Subj: Integration of Yoneda Date: Tue, 1 Sep 92 14:06:05 +1000 From: street@macadam.mpce.mq.edu.au In 1968, I was a postdoc at the University of Illinois auditing John Gray's graduate course on categories. I believe John had learned that every functor into sets was a canonical colimit of representables from Gabriel-Zisman: I don't think they give a reference. Coming out of his work on extraordinary naturality with Sammy and on enriched functor categories, Max Kelly had defined (sometime between La Jolla and 1967) the notion of "end" for enriched categories. While browsing in the Illinois Math Dept Library I actually READ Yoneda's paper "On Ext and exact sequences" J Fac Sci U Tokyo 8 (1960) 507-576. I saw that Yoneda had already discovered ends for additive categories, but had not called them that. Yoneda used the integral notation for ends, which I recommended to Brian Day and Max, and they adopted it for their SLNM 106 article. I was learning lots about comma categories from Gray and enjoying translating it into ends to extend the results to enriched categories. Therefore, I claim that Yoneda pre 1960 was well aware of: (i) the end formula nat(f,g) = end(f->g) for the set (or object) of natural transformations; (ii) the coend formula f = coend(A(a,-)*f) expressing each functor into sets (the base) as a canonical colimit of (generalised) representables ("Fourier-like theorem"); (iii) the Lemma which category theorists associate with his name, viz, end(A(a,-)->f) = fa. [I write *, -> for tensor, cotensor, resp.] I also suspect Isbell must have known of the adequacy of the canonical embedding into presheaves in "Adequate subcats" Ill J 4 (1960) 541-552 but have no time now to check. Best regards, _________________________________________________________ / Ross STREET, Professor of Mathematics \ / School of Mathematics, Physics, Computing and Electronics \ / Macquarie University, New South Wales 2109, AUSTRALIA \ / Telephone: 61-2-805-8946 Facsimile: 61-2-805-8241 \ /-----------------------------------------------------------------\ ============================================================================== Subj: CTCS Conference Date: Wed, 2 Sep 92 15:44:45 BST From: David Rydeheard 1993 ********* Preliminary Announcement *********** * * * CATEGORY THEORY AND COMPUTER SCIENCE * * ------------------------------------ * * * * Fifth Biennial Meeting * * * ********************************************** CTCS-5 Dates: 7th-10th September 1993. Venue: CWI, Amsterdam, The Netherlands. The fifth of the biennial conferences on category theory and computer science is to be held in Amsterdam in 1993. The purpose of the conference series is the advancement of the foundations of computing using the tools of category theory, algebra, geometry and logic. Whilst the emphasis is upon applications of category theory, it is recognised that the area is highly interdisciplinary and the organising committee welcomes submissions in related areas. Topics central to the conference include: * The semantics of computation * Program logics and specification * Type theory and its semantics * Domain theory * Linear logic and its semantics * Categorical programming Submissions purely on category theory are also acceptable as long as the applicability to computing is evident. Previous meetings have been held in Guildford (Surrey), Edinburgh, Manchester and Paris. The format of this fifth meeting is to differ from previous meetings. Abstracts of talks are to be submitted to the organiser (details below). These will undergo a preliminary selection procedure and authors will be notified of the result. Proceedings of the conference will appear in a special issue of the journal Mathematical Structures in Computer Science. All contributors to the conference will be invited to submit full papers to the special issue. Submissions will undergo the usual refereeing process for MSCS, which accepts only very high standard contributions. Organising and program 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, E. Wagner. IMPORTANT DATES Submission of abstracts of talks 25th May 1993 Notification of acceptance 1st July 1993 Submission of Abstracts. Authors should send 3 hard copies of an abstract and a cover page (preferably in 11pt LaTeX format) to: Dr. David Pitt, Department of Mathematics University of Surrey, Guildford, Surrey GU2 XH United Kingdom. email: d.pitt@mcs.surrey.ac.uk Authors without access to reproduction facilities may submit a single copy of their submission. The cover page of the submission should include the title, authors, a brief synopsis, and the corresponding author's name, address, phone number, fax number, and e-mail address if available. Abstracts should consist of no more than 3 (three) A4 sides (not including references). They must be in English, clearly written, and provide sufficient detail to allow the program committee to assess the merits of the paper. Each submission should make clear the advances made by the authors, the relevance to the subject, the background involved and the relationship to other work in the area. If the authors believe that more details are essential to substantiate the main claims of the paper, they may include a clearly marked appendix to be read at the discretion of the committee. Late abstracts, or those departing significantly from these guidelines, run a high risk of rejection. Local Arrangements: These will be notified later. The local co-ordinator is: Dr. Fer-Jan de Vries Department of Software Technology CWI Kruislaan 413 1098 SJ Amsterdam The Netherlands email: F.J.de.Vries@cwi.nl ============================================================================== Subj: paper available by ftp Date: Thu, 3 Sep 1992 14:18:24 +0100 (BST) From: Samson Abramsky The following paper is now available by anonymous ftp from theory.doc.ic.ac.uk, in papers/Abramsky. Anyone needing a hard copy should send me their postal address. -------- \documentstyle[11pt]{article} \begin{document} \bibliographystyle{alpha} \title{Games and Full Completeness for Multiplicative Linear Logic} \author{Samson Abramsky and Radha Jagadeesan \\ Imperial College.} \maketitle \begin{abstract} We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove {\em full completeness} for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of {\em history-free} strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass {\it et al}. \end{abstract} \end{document} ============================================================================== Subj: Re: 4 colours From: Richard Wood Date: Tue, 8 Sep 1992 11:43:01 -0300 Please also note "A categorical characterization of the four colour theorem"by Barry Fawcett, Canad. Math. Bull. Vol. 29 (4), 1986, perhaps the main result of which is the equivalence of the four colour theorem with the statement that epimorphisms are surjective in the category of planar graphs. Wood ============================================================================== Subj: Commutative Diagrams in TeX - NEWS Date: Mon, 07 Sep 92 17:14:06 ADT From: Paul Taylor =============================================== Commutative Diagrams in TeX - towards version 4 =============================================== This message brings news of the development of my TeX package for drawing "commutative" diagrams, which is now widely used in the category theory and theoretical computer science communities. It is being sent directly to all of the users I know of (who have requested it by electronic mail or FTP from me, or asked questions about it), but as I know the package has been passed on, I would be grateful if you would ************************************************************************** * copy this message to anyone to whom you have given the package itself. * ************************************************************************** The package was originally advertised on the "types" and "categories" electronic mailing lists in July 1990. In the following eighteen months some fixing of bugs took place, but there was little substantial change. Since April 1992, I have re-written most of the code, largely with a view to improving the geometrical layout of the diagrams. Before completing this work and calling it version 4, I would like some feedback from users. One of the areas which I have neglected in the past (largely because TeX makes it so difficult) is diagonal arrows. The code for drawing these using LaTeX line segments has been re-written: now the closest available slope is chosen automatically and the commands have names similar to the horizontals and verticals. However to do a better job of diagonals (and in future to support curved lines) some extension to TeX is needed. Being extensions they are necessarily not standard. Three possibilities are: (1) additional fonts (as, for example, used by Spivak's Lamstex). However my experience of design-size fonts and linear logic symbols suggests that for users without expert knowledge or control of their local TeX systems this is more trouble than it's worth. (2) PostScript is, I believe, now almost universally used as the language in which TeX documents are sent to a printer. PS commands can be embedded in DVI files and incorporated in the PS translation without extra system or user files or any user intervention. This is to some extent dependent on which DVI->PS translator is used. In the new version this is exploited in an option to implement diagonals by rotating horizontals, which works with Tomas Rokicki's "dvips". (3) TPIC is a graphics extension of TeX which uses a simpler set of embedded commands. These can be used to draw diagonal lines and curves but not to perform rotations; they are, however, understood by Vojta's "xdvi" as well. Another option in the new version uses these to draw diagonal lines. Besides diagonals, the code for adjusting horizontal and vertical arrows has been completely rewritten and does a much better job of the geometry. Many of the problems with alignment, positioning and gaps have been fixed automatically, and greater control is given to the user to adjust those which cannot be. There are also several new options for the placement of the finished diagram on the page. Arrow commands are now declared in a much simpler way. The declaration \newarrow{CrossedInto}{hook}-+-> is now all that is needed to define the example \rCrossedInto in the manual, along with the corresponding left, down, up and diagonal commands. Another option makes a consistent selection of arrowheads for all arrows, from a choice of vee, LaTeX, curlyvee, triangle and blacktriangle. So much for selling you the new version. The reason for mailing you and asking for comments before completing what I intend to do for version 4 is that I want to get feedback on the following questions: (1) Can you use FTP (file transfer protocol)? This is the easiest method of distribution for me and for you, and there is now a huge volume of public domain software available by this method. My archive is called theory.doc.ic.ac.uk (146.169.2.37) and the diagrams package is in the directory /tex/contrib/Taylor/tex. Please try to fetch the new version and manual by this method. (2) If you can't use FTP, and your electronic mail passes via non-ASCII machines (particularly BITNET), what characters tend to get corrupted? The new version uses a restricted character set to avoid this problem. (3) Do you have available for printing final copy a printer which understands Adobe PostScript, for example an Apple or Sun laserwriter? Who is the author of the DVI->PS translation program you use? Please fetch the new version, try the PostScript option and tell me if you have any difficulty printing. (You may have to change the \verbatim@ps@special macro if you don't use Rokicki's dvips: if so, please send me details.) You can preview with a PS previewer such as PageView under OpenWindows or GhostView/GhostScript under Xwindows. (4) Do your DVI translators and previewers understand TPIC \specials (as used in eepic.sty)? Please try the TPIC option. I would like to know whether it is worth putting effort into PostScript, TPIC or some other method. (5) Have you defined your own arrow commands using \HorizontalMap, \VerticalMap or \DiagonalMap? Please use "grep alMap *.tex *.sty" or some similar command to find out, and tell me if you have used any components other than those in the source of version^3. It is in your interests to do this, because \newarrow defines arrow commands in a different way. (6) Please tell me if you have any difficulty adapting to the new version, or any general comments about doing so which might be of benefit to other users. (7) Other comments: have you used other packages for drawing diagrams? Do you have applications for my package other than the categorical diagrams for which it was designed? What do you see as the major limitations of the package? What persuaded you to use it, or not to use it? Version numbers: 2 was circulated to some people in September 1989 3.16 was advertised on types & categories in July 1990 and emailed to those who asked for it. 3.18 was the final bug-fix before the re-write began in April 1992. 3.20 introduced error-recovery, and \newarrow for horizontals & verticals 3.22 completely rewrote the reformatting program for h & v and corrected numerous alignment errors; introduced options in square brackets 3.23 fixed a catastrophic error in nested diagrams 3.24 (current) extended \newarrow to diagonals, added trigonometry code, rewrote code for drawing LaTeX diagonals, introduced PostScript and TPIC diagonals, consistent choice of arrowheads. It will be called "version 4.0" when the diagonals are adjusted to meet their endpoints (the one remaining big project, which could not be done before the others above) and I have dealt with my list of minor quibbles. Paul Taylor, 7 September 1992 Department of Computing, Imperial College of Science, Technology & Medicine, +44 71 589 5111 x 5057 180 Queen's Gate, +44 71 581 8024 (FAX) South Kensington, London SW7 2BZ, UK pt@doc.ic.ac.uk ============================================================================== Subj: Extraductions from topos theory Date: Wed, 09 Sep 92 12:49:48 From: sjv@doc.ic.ac.uk (Steve Vickers) It's common and not unreasonable to give an introduction to topos theory by starting from motivations in various subjects including algebraic topology and geometry. MacLane and Moerdijk explain these motivations in some detail, and of course the original papers on toposes presupposed knowledge of algebraic geometry. But I feel that I understand toposes better than the motivating subjects. Does anyone know of any good "extraductions from" topos theory that use topos intuitions to help introduce the concepts and methods of algebraic topology and algebraic geometry? Steve Vickers. ============================================================================== Subj: Re: 4 colours Date: Tue, 8 Sep 92 13:31:07 EDT From: pavlovic@triples.Math.McGill.CA (Dusko Pavlovic) Thanks, Richard. As you could see if you FTP-ed my note, I did quote Barry Fawcett's characterisation (and thanked Robert Pare for mentioning it to me). I also explained that Fawcett's approach is, in a sense, dual to mine. I hope I'll be able to say more about this in another note. Best regards, Dusko Pavlovic ============================================================================== Subj: Re: Extraductions from topos theory From: ramu@cadsun.corp.mot.com Date: Wed, 9 Sep 92 09:25:17 CDT -> Date: Wed, 09 Sep 92 12:49:48 -> From: sjv@doc.ic.ac.uk (Steve Vickers) -> Does anyone know of any good "extraductions from" topos theory that use -> topos intuitions to help introduce the concepts and methods of algebraic -> topology and algebraic geometry? A. Koch, Synthetic Differential Geometry, Cambridge University Press, 1980 (?). --Ramu Iyer Email: ramu@cadsun.corp.mot.com ============================================================================== Subj: FMCS workshop report Date: Tue, 15 Sep 92 11:15:05 -0700 From: David B. Benson Foundational Methods in Computer Science: A workshop on applications of categories in computer science 1992 May 30-31 Sloan Hall School of Electrical Engineering and Computer Science Washington State University, Pullman WA 99164-2752 The workshop was organized by David B. Benson with assistance from Purandar Bhaduri. Robin Cockett and Dwight Spencer helped set the workshop scope and atmosphere. Notably, the workshop was almost entirely organized via email. Essentially only the preregistration fees passed through the postal services. The invited hour speakers were Robin Cockett, Steve Bloom, Ernie Manes, and Bob Walters. The workshop was preceeded by a one day introduction to categories. A majority of the workshop participants attended these tutorials. The workshop was informal, even casual. Everyone seemed to enjoy and profit from this organization of a workshop around an idea and style which has many applications. In keeping with the spirit of sharing the latest scientific and technical information, there will be no workshop proceedings. We expect many of the presentations to develop into monographs and journal papers. Scientific program ------------------ Friday, 1992 May 29 0900-1130 Introduction to categories for computer science (D. B. Benson) 1300-1430 Introduction to monads (E. G. Manes) 1530-1700 Introduction to categorical logic (P. Bhaduri) Saturday, 1992 May 30 0900-1000 Robin Cockett: Distributive Matters, Data, Charity and Programming 1020-1040 Tom Fukushima Monads in Charity 1040-1100 Dick Kieburtz Involution and Duality 1120-1140 Jim Hook Program Development inspired by monads 1140-1200 Juergen Koslowski Computational monads in programming languages 1400-1500 Steve Bloom: Iteration Theories and Initiality 1520-1540 Francoise Bellegarde ASTRE: Program transformation and rewriting 1540-1600 Purandar Bhaduri Functorial view of concurrency 1600-1620 Mike Levy (for Bill Wadge) Monads and intensionality 1620-1640 Rakesh Dubey On a general definition of safety and liveness Sunday, 1992 May 31 0900-1000 Ernie Manes: Boolean categories 1020-1040 John MacDonald Soft Adjunction I, Introduction 1040-1100 Art Stone Soft Adjunction II, Examples 1300-1400 Bob Walters: An imperative language based on distributive categories 1420-1440 Carolyn Brown A categorical approach to concurrency 1440-1500 Wafaa Khalil A universal IMP(G) program 1500-1520 Eric Wagner About and around distributive categories Participants (27) ------------ Karl Abrahamson Francoise Bellegarde David B. Benson Purandar Bhaduri Steve Bloom Carolyn Brown Robin Cockett Rakesh Dubey Tom Fukushima Mike Herman James Hook Wafaa Khalil Dick Kieburtz J"urgen Koslowski Mike Levy John C. MacDonald Ernest G. Manes Neal Nelson Raja Nagarajan Tom Rigles Marc Schroeder Tim Sheard Dwight L. Spencer Art Stone Eric Wagner Bob Walters Barry Yee ftp sites --------- Here are ftp sites at which preprints of presented or related papers may be obtained. maths.su.oz.au anonymous login your email address as password directory sydcat cpsc.ucalgary.ca anonymous login ident as password directories: pub/charity for the charity system pub/charity/PAPERS for papers related to charity ftp.eecs.wsu.edu anonymous login your email address as password directory pub/papers -- submitted by David B. Benson School of Electrical Engineering and Computer Science Washington State University Pullman WA 99164-2752 ============================================================================== Subj: operads and n-cats Date: Thu, 17 Sep 92 13:54:24 GMT-0400 From: jds@rademacher.math.upenn.edu \magnification = \magstep2 \centerline{GRADUATE STUDENT ALGEBRA SEMINAR} \centerline{CONFORMAL FIELD THEORY } \vskip3ex \centerline{Tuesdays at Noon} \vskip2ex \centerline{Room ?4E17? DRL} \vskip2ex \centerline{Introduction and Overview} \vskip2ex \centerline{Y.-Z. Huang} \vskip2ex \centerline{Tuesday September 15} \vskip3ex \vskip3ex \centerline{CONFORMAL FIELD THEORY SEMINAR} \vskip3ex \centerline{Tuesdays at 3:10} \vskip2ex \centerline{Room ?4E17? DRL} \vskip2ex \centerline{Introduction to Operads} \vskip2ex \centerline{Jim Stasheff - UNC and U Penn} \vskip2ex \centerline{Tuesday September 22} \vskip3ex An operad is an algebraic/topological gadget for keeping track of multiparameter families of maps $X^n\rightarrow X^k$. Originally invented for the homotopical characterization of iterated (based) spaces of loops, the special example of disks within disks within disks... has recently been observed within conformal field theory. There is also some relevance to n-categories. \end ============================================================================== Subj: LICS 93 call for papers Date: Tue, 22 Sep 92 02:55:05 ADT From: "daniel leivant" Eight Annual IEEE Symposium on LOGIC IN COMPUTER SCIENCE June 20-23, 1993, Montreal, Quebec (Canada) CALL FOR PAPERS The LICS Symposium aims for wide coverage of theoretical and practical issues in computer science that relate to logic in a broad sense, including algebraic, categorical and topological approaches. Suggested, but not exclusive, topics of interest include: abstract data types, automated deduction, concurrency, constructive mathematics, data base theory, finite-model theory, knowledge representation, lambda and combinatory calculi, logical aspects of computational complexity, logics in artificial intelligence, logic programming, modal and temporal logics, program logic and semantics, rewrite rules, logical aspects of symbolic computing, problem solving environments, software specification, type systems, verification. PROGRAM CHAIR: Moshe Y. Vardi IBM Research Almaden Research Center, K53-802 650 Harry Rd. San Jose, CA 95120-6099, USA vardi@almaden.ibm.com, vardi@almaden.bitnet Phone: (408) 927-1784 Fax: (408) 927-2100 PROGRAM COMMITTEE: M. Abadi (DEC SRC), S. Abramsky (Imperial Coll.), B. Bloom (Cornell), P. Clote (Boston Coll.), P.J. Freyd (Univ. of Pennsylvania), D. Harel (Weizmann Inst.), K.G. Larsen (Aalborg Univ.), P. Lescanne (CRIN and INRIA-Lorraine), D. McAllester (MIT), J. Meseguer (SRI), D. Miller (Univ. of Pennsylvania), Y. Moschovakis (UCLA), N. Shankar (SRI), C. Talcott (Stanford), M.Y. Vardi (IBM Almaden), and P. Wolper (Univ. of Liege). LICS GENERAL CHAIR: Robert Constable Department of Computer Science Cornell University Ithaca, NY 14253 rc@cs.cornell.edu CONFERENCE CO-CHAIRS: Mitsu Okada Prakash Panangaden Department of Computer Science School of Computer Science Concordia University McGill University Montreal Montreal H3G 1M8 Quebec, Canada H3A 2A7 Quebec, Canada okada@concour.cs.concordia.ca prakash@cs.mcgill.ca PAPER SUBMISSION: 10 hard copies of a detailed abstract (not a full paper) and 20 additional copies of the cover page should be RECEIVED by DECEMBER 8, 1992 by the PROGRAM CHAIR (Attn: LICS). Authors without access to reproduction facilities may submit a single copy of their submission. Authors will be notified of acceptance by February 14, 1992. Accepted papers in the specified format for the symposium proceedings will be due April 6, 1993. The cover page of the submission should include the title, authors, a brief synopsis, and the corresponding author's name, address, phone number, fax number, and e-mail address if available. Abstracts must be in English, clearly written, and provide sufficient detail to allow the program committee to assess the merits of the paper. References and comparisons with related work should be included. It is recommended that each submission begin with a succinct statement of the problem, a summary of the main results, and a brief explanation of their significance and relevance to the conference, all suitable for the non-specialist. Technical development of the work, directed to the specialist, should follow. Abstracts of fewer than 1500 words are rarely adequate, but the total abstract, should not be longer than 10 typed pages with roughly 35 lines per page. If the authors believe that more details are essential to substantiate the main claims of the paper, they may include a clearly marked appendix to be read at the discretion of the committee. Late abstracts, or those departing significantly from these guidelines, run a high risk of rejection. The results in the submission must be unpublished and not submitted for publication elsewhere, including proceedings of other symposia or workshops. All authors of accepted papers will be expected to sign copyright release forms, and one author of each accepted paper will be expected to present the paper at the conference. The symposium is sponsored by the IEEE Technical Committee on Mathematical Foundations of Computing in cooperation with the Association for Symbolic Logic and the European Association of Theoretical Computer Science. Cooperation with the ACM is anticipated. ORGANIZING COMMITTEE: M. Abadi, S. Abramsky, S. Artemov, J. Barwise, M. Blum, A. Bundy, S. Buss, E. Clarke, R. Constable (Chair), E. Engeler, J. Gallier, U. Goltz, Y. Gurevich, S. Hayashi, G. Huet, G. Kahn, D. Kapur, C. Kirchner, R. Kosaraju, J. W. Klop, P. Kolaitis, D. Leivant, A. Meyer, G. Mints, J. Mitchell, Y. Moschovakis, A. Pitts, G. Plotkin, S. Ronchi Della Rocca, G. Rozenberg, A. Scedrov, D. Scott, J. Tiuryn, M. Vardi, R. de Vrijer PUBLICITY CHAIR: Daniel Leivant Department of Computer Science Indiana University Bloomington, IN 47405, USA lics@cs.indiana.edu ============================================================================== Subj: From triposes to assemblies Date: 21 Sep 92 16:58:04 EST From: Colin McLarty ------------- Jaap van Oosten's dissertation uses triposes to develop numerous realizability toposes, and incidentally raises the question as to which of them can be presented as the effective reflection of a category of assemblies. The answer is that all of his tripos constructions actually include constructing such assemblies. I will spell this out in some generality. Let N be any partial combinatory algebra (see Jaap's dissertation p.35). Write e.m for the result of applying an element e of N to another element m. The notation suggests the natural numbers taken as codes of partial recursive functions, but everything here works for any PCA. For any subsets W and V of N, we call an element e of N a "modulus" mapping W to V iff for every m in W the value e.m is defined and is in V. More generally, we say e maps one list of subsets W1, W2, W3... to another V1, V2, V3... iff e simultaneously maps each Wi into Vi. By a "tripos based on N" I mean a tripos such that for some subset Sub of some cartesian power of the powerset of N: 1) The fiber Sigma(X) over any set X consists of suitable pairs with alpha a function from X to Sub and t coding some further structure. The pre-order says is smaller than iff there is a modulus mapping alpha to alpha' and that modulus meets conditions which may depend on t and t'. 2) The action of any function f:X___>Y is by composition. Given any function from Y to Sigma, compose with f to get one from X to Sigma. For many realizabilities the tags "t" are superfluous. But in extensional realizability, for example, a member of Sigma(X) is an X-tuple of subsets of N, tagged by an equivalence relation on each one and moduli must preserve those relations. We say an element x of X is "minimally realized" by a given alpha in Sigma(X) iff alpha(x) is minimal in Sigma(1). As in any tripos we define a "relation in the sense of the tripos" on a set X to be a member of the fiber over XxX. The tripos construction of the topos works by taking as objects all pairs (X,=) where "=" is a partial equivalence relation in the sense of the tripos on the set X. The arrows are all functional relations between these pers, and the result is a topos. But in a tripos based on N we can instead take only those pairs (X,=) such that is minimally realized by = unless x=y. These generalize the canonically separated objects in the effective topos. Together with the same arrows they form a regular category. Every partial equivalence relation on X in the sense of the tripos contains a canonically separated = such that, for any pair non-minimally realized in the relation both and are non-minimally realized in =. By the soundness of tripos logic, equivalence relations and functional relations IN THE CATEGORICAL SENSE in the regular category correspond exactly to those in the tripos sense. I.e. the tripos's topos is also the effective reflection of this regular category. It only remains to note that a canonically separated = virtually is an assembly and for this special case the functional relations correspond to actual functions between carriers with moduli. A canonically separated = on X amounts to a function from X to Sigma taking each x in X to the value =. If Sub is contained in the powerset of N then such a function gives a relation from X to N and can be given in terms of caucuses: for each n in N the n-th caucus of the assembly is the set of all x in X related to n. If Sub is contained in the I-fold cartesian power of the powerset then we get a distinct series of caucuses for each i in I. For each i we will call these the i-caucuses, so that for each n in N there is an n-th i-caucus. The tags "t" in the fibers can be kept as tags on the assemblies. The "carrier" of = is the set of all x in X with non-minimally realized by =. A functional relation F in the sense of the tripos from (X,=) to (Y,=') amounts to an ordinary function F from the carrier of = to the carrier of =', which has a modulus. Here "modulus" means an element e of N which, for each x in the carrier of =, maps the Sigma element = to the Sigma element =' compatibly with the tags. In other words, let Sub be a subset of the I-fold cartesian power of the powerset of N. Then a modulus for F is a member e of N such that: for every i in I and n in N, if a member x of X is in the n-th i-caucus of = then e.n is defined and Fx is in the e.n-th i-caucus of ='; and e meets any conditions set by tags on = and ='. On his p.4 Jaap gives data for a tripos for each of some dozen kinds of realizability. His Sigma for each kind is our Sigma, and specifies what PCA each kind uses. His material implications tell which moduli are allowed. The present construction applied to his Kleene tripos gives the familiar assemblies for the effective topos. It should be clear that such assemblies and tags restricting the moduli can be described directly. On that approach, you have to show that they give a cartesian closed regular category with weak subobject classifier. That verification will be a lot like checking the Beck condition and the generic predicate in a tripos. In effect, the regular category uses the same data as the tripos only without the indexed pre-order structure. Any such regular category has a topos as effective reflection, and forms the separated objects of the topos. The method is due to Freyd and can be gleaned from Freyd and Scedrov _Categories, Allegories_. The actual statement and proof are given in Ch.25 of my _Elementary categories, elementary toposes_. Again, it is a close parallel to part of the proof that triposes give toposes, but omits much. ============================================================================== Subj: Comparison between Functor Categories Date: Wed, 23 Sep 92 14:17 GMT From: MAS034@BANGOR.AC.UK Dear Colleagues, let F:C -> D be a functor between two small categories. Its induced functor F^*:Fun(D,Set) -> Fun(C,Set) has both a left adjoint L and a right adjoint R. 1. Under which precise conditions on F is L F^* = Id. 2. Under which precise conditions on F is R F^* = Id. 3. Under which precise conditions on F is F^* R = Id. 4. Under which precise conditions on F is F^* L = Id. Does anybody know any answers to one of these queries? What happens if equality is replaced by natural equivalence? Does anybody know of good references to these or similar problems? Markus Pfenniger Andy Tonks School of Mathematics School of Mathematics Dean Street Dean Street Bangor, LL57 1UT Bangor, LL57 1UT UK UK ============================================================================== Subj: Re: Comparison between Functor Categories Date: Wed, 23 Sep 92 14:19:16 EDT From: barr@triples.Math.McGill.CA (Michael Barr) The first thing to note is that your question, as stated, is meaningless. The proper statement is that the induced functor has left and rights adjoints. If for one of them, one of the composites you mention is naturally equivalent to the identity, then another could be chosen (using AC) for which the composite in question was equal to the identity. On the other hand, in the usual set theory, the usual construction that shows that adjoint exists will use more complicated sets than the functor and thus the composite will never be the identity. Thus the only question that can meaningfully raised is when one or the other of the composites is naturally equivalent to the identity. I think sufficient conditions are known, although I don't recall them offhand, but it seems awfully unlikely to me that useful necessary and sufficient conditions are known. Michael Barr ============================================================================== Subj: Re: Comparison between Functor Categories Date: Thu, 24 Sep 92 12:40:45 +1000 From: street@macadam.mpce.mq.edu.au Since L and R are only determined up to isomorphism, how can one ask for equalities like F^* R = Id? Ross ============================================================================== Subj: Diagrams in LaTeX Date: Thu, 24 Sep 92 14:51:44 EDT From: barr@triples.Math.McGill.CA (Michael Barr) A new version of LaTeX is being produced and I have agreed to participate in a subproject to decide on the best syntax for commutative diagrams. I am not at all sure that I know what the best syntax is. Mine is good for some purposes; not so good for others. There are a set of macros by Kris Rose (not specifically for LaTeX, but they will work with it) that really produce good results, but not if you put long labels on diagrams. They can be modified to work better in that case, but then get much more complicated. Then there are Paul Taylor's macros, which have advantages, but are certainly inferior to Rose's (the latter use their own fonts, which is a large reason for their superiority). Anyway, I welcome all suggestions. Michael Barr ============================================================================== Subj: Re: Comparison between Functor Categories Date: Thu, 24 Sep 92 11:09:50 From: sjv@doc.ic.ac.uk (Steve Vickers) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> let F:C -> D be a functor between two small categories. Its induced functor F^*:Fun(D,Set) -> Fun(C,Set) has both a left adjoint L and a right adjoint R. 1. Under which precise conditions on F is L F^* = Id. 2. Under which precise conditions on F is R F^* = Id. 3. Under which precise conditions on F is F^* R = Id. 4. Under which precise conditions on F is F^* L = Id. Does anybody know any answers to one of these queries? What happens if equality is replaced by natural equivalence? Does anybody know of good references to these or similar problems? Markus Pfenniger Andy Tonks <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< I don't know the answers. However, I can describe an analogous context (ring theory) where there is a surprising answer to 2: the appropriate analogue holds iff F is an epimorphism. Perhaps someone already knows whether the corresponding conjecture is true in the category (or monoid) context, and anyway perhaps the different perspective will cast light on the problem. First, as Mike Barr points out, natural equivalence is the appropriate property to look for, and in fact I don't know how you might go beyond the question of when the units or counits of the adjunctions are natural isos. If R is a ring, then the category of (right, say) modules over R is very like a functor category. R _is_ a category (one object, lots of morphisms), and a module is a functor from R to Abelian Groups. Of course, it's more special than that, because we also require that the functor should preserve the additive structure. However, the idea that a functor F from a category C to Sets is a kind of module is quite a natural one. The "elements of the module" are the elements of the sets F(i) where i ranges over objects of C, and a morphism f: i -> j of C "acts on" the elements (at least, those in F(i)) by xf = F(f)(x). You'll see this more clearly in the case where C is a monoid (only one object). This perspective is explained in Popescu, "Abelian categories with applications to rings and modules" (LMS Monographs 3, Academic Press, 1973). It extends readily to "ringoids", i.e. categories enriched over Abelian groups (Barry Mitchell "Rings with several objects", Advances in Mathematics 37 (1972) 1-161). If F: R -> S is a ring homomorphism, then you get a functor F^*: Mod-S -> Mod-R ("restriction of scalars") with left adjoint L, M |-> M tensor_R S, and right adjoint R, M |-> S hom_R M. It is known that the counit of the adjunction L -| F^* is a natural iso if and only if F is an epimorphism in the category of rings. (See, e.g., Stenstro"m "Rings of Quotients", Springer 1975.) (Note that epimorphisms are not always surjective - e.g. the embedding of the integers in the rationals is epi.) This is also true for ringoids (I believe it's in Mitchell's paper). It's conceivable that the direct analogue holds for categories, i.e. (2) (in suitable form) iff F is an epimorphism in the category of (small) categories, though at first glance the proof in Stenstro"m doesn't seem to generalize. If you look into this, I'd suggest you study the monoid case first. Steve Vickers ============================================================================== Subj: Re: Diagrams in LaTeX From: cbj@dcs.ed.ac.uk Date: Fri, 25 Sep 92 11:11:13 +0100 Michael Barr writes: > A new version of LaTeX is being produced and I have agreed to > participate in a subproject to decide on the best syntax for > commutative diagrams. ..... > (the latter [Rose's macros] use their own fonts, which is a > large reason for their superiority). Without having any knowledge of Rose's package, I would much prefer to have complete flexibility in the choice of labels. Two reasons for this are: (i) labels in the diagrams should agree with those in the text, and; (ii) fonts change more rapidly than diagram styles. One further point comes to mind. If other fonts can't be imported, then perhaps other macros can't be used either. For example, one might use arrays to label linear transformations. I suggest as a design criterion that Any box should be admissible as a label. Barry Jay ============================================================================== Subj: Re: Comparison between Functor Categories Date: Fri, 25 Sep 92 15:37:23 EDT From: pavlovic@triples.Math.McGill.CA (Dusko Pavlovic) >1. Under which precise conditions on F is L F^* = Id. >2. Under which precise conditions on F is R F^* = Id. >3. Under which precise conditions on F is F^* R = Id. >4. Under which precise conditions on F is F^* L = Id. (I guess some people know the precise answers at once, but don't bother to tell us. Here is what I figured.) Suppose, as M.Barr and R.Street promptly suggested, that the equalities in the query denote the natural isos. Of course, * 1 and 2 are equivalent: they mean that F^* is fully faithful; and * 3 and 4 are equivalent: they say that the essential morphism of toposes R: (D,Set)-->(C,Set) is an injection. A sufficient condition for 1 and 2 is that F:C-->D is a stably initial functor. (Initial functors are orthogonal to discrete opfibrations: cf. e.g. "The comprehensive factorisation of a functor" by Street and Walters, early seventies. Now F^* is pulling back of discrete opfibrations along F. Use orthogonality to show that it is fully faithful.) A sufficient condition for 3 and 4 is that F is fully faithful. (Just write down the pointwise Kan-extension formula for L (or R): to calculate F^*L(G) at X from C, we use the diagram obtained by projecting the comma category F/FX to C; we apply G on this diagram and calculate the colimit. When F is fully faithful, this diagram is just a cocone to X - so the colimit is GX.) Now what are the necessary conditions? Best regards, Dusko Pavlovic ============================================================================== Subj: Re: Diagrams in LaTeX From: Paul Taylor Date: Fri, 25 Sep 1992 18:54:07 +0100 Mike Barr has pointed out that some La/TeX packages for commutative diagrams or general graphics use specially provided fonts. My own experience with *any* divergence from the fonts provided in bog-standard TeX distribution causes more trouble than it's worth: witness the design size (Sauter) fonts in particular. (Incidenatlly, these are now no longer used by default in our local LaTeX implementation, and so should not be needed when fetching dvi files from our archive at theory.doc.ic.ac.uk). *Not* using extra fonts is listed as one of the design criteria of my diagrams package. Nevertheless, unlike any of the others I've seen (including Mike's) it does provide facilities for you to define arrow heads in terms of whatever fonts you like. If there is demand (which I haven't heard) I am willing to include macros for using other fonts within my package. However fonts are a red herring: where Mike came in was on the *syntax* of macros for expressing diagrams. With respect, but since he has thrown down this gauntlet, I find it amazing that the author of a package with as bizzare a syntax as his, who has submitted to this list a LaTeX document which directly conflicts with the recent developments in LaTeX, should be involved in proposing an authoritative syntax. A great deal of thought and work has gone into the design of the language in my package. Indeed there are many features I could have added but haven't because I haven't thought of the right language for expressing them. ============= Last week I was reading a recent paper which used my diagrams (I'm not going to say whose). The version used must have been about five years old and really made me cringe, because of the bugs in it which were fixed long ago. If you have a version dated before July 1990 or which is undated *PLEASE* throw it away and use a new one. The diagrams look so much better nowadays! If there are any problems, I shall be happy to help. Paul ============================================================================== Subj: Re: Diagrams in LaTeX Date: Fri, 25 Sep 92 20:11:01 EDT From: barr@triples.Math.McGill.CA (Michael Barr) I should clarify, for the benefit of those who think I meant that xypic (the package in question) uses its own fonts for the arrow sources or labels, I should clarify that it uses the ordinary TeX fonts except for the arrows. These are built from line segments at 256 different slopes, giving angular resolution of 256/180 = .7 deg and 512 half arrowheads giving arrowheads at that many angles. The arrowheads are extremely graceful, more elegant than those in ordinary Tex and massively more elegant than those of latex. Other than that, it uses whatever your symbol font is. For example, you could substitute the concrete math fonts, if you wanted. I have gotten several replies, the most interesting from David Benson who feels that xypic is probably the best, but that there shouldn't be just one, but a choice. He feels that a package that makes it hard to draw a complicated diagram that *also* has complicated arrow labels should be encouraged, so that people tempted to use such a horror be discouraged. Perhaps I have stated it a bit more strongly than he would, but that is the thrust of his remarks. Michael ============================================================================== Subj: Diagrams in (La)TeX From: koslowj@math.ksu.edu (Juergen Koslowski) Date: Sat, 26 Sep 92 11:54:27 CDT (Yes, I'm still around --- and hope to be until December.) Concerning the recent discussion of diagrams in LaTeX: I hope that any solution that is eventually adopted is not limited to LaTeX, but can become the standard for users of plain TeX as well as of AMS(La)TeX. Moreover, the syntax should be flexible enough to allow the design of various kinds of diagrams, *and* allow customization: i.e., if I want a square like f A -------------> B | | | | g | | h | | V V C -------------> D k I want to be able to *define* a macro \square with (maybe) 8 parameters that can be called like \square AfBghCkD on top of whatever underlying syntax is provided. Moreover, I want to be able to chose the direction of the arrows (and maybe even the type of arrow) when specifying its label. E.g., \square AfB{\dl g}{\dl h}CkD should reverse the arrows g and h. Even if the new standard comes with certain pre-defined diagrams, it should be easily extensible with user-defined ones. There is another issue: 2-cells! Just look at the ever-growing importance of 2-categorical ideas. It would be highly desirable to have support for pasting 2-cells. After some experiments trying to add these (or trying to write a \cube macro) I very soon ran out of macro parameters (for a cube you need 20, while TeX only allows you 9). One can get around this limitation, but the end user should not have to bother with this. What we need is a 2 level syntax. The base level should take care of the proper placement of objects and arrow labels, while the second level should give the user easy access to a standard repertoire of diagrams that can be extended if necessary. cheers, J"urgen - J"urgen Koslowski | If I don't see you no more in this world Department of Mathematics | I meet you in the next world Kansas State University | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) ============================================================================== Subj: Re: Comparison between Functor Categories Date: Sat, 26 Sep 92 14:43:15 +1000 From: street@macadam.mpce.mq.edu.au There is a need to say a few more elementary things on this question. My initial reply was short because I was busy teaching an intensive on-campus distance course, and I hoped someone else would respond in more detail. The equality question is meaningless. Also, the term "natural equivalence" is a term used "classically" by category theorists, but not a very good one; I suggest we should try to avoid it since "equivalence" means something else. Let F : C --> D be a functor and let L --| F* --| R as in the question. Let e : L F* --> 1, n : 1 --> F* L, e' : F* R --> 1, n' : 1 --> R F* be the counits and units for the adjunctions. Fact 1: n invertible iff L fully faithful iff R fully faithful iff e' invertible. Fact 2: e invertible iff F* fully faithful iff n' invertible. So there are really only two problems (as I just notice Dusko Pavlovic has pointed out): (a) when is L fully faithful? (b) when is F* fully faithful? Answer to (a): L fully faithful iff F is fully faithful. Proof: Since we are taking Kan extensions along F of functors into Set, there are formulas for L (pointwise left Kan extension). Any formula can be used to show F fully faithful implies n invertible, so L is fully faithful. This must be in all the textbooks (eg Mac Lane). Conversely, there is a square which commutes up to isomorphism (or equality if we choose L suitably on representables) involving L, two Yoneda embeddings, and F^op : C^op --> D^op. If L is fully faithful, so is F^op (since the Yon embs are), and so, so is F.///// Now some comments on (b): (i) Let's call F a "localization" when there exists a set S of arrows in C for which F is the universal functor out of C inverting the arrows of S. Given F, if there is an S, the set of arrows inverted by F is the largest such S. Localizations are bijective on objects. Localizations are coinverters of natural transformations between functors into C. If F is a localization, it follows that F* is an inverter of some natural transformation and hence is fully faithful. (ii) F* fully faithful does NOT imply F localization. For, if F induces an equivalence of categories on the cauchy (idempotent splitting) completions of C, D it will still have F* fully faithful, but need not have F bijective on objects. (iii) F* conservative (= reflects isos) iff e epic iff each object d of D is a retract of an object Fc for c in C. (iv) These things suggest to me that the answer to (b) could be: "F* fully faithful iff each object of D is a retract of an object in the image of F, and F = G H with G fully faithful and H a localization". However, localizations in Cat are notoriously difficult to characterize (easier in Lex, Rex, Pb, . . .). The above goes over to enriched categories with appropriate change in the notion of cauchy completion (and epic in (iii) becomes extremal epic); eg, additive categories for the context of Steve Vickers' response. Regards, Ross ============================================================================== Subj: Re: Diagrams in LaTeX Date: 26 Sep 92 15:36:10 PDT (Sat) From: pratt@cs.stanford.edu I didn't reply to the diagram syntax question because I couldn't come up with a phrasing of my druthers that adequately hid how much larger my eyes were than my mouth. But J"urgen just now expressed what I really wanted beautifully, so let me add my vote to his request. -Vaughan Pratt ============================================================================== Subj: Re: Diagrams in LaTeX (3 postings) Date: Sun, 27 Sep 92 12:13:12 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Ok, but that is not really a reply to a request for what is the best syntax. I tend towards the syntax of xypic, even though I think the program itself has a flaw. Obviously, you can then go on to create your own macros starting with it. I don't actually use xypic, but if I were starting fresh.... Michael +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: Mon, 28 Sep 92 09:53:52 +1000 From: street@macadam.mpce.mq.edu.au I'll sit alongside Vaughan on the J"urgen Koslowski driven 2D-bandwaggon. Might I also mention strings, links, and Penrose tensor notation. I have seen papers where these are done well, but I imagine with great effort. It should be possible to describe horizontal layers with plugs for vertical stacking during which the joins are smoothed. --Ross +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: Mon, 28 Sep 92 09:45:29 BST From: Robert Tennent I'd just like to put in a plug for the philosophy behind John Reynolds's package. It doesn't attempt to perform miracles (such as global layout), but does look after messy low-level details such as shading of arrows by vertex labels. Furthermore, it has a two-level structure which allows an expert to do non-standard things without complicating life for the beginner. Bob Tennent ============================================================================== Subj: Re: Comparison between Functor Categories From: Al Vilcius Date: Mon, 28 Sep 1992 00:51:48 -0400 In the Comparison between Functor Categories question from Bangor Sept 23, we had a functor F:C->D, but now instead of Set, consider an arbitrary category A and the induced functor A^F:A^D->A^C. Left and right adjoints, when they exist, are patched together from Kan extensions. For any particular T in A^C, we could have Ran_F(T), Lan_F(T) in A^D with n.t.(S,Ran_F(T)) iso n.t.(A^F(S),T) or n.t(Lan_F(T),S) iso n.t.(T,A^F(S)) which occurrences we call right or left Kan extensions of T along F. These are approximations of the variable object T by images of A^F, which I think of heuristically as a sort of Dedekind cut situation, in my own naive way, as follows: think of those S in A^D for which SF<=T and call the Lower also think of those S for which T<=SF and call them Upper here SF=A^F(S) of course, and X<=Y just means there is some n.t. X->Y. Then A^F images of Lowers approximate T from below, and Uppers from above. Of course we proceed to look for best approximations. Among Lowers, the closest image to T (w.r.t. <=) would be the sup, and among Uppers it's the inf, corresponding to pointwise Kan ext. as limits. So T gets caught in a squeeze play (like a real number), but still might have lots of room to bounce around. Back to the Bangor question (as least partly), we could ask what happened if T got hit on the nose by A^F with some Z in A^D? ie. ZF->T is identity n.t. Surely Z is then the best approx. to T from above and below since it is bang on. But does Z have to be (iso to) a Kan ext. of T along F? (which it needs to be if it is going to participate in any adjoint for A^F). Consider left extensions (inf of Uppers): from ZF=T<=SF we need to produce a n.t. Z->S. Again heuristically, the temptation is to cross off F on the right in a kind of epi maneuver, which relates to Steve Vickers' comment for rings. By the way, the very nice perspective on modules SJV mentioned also appears as Exercise 3(b) Chap VII p.415 in Mac Lane/Moerdijk '92 "Sheaves..". But epi for F seems too strong to be necessary, whereas the nice suggestion of Ross Street that each object of D should be a retract of an object in the image of F with F factoring in a certain friendly way looks really neat. Still, not clear it should be necessary. Anyway, might help to think about it in different ways. Cheers ........................................Al Vilcius, Toronto ============================================================================== Subj: Re: Comparison between Functor Categories Date: Tue, 29 Sep 92 10:38:15 From: sjv@doc.ic.ac.uk (Steve Vickers) Some not very exciting developments: If F: C -> D is a functor, then the question was about when F*: S^D -> S^C was full and faithful. (Incidentally, it took a lot of headache before I convinced myself that "fully faithful" just meant "full and faithful". I couldn't find the phrase defined in any of the standard texts. Is its mellifluousness really enough to justify its use?) My conjecture was that that this happens iff F is an epimorphism of categories. This can't be correct. An epimorphism must be surjective on objects (otherwise you can map the objects not in the image to distinct isomorphic copies), but if F is any equivalence then F* is as well, and hence full and faithful. (So my account of Mitchell's results for ringoids as opposed to rings was probably oversimplified.) I assume that "epimorphism", with its demands of on-the-nose equality, is simply not a good notion in the 2-categorical context of categories. In the monoid case, we do have an implication in at least one direction (adapted from the argument for rings): Suppose C and D are monoids. If F* is full and faithful, then F is an epimorphism (of monoids). Proof: Let G,H: D => E with F;G = F;H. E is a D-set, with action ed = e.H(d) Consider the function G: D -> E. This is a homomorphism of C-sets, for G(dc) = G(d).G(F(c)) = G(d).H(F(c)) = G(d)F(c) = G(d)c Hence by fullness of F* (note that faithfulness is automatic in this context), it is a homomorphism of D-sets, so for any d in D we have G(d) = G(1d) = G(1)d = 1.H(d) = H(d) Remaining questions: * Is the converse true in the monoid case? * Is there a 2-categorical generalization of epi that (includes equivalences and) restores the original conjecture? Steve Vickers. ============================================================================== Subj: Re: Comparison between Functor Categories Date: Tue, 29 Sep 92 11:30:25 EDT From: barr@triples.Math.McGill.CA (Michael Barr) Just a quick comment on Steve Vickers' post. This ought to be well known, but either it isn't as well known as I thought, or people aren't thinking of it in this connection. If C --> D induces an equivalence between idempotent completions (for example, if it is inclusion of C into its idempotent completion), then the induced Set^D --> Set^C is an equivalence. In fact, the analagous statement is true for any base that is itself idempotent complete. Now of course an inclusion of monoids that had that property would already be an equivalence since the effect of idempotent completion is to add more objects, but (effectively) no more arrows. On the other hand, for a monoid with many objects (aka a category), the situation is quite different. On the other hand, it might be useful to confine the discussion to idempotent complete categories to avoid this particular problem. Michael ============================================================================== Subj: Re: Comparison between Functor Categories (2 postings) Date: Wed, 30 Sep 92 12:00:29 +1000 From: street@macadam.mpce.mq.edu.au Dear Steve Sorry to cause you grief with the "fully faithful" terminology which is common in categorical papers, but perhaps not textbooks. The French use "pleinement fidele". The point about Cauchy-Morita completion is this: we cannot recapture a category A from its presheaf category P(A); we can only capture, up to equivalence, the Cauchy completion Q(A) of A. Given F : A --> B, if F* (or P(F) : P(B) --> P(A)) is fully faithful then the same will be true for Q(F) replacing F. For ordinary categories, Q(A) is the completion of A wrt splitting idempotents; ie, the full subcat of P(A) consisting of retracts of representables. Hence my point about F being surjective on objects up to retraction. [For additive categories, Q(A) is the full subcat of P(A) (additive ab-gp-valued presheaves) consisting of retracts of finite direct sums of representables.] +++++++++++++++++++++++++++++++++++++++++++++ From: Paul Taylor Date: Wed, 30 Sep 1992 11:45:33 +0100 Be careful about epimorphisms of categories.