Subject: Re: Beck's definition of module Date: Thu, 2 Sep 93 07:02:50 -0700 From: rowan@garnet.berkeley.edu I have been using modules of universal algebras in my work, and plan to continue doing so. The following papers of mine use the concept: Enveloping ringoids of universal algebras. Dissertation, University Microfilms International, 1992. The modules of a universal algebra A are the same as the left modules of the enveloping ringoid Z[A] of the universal algebra. An appendix treats modules in detail, including restriction and induction functors, the centralizer of a module, and some important examples. Enveloping ringoids. To appear in Algebra Universalis in Day conference special issue. Summary of thesis. Maximal subalgebras of CM algebras. In review. Classifies maximal subalgebras of CM algebras (e.g. groups, rings, Lie algebras, lattices) into 7 types. I was disappointed recently when the concept, which I feel is very important, was never mentioned at the July UACT conference at MSRI. (Perhaps if Dr. Barr had attended...) William H. Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Beck's definition of module Date: Fri, 03 Sep 93 16:09:11 +0100 From: Valeria de Paiva Jon Beck told me that his thesis was never published and that he wrote quite a bit about the "modules" that never appeared even in the thesis. I guess if someone wanted to publish it as a technical report of some description, Beck might be persuaded to give them the manuscript. But this is off the top of my head, one would have to contact him and ask. Unfortunately I couldn't even try do this in Cambridge, as only the Computer Laboratory produces technical reports... Valeria de Paiva ------------------------------------------------------------------------------ Valeria de Paiva, | University of Cambridge | Phone: +44 (0)223 334418 Computer Laboratory | Fax: +44 (0)223 334678 New Museums Site, Pembroke Street | JANET: Valeria.Paiva@uk.ac.cam.cl Cambridge CB2 3QG, England, UK | Internet: Valeria.Paiva@cl.cam.ac.uk ------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: About an Outline (1) Date: Fri, 3 Sep 93 18:37:55 +0200 From: lair@frunip62.bitnet (Christian Lair) COMMENTAIRES SUR UN " OUTLINE " (1) C. Wells a recemment rendu public (et disponible par FTP, en ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte intitule: " Sketches : Outline with References ". La version multigraphiee, du 6 juillet 1993, qui m'est parvenue necessite quelques commentaires ... En voici un premier. La PREMIERE esquisse des categories - completement DETAILLEE et en termes tout a fait EXPLICITES - se trouve en [Ehresmann, 1966]. Il est donc pour le moins SURPRENANT que l'Outline affirme peremptoirement (en son point 4.2, lignes 2 et 3) que: " The ideas, not in sketch language, date back to Lawvere [1966] ". Et il est tout a fait INSUFFISANT que l'Outline se limite a signaler innocemment (en son point 4.2, lignes 1 et 2) que: " The sketch for categories is given in detail in [Barr and Wells, 1990] ... and in [Coppey and Lair, 1988] ... ". REFERENCES (autres que celles figurant dans le Outline ) [Ehresmann, 1966] : C. Ehresmann, Introduction to the theory of structured categories, Techn. Rep. 10, Univ. of Kansas, Lawrence (1966). Christian LAIR ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: *** Publication Announcement *** Date: Tue, 7 Sep 93 19:05:21 +1000 From: domv@macadam.mpce.mq.edu.au (Dominic Verity) The following work has recently been re-printed as Macquarie University Mathematics Report no. 93-123: Enriched Categories, Internal Categories and Change of Base. By Dominic Verity (Doctoral Thesis, prepared under the supervision of Dr. Martin Hyland, submitted to Cambridge University in April 1992 and sucessfully examined in July 1992) Copies may be obtained by sending a request to me at the following address: domv@macadam.mpce.mq.edu.au A brief description follows: Part I: Change of Base. ----------------------- As the utility of generalised category theories has become more apparent, a coherent description of what happens when we change our base category has become more and more important. For instance suppose we have two monoidal categories V and W and a homomorphism H:V--->W which admits a right adjoint, then we might ask for a full description of the structures that H induces between the bicategories of V- and W- enriched categories and functors (or profunctors). A closely related problem asks for a description of the analogous structures induced on bicategories of internal categories, fibrations or stacks by a geometric morphism of toposes f:E--->F. Some partial results in this direction are well known, for instance in all of the situations described above we will get biadjoints between the bicategories of generalised categories and functors. In general the action of base change on bicategories of profunctors (aka bimodules) is more difficult to describe. Various "local adjointness" notions, of differing utility and complexity, have been introduced to cope with this problem, but none of these have been fully satisfactory. A significant difficulty with this approach is that a given morphism of bicategories may admit many different local adjoints. Here we give a complete solution to this problem by considering base change structures between bicategories of functors and profunctors at one in the same time. To do this we consider proarrow equipments (M,K,*) (in the sense of Wood [4]), which we show are the objects of a family of closely related "bicategory enriched" categories, called EHom, EMor, EcoMor and EMap. These are bicategory enrichment in the sense that they are enriched in the strict traditional way over the closed category of (small) bicategories and normalised homomorphisms, wherein B^C is the bicategory of normalised homomorphisms, strong transformations and modifications form C to B. Just as in a 2-category we may describe adjunctions equationally in terms of unit and counit 2-cells, we may interpret analogously the notion of (normalised) 'biadjunction' in such bicategory enriched categories. It turns out that base change gives rise to biadjoints of this type in the enriched categories of proarrow equipments introduced above. This fact is proved in detail for all of the (closely related) examples above, culminating in a "comparison lemma" for equipments of stacks. An important application of this framework is a precise general result about the effect of base change on the weighted colimits (or limits) that a generalised category possesses. This follows directly from the description of weighted limits in terms of kan extensions/liftings of profunctors and representability. It is worth pointing out that this approach to base change is very close in spirit to that of Carboni, Kelly and Wood [2] (wherein all bicategories are locally ordered) which it generalises. Part II: Double Limits. ----------------------- At the International Category Theory meeting at Bangor North Wales in 1989, Bob Pare introduced an alternative approach to describing limits in 2-categories which used Double Categories rather than 2-weights (cf [3]). He went on to describe a particularly nice class of well behaved 2-limits, which he dubbed "Persistent" and defined in terms of stability properties with respect to equivalences. These he characterised in terms of structural properties of the double categories that parameterise them. However, while he was able to construct a double category which parameterised the same limits as any given 2-weight, he admitted that constructing a 2-weight from a double category was more problematic. We might think of this as a change of base problem, from Cat-enriched category theory to the theory of categories internal to Cat (which is exactly what double categories are). To make this precise we must first recognise that coequalisers in Cat are not well behaved, being unstable under pullback, so we cannot construct a bicategory of profunctors internal to it. However we can embed Cat in the far better behaved category of simplicial sets and do our work there. All this we make precise and then apply the work of part I to relate closed classes of 2-weights to corresponding classes of double categories satisfying certain closure properties. Once all this has been done it becomes a triviality to provide a "basis" of fundamental double limits which collectively generate the closed class of persistent limits. In fact products and splitting of idempotents along with constructions called inserters and equifiers are enough, which demonstrates that Pare's persistent limits coincide with the Flexible ones defined (in the context of 2-weights) by Street and Kelly et al [1]. References: [1] Bird, Kelly, Power and Street, Flexible limits for 2-categories, JPAA 61, November 1989. [2] Carboni, Kelly and Wood, A 2-categorical approach to change of base and geometric morphisms I, Cahiers de Top. et Geom. Diff. Categoriques 32(1991), 47-95. [3] Pare, Double Limits, July 1989, Unpublished notes from Bangor summer meeting. [4] Wood, Abstract proarrows I, Cahiers de Top. et Geom. Diff. 23-3(1982), 279-290. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: About an Outline (2) Date: Wed, 8 Sep 93 18:29:53 +0200 From: lair@frunip62.bitnet (Christian Lair) COMMENTAIRES SUR UN " OUTLINE " (2) C. Wells a recemment rendu public (et disponible par FTP, en ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte intitule: " Sketches : Outline with References ". La version multigraphiee, du 6 juillet 1993, qui m'est parvenue necessite quelques commentaires ... En voici un deuxieme. Les PREMIERES esquisses de categories munies de choix de limites et colimites de "types" (i. e. de formes d'indexations) "varies" se trouvent en [Lair, 1970] et [Burroni, 1970]. Pour traiter de questions de monadicite que ces premieres esquisses ne permettaient pas de resoudre directement, d'autres esquisses pour ces memes categories (mais mieux adaptees) ont ete construites et tout aussi explicitement detaillees en [Lair, 1977] et [Lair, 1979] (voir aussi [Lair, 1975]). On y trouve egalement les esquisses de categories munies d'extensions de Kan a droite et a gauche de "types" (i. e. le long de foncteurs) "varies". L'Outline fait donc preuve d'une grande NEGLIGENCE en se limitant a signaler INGENUMENT (en son point 4.2, lignes 4 a 8) que : " Many types of categories with extra-structure ... can be sketched by an FL sketch ... These include categories with various types of canonically-chosen limits and colimits ... Some examples are in [Coppey and Lair, 1988], [Barr and Wells, 1985] ... and [Wells, 1990] ". REFERENCES (autres que celles figurant dans le Outline) [Lair, 1970] : C. Lair, Constructions d'esquisses et transformations naturelles generalisees, Esquisses Math. 2, Paris (1970). [Burroni, 1970] : A. Burroni, Esquisses des categories a limites et des quasi-topologies, Esquisses Math. 5, Paris (1970). [Lair, 1977] : C. Lair : These de Doctorat es Sciences, Amiens (1977). Christian LAIR ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: My new address From: Hongde Hu Date: Fri, 10 Sep 1993 08:44:36 -0400 My new address is Hongde Hu Dept. of Math. and Stat. York Univ. 4700 keele St. North York, Ont. Canada M3J 1P3 e-mail: hhu@mathstat.yorku.ca Please send any correspondance to the above address. Hongde Hu ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: About an Outline (3) Date: Mon, 13 Sep 93 17:26:27 +0200 From: lair@frunip62.bitnet (Christian Lair) COMMENTAIRES SUR UN " OUTLINE " (3) C. Wells a recemment rendu public (et disponible par FTP, en ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte intitule: " Sketches : Outline with References ". La version multigraphiee, du 6 juillet 1993, qui m'est parvenue necessite quelques commentaires ... En voici un troisieme, concernant l'esquisse des 2-categories, dont on peut se douter par avance qu'elle ne figure pas qu'en [Power et Wells, 1992]. En [E. Burroni, 1970] sont etudiees les "categories discretement structurees" par une categorie (cartesienne) V : en termes "actuels", elles s'identifient aux categories enrichies par V . On y trouve, notamment, la description detaillee d'une FP-esquisse petite Ecatenrich(x) de sorte que : - pour tout ensemble x et pour toute categorie cartesienne V , la categorie Mod ( Ecatenrich(x) , V ) est equivalente a la categorie des V-categories (petites) qui ont x pour ensemble d'objets (si, E etant une esquisse et C une categorie, on convient de noter Mod ( E , C ) la categorie des modeles de E dans C ). En particulier, si V = Cat , la categorie Mod ( Ecatenrich(x) , Cat ) est donc equivalente a la categorie des 2-categories (petites) qui ont x pour ensemble d'objets. En [Ehresmann, 1963a et 1963b] sont introduites les (petites) categories structurees par une categorie (localement petite et finiment complete) C : en termes qui se veulent "modernes", elles s'identifient aux categories internes a C . De la sorte, la categorie des petites categories structurees par C est equivalente a la categorie Mod ( Ecat , C ) (si Ecat designe l'esquisse des categories). Par exemple, les categories doubles de [Ehresmann, 1963a et 1963b] sont les categories structurees par C = Cat : ainsi, une categorie double a une "categorie des objets", une "categorie des fleches" ... En particulier, les 2-categories s'identifient donc aux categories doubles dont la "categorie des objets" est une categorie discrete. De [Lair, 1975b] ressort (notamment) que la categorie des FP- esquisses petites est munie d'une structure monoidale, symetrique et fermee "canonique" (parmi d'autres). De sorte que le produit tensoriel "canonique" EoE' , d'une FP-esquisse petite E par une autre FP-esquisse petite E' , verifie : - si C est une categorie localement petite et finiment complete, alors les trois categories Mod ( EoE' , C ) , Mod ( E , Mod(E',C) ) et Mod ( E' , Mod(E,C) ) sont "canoniquement" quivalentes. En particulier, si C = Ens , si E = Ecatenrich(x) et si E' = Ecat , alors on voit IMMEDIATEMENT que Ecatenrich(x) o Ecat est une esquisse pour les 2-categories qui ont x pour ensemble d'objets. De meme, si C = Ens et si E = E' = Ecat , alors on voit IMMEDIATEMENT que Ecat o Ecat est une esquisse pour les categories doubles. Comme il est PLUS QUE TRIVIAL de construire une esquisse Ecatdis pour les categories discretes, il est PLUS QU'ELEMENTAIRE d'en deduire qu'une somme (fibree) convenable "au dessus de Ecat " : (Ecat o Ecat) + Ecatdis Ecat est une esquisse pour les 2-categories. L'Oultine est donc PLUS QU'INCONSEQUENT en se contentant de proclamer NAIVEMENT (en son point 4.2, lignes 3 et 4 ) que: " The sketch for 2-categories is given in [Power and Wells,1992]" . REFERENCES (autres que celles figurant dans le Outline) [Ehresmann, 1963a] : C. Ehresmann, Categories structurees, Ann. Sc. Ec. Norm. Sup., t. 80, pp. 349-426, Paris (1963). [Ehresmann, 1963b] : C. Ehresmann, Categories doubles et categories structurees, Note aux C.R.A.S., t. 256, pp. 1198-1201, Paris (1963). [E. Burroni, 1970] : E. Burroni, Categories discretement structurees et triples, Esquisses Math. 5, Paris (1970). Christian LAIR ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Modules mailing list Date: Mon, 13 Sep 93 18:43:44 -0700 From: rowan@garnet.berkeley.edu I am thinking of establishing a modules mailing list, and account where a bibliography of articles on modules (as apparently first defined by Jon Beck, unless it was folklore before that) could be found. If anyone is interested in being on such a mailing list, please send mail to rowan@garnet.berkeley.edu along with any comments. Best regards, Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: The paper `Outline of Sketches' Date: Tue, 14 Sep 93 20:00:39 -0400 From: cfw2@po.CWRU.Edu (Charles F. Wells) C. Lair has been sending messages to this mailing list with additions and corrections to my article, "Sketches : Outline with References", which I prepared for the joint meeting on universal algebra and category theory in Berkeley last July. It is available by ftp from ftp.cwru.edu in the math/wells directory (the file is sketch.dvi). The paper is a preliminary version, and in the next few weeks I intend to post a list of addenda and corrections to the paper in the same directory. They will include the papers Lair has mentioned, although I do not have most of them. (I expect to request some of them by interlibrary loan.) They will also include some other papers that I am embarrassed at having omitted because I DO own them. (These include papers by T. Fox and P. Johnstone and at least one by C. Lair.) Perhaps sometime next year I will have time for an extensive rewrite of the paper. It has expository shortcomings as well as bibliographical ones and I would hope to remedy some of them. Any suggestions concerning expositions and bibliography will be welcome. The paper begins, "This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given." It is NOT A HISTORY OF THE DEVELOPMENT OF SKETCHES. I did not claim that it was a history and I do not intend to make it one. The main emphasis is to describe papers in the literature that I think will be usable by those who want to learn about the subject, including people in universal algebra, theoretical computer science and other areas who would perhaps find expository papers and books more usable than the original sources. Even so, I want the bibliography to be inclusive and so appreciate being told of papers not mentioned. One specific comment: Any category theorist with some experience with sketches could derive the sketch for 2-categories. The paper by Power and me mentioned by Lair includes that sketch explicitly because our paper is written for theoretical computer scientists, who might not find it so easy to come up with the sketch. -- 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: categories & baseball Date: Wed, 15 Sep 93 10:36:37 EDT From: fox@triples.Math.McGill.CA (Thomas F. Fox) The Montreal Expos are four and a half games out of first. Why should category theorists care about this? Because that means they are threatening to play baseball in October, which will make it difficult to find hotel rooms for the CATEGORY THEORY OCTOBERFEST at MCGILL, OCT 9-10. So be sure you make your reservations as soon as possible. If you need a list of hotels or further information, please let me know. - Tom Fox fox@triples.math.mcgill.ca ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Is this result published somewhere? Date: Wed, 15 Sep 93 16:33:27 +0200 From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens) Can somebody give me a reference to a proof of the following result (or some generalisation of it)? (I need this in my research, and if a proof is published somewhere, I don't have to do the proof myself) Let C be a small category and let F:C -> Set be a functor. Then, the slice category Fun(C,Set)/F is equivalent with Fun(G(C,F),Set) where G is the Grothendieck-construction for Set-valued functors as described in "Category theory for computing science" (Barr and Wells) chapter 11. As a simple example, take C=1, and one becomes the well-known fact that Set/A is equivalent with Fun(A,Set) (A regarded as a discrete category). If this is "well-known" among category-theorists, without any published proof around, please let me know this too. Thanks in advance, Frank Piessens Dept. of Computing Science Katholieke Universiteit Leuven ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Is this result published somewhere? (3 posts) Date: Thu, 16 Sep 93 12:25:37 +1000 From: street@macadam.mpce.mq.edu.au >From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens) >Can somebody give me a reference to a proof of the following >result: >Let C be a small category and let F:C -> Set be a functor. >Then, the slice category Fun(C,Set)/F is equivalent with >Fun(G(C,F),Set) where G is the Grothendieck-construction . . . This result has been known to category theorists for a long time, and probably appears somewhere in SGA 4. For a published reference with two-line proof (given a little knowledge of fibrations) see my Proposition (7.3) page 293 of "Cosmoi of internal categories" Transactions AMS 258 #2 (1980) 271 - 318. This result was used to produce an algorithm for finding all internal full subcategories of a presheaf category. Moreover, I pushed up the equivalence to the case where C supports a sketch (I called sketches "Gabriel theories" in that paper with a footnote giving the other terminology). If F is a model of the sketch, one obtains a sketch on your G(C,F), and an equivalence Mod(C,Set)/F ~~~ Mod(G(C,F),Set). See Proposition 7.21 on page 297. I used this result to produce an algorithm for finding all internal full subcategories of certain locally presentable categories (eg, Cat). I also used it to characterize the sketches whose model categories had cartesian closed slice categories (Theorem 7.25). [I mention the sketch result because sketches seem to be of interest to computer scientists.] An interesting example of the original equivalence is the case where C is a group and F is a G-set, so that G(C,F) is the wreath product; or, if F is a G-module, G(C,F) is the semidirect product. Regards, Ross ++++++++++++++++++++++ Date: Thu, 16 Sep 1993 12:11:30 +0100 (BST) From: Roy Crole In reply to Frank Piessen's question about the equivalence of [C,Set]/F and [G(F),Set]. I would say that this result is well known, though I am not so familiar with the original literature. A discrete fibration p : E--->C (over C) is a functor for which given any morphism f : A---> A' in C and X' in E with p(X') = A', there is a unique v : X---> X' in E with p(v) = f. Then there is a category DFib/C of discrete fibrations over C, and it is well known that DFib/C <--equiv--- [C^op, SET] : G , this being the analogue of the more commonly cited result for split fibrations. One half of the above equivalence is given by the Grothendieck construction, which induces an equivalence on the slices ( DFib/C ) / ( G(F)---> C) <---equiv--- [C^op,Set] / F. If p = p'' o p' is a composition of discrete fibrations, then p' is a discrete fibration; thus ( DFib/C ) / ( G(F)---> C) equiv DFib / G(F) and this gives the result (modulo op's !) Of course, proving the result directly is easy enough, I guess. Roy Crole +++++++++++++++++++ From: Richard Wood Date: Thu, 16 Sep 1993 08:39:57 -0300 The result appears in Diaconescu's thesis (Dalhousie '73) with `Set' generalized to an elementary topos, Proposition I.1.5 . From the preceding text one gets the impression that the result was already well known for `Set' but none of Diaconescu's references are likely sources. Diaconescu's proofs of results about "internal category theory" contain truly marvelous diagrams the like of which we will probably not see again until various TeX issues are settled. At the time it seems that rigor demanded them but that was before indexed categories, fibrations and languages were well understood. It is now clear that a careful proof of the result in question, written as for sets, suffices. It would be useful to have the result and similar ones proved in a text or expository paper. I do not know of one. RJ Wood ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Octoberfest Date: Sun, 19 Sep 93 17:15:03 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Here is the final list of speakers. No more will be added. As you see we have 19 and that is really too many already. The talks will begin at 9:00 each morning and the Sunday session will end at 2:40. A detailed schedule will be distributed in a few days. Michael Giulio Katis 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 <> 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 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 Tau Categories and Logic Programming 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!) Jim Otto Categories and complexity Phil Scott Coherence and Undecidability for CCC's Abstract: (Joint Work with M. Okada) We show the equational theory of simply typed lambda calculus with strong natural numbers object is undecidable, thus the coherence problem for equality of arrows in the free ccc with NNO is undecidable. We study the rewriting theory (made equational by Lambek's use of Mal'cev operators) and prove in fact the appropriate lambda calculus is not Church-Rosser, but is Strongly Normalizing. The latter proofs require heavy rewriting techniques. Jonathon Funk The display locale of a cosheaf Peter Freyd Hardware design and free allegories. Kimmo Rosenthall TBA Martin Markl <> TBA Wim Ruitenburg Yet another constructive logic Andre Joyal How to complete a category by freely adjoining all limits and colimits ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Modules list Note from Moderator: Apologies to subscribers and the network gods for the typing error which resulted in resending the categories and baseball message (the Expos were four games back this morning.) ++++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: Sun, 19 Sep 93 13:59:16 -0700 From: rowan@garnet.berkeley.edu The response to my proposed modules mailing list has been very gratifying (almost 20 people). I have decided to go ahead with it, at least for a trial period. The focus of the mailing list will be modules (Beck's definition of an A-module, for an object A in a category C, is an abelian group object in the category of objects of C over A) and the corresponding pointed set objects. I have my own equivalent definition, and call the pointed set objects pointed overlaying algebras. I am personally most interested in modules and pointed overlaying algebras over universal algebras A, and interactions with the structure of A. (For example, congruences of A naturally give rise to pointed overlaying algebras, and frequently to modules.) However, the subject where this theory first got started was cohomology theories. To start with, there are two activities I want to pursue: (1) Discussion. Please contribute opinions, queries, announcements, whatever, concerning modules and pointed overlaying algebras. (2) Creating a bibliography of articles on this subject, with brief abstracts. Please let me know of your work in this area, and how preprints may be obtained. Hopefully, I will be making this bibliography available on the network. To become a charter subscriber, send e-mail to rowan@garnet.berkeley.edu. I am not yet ready to receive contributions, and will make another announcement when I am ready. Thank you, Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: This attached memo Date: Thu, 23 Sep 93 08:02:22 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Can someone help this guy out with this? > Date: Thu, 23 Sep 93 12:45:19 +0200 > From: ldup@alcbel.be (Luc Duponcheel) > To: barr@Math.McGill.CA > > Michael, > > > I have a simple question. I work in the following `framework' : > > > A category which is such that all Hom(A,B) are themselves categories > and having the following properties : > > > first some notation > ------------------- > > morphisms are denoted as F : A -> B, G : C -> D, ... > and their composition is denoted as GF > > morphisms in Hom(A,B) (called transformations) > are denoted as alpha : F -> G, beta : H -> K > and their composition is denoted as beta . alpha > > > here come the actual properties > ------------------------------- > > 1) for all transformations alpha : F -> G where F : A -> B and G : A -> B, > and all morphisms H : B -> Y there exists a transformation > alpha H : FH -> GH. > > 1a) GF alpha = G (F alpha) > 1b) F (beta . alpha) = F beta . F alpha > > 2) for all transformations alpha : F -> G where F : A -> B and G : A -> B, > and all morphisms H : X -> A there exists a transformation > H alpha : HF -> HG. > > 2a) alpha GF = (alpha G) F > 2b) (beta . alpha) F = beta F . alpha F > > 3) G (alpha F) = (G alpha) F > > > > > > BTW > --- > > a transformation beta : H -> K is natural if for all transformations > alpha : F -> G one has K alpha . beta F = beta G . H alpha > > > > > > These axioms are a subset of the ones used for 2-categories. > The transformations do not need to be (but may, of course, be) natural. > I do not, a priory, need any `horizontal' composition "*" of transformations. > > > > One of the results which I want to prove in this framework > is just the fact that certain transformations (who do not a priory need to > be natural at all) are nevertheless, under certain conditions (of a > different nature) natural. > > > Is there any *name* for this `framework'? > > > I could call it *categories with transformations* but if there is > any other name which is commonly used, then I would appreciate if you > can inform me about it. > > > > Thanks! > > > > Luc. > ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Memo of Luc Duponcheel Date: Fri, 24 Sep 93 11:04:09 +1000 From: street@macadam.mpce.mq.edu.au Luc seems to be describing what, in our work on rewrites etc, Eilenberg wanted to call a one-and-a-half-category, and I changed (and I'm not sure how Sammy feels about it) to sesquicategory when writing the article "Categorical structures" for "Handbook of Algebra" Volume 2 (Elsevier, North Holland). [The preprint is dated Nov 1992 but the volume probably won't appear until late 1994!] There are two symmetric monoidal closed structures on the category Cat of categories. One is the usual cartesian closed structure where the internal hom [A,B] is the usual category of functors A --> B and NATURAL transformations between these. The other closed structure (which I call the "funny" one) has internal hom [[A,B]] the category of functors and transformations between these (which consist of the data for a nat trans without the naturality requirement). This funny structure WAS an example in Eil-Kelly "Closed categories" La Jolla 1965. Carolyn Brown has used the funny structure in her work on Petri nets. Sesquicategories are V-categories where V is Cat with the funny monoidal structure. John G. Stell tells me he has used sesquicategories in computer science, and apparently independently came up with the same name for them. Perhaps of future interest to those interested in sesquicats are the various monoidal structures on 2-Cat. For example, there is John Gray's tensor product of 2-categories, where "natural", instead of being dropped, is replaced by "lax natural". A recent paper of Gordon, Power, Street looks at the case where V = 2-Cat with the monoidal structure obtained by replacing "natural" by "pseudonatural"; we prove that every tricategory is equivalent (in the approp sense) to a V-category. Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: tentative timetable for Octoberfest Date: Fri, 24 Sep 93 07:27:44 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Saturday, October 9 ---------------------------------------------------------------------------- Chair: J. Lambek 9:00 - 9:45 Andre Joyal, How to complete a category by freely adjoining all limits and colimits 9:50 - 10:20 Martin Markl, Deformations of everything 10:20 - 10:50 Break Chair: A. Joyal 10:50 - 11:20 Wim Ruitenburg, Yet another constructive logic 11:25 - 11:55 L Gaunce Lewis Jr, Equivariant Freudenthal suspension theorem 12:00 - 12:30 Robin Cockett, Copy Categories 12:30 - 2:00 Lunch Chair: R. Wood 2:00 - 2:30 Richard Wood, Distributive adjoint strings 2:35 - 3:05 Phil Scott (and M. Okada), Coherence and Undecidability for CCC's 3:10 - 3:40 Rick Blute, (Robert Seely and Robin Cockett) Contextual Logic 3:40 - 4:10 Break Chair: R. Rosebrugh 4:10 - 4:40 Andreas Blass, TBA 4:45 - 5:15 Bob Gordon (and John Power), Enrichment Through Variation 5:20 - 5:50 Jon Beck, TBA 6:30 - Reception Sunday, October 10 --------------------------------------------------------------------------- Chair: T. Fox 9:00 - 9:45 Till Plewe, When a locale product of metrizable spaces is spatial 9:50 - 10:20 Jonathan Smith (and A. Romanowska), Duality for semilattice representations 10:20 - 10:50 Break Chair: P. Scott 10:50 - 11:20 Peter Freyd, Hardware design and free allegories 11:25 - 11:55 Stacy Finkelstein, Tau Categories and Logic Programming 12:00 - 12:30 Kimmo Rosenthall, TBA 12:30 - 1:00 Break Chair: R. A. G. Seely 1:00 - 1:30 Jonathon Funk, The display locale of a cosheaf 1:35 - 2:05 Djordje Cubric, Interpolation property for bicartesian closed categories 2:10 - 2:40 Jim Otto, Categories and complexity Jon Beck TBA Andreas Blass TBA Rick Blute Contextual Logic (joint with Robert Seely and Robin Cockett) 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!) Djordje Cubric Interpolation property for bicartesian closed categories Stacy Finkelstein Tau Categories and Logic Programming Peter Freyd Hardware design and free allegories Jonathon Funk The display locale of a cosheaf Bob Gordon Enrichment Through Variation (joint with John Power) Andre Joyal How to complete a category by freely adjoining all limits and colimits L Gaunce Lewis Jr Equivariant Freudenthal suspension theorem One of those nice situations when just a little touch of category theory cleans up a mess in topology. Martin Markl <> Deformations of everything Jim Otto Categories and complexity Till Plewe <> When a locale product of metrizable spaces is spatial Kimmo Rosenthall TBA Wim Ruitenburg Yet another constructive logic Phil Scott Coherence and Undecidability for CCC's (joint with M. Okada) We show the equational theory of simply typed lambda calculus with strong natural numbers object is undecidable, thus the coherence problem for equality of arrows in the free ccc with NNO is undecidable. We study the rewriting theory (made equational by Lambek's use of Mal'cev operators) and prove in fact the appropriate lambda calculus is not Church-Rosser, but is Strongly Normalizing. The latter proofs require heavy rewriting techniques. 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. Richard Wood Distributive adjoint strings ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: change in chairs Date: Sat, 25 Sep 93 09:37:24 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Please make the following changes in chair assignments: Wood --> Sunday at 9 Fox --> Saturday at 2 Sorry about that. --MB ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: What do you call this? Date: Fri, 24 Sep 93 23:51:34 -0700 From: rowan@garnet.berkeley.edu I call a functor F, such that F is one-one and onto on objects, and F sends each hom(a,b) ONTO hom(Fa,Fb), a _cofaithful_ functor. I call it that because every functor has an essentially unique decomposition as a faithful functor, followed by a cofaithful one. The question is, what do YOU call this? I would like to use the standard terminology if there is one. Thanks, Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: On UA-CAT Workshop at MSRI Note from moderator: The following was received as a typescript from Saunders Mac Lane, with a request to post it, so typographical errors are mine. He mentions that his e-mail does not work well, and I will forward a copy of any discussion which ensues here to him, but of course he can also be contacted directly by mail. Bob Rosebrugh +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ From: Saunders@math.uchicago.edu Date: Sept. 25, 1993 RANDOM THOUGHTS ON THE RECENT UA-CAT WORKSHOP AT MSRI Saunders Mac Lane Dept. of Math, Univ. of Chicago, Aug, 1993 In addition to drafting a final report on this workshop, I'll venture here to ask some provocative questions about the interrelation between Universal Algebra (UA), Category Theory (CAT) and the connections of both subjects to all (!) the rest of mathematics. I hope thereby to provoke both strong dissent and some progress and innovation. First those minor remarks about terminology. CLONE vs. ALGEBRAIC THEORY. The notion of a clone - for each n, all n-ary operations of the intended type of algebra - is due to Philip Hall (apparently unpublished by him). Lawvere's notion of an "algebraic theory" (his thesis, 1963) is a category, objects the natural numbers, maps n-->m given by all m-tuples of term operations. These two notions are essentially equivalent; the word "clone" is short and convenient, while the Lawvere notion includes more composites and so is more flexible and easier to formulate and to generalize. It follows one of the tacit slogans of CAT: With every map, present at once both the domain and the codomain, by this effective presentation making the whole situation more visible and showing what composites are possible. This is the useful adage "Draw Diagrams! This art should be easy to learn; but as for the choice of words, I suppose that both "clone" and "Algebraic theory" will continue in use, hopefully with a more general recognition of their equivalence. "GROUPOID" as the name of a category with every arrow an isomorphism, apparently appeared first in 1926 in a study of the ideals of algebraic integers in a non-commutative algebra (Brandt). This notion has vital uses in many parts of mathematics, as for example for the fundamental groupoid of a space, and so combining the fundamental groups at each base point. It appears also in differential geometry and in mechanics, as a means of transporting structures along one of the paths of the space at hand. In UA, on the other hand, a groupoid has meant a set with a single binary operation--and no required identities; it may be that this usage first appeared in a book by R. H. Bruck. Such objects do need a name, but clearly the categorical use of "groupoid" is the widest and most important. I hope that the other use can be replaced by some other suitable term; possibly "binary algebra" or perhaps even "binoid". The "empty algebra" has been proscribed in UA. Since there is now occasion to consider models of algebras not just in sets (possibly in topoi or in regular categories) this early proscription seems too narrow. WHAT IS AN ALGEBRA? CAN WE LOOK BEYOND THE PRESENT? The Clone-algebraic theory notion is now evidently too narrow; no infinitary operations. The algebras for a Monoid (=triple, as in Linton's lecture) are somewhat more general and much neater (adjoints again!, cf. my "Categories Work:"). The algebras described by sketches are considerably more general. Recall the lecture by Wells, where a sketch together with its models is described by a graph, together with a set of diagrams there (to commute in the models), a set of cones (to be limits) and a set of cocones (colimits). This notion is flexible, especially because it comprises colimits as well as limits. The lecture by John Gray indicated how this works for computer science; see also recent books e.g., by Barr-Wells. The Workshop did not get to consider how standard concepts of UA would work out for sketches. Much more might develop here! WHERE DO THE MODELS OF ALGEBRAS LIE? Classical model theory says: "in sets," with a spectrum. There have been a few starts at taking models in a general topos. Does this provide for greater "Variety"? MALTSEV ALGEBRAS are conventionally described by those familiar term identities--but their bearing on the composition of relations is relevant, while a recent paper of Carboni, Lambek and Pedicchio (JPAA, 69, 1991) indicates how a category of such algebras can be characterized as a kind of regular category. This offers a striking connection; see also a preprint, Kelly et al, on Goursat categories. Many prospects might open up here. TAME CONGRUENCE THEORY provides remarkable and surprising results for finite algebras (those astonishing five types of possible quotients for a covering). But finite algebras have a somewhat limited scope; perhaps something equally "Tame" could be managed for some classes of infinite algebras, possibly by the well-tried method of introducing a topology? FINITE CLONES on two elements (E. L. Post) bear on the problems of "truth values"; but there appear to be too many clones on bigger finite sets; it is not yet clear to me how they relate to other mathematics. COMMUTATOR THEORY is a fascinating development, especially in view of its relation to commutators in group theory--and what about Lie algebras and the Lie bracket here? At the end of the workshop M. C. Pedicchio mentioned to me a somewhat more categorical description of such commutator theory. SUBOBJECT LATTICES and CONGRUENCE LATTICES seem to arise almost separately in UA. One merit of categories is that they bring together the inclusions (subobjects) and the epimorphisms (quotients), both appearing as arrows. Does this view have consequences? Is it simpler? INTERACTIONS. Some of the most striking developments of mathematics come from unexpected connections (collisions) between different subjects. I know of several such collisions in CAT: The use of categories in the drastic reformulation of algebraic geometry by Grothendieck, the connection to Quillen Model structures in topology (Tierney's lecture) and the recent use of coherence theorems for commuting diagrams as they appear in quantum field theory and quantum groups (which are really not groups but deformed Hopf algebras). Then there is computer science: sketches, categories expressing data types and the newer uses of topoi there. I do not know of similar outside connections for UA. How should one search and in which other fields? In Logic? SIMPLICIAL SETS were designed for calculations for Eilenberg-Mac Lane spaces, defined as functors, and now are used as surrogate spaces! CONCEPTS: Emmy Noether's "modern algebra" achieved much, and in particular emphasized the astute use of general concepts--the homomorphism theorems, for example, to illuminate and make understandable known mathematical results. This was in particular the case with Galois theory, which before had been expressed in terms of obscure "substitutions", but was then organized in terms of Automorphisms to become more perspicuous, as later in Emil Artin's presentation. The same clarification came to ideal theory, done jointly for algebraic numbers and for algebraic varieties. I personally hold that a major role of "abstract" mathematics is this sort of clarification and understanding--both for its own sake and for the new results it can and does encourage. The recent reformulation of algebraic geometry is a striking example of this. The notion of an ADJOINT FUNCTOR (for posets, Galois correspondence) is one of the most pervasive and effective notions of CAT (recall Joyal's lecture on the Witt vectors as an adjoint). TWO-DIMENSIONAL categories (with 2-cells to represent things such as homotopies) and the related higher dimensional categories, originally seemingly complicated, have now found much use (Street's lecture), as in the solutions of Yang-Baxter equations, and in coherence theorems. PROBLEMS vs. CONCEPTS. Sometimes mathematics is viewed as the construction and astute solution of hard problems. When they are indeed of central interest and make use of powerful methods, as in the recent case of Fermat, this is splendid. On the other hand, not every hard problem is necessarily relevant to the progress of knowledge. How to select? The PARTICULAR vs. the GENERAL. Some categorical studies are troubled by a sort of "dual" objective--the study of special problems vs. the formulation of general notions directed at wide aspects of mathematics - what is a theory, for example. Lawvere's lecture at the Workshop emphasized the general, in the guise of a Doctrine (a category with structure) or a hyperdoctrine. These questions may have some bearing on education: in teaching, should we present just the particulars and just the corresponding skills, or somehow try to convey the general in its particular guise. Such questions are provocative and contentious, but nevertheless useful. Learn it all young! This is now possible; recently, Lawvere and Schanuel have prepared an elementary text on "Conceptual mathematics." A TOPOS is a category with good structure (finite limits, exponentials. subobject classifier). It encompasses sheaf theory, yields an internal logic with a Heyting algebra of truth values, while sheafification gives a formulation of Cohen forcing, Topoi have recently been used in modelling computer science; a topos thus exemplifies many connections. Thus arises my hope that we can search here and elsewhere for decisive new interactions. The growth of ideas can be splendid! Saunders Mac Lane ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: one minor addition Date: Tue, 28 Sep 93 21:02:08 EDT From: James Stasheff to Saunder's description of clones,etc. recent work in String Field Theory has found use not only for operads (the pieces which assemble into a monoid) corresponding to n-ary ops but something that is new (at least to me): think of rooted trees as desribing n-ary ops with composition given by grafting root to branch e.g. associative algebras can be described by planar trees and Lie algebras by abstract trees then trees with edge lengths on the internal edges can be used to described strong homotopy analogs now what if we allow more general graphs with grafting allowed between any two branches as well has anyone other than a string field theorist ever seen such algebra?? jim ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Announcement Date: Wed, 29 Sep 93 11:58:42 EDT From: SCPSG@acadvm1.uottawa.ca Dear Colleagues: The annual winter meeting of the Canadian Mathematical Society is being held this year in Ottawa, Ontario, December 11- 13. I was asked to organize one of the special sessions, entitled Categorical Logic and Theoretical Computer Science. The speakers and titles are: M. Barr: Pontrjagin Duality Revisited A. Blass: Geometric Morphisms R. Cockett: Categorical Recursion Theory P. Freyd: tba A. Joyal: Completion of Categories and Communication Games M. Makkai: Contributions to the Theory of Doctrines. R. Pare: Dinatural Numbers. A. Pitts (Plenary Speaker): Category Theory and the Semantics of Programming Languages. R. Rosebrugh: Functorial Aspects of Relational Databases. A. Scedrov: Relators R. Seely: Coherence of Bimonoidal Categories In case anyone is thinking of coming, I enclose some hotel data. The fee schedule, other technical information about the meeting, other special sessions, etc. are available in the CMS Notes (September issue), or you can contact me. Fees are cheaper if you register soon. Philip J. Scott Dept. of Mathematics University of Ottawa 585 King Edward Ottawa, Ont. Canada K1N 6N5 e-mail: scpsg@acadvm1.uottawa.ca FAX: 613-564-3822 --------------------------------------------------------- THE MEETING The meeting is being held in the Westin Hotel, a somewhat upscale hotel in downtown Ottawa, connected to Rideau Centre Mall. There are lots of shops and restaurants in the mall and no need to wear heavy clothes if you stay indoors. Rideau Mall is right downtown (but, for walkers, December weather can be quite unpredictable). For American visitors: all prices quoted are CANADIAN DOLLARS, currently 75 cents U.S. Hotels: Westin Hotel (1-800-228-3000): Conference Centre. Winter Canadian Math Society (CMS) prices: $99.00 Can. (single), $109.00 Can. (double) per night. Make sure you get the CMS prices. The following hotels/BB are very close by (note the Novotel and B & B below): NOVOTEL (1-800-221-4524): across the street from the Rideau Centre: 2 minutes walk to the Westin. Not as fancy as the Westin, but modern. I got a SPECIAL PRICE for rooms: (must be booked before Nov. 10: Ask for Canadian Math. Society rate: Jim Tanguay, Manager) $75.00 Can. (single or double) per night. LES SUITES (1-800-267-1989): Executive Apartment Suites. These have full apartments, and are brand new. Across the street from Rideau Centre, next door to Novotel. Weekend Specials: 1 bedroom apartment: $85.00 Can. per night. 2 bedroom apartment: $125.00 Can. per night. B&B: Gasthouse Switzerland: (613-237-0335) A European-run Bed and Breakfast. About 3 minute walk to Rideau Centre, less than 5 minutes to the Westin hotel. $58.00 single room, with breakfast and private bath. $68.00 double room, with breakfast and private bath. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Memo of Luc Duponcheel Date: Wed, 29 Sep 93 08:58:15 +0100 From: grandis@cartesio.dima.unige.it. (Marco Grandis) I used recently the following hierarchy of notions (one of them coinciding with L.D.'s one) as abstract settings for homotopical algebra. A synthesis is given in Cahiers Top. Geom. Diff. Cat. 33 (1992, 135-175 (ch. 5-6); a longer 1991 preprint on the subject is no longer available, but a new version will soon be ready. - "h-category", or "category with homotopies". (This notion goes back to K.H. Kamps (Manuscripta Math. 3 (1970), 237-255), who used it in a slightly different, equivalent form under the name of "generalized homotopy system"; it is an extension of a Kan "homotopy system", or category with cylinder endofunctor) Formally, an h-category is a category enriched over Reflexive Graphs, with a suitable monoidal closed structure. Concretely, an h-category has objects, maps f: A -> B and "cells" (or homotopies, or transformations) alpha: f -> g (for f: A -> B). Objects and maps form a category; further, there is a "vertical identity" id f: f -> f for every map and a "reduced horizontal composition" of cells with maps: k alpha h (for h: A' -> A, k: B -> B') under the obvious axioms for identities and associativity (id B) alpha (id A) = alpha, k (id f) h = id (kfh), (k'k) alpha (hh') = k' (k alpha h) h'. Of course one may separate the left and right composition of cells with maps: k alpha, alpha h. Note that there is no vertical composition. - h1-category = h-category + vertical involution (under some weak axioms) - h2-category = h-category + vertical composition (under some weak axioms; for instance the vertical comosition is not required to be associative) (strict h2-category = h2-category with axioms for vertical identities and vertical associativity = sesquicategory in the sense of R. Street's reply = (probably) "category with transformations" in the sense of L.D.'s message - h3-category = h-category + vertical involution and composition (under some weak axioms) (strict h3-category = h3-category with axioms for vertical identities, vertical inverses and vertical associativity = category enriched over groupoids, with the "funny" structure = sesquigroupoid? - h4-category = h3-category + second-order homotopy relation making it a sort of relaxed 2-category. To write down the complete definitions would be too long here. But the following examples should be sufficient to make them clear, and also to motivate them a) Top = Topological spaces, continuous maps and homotopies. It is h4, not strict - the vertical identities, involution and composition just "work well" up to second-order homotopy. b) C*A = Chain complexes (over a preadditive category A), chain maps, homotopies. It is h4 and strictly h3 (the vertical composition of homotopies is obtained from the sum, and works well in the strict sense). This sesquicategory could be of interest for L.D., as an algebraic model of his situation. The presence of a vertical involution (strictly well behaved) should also be of interest. c) Dga = Differential graded algebras (over some unital ring), their homomorphisms and homotopies. It is just an h-category - the multiplicativity conditions on homotopies prevent to reverse or add them. Best regards, Marco Grandis ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: AMAST Workshop on Real-Time Systems Date: Wed, 29 Sep 93 17:14:59 BST From: Joseph.Goguen@prg.oxford.ac.uk The following seems relevant to Mac Lane's remark that he does not know of applications of universal algebra. In fact, in Computer Science, the line between UA and CATH is not so firmly drawn as in Mathematics, and there is actually a bias towards using the least sophisticated formalism possible for a given application. Joseph &&&&&&&&&&&&&&&&&&&&&&&&&&&&& Signature File &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Joseph A. Goguen, Professor of Computing Science, Programming Research Group, University of Oxford, 11 Keble Road, Oxford OX1 3QD, United Kingdom. email: Joseph.Goguen@prg.ox.ac.uk [internet] -- usually also works in the UK, but if not, try Joseph.Goguen@uk.ac.ox.prg phone: 272567 [my office]; 272568 [secy]; 273838 [PRG office]; 273839 or 272582 [FAX]. From USA, dial 011-44-865-...; from UK, dial (0865)-... ****************************************************************************** ****************************************************************************** Date: Tue, 28 Sep 1993 17:57:23 +0100 To: concurrency@nl.cwi Subject: AMAST Workshop on Real-Time Systems From: Teodor Rus Sender: fritsv@nl.cwi \documentstyle [11pt]{article} \textwidth = 6.5in \textheight = 8.5in \topmargin = -0.3in \topskip = 0in \oddsidemargin = -0.1in \evensidemargin = -0.1in \addtolength{\parskip}{0.5ex} \begin{document} \begin{center} {\LARGE {\bf PROGRAM}}\\ \ \ \\ {\Large {\bf First AMAST International Workshop on Real-Time Systems}}\footnote {This conference is sponsored by grants from the National Science Foundation, Office of Naval Research, ESPRIT Basic Research Programme, University of Iowa, and University of Twente. }\\ {\Large {\it 1--3 November 1993, Iowa City, Iowa, USA}} \end{center} \medskip\noindent{\bf Organizing Committee:} \begin{quote} \begin{tabbing} Maurice Nivat, University of Paris VII, France\\ Charles Rattray, University of Stirling, Scotland\\ Teodor Rus, University of Iowa, Iowa City, IA, USA\\ Giuseppe Scollo, University of Twente, The Netherlands \\ \end{tabbing} \end{quote} \noindent {\bf Aim:} Dedicated real-time applications form one of the areas of great practical accomplishment of current computer technology. Real-time applications, however, bring to the fore new and intriguing questions regarding program specification, verification, and development. Correctness of solutions to the problems raised by real-time programming is particularly important due to the catastrophic nature of failure in real-time systems. This motivates the extensive work in the past decade on the formal theory of specification, verification, and development of real-time systems. At the same time, the AMAST movement, initiated in 1989 and aiming to use algebraic methodology for the development of software technology, has started to show practical results. The goal of this workshop is to expand the AMAST results to real-time system development, by: \begin{enumerate} \item Providing a forum for a dialog on the suitability of using algebraic methodology for real-time system development. \item Tracing the directions of a unifying approach for real-time system development within the framework provided by universal algebra. \item Promoting the integration of real-time system development within software technology based on the new algebraic methodology which is emerging from an AMAST approach. \end{enumerate} It is the intention of the organizers to publish the research reported at this workshop in a {\it Handbook on Real-Time System Development} in the AMAST Series in Computing. The feasibility of this project will be discussed in the special sessions scheduled during the workshop. We invite contributions to these discussions and submissions to the handbook from all attendees of the workshop. \medskip\noindent All meetings of this workshop will take %place in the room 345 (Northwestern) place at the Iowa Memorial Union. Each talk presented at this workshop will be 50 minutes long, followed by 10 minutes discussion. Supplementary discussion time will be provided in special sessions. \medskip\noindent Continental breakfast will be served each morning 8:30--9:00 at the meeting room. Lunch will be served each day 12:30--1:30 in BF 236, Second Floor, Iowa Memorial Union. \newpage {\small \noindent {\bf Monday, November 1-st, 9:00--12:30 Session 1} \medskip\noindent $\spadesuit$ 8:00--8:30 Registration and breakfast \medskip\noindent$\spadesuit$ 8:30-9:00 Opening address by Prof. David J. Skorton, Vice President for Research, The University of Iowa. \medskip\noindent 1. 9:00--10:00 {\it Finite Automata, Omega-Languages and Distributed Systems} by Maurice Nivat, Universit\'{e} Paris 7, France. \medskip\noindent 10:00--10:15 Coffee break \medskip\noindent 2. 10:15--11:15 {\it Issues in the Specification and Verification of Telephone Systems} by Luigi Logrippo, Department of Computer Science, University of Ottawa, Ottawa, Ont, Canada K1N 6N5. \medskip\noindent 11:15--11:30 Coffee break \medskip\noindent 3. 11:30--12:30 {\it On the Design of Timed Systems} by Juan Quemada, Departmento de Ingineria Telematica, Universidad Politechinica de Madrid, Spain. \medskip\noindent 12:30--1:30 Lunch break %\newpage \medskip\noindent {\bf Monday, November 1-st, 1:30--5:00 Session 2} \medskip\noindent 4. 1:30--2:30 {\it Visual Tools for Verifying Real-Time Systems} by Jonathan Ostroff, Department of Computer Science, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3. \medskip\noindent 2:30-2:45 Coffee break \medskip\noindent 5. 2:45--3:45 {\it Integrating State Machines, Temporal Logic, and Algebraic Models of Data} by Armen Gabrielian, UniView Systems, Mountain View, California, USA. \medskip\noindent 3:45-4:00 Coffee break \medskip\noindent 6. 4:00--5:00 {\it Towards Full Timed LOTOS} by Tommaso Bolognesi, C.N.R. Istituto CNUCE, 36, Via S. Maria, 56100 - Pisa, Italy. \medskip\noindent 5:00--8:00 Dinner \medskip\noindent$\bullet$ 8:00--10:00 Special session %\newpage \medskip\noindent {\bf Tuesday, November 2-nd, 9:00--12:30 Session 3} \medskip\noindent 7. 9:00--10:00 {\it Refining and Abstracting Time Information} by Steve Schneider, Oxford University, England. \medskip\noindent 10--10:15 Coffee break \medskip\noindent 8. 10:15--11:15 {\it Real-Time System = Discrete System + Clock Variables}, Part I by Rajeev Alur, AT\&T Bell Labs, Murray Hill, New Jersey, USA and Tom Henzinger, Department of Computer Science, Cornell University, Ithaca, New York, USA. \medskip\noindent 11:15--11:30 Coffee break \medskip\noindent 9. 11:30--12:30 {\it Real-Time System = Discrete System + Clock Variables}, Part II by Rajeev Alur, AT\&T Bell Labs, Murray Hill, New Jersey, USA and Tom Henzinger, Department of Computer Science, Cornell University, Ithaca, New York, USA. \medskip\noindent 12:30--1:30 Lunch %\newpage \medskip\noindent {\bf Tuesday, November 2-nd, 1:30--5:00 Session 4} \medskip\noindent 10. 1:30--2:30 {\it An Experience with the Formal Description in LOTOS and Prototyping of the Airbus A320 Flight Warning Computer} by Hubert Garavel, VERIMAG, Miniparc-ZIRST, rue Lavoisier, 38330 Montbonnot St Martin, France and Rene-Pierre Hautbois, Aerospatiale A/DL/EP, M 8621, 316 route de Bayonne, 31060 Toulouse cedex 03 France. \medskip\noindent 2:30--2:45 Coffee break \medskip\noindent 11. 2:45--3:45 {\it Specification and Proof in Real-time CSP} by Jim Davies, Department of Computer Science, University of Reading, Reading RG6 2AH, England. \medskip\noindent 3:45--4:00 Coffee break \medskip\noindent 12. 4:00--5:00 {\it The Priority Inversion Problem and Real-Time Symbolic Model Checking} by Edmund Clarke and Sergio V. Campos, Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. \medskip\noindent 5:00--8:00 Dinner \medskip\noindent 8:00--10:00 Special Session %\newpage \medskip\noindent {\bf Wednesday, November 3-rd, 9:00--12:30 Session 5} \medskip\noindent 13. 9:00--10:00, {\it Using Synchronized Transition Systems to Develop Real-Time Software: An Experiment} by Didier Begay, Universit\'e Bordeaux I, LaBRI, 351, cours de la Lib\'eration 33405 Talence, France. \medskip\noindent 10:00--10:15 Coffee break \medskip\noindent 14. 10:15--11:15 {\it Verification of the Easylink Protocol} by Frits Vaandrager and Indra Polak, CWI and University of Amsterdam, The Netherlands. \medskip\noindent 11:15--11:30 Coffee break \medskip\noindent 15. 11:30--12:30 {\it Performance Analysis and True Concurrency Semantics} by Ed Brinksma, Joost-Pieter Katoen, Rom Langerak, and Diego Latella, Department of Computer Science, University of Twente, The Netherlands. \medskip\noindent 12:30--1:30 Lunch %\newpage \medskip\noindent {\bf Wednesday, November 3-rd, 1:30--5:00 Session 6} \medskip\noindent 16. 1:30-2:30 {\it Using Iterative Symbolic Approximation for Timing Verification} by David Dill and Howard Wong-Toi, Department of Computer Science, Stanford University, Stanford, CA, USA. \medskip\noindent 2:30--2:45 Coffee break \medskip\noindent 17. 2:45--3:45 {\it Analysis, Synthesis, and Optimization of Real-Time Systems in a Temporal Logic Framework} by Dan Ionescu, Department of Electrical Engineering, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5. \medskip\noindent 3:45--5:00 Administrative matters and departure } \newpage \begin{center} {\Large {\bf General Information}} \end{center} \medskip\noindent All speakers at this workshop have been invited. Their presentations represent some of the best known research directions in real-time system development and we hope that their work will be of interest to a large audience. So, we would encourage all those who believe that they can benefit from these presentations to attend this workshop, to contribute to the discussions, and to further the development of real time systems. % by discussions during these three days of presentations. \medskip\noindent {\bf Location:} The conference will be held at the Conference Center of the University of Iowa\footnote{The University of Iowa does not discriminate in its educational programs and activities on the basis of race, national origin, color, sex, age, or disability. The University also affirms its commitment to providing equal opportunities and equal access to University facilities without reference to affectional preference. For additional information on nondiscrimination policies, contact the Coordinator of Title IX and Section 504 in the Office of Affirmative Action, telephone (319)335-0705, 202 Jessup Hall, The University of Iowa, Iowa City, Iowa 52242-1316. If you are a person with disability who requires reasonable accommodations in order to participate in this program, please contact the sponsoring department at (319)335-3231 to discuss your needs.}, Iowa Memorial Union. All meetings will be held in Room 345, Northwestern, at that location. \medskip \noindent {\bf Transportation} \begin{enumerate} \item The airport that services Iowa City is at Cedar Rapids, 25 miles from Iowa City. The closest international airport from Cedar Rapids is Chicago. Limousine services between Cedar Rapids airport and Iowa City are available. \item Interstate 80 is the easiest access route to Iowa City. Exit 244, Dubuque Street, leads you to downtown Iowa City. \end{enumerate} \noindent {\bf Climate:} It usually rains in Iowa City on November 1-st. However, considering the amount of rain we have had so far maybe it will be sunny this time. \medskip\noindent {\bf Registration fees}: \$150; this includes breakfast, lunch, coffee and refreshments, and the program and other documents distributed at the conference site. \medskip\noindent {\bf Hotel Reservation:} For hotel reservation please call 319-335-3513, Iowa House, indicating that you are attending the First AMAST International Workshop on Real-Time Systems. A block of rooms have been already reserved for you at \$52-single and \$58 double, a night. They will be assigned to the attendees on the basis of first come first served. The alternative is Holiday Inn -- downtown Iowa City -- which is within walking distance from the Iowa Memorial Union. The number to call is 319-337-4058, reservations. The Center for Conferences and Institutes is handling the registration and the other arrangements. For more information about reservation and registration contact: \begin{tabbing} Bobby C Davis or Lisa Barnes\\ Center for Conferences and Institutes \\ The University of Iowa, Iowa Memorial Union \\ Iowa City, Iowa 52242 \\ Phone (319)335-3220 \end{tabbing} \end{document}