with 0 or 1 elements. I guess this is familiar from topos theory. But I don't know if there's an extra twist due to all the "higher-dimensional" stuff going on in my reasoning above. Are truth values for an omega-categorical constructivist still more interesting than for an ordinary constructivist? The ordinary constructivist may not know whether two things are equal. The omega-categorical constructivist may not know whether all morphisms between two things are related by a 2-morphism, or whether all such 2-morphisms are related a 3-morphism, and so on.... From rrosebru@mta.ca Sat Dec 16 12:29:49 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBGFUKJ18798 for categories-list; Sat, 16 Dec 2000 11:30:20 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 15 Dec 2000 17:01:17 GMT From: Marta Z Kwiatkowska Message-Id: <200012151701.RAA01777@chip.cs.bham.ac.uk> To: categories@mta.ca Subject: categories: Job: Lectureships in Computer Science at Birmingham Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 32 [Apologies for multiple postings.] ----------------------------------------------------------------------- SCHOOL OF COMPUTER SCIENCE THE UNIVERSITY OF BIRMINGHAM England Three Lecturership Posts Applications are invited for three lectureship posts in Computer Science, starting as soon as possible (September 2001 latest). Postholders will be expected to contribute strongly to research, teaching and administration within the School. As an exception, consideration will be given for one of the posts to applicants whose strength is mainly in teaching in an area of special benefit to the School, for example systems analysis, software design, information systems, human-computer interaction or natural computation (taken to include evolutionary, neurally-inspired, and other nature-based computational styles). We intend one of the positions to be filled by someone able to contribute to teaching in our new EPSRC-funded MSc in Natural Computation. Applicants may be in any area of research in computational science. Generally, applicants should have, or soon expect to have, a PhD in computer science or an appropriate, closely related field, together with research experience as evidenced by publications in leading international journals or conference proceedings. However, teaching-orientated applicants may be acceptable if they have an appropriate, strong educational or industrial background and have a good university degree in an appropriate field. ---------------------------------------------------------------------- For further information please see http://www.cs.bham.ac.uk/research/jobs/lect01/ For the formal application procedure please see that webpage. CLOSING DATE: 31 January 2001. (Late applications may be considered.) Interviews are tentatively planned for late February / early March. It may be possible to negotiate part-time work in special cases. Additional research posts in Computer Science and Artificial Intelligence in this School can be found at: http://www.cs.bham.ac.uk/school/jobvacancies Informal enquiries may be made to Prof. John Barnden (Head of School) Tel: (+44) (0)121 414-3711 Email: J.A.Barnden@cs.bham.ac.uk From rrosebru@mta.ca Sun Dec 17 10:11:55 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBHDX0e24555 for categories-list; Sun, 17 Dec 2000 09:33:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Sat, 16 Dec 2000 17:21:18 -0800 (PST) From: Bill Rowan Message-Id: <200012170121.eBH1LIP10578@transbay.net> To: categories@mta.ca Subject: categories: Picard group of a ringoid Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 33 Tensor product gives a monoid structure on the class of isomorphism types of R,R-bimodules, for a ring or ringoid R. Restricting to those elements for which there is a two-sided inverse yields a group. I am inclined to call this the nonabelian Picard group and denote it by NPic(R). If we start with a commutative ring R, then the usual Picard group of R, Pic(R), can be viewed as an abelian subgroup of NPic(R). Has anyone seen this before? Does anyone have some other idea about what this should be called? Bill Rowan From rrosebru@mta.ca Sun Dec 17 10:18:43 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBHDVtK31682 for categories-list; Sun, 17 Dec 2000 09:31:55 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Sat, 16 Dec 2000 11:22:34 -0500 (EST) From: Peter Freyd Message-Id: <200012161622.eBGGMY222073@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: whoops Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 34 There was a little trap waiting for Mike and me to fall into. We were seeking conditions on a category C that force Ab[C] to be abelian and we came up with the same condition. I wrote, "Note that the conclusion (abelian) is self-dual but the condition (effective regular) is not." The trouble is: the definition of Ab[C] is not self-dual. I suppose my assertion is true as it stands but the implicit message is wrong. Mike had no such luck; he made it explicit: "[I]t is sufficient that either C or C^op be exact." (Yes, my "effective regular" and Mike's "exact" are equivalent -- it follows just from regularity that the pullback of a cover against itself is a pushout.) It took me a while to find a counterexample and I'm not happy with the one I found. Adjoin to the equational theory of abelian groups a new constant and no further axioms. Let C be the category of finite models. As is the case for the finite models of any equational theory, C is effective regular (and Ab[C] is isomorphic to the category of finite abelian groups). But Ab[C^op] is not abelian. It's empty. (C^op doesn't have a terminator.) From rrosebru@mta.ca Mon Dec 18 09:27:00 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBICXMY23191 for categories-list; Mon, 18 Dec 2000 08:33:22 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Sun, 17 Dec 2000 14:10:04 -0500 (EST) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: categories@mta.ca Subject: categories: Re: whoops In-Reply-To: <200012161622.eBGGMY222073@saul.cis.upenn.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 Sigh! Peter is, of course, correct. It actually occurred to me a couple days after I wrote my reply that Ab[C^op] is not the opposite of Ab[C] but I never looked into it. Michael On Sat, 16 Dec 2000, Peter Freyd wrote: > There was a little trap waiting for Mike and me to fall into. We were > seeking conditions on a category C that force Ab[C] to be abelian > and we came up with the same condition. I wrote, "Note that the > conclusion (abelian) is self-dual but the condition (effective > regular) is not." The trouble is: the definition of Ab[C] is not > self-dual. I suppose my assertion is true as it stands but the > implicit message is wrong. Mike had no such luck; he made it explicit: > "[I]t is sufficient that either C or C^op be exact." (Yes, my > "effective regular" and Mike's "exact" are equivalent -- it follows > just from regularity that the pullback of a cover against itself is > a pushout.) > > It took me a while to find a counterexample and I'm not happy with the > one I found. Adjoin to the equational theory of abelian groups a new > constant and no further axioms. Let C be the category of finite > models. As is the case for the finite models of any equational theory, > C is effective regular (and Ab[C] is isomorphic to the category of > finite abelian groups). But Ab[C^op] is not abelian. It's empty. > (C^op doesn't have a terminator.) > > From rrosebru@mta.ca Mon Dec 18 10:45:08 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBICYEP15150 for categories-list; Mon, 18 Dec 2000 08:34:14 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 18 Dec 2000 13:41:51 +1100 (EST) From: maxk@maths.usyd.edu.au (Max Kelly) Message-Id: <200012180241.NAA24289@milan.maths.usyd.edu.au> To: categories@mta.ca Subject: categories: reprinting my book Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 36 Allow me to express my thanks to the numerous colleagues who have written during the last couple of days, urging me to re-issue the book and suggesting ways and means. I have replied in person to a few who have made specific offers to participate in the process; but I am grateful to all, both for their useful suggestions and for their encouragement. I shall give more thought to the possibilities early in the new year. Max Kelly. From rrosebru@mta.ca Mon Dec 18 17:24:19 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBIKcsF06455 for categories-list; Mon, 18 Dec 2000 16:38:54 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Mon, 18 Dec 2000 08:33:46 -0500 (EST) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: categories@mta.ca Subject: categories: Re: Picard group of a ringoid In-Reply-To: <200012170121.eBH1LIP10578@transbay.net> 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: 37 I have never seen a name for this. I think, if it hasn't been defined before, I would be inclined to call it the Morita group. There is a large groupoid, let me call it the Morita groupoid, whose objects are rings and for which a morphism R --> S is a left S, right R bimodule M such that tensoring with M gives an equivalence between the category of left R modules and left S modules. This is locally small since M must be a finitely generated projective left S module and the group you are dealing with is simply the group of endomorphisms of R in that groupoid. the whole theory is due to Morita (and the main theorem, the Morita theorem). This is for rings, of course. I assume that a ringoid is a small preadditive category. A preadditive category with finitely many objects is Morita equivalent to a ring so it will be true for them. Beyond that, it would have to be examined because I am not sure what corresponds to finitely generated. On Sat, 16 Dec 2000, Bill Rowan wrote: > Tensor product gives a monoid structure on the class of isomorphism types > of R,R-bimodules, for a ring or ringoid R. Restricting to those elements > for which there is a two-sided inverse yields a group. I am inclined to call > this the nonabelian Picard group and denote it by NPic(R). If we start with > a commutative ring R, then the usual Picard group of R, Pic(R), can be viewed > as an abelian subgroup of NPic(R). > > Has anyone seen this before? Does anyone have some other idea about what this > should be called? > > Bill Rowan > From rrosebru@mta.ca Tue Dec 19 13:39:02 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBJGl5M13328 for categories-list; Tue, 19 Dec 2000 12:47:05 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: (-2)-categories To: categories@mta.ca Date: Tue, 19 Dec 2000 11:24:17 -0400 (AST) X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20001219152417.49F8A6668D@chase.mathstat.dal.ca> From: pare@mscs.cs.Dal.CA (Robert Pare) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 38 Here are some thoughts I had on (-2)-categories. When I read John Baez's posting I thought I had missed the boat, but I guess the point is that strict n-categories are not the same as weak n-categories (surprise!). A *strong* n-category (henceforth called n-category) is a category enriched in (n-1)-Cat, the cartesian closed *category* of small (n-1)-categories. In the time honoured way, start with the empty category. It's cartesian closed (it doesn't have a terminal object but let's glance over that). A category enriched in it can have no objects, for where would their homs land. So there is only one, the empty category itself. So the category of small categories enriched in the empty category is 1, the category with one morphism, which is cartesian closed of course. A category enriched in 1 is just a class of objects, and a small one is just a set. So now the category of small categories is just Set, and we are on our way. So what's the point? A set is a 0-category, there is only one (-1)-category, the empty category, and there are no (-2)-categories. There *is* a notion of (-2)-category, viz. an object of the empty category, but there aren't many of those! I think it should be clear from this that there is not even a notion of (-3)-category so it doesn't even make sense to say there are none. Bob Pare From rrosebru@mta.ca Wed Dec 20 15:33:49 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBKIPLe02263 for categories-list; Wed, 20 Dec 2000 14:25:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: coltrane.dimi.uniud.it: miculan owned process doing -bs Date: Tue, 19 Dec 2000 19:25:06 +0100 (CET) From: Marino Miculan To: Subject: categories: FoSSaCS 2001 Accepted Papers Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1 X-MIME-Autoconverted: from 8bit to quoted-printable by ten.dimi.uniud.it id TAA04439 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id eBJIOxt24427 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 39 Please post. Regards, - Marino ----------------------------------------------------------------- FoSSaCS 2001 Accepted Papers Please find enclosed the list of papers accepted at FoSSaCS 2001. The list is available also on the FoSSaCS web site at http://fossacs.dimi.uniud.it/accepted.html Furio Honsell FoSSaCS'01 Chair ----- 1.Verified Bytecode Verifiers Tobias Nipkow 2.Axioms for Recursion in Call-by-Value Masahito Hasegawa, Yoshihiko Kakutani 3.Decidability of weak bisimilarity for a subset of basic parallel processes Colin Stirling 4.On the complexity of parity word automata Valerie King, Orna Kupferman, Moshe Vardi 5.Secrecy Types for Asymmetric Communication Martin Abadi, Bruno Blanchet 6.Higher-Order Abstract Syntax with Induction in Isabelle/HOL: Formalizing the Pi-Calculus and Mechanizing the Theory of Contexts Christine Röckl, Daniel Hirschkoff, Stefan Berghofer 7.An axiomatic semantics for the synchronous language Gentzen Simone Tini 8.Temporary Data in Shared Dataspace Coordination Languages Nadia Busi, Roberto Gorrieri, Gianluigi Zavattaro 9.Type Isomorphisms and Proof Reuse in Dependent Type Theory Gilles Barthe, Olivier Pons 10.Coalgebra of Abstract Processes Sava Krstic, John Launchbury, Dusko Pavlovic 11.On the Modularity of Deciding Call-by-Need Irène Durand, Aart Middeldorp 12.On Garbage and Program Logic Peter W. O'Hearn, Cristiano Calcagno 13.Axiomatizing tropical semirings Luca Aceto, Zoltan Esik, Anna Ingolfsdottir 14.On the Duality between Observability and Reachability Michel Bidoit, Rolf Hennicker, Alexander Kurz 15.The Complexity of Model Checking Mobile Ambients Witold Charatonik, Silvano Dal Zilio, Andrew D. Gordon, Supratik Mukhopadhyay, Jean-Marc Talbot 16.Class Analysis of Object-Oriented Programs through Abstract Interpretation Thomas Jensen, Fausto Spoto 17.Foundations for a graph-based approach to the specification of Access Control Policies Manuel Koch, Luigi Vincenzo Mancini, Francesco Parisi-Presicce 18.High-Level Petri Nets as Type Theories in the Join Calculus Maria Grazia Buscemi, Vladimiro Sassone 19.On Rational Message Sequence Chart Languages and Relationships to Mazurkiewicz Trace Theory Remi Morin 20.On the Decidability of the Finite Model Problem Mikolaj Bojanczyk 21.The Rho Cube Horatiu Cirstea, Claude Kirchner, Luigi Liquori 22.Computational Completeness of Programming Languages Based on Graph Transformation Detlef Plump 23.Synchronized Tree Languages Revisited and New Applications Valerie Gouranton, Helmut Seidl, Pierre Rety 24.Type inference with recursive type equations Mario Coppo 25.Model checking CTL+ and FCTL is hard Francois Laroussinie, Nicolas Markey, Philippe Schnoebelen ----------------------------------------------------------------- From rrosebru@mta.ca Thu Dec 21 16:43:04 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBLJo0817687 for categories-list; Thu, 21 Dec 2000 15:50:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3A4249FF.76B23EBD@comlab.ox.ac.uk> Date: Thu, 21 Dec 2000 18:20:47 +0000 From: Samson Abramsky Organization: Oxford University Computing Laboratory, UK X-Mailer: Mozilla 4.75 [en] (X11; U; SunOS 5.7 sun4u) X-Accept-Language: en MIME-Version: 1.0 To: types@cis.upenn.edu, categories@mta.ca, linear-request@cs.stanford.edu, cav-all@csa.cs.technion.ac.il Subject: categories: TLCA 2001 Accepted Papers 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: 40 The list of accepted papers for TLCA 2001 is now available at http://www.ii.uj.edu.pl/zpi/tlca2001/acc_papers.html Samson Abramsky From rrosebru@mta.ca Sun Dec 24 12:59:27 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBOGHil20512 for categories-list; Sun, 24 Dec 2000 12:17:44 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3A43955F.463926CE@comlab.ox.ac.uk> Date: Fri, 22 Dec 2000 17:54:39 +0000 From: David Nowak Organization: Oxford University Computing Laboratory, UK X-Mailer: Mozilla 4.72 [en] (X11; I; SunOS 5.7 i86pc) X-Accept-Language: fr, en MIME-Version: 1.0 Subject: categories: AVoCS'01 announcement 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: 41 ANNOUNCEMENT (Apologies if you receive multiple copies) AVoCS'01 Workshop on Automated Verification of Critical Systems Oxford, April 19-20, 2001 http://web.comlab.ox.ac.uk/oucl/conferences/wavcs2001/ This workshop continues the annual DERA/OUCL workshop series. It will combine invited presentations with accepted submissions. Motivation The aim of this workshop is to foster a research community in verification in United Kingdom through encouraging communication among researchers. Specific objectives include concrete efforts at integration as well as the transfer of methods between different groups. This year, a special session featuring security will be held. Organising Committee Marta Kwiatkowska, University of Birmingham Michael Leuschel, University of Southampton David Nowak (contact person), Oxford University Joy Reed, Oxford Brookes University Mike Reed, Oxford University Bill Roscoe, Oxford University Ulrich Ultes-Nitsche, University of Southampton Irfan Zakiuddin, DERA Contact Person David Nowak Oxford University Computing Laboratory Wolfson Building Parks Road Oxford, OX1 3QD United Kingdom David.Nowak@comlab.ox.ac.uk Sponsors Defence Evaluation and Research Agency (DERA) and Office of Naval Research (ONR) Topics Topics include but are not limited to: * Abstraction methods (symmetry, ...) * Applications to security * Applications to hardware * Case studies * Connections with abstract interpretation * Data independence * Integrating theorem proving and model checking * Model checking * Model checking of infinite systems * Model checking of real-time systems * Performance and dependability evaluation * Refinement checking * Software verification * Temporal and modal logic * Theorem Proving We also encourage submissions which integrate or compare these different approaches. Submissions Authors are requested to submit an abstract to the contact person (cf. above). If possible, submissions should be made electronically in postscript (or PDF format). Additional details on submission guidelines will be posted at the workshop home page. Abstracts relevant to the scope of this workshop will be selected for presentation. Submissions of work in progress talks are encouraged. Contributions will not be peer-reviewed and can thus still be submitted for publication in other fora. Proceedings Informal proceedings will be available at the workshop as a OUCL technical report. Participants are thus invited to submit working notes or abstracts for inclusion in proceedings. After the workshop, proceedings upon invitation will be published. Registration Advance enrolment is required. To enrol please send your name, affiliation, e-mail, address, phone number and fax number to the contact person (cf. above). There will be no registration fee for the workshop. Note that attendance could be restricted or limited, therefore travel plans should not be made until confirmation of your registration has been received. From rrosebru@mta.ca Sun Dec 24 12:59:27 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBOGGcs20792 for categories-list; Sun, 24 Dec 2000 12:16:38 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: X-Mailer: Novell GroupWise Internet Agent 5.5.3.1 Date: Fri, 22 Dec 2000 17:23:05 -0500 From: "Ajay Gupta" Subject: categories: job: faculty position at Western MIchigan University Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-7 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id eBMMOcg26430 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 Our sincere apologies, if by mistake you receive it multiple times. ¯--------------------------------------------------------------------------- WESTERN MICHIGAN UNIVERSITY The Department of Computer Science at Western Michigan University (www.cs.wmich.edu) seeks applications for a tenure-track assistant professor position in computer science beginning August 2001. A Ph.D. in computer science or a closely related field is r equired. The development of externally funded research and participation in the department's doctoral program, as well as teaching at all levels, are expected. Applications from persons with research interests in all areas of computer science will be cons idered. Candidates with research interests in networks and systems are especially encouraged to apply. Western Michigan University, a student-centered research university, encourages applications from underrepresented groups. Send letter of application, v ita, transcripts, statements of research and teaching plans and three letters of reference to: Ajay Gupta, Chair, Department of Computer Science, Western Michigan University, 1903 W. Michigan Avenue., Kalamazoo, MI 49008-5371. Fax: (616) 387-3999; Email: ajay.gupta@wmich.edu. Review of applications will begin November 1, 2000, and applications will be accepted until the position is filled. From rrosebru@mta.ca Thu Dec 28 20:20:43 2000 -0400 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBSNUD414039 for categories-list; Thu, 28 Dec 2000 19:30:13 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Mime-Version: 1.0 X-Sender: ugo@mailserver.di.unipi.it Message-Id: Date: Tue, 26 Dec 2000 10:11:48 +0100 To: categories@mta.ca From: Ugo Montanari Subject: categories: CfP CMCS2001 - Extended deadline Content-Type: text/plain; charset="us-ascii" ; format="flowed" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 43 DEADLINE EXTENDED to Jan 8. ============================================================ C A L L F O R P A P E R S 4th International Workshop on Coalgebraic Methods in Computer Science (CMCS2001) Genova, Italy 6-7 April 2001 A satellite workshop of ETAPS 2001 ------------------------------------------------------------ Aims and Scope During the last few years, it is becoming increasingly clear that a great variety of state-based dynamical systems, like transition systems, automata, process calculi and class-based systems can be captured uniformly as coalgebras. Coalgebra is developing into a field of its own interest presenting a deep mathematical foundation, a growing field of applications and interactions with various other fields such as reactive and interactive system theory, object oriented and concurrent programming, formal system specification, modal logic, dynamical systems, control systems, category theory, algebra, analysis, etc. The aim of the workshop is to bring together researchers with a common interest in the theory of coalgebras and its applications. The topics of the workshop include, but are not limited to: * the theory of coalgebras (including set theoretic and categorical approaches); * coalgebras as computational and semantical models (for programming languages, dynamical systems, etc.); * coalgebras in (functional, object-oriented, concurrent) programming; * coalgebras and data types; * (coinductive) definition and proof principles for coalgebras (with bisimulations or invariants); * coalgebras and algebras; * coalgebraic specification and verification; * coalgebras and (modal) logic; * coalgebra and control theory (notably of discrete event and hybrid systems). The workshop will provide an opportunity to present recent and ongoing work, to meet colleagues, and to discuss new ideas and future trends. Previous workshops of the same series have been organized in Lisbon, Amsterdam and Berlin. The proceedings appeared as ENTCS Vols. 11,19 and 33. ------------------------------------------------------------ Location CMCS2001 will be held in Genova on 6-7 April 2001, just after ETAPS2001 (European Joint Conferences on Theory and Practice of Software). ------------------------------------------------------------ Program Committee Alexandru Baltag (Amsterdam), Andrea Corradini (Pisa), Bart Jacobs (Nijmegen), Marina Lenisa (Udine), Ugo Montanari (chair, Pisa), Larry Moss (Bloomington, IN), Ataru T. Nakagawa (Tokyo), Dusko Pavlovic (Palo Alto), John Power (Edinburgh), Horst Reichel (Dresden), Jan Rutten (Amsterdam). ------------------------------------------------------------ Submissions Submissions will be evaluated by the Program Committee for inclusion in the proceedings, which will be published in the ENTCS series. Papers must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. Papers (of at most 15 pages) should be submitted electronically as uuencoded PostScript files at the address cmcs2001@di.unipi.it. A separate message should also be sent, with a text-only one-page abstract and with mailing addresses (both postal and electronic), telephone number and fax number of the corresponding author. ------------------------------------------------------------ Important Dates EXTENDED, HARD DEADLINE for submission: 8 January 2001, midnight MET. Notification of acceptance: 20 February 2001. Final version due: 10 March 2001. Workshop dates: 6-7 April 2001. ------------------------------------------------------------ Organizers Andrea Corradini (Pisa), Marina Lenisa (Udine), Ugo Montanari (Pisa). ------------------------------------------------------------ For more information: http://www.di.unipi.it/~ugo/CMCS2001.html mailto:cmcs2001@di.unipi.it ============================================================ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ Prof. Ugo Montanari Phone: +39 050 887221 Dipartimento di Informatica Fax: +39 050 887226 Universita' di Pisa Note the zero prefix in the area code. Corso Italia, 40 Email: ugo@di.unipi.it I-56100 Pisa, Italy http://www.di.unipi.it/~ugo/ugo.html @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@