From MAILER-DAEMON Fri Nov 7 15:32:18 2003 Date: 07 Nov 2003 15:32:18 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1068233538@mta.ca> X-IMAP: 1065141141 0000000042 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Thu Oct 2 11:20:12 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:20:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54HU-0001Iy-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:18:12 -0300 From: Jpdonaly@aol.com Message-ID: <2d.348a2564.2cabc133@aol.com> Date: Wed, 1 Oct 2003 01:33:39 EDT Subject: categories: Re: Categories of elements To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 1 Dear Professor Lawvere, Thanks for your clarifications and views in response to my latest note. Coming from an applications-oriented environment, I do assume a set of Zermelo-Fraenkel axioms with a universe of small sets (as prescribed in CWM) in order to ensure access to a fully viable arithmetic of natural transformations. This seems to allow for more than enough categories for my purposes, but it certainly does give the category of small functions a prominence which can feel artificially restrictive at times. Thus I would be especially attentive to any comments which you might make specifically on the functorial isomorphism (I presume to call it a "Lawvere isomorphism" ) which, in converting the Yoneda picture (function-valued natural transformations) of categorical duality into the Lawvere picture (cocompatible functors), represses the category of small functions and, as I do realize, moves things into the context of the general existence theory of adjunctions and Kan extensions, possibly providing a functorial interpretation of your explanation of the origin of comma categories. By now this isomorphism seems to me to be more of a perspicuous relabelling than a redefiner of concepts, so that I have to plead innocent to your apparent conviction that I agonize over the definition of elements. I am in full accord with the doctrine of elements as you have described it, and the Lawvere isomorphism actually relieves some conceptual agony in this regard by smoothly ensuring that, to within a label, the elements of a function-valued functor constitute a (limit) object which is in the functor's codomain category. But I have to restate my belief that the otherwise perfectly redeemable sentence, "An element of a functor is an attaching functor into the category of elements of the functor," is unacceptably confusing due to the fact that the category of elements of a functor does not in any sense consist of the elements of the functor (as you would describe them). So I would rename it. Pat Donaly From rrosebru@mta.ca Thu Oct 2 11:20:12 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:20:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54FG-000176-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:15:54 -0300 Date: Tue, 30 Sep 2003 14:50:47 -0400 (EDT) From: F W Lawvere Reply-To: wlawvere@acsu.buffalo.edu To: categories@mta.ca Subject: categories: Re: Categories of elements (Pat Donaly) In-Reply-To: <191.1f8940ae.2ca4976e@aol.com> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 The motivation for introducing 40 years ago the construction, of which "categories of elements" is a special case, was to make clear the elementary nature of the notion of adjointness. Given an opposed pair of functors between two arbitrary given categories, one obviously elementary way of providing them with an adjointness is to give two natural transformations satisfying two equations; but very useful also is the definition in terms of bijections of hom sets which should be equivalent. The frequent mode for expressing the latter in terms of presheaf categories involved the complicated logical notion of "smallness" and the additional axiom that a category of small sets actually exists, but had the disadvantage that it would therefore not apply to arbitrarily given categories. By contrast, a formulation of this bijection in terms of discrete fibrations required no such additional apparatus and was universally applicable. Unfortunately, since I had given the construction no name, people in reading it began to use the unfortunate term "comma". It would indeed be desirable to have a more objective name for such a basic construction. (The notation involving the comma was generalized from the very special case when the two functors to B, to which the construction is applied, both have the category 1 as domain, and the result of the construction is the simple hom set in B between the two objects, which is often denoted by placing a comma between the names of the objects and enclosing the whole in parentheses.) One habit which it would be useful to drop is that of agonizing over the true definition of elements. In any category the elements of an object B are the maps with codomain B, these elements having various forms which are their domains. For example, if the category has a terminal object, we have in particular the elements often called punctiform. On the other hand, it is often appropriate to apply the term point to elements more general than that, for example, in algebraic geometry over a non-algebraically closed field, points are the elements whose forms are the spectra of extensions of the ground field. As Volterra remarked already in the 1880s, the elements of a space are not only points, but also curves, etc.; it is often convenient to use the term "figure" for elements whose forms belong to a given subcategory. Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Thu, 25 Sep 2003 Jpdonaly@aol.com wrote: > To all category theorists: > > In various textbooks, I see reference to the common comma category Elts(G), > which is called the "category of elements of functor G". This category seems to > be drastically misnamed. Does anyone agree? Here is my side of the story, > beginning with a review of the nature of Elts(G) and some of its significance. > > G is a functor from a small category C into the category F of small > functions. Denoting the singleton {0} of the void set 0 by 1 (as usual), Elts(G) > consists of all triples (g,a,f) with a in the domain category C and > g:1--->G(codomain a), f:1--->G(domain a) such that g = G(a) o f, where "o" denotes function > composition. (Warning: By my conventions, a.domain a = a; codomain a.a = a.) The > composition of Elts(G) is defined by (h,b,g)(g,a,f) = (h,ba,f). The objects > are the Elts(G)-morphisms of the form (f,u,f), where u is a C-object, and the > map (f,u,f)-->f(0) identifies each of these with an element of the set G(u), so > that the convention of naming categories after their objects (to the extent > possible) is what presumably leads to calling Elts(G) "the category of elements > of the values of G at objects" or, for short, "the category of elements of G". > > Several basic features of Elts(G) are exposed by treating it as a subcategory > of a product BxC of a transition category B with the domain category C of G. > To define B, let X be the set of functions f:1--->G(u) as u varies over the > objects of C; B is then the full transition category or groupoid of X, that is, > the self-product XxX with the transition composition (h,g)(g,f) = (h,f). But > rather than taking the ordinary cartesian product for BxC, one uses the > attachment product consisting of those triples (g,a,f) with (g,f) in B and a in C, so > that C-morphism a is viewed as being attached on its left to g and on its > right to f. Then Elts(G) inherits by restriction the projection functor > (g,a,f)-->a, which is reasonably called the detaching functor from Elts(G) into > C---this functor will be generically denoted by "det". There is also the transition > projection (g,a,f)-->(g,f) which maps Elts(G) functorially onto a transitive > relation on X, and the rule (g,a,f)-->g defines the entwining function of the > canonical natural transformation which entwines the constant functor > (g,a,f)-->1 on Elts(G) with the function composite functor G o det: Elts(G)--->F. (A > constant functor with value 1 will be denoted generically by "delta(1)".) > > There are many important examples: If C is a group with object e so that G is > group action with action set Y=G(e), then, to within the identification > (g,a,f)-->(g(0),a,f(0)), Elts(G) is the traditional idea of a G-action as a > function from CxY into Y after correction to remove the categorically problematic > product CxY. If C is actually the group RxR, R being the additive group of real > numbers and G the action of C on the real affine plane by translation, then > Elts(G) is essentially the category of attached planar vectors as used in > Engineering Statics 101, which is why it seems appropriate to continue to use the > word "attach" in the context of a more general function-valued functor G:C--->F. > If C is a certain type of monoid, then Elts(G) is a semiautomaton. Among its > theoretical services is the fact that Elts(G) plays a role in the construction > of Kan extensions along inclusion functors, thus in particular in the theory > of induced group actions. It plays an analogous part in sheafification relative > to a Grothendieck site, and it is used to show that representable functors > are dense in the set of function-valued functors on C, providing, according to > Mac Lane and Moerdijk, "a plethora of tensor products". Even more basically, if > the domain C of G is discrete, then Elts(G) is a coproduct a.k.a. a disjoint > union of (the object values) of G. As will be noticed in a moment, the set of > functors A:C--->Elts(G) which are right inverse to the detaching functor det > on Elts(G)---that is, the "attaching functors" into Elts(G)---constitute a > (small) limit object of G, thus, in the case of discrete C, a product of the sets > G(u). From these examples it appears that Elts(G) is sufficiently important to > require a unique and unambiguous nomenclature, but, unfortunately, the things > in Elts(G) are really not the elements of G. > > The (global) elements of an object u in a category are generally agreed to be > the morphisms from a given terminal object t to u. This convention > terminologically extends the observation that the functions f from the terminal object 1 > into a (small) set X can be identified with the elements of X by the mapping > f-->f(0). The general definition has the virtue that each terminal object has > only one element, as should surely be the case, and the representable functor > of t provides a plausible (but not necessarily effective) attempt to convert a > given category into a category of functions between sets of elements. In > fact, this language seems to have found broad acceptance. But then the functor G > is an object in the morphismwise (i.e. "vertical") composition category F^C of > natural transformations whose (fully extended) entwining functions map from C > into F, and, because 1 is terminal in F, the constant functor delta(1) on C is > terminal in F^C. So G already has a set of elements, namely, those natural > transformations which entwine delta(1) with G. Such elements of G are not in > Elts(G) in any sense. > > At first sight this terminological conflict might seem to be innocuous, since > Elts(G) and global elements of G occur in somewhat disparate contexts, but > the apparent separation does not hold up well when one considers how close the > set of global elements of G is to being a limit object of G. The only problem > with it is that it is not small; that is, it is not in the codomain category F > of G, and the only reason for this defect is that F, the common codomain of > the entwining functions of the things in F^C, is not small. Mac Lane in CWM > gives an ad hoc workaround which replaces, for a given G, the category F with a > small, G-dependent category of small functions, but this approach effectively > isolates G by artificially depriving it of morphisms into functors which do not > happen to map into Mac Lane's ad hoc replacement category; so one needs a more > perspicuous method of eliminating F and its untoward largeness. > > F. W. Lawvere was apparently motivated by such considerations to introduce > comma categories in his thesis, an approach which works very well in addressing > the present awkwardnesses. One defines a category Law(C) whose objects are the > categories Elts(G) as G ranges through the functors in F^C and whose > morphisms are the cocompatible functors S:Elts(G)--->Elts(H) between such objects. The > composition is function composition of functors, and "cocompatible" means > that S does not disturb middle components of attached C-morphisms or, > alternatively put, det o S = det, where "det" continues to be the generic symbol for a > detaching functor. Then, if s in F^C entwines functor G with functor H, there is > a cocompatible functor S:Elts(G)--->Elts(H) which is evaluated at an attached > C-morphism (y*,a,x*) by > > S(y*,a,x*)=(s(codomain a)(y)*, a, s(domain a)(x)*), > > where I use y*, for example, to denote that function f:1--->G(codomain a) > whose value is y. Then the assignments s-->S define a functorial > isomorphism---which I call the Lawvere isomorphism (but should this be attributed to someone > else?) from F^C onto Law(C). Moreover, the objects Elts(G) of Law(C) are small. > This implies, of course, that the homset of cocompatible functors from > Elts(G) into Elts(H) is also small. > > Elts(delta(1)) evidently consists of triples of the form (0*,a,0*) and can > thus be identified with C by the detaching functor (0*,a,0*)-->a on > Elts(delta(1)). With this identification, a cocompatible functor from Elts(delta(1)) into > Elts(G) becomes an attaching functor into Elts(G), so that, in the Lawvere > picture, the global elements of G are the attaching functors into the category > of...uh...elements of G. The set of such global elements is plainly small and > therefore must be what God intends to be the standard limit object of G, except > that it is difficult to believe that God would use such a verbal collision to > say what a global element is. These are my grounds for believing that Elts(G) > has to be renamed and redenoted. > > As a related suggestion, I might recommend dropping the habit of referring to > categories by the names of their objects. This illogicality immediately > inhibits use of the subcategory concept (I still don't know what categorists use to > refer to the subcategory formed by the monomorphisms in an abstractly given > category), and then it just goes looking for the sort of trouble which has > turned up as "the category of elements of G". Besides this, the terminology "comma > category" is disrespectful of category theory, itself, due to the > inappropriateness of naming a fundamental, overarching categorical concept after a > punctuation mark. ("Slice category" doesn't seem to be any better.) Given the > precedent of attached vectors, which are used in a rough sense even by sophisticated > diagrammaticists, the category Elts(G) is obviously some kind of attachment > category, and since the transition components of any of its morphisms all have > domain 1, it is a based attachment category with base 1 or just a basement > category---or even just a basement denoted by something like G/1, if you're used > to placing domains on the right. Anyway, this is approximately what I use in > my study notes, and so far it works fine. > > At the same time, I would be interested in seeing sharp, well reasoned > criticisms of this note provided that they are written at about the same technical > level so that I can understand them. I would like to emphasize that, aside from > what may be terminologically or notationally novel here, I am not making a > substantial research proposal or claiming priority for any discoveries. I have > no reason at all to doubt that all of the mathematics here is well known. > > Pat Donaly > > > > > > From rrosebru@mta.ca Thu Oct 2 11:22:05 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:22:05 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54JY-0001Po-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:20:20 -0300 Date: Wed, 1 Oct 2003 16:51:13 +0100 From: Paul Taylor Message-Id: <200310011551.h91FpDR01952@primrose.cs.man.ac.uk> To: categories@mta.ca Subject: categories: new job & address for me too X-Spam-Score: -4.9 (----) X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *1A4jFv-0004KX-W9*5ZoX00up9kg* Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 3 As Tom Leinster has told you about his new job, maybe I should do the same... A project to study "ABSTRACT STONE DUALITY" has been funded by the EPSRC (the UK funding agency for the exact sciences), as a result of which I have a job for three years in the Mathematical Foundations Group, Department of Computer Science, University of MANCHESTER and so have new email and web addresses pt@cs.man.ac.uk www.cs.man.ac.uk/~pt I am very much looking forward to working with my new colleagues there, including Peter Aczel, David Rydeheard, Andrea Schalk and Harold Simmons. Please note that, whilst I shall be making regular visits to Manchester, for the time being I shall continue to live and work in LONDON (four hours away by train). If, therefore, you want to send me anything by post, please see my web page or ask me by email for my home address. My email is forwarded from pt@dcs.qmw.ac.uk, pt@dcs.qmul.ac.uk and pt@di.unito.it, so please do not send it to multiple addresses. -------------------------------- Although funding for this project was in fact confirmed two months ago, I delayed making this announcement because I wanted to say at the same time that all of my WEB PAGES have been thoroughly revised, (though, needless to say, they haven't yet). Nevertheless, amongst my PAPERS linked from www.cs.man.ac.uk/~pt are - the ABSTRACT STONE DUALITY papers and proposal, - the HTML version of my book, "PRACTICAL FOUNDATIONS", - my older papers on CONTINUOUS, STABLE & SYNTHETIC DOMAIN THEORY - also on INTUITIONISTIC SETS AND ORDINALS and CATEGORICAL RECURSION PLUS *** NEW!!! *** - my teaching materials for the first year computer science course INTRODUCTION TO ALGORITHMS that I taught at QMW. - the complete text of Jean-Yves Girard's book PROOFS AND TYPES, as this is now out of print. - a translation of Gauss's second proof (1815) of the "fundamental theorem of algebra" (every polynomial has a complex root) Then of course there are (La)TeX MACROS (still in need of new web pages) - my famous DIAGRAMS package (which now supports PDFTEX) - PROOF TREES and BOXES - QED macros (with new documentation) - supermarket bar codes (OK, nothing to do with category theory) The papers are available in the usual variety of formats, whilst their web pages have been designed to allow navigation entirely in DVI or PDF format, using XDVI, XPDF or ACROREAD. This of course is prone to bugs, so please tell me if any of the links go astray. If there are any other (mathematical or programming) materials of mine that you still consider useful, but which haven't been included or updated in the new web pages, please ask. -------------------------------- Amongst 48 proposals that they considered in that quarter, the EPSRC Computer Science panel ranked this one equally amongst the top three, on the basis of four outstanding referees' reports. My two nominated referees were GRAHAM WHITE and RICHARD WOOD, to whom I would like to express my appreciation for their support. Graham had in fact also helped me to write the proposal in the first place. EPSRC rules did not allow me to submit this proposal in my own name, as I did not have a teaching (or indeed any) job. (In fact, they don't even allow people to buy themselves out of their teaching, just to employ other people to do the research for them, which makes some sense in experimental sciences, but not really in mathematics.) The success of this proposal follows a period of 20 months during which I continued to work on this research programme and attend conferences funded from my own savings --- and, I would like to make clear, NOT from social security. My attendance at SOME of the meetings was paid for by Birmingham, Dalhousie, St. Mary's and Turin Universities and the EU APPSEM project. I would not, however, have had the emotional energy to do this without the constant support of my partner, RICHARD SYMES, who let me pursue Plan A, even though (being a modern languages graduate) he had no clue what it was about, or - until he saw the referees' reports - whether it had any prospects of success. I would also like to thank MARIANGIOLA DEZANI of the Universita` degli Studi di Torino for her offer of funding, even though it turned out that certain Italian bureaucracy didn't allow it. Her support, and her personal hospitality when I visited Turin, meant a lot to me. -------------------------------- Finally, I feel that having paid for myself to attend conferences gives me the prerogative to make a comment on the way that they are organised: I deplore the practice of charging those like me (and numerous others, some of them important members of our community) who have had to pay for their own research, to subsidise "senior" people as guest speakers, with longer time-slots at conferences. These people have research grants and professorial salaries from rich universities. I have noticed, however, that it is the guest speakers who most commonly over-run their allotted time, and sometimes present material that would not have been accepted from others by the referees. Paul Taylor From rrosebru@mta.ca Thu Oct 2 11:22:20 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:22:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54K3-0001Sp-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:20:52 -0300 Message-Id: <200310011738.h91Hcq022554@math-ws-n09.ucr.edu> Subject: categories: Euler characteristic versus homotopy cardinality To: categories@mta.ca (categories) Date: Wed, 1 Oct 2003 10:38:52 -0700 (PDT) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 4 Dear Categorists - Some of you might be interested in this talk, since it's secretly about attempts to categorify the rational numbers. http://www.math.ucr.edu/home/baez/cardinality/ Euler Characteristic versus Homotopy Cardinality Abstract: Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups. The two quantities have many of the same properties, but it's hard to tell if they're the same, since like Jekyll and Hyde, they're almost never seen together: there are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series - and the two then agree! We give examples of this phenomenon and beg the audience to find some unifying concept which has both Euler characteristic and homotopy cardinality as special cases. From rrosebru@mta.ca Thu Oct 2 11:22:45 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:22:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54Ki-0001YM-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:21:33 -0300 Message-ID: <3F7C2049.8C1896A3@email.unc.edu> Date: Thu, 02 Oct 2003 08:55:37 -0400 From: jim stasheff X-Mailer: Mozilla 4.8 [en] (Win98; U) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: query Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 5 Apparently Graeme's approach to infinite sloop spaces or rather to one fold loop spaces has be further abstracted to produce what is known as a `Segal category' at a quick glance, it seems to me these are related to Fukaya's A_\infty cats as my approach to \Omega X is related to Graeme's anyone seen this worked out or even commented on? jim From rrosebru@mta.ca Thu Oct 2 21:37:24 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 21:37:24 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A5DtL-0001TV-00 for categories-list@mta.ca; Thu, 02 Oct 2003 21:33:55 -0300 Date: Thu, 2 Oct 2003 09:42:54 -0500 (CDT) From: Peter May Message-Id: <200310021442.h92EgsNI002556@tachyon.uchicago.edu> To: cat-dist@mta.ca Subject: categories: Stasheff's question Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 6 The comparison between Segal categories and A infinity categories works in close analogy with the comparison between A infinity spaces (for any A infinity operad) and Segal's special Delta spaces. The topological comparison was worked out in papers of Thomason and Fiedorowicz, themselves analogues of earlier work of Thomason and myself comparing infinite loop space machines. The categorical comparison will appear in a paper I'm writing --- I've talked about it at MSRI and the Newton Institute. Peter May From rrosebru@mta.ca Fri Oct 3 15:33:33 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Oct 2003 15:33:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A5UgP-00019P-00 for categories-list@mta.ca; Fri, 03 Oct 2003 15:29:41 -0300 Date: Fri, 3 Oct 2003 14:16:52 -0300 (ADT) From: jim stasheff To: categories Subject: categories: Re: Stasheff's question Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 7 Peter May wrote: > > The comparison between Segal categories and A infinity categories > works in close analogy with the comparison between A infinity spaces > (for any A infinity operad) and Segal's special Delta spaces. The > topological comparison was worked out in papers of Thomason and > Fiedorowicz, themselves analogues of earlier work of Thomason and > myself comparing infinite loop space machines. The categorical > comparison will appear in a paper I'm writing --- I've talked about > it at MSRI and the Newton Institute. Peter May Yes, that's what I had in mind question is: does it carry over to (Fukaya) A_\infty cats and Segal cats? jim From rrosebru@mta.ca Fri Oct 3 15:33:33 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Oct 2003 15:33:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A5Ueu-00013M-00 for categories-list@mta.ca; Fri, 03 Oct 2003 15:28:08 -0300 Mime-Version: 1.0 X-Sender: street@icsmail.ics.mq.edu.au Message-Id: Date: Fri, 3 Oct 2003 14:20:05 +1000 To: categories@mta.ca From: Ross Street Subject: categories: Categories of elements Content-Type: text/plain; charset="us-ascii" ; format="flowed" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 > "An element of a functor is an attaching functor into the category of > elements of the functor," is unacceptably confusing due to the fact that > the category of elements of a functor does not in any sense consist of > the elements of the functor (as you would describe them). I'm not sure where the above quote is taken from but I agree it is confusing. Here is my argument in favour of the traditional name. As Bill says, an element of an object F in a category is generally any morphism A --> F into F. It just happens that in many categories F is determined by elements with a restricted class of domains A. In Set, we can restrict A to be terminal. In a presheaf category, we can restrict A to be representable. The objects of the category elF of elements of F are (up to isomorphism) elements A --> F with A representable. It is also conventional to name categories after their objects (although the Ehresmann convention of naming them after their morphisms is more precise). Hence elF is the category of elements of F. Regards, Ross From rrosebru@mta.ca Sun Oct 5 12:03:37 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 05 Oct 2003 12:03:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A6AEf-0005wi-00 for categories-list@mta.ca; Sun, 05 Oct 2003 11:51:49 -0300 Date: Fri, 3 Oct 2003 15:06:33 -0400 (EDT) From: Susan Niefield To: categories@mta.ca Subject: categories: Union College Conference Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-RAVMilter-Version: 8.4.3(snapshot 20030212) (nott.union.edu) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 9 UNION COLLEGE MATHEMATICS CONFERENCE Saturday and Sunday November 8-9, 2003 This is an updated announcement for the eleventh Union College Mathematics Conference. This year the conference topics are category theory, algebraic topology, and differential geometry. The plenary speakers for the conference are: Andre Joyal, UQAM Claude LeBrun, SUNY Stony Brook Ulrike Tillmann, Oxford University There will also be shorter contributed talks in parallel sessions. Anyone interested in giving such a talk should indicate this on the registration form on the conference website (see URL below). The deadline for abstract submission is October 17th. The registration deadline is October 24th. For more information about the conference, including registration, submission of abstracts, housing and transportation, please visit our website at: http://www.math.union.edu/~leshk/03Conference/ We hope to see you in November! ORGANIZERS Category Theory Susan Niefield niefiels@union.edu Kimmo Rosenthal rosenthk@union.edu Algebraic Topology Brenda Johnson johnsonb@union.edu Kathryn Lesh leshk@union.edu Differential Geometry Christina Tonnesen-Friedman tonnesec@union.edu From rrosebru@mta.ca Mon Oct 6 16:52:45 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Oct 2003 16:52:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A6bLa-0005uQ-00 for categories-list@mta.ca; Mon, 06 Oct 2003 16:48:46 -0300 Message-Id: <200310061349.h96DnlJ02194@math.u-strasbg.fr> Date: Mon, 6 Oct 2003 15:49:47 +0200 (MEST) From: Philippe Gaucher Reply-To: Philippe Gaucher Subject: categories: email address: Philippe Gaucher To: categories@mta.ca X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Content-Type: text X-Sun-Text-Type: ascii X-Antivirus: scanned by sophos at u-strasbg.fr Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 Dear all, I am going to move for Paris in a few days. By the end of October, my email address will be Philippe.Gaucher@pps.jussieu.fr and you can already use it if you want to send me an email. pg. From rrosebru@mta.ca Mon Oct 6 16:52:45 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Oct 2003 16:52:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A6bKr-0005rO-00 for categories-list@mta.ca; Mon, 06 Oct 2003 16:48:01 -0300 Date: Mon, 6 Oct 2003 09:31:41 -0400 Mime-Version: 1.0 (Apple Message framework v552) Content-Type: text/plain; charset=US-ASCII; format=flowed Subject: categories: Question on standard terminology From: Steve Stevenson To: Categories List Content-Transfer-Encoding: 7bit Message-Id: <6B79B655-F801-11D7-A52E-000A959EB774@cs.clemson.edu> X-Mailer: Apple Mail (2.552) X-CPSC-Clemson-MailScanner: Found to be clean Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 Good Morning: I have a question about "standard terminology" or "standard methodology". Is there a standard technique to generate the closure of a set through generating the next element from the current? The motivating idea is a simple one: generate a free language from the list of characters. This is a standard inductive process seen in lots of automata and formal language books. Seems like there is product followed by a co-product sort of action. best regards, steve -------- D. E. Stevenson, Department of Computer Science Clemson University, Clemson, SC 29634-0974 864.656.6880 http://www.cs.clemson.edu/~steve From rrosebru@mta.ca Mon Oct 6 16:52:45 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Oct 2003 16:52:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A6bLy-0005wF-00 for categories-list@mta.ca; Mon, 06 Oct 2003 16:49:10 -0300 Message-ID: <3F817329.4@csc.liv.ac.uk> Date: Mon, 06 Oct 2003 14:50:33 +0100 From: Peter McBurney User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.2) Gecko/20030708 X-Accept-Language: en-us, en MIME-Version: 1.0 To: CATEGORIES LIST Subject: categories: The Calculus Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *1A6Vkw-0007NG-Rv*qTQ4l8pReMA* Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 12 Hello all -- Apologies for what may be an elementary question: Does anyone know of a fully-worked-through, category-theoretic treatment of the differential calculus? Thanks, -- Peter McBurney University of Liverpool From rrosebru@mta.ca Wed Oct 8 15:35:44 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Oct 2003 15:35:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A7J69-0004FY-00 for categories-list@mta.ca; Wed, 08 Oct 2003 15:31:45 -0300 Date: Mon, 6 Oct 2003 15:43:13 -0500 Subject: categories: re: The Calculus Mime-Version: 1.0 (Apple Message framework v552) To: From: Larry Stout In-Reply-To: <3F817329.4@csc.liv.ac.uk> Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 14 I taught a course on this two years ago using J.L. Bell A Primer of Infinitesimal Analysis , Cambridge University Press, 1998 . It was great fun. The syllabus is at http://www.iwu.edu/~lstout/InfinitesimalAnalysis/syl489s01.html Lawrence Stout Department of Mathematics and Computer Science Illinois Wesleyan University On Monday, October 6, 2003, at 08:50 AM, Peter McBurney wrote: > Hello all -- > > Apologies for what may be an elementary question: Does anyone know of > a > fully-worked-through, category-theoretic treatment of the differential > calculus? > > Thanks, > > > > > > -- Peter McBurney > University of Liverpool > From rrosebru@mta.ca Sat Oct 11 15:50:51 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 11 Oct 2003 15:50:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A8Olm-0002hY-00 for categories-list@mta.ca; Sat, 11 Oct 2003 15:47:14 -0300 Date: Fri, 10 Oct 2003 09:47:51 +0200 From: Alberto Peruzzi Subject: categories: 2nd announcement: Workshop and Symposium RAMIFICATIONS OF CATEGORY THEORY To: categories@mta.ca Message-id: MIME-version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 15 Re: Workshop and Symposium RAMIFICATIONS OF CATEGORY THEORY, November 18-22, 2003 At the University of Florence, Italy The abstracts of talks and the time table can be found on the web page http://ramcat.scform.unifi.it/ Alberto Peruzzi Departmento of Philosophy Via Bolognese 52 50139 Florence Italia From rrosebru@mta.ca Sun Oct 12 14:05:32 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 12 Oct 2003 14:05:32 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A8jcj-0000Ys-00 for categories-list@mta.ca; Sun, 12 Oct 2003 14:03:17 -0300 Message-Id: <200310120057.h9C0vK816608@math-cl-n01.ucr.edu> Subject: categories: quantum logic To: categories@mta.ca (categories) Date: Sat, 11 Oct 2003 17:57:20 -0700 (PDT) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 16 Dear Categorists - Do any of you know particularly insightful treatments of quantum logic via category theory? I'm more or less familiar with quantum logic as the theory of the complete orthocomplemented lattice of closed subspaces of a given Hilbert space. But now I'm interested in developing quantum logic starting as much as possible from general properties of and structures on the category of Hilbert spaces and bounded linear maps - for example, the fact that it's an abelian category, and becomes a *-category and symmetric monoidal category in a nice way (with Hilbert tensor product as the monoidal structure). And I'm interested in things like how the 2-dimensional Hilbert space acts a bit like a subobject classifier. I don't mind sticking with finite-dimensional Hilbert spaces for now to avoid certain subtleties. On a related note: I've repeatedly heard people say something like "the multiplicative fragment of linear logic is the internal logic of (closed symmetric?) monoidal categories", but I've never heard a precise result along these lines. Has anyone worked out a sufficiently general concept of "the internal logic of a category" or "the internal logic of a certain 2-category of categories" so that one could take something like a monoidal category, or a symmetric monoidal category, or a closed symmetric monoidal category - or maybe the 2-category of all such - and extract by some systematic method the corresponding "internal logic"? I'm vaguely imagining some class of generalizations of the Mitchell-Benabou language of a topos, or something like that - but I'm really interested in the nonCartesian case. The reason I ask this is that it would be nice if you could throw the (closed, symmetric, monoidal, *, etcetera...) category of Hilbert spaces into some big machine and have "quantum logic" pop out - and then throw in other similar categories, and have other kinds of logic pop out. Best, jb From rrosebru@mta.ca Mon Oct 13 09:41:48 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Oct 2003 09:41:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A91xO-00061y-00 for categories-list@mta.ca; Mon, 13 Oct 2003 09:37:50 -0300 To: categories@mta.ca Subject: categories: FMCO 2003: CALL FOR PARTICIPATION Message-Id: From: etaps02 VERIMAG Date: Thu, 09 Oct 2003 16:59:03 +0200 X-IMAG-MailScanner: Found to be clean X-IMAG-MailScanner-Information: Please contact the ISP for more information Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 (We apologize for the reception of multiple copies) *********************** CALL FOR PARTICIPATION ********************** Second International Symposium on Formal Methods for Components and Objects (FMCO 2003) DATES 4 - 7 November 2003 PLACE Lorentz Center, Leiden University, Leiden, The Netherlands REGISTRATION FORM http://fmco.liacs.nl/fmco03.html REGISTRATION FEES 400 euro for regular participants and 275 euro for students PRELIMINARY PROGRAM Tuesday 4th, November 2003 8:45 - 9:00 Welcome 9:00 - 10:00 Keynote: David Parnas (University of Limerick, IE) Mathematical Documentation of Software 10:00 - 10:30 Break 10:30 - 11:15 Razvan Diaconescu (IMAR, RO) Behavioural specification for hierarchical object composition 11:15 - 12:00 Heike Wehrheim (University of Oldenburg, DE) Preserving Properties under Change 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Andrew D. Gordon (Microsoft Research, UK) Formal Tools for Securing Web Services 14:30 - 15:00 Break 15:00 - 15:45 Jeannette Wing (Carnegie Mellon University, USA) Vulnerability Analysis Using Attack Graphs 15:45 - 16:00 Break 16:00 - 16:45 Albert Benveniste (IRISA/INRIA - Rennes, FR) Heterogeneous reactive systems formal modeling 16:45 - 17:30 Yassine Lakhnech (University of Grenoble, FR) t.b.a. WEDNESDAY 5th, November 2003 9:00 - 10:00 Keynote: Tony Hoare (Microsoft Research Cambridge, UK) The Verifying Compiler: a Grand Challenge for Computing Research 10:00 - 10:30 Break 10:30 - 11:15 Willem-Paul de Roever (University of Kiel, DE) Data Refinement: model-oriented proof methods and their comparison 11:15 - 12:00 Frank de Boer (CWI, Amsterdam, NL) Hoare Logics for Object-Oriented Programming: State of the Art 12:00 - 13:30 Lunch break 13:30 - 14:15 Jean-Marc Jezequel (IRISA, Rennes, FR) Model-Driven Engineering: Basic Principles and Open Problems 14:15 - 15:00 Jan Friso Groote (Eindhoven University, NL) Visualisation of HUGE state spaces 17:00 - 19:15 Social Event 19:30 - Dinner THURSDAY 6th, November 2003 9:00 - 10:00 Keynote: Yuri Gurevich (Microsoft Research Redmond, USA) The Semantics of AsmL 10:00 - 10:30 Break 10:30 - 11:15 Egon Boerger (Pisa University, IT) Exploiting the "A" in Abstract State Machines for Specification Reuse. A Java/C# Case Study. 11:15 - 12:00 Werner Damm (University of Oldenburg, DE) t.b.a. 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Joseph Sifakis (Verimag, FR) Component-based construction of deadlock-free systems 14:30 - 15:00 Break 15:00 - 15:45 Philippe Schnoebelen (CNRS, Cachan, FR) The Verification of Lossy Channel Systems 15:45 - 16:30 Bengt Jonsson (Uppsala University, SE) t.b.a. 16:30 - 16:45 Break 16:45 - 17:30 Jan Rutten (CWI, Amsterdam, NL) A case study in coinductive stream calculus: signal flow graphs for dummies FRIDAY 7th, November 2003 9:00 - 10:00 Keynote: E. Allen Emerson (University of Texas, USA) Model checking many components 10:00 - 10:30 Break 10:30 - 11:15 Amir Pnueli (The Weizmann Institute of Science, ISR) t.b.a. 11:15 - 12:00 Natalia Sidorova (Eindhoven University, NL) Practical approaches for the verification of asynchronous components: model checking, abstraction and static analysis 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Desmond D'Souza (Kinetium, Austin, USA) Component Architectures - Some meeting points of practice, trend, and theory 14:30 - 15:00 Break 15:00 - 15:45 Jose Luiz Fiadeiro (University of Leicester, UK) CommUnity on the move: architectures for distribution and mobility 15:45 - 16:30 Gregor Engels (University of Paderborn, DE) Consistent interaction of components 16:30 - 17:15 Rob van Ommering (Philips Research Laboratories, NL) Component Based Architectures and Formalization MOBI-J AFFILIATED WORKSHOP On Monday 3rd, November 2003, there will be a one-day Mobi-J workshop on "Assertional Methods for Java and its Extension with Mobile Asynchronous Channels". REGISTRATION Participation is limited to about 80 people, using a first-in first-served policy. To register, please fill in the registration form at http://fmco.liacs.nl/fmco03.html. The registration fee is 400 euro for regular participants and 275 euro for students It includes the participation to the symposium, a copy of the proceedings, all lunches and refreshments, and a social event (with dinner). ORGANIZING COMMITTEE F.S. de Boer (CWI and Utrecht University) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.P. de Roever (CAU) For more information about participation and registration see the FMCO site above or consult either F.S. de Boer (frb@cwi.nl) or M.M. Bonsangue (marcello@liacs.nl). ----------- you received this e-mail via the address categories@mta.ca From rrosebru@mta.ca Mon Oct 13 09:43:27 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Oct 2003 09:43:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A922k-0006DU-00 for categories-list@mta.ca; Mon, 13 Oct 2003 09:43:22 -0300 Date: Sun, 12 Oct 2003 14:31:11 -0400 (EDT) From: Robert Seely To: categories Subject: categories: Re: quantum logic In-Reply-To: <200310120057.h9C0vK816608@math-cl-n01.ucr.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: 18 On Sat, 11 Oct 2003, John Baez wrote: > On a related note: I've repeatedly heard people say something > like "the multiplicative fragment of linear logic is the internal > logic of (closed symmetric?) monoidal categories", but I've never heard > a precise result along these lines. Has anyone worked out a sufficiently > general concept of "the internal logic of a category" or "the > internal logic of a certain 2-category of categories" so that one > could take something like a monoidal category, or a symmetric monoidal > category, or a closed symmetric monoidal category - or maybe the > 2-category of all such - and extract by some systematic method the > corresponding "internal logic"? I'm vaguely imagining some class > of generalizations of the Mitchell-Benabou language of a topos, or > something like that - but I'm really interested in the nonCartesian > case. Hi John - You might want to take a look at the paper by Robin Cockett and me "Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories " in TAC Vol 3 No 5. ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1997/n5/n5.ps As for a general theory - there are plenty of examples, though I don't know if anyone has really made a general theory of this notion of a categorical doctrine, often referred to, and based on a paper of Kock and Reyes from the 70's. But there are many examples (many in the papers Robin and I have written on linearly distributive categories and related structures - visit my webpage if you're interested), which make clear how to go from the internal logic of a category to the category and back. I suggest also you look at our "Introduction to linear bicategories" (MSCS:10(2000)2 pp 165-203), also available on my webpage, for a higher dimensional approach. - all the best, Robert -- From rrosebru@mta.ca Mon Oct 13 09:45:19 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Oct 2003 09:45:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A924W-0006Ix-00 for categories-list@mta.ca; Mon, 13 Oct 2003 09:45:12 -0300 Date: Sun, 12 Oct 2003 16:49:12 -0400 (EDT) From: Michael Barr To: categories Subject: categories: Re: quantum logic In-Reply-To: <200310120057.h9C0vK816608@math-cl-n01.ucr.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: 19 I will let others answer about the connection between closed monoidal categories and MLL, but I just wanted to say that I am not sure what you mean by the category of Hilbert spaces. If you want the inner product preserved, then only isometric injections are permitted. If you want just bounded linear maps then you are not making any real use of the inner product. And the spaces are self-dual, so it is not a good model of *-autonomy. Perhaps of compact categories, I would have to think about it. But anyway, you have to say what category is meant. Another possibility is partial isometries (which can be thought of as total by being zero on the subspace orthognal to the domain). This is a lot like sets and partial injections. Michael From rrosebru@mta.ca Mon Oct 13 09:46:43 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 13 Oct 2003 09:46:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A925w-0006Ou-00 for categories-list@mta.ca; Mon, 13 Oct 2003 09:46:40 -0300 Message-Id: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> Subject: categories: re: quantum logic To: categories@mta.ca (categories) Date: Sun, 12 Oct 2003 15:08:54 -0700 (PDT) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 20 Michael Barr wrote: > I will let others answer about the connection between closed monoidal > categories and MLL, but I just wanted to say that I am not sure what you > mean by the category of Hilbert spaces. If you want the inner product > preserved, then only isometric injections are permitted. If you want just > bounded linear maps then you are not making any real use of the inner > product. Right. I wanted to leave things flexible so different readers could interpret my question in different ways, but I also tried to hint that I think it's crucial to work with the *-category Hilb whose objects are Hilbert spaces, whose morphisms are bounded linear maps, and whose *-structure sends the bounded linear map f: H -> H' to its Hilbert space adjoint f*: H' -> H. This *-structure can be used to define concepts crucial for quantum mechanics, like "self-adjoint" and "unitary" operators, as well as "isometric injections". Isometric injections are a nice way to study subobjects in Hilb, but they're not good enough for doing full-fledged quantum mechanics, nor is ignoring the inner product altogether. Category theorists are often a bit uncomfortable with *-categories because they prefer "adjoints" that are defined using other structure rather than put in by brute force. However, I'm convinced that we can only understand how quantum field theory exploits the analogy between differential topology and Hilbert space theory if we think about *-categories. For example, a topological quantum field theory is a symmetric monoidal functor from some *-category of cobordisms to the *-category Hilb - but the most physically realistic TQFTs are the "unitary" ones, which preserve the *-structure. I've talked about this *-stuff and the nascent concept of "n-categories with duals" in my papers on 2-Hilbert spaces http://math.ucr.edu/home/baez/2hilb.ps and 2-tangles http://math.ucr.edu/home/baez/hda4.ps and now I want to say a bit about how it impinges on quantum logic - but to avoid reinventing the wheel, I'd like to hear anything vaguely relevant anyone knows about approaching quantum logic with an eye on category theory. (I know a bit about quantales, but maybe there's other stuff I've never heard of.) From rrosebru@mta.ca Thu Oct 16 16:46:09 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:46:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE1B-00031U-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:42:41 -0300 Date: Mon, 13 Oct 2003 14:01:15 +0100 Subject: categories: Re: quantum logic Content-Type: text/plain; charset=US-ASCII; format=flowed Mime-Version: 1.0 (Apple Message framework v552) To: categories@mta.ca From: Pedro Resende In-Reply-To: <200310120057.h9C0vK816608@math-cl-n01.ucr.edu> Message-Id: <5431C21E-FD7D-11D7-8C34-0003934B6278@math.ist.utl.pt> Content-Transfer-Encoding: 7bit X-Mailer: Apple Mail (2.552) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 Hi John, One of the ideas behind the theory of quantales is that the category of "sheaves" on a given quantale should be a topos in *some* generalized sense, whose subobject classifier would be a quantale (which is then related to the multiplicative fragment of a noncommutative linear logic). There are a few papers by various authors addressing sheaves on quantales, however none getting near a satisfactory definition of "quantum topos", but I don't think the field is exhausted. In particular Chris Mulvey wrote a paper with his student Nawaz (you can download it from Chris' web page), but restricted to idempotent right-sided quantales, which form a rather limited class. Nevertheless that paper gives you a category of sheaves which actually is a topos in the classical sense, but equipped with additional structure that provides the "quantum" part. I know that currently he has been working with another student on an extension of this to a more general situation encompassing all involutive quantales (= involutive monoids in the monoidal category of sup-lattices), and last time I heard about it the results looked promising. The significance of this wrt Hilbert spaces is that once you consider involutive quantales of the form "Max(A)" (ie, those consisting of all the closed linear subspaces of a unital C*-algebra A), there is a notion of "irreducible representation" of Max(A) that classifies up to unitary equivalence the irreducible representations of A, and, to a certain extent still in need of further clarification (very preliminary material is in a paper of mine which is due to appear in the J. Algebra and is downloadable from my web page - still a couple of typos and minor bugs in the on-line version, I'm afraid), the category of representations of A is approximated by the corresponding category of quantale modules over Max(A). (Each representation of A on a Hilbert space H induces in a natural way an action of Max(A) on the lattice of closed linear subspaces of H.) By all of this I mean that ultimately the category of sheaves on Max(A) should provide a logical handle on the category of representations of A, and it seems reasonable to expect that what you are saying about the category of Hilbert spaces and bounded linear maps may relate to this general scheme. Best, Pedro. On Sunday, October 12, 2003, at 01:57 AM, John Baez wrote: > Dear Categorists - > > Do any of you know particularly insightful treatments of > quantum logic via category theory? I'm more or less familiar > with quantum logic as the theory of the complete orthocomplemented > lattice of closed subspaces of a given Hilbert space. But now I'm > interested in developing quantum logic starting as much as possible > from general properties of and structures on the category of > Hilbert spaces and bounded linear maps - for example, the fact > that it's an abelian category, and becomes a *-category and symmetric > monoidal category in a nice way (with Hilbert tensor product as the > monoidal structure). And I'm interested in things like how the > 2-dimensional Hilbert space acts a bit like a subobject classifier. > > I don't mind sticking with finite-dimensional Hilbert spaces for now > to avoid certain subtleties. > > On a related note: I've repeatedly heard people say something > like "the multiplicative fragment of linear logic is the internal > logic of (closed symmetric?) monoidal categories", but I've never heard > a precise result along these lines. Has anyone worked out a > sufficiently > general concept of "the internal logic of a category" or "the > internal logic of a certain 2-category of categories" so that one > could take something like a monoidal category, or a symmetric monoidal > category, or a closed symmetric monoidal category - or maybe the > 2-category of all such - and extract by some systematic method the > corresponding "internal logic"? I'm vaguely imagining some class > of generalizations of the Mitchell-Benabou language of a topos, or > something like that - but I'm really interested in the nonCartesian > case. > > The reason I ask this is that it would be nice if you could > throw the (closed, symmetric, monoidal, *, etcetera...) category > of Hilbert spaces into some big machine and have "quantum logic" > pop out - and then throw in other similar categories, and have other > kinds of logic pop out. > > Best, > jb > > > > From rrosebru@mta.ca Thu Oct 16 16:46:09 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:46:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE3o-0003D7-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:45:24 -0300 Date: Thu, 16 Oct 2003 15:41:16 +0200 (CEST) From: Riccardo Focardi To: Riccardo Focardi Subject: categories: CFP: 17th IEEE Computer Security Foundations Workshop Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1 Content-Transfer-Encoding: QUOTED-PRINTABLE Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 22 (Apologies for multiple copies) ----------------------------------------------------------------------- First Call For Papers 17th IEEE Computer Security Foundations Workshop June 28 - 30, 2004 Asilomar, Pacific Grove, CA, USA ----------------------------------------------------------------------- Sponsored by the Technical Committee on Security and Privacy of the IEEE Computer Society This workshop series brings together researchers in computer science to examine foundational issues in computer security. We are interested both in new results in theories of computer security and also in more exploratory presentations that examine open questions and raise fundamental concerns about existing theories. Both papers and panel proposals are welcome. Possible topics include, but are not limited to: Access control Authentication Data and system integrity Database security Network security Distributed systems security Anonymity Intrusion detection Security for mobile computin= g Security protocols Security models Decidability issues Privacy Executable content Formal methods for security Information flow Language-based security For background information about the workshop, and an html version of this Call for Papers, see http://www.csl.sri.com/csfw/index.html (the CSFW home page). This year the workshop will be held in Pacific Grove, CA, USA. Information about the location and the organization will be soon available on the web page. (http://www.csl.sri.com/csfw/csfw17). The proceedings are published by the IEEE Computer Society Press and will be available at the workshop. Selected papers will be invited for submission to the Journal of Computer Security. Instructions for Participants ----------------------------- Submission is open to anyone. Workshop attendance is limited to about 50 participants. Submitted papers must not substantially overlap papers that have been published or that are simultaneously submitted to a journal or a conference with proceedings. Papers should be at most 20 pages long excluding the bibliography and well-marked appendices (using 11-point font, single column format, and reasonable margins on 8.5"x11" paper), and at mos= t 25 pages total. Alternatively, papers can be submitted using the two-colum= n IEEE Proceedings style available for various document preparation systems a= t ftp://pubftp.computer.org/Press/Outgoing/proceedings/. Papers in this styl= e should be at most 12 pages long (at most 15 pages including bibliography an= d appendices). The page limit will be strictly adhered to. Committee member= s are not required to read the appendices, and so the paper should be intelligible without them. Proposals for panels should be no more than five pages in length and should include possible panelists and an indicatio= n of which of those panelists have confirmed participation. Instructions about how to submit a paper will be soon available on the web page (http://www.csl.sri.com/csfw/csfw17). If for any reason you cannot conform to those submission guidelines, please contact the program chair at focardi@dsi.unive.it. Papers should be submitted in Postscript or Portable Document Format (PDF). Papers submitted in a proprietary word-processor format such as Microsoft Word cannot be considered. At least one coauthor of each accepted paper is expected to attend CSFW-17. Papers that do not adhere to this policy will be removed from the proceedings. Important Dates --------------- Submission deadline: January 27, 2004 Notification of acceptance: March 12, 2004 Camera-ready papers: April 6, 2004 Program Committee ----------------- Michael Backes, IBM Zurich Research Lab, Switzerland Agostino Cortesi, University of Venice, Italy Pierpaolo Degano, University of Pisa, Italy Riccardo Focardi (chair), University of Venice, Italy Dieter Gollmann, TU Hamburg-Harburg, Germany Andrew Gordon, Microsoft Research, UK Joshua Guttman, The MITRE Corporation, USA Masami Hagiya, University of Tokyo, Japan Chris Hankin, Imperial College London, UK Gavin Lowe, Oxford University, UK Heiko Mantel, ETH Z=FCrich, Switzerland Catherine Meadows, Naval Research Laboratory, USA Jonathan Millen, SRI International, USA John Mitchell, Stanford University, USA Andrew Myers, Cornell University, USA Mike Reiter, Carnegie Mellon University, USA Pierangela Samarati, University of Milan, Italy Andre Scedrov, University of Pennsylvania, USA Andrei Serjantov, University of Cambridge, UK Geoffrey Smith, Florida International University, USA Workshop Location ----------------- The 17th IEEE Computer Security Foundations workshop will be held at the Asilomar Conference Center, located on the beautiful Monterey Peninsula in Pacific Grove California. Asilomar, meaning "refuge by the sea", is a tranquil environment surrounded by forest and white sand beaches. As a member of the California State Park system, it offers 107 extraordinary acres of forests, dunes, and coastline situated on the Monterey Bay National Marine Sanctuary. Founded in 1913 as the western conference center for the Young Women's Christian Association (YWCA), it is the ideal conference setting. Asilomar offers secluded guest rooms with forest or marine views. Rooms are clustered into quaint lodges, some of which feature fireplaces. Sunset walks along the beach and coastal trails are a great way to unwind. On-sit= e recreation includes a heated swimming pool, volleyball and billiard tables. Just minutes away is Pebble Beach, featuring world-class golf courses and scenic 17-Mile Drive. And some of the most breathtaking coastline in the world can be found just 20 minutes to the south toward Big Sur along Hwy 1. Also nearby is the Monterey Bay Aquarium, featuring spectacular, deep-sea and kelp forest exhibits. Monterey Bay hosts a unique deep-sea environment close to shore. There is an underwater canyon over 2km deep at around 15km from shore. Asilomar is 2.3 hours by car from San Francisco International Airport (SFO). There are direct flights between San Francisco and most major European and American cities. The Monterey regional airport (MRY) is 10 minutes by car from Asilomar. There are direct flights to MRY from Los Angeles International Airport (LAX) and SFO about every 2 hours until 10pm. More travel information can be found on the CSFW17 website. Additional Information ---------------------- For further information contact: General Chair George Dinolt Naval Postgraduate School 1 University Circle Monterey, CA 93943-5001, USA gwdinolt@nps.navy.mil Program Chair Riccardo Focardi Dipartimento di Informatica Universita' di Venezia via Torino 155, I-30172 Mestre (Ve), Italy +39-041-2348438 focardi@dsi.unive.it Publications Chair Jonathan Herzog The MITRE Corporation 202 Burlington Road Bedford, MA 01730-1420 USA jherzog@mitre.org From rrosebru@mta.ca Thu Oct 16 16:46:09 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:46:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE4U-0003GP-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:46:06 -0300 From: "M.M. Bonsangue" Date: Thu, 16 Oct 2003 16:01:47 +0200 Message-Id: <200310161401.h9GE1l206890@pc157aa.liacs.nl> To: categories@mta.ca Subject: categories: Final call: Formal Methods for Components and Objects Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 23 (We apologize for the reception of multiple copies) ********************LAST CALL FOR PARTICIPATION ********************** Second International Symposium on Formal Methods for Components and Objects (FMCO 2003) DATES 4 - 7 November 2003 PLACE Lorentz Center, Leiden University, Leiden, The Netherlands REGISTRATION FORM http://fmco.liacs.nl/fmco03.html REGISTRATION FEES 400 euro for regular participants and 275 euro for students FINAL PROGRAM TUESDAY 4th, November 2003 8:45 - 9:00 Welcome 9:00 - 10:00 Keynote: David Parnas (University of Limerick, IE) Mathematical Documentation of Software 10:00 - 10:30 Break 10:30 - 11:15 Razvan Diaconescu (IMAR, RO) Behavioural specification for hierarchical object composition 11:15 - 12:00 Heike Wehrheim (University of Oldenburg, DE) Preserving Properties under Change 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Andrew D. Gordon (Microsoft Research, UK) Formal Tools for Securing Web Services 14:30 - 15:00 Break 15:00 - 15:45 Jeannette Wing (Carnegie Mellon University, USA) Vulnerability Analysis Using Attack Graphs 15:45 - 16:30 Yassine Lakhnech (University of Grenoble, FR) Security protocols, their modes and analysis: a survey 16:30 - 16:45 Break 16:45 - 17:30 Albert Benveniste (IRISA/INRIA - Rennes, FR) Heterogeneous reactive systems formal modeling WEDNESDAY 5th, November 2003 9:00 - 10:00 Keynote: Tony Hoare (Microsoft Research Cambridge, UK) The Verifying Compiler: a Grand Challenge for Computing Research 10:00 - 10:30 Break 10:30 - 11:15 Willem-Paul de Roever (University of Kiel, DE) Data Refinement: model-oriented proof methods and their comparison 11:15 - 12:00 Frank de Boer (CWI, Amsterdam, NL) Hoare Logics for Object-Oriented Programming: State of the Art 12:00 - 13:30 Lunch break 13:30 - 14:15 Jean-Marc Jezequel (IRISA, Rennes, FR) Model-Driven Engineering: Basic Principles and Open Problems 14:15 - 15:00 Jan Friso Groote (Eindhoven University, NL) Visualisation of HUGE state spaces 17:00 - 19:15 Social Event 19:30 - Dinner THURSDAY 6th, November 2003 9:00 - 10:00 Keynote: Yuri Gurevich (Microsoft Research Redmond, USA) The Semantics of AsmL 10:00 - 10:30 Break 10:30 - 11:15 Egon Boerger (Pisa University, IT) Exploiting the "A" in Abstract State Machines for Specification Reuse. A Java/C# Case Study. 11:15 - 12:00 Werner Damm (University of Oldenburg, DE) t.b.a. 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Joseph Sifakis (Verimag, FR) Component-based construction of deadlock-free systems 14:30 - 15:00 Break 15:00 - 15:45 Philippe Schnoebelen (CNRS, Cachan, FR) The Verification of Lossy Channel Systems 15:45 - 16:30 Bengt Jonsson (Uppsala University, SE) t.b.a. 16:30 - 16:45 Break 16:45 - 17:30 Jan Rutten (CWI, Amsterdam, NL) A case study in coinductive stream calculus: signal flow graphs for dummies FRIDAY 7th, November 2003 9:00 - 10:00 Keynote: E. Allen Emerson (University of Texas, USA) Model checking many components 10:00 - 10:30 Break 10:30 - 11:15 Amir Pnueli (The Weizmann Institute of Science, ISR) t.b.a. 11:15 - 12:00 Natalia Sidorova (Eindhoven University, NL) Practical approaches for the verification of asynchronous components: model checking, abstraction and static analysis 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Desmond D'Souza (Kinetium, Austin, USA) Component Architectures - Some meeting points of practice, trend, and theory 14:30 - 15:00 Break 15:00 - 15:45 Jose Luiz Fiadeiro (University of Leicester, UK) CommUnity on the move: architectures for distribution and mobility 15:45 - 16:30 Gregor Engels (University of Paderborn, DE) Consistent interaction of components 16:30 - 17:15 Rob van Ommering (Philips Research Laboratories, NL) Component Based Architectures and Formalization MOBI-J AFFILIATED WORKSHOP On Monday 3rd, November 2003 from 13:30 till 17:00 there will at the Lorentz Center be a half-day Mobi-J workshop on "Assertional Methods for Java and its Extension with Mobile Asynchronous Channels". REGISTRATION Participation is limited to about 80 people, using a first-in first-served policy. To register, please fill in the registration form at http://fmco.liacs.nl/fmco03.html. The registration fee is 400 euro for regular participants and 275 euro for students It includes the participation to the symposium, a copy of the proceedings, all lunches and refreshments, and a social event (with dinner). ORGANIZING COMMITTEE F.S. de Boer (CWI and Utrecht University) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.P. de Roever (CAU) For more information about participation and registration see the FMCO site above or consult either F.S. de Boer (frb@cwi.nl) or M.M. Bonsangue (marcello@liacs.nl). From rrosebru@mta.ca Thu Oct 16 16:47:35 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:47:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE5r-0003O9-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:47:31 -0300 Message-ID: <3F8AA6DA.6090503@csc.liv.ac.uk> Date: Mon, 13 Oct 2003 14:21:30 +0100 From: Peter McBurney X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories Subject: categories: Re: quantum logic References: <200310120057.h9C0vK816608@math-cl-n01.ucr.edu> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 24 John -- Although not a categorical treatment, a recent paper by Kurt Engesser and Dov Gabbay in discusses a connection between Quantum Logic and Hilbert spaces. (The reason the work appeared in the leading AI journal is that there are applications to nonmonotonic reasoning, which is a major area of research in AI.) Citation details and abstract below. -- Peter ================================================================== Artificial Intelligence Volume 136, Issue 1 , March 2002 , Pages 61-100 "Quantum logic, Hilbert space, revision theory" Kurt Engesser and Dov M. Gabbay a Birkenweg 3, 78573 Wurmlingen, Germany b Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK Abstract Our starting point is the observation that with a given Hilbert space H we may, in a way to be made precise, associate a class of non-monotonic consequence relations in such a way that there exists a one-to-one correspondence between the rays of H and these consequence relations. The projectors in Hilbert space may then be viewed as a sort of revision operators. The lattice of closed subspaces appears as a natural generalisation of the concept of a Lindenbaum algebra in classical logic. The logics presentable by Hilbert spaces are investigated and characterised. Moreover, the individual consequence relations are studied. A key concept in this context is that of a consequence relation having a pointer to itself. It is proved that such consequence relations have certain remarkable properties in that they reflect their metatheory at the object level to a surprising extent. The tools used in the investigation stem from two different areas of research, namely from the disciplines of non-monotonic logic on the one hand and from Hilbert space theory on the other. There exist surprising connections between these two fields of research the investigation of which constitutes the purpose of this paper. Author Keywords: Quantum logic; Hilbert space; Revision theory; Consequence relation; Non-monotonic logic ==================================================================== From rrosebru@mta.ca Thu Oct 16 16:48:55 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2003 16:48:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AAE77-0003T9-00 for categories-list@mta.ca; Thu, 16 Oct 2003 16:48:49 -0300 Date: Mon, 13 Oct 2003 11:10:03 -0400 (EDT) From: Michael Barr X-X-Sender: barr@triples.math.mcgill.ca To: categories Subject: categories: re: quantum logic In-Reply-To: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 25 I think Rick Blute (+ collaborators) has done some things with this. It is not clear whether you want a self-duality or a *-autonomous category. If you stick to finite dimensional Hilbert spaces, the situation seems simple. If V and W are inner product spaces, then for f, g: V --> W, let f.g = \sum f(v_i).g(v_i) the sum taken over an orthonormal basis. I believe this is invariant to an orthonormal base change and it is obviously positive definite. For infinite dimensional spaces, you would have to stick to f for which \sum f(v_i)^2 < oo. But this isn't a category. It is closed under composition (I think) but certainly lacks identities. This gives rise to something called a nuclear category. The category has all maps and there is sub-non-category of nuclear maps. This all goes back (needless to say) to Grothendieck. If by *-category you just mean self dual, well then Hilbert spaces certainly are that. Self dual categories are a dime a dozen. Just take C x C^op. The amazing thing is that if C is closed, C x C^op is *-autonomous, (assuming C has binary cartesian products). Michael From rrosebru@mta.ca Fri Oct 17 09:22:21 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 17 Oct 2003 09:22:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AATaq-0000mV-00 for categories-list@mta.ca; Fri, 17 Oct 2003 09:20:32 -0300 Date: Thu, 16 Oct 2003 17:39:27 -0400 (EDT) From: James Stasheff To: dmd1@lehigh.edu, categories@mta.ca Subject: categories: terminology 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: 26 In `higher homotopy theory', terminology has not setled down nor is it transparent homotopy ___________ algebra can mean a variety of things letting ______________ = associative it can mean JUST that there is a homtopy for associaitivity or some authors use it to mean A_\infty which I initially tried to indicate by strongly homtopy associative _\infty seems to have caught on to mean the presence of higher homtopies of all orders in most but not all cases, such algebras have a homtopy invariant defintion so I would suggest the following revisionist terminology 1-homotopy associative means JUST that there is a homotopy for associaitivity similarly n-homotopy associative would mean homotopies of homotopies of... homotopy invariant ___ algebra would mean just what it says so far so good but now what about e.g. 1-homotopy associaitve satisfying a STRICT pentagon?? perhaps strict 1-homotopy open to suggestions Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds As of July 1, 2002, I am Professor Emeritus at UNC and I will be visiting U Penn but for hard copy the relevant address is: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 From rrosebru@mta.ca Mon Oct 20 14:49:09 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 20 Oct 2003 14:49:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ABe5L-0006EC-00 for categories-list@mta.ca; Mon, 20 Oct 2003 14:44:51 -0300 X-Sender: grandis@pop4.dima.unige.it Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 17 Oct 2003 17:19:09 +0200 To: categories@mta.ca From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Re: terminology X-OriginalArrivalTime: 17 Oct 2003 15:14:06.0937 (UTC) FILETIME=[4EDF6C90:01C394C1] Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 27 In reply to Stasheff's question on terminology for homotopy coherent algebras: >but now what about e.g. 1-homotopy associaitve satisfying a STRICT >pentagon?? >perhaps strict 1-homotopy I would say: "2-strict sha-algebra", as motivated below. (sha = strongly homotopy associative) However: after the strict pentagon, this structure has a second coherence condition for the associativity homotopy (which disappears for monoidal categories, just because their 2-morphisms are trivial) ____________ In a paper [*] on strongly homotopy associative (differential) algebras, I proposed this definition (4.2; pages 38-39). Notation: a sha-algebra is a graded module A with morphisms (sort of components of a global differential d of bar coalgebras) d_1: A --> A (degree - 1; the differential) d_2: AoA --> A (degree 0; the product) d_3: AoAoA --> A (degree 1; the associativity 1-homotopy) ........ d_n: A^n --> A (degree n - 2; the coherence n-homotopy) ........ ( o = tensor product; ^n = tensor power) under axioms (1) d_1.d_1 = 0 (2) .... (expressing dd = 0 for the global differential). DEF. This is called an *n-strict sha-algebra* if d_p = 0 for p > n. Equivalently, the morphisms d_1,..., d_n have to satisfy the original axioms (1) ... (n) plus n - 1 conditions obtained from the axioms (n+1) ... (2n - 1), cancelling the null d_p's (the remaining axioms become trivial). This gives: 1-strict = differential module 2-strict = associative differential algebra 3-strict = 1-homotopy associative differential algebra with strict pentagon (from axiom (3)) and axiom (4) reduced to: (4) d3 (1o1od3 + 1od3o1 + d3o1o1) = 0. _______ So far in that paper. The name is chosen to make d_n the last relevant component, in the n-strict case. I might now (more geometrically) prefer a - 1 shift in these names, so that the last example would be named 2-strict, in accord with the fact that the last relevant homotopy is a an ordinary ("one-dimensional") homotopy and everything becomes strict starting with "dimension 2". _______ Reference: [*] M. Grandis, On the homotopy structure of strongly homotopy associative algebras, J. Pure Appl. Algebra 134 (1999), 15-81. _______ Regards MG From rrosebru@mta.ca Mon Oct 20 14:51:18 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 20 Oct 2003 14:51:18 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ABeBR-0006nf-00 for categories-list@mta.ca; Mon, 20 Oct 2003 14:51:09 -0300 Date: Sat, 18 Oct 2003 16:57:00 -0400 (EDT) From: Michael Barr To: categories Subject: categories: Re: quantum logic In-Reply-To: <200310122208.h9CM8sf26075@math-cl-n01.ucr.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: 28 After giving the matter some thought, I finally decided that the category of Hilbert spaces should have as its morphisms norm-reducing linear maps. At the very least that will ensure that an isomorphism is an isometry. And yes the category of finite dimensional Hilbert spaces is *-autonomous. The internal hom of two is the space of all linear maps and the inner product = \sum f(u_i)g(u_i), taken over an orthonormal basis of the domain. This can be shown to be invariant under orthogonal change of basis. The norm of the identity on an n-dimensional space is sqrt(n). The dual of a space is itself, of course with the duality being adjunction (or transpose). Then the tensor product H # G = (H --o G^*)^*. Here is another approach to the same structure. Consider a pair (V,\phi) where V is a finite dimensional space and \phi is an isomorphism of V with its dual space. You have to add positive definiteness and symmetry, but that is no problem. Maps again are norm reducing. Now we can define (V,\phi) # (W,\psi) = (V # W,\phi # \psi), (V,\phi)^* = (V,\phi^{-1}), and (V,\phi) --o (W,\psi) = (V # W,\phi^{-1} # \psi). The resultant category is exactly the same as before. BTW, it is easy to see that the transpose of a norm-reducing map is norm reducing. On Sun, 12 Oct 2003, John Baez wrote: > Michael Barr wrote: > > > I will let others answer about the connection between closed monoidal > > categories and MLL, but I just wanted to say that I am not sure what you > > mean by the category of Hilbert spaces. If you want the inner product > > preserved, then only isometric injections are permitted. If you want just > > bounded linear maps then you are not making any real use of the inner > > product. > > Right. I wanted to leave things flexible so different readers could > interpret my question in different ways, but I also tried to hint > that I think it's crucial to work with the *-category Hilb whose objects > are Hilbert spaces, whose morphisms are bounded linear maps, and whose > *-structure sends the bounded linear map f: H -> H' to its Hilbert > space adjoint f*: H' -> H. This *-structure can be used to define > concepts crucial for quantum mechanics, like "self-adjoint" and > "unitary" operators, as well as "isometric injections". Isometric > injections are a nice way to study subobjects in Hilb, but they're > not good enough for doing full-fledged quantum mechanics, nor is > ignoring the inner product altogether. > > Category theorists are often a bit uncomfortable with *-categories > because they prefer "adjoints" that are defined using other structure > rather than put in by brute force. However, I'm convinced that we > can only understand how quantum field theory exploits the analogy > between differential topology and Hilbert space theory if we think > about *-categories. For example, a topological quantum field theory > is a symmetric monoidal functor from some *-category of cobordisms > to the *-category Hilb - but the most physically realistic TQFTs are > the "unitary" ones, which preserve the *-structure. > > I've talked about this *-stuff and the nascent concept of "n-categories > with duals" in my papers on 2-Hilbert spaces > > http://math.ucr.edu/home/baez/2hilb.ps > > and 2-tangles > > http://math.ucr.edu/home/baez/hda4.ps > > and now I want to say a bit about how it impinges on quantum > logic - but to avoid reinventing the wheel, I'd like to hear > anything vaguely relevant anyone knows about approaching quantum > logic with an eye on category theory. > > (I know a bit about quantales, but maybe there's other stuff > I've never heard of.) > > > > > From rrosebru@mta.ca Wed Oct 22 11:36:16 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Oct 2003 11:36:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ACK0U-0002YF-00 for categories-list@mta.ca; Wed, 22 Oct 2003 11:30:38 -0300 Date: Mon, 20 Oct 2003 12:51:07 -0700 From: Toby Bartels To: categories Subject: categories: Re: quantum logic Message-ID: <20031020195106.GA2487@math-rs-n03.ucr.edu> References: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 29 Michael Barr wrote in part: >After giving the matter some thought, I finally decided that the >category of Hilbert spaces should have as its morphisms norm-reducing >linear maps. At the very least that will ensure that an isomorphism is >an isometry. True, but are you begging the question by trying to ensure that? After all, an invertible bounded linear map is enough to deduce that Hilbert spaces are isomorphic (even in the sense of isometric), so why not count those maps as isomorphisms themselves? This matter is much bigger than Hilbert spaces, of course; moving to Banach spaces (a closed category even for arbitrary dimension), we can even see how, /as/ a closed category, it doesn't really matter! The question is, what is the forgetful functor from Ban to Set? Do we take the set of all vectors? or do we take the closed unit ball? The former corresponds to allowing all bounded linear maps as morphisms, while the latter corresponds to requiring norm-reducing linear maps. But in the closed category Ban, the Banach space of morphisms is, whatever your conventions, the space of all bounded linear maps. Still, this can be consistent with either choice of hom-SET, since the closed unit ball in the Banach space of bounded linear maps is none other than your preferred hom-set of norm-reducing maps. Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category than a category in the first place. We can do this on a more elementary level with metric spaces; is the hom-set the set of all Lipschitz continuous functions, or is it only the set of distance-reducing functions? But unlike with Banach (or Hilbert) spaces, this makes a difference even to the classification of metric spaces into isomorphism classes. The question becomes, is an isomorphism of metric spaces merely a relabelling of points keeping all distances the same, or does it also allow for a recalibration of ones ruler? Which is the correct interpretation may depend on the application, and how absolute -- rather than measured in some unit -- the distances are. (One can even recalibrate more generously to allow as morphisms all uniformly continuous maps, or even all continuous maps. Thus classically one speaks of variously "equivalent" metric spaces, such as "uniformly equivalent" or "topologically equivalent".) To get closed categories here, one must restrict to bounded metric spaces; the analysis is a little more fun than for Banach spaces, especially with the degeneracy surrounding the initial and terminal spaces. -- Toby From rrosebru@mta.ca Thu Oct 23 11:11:00 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Oct 2003 11:11:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ACg9D-00047P-00 for categories-list@mta.ca; Thu, 23 Oct 2003 11:09:07 -0300 Date: Wed, 22 Oct 2003 13:14:38 -0700 From: Toby Bartels To: categories Subject: categories: Re: quantum logic Message-ID: <20031022201437.GG22371@math-rs-n03.ucr.edu> References: <20031020195106.GA2487@math-rs-n03.ucr.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: User-Agent: Mutt/1.4i Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 30 Michael Barr wrote: >For Banach spaces, if you take as underlying functor the closed unit ball, >it has an adjoint. It is not tripleable, however, but C^*-algebras are >(with the unit ball underlying functor). OK, that's a good point. I agree (with the L-1 norm on the free space). -- Toby From rrosebru@mta.ca Thu Oct 23 11:11:00 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Oct 2003 11:11:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ACg7r-00041x-00 for categories-list@mta.ca; Thu, 23 Oct 2003 11:07:43 -0300 X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Wed, 22 Oct 2003 12:01:26 -0400 (EDT) From: Michael Barr X-X-Sender: barr@triples.math.mcgill.ca To: categories Subject: categories: Re: quantum logic In-Reply-To: <20031020195106.GA2487@math-rs-n03.ucr.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: 31 I will stick to my perception that if you dealing with Hilbert or Banach spaces isomorphisms should be just that. It makes no difference to the *-autonomous structure anyway. For Banach spaces, if you take as underlying functor the closed unit ball, it has an adjoint. It is not tripleable, however, but C^*-algebras are (with the unit ball underlying functor). On Mon, 20 Oct 2003, Toby Bartels wrote: > Michael Barr wrote in part: > > >After giving the matter some thought, I finally decided that the > >category of Hilbert spaces should have as its morphisms norm-reducing > >linear maps. At the very least that will ensure that an isomorphism is > >an isometry. > > True, but are you begging the question by trying to ensure that? > After all, an invertible bounded linear map is enough to deduce > that Hilbert spaces are isomorphic (even in the sense of isometric), > so why not count those maps as isomorphisms themselves? > > This matter is much bigger than Hilbert spaces, of course; > moving to Banach spaces (a closed category even for arbitrary dimension), > we can even see how, /as/ a closed category, it doesn't really matter! > The question is, what is the forgetful functor from Ban to Set? > Do we take the set of all vectors? or do we take the closed unit ball? > The former corresponds to allowing all bounded linear maps as morphisms, > while the latter corresponds to requiring norm-reducing linear maps. > But in the closed category Ban, the Banach space of morphisms > is, whatever your conventions, the space of all bounded linear maps. > Still, this can be consistent with either choice of hom-SET, > since the closed unit ball in the Banach space of bounded linear maps > is none other than your preferred hom-set of norm-reducing maps. > > Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category > than a category in the first place. > > We can do this on a more elementary level with metric spaces; > is the hom-set the set of all Lipschitz continuous functions, > or is it only the set of distance-reducing functions? > But unlike with Banach (or Hilbert) spaces, this makes a difference > even to the classification of metric spaces into isomorphism classes. > The question becomes, is an isomorphism of metric spaces > merely a relabelling of points keeping all distances the same, > or does it also allow for a recalibration of ones ruler? > Which is the correct interpretation may depend on the application, > and how absolute -- rather than measured in some unit -- the distances are. > (One can even recalibrate more generously to allow as morphisms > all uniformly continuous maps, or even all continuous maps. > Thus classically one speaks of variously "equivalent" metric spaces, > such as "uniformly equivalent" or "topologically equivalent".) > To get closed categories here, one must restrict to bounded metric spaces; > the analysis is a little more fun than for Banach spaces, > especially with the degeneracy surrounding the initial and terminal spaces. > > > -- Toby > > > From rrosebru@mta.ca Thu Oct 23 11:11:00 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Oct 2003 11:11:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ACg8Y-00044Z-00 for categories-list@mta.ca; Thu, 23 Oct 2003 11:08:26 -0300 Date: Wed, 22 Oct 2003 14:07:05 -0400 From: Fred E.J. Linton To: categories Subject: categories: Re: quantum logic Mime-Version: 1.0 Message-ID: <856HJVsHF4064S16.1066846025@uwdvg016.cms.usa.net> Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 32 I'll address two of these questions. The first: > The question is, what is the forgetful functor from Ban to Set? > Do we take the set of all vectors? or do we take the closed unit ball? > The former corresponds to allowing all bounded linear maps as morphisms= , > while the latter corresponds to requiring norm-reducing linear maps. Actually, when the "underlying-set functor" for Banach spaces is taken to be the unit disk functor, and the morphisms are taken as the norm-decreasing maps, the situation is really great, because the norm-decreasing maps DO constitute the unit disk of the Banach space of bounded linear transformations, as you know. And products and coproducts are as Banach spacists like to see them (the familiar L-infinity style "full direct product" and and L-1 = style "weak direct product", respectively). When the underlying-set functor is taken to be ALL the vectors = of the Banach space, on the other hand, products and coproducts = misbehave quite badly. = As for the question, > After all, an invertible bounded linear map is enough to deduce > that Hilbert spaces are isomorphic (even in the sense of isometric), > so why not count those maps as isomorphisms themselves? I'd answer by saying that unless the invertible bounded linear map in the question IS an isometry I'd never dare call it one. -- Fred (usually ) Toby Bartels wrote: > Michael Barr wrote in part: > = > >After giving the matter some thought, I finally decided that the > >category of Hilbert spaces should have as its morphisms norm-reducing > >linear maps. At the very least that will ensure that an isomorphism i= s > >an isometry. > = > True, but are you begging the question by trying to ensure that? > After all, an invertible bounded linear map is enough to deduce > that Hilbert spaces are isomorphic (even in the sense of isometric), > so why not count those maps as isomorphisms themselves? > = > This matter is much bigger than Hilbert spaces, of course; > moving to Banach spaces (a closed category even for arbitrary dimension= ), > we can even see how, /as/ a closed category, it doesn't really matter! > The question is, what is the forgetful functor from Ban to Set? > Do we take the set of all vectors? or do we take the closed unit ball? > The former corresponds to allowing all bounded linear maps as morphisms= , > while the latter corresponds to requiring norm-reducing linear maps. > But in the closed category Ban, the Banach space of morphisms > is, whatever your conventions, the space of all bounded linear maps. > Still, this can be consistent with either choice of hom-SET, > since the closed unit ball in the Banach space of bounded linear maps > is none other than your preferred hom-set of norm-reducing maps. > = > Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed cate= gory > than a category in the first place. > = > We can do this on a more elementary level with metric spaces; > is the hom-set the set of all Lipschitz continuous functions, > or is it only the set of distance-reducing functions? > But unlike with Banach (or Hilbert) spaces, this makes a difference > even to the classification of metric spaces into isomorphism classes. > The question becomes, is an isomorphism of metric spaces > merely a relabelling of points keeping all distances the same, > or does it also allow for a recalibration of ones ruler? > Which is the correct interpretation may depend on the application, > and how absolute -- rather than measured in some unit -- the distances = are. > (One can even recalibrate more generously to allow as morphisms > all uniformly continuous maps, or even all continuous maps. > Thus classically one speaks of variously "equivalent" metric spaces, > such as "uniformly equivalent" or "topologically equivalent".) > To get closed categories here, one must restrict to bounded metric spac= es; > the analysis is a little more fun than for Banach spaces, > especially with the degeneracy surrounding the initial and terminal spa= ces. > = > = > -- Toby > = > = > = > = From rrosebru@mta.ca Fri Oct 24 11:03:59 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Oct 2003 11:03:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AD2TM-0004xQ-00 for categories-list@mta.ca; Fri, 24 Oct 2003 10:59:24 -0300 Message-Id: <200310232333.h9NNXTZ24668@math-ws-n09.ucr.edu> Subject: categories: regular, geometric and coherent categories To: categories@mta.ca (categories) Date: Thu, 23 Oct 2003 16:33:29 -0700 (PDT) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 33 Dear Categorists - In my quest to understand how various flavors of monoidal category can be seen as having various flavors of "internal language" and "internal logic", I've been enjoying the section in Johnstone's "Sketches of an Elephant" where he discusses different fragments of first-order logic and how they can be interpreted in categories with different properties. Of course, this being a book on topos theory, none of this deals with monoidal categories where the tensor product is not cartesian - my main interest, for applications to quantum logic. But, it's still lots of fun. I'd like to get a better feel for some of these things. For example, he talks about "cartesian categories" "regular categories", "geometric categories", "coherent categories" and describes which fragment of first-order logic can be interpreted in each of these things: "cartesian logic", "regular logic", "geometric logic" "coherent logic". Here's some stuff I think I know. I know the definitions of the above concepts, as long as I have the book open to the right page... but I left it at home, so these could be wrong! Cartesian categories have finite limits. Regular categories are cartesian categories with regular epi/mono factorizations, which must be stable under pullbacks. Geometric categories are regular categories admitting arbitrary unions of subobjects, which must be stable under pullbacks. Coherent categories are geometric categories where pullback of subobjects has a right adjoint (which plays the role of "for all"). I have a fairly good feel for categories with finite limits and "finite limits theories"; the others seem more mysterious to me, since I don't know enough examples illustrating the distinctions. Categories monadic over Set are regular, so AbGp is regular - but it's not coherent, since in a coherent category every morphism to the initial object is an isomorphism. What are some other examples of all these things? From rrosebru@mta.ca Fri Oct 24 11:03:59 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Oct 2003 11:03:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AD2V2-00055y-00 for categories-list@mta.ca; Fri, 24 Oct 2003 11:01:08 -0300 Message-ID: <3F98C16E.C4969E4A@itee.uq.edu.au> Date: Fri, 24 Oct 2003 16:06:38 +1000 From: Antonio Cerone Organization: The University of Queensland MIME-Version: 1.0 To: categories@mta.ca Subject: categories: QAPL 2004: Call for Papers Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 34 ------------------------------------------------------------------------- Call for Papers ------------------------------------------------------------------------- Q A P L 2 0 0 4 ------------------------------------------------------------------------- 2nd Workshop on QUANTITATIVE ASPECTS OF PROGRAMMING LANGUAGES Barcelona - Spain 27th - 28th March, 2004 Satellite Event of ETAPS 2004 http://qapl04.di.unipi.it ------------------------------------------------------------------------- SCOPE Quantitative aspects of computation are important and sometimes essential in characterising the behaviour and determining the properties of systems. They are related to the use of physical quantities (storage space, time, bandwidth, etc.) as well as mathematical quantities (e.g. probability and measures for reliability, risk and trust). Such quantities play a central role in defining both the model of systems (architecture, language design, semantics) and the methodologies and tools for the analysis and verification of system properties. The aim of this workshop is to discuss the explicit use of quantitative information such as time and probabilities either directly in the model or as a tool for the analysis of systems. In particular, the workshop focuses on - the design of probabilistic and real-time languages and the definition of semantical models for such languages; - the discussion of methodologies for the analysis of probabilistic and timing properties (e.g. security, safety, schedulability) and of other quantifiable properties such as reliability (for hardware components), trustworthiness (in information security) and resource usage (e.g., worst-case memory/stack/cache requirements); - the probabilistic analysis of systems which do not explicitly incorporate quantitative aspects (e.g. performance, reliability and risk analysis); - applications to safety-critical systems, communication protocols, control systems, asynchronous hardware, and to any other domain involving quantitative issues. ------------------------------------------------------------------------- TOPICS Topics include (but are not limited to) probabilistic, timing and general quantitative aspects in Language design Performance analysis Language extension Program analysis Language expressiveness Verification Hardware description languages Asynchronous hardware analysis Logic Refinement Semantics Automated reasoning Coordination models Model-checking Distributed systems Security Time-critical systems Safety Embedded systems Risk and Hazard Analysis Multi-tasking systems Scheduling theory Information systems Testing ------------------------------------------------------------------------- INVITED SPEAKERS R. Gorrieri (University of Bologna, Italy) P. Harrison (Imperial College London, UK) P. Panangaden (McGill University, Canada) W. Yi (Uppsala University, Sweden) ------------------------------------------------------------------------- SUBMISSION Authors are invited to submit papers up to 15 pages long in the ENTCS style format. Papers should clearly state the topics covered. Electronic submission is highly recommended. Detailed information is available on the web site http://qapl04.di.unipi.it. In case of problems with access to internet, it is possible to submit 3 copies of the paper to one co-chairperson of the program committee. ------------------------------------------------------------------------- IMPORTANT DATES Deadline for submission: 14 November, 2003 Notification to authors: 13 January, 2004 Final version: 13 February, 2004 Workshop: 27-28 March, 2004 ------------------------------------------------------------------------- PROCEEDINGS Accepted papers will be published in Elsevier's ENTCS (Electronic Notes in Theoretical Computer Science). Publication of a selection of the papers in a special issue of Theoretical Computer Science is currently under negotiation. ------------------------------------------------------------------------- ORGANISERS Antonio Cerone Alessandra Di Pierro School of ITEE Dipartimento di Informatica The University of Queensland University of Pisa Australia Italy Phone: +61 7 33651651 Phone: +39 050 2212779 Fax: +61 7 33651533 Fax: +39 050 2212726 ------------------------------------------------------------------------- PROGRAM COMMITTEE G. Bernat (York, UK) F. de Boer (Utrecht, The Netherlands) A. Cerone (PC co-chair) L. de Alfaro (Santa Cruz, USA) A. Di Pierro (PC co-chair) C. Fidge (Queensland, Australia) M. Gabbrielli (Bologna, Italy) M. Huth (IC London, UK) S.D. Johnson (Indiana, USA) M.Z. Kwiatkowska (Birmingham, UK) J. Ostroff (York, Canada) H. Wiklicky (IC London, UK) W. Yi (Uppsala, Sweden) ------------------------------------------------------------------------- -- Antonio Cerone antonio@itee.uq.edu.au School of ITEE The University of Queensland, QLD 4072 Ph. +61-7-33651651 Australia Fax +61-7-33651533 From rrosebru@mta.ca Sat Oct 25 08:48:39 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Oct 2003 08:48:39 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ADMoU-0001tX-00 for categories-list@mta.ca; Sat, 25 Oct 2003 08:42:34 -0300 Message-Id: <200310250108.h9P18lo20637@math-cl-n01.ucr.edu> Subject: categories: regular, geometric, and coherent categories To: categories@mta.ca (categories) Date: Fri, 24 Oct 2003 18:08:47 -0700 (PDT) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 I wrote: > Here's some stuff I think I know. I know the definitions of > the above concepts, as long as I have the book open to the > right page... but I left it at home, so these could be wrong! Some were. > Cartesian categories have finite limits. Regular categories > are cartesian categories with regular epi/mono factorizations, > which must be stable under pullbacks. Fine. > Geometric categories are > regular categories admitting arbitrary unions of subobjects, > which must be stable under pullbacks. Coherent categories > are geometric categories where pullback of subobjects has a > right adjoint (which plays the role of "for all"). These were mixed up. From rrosebru@mta.ca Sat Oct 25 08:48:39 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Oct 2003 08:48:39 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ADMrp-0001xM-00 for categories-list@mta.ca; Sat, 25 Oct 2003 08:46:01 -0300 Message-ID: <3F98CF20.6050907@usa.net> Date: Fri, 24 Oct 2003 03:05:04 -0400 From: "Fred E.J. Linton" User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.0.1) Gecko/20020823 Netscape/7.0 (nscd2) MIME-Version: 1.0 To: categories Subject: categories: Re: quantum logic References: <856HJVsHF4064S16.1066846025@uwdvg016.cms.usa.net> <20031022201258.GF22371@math-rs-n03.ucr.edu> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 36 Toby Bartels wrote: > For finitary products/coproducts, the L-p style norm will work for any p, > which is no surprise since the results are isomorphic (in either category). > In fact, we get a biproduct diagram that works regardless of norm > (so long as the projections and injections are norm-reducing). Sorry, in the maps-norm-decreasing, disk-as-underlying-set category, even RxR gives you isometrically different Banach spaces for different values of p .(*) Only the L-1 style norm gives you a coproduct, only the L-infinity style norm gives you a product (using the usual "as-vector-space" injections and projections); the other choices of p give you god-only-knows-what. As regards another point, I think this is dead wrong: > The L-oo style full direct product and the L-1 style weak direct product > work as (respectively) product and coproduct using /either/ hom-set > (and hence using either corresponding choice of underlying-set functor). > This is because |f| <= sup_i |f_i| holds (for both product and coproduct, > albeit by a different calculation for L-oo product than for L-1 coproduct). Here's why: a bounded linear transformation to the L-oo style product of a bunch of real lines, say (in the real Banach space case) arises from a BOUNDED family of bounded linear functionals. An UNbounded family of bounded linear functionals WILL give you a continuous linear transformation, of course, but NOT to the L-oo style product of R 's -- it will be taking values in the topological-vector-space product of those R 's. Same problem, in reverse, for the L-1 style weak direct product as coproduct: the bounded maps from, say, l_1(aleph-0) to a Banach space B correspond, after composing with the injections, to BOUNDED families of maps R --> B (i.e., bounded families of vectors in B ). But ARBITRARY families of maps R --> B should have a common extension to a continuous map from the coproduct of those R 's. So their L-1 style weak direct product (which is what l_1(aleph-0) is) won't be the coproduct in the continuous-linear-transformation category. Eilenberg, may he rest in peace, once summed up the dilemma: are you talking about Banach spaces? or about Banachable spaces? (Banachable spaces are topological vector spaces, complete in their (uniform) topology, whose topology can come from a norm.) In the latter case, continuous linear transformations are all there is. And if you want invertible bounded linear transformations to be isomorphisms, Banachable spaces is all you can be capturing. But products, as topological vector spaces, of too many Banachable spaces are no longer Banachable; and coproducts ... are no longer even uniformly complete. So if you want to talk about Banach spaces, with the expected L-1 style weak products as coproducts and the expected L-oo style products as products, then you are obviously focussed on the norms, and you've got to be focussed on maps that don't increase the norms, for otherwise you're only focussing on the Banachable aspect of the topological vector spaces underlying your Banach spaces. As to other remarks: > If I were talking with John Baez, and he had just said > that he was accepting all bounded linear maps as morphism, > then I /would/ dare call an invertible bounded linear map an isomorphism, > because it would in fact /be/ an isomorphism in that category. And I'd understand he was interested only in Banachable TVSes, and not actually in Banach spaces. > (But in a general context, I would call /only/ isometries isomorphisms, > because otherwise people might get confused about what I meant!) This would tell me you're interested not merely in Banachable spaces, but in actual Banach spaces. > I say this just to remind us that we're discussing which category is /best/, > not which category is /correct/. Both categories (Banach spaces -- with norm-non-increasing maps, and Banachable spaces, with continuous linear transformations) are useful categories. But even their objects are different, not just the maps allowed between two particular Banach spaces. > (There's the additional matter that the "unit disk functor" > isn't a functor at all if all bounded linear maps are morphisms, > but the correspondence is stronger than that.) That's because Banachable spaces have no unit disks -- it's not that "the 'unit disk functor' isn't a functor at all," it's that there isn't even a CANDIDATE for object-function of a putative unit disk functor! Nonetheless, both categories -- Ban , and Banachable -- though far from equivalent, have their uses. And Ban , though not monadic over Sets via its unit disk functor, as Mike Barr has correctly pointed out, IS a full reflective subcategory of the category of algebras over the monad for that unit-disk functor; in that regard, it somewhat resembles the category of torsion-free abelian groups (likewise not monadic, yet fully reflective in the category of algebras for its underlying set functor, viz., in Ab.Gps). Hope these comments help. (*)PS: by a fluke, l_1(n) and l_oo(n) can be made isometric, for n=2: send (1, 0) in l_1(2) to (1, 1) in l_oo(2), and send (0, 1) in l_1(2) to (-1, 1) in l_oo(2), and extend by linearity. This is the linear map R^2 --> R^2 that rotates by 45 degrees and then multiplies by square.root(2) , and it carries the l_1 unit diamond onto the l_oo unit square. I don't think anything like this can work for exhibiting isometries between l_1(2) and l_p(2) for any other p, and I don't think anything like this can work for l_1(n) and l_oo(n) for any n > 2. But enough for now. -- F. From rrosebru@mta.ca Sat Oct 25 08:48:40 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Oct 2003 08:48:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ADMpj-0001uy-00 for categories-list@mta.ca; Sat, 25 Oct 2003 08:43:51 -0300 X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Fri, 24 Oct 2003 21:03:52 -0400 (EDT) From: Michael Barr X-X-Sender: barr@triples.math.mcgill.ca To: Categories list Subject: categories: Upgrade of diagxy 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 just posted a minor upgrade of diagxy. At the suggestion of a graduate student at Kent State named Gerd Zeibig, I have added a new feature that allows to identify nodes of a diagram with short identifiers and then draw arrows between those nodes. He had implemented this and asked me what I thought. I found a simpler way than he (he had defined over 100 new counters and the number of counters that tex allows is 256, so they are considered a rare commodity; my code uses only 2 new counters). This kind of reintroduces the matrix mode except with absolute positioning of the vertices. The results are found, as usual, in ftp.math.mcgill.ca/pub/barr/diagxy.zip. From rrosebru@mta.ca Sun Oct 26 11:10:00 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Oct 2003 11:10:00 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ADmSV-0001jj-00 for categories-list@mta.ca; Sun, 26 Oct 2003 11:05:35 -0400 Date: Sat, 25 Oct 2003 22:56:32 +0100 From: Doron Peled TMP ACCT To: categories@mta.ca Subject: categories: CFP: Computer Aided Verification (CAV) 2004, Boston, MA Message-ID: <20031025225632.A11641@gem.dcs.warwick.ac.uk> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.2.5i Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 38 CALL FOR PAPERS COMPUTER AIDED VERIFICATION (CAV) 16th International Conference July 13 -- 17 , 2004, Omni Parker House Hotel, Boston, USA http://www.dcs.warwick.ac.uk/CAV Aims and Scope: CAV'04 conference is the 16th in a series dedicated to the advancement of the theory and practice of computer-assisted formal analysis methods for software and hardware systems. The conference covers the spectrum from theoretical results to concrete applications, with an emphasis on practical verification tools and the algorithms and techniques that are needed for their implementation. The proceedings of the conference will be published in the Springer-Verlag Lecture Notes in Computer Science series. Sample topics of interest include: o Algorithms and tools for verifying models and implementations o Deductive, compositional, and abstraction techniques for verification o Modeling and specification formalisms o Program analysis and software verification o Testing and runtime analysis based on verification technology o Applications and case studies o Verification in industrial practice Special Events: CAV'04 is colocated with the International ACM Symposium on Software Testing and Analysis, ISSTA'04. Invited speakers for CAV'04 are David Harel (Weizmann Institute, plenary speaker for the joint CAV-ISSTA session), Mary Jean Harrold (Georgia Institute of Technology), and Tom Reps (University of Wisconsin). CAV will be preceded by an invited tutorial on processor verification by Randy Bryant (Carnegie Mellon University), David Dill (Stanford University), and Warren Hunt (University of Texas, Austin). The conference will be followed by special workshops. Paper submission: There are two categories of submissions: A. Regular papers. Submissions, not exceeding thirteen (13) pages using Springer's LNCS format, should contain original research, and sufficient detail to assess the merits and relevance of the contribution. For papers reporting experimental results, authors are strongly encouraged to make their data available with their submission. Simultaneous submission to other conferences with proceedings or submission of material that has already been published elsewhere is not allowed. B. Tool presentations. Submissions, not exceeding four (4) pages using Springer's LNCS format, should describe the implemented tool and its novel features. A demonstration is expected to accompany a tool presentation. Papers describing tools that have already been presented in this conference before will be accepted only if significant and clear enhancements to the tool are reported and implemented. Information concerning the procedure for submissions is available on the conference home page http://www.dcs.warwick.ac.uk/CAV Important dates: Paper submission (strict): January 23, 2004 Notification of acceptance/rejection: March 29, 2004 Final version due: April 30, 2004 Program Chairs: Rajeev Alur, University of Pennsylvania alur@cis.upenn.edu Doron A. Peled, The University of Warwick doron@dcs.warwick.ac.uk Program Committee: Rajeev Alur, U Pennsylvania David Basin, ETH Zurich Armin Biere, ETH Zurich Randy Bryant, CMU Dennis Dams, Bell Labs Luca de Alfaro, UC Santa Cruz David Dill, Stanford U Allen Emerson, UT Austin Kousha Etessami, U of Edinburgh Steven German, IBM Rob Gerth, Intel Mike Gordon, U of Cambridge Aarti Gupta, NEC Labs Klaus Havelund, NASA Ames Holger Hermanns, Saarland U Pei-Hsin Ho, Synopsis Alan Hu, U of British Columbia Bengt Jonsson, Uppsala U Andreas Kuehlman, Cadence Labs Salvatore La Torre, U of Salerno Oded Maler, Verimag Pete Manolias, Georgia Tech Ken McMillan, Cadence Labs Anca Muscholl, U of Paris 7 Chris Myers, U of Utah Doron Peled, U of Warwick Fabio Somenzi, U of Colorado Amir Pnueli, NYU Shaz Qadeer, Microsoft Research Jun Sawada, IBM Frits Vaandrager, U of Nijmegen Pierre Wolper, U of Liege Sergio Yovine, Verimag\\[1mm] Steering Committee: Edmund M. Clarke, CMU Mike Gordon, U of Cambridge Robert P. Kurshan, Cadence Amir Pnueli, NYU From rrosebru@mta.ca Mon Oct 27 13:40:22 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Oct 2003 13:40:22 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AEBBh-0002cU-00 for categories-list@mta.ca; Mon, 27 Oct 2003 13:29:53 -0400 Message-ID: <3F9D49F6.5090708@itu.dk> Date: Mon, 27 Oct 2003 17:38:14 +0100 From: Thomas Hildebrandt X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: CFP: 10th Conference on Category Theory and Computer Science (CTCS 2004) and Summer School Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Virus-Scanned: by amavisd-new at itu.dk Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 39 10th CONFERENCE ON CATEGORY THEORY AND COMPUTER SCIENCE (CTCS'04) AUGUST 12-14, 2004 AND SUMMER SCHOOL AUGUST 9-11, 2004 IT University of Copenhagen (ITU) Copenhagen, Denmark FIRST CALL FOR PAPERS CTCS'04 is the 10th Conference on Category Theory and Computer Science. The purpose of the conference series is the advancement of the foundations of computing using the tools of category theory. The emphasis is upon applications of category theory, but it is recognized that the area is highly interdisciplinary. Typical topics of interest include, but are not limited to, category-theoretic aspects of the following: coalgebras and computing concurrent and distributed systems constructive mathematics declarative programming and term rewriting domain theory and topology foundations of computer security linear logic modal and temporal logics models of computation program logics, data refinement, and specification programming language semantics type theory Previous meetings have been held in Guildford (Surrey), Edinburgh (twice), Manchester, Paris, Amsterdam, Cambridge, S. Margherita Ligure (Genova), and Ottawa. The proceedings of the conference will be published as a special issue of ENTCS (Electronic Notes in Theoretical Computer Science). Invited Speakers: Francois Bergeron Martin Hyland Robin Milner Andrew Pitts Thomas Streicher SUMMER SCHOOL Inspired by the success of the graduate student preconference of CTCS'02 in Ottawa, the CTCS of this year will have a similar event: A summer school from August 9-11. The goal is to prepare students - both graduate and undergraduate, with basic knowledge of category theory - for CTCS, through mini-courses in the basic areas underlying some of the fields of the conference. We anticipate offering courses in among others the following areas: Coalgebras Game Semantics Categorical Models for Concurrency Operational Semantics in Concurrency PROGRAMME COMMITTEE Lars Birkedal, Chair (IT University of Copenhagen) Marcelo Fiore (University of Cambridge) Masahito Hasegawa (Kyoto University) Bart Jacobs (University of Nijmegen) Ugo Montanari (University of Pisa) Valeria de Paiva (Palo Alto Research Center) Dusko Pavlovic (Kestrel Institute) John Power (University of Edinburgh) Edmund Robinson (University of London) Peter Selinger (University of Ottawa) ORGANIZING COMMITTEE E. Moggi, Chair, (Genova) S. Abramsky (Oxford) P. Dybjer (Chalmers) B. Jay (Sydney) A. Pitts (Cambridge) LOCAL ORGANIZING COMMITTEE C. Butz T. Hildebrandt A.L. Moerk SUBMISSION OF PAPERS Papers should be submitted, preferably in electronic form, to ctcs04@itu.dk. Papers are limited to 15 pages, and must be submitted in dvi, postscript, or pdf format, possibly gzipped and/or uuencoded, or sent as a standard email attachment. All submissions must be received by April 9th, 2004. If you cannot submit your paper electronically, please contact the program chair at ctcs04@itu.dk. IMPORTANT DATES April 9th, 2004: Submission deadline June 1st, 2004: Notification of authors of accepted papers July 1st, 2004: Revised Papers Due CONFERENCE HOMEPAGE Updated information is available from http://www.itu.dk/research/theory/ctcs2004 SPONSORSHIP The conference and summer school are sponsored by the FIRST graduate school (www.first.dk) and the Theory Department at the IT University of Copenhagen (www.itu.dk/English/research/theory/ From rrosebru@mta.ca Mon Oct 27 17:01:14 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Oct 2003 17:01:14 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AEESY-0000zi-00 for categories-list@mta.ca; Mon, 27 Oct 2003 16:59:30 -0400 Date: Mon, 27 Oct 2003 14:11:55 -0600 (CST) From: Peter May To: categories@mta.ca Subject: categories: Announcement: Workshop: n-Categories: Foundations and Applications. Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 There will be a two week workshop, June 7 - 18, 2004, on the topic: n-Categories: Foundations and Applications. This is the 2004 Summer Program of the Institute for Mathematics and its Applications, at the University of Minnesota. Further information may be found at: http://www.ima.umn.edu/categories Our main goal is to make progress in sorting out the basic foundational issues concerning definitions of "weak n-category", but the main areas of current and potential application will also be discussed. We have generous but limited funding from the IMA, and we have applied to the NSF for supplemental funding. Those interested are invited to register on-line at: http://www.ima.umn.edu/docs/reg_form1.html We will do the best that we can to find (partial) support. Graduate students and postdocs who are seriously interested in learning about this subject and who have some background in category theory and/or categorical homotopy theory are especially invited to register. John Baez and Peter May From rrosebru@mta.ca Fri Oct 31 11:37:57 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Oct 2003 11:37:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AFbFf-00004w-00 for categories-list@mta.ca; Fri, 31 Oct 2003 11:31:51 -0400 Date: Thu, 30 Oct 2003 12:32:16 GMT Message-Id: <200310301232.h9UCWGi07345@cuillin.inf.ed.ac.uk> To: categories@mta.ca From: Alex Simpson Subject: categories: LICS 2004 - Call for Papers Reply-To: als+lics-junk@dcs.ed.ac.uk Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 41 CALL FOR PAPERS Nineteenth Annual IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS 2004) July 14th - 17th, 2004, Turku, Finland http://www.lfcs.informatics.ed.ac.uk/lics/ The LICS Symposium is an annual international forum on theoretical and practical topics in computer science that relate to logic in a broad sense. We invite submissions on that theme. Suggested, but not exclusive, topics of interest for submissions include: automata theory, automated deduction, categorical models and logics, concurrency and distributed computation, constraint programming, constructive mathematics, database theory, domain theory, finite model theory, proof theory, formal aspects of program analysis, formal methods, hybrid systems, lambda and combinatory calculi, linear logic, logical aspects of computational complexity, logics in artificial intelligence, logical representation of knowledge, logics of programs, logic programming, modal and temporal logics, model checking, programming language semantics, reasoning about security, rewriting, specifications, type systems and type theory, and verification. Important Dates: Authors are required to submit electronically a paper title and a short abstract of about 100 words before submitting the extended abstract of the paper. Titles & Short Abstracts Due : January 26, 2004 Extended Abstracts Due : February 2, 2004 Author Notification : March 27, 2004 Camera-ready Papers Due : April 25, 2004 All deadlines are firm; late submissions will not be considered. Detailed information about electronic paper submission will be posted at the LICS website. Submission Instructions: Extended abstracts must be submitted electronically in the IEEE Proceedings two-column camera-ready format. Each abstract must be in English and provide sufficient detail to allow the program committee to assess the merits of the paper. It should begin with a succinct statement of the issues, a summary of the main results, and a brief explanation of their significance and relevance to the conference and to computer science, all phrased for the non-specialist. Technical development directed to the specialist should follow. References and comparisons with related work should be included. Extended abstracts may be no longer than 10 pages including references, and must be formatted in the IEEE Proceedings two-column camera-ready style (IEEE style files will be accessible from the LICS website). If necessary, detailed proofs of technical results can be included in a clearly-labelled appendix in the same two-column format following the 10-page extended abstract. This material may be read at the discretion of the program committee. Extended abstracts not conforming to the above requirements concerning format and length may be rejected without further consideration. The results must be unpublished and not submitted for publication elsewhere, including the proceedings of other symposia or workshops. All authors of accepted papers will be expected to sign copyright release forms. One author of each accepted paper will be expected to present it at the conference. Short Presentations: LICS 2004 will have a session of short (5--10 minutes) presentations. This session is intended for descriptions of work in progress, student projects, and relevant research being published elsewhere; other brief communications may be acceptable. Submissions for these presentations, in the form of short abstracts (1 or 2 pages long), should be entered at the LICS 2004 submission site between March 27th and April 4th, 2004. Authors will be notified of acceptance or rejection by April 17th, 2004. Kleene Award for Best Student Paper: An award in honor of the late S.C. Kleene will be given for the best student paper, as judged by the program committee. For a submission to be eligible, the research presented in the paper must have been carried out while all authors were full-time students. The program committee may decline to make the award or may split it among several papers. Affiliated Workshops: As in previous years, there will be a number of workshops affiliated with LICS 2004; information will be posted at the LICS website. Program Chair: Harald Ganzinger MPI Informatik, Saarbruecken, Germany http://www.mpi-sb.mpg.de/~hg/ Program Committee: Rajeev Alur, U. of Pennsylvania Andrew Appel, Princeton U. Albert Atserias, UPC, Barcelona Franz Baader, Dresden U. Samuel Buss, U. of California, San Diego Roberto Di Cosmo, U. de Paris VII Gilles Dowek, Ecole Polytechnique, Paris Harald Ganzinger, MPI, Saarbruecken (chair) Martin Hofmann, LMU Muenchen Achim Jung, U. of Birmingham Leonid Libkin, U. of Toronto Kim Larsen, Aalborg U. Rocco de Nicola, U. di Firenze Damian Niwinski, Warsaw U. Prakash Panangaden, McGill U., Montreal Albert Rubio, UPC, Barcelona Vitaly Shmatikov, SRI International Moshe Vardi, Rice U., Houston Helmut Veith, TU Wien Andrei Voronkov, U. of Manchester Conference Chair: Lauri Hella Department of Math., Stat., and Phil. Kanslerinrinne 1 33014 University of Tampere, Finland Email: lauri.hella@uta.fi Workshops Chair: Phil Scott, U. of Ottawa Email: phil@site.uottawa.ca Publicity Chair: Alex Simpson, U. of Edinburgh Email: Alex.Simpson@ed.ac.uk General Chair: Phokion G. Kolaitis, UC Santa Cruz Email: kolaitis@cse.ucsc.edu Organizing Committee: S. Abramsky, A. Broder, E. Clarke, A. Felty, U. Furbach, H. Ganzinger, H. Gabow, J. Halpern, L. Hella, U. Kohlenbach, P. Kolaitis (chair), D. Leivant, G. Longo, H. Mairson, A. Middeldorp, J. Mitchell, M. Nielsen, P. Panangaden, G. Plotkin, P. Scott, R. Shore, A. Simpson, I.A. Stewart. Advisory Board: Y. Gurevich, C. Kirchner, D. Kozen, U. Martin, L. Pacholski, V. Pratt, A. Scedrov, M.Y. Vardi, G. Winskel. Sponsorship: The symposium is sponsored by the IEEE Technical Committee on Mathematical Foundations of Computing in cooperation with the Association for Symbolic Logic, and the European Association for Theoretical Computer Science. Collocated events: ICALP'04 will be collocated with LICS'04; for details see http://www.math.utu.fi/ICALP04/. From rrosebru@mta.ca Fri Oct 31 11:37:57 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Oct 2003 11:37:57 -0400 Message-ID: <3F9FEEEE.9070601@bath.ac.uk> Date: Wed, 29 Oct 2003 16:46:38 +0000 From: "David J. Pym " To: categories Subject: categories: Paper Announcement: On the Geometry of Interaction for Classical Logic Content-Type: text/plain; charset=3DISO-8859-1; format=3Dflowed Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 42 The following may be of interest to readers of this list: Carsten F=FChrmann and David Pym, ''On the Geometry of Interaction for Classical Logic''. Abstract. It is well-known that weakening and contraction cause na=EFve categorical models of the classical sequent calculus to collapse to Boolean lattices. In a previous paper, summarized herein, we provided sound and complete models that avoid this collapse by interpreting cut-reduction by a partial order between morphisms. In this article, we provide concrete examples of such models, based on geometry of interaction and data-flow. Our models provide detailed analyses of the relationships between negation, weakening, and contraction under cut-reduction. Manuscript available at http://www.cs.bath.ac.uk/~pym/classical-GoI.pdf We'd be very pleased to receive comments. This paper follows on from Carsten F=FChrmann and David Pym, "Order-enriched Categorical Models of the Classical Sequent Calculus". Abstract. It is well-known that weakening and contraction cause na=EFve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations. This paper is in submission. Manscript available at http://www.cs.bath.ac.uk/~pym/oecm.pdf Again, we'd be very pleased to receive comments. Regards, David --=20 Prof. David J. Pym Telephone: +44 (0)1 225 38 3246 Professor of Logic & Computation Facsimile: +44 (0)1 225 38 3493 University of Bath Email: d.j.pym@bath.ac.uk Bath BA2 7AY, England, U.K. Web: http://www.bath.ac.uk/~cssdjp