Subject: Re: question about distributive categories From: koslowj@math.ksu.edu (Juergen Koslowski) Date: Mon, 2 Aug 93 14:29:26 CDT Vaughan Pratt's reply confirms my suspicion that for a symmetric monoidal closed category with tensor @ and exponential -o the Keisli category for the (strong) monad induced by the functor T that maps X to (X -o A) -o A (A fixed) is highly unlikely to be cartesian closed AND non-trivial. This probably is a well-known result. I'd appreciate pointers to the literature. -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: On calculuses of fractions Date: Tue, 3 Aug 93 12:37:58 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Can anyone supply a reference to the fact that if you add to the hypotheses of a calculus of right fractions the assumption that if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude that if the original category has all limits, so does the fraction category and the canonical functor to the fraction category preserves them. Michael Barr +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: New book: Theory and Formal Methods 1993 Date: Tue, 03 Aug 93 14:55:18 +0100 From: Mark Ryan Announcing: "Theory and Formal Methods 1993" Eds. Geoffrey Burn, Simon Gay and Mark Ryan A new title in Springer's "Workshops in Computing" Series. Proceedings of the First Imperial College Department of Computing Workshop on Theory and Formal Methods, Isle of Thorns Conference Centre, Chelwood Gate, Sussex, UK, 29--31 Match 1993. The Theory and Formal Methods Section of the Imperial College Department of College has an international reputation for research into the foundations of computer science, and the application of this theory to real computing problems. In March 1993 it held the first in a proposed series of workshops on theory and formal methods at the Isle of Theory Conference Centre in Sussex, UK. This volume contains revised versions of the papers presented at the workshop. They cover four main areas --- semantics, concurrency, logic, and specification -- with several papers spanning a variety of disciplines. The contributions fall into two main categories: review papers which provide the reader with an introduction to some specific areas being studied by the Section, and research papers which give details of the latest results in these areas. THEORY AND FORMAL METHODS 1993 provides a comprehensive overview of the work being carried out by one of the world's leading research centres in theory and formal methods. It will be of interest to practitioners and researchers, as well as post- and undergraduate students. OVERVIEW PAPERS Geoffrey Burn The Abstract Interpretation of Functional Languages Roy Crole Deriving Category Theory from Type Theory Steve Vickers Geometric Logic in Computer Science Chris Hankin Graph Rewriting Systems and Abstract Interpretation RESEARCH PAPERS Samson Abramsky Interaction Categories Mark Dawson Animating LU Abbas Edalat Dynamical Systems, Measures and Fractals via Domain Theory Abbas Edalat Self-Duality, Minimal Invariant Objects and Karoubi Invariance in Information Categories Lindsay Errington, Chris Hankin and Thomas Jensen Reasoning about GAMMA Programs Jose Fiadeiro and Tom Maibaum Generalising Interpretation between Theories in the Context of (pi-)Institutions Simon Gay and Raja Nagarajan Modelling SIGNAL in Interaction Categories Reinhold Heckmann Product Operations in Strong Monads Michael Huth On the Equivalence of State-transition Systems Stuart Kent Towards a Modal Logic of Durative Actions Marta Kwiatkowska Concurrency, Fairness, and Logical Complexity Marta Kwiatkowska and Iain Phillips Concurrency and Conflict in CSP Sarah Liebert A Complete Axiom System for CCS with a Stability Operator Ian Mackie, Leopoldo Roman and Samson Abramsky An Internal Language for Autonomous Categories Juarez Muylaert Filho and Geoffrey Burn Continuation-passing Transformation and Abstract Interpretation Iain Phillips A Note on the Expressiveness of Process Algebra Mark Ryan Prioritising Preference Relations David Sands Laws of Parallel Synchronised Termination Zvi Schreiber Implementing Process Calculi in C Paul Taylor An Exact Interpretation of While Irek Ulidowski Congruences for tau-respecting Formats of Rules Theory and Formal Methods 1993 Eds. Geoffrey Burn, Simon Gay and Mark Ryan Springer-Verlag London, 1993. 336pp (approx), 32 figures, 12 tables. Soft cover. Publication due: September 1993. Price: 37 pounds (30 pounds to members of the BCS). ISBN 3-540-19842-3. ORDER FORM Please order from your bookseller or from: Springer Verlag London Ltd Sales Office Springer House 8 Alexandra Road London SW19 7JZ Tel: +44 81 947 1280 Fax: +44 81 947 1274 Please send me ____ copies of ISBN 3-540-19842-3 (Theory and Formal Methods 1993, eds Burn, Gay, Ryan) at 37 / 30 pounds (delete as necessary). TOTAL ___________ Please charge my eurocard/access/mastercard/visa/bankamericard (delete as necessary) number |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_| valid until_____________________ Today's date____________________ Signature_______________________ OR Cheque payable to Springer Verlag London Ltd OR Proforma Invoice (delete as necessary) Name________________________________________________________ Address_____________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Lie? From: Kirill Mackenzie Date: 4 Aug 93 14:13:48 BST This is in response to Andre Joyal's message of Jul 13, and in enlargement of Ross Street's of last week. I have been away for the last three weeks. As Ross already intimated, quite a lot of work has been done on the infinitesimal theory associated with Lie groupoids since Pradines' notes of the 1960s. I just add a few comments. (1) The object which Joyal describes: > It is a pair (A,D) where A is a commutative > R-algebra and D is a Lie algebra (over R) such that > > 1) D is acting on A (as derivations): > X(fg) = X(f)g + fX(g) , [X,Y](f) = X(Y(f)) - Y(X(f)) > 2) D is equipped with an A-module structure such that > (fX)(g) = fX(g) and [X,fY] = X(f)Y + f[X,Y] > (for any f,g in A and X,Y in D) was defined abstractly by Herz in 1953 and called a Lie pseudo-algebra. It has since been reinvented by a very large number of people, most of whom also invented a new name; there are some 14 different terminologies in the literature. Pradines was unique in defining the narrower concept of Lie algebroid: a vector bundle $A$ on base $M$ (the set of identities when the Lie algebroid comes from a Lie groupoid) together with a vector bundle morphism $a\colon A \to TM$ and a bracket $[\ ,\ ]$ of global sections of $A$ which makes the real vector space of global sections into a real Lie algebra and obeys $a[X,Y] = [aX,aY]$ and $[X,fY] = f[X,Y] + a(X)(f)Y$ for global sections $X,Y$ of $A$ and smooth functions $f$ on $M$. This is the object which matters for the Lie theory of Lie groupoids. Note that the partiality of the multiplication for groupoids gets transformed into the module structure for the global sections and the map $a$, rather than into a "partial bracket". It can be interesting to think of the associated Lie pseudo-algebra as an infinite-dimensional Lie algebra associated to the infinite-dimensional group of admissible sections (H in Joyal's notation). But so far as I know there is no sufficently developed general theory that one could then apply. Rather, the Lie groupoid structure enables one to do directly in this case (most of) what one wishes a general theory of infinite-dimensional Lie groups could do. (For a groupoid which comes from a principal bundle, the group H of admissible sections is the group of gauge transformations.) The Lie groupoid/Lie algebroid formalism allows one (amongst many other things) to pretend that any manifold is a Lie group, any (sufficiently smooth) equivalence relation arises from a smooth Lie group action, and so on. (2) Pradines' notes of the 1960s sketched a Lie theory for Lie groupoids and Lie algebroids, but gave very few details. In the 1987 book which Ross mentioned I gave a full account for locally trivial Lie groupoids and transitive Lie algebroids (= the map $a\colon A\to TM$ is surjective). In this case, one can use methods quite different from those which Pradines sketched. In the general case quite a lot is known but there is no systematic account and some things are still open. Since my book was written Weinstein and, independently, Karasev, have created an extensive theory of symplectic groupoids, and special techniques are available in this case also. Symplectic groupoids are very far from being locally trivial. (Terminology: In my book, a Lie groupoid is always taken to be locally trivial, and the general concept is called a differentiable groupoid.) (3) Functoriality can be achieved without any dualization. Philip Higgins and myself (J. Alg. 129, 1990, 194--230) developed a means of working with general morphisms of Lie algebroids. The basic categorical constructions with groupoids (equivalence between actions and action morphisms, semi-direct products, quotients, etc) then also hold for abstract Lie algebroids. It is a curious fact that, although there is a very natural definition of morphism for Lie pseudo-algebras, it does not correspond to the infinitesimal map induced by a morphism of Lie groupoids. This is another reason for distinguishing between Lie algebroids and Lie pseudo-algebras. The dual of a Lie algebroid has a natural Poisson structure, extending the linear Poisson structure on the dual of a Lie algebra (Courant, Trans AMS, 319, 1990, 631--661). One knows that a linear map between Lie algebras is a morphism iff its dual is a Poisson map. Extending this to Lie algebroids runs into the problem of how to dualize a base-changing morphism of vector bundles. Higgins and I found a way around this ("Duality for base-changing morphisms...", Math. Proc. Camb. Phil. Soc. to appear very shortly). (4) > It is easy to see that we have a contravariant functor > > {Lie groupoids}--->{de Rham complexes} > > and it follows from the work of Pradine that it has a left adjoint Exp > (the exponential functor) > > Exp:{de Rham complexes}--->{Lie groupoids} >> when it is restricted to complexes (A,F,D) in which A is an algebra of > smooth functions on a manifold. > > The functor Exp is full and faithful. Its image should consist > of some kind of simply connected Lie groupoids. I guess (I should really > read Pradine ...) these groupoids are those for which the fibers of d0 are > simply connected. ?? Not all Lie algebroids arise as the Lie algebroids of Lie groupoids, even in the transitive case. Counterexamples were given by Almeida and Molino (CRAS (Paris), 300, 1985, 13--15). For transitive Lie algebroids I gave a cohomological obstruction to integrability; this is a kind of nonabelian first Chern class. See Chapter 5 of the book already mentioned for the case where the base is simply-connected and Cahiers 28, 1987, 29--52, for the general case. A general framework for thinking of this obstruction is given in JPAA 58, 1989, 181--208. (5) There is also a Lie theory associated with double Lie groupoids. Here a double groupoid is a groupoid object in the category of groupoids: it is a double Lie groupoid if it has a smooth structure making all four groupoid structures Lie and such that the map which sends any square to (say) its right side and its bottom side is a surjective submersion. Double Lie groupoids arise by natural games with ordinary groupoids, and in homotopy theory, but also in the integration of Poisson Lie groups (Lu and Weinstein, CRAS (Paris), 309, 1989, 951--954) and Poisson groupoids. Taking the infinitesimal object associated to a double Lie groupoid is a two-step process; the first step yields an LA-groupoid, that is, a groupoid object in the category of Lie algebroids. These are of interest in themselves: the cotangent bundle of a Poisson Lie groupoid is an LA-groupoid, for example. See Adv. Math. 94, 1992, 180--239. QUESTION: What is the infinitesimal object corresponding to a (Lie) 2-category? Is there a Lie theory here also? Lastly, to revert to Jim Stasheff's original question, given all the above and given that there is also a theory of Lie semigroups (Hilgert-Hofmann-Lawson), it should be straightforward to define an infinitesimal object associated to a "Lie category". How far one could get with a Lie theory is another matter entirely. My impression of the semigroup theory is that the absence of inverses causes plenty of difficulties there already. Kirill Mackenzie +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On calculuses of fractions Date: Thu, 5 Aug 93 10:10:44 +1000 From: kelly_m@maths.su.oz.au (Max Kelly) Michael Barr asks, in a letter dated 3 Aug, ``Can anyone supply a reference to the fact that if you add to the hypotheses of a calculus of right fractions the assumption that if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude that if the original category has all limits, so does the fraction category and the canonical functor to the fraction category preserves them." For closely-related results, see Kelly, Lack, & Walters, Coinverters and categories of fractions for categories with structure, Applied Categorical Structures, to appear in first issue; and a paper (ibid) by Kelly and Lack to which this appeals. Max Kelly. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On calculuses of fractions Date: Thu, 5 Aug 93 08:53:57 +1000 From: street@macadam.mpce.mq.edu.au >Can anyone supply a reference to the fact that if you add to the >hypotheses of a calculus of right fractions the assumption that >if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then >so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude >that if the original category has all limits, so does the fraction >category and the canonical functor to the fraction category >preserves them. A relevant reference in the case of finite products is Brian Day, Note on monoidal localisation, Bulletin Australian Math Soc Volume 8 (1973) 1 - 16. Brian called a class S of arrows s in a monoidal category V a monoidal class when s * X and X * s are both in the class when s is. Then the localisation V[S^-1] becomes monoidal with tensor-product preserving projection. The result you state was surely known to Brian and others (such as Harvey Wolff) and may even occur in Brian's work, but I don't have time now to scan his work. I'll ask him when next I talk to him. Regards, Ross +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: ?lie Date: Thu, 5 Aug 93 15:54 GMT From: MAS010@BANGOR.AC.UK (Ronnie Brown) \documentstyle{article} \begin{document} This is a further point in reply to the report on Lie groupoids by Andre Joyal. Pradines' groupoid analogue of the functor {Lie algebras}$\rightarrow$ {simply connected Lie groups} has subtleties which do not appear in the group case. These are stated in his CR Note of 1966, and an outline of the constructions was explained by him to me in the years from 1981. 1)Holonomy: From a Lie algebroid there may be obtained a locally Lie groupoid, i.e. a groupoid $G$ with a Lie structure on a subset $W$ of $G$ containing the identities of $G$. However, unlike the group case, the simple conditions which arise naturally for this situation do not imply that the Lie structure extends to give a Lie groupoid structure on $G$. Instead, under suitable conditions, a new groupoid, the holonomy groupoid $Hol(G,W)$ is obtained which has a Lie structure and which maps to $G$. This groupoid is the minimal "covering" of $(G,W)$ which has a Lie structure. Complete statements and proofs for the topological case are given in Aof and Brown, "The holonomy groupoid of a locally topological groupoid" Top. Appl. 47 (1992) 97-113. (This is essentially an account of Th\'eor\`eme 1 of Pradines Note.) 2) The natural analogue of a simply connected toplogical group is a topological groupoid whose stars (the inverse images of the source map) are simply connected. This we call star simply connected. The problem is then to construct from a Lie groupoid $G$ a star simply connected Lie groupoid $\tilde{G}$ and morphism $p:G\rightarrow \tilde{G}$ which is a universal covering morphism on stars. The existence of this is a part of the statement of Th'eor\`eme 2 of the same Note. Full details of the construction and full prooofs of its properties are in a Bangor preprint 93.09, R Brown and O Mucuk, ``The monodromy groupoid of a Lie groupoid". This also discusses the Lie case of holonomy (in line with Pradines' Note). The point is that is not hard to construct a groupoid which is the universal cover of each star; the problem is to get a topology making it a Lie groupoid. The proof given follows Pradines' outline (given verbally) in using holonomy arguments. 3) For applications to foliations, one needs to recognise that a foliation on a paracompact manifold gives rise to a locally Lie groupoid. This is proved in Bangor preprint 93.10 by Brown and Mucuk. These two preprints have not yet been duplicated, but I can send a TEX file of 93.09, a postscript file of 93.10 (3MB) (this has some pictures), or hard copies to anyone interested. These results formed a part of Mucuk's thesis (Jan, 1993). The locally trivial case of the monodromy construction is dealt with in Mackenzie's book ``Lie groupoids and Lie algebroids in differential geometry" (Cambridge, 1987), by a different method. Kock and Moerdijk also have work on related ideas for local equivalence relations. \end{document} Ronnie Brown +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: xypic Date: Fri, 6 Aug 93 13:58:53 -0400 From: cfw2@po.CWRU.Edu (Charles F. Wells) Has anyone successfully used xypic with LaTeX on an MS DOS machine? I am having some very puzzling problems I would like to discuss with someone, but I didn't want to take up the category network's time by listing them all to everyone. --Charles -- Charles Wells, Department of Mathematics, Case Western Reserve University 10900 Euclid Avenue, Cleveland OH 44106-7058, USA Phone 216 368 2880 or 216 774 1926 FAX 216 368 5163 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: lie? Date: Fri, 6 Aug 93 9:35 GMT From: MAS010@BANGOR.AC.UK Damn: I just noticed my note got the map $p: \tilde{G} \to G$ the wrong way round, though I am sure the correction was spotted easily. I should also say that aim of the Brown-Mucuk paper (again following Pradines) is to use methods of free groupoids to get the monodromy principle as well: the globalisation of local morphisms to the star universal cover. The nice point is that if (G,W) is a locally Lie groupoid one gets star covering morphisms \tilde G \to Hol(G,W) \to G, and there may be interesting Lie groupoids sandwiched between the first two. There is a problem of terminology. From a topological groupoid G one obtains a groupoid $\tilde{G}$ which is the star universal cover. If $G = X \times X$, then \tilde G is the fundamental groupoid of X. If G is the equivalence relation of a foliation, then \tilde G is the socalled ``homotopy groupoid'' of the foliation, i.e. the fundamental groupoid of X with the leaf topology. There is a temptation to call \tilde G the ``fundamental groupoid of G'', but this conflicts with the fundamental groupoid of the underlying space of (the arrows of ) G. Ronnie +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: compactly generated locales. From: Francois Lamarche Date: Mon, 9 Aug 93 15:58:29 BST There is a well-known cartesian closed category of topological spaces, described in Mac Lane's CWM. A Hausdorff space is compactly generated if a set is closed iff its intersection with every compact subspace is closed. Question: does this construction generalize to an arbitrary topos? In other words is there a notion of compactly generated locale, such that we get a cartesian closed full subcategory of the category of locales over that topos? Ideally all separation axioms should be dropped. Also one would (well I would) like to have these locales to have enough points. Francois Lamarche, Imperial College. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: CTCS-5 Program Date: Mon, 9 Aug 1993 13:52:27 +0200 From: F.J.de.Vries@cwi.nl CTCS 5 Category Theory in Computer Science 7th-10th September 1993 CWI, Amsterdam, The Netherlands Program and Registration Form PROGRAM Tuesday 7th September 9.15 Opening 9.30 - 10.30 Saunders Mac Lane (Invited Speaker) 10.30 - 11.00 Tea/Coffee 11.00 - 11.40 Pietro Cenciarelli and Eugenio Moggi A Syntactic Approach to Modularity in Denotational Semantics 11.40 - 12.20 Claudio Hermida and Bart Jacobs Fibrations with Indeterminates: Contextual and Functional Completeness for Polymorphic Lambda Calculi 12.20 - 13.00 G. Michele Pinna and Axel Poign\'e The Mathematics of Event Automata 13.00 - 14.00 Lunch 14.00 - 14.40 A.J. Power Why Tricategories? 14.40 - 15.20 N. Sabadini, R.F.C. Walters and Henry Weld On Distributive Automata and Asynchronous Circuits 15.20 - 16.10 Tea/Coffee 16.10 - 17.00 Alain Prout\'e ``Substitution Should Not Respect Equality'' 17.00 - 17.40 Adam Obtulowicz Graphical Sketches, a Finite Presentation of Infinite Graphs 18.00 - 19.30 Reception Wednesday 8th September 9.30 - 10.30 N. Shanin (Invited Speaker) An Finitary Version of Mathematical Analysis Oriented to Computer Science 10.30 - 11.00 Tea/Coffee 11.00 - 11.40 Paul Taylor Intuitionistic Ordinals and Tarski's Theorem 11.40 - 12.20 Marcelo P. Fiore Cpo Categories of Partial Maps 12.30 Conference Outing 18.30 Conference Dinner Thursday 9th September 9.30 - 10.30 G. Rosolini (Invited Speaker) 10.30 - 11.00 Tea/Coffee 11.00 - 11.40 Daniele Turi and Bart Jacobs On final Semantics for Applicative and Non-deterministic Languages 11.40 - 12.20 S. Soloviev Reductions in Intuitionistic Linear Logic 12.20 - 13.00 B.P. Hilken and D.E. Rydeheard A Theory of Classes: Proofs and Models 13.00 - 14.00 Lunch 14.00 - 14.40 A. Carboni and P. Johnstone Connected Limits, Familial Representability and Artin Glueing 14.40 - 15.20 B.P. Hilken and D.E. Rydeheard Computing Colimits 15.20 - 16.10 Tea/Coffee 16.10 - 17.00 John G. Stell Sesqui-Categories and their Applications to Rewriting Systems 17.00 - 17.40 S. Kasangian, G. Mauri and N. Sabadini (presented by Sebastiano Vigna) Trees of Traces: A Categorical View Friday 10th September 9.30 - 10.30 A. Joyal (Invited Speaker) 10.30 - 11.00 Tea/Coffee 11.00 - 11.40 Akira Mori and Yoshihiro Matsumoto Unification in Categories and Proof Search in Intuitionistic Propositional Calculus 11.40 - 12.20 Martin Hofmann Sound and Complete Axiomatisations of Call by Value Control Operators 12.20 - 13.00 Eike Ritter and Valeria de Paiva Syntactic Multicategories and Categorical Combinators for Linear Logic 13.00 - 14.00 End/Lunch PROGRAM AND ORGANIZING COMMITTEE S. Abramsky, P.-L. Curien, P. Dybjer, G. Longo, G. Mints, J. Mitchell, E. Moggi, D. Pitt, A. Pitts, A. Poigne, D. Rydeheard, F.J. de Vries and E. Wagner. LOCAL ARRANGEMENTS Fer-Jan de Vries Department of Software Technology, CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands CONFERENCE ADMINISTRATOR CTCS-5, c/o CWI Ms. Anna Baanders P.O. Box 4079 1009 AB Amsterdam The Netherlands Tel. +31-20-5924048 Fax. +31-20-5924199 email: anna@cwi.nl REGISTRATION Register and make your hotel reservation in advance using the Registration and Accommodation Form. Mail or email the form to the Conference Administrator. No registration or reservation will be effective until full payment is received. CONFERENCE FEE The conference fee for CTCS-5 is NLG 500 for advance registration (to be paid by 10 August 1993) and NLG 600 after 10 August 1993. The fee includes admission to all sessions, lunches and coffee breaks, the conference dinner and the conference excursion. No refunds of payments will be made for cancellations received less than two weeks before the start of the conference. In other cases NLG 25 administrative charges and hotel no-show costs, if any. ACCOMMODATION A (limited) number of hotelrooms have been reserved at one of the AMS-group Hotels in centre town. Roomrates are NLG 130-170 for single and NLG 190 for double use (sharing a room is the responsibility of the registrant though: no matching service will be provided). To book accommodation, fill out the hotel reservations part of the Registration and Accommodation Form. A deposit of NLG 300 is to be paid in advance to make any reservation effective. This deposit will be deducted from your account when leaving. PAYMENTS Payments, net of all charges, are to be made in Dutch Guilders (NLG) and can be made by either -Banker's draft (bank check), made payable to "Stichting Wiskunde en Informatica Conferenties", and mailed to the conference administrator -Money transfer to account no. 31.35.57.977 of "Stichting Wiskunde en Informatica Conferenties" at the RABO-bank, Middenweg 88, Amsterdam (Postal giro of bank: 187744) Make sure that payments mention "CTCS-5" and your name. REGISTRATION AND ACCOMMODATION CTCS-5 Please type or print Name (last)____________________________ (first)______________ Affiliation_______________________________________________ ______________________________________________________ Address________________________________________________ Postal Code________________ City__________________________ Country________________________________________________ Email__________________________ Fax_____________________ Telephone______________________________________________ Special Requests_________________________________________ registers for CTCS-5. Registration fee: o NLG 500 o NLG 600 (late fee) Hotel Reservations Please reserve a o single/o double room in AMS Hotel Date of arrival:_________________ departure:___________________ Number of nights:_________________________________________ Payment : o NLG 300 Hotel deposit Payment: o I enclose a banker's draft o I transferred the registration fee to your bankaccount Mail to: CTCS-5 Secretariaat, CWI/Ms. Anna Baanders P.O. Box 4079, NL 1009 AB Amsterdam +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Group photograph Date: Mon, 09 Aug 93 16:38:58 ADT From: es@math.mcgill.ca (Elaine Swan) Can anyone please supply me with the correct email address for Dr. Chiment? The one I have, seems to be incorrect. This request is for Prof. J. Lambek. Thank you +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: New Textbook Date: Tue, 10 Aug 1993 10:30:51 +0100 (BST) From: Roy Crole PRELIMINARY ANNOUNCEMENT ------------------------------- The following book will be available shortly: CATEGORIES FOR TYPES Cambridge Mathematical Textbooks, Cambridge University Press Roy L. Crole, Imperial College, University of London. Abstract: This textbook explains the basic principles of categorical type theory and illustrates some of the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory. Categories, functors and natural transformations are covered, along with the Yoneda Lemma, cartesian closed categories, limits and colimits, adjunctions and indexed categories. Four kinds of formal system are presented in detail, namely algebraic, functional, second order polymorphic and higher order polymorphic type theories. For each of these type theories a categorical semantics is derived from first principles, and soundness and completeness results are proved. Correspondences between the type theories and appropriate categorical structures are formulated, along with a discussion of internal languages. Specific examples of categorical models are given, and in the case of polymorphism both PER and domain-theoretic structures are considered. Categorical gluing is used to prove results about type theories. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians, and mathematicians specialising in category theory. Contents: (1) Order, Lattices and Domains [1--36] (2) A Primer on Category Theory [37--119] (3) Algebraic Type Theory [120--153] (4) Functional Type Theory [154--196] (5) Polymorphic Functional Type Theory [197--268] (6) Higher Order Polymorphism [269--308] Bibliography [309--313] Index [314--335] ISBN 0521 450926 (HB) The Edinburgh Building, Shaftesbury Road, CAMBRIDGE, CB2 2RU, England, UK. 40 West 20th Street, New York, NY 10011-4211, USA. 10 Stamford Road, Oakleigh, Victoria 3166, Australia. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: change of address Date: Thu, 12 Aug 1993 09:53:10 -0400 From: Reinhard.Boerger@FERNUNI-HAGEN.DE Please send all further messages for me to Hagen, where I returned after six months at York. My internet address is: Reinhard.Boerger@Fernuni-Hagen.de My postal address is: Fachbereich Mathematik Fernuniversit"at Postfach 940 58084 Hagen Germany Please note the new postal code according to the new German system. I should appreciate if the other network participants in Germany also posted their new postal codes by the network. Greetings Reinhard B"orger +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: categories and complexity Date: Thu, 12 Aug 93 09:31:06 EDT From: otto@triples.Math.McGill.CA (James Otto) Kalmar, Linear Space, and P J. Otto Department of Mathematics McGill University otto@math.mcgill.ca August 12, 1993 is available by anonymous ftp from triples.math.mcgill.ca. It is in uuencoded compressed postscript in pub/otto/0-k-ls-p.uu. Thus ftp ftp> open triples.math.mcgill.ca ftp> Name (...): anonymous ... Guest login ok, send e-mail address as password. Password: ftp> cd pub ftp> cd otto ftp> get 0-k-ls-p.uu ftp> quit uudecode 0-k-ls-p.uu uncompress 0-k-ls-p.ps.Z We use 2-simplices to cut down the primitive recursive functions to the Kalmar elementary functions, and 1-simplices to cut down the primitive recursive functions to the functions of complexities linear space and P time. This translates work of [Leivant Marion], [Bellantoni], and [Bellantoni Cook]. As 2-simplices are less degenerate than 1-simplices, we first consider Kalmar elementary. Then, successively modifying characterizations, we consider linear space and P time. Remarks. 1. Kalmar elementary and linear space are levels 3 and 2 of the Grzegorczyk Hierarchy. E.g. see [Rose]. 2. The characterizations can be viewed as programming languages whose typing guarantees, without explicit bounds checking, that programs represent precisely the functions of the complexity class. Most of the category theory we need is in [Barr Wells]. For the little that we need of 2-categories, e.g. see [Kelly Street], [Makkai Pare]. This paper revises and expands the April 18, 1993 paper `Kalmar Elementary and 2-Simplices'. In particular, besides minor changes and corrections, 1. Linear time is omitted, as the characterization I had translated may need some repair. 2. Machine based soundness and completeness proofs for the linear space and P time characterizations are added. The machine based proofs unfold [Ritchie], [Cobham] from the proofs of [Bellantoni], [Bellantoni Cook] and make this paper largely self contained. [Bloch] has related work on machines. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Lie? Date: Thu, 12 Aug 1993 16:59:28 -0400 From: chase@math.cornell.edu (Stephen Chase) I would like to comment on the considerations raised in the recent messages of Joyal, Street, Mackenzie, and Brown, but from a somewhat different perspective. Although I accept the fact that the notion of Lie algebroid is what matters for the Lie theory of Lie groupoids, I think there are a couple of reasons why the general concept of Lie pseudo-algebra should not be dismissed. The first is that examples arise from a variety of sources in algebra. A fundamental one comes from any k-algebra A (k a field, say) with commutative subalgebra K. Namely, let A+ be the "differential normalizer" of K in A: The set of all u in A with ux - xu in K for all x in K. Then (K,A+) is a Lie pseudo-algebra over k; moreover, for fixed K/k, the functor A ------> (K,A+) has a left adjoint (K,L) ------> U(K.L), the universal enveloping algebra of a Lie pseudo- algebra constructed by Rinehart [TAMS 108 (1962)]. (Actually, this is not quite the whole story, since in general one must consider algebra maps K ----> A that are not necessarily embeddings). Another important example arises from a Lie k-algebra L acting on a commutative k-algebra K by derivations: (K,K#L) is then a Lie pseudo-algebra, where K#L is the tensor product of K with L over k with a suitable twisted bracket operation. My other reason to keep Lie pseudo-algebras in mind is that they should form part of a larger theory of groupoid schemes and formal groupoids. Although some years ago I explored and used some special cases of these notions (see below), I am not aware of any systematic development of them in the literature. That is, with the exception of the appendix on groupoid schemes and Hopf algebroids in Ravenel's 1986 book on stable homotopy. ;I would be interested in hearing of other references. (In this regard I should also mention Huebschmann's paper on Poisson algebras [Crelle 408 (1990)], in w;hich Lie pseudo-algebras play an important role). As is suggested by the examples discussed above, Lie pseudo-algebras have a long history in the Galois theory and Brauer group literature. Jacobson's Galois correspondence for purely inseparable extensions of exponent one is between intermediate fields in such an extension and restricted Lie pseudo-subalgebras of (K,L), with L the restricted Lie pseudo-algebra of k-derivations of K [Amer. J. Math. 66 (1944)]. Hochschild classified Brauer classes of central simple k-algebras split by K (with K/k as above) in terms of extensions of restricted Lie pseudo-algebras [TAMS 76 (1954) and 79 (1955)]. In my paper [Amer. J. Math 98 (1976)] I generalized Jacobson's theorem to modular purely inseparable extensions of arbitrary exponent by a method that rests on the notion of the group scheme G# of admissible sections of a finite groupoid scheme. Although at the time it seemed clear to me that an analogous theory of Lie groupoids should exist, until 1988 I was unaware of the work that had been done in that area, and I am indebted to Ronnie Brown for informing me of it. It is not surprising that the group of admissible sections should play a role in Galois theory, since it provides a very natural definition of the automorphism group (or "group of symmetries") of a collection of objects, in a manner which takes into account not only the symnmetries of the individual objects but also their isomorphisms with each other. The literature described above also suggests the appropriate groupoid analogue of the cocommutative Hopf algebra that arises as the continuous dual of the algebra of representative functions of a formal group, or the Hopf-Sweedler dual of the algebra of representative functions of a group scheme. In my view, this analogue should be Sweedler's notion of a xK-bialgebra [Groups of simple algebras, Publ. IHES no. 44 (1975)] (although presumably with some sort of antipode). To oversimplify a bit, a xK-bialgebra is a k-algebra A which is also a K-coalgebra, with suitable axioms linking the two structures (the module of primitive elements of A then yields a Lie pseudo-algebra for K/k). These algebras are a ubiquitous as Lie pseudo-algebras: In addition to the universal enveloping algebras mentioned above, they include the classical smash product K#G (G a group acting on K by k-algebra automorphisms) and the algebra of differential operators on K. But logically (if not historically) the paradigm here is the k-algebra A = kC of a small category C, with K the k-algebra of k-valued functions on the set X of objects of C; the coalgebra structure on A is defined in a manner entirely analogous to that of a group algebra (it also satisfies a similar universal property). Finally, the grouplike elements of A constitute the monoid of admissible sections of C. Steve Chase +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: noncomm geom/n-cats and quantization Date: Sat, 14 Aug 93 08:38:19 -0400 From: jds@math.upenn.edu In cae you haven't noticed, Penkov's review of Manin's book in July Bull AMS says: Manin explains also why monoidal categories can be viewed as a unification base (at least) of quantum geom and supergeom. p.108 jim stasheff +++++++++++++++++++++++++++++++ Date: Sat, 14 Aug 93 08:40:20 -0400 From: jds@math.upenn.edu Dan Freed (dafr@math.utexas.edu) has a preprint Higher algebraic structures and quantization where the higher alg structures are in fact n-cats or n-ple cats or whichever variant is appropriate jim +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Diaconescu for sheaves Date: Fri, 13 Aug 1993 14:20:00 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) Does anyone know where there's a proof of a relativized version of the well-known fact that the topos of sheaves over a locale classifies the points of the locale? What I imagine is something like the following: Let E be a topos, and A a frame in it. Let E' be the topos of internal (in E in some suitable sense) sheaves over the locale of A. Then for any topos F there is an equivalence between - * the category of geometric morphisms from F to E' and * the category whose objects are pairs (f, x) where f: F -> E is a geometric morphism and x: A -> f_*(Omega_F) is a frame homomorphism (with suitable morphisms between these pairs). (Presumably this could be deduced from a more general theorem that generalizes Diaconescu by dealing with toposes of sheaves over sites, not just presheaves over categories.) I feel a bit stupid, because I know topos theorists take results like this for granted all the time. But I can't track down any account of the details, nor quite convince myself that methods such as Joyal and Tierney's ("pretend E is ordinary sets but take care to reason constructively") do the trick in this case. Steve Vickers. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: UACT Participants List-posting corrected Date: 18 Aug 93 01:15:43 EDT From: "Paul H. Palmquist" <76600.1050@CompuServe.COM> Another correction to the UACT address list. My Zip Code got mashed into my email address. The correction follows. Paul H. Palmquist 13112 Dewey St Los Angeles, CA 90066 Email: 76600.1050@compuserve.com +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: change of address Date: Thu, 19 Aug 93 19:02:50 BST From: Thorsten Altenkirch Reinhard Boerger recently wrote: Please note the new postal code according to the new German system. I should appreciate if the other network participants in Germany also posted their new postal codes by the network. Greetings There are some servers set up which you can use to find out the new german zipcodes via the network. The easiest way is to do rlogin -l plz plz.isr.uni-stuttgart.de You are connected to a program which translates addresses. Unluckily the help text is in german. However the usage is simply to type in an address as in the example followed by control-D. I would appreciate it if the noise on the network created by change of address mails could be kept on a minimum level. If there is serious interest in having addresses available somebody should collect them centrally and make them available by ftp (like Vaughan Pratt's structdir). Another way to distribute address changes is to make it available to the finger program by putting the new address into the .project file (e.g. try "finger alti@yell.dcs.ed.ac.uk"). Cheers, Thorsten Thorsten Altenkirch Kennst du das Land, wo die Zitronen blu"hn, Laboratory for Foundations Im dunkeln Laub die Gold-Orangen glu"hn, of Computer Science Ein sanfter Wind vom blauen Himmel weht, Die Myrte still und hoch der Lorbeer steht, University of Edinburgh Kennst du es wohl? Dahin! Dahin Mo"cht ich mit dir, o mein Geliebter, ziehn. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: McGill Octoberfest Date: Sun, 22 Aug 93 19:42:24 ADT From: fox@triples.math.mcgill.ca CATEGORY THEORY MEETING: OCT 9-10, 1993 CATEGORY THEORY RESEARCH CENTER, MCGILL UNIVERSITY Dear Colleague, We look forward to seeing you again this fall. We will meet in the basement of Burnside Hall (805 Sherbrooke St W) at 8:30 Saturday morning for coffee, and the first talk will be at 9:00. If you wish to speak, please contact Michael Barr as soon as possible. A final schedule will be drawn up Saturday morning. In the past these meetings have been run without registration fees. However, the combination of increasing costs and decreasing grants has made it impossible to continue on this basis. In order to recover part of our costs, this year there will be a registration fee of $25 for professors, $15 for students. Below you will find a list of hotels and tourist rooms within easy walking distance of McGill. To obtain the quoted price you should mention McGill when making your reservation. The area code for Montreal is 514. If you have any further questions, contact Tom Fox. Hotels: L'Appartement, 455 Sherbrooke W, 284-3634, $72 Howard Johnson Plaza, 475 Sherbrooke W, 842-3961, $79 Citadelle, 410 Sherbrooke W, 844-8851, $79 Four Seasons, 1050 Sherbrooke W, 284-1110, $120 Holiday Inn, 420 Sherbrooke W, 842-6111, $94 Journey's End, 3440 Park Ave, 849-1413, $79-88 Hotel du Parc, 3625 Park Ave, 288-6666, $90 Versailles*, 1659 Sherbrooke W, 933-3611, $89 Tourist Rooms: Ambrose, 3422 Stanley, 288-6922, $45-50 Armor*, 151 Sherbrooke E, 285-0140, $55-62 Casa Bella, 258 Sherbrooke W, 849-2777, $35-70 Pierre*, 169 Sherbrooke E, 288-8519, $35-55 *20 minute walk from McGill Dept of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, Quebec CANADA H3A 2K6 Michael Barr barr@triples.math.mcgill.ca Tom Fox fox@triples.math.mcgill.ca +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Lie? From: Kirill Mackenzie Date: 17 Aug 93 16:50:54 BST > Date: Thu, 12 Aug 1993 16:59:28 -0400 > From: chase@math.cornell.edu (Stephen Chase) > Although I accept the fact that the notion of Lie algebroid is what matters > for the Lie theory of Lie groupoids, I think there are a couple of reasons > why the general concept of Lie pseudo-algebra should not be dismissed. I certainly agree that Lie pseudo-algebras deserve independent study. I have a survey article "Generalized Lie theories: Lie algebroids and Lie pseudo-algebras as algebraic invariants in differential geometry" in near-final stage, with a bibliog of about 100 references; this is available to people who have an interest in the subject. The orientation is towards the use of Lie algebroids and Lie pseudo-algebras as a unifying concept in first-order differential geometry, but the bibliog is intended to be complete and covers purely algebraic work that I know of. > Another important > example arises from a Lie k-algebra L acting on a commutative k-algebra > K by derivations: (K,K#L) is then a Lie pseudo-algebra, where K#L is > the tensor product of K with L over k with a suitable twisted > bracket operation. > That is, with the exception of the appendix on > groupoid schemes and Hopf algebroids in Ravenel's 1986 book on stable > homotopy. If one considers Lie algebroids over a fixed base manifold, or if one considers Lie pseudo-algebras over a fixed (commutative) algebra, then everything in both cases is quite similar to the case of finite-dimensional Lie algebras over a field, at least as regards basic definitions and constructions. If one allows arbitrary base manifolds or base algebras things become very different. Firstly, morphisms of Lie algebroids and morphisms of Lie pseudo-algebras no longer correspond: the concept of morphism of Lie algebroid which arises by differentiating a morphism of Lie groupoids does not corrrespond to the natural concept of morphism of Lie pseudo-algebra. Higgins and I (Math. Proc. Camb. Phil. Soc., to appear) defined concepts of comorphism, so that a comorphism of Lie algebroids induces a morphism of their Lie pseudo-algebras, and a morphism of Lie algebroids induces a comorphism of Lie pseudo-algebras; this in particular enables a duality to be defined in both categories. From this perspective, the point of the # construction is that it enables general Lie pseudo-algebra morphisms to be reduced to the algebra-preserving case. In (K,K#L) above, L itself can be a Lie pseudo-algebra. Now if L --> L' is a morphism of Lie pseudo-algebras over a morphism of k-algebras K --> K', there is an action of L on K', and the morphism can be lifted to (K',K'#L), and is now a morphism over K'. [MPCPS, cited above.] In the Lie algebroid context the analogue of the # construction was found independently and called an "action Lie algebroid"; it is the infinitesimal analogue of the Ehresmann construction of a groupoid from an action of a group(oid) and its characterization in terms of covering morphisms. See J. Alg. 129, 1990, 194--230 and refs there. Incidentally, morphisms of Lie pseudo-algebras also arise in Poisson geometry: a Poisson morphism induces not a morphism of the cotangent Lie algebroids but a morphism of the Lie pseudo-algebras of 1-forms. Quotients of Lie algebroids in the base-varying case are more complicated and need a kind of reduction process. I am not sure what the situation is with quotients of Lie pseudo-algebras. See J. Alg. 129, 1990, 194--230 again. The corresponding general quotient for (Lie) groupoids [Jour LMS, 42, 1990, 101--110] should have implications for Ravenel's 1986 Appendix. Kirill Mackenzie +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: From: Oege de Moor Date: Wed, 25 Aug 93 11:25:00 JST Let F be an endofunctor on a category C that has finite products. A strength of F is a natural transformation phi(A,B) : FA x B -> F(A x B) satisfying the obvious coherence conditions with respect to the terminal object and associativity of products. Is anything known about uniqueness of such strengths, if C is not well-pointed? +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Notice of Workshop Date: Tue, 24 Aug 93 21:22:11 EST From: La Monte H Yarroll Forwarded message: >From H.GUSTAFSON@qut.edu.au Tue Aug 24 14:05:16 1993 Date: Tue, 24 Aug 93 13:49 +1000 ****************************************************************************** ** 3C WORKSHOP 93 CATEGORIES, COMPUTING AND COMBINATORICS September 1-3, 1993 School of Computer Science and Engineering University of New South Wales * Second announcement and Programme * This is the second in a series of _informal_ workshops exploring the connections between category theory, logic theoretical computer science and algebraic combinatorics. It will provide a forum at which experts and interested workers from outside the field can exchange ideas. The scope of the workshop has been significantly widened. There will be talks in the following general areas: 1. Categorical logic and categorical models of computation (e.g. imperative programming, distributed computing) 2. The role of category theory in computer system design (e.g. object-oriented design, information systems and database theory, design process automation) 3. Applications of category theory to combinatorics and combinatorial topology 4. Algebraic combinatorics Our emphasis will be on the connections between different strands of research, and on new ways of thinking about old problems. We hope to promote cross-fertilisation between areas. There will be a series of tutorials on each of these areas during the afternoon of Wednesday, September 1; these will assume only a passing acquaintance with category theory and/or combinatorics. They are meant to provide a suitable background for the more specialised talks on Thursday and Friday. We expect a number of international visitors to be participating: these will include Takayuki Hibi (Hokkaido; algebraic combinatorics). Arrangements for other visitors have not yet been finalised, but they _tentatively_ include Nicoletta Sabadini (Milan), Lin Ying Ming(Szechuan), Eric Wagner (IBM Yorktown Heights) and Rod Burstall (Edinburgh). Anyone is welcome to participate. We intend the talks to be accessible to people without an extensive background in category theory and/or combinatorics; one of our major goals is to make the most recent ideas and results in these areas available to a wider community. In particular we would hope to have a number of industrial participants, as we did last year. Research students are also especially welcome. The programme will be finalised in August. Prospective speakers should contact Amitavo Islam as soon as possible. A limited amount of financial assistance is available to assist speakers travelling from interstate. Organisers: Wesley Phoa (UNSW) general organiser Karl Wehrhahn (Sydney) combinatorics Dominic Verity (Macquarie) category theory Amitavo Islam (Sydney) administration Linda Milne (UNSW) UNSW arrangements This workshop is being run in conjunction with the Software Engineering Research Group (UNSW), the Sydney Category Group, the CATACOMB Group (Sydney) and the Theory Group (Macquarie). ****************************************************************** WORKSHOP ON CATEGORIES, COMPUTING AND COMBINATORICS School of Computer Science and Engineering University of New South Wales Meeting Room 1, Samuels Building September 1-3, 1993 TENTATIVE PROGRAMME WEDNESDAY, September 1 -- tutorials 1:00 "Combinatorial Species" (Bill Unger) 2:00 "Ehrhart Polynomials" (Takayuki Hibi) 3:00 [coffee] 3:30 "Natural transformations and coherence" (Nick Verne) 4:30 "Exactness for datatypes" (Barry Jay) THURSDAY, September 2 9:00 "Graphs, Hall Algebras and quantum groups" (Jie Du) 9:55 [coffee] 10:15 "Toric varieties in combinatorics" (Vladimir Popov) 11:15 "Cyclotomic identities in combinatorics" (Adrian Nelson) 12:10 [lunch] 1:30 "Insight into information system structures using category theory" (Kit Dampney and Mike Johnson) 2:30 "Consistency for SML: an operational perspective" (Ed Kazmierczak) 3:25 [coffee] 3:45 "A categorical description and justification for a synthesizer" (Trudy Weibel) 4:45 "The programming language dr" (Desmond Fearnley-Sander) FRIDAY, 3 September 9:00 "Rewrite systems and 2-categories" (Ross Street) 9:55 [coffee] 10:15 "Distributive categories and parallel computation" (Henry Weld, Bob Walters and Nicoletta Sabadini) 11:15 "Categorical term rewriting" (Barry Jay) 12:10 [lunch] 1:30 "There is an anti-intuitionistic co-implication operator in Set^op" (La Monte Yarroll) 2:30 "Survey of Face Vectors and Ehrhart polynomials of convex polytopes" (Takayuki Hibi) 3:25 [coffee] 3:45 "Survey of Fibonacci lattices" (Rowan Kemp) 4:45 "Categories, dependent types and donkey sentences" (Wesley Phoa) ****************************************************************************** ****** This letter was sent to us by Karl Wehrhahn whose e-mail address is: wehrhahn_k@maths.su.oz.au It has been forwarded by Helen Gustafson, CMSA Membership Secretary +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Uniqueness of strengths Date: Wed, 25 Aug 93 13:41:30 -0600 From: robin@cpsc.ucalgary.ca (Robin Cockett) In partial answer to Oege de Moor's query concerning the uniqueness of strength in non-well-pointed finite product categories ... there is an, admittedly, special case when the strength is unique: namely when the type F is a strong initial (or final) datatype. The strength can be described as a fold (or unfold) and the uniqueness of this guarantees the uniqueness of the strength (actually this is dependent on the uniqueness of the strengths of the functors over which the datatype is defined ... which inductively one assumes are datatypes). Strong datatypes were described in "Strong Catgeorical Datatypes I" by myself and Dwight Spencer (p 141-169, Category Theory 1991) - also see Dwights thesis. More recently Bart Jacobs has given an alternative treatment which elides much of the fibrational and 2-categorical aspects present in our treatment. There are besides various other circumstances which force strengths to be unique. E.g. if F is cartesian over a functor G with a unique strength (that is there is a strong natural transformation F --> G with all naturality squares pullbacks) then its strength is uniquely determined. These observations were made in "Data III" (with Dwight: an unpublished follow on to the yet unpublished "Data II"!!) and were recently sharpened when Barry Jay visited here (Calgary). BUT does anyone have an example of a functor with a non-unique strength? -robin (Robin Cockett) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Beck's definition of module Date: Mon, 30 Aug 93 12:42:02 EDT From: barr@triples.Math.McGill.CA (Michael Barr) In his 1967 thesis (but actually essentially completed by 1964) Beck defined an A-module, for an object A of a category _A_ to be an abelian group object in the slice _A_/A. This turned out to mean 2-sided A-modules, left A-modules, left A-modules and left A-modules in the categories of associative algebras, commutative (associative) algebras, groups and Lie algebras, resp., that is in all cases the desired coefficients for cohomology. This is all widely known among categorists. Has a full exposition of this ever been published and, if so, where? --Michael +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Beck's definition of module Date: Tue, 31 Aug 1993 08:57:54 -0400 (EDT) From: MTHFWL@ubvms.cc.buffalo.edu I too would like such an exposition. Hopefully it will also explain why modules may have tensor products. Also what about the dual formulation for say distributive categories, in particular explaining why the infinitesimal neighborhoods of the diagonal " can't be defined" for arbitrary objects in the gros Zariski topos. Bill +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: october fest Date: Tue, 31 Aug 93 09:33:10 EDT From: barr@triples.Math.McGill.CA (Michael Barr) In the past, most people who wanted to talk at the meeting told me about it at the last minute. Increasingly, people have asked in advance for a slot. Unless I have lost some, I have so far 11 requests. There is a practical limit of about 15, since we have to finish early enough on Sunday to allow people to return home. Below are the requests I have so far, names, email addresses, titles or descriptions and abstracts or any other relevant information on the talks. If you have made a request, be sure to check if you are on the list. If you want to request a slot, do not tarry. The order below is in the order I received them, BTW. Michael Giulio Katis working on Cauchy completion Jonathan Smith Duality for semilattice representations (with A. Romanowska) We present general machinery for extending a duality between complete, cocomplete concrete categories to a duality between corresponding categories of semilattice representations. This enables known dualities to be regularised. Among the applications, regularised Lindenbaum-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of PASCAL-like programming languages) is the logic for semilattice-ordered systems of sets. Another application enlarges Pontryagin duality by regularising it to obtain duality for commutative inverse Clifford monoids. Till Plewe <> on when a locale product of metrizable spaces is spatial Rick Blute Contextual Logic (joint with Robert Seely and Robin Cockett) Andreas Blass TBA Djordje Cubric Interpolation property for bicartesian closed categories Bob Gordon Enrichment Through Variation (joint with John Power) L Gaunce Lewis Jr a talk about the equivariant Freudenthal suspension theorem One of those nice situations when just a little touch of category theory cleans up a mess in topology. Richard Wood Distributive adjoint strings Stacy Finkelstein TBA Robin Cockett Copy Categories. These are symmetric monoidal categories in which every object has a natural coassociative cocommutative comultiplication -- but no (natural) counit. Examples include the category of partial maps of a finitely complete category, the Kleisli category of the exception monad of a distributive category, ... I shall describe the category of "formal propositions" of a copy category and why this gives insight into the embedding of a distributive category into an extensive category (its the 2-category theory behind it!)