February 1994 - Categories - categories.org.au

new book
by Jiri Rosicky 18 Feb '94

18 Feb '94

Preliminary announcement The following book will be available shortly: Locally Presentable and Accessible Categories J.Adamek, J.Rosicky Cambridge University Press London Mathematical Society Lecture Notes Series 189 ISBN 0 521 42261 2 330 pp. Abstract: The concepts of a locally presentable category and an accessible category have turned out to be useful in formulating connections between universal algebra, model theory, logic and computer science. The aim of this book is to provide an exposition of both the theory and the applications of these categories at a level accessible to graduate students. Firstly the properties of lambda-presentable objects, locally lambda-presentable categories, and lambda-accessible categories are discussed in detail, and the equivalence of accessible and sketchable categories is proved. The authors go on to study categories of algebras and prove that Freyd's essentially algebraic categories are precisely the locally presentable categories. In the final chapters they treat some topics in model theory and some set theoretical aspects. Contents: 0. Preliminaries, 1. Locally presentable categories, 2. Accessible categories, 3. Algebraic categories, 4. Injectivity classes, 5. Categories of models, 6. Vopenka's principle, Appendix: Large cardinals, Open problems.

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Re: cantor-bernstein
by Dusko Pavlovic 14 Feb '94

14 Feb '94

Hi. I was away for a while and read the postings from the last couple of months only now. Hence this delayed reaction on Peter Freyd's comments from January on the Cantor-Schroeder-Bernstein theorem. > I trust that CSB holds for any boolean topos. (Anyone want to confirm?) The other way around, the validity of CSB in a topos ALMOST implies that the topos is boolean. "Almost" means a slightly stronger version needs to be considered: (CSB*) if f:A->B and g:B->A are monics then there is an iso h:A->B such that for every x in A holds h(x) = f(x) or h(x) = g^{-1}(x) The usual set-theoretical proof of CSB (which goes through in boolean toposes) actually yields (CSB*). In return, (CSB*) implies booleanness. We assume that every topos has an object of natural numbers. PROPOSITION. A topos is boolean if and only if it satisfies (CSB*). PROOF of the "if" part: Let X be a subobject of Y. We use (CSB*) to construct its complement. Define A = YxN B = X+(YxN) (where N is the object of natural numbers). Let f:A-->B be the inclusion, and g:B-->A the unique arrow which maps X into the first copy of Y in A, while it takes the copies of Y contained in B into the second one, the third one and so on. (Formally, g is defined using the universal property of coproducts, and the structure of N.) It is clear that both f and g are monics. By (CSB*), there is a bijection h:A-->B with h(x)=f(x) or h(x)=g^{-1}(x). This means that the union of C = {z | h(z)=f(z)} and D = {z | h(z)=g^{-1}(z)} covers A. On the other hand, the definitions of f and g imply that C and D are disjoint. So they are complements in A. By pulling back C and D along the inclusion of Y iso Yx{0} in A, we get two complementary subobjects of Y. The claim is now that the intersection (pullback) X' of D and Yx{0} is isomorphic to X -- so that the intersection of C and Yx{0} yields the complement of X in Y. First of all, each element <y,n> of D must surely be in the image of g. Thus, if <y,0> is in D, then y must be in X. So X' is contained in Xx{0}. The other way around, since h(C) is contained in f(C), the intersection in B of X and h(C) must be empty. Therefore, X must be contained in h(D): indeed, h is an iso, and the direct images h(C) and h(D) must be complementary in B. But h(D) is contained in g^{-1}(D). So X is in g^{-1}(D), i.e. g(X) is in D. Hence, Xx{0}=g(X) is contained in X'. Regards to all, Dusko --

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categories of graphs
by Prof R. Brown 05 Feb '94

05 Feb '94

We are starting to follow up the work of John Shrimpton' Bangor thesis, whose aim was the application of categorical methods in group theory, and was partially published as "Some groups related to the symmetry of a directed graph" JPAA 72 (1991) 303-318, and would be grateful for information on publications in this area. We know of the paper by Bumby and Latch, Internat J Math Sci 9 (1986) 1-16, P Ribenboim Algebraic structures on graphs, Alg. Univ. 16 (1983) 105-123; Bill Lawvere, Qualitative distictions between some toposes of generalized graphs, Cont. Math. 92, 261-299, (1989) and would be grateful for further information. The aim of the project is to use categorical methods to give insight into combinatorial problems, constructions and questions. A further aim was to understand Grothendieck's notion of "Teichmuller groupoid" referred to in "Pursuing stacks", in the sense that if what he asserted could be done was actually written up for surfaces, one should also be able to do it for graphs, (or vice versa), and so obtain presentations of a "symmetry groupoid" of a graph from some decomposition into smaller pieces. I have no idea how to do this. Grothendieck's observation in this area is quoted in my survey "From groups to groupoids" Bull London Math Soc. 19 (1987) 113-134. Part of this programme is "what should be the symmetry object of a given structure?". New answers should be of general interest, for obvious reasons. Shrimpton's work showed that these questions lead to that of, for example, inner automorphism of a directed graph, and the interpretation of this in graph theoretic terms. There should be more in this area. Ronnie Brown Prof R. Brown School of Mathematics, University of Wales, Bangor Gwynedd LL57 1UT UK

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54th PSSL: preliminary announcement (text)
by Edmund Robinson 04 Feb '94

04 Feb '94

Preliminary Announcement 54th PSSL ================================== The 54th meeting of the Peripatetic Seminar on Sheaves and Logic will be held over the weekend of the 26th--27th March 1994 at the University of Sussex. As usual, we welcome talks on topics in logic, category theory and related areas. Please publicise this meeting amongst your colleagues. Further information about the exact venue, arrangements for meeting, etc. will be distributed later. Information about travel, local accommodation, participants and talks offered will be placed in the Sussex anonymous ftp archive (ftp.cogs.sussex.ac.uk, subdirectory pub/pssl) as it becomes available. There is also a mailbox "pssl(a)cogs.sussex.ac.uk", specifically for conference mail. Use of this will considerably simplify our administration. We hope to arrange some sponsorship for the meeting, but participants should note that we may be forced to charge a small registration fee to cover the provision of food at the meeting. Return, preferably by email to "pssl(a)cogs.sussex.ac.uk", or to Edmund Robinson, School of Cognitive and Computing Sciences, University of Sussex, Brighton, BN1 9QH ENGLAND Tel: +44 273 678593 Fax: +44 273 671320 We look forward to seeing you in Sussex. The local organizers, Carolyn Brown, Chris Mulvey, Edmund Robinson, Gavin Wraith -------------------------------------------------------------------------- I would like to attend the 54th PSSL: Name: Return address: Email: [] I wish to give a talk entitled: of 20 / 30 / 45 minutes. (Please send an abstract if possible).

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54th PSSL: preliminary announcement (latex)
by Edmund Robinson 04 Feb '94

04 Feb '94

\documentstyle[12pt,a4]{article} \pagestyle{empty} \begin{document} \begin{center} {\Large\bf Preliminary Announcement 54th PSSL} \end{center} The 54th meeting of the {\em Peripatetic Seminar on Sheaves and Logic} will be held over the weekend of the 26th--27th March 1994 at the University of Sussex. As usual, we welcome talks on topics in logic, category theory and related areas. Please publicise this meeting amongst your colleagues. \medskip Further information about the exact venue, arrangements for meeting, etc{.} will be distributed later. Information about travel, local accommodation, participants and talks offered will be placed in the Sussex anonymous ftp archive ({\tt ftp.cogs.sussex.ac.uk}, subdirectory {\tt pub/pssl}) as it becomes available. There is also a mailbox {\tt pssl(a)cogs.sussex.ac.uk}, specifically for conference mail. Use of this will considerably simplify our administration. \medskip We hope to arrange some sponsorship for the meeting, but participants should note that we may be forced to charge a small registration fee to cover the provision of food at the meeting. \medskip Return, preferably by email to {\tt pssl(a)cogs.sussex.ac.uk}, or to \begin{tabbing} aaaaaa\=aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\=\kill \> Edmund Robinson, \\ \> School of Cognitive and Computing Sciences,\\ \> University of Sussex,\\ \> Brighton, BN1 9QH\\ \> ENGLAND\\[1ex] \> Tel: +44 273 678593\\ \> Fax: +44 273 671320 \end{tabbing} We look forward to seeing you in Sussex. The local organizers, \\ Carolyn Brown,\\ Chris Mulvey,\\ Edmund Robinson,\\ Gavin Wraith \medskip \noindent\hrulefill \medskip I would like to attend the 54th PSSL: \medskip \noindent Name \dotfill \noindent Return address \dotfill\\ \mbox{}\dotfill \noindent Email \dotfill \medskip \begin{itemize} \item[$\Box$] I wish to give a talk entitled \noindent\mbox{}\dotfill \noindent of 20 / 30 / 45 minutes. (Please send an abstract if possible). \end{itemize} \end{document}

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Is *Loc* regular?
by Todd Wilson 02 Feb '94

02 Feb '94

I have been wondering lately whether every regular monomorphism of the category of frames is universal (= pushout stable). But it is not hard to see (using, e.g., Kelly's "Monomorphisms, epimorphisms, and pull-backs" J.Austr.Math.Soc. 9, 1969) that this is equivalent to the question of whether the category of locales is a regular category. Does anyone have an answer to this question? --Todd Wilson

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Union Conference
by NIEFIELD, SUSAN 02 Feb '94

02 Feb '94

NINTH BIENNIAL UNION COLLEGE MATHEMATICS CONFERENCE Saturday and Sunday April 9-10, 1994 Union College Schenectady, New York PRINCIPAL SPEAKERS Zoltan Balogh Andre Joyal J. Peter May The Mathematics Department of Union College is pleased to announce its Ninth Biennial Mathematics Conference. In addition to the lectures by the principal speakers, there will be concurrent sessions in Algebraic Topology, Category Theory, and Set-Theoretic Topology. SCHEDULE A detailed schedule will be mailed in March. Friday Evening Reception 9:00 - 11:00 PM (Bailey Hall, 2nd Floor) Saturday 9:30 Registration, coffee, doughnuts 10:00 Parallel Sessions 11:30 Invited Address I: Andre Joyal 12:30 Lunch 2:00 Parallel Sessions 4:00 Coffee Break 4:30 Invited Address II: Peter May 6:00 Banquet and Party Sunday 9:00 Coffee and doughnuts 9:30 Invited Address III: Zoltan Balogh 10:30 Parallel Sessions 12:00 Lunch 1:00 Parallel Sessions 3:30 Conference ends REGISTRATION Please pre-register for the conference before April 1, by contacting one of the organizers listed below. There will be a $25 registration fee ($15 for students) payable by mail or when you register at the conference. Checks should be made out to "Union College Mathematics Conference". CONTRIBUTED TALKS If you wish to give a talk in one of the sessions, please contact one of the organizers of that session as soon as possible. Such talks should be 25 minutes long. ORGANIZERS Algebraic Topology: Brenda Johnson (johnsonb(a)unvax.union.edu) Gaunce Lewis (gaunce(a)ichthus.syr.edu) Category Theory: Susan Niefield (niefiels(a)gar.union.edu) Kimmo Rosenthal (rosenthk(a)gar.union.edu) Set-Theoretic Topology: Tim LaBerge (laberget(a)gar.union.edu) ACCOMODATIONS Blocks of rooms have been reserved at the Days Inn and the Holiday Inn in Schenectady. To obtain one of these rooms at the rates listed below, mention the Union College Mathematics Department when you call. Both are within easy walking distance of the campus. The Ramada Inn is also within walking distance of campus. 1. Holiday Inn (518) 393-4141 Single $55 Double $55 2. Days Inn (518) 370-0851 Single $45 Double $50 3. Ramada Inn (518) 370-7151 Single $63 Double $73 Department of Mathematics Union College Schenectady, NY 12308 (518) 388-6246

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Re: question on homotopies
by James Stasheff 02 Feb '94

02 Feb '94

> From: Michael Barr <barr(a)triples.Math.McGill.CA> > > In the category of chain complexes (over some abelian category) > it is clear that if you identify homotopic arrows, you will also > invert homotopy equivalences. Does anyone know if the converse is > true? If you invert homotopy equivalences, do you wind up identifying > homotopic arrows? What if you replace homotopy by homology? Does > inverting maps that induce isomorphism on homology have the effect of > identifying maps that induce the same map on homology? > > Michael > In the category of chain algebras, this has been well studied the point being that the inverse (up to homotpy) of a multiplicative homotopy equivalence is generally NOT mutiplicative the new category is known as DASH and has an extensive literature jim

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