November 1995 - Categories - categories.org.au

local set theory
by Andreas Blass 28 Nov '95

28 Nov '95

This is in response to Michelbrink's question, posted on the network by Koslowski, about the footnote on page 86 of Bell's "Toposes and Local Set Theories." I agree with Michelbrink that the footnote is incorrect, and I pointed out the problem in my review of the book for Mathematical Reviews (90e:18002 --- see Reviewer's Remark (3) near the end of the review). Andreas Blass

1 0

local_set_theory
by koslowj＠iti.cs.tu-bs.de 28 Nov '95

28 Nov '95

The following question concerning the book "Toposes and Local Set Theories" by J. L. Bell was posed by Markus Michelbrink, a logician from the University of Hannover. I am just forwarding it to the list. [Remark. The book has been reviewed for the Journal of Symbolic Logic by G. C. Wraith shortly after it appeared. The reviewer comments on the author's unfortunate bias towards locally small categories.] Markus has a problem with the footnote on p. 86. For a local set theory S, Bell proceeds to define a category C(S). The footnote says: "Strictly speaking, for this to be the case we need also show that the collection of S-maps between any pair of S-sets forms an (intuitive) set. Actually, this follows easily from the assumption made initially that the collection of function symbols of \cal L of a given signature forms a set: we leave the proof to the reader." Markus' argument, why this may not be so obvious, is as follows: Form a local language \cal L with a proper class of ground type symbols, and fix two arbitrary ground type symbols A' and B'. For every ground type symbol A let there be exactly one function symbol f_A:PA --> P(A' x B'). The function symbols for any signature then form a set, namely either the empty set, or a singleton. Now for any ground type symbol A there is a closed term f_A({x_A:true}) of type P(A' x B'), and for distinct ground type symbols A,B these closed terms are distinct. Hence we have a proper class of closed terms of type P(A' x B'). Consider the local set theory S that is generated by the proper class of sequents <\emptyset, f_A({x_A:true})\in {x_B':true}^{x_A':true}> where A is a ground type symbol. [I.e., we require the f_A-image of A to be a function from A' to B'.] S induces an equivalence relation ~S via X ~S Y iff |-_S X=Y This equivalence relation partitions the class of closed terms \tau of type P(A' x B') with |-_S \tau \in {x_B':true}^{x_A':true}, i.e., those terms which represent functions from A' to B' in S. Question 1: why should this partition have a *set* of representatives? Question 2: can there be any distinct ground type sumbols A and B such that f_A({x_A:true} and f_B({x_B:true} are S-equivalent? [I hope the ASCII transcriptions and the occasional quasi-TeX are self-explanatory. Whoever has access to the book should be able to follow the argument.] Best regards, -- J"urgen Koslowski Institut f\"ur Theoretische Informatik TU Braunschweig email: koslowj(a)iti.cs.tu-bs.de

2 1

adjunction question
by Andrew Ensor 27 Nov '95

27 Nov '95

I was wondering if anyone could provide me with some information on the following: Suppose <F,G,eta,epsilon>:A -> B is an adjunction and let C denote the collection of all possible finite horizontal compositions of F,G (as identity natural transformations),eta, and epsilon. So, for example the horizontal composite epsilon.F.eta is a natural transformation in C. I am interested in knowing more about vertical compositions using members of C. For example the vertical composite of epsilon.F with F.eta is epsilon.F.eta, and the vertical composite of epsilon.F.G.F with epsilon.F.eta.eta is epsilon.epsilon.F.eta.eta. Must C be closed under all possible vertical compositions? Any assistance or reference would be appreciated. Sincerely, Andrew Ensor.

1 0

Toronto Spring Meeting
by Walter Tholen 27 Nov '95

27 Nov '95

TORONTO SPRING MEETING We are planning to host a meeting on Category Theory on the weekend of April 13/14, 1996, at York University, Toronto. It is our intention to follow the style of the Montreal Oktoberfest, although we shall also have a couple of invited speakers. Hence we call for contributed talks from all areas of mathematics (incl. theoretical computer science and physics) using category theory, and especially encourage graduate students to participate. There will be a registration fee sufficient to cover expenses for the Saturday night dinner. Graduate students will pay a reduced fee. More detailed information will be given in the second announcement to be sent in January. Meanwhile, if you have any questions, please write to yorkcat(a)mathstat.yorku.ca We hope to see you here in April! Xiaomin Dong Sandro Fusco Joan Wick Pelletier Walter Tholen -- Walter Tholen Department of Mathematics and Statistics York University, North York, Ont., Canada M3J 1P3 tel. (416) 736 5250 or 736 2100, ext. 33918 fax. (416) 736 5757 http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html

1 0

60th PSSL
by Dr. P.T. Johnstone 20 Nov '95

20 Nov '95

PERIPATETIC SEMINAR ON SHEAVES AND LOGIC 60th meeting - Preliminary announcement The sixtieth meeting of the PSSL will be held in Cambridge over the weekend of 27-28 January 1996. The seminar welcomes talks using or addressing category theory or logic, either explicitly or implicitly, in the study of any aspect of mathematics or science. If you wish to attend, please return the form below to Peter Johnstone, either electronically or by ordinary mail. We shall attempt to arrange accommodation in College guest rooms (or, failing that, in local guest houses) for those who request it by 17 January 1996. A provisional programme will be circulated in the week before the meeting. Martin Hyland Peter Johnstone P.S. -- This announcement is being sent out via the `categories' mailing list, and in hard copy to a number of people who are known not to receive electronic mail. If you know of anyone who would like to come to the meeting, but who has not received the announcement, please feel free to copy it to him/her. ................................................................................ Please return to Dr P.T. Johnstone, DPMMS, 16 Mill Lane, Cambridge CB2 1SB (e-mail: ptj(a)pmms.cam.ac.uk) by 17 January 1996. I intend to come to the 60th meeting of the PSSL * I should like to give a talk entitled lasting about minutes. * Please reserve accommodation for Friday/Saturday/Sunday night(s). Name : Address : E-mail : * Delete whatever does not apply.

1 0

Abstracts "Descent Theory", Oberwolfach '95
by Walter Tholen 10 Nov '95

10 Nov '95

Anybody who is interested in getting the files for the Abstracts of talks given at the meeting on "Geometric and Logical Aspects of Descent Theory" in Oberwolfach (September '95) may access these by contacting my home page on the WWW (address below) and clicking on the respective item. Participants will receive a hardcopy of these abstracts automatically, sent to them by the Institute. -- Walter Tholen Department of Mathematics and Statistics York University, North York, Ont., Canada M3J 1P3 tel. (416) 736 5250 or 736 2100, ext. 33918 fax. (416) 736 5757 http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html

1 0

Book Announcement
by Walter Tholen 10 Nov '95

10 Nov '95

The book "Categorical Structure of Closure Operators" by D. Dikranjan and W. Tholen has appeared in the "Mathematics and Its Applications" series of Kluwer Academic Publishers (Dordrecht, Boston, London 1995; ISBN 0-7923-3772-7). Abstract: The book provides a comprehensive categorical theory of closure operators, with applications to topological and uniform spaces, groups, R-modules, fields and topological groups, as well as to partially ordered sets and graphs. In particular, closure operators are used to give solutions to the epimorphism and cowellpoweredness problem in many concrete categories. The material is illustrated with many examples and exercises, and open problems are formulated in order to stimulate further research. -- Walter Tholen Department of Mathematics and Statistics York University, North York, Ont. Canada M3J 1P3 tel. (416) 736 5250 fax. (416) 736 5757

1 0

Robert Thomason
by James Stasheff 08 Nov '95

08 Nov '95

you may already have recived this Jim Stasheff jds(a)math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 May 15 - August 15: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 ---------- Forwarded message ---------- Date: Mon, 06 Nov 1995 20:13:25 EST From: DONALD M. DAVIS <dmd1(a)Lehigh.EDU> Subject: Robert Thomason Mon, 6 Nov 1995 16:30:40 -0500 From: karoubi(a)math.wisc.edu To: dmd1(a)lehigh.edu Subject: Thomason I just received the following sad message from the secretary of Paris 7 : >Date: Mon, 6 Nov 1995 15:36:12 +0100 >J'ai la penible charge de vous faire part du deces de Robert THOMASON. >Cette nouvelle m'a ete communiquee par le Commandant du poste de police >"Picpus Bercy". Il est decede hier (jour de son anniversaire) des suites de sa >maladie. Son corps a ete transfere a l'Institut medico legal. > > >________________________________ >Liliane Barenghi >Institut de Mathematiques de Jussieu-UMR CNRS 9994 >Equipe "Theories Geometriques" >Universite Paris 7 - Case 7012 >2, Place Jussieu, 75251 Paris cedex 05 - FRANCE >Tel 33 (1) 44 27 69 32 ; Fax 33 (1) 44 27 63 66

1 0

announcement
by Michael Makkai 08 Nov '95

08 Nov '95

This is to announce a research monograph, First Order Logic with Dependent Sorts, with Application to Category Theory by M. Makkai (McGill Univ.) (Preliminary version) Abstract J. Cartmell [2] introduced a syntax of variable types, which I call dependent sorts, for the purposes of presenting generalized algebraic theories. Cartmell's syntax was "abstracted from ... Martin-Lof type theory". I add propositional connectives and quantification to a simplified version of Cartmell's syntax, to obtain what I call First-Order Logic with Dependent Sorts (FOLDS). The simplification consists in the exclusion of operation symbols, and a severe restriction on the use of equality. Quantification is subject to the natural restriction that a quantifier "for all x " or "there is x " cannot be applied if in the resulting formula there is a free variable whose sort depends on x . An important special case of FOLDS was introduced by G. Blanc [1] for the purpose of characterizing first-order formulas in the language of categories that are invariant under equivalence of categories. P. Freyd's earlier characterization [3], although not explicitly coached in an instance of FOLDS, is essentially the same as Blanc's. A. Preller [7] gives an explicit comparison of Blanc's and Freyd's contexts. The main aim of the present work is to extend Blanc's and Freyd's characterization from statements about categories to statements about more complex categorical structures. A similarity type for structures for FOLDS is given by a one-way category of sort-forming symbols and relation symbols. One-way categories were isolated by F. W. Lawvere [4], and were subsequently shown by him to be relevant for the generalized sketch-syntax of [5]. The basic metatheory of FOLDS is a simple extension of that of ordinary multisorted first-order logic. There are simply formulated complete formal systems for both classical and intuitionistic FOLDS, with Kripke-style completeness for the intuitionistic case. The systems use entailments-in-contexts as their basic units; contexts are systems of typings of variables as usual in Martin- Lof-style systems. We have Gentzen-style systems admitting cut- elimination. Natural forms of Craig Interpolation and Beth Definability are true in both classical and intuitionistic FOLDS. Much of the basic metatheory is done through the formalism of appropriate fibrations (hyper-doctrines). The main new concept is a notion of equivalence of structures for FOLDS. Equivalent structures satisfy the same sentences of FOLDS. The main general result is that conversely, first order properties invariant under equivalence are expressible in FOLDS. A stronger version of the result takes the form of an interpolation theorem. Two categories are equivalent in the usual sense iff they are equivalent as structures for FOLDS. This connection between the categorical concept of equivalence and FOLDS-equivalence persists for more complex categorical structures such as (1) a diagram of categories, functors and natural transformations, or (2) a bicategory, or (3) a diagram of bicategories, etc., if we pass to "ana"-versions of the concepts of functor, bicategory, etc.; the latter were introduced in [6]. Every functor, bicategory, etc., has its so-called saturation, a simply defined saturated anafunctor, saturated anabicategory, etc., respectively. A property written in FOLDS of the saturation is a particular, "good", kind of first- order property of the original. Applications of the foregoing give syntactical characterizations of properties invariant under equivalence in the contexts mentioned. E.g., a first-order property of a variable bicategory is invariant under biequivalence iff it is expressible in FOLDS as a property of the saturated anabicategory canonically associated with the given bicategory. References [1] G. Blanc, Equivalence naturelle et formules logiques en theorie des categories. Archiv math. Logik 19 (1978), 131-137. [2] J. Cartmell, Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 (1986), 209-243. [3] P. Freyd, Properties invariant within equivalence types of categories. In: Algebra, Topology and Category Theories, ed. A. Heller and M. Tierney, Academic Press, New York, 1976; pp. 55-61. [4] F. W. Lawvere, More on graphic toposes. Cah. de Top. et Geom. Diff. 32 (1991), 5-10. [5] M. Makkai, Generalized sketches as a framework for completeness theorems. To appear in J. Pure and Applied Algebra. [6] M. Makkai, Avoiding the axiom of choice in general category theory. To appear in J. Pure and Applied Algebra. [7] A. Preller, A language for category theory in which natural equivalence implies elementary equivalence of models. Zeitschrift f. math. Logik und Grundlagen d. Math. 31 (1985), 227-234. (End of Abstract) A manuscript copy of this work was placed on exhibit at the Category Theory Meeting in Halifax, 1995; my talk was about the same subject. I promised to send copies to people who signed up for them. I would appreciate if those who have been waiting for this, and now find this announcement, would let me know. I will try to contact those on the list with me who do not respond. The manuscript has been placed on anonymous ftp at triples.math.mcgill.ca in directory /pub/makkai/folds in several files. You may consult the README file in the directory /pub/makkai.

1 0

Electronic supplement to ctcs
by Michael Barr 03 Nov '95

03 Nov '95

The electronic supplement is now in my ftp directory under the name that is given in ctcs, namely ctcs.elec.supp.??. There are actually six forms: {ps,dvi}{ ,gz,zip}.

1 0