Subject: Re: Name for this concept? Date: Mon, 1 Nov 93 15:58:46 +1100 From: kelly_m@maths.su.oz.au (Max Kelly) Bill Rowan asks: "What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g. A functor C->C' which is one leg of an equivalence takes such a set to another one X' in C', and any functor which is the other leg of the equivalence takes X' back to X." Well, I have called it an IDEAL; see G.M.Kelly, On the radical of a category, Jour. Austral. Math. Soc. 4 (1964), 299-307 and G.M.Kelly and F.W.Lawvere, On the complete lattice of essential localizations, Bull. Soc. Math. Belgique 41 (1989), 289-319. Max Kelly. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Name for this concept? Date: Mon, 1 Nov 1993 12:02:07 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) >What do you call it if you have a category C, and you have a class X of >arrows of C such that if x in X, then gxf in X for all composable isomorphisms >f and g? I haven't seen a name, but can I suggest calling X a "2-sided sieve"? (Or "2-sided crible"?) Conceivably, people who know what a sieve is could hazzard a guess at what "2-sided sieve" means. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Name for this concept? Date: Sun, 31 Oct 93 14:05:08 PST From: pratt@CS.Stanford.EDU Date: Thu, 28 Oct 1993 21:30:45 -0700 From: "William H. Rowan" What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g. This rolls connectedness and abstractness into the one condition, as follows. It had therefore better have some independently justifiable redeeming social value before burdening math dictionaries with its own name. 1. Dropping "iso" from your condition strengthens it to the notion of *connected component*. 2. Requiring that X be closed under all automorphisms F:C->C, in the sense that x in X implies F(x) in X, strengthens your condition to what might reasonably be called an *abstract class*. A category with two distinct isomorphic connected components (e.g. *-->* *-->*) witnesses the strictness of this strengthening, in that a single component does not form an abstract class but does satisfy your condition. Therefore, as a strong common weakening of "connected component" and "abstract class" (but not the strongest, being strictly weaker than their disjunction), it would seem that your condition deserves nothing shorter than "connected-abstract class." A functor C->C' which is one leg of an equivalence takes such a set to another one X' in C', [...] No, F(X) need not be a connected-abstract class even if we assume of X not the disjunction but the conjunction, that X is *both* a connected component and an abstract class. Witness any full embedding F:G->H of a group G (as a one-object groupoid) in a connected groupoid H having more than one object. Here F is an equivalence, and the set X of all morphisms of G is both a connected component and an abstract class. But not only is F(X) neither a connected component nor an abstract class of H, it is not even a connected-abstract class of H. What interesting theorem justifies adding "connected-abstract" to the lexicon? -- Vaughan Pratt (FTPables: boole.stanford.edu:/pub/ABSTRACTS.) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Name for this concept? Date: Tue, 02 Nov 93 12:51:22 SET From: Reinhard.Boerger@FernUni-Hagen.de I think "ideals" should be closed under composition with all morphisms (not just isos). I suggest the adjective "isomorphism-closed" or "replete", which coincide with the common terminology for full sucategories. Reinhard Boerger ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Names of these concepts Date: Tue, 02 Nov 1993 15:16:52 -0400 (EDT) From: MTHISBEL@ubvms.cc.buffalo.edu Max says he has called these globs 'ideals', and I wouldn't question that. Except that he seems to say that Bill Lawvere has joined him in so calling them. (Boy! Call me a taxi!) Bill is not available at the moment. Steve Schanuel and I doubt that Bill has called them ideals singularly or plurally. We suggest that Max and Bill have called a class of morphisms closed under pre- or postmultiplication by ANYBODY ideals. That is not Rowan's question. John Isbell ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Name for this concept? Date: Tue, 2 Nov 1993 12:56:57 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) >This rolls connectedness and abstractness into the one condition, as >follows. It had therefore better have some independently justifiable >redeeming social value before burdening math dictionaries with its own >name. A category is a ring with many objects but no additive structure, and a presheaf is then a left module. A representable presheaf is the ring considered as a left module over itself (but there are lots of them because of the many objects), and a sieve - a subpresheaf of a representable presheaf - is a left ideal. The category is also a - just one - bimodule (or profunctor) over itself just as a ring is, and the concept under discusion, a subbimodule, is an ideal exactly as Max Kelly said (a 2-sided ideal, or 2-sided sieve). That's a justification by analogy with ring theory, though there is a gap: ideals of rings are good not just because they are subbimodules. They are also kernels of quotients, and once the additive structure is dropped then it is no longer true that quotients are equivalent to subbimodules. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Modules list Date: Mon, 1 Nov 1993 23:03:45 -0800 From: "William H. Rowan" Quite a number of people (about 30) responded to my proposed Beck modules mailing list. I now have an account that will support this. However, I have decided to do things a little differently. The first change is to broaden the scope a little bit. It will now be about Beck modules, their pointed set object counterparts, and in addition, ringoids. Ringoids are rings with several objects, or, small additive categories. Ringoids come into it naturally because the natural categories of A-modules, for A a universal algebra, are identical with the categories of left modules over certain ringoids, called the enveloping ringoids. (I wrote my thesis on this.) The other change I am making is, instead of having a mailing list, to use the Usenet news system, in particular the newsgroup sci.math.research. This is a moderated newsgroup, the (slightly edited) announcement for which follows: From dan@math.uiuc.edu Sat Oct 2 18:31:11 PDT 1993 Article: 1298 of sci.math.research From: "Daniel R. Grayson" Newsgroups: sci.math.research Subject: Welcome to sci.math.research Date: Wed, 15 Sep 1993 07:02:22 -0500 Organization: University of Illinois at Urbana Approved: Daniel Grayson Originator: dan@symcom.math.uiuc.edu Welcome to sci.math.research. Here is our charter. CHARTER: This newsgroup is a forum for the discussion of current mathematical research. You are also encouraged to post announcements of - mathematics conferences - preprints available - new mathematics journals - online services for distribution of preprints A full archive maintained by Michael Boardman and Lake Forest College is available via gopher to math.lfc.edu, under the heading - Mathematics Related Items and its subheading - Archive of sci.math.research USENET newsgroup. It is indexed for fast retrieval of articles based on words within the text of the articles. Readers should also be aware of the mathematical information services available via gopher to e-math.ams.com. (end of announcement) In order to help ourselves (and others) recognize messages to our subgroup, we should all use the key phrase "Beck modules" (two words) in our subject line, and also in the text of the message. If you are new to the Usenet news system, note that the threaded news reader trn will allow you to select only those messages that have this key phrase, if that is what you want to do. There are several clear advantages to doing this instead of a mailing list, as well as a few disadvantages. I find the Usenet news system to be quite informative and convenient. It is important to use a threaded news reader such as trn, so that you will see discussion threads in sequence. It is _essential_ to learn how to use a news reader well so that you can filter out stuff you don't want to read. I imagine many of you have been doing this for much longer than I have, and will be quite comfortable with it. On the other hand, if anyone has difficulty with this system (for example, they have access to e-mail but not the newsgroup) please let me know and I will try to help you stay in the loop somehow. The discussion can begin at any time. I screwed up this announcement the first time, and no one got it. In the mean time, I posted a so-called "FAQ" to the news group to answer Frequently Asked Questions, and will post a new version of it when I have accumulated some substantial changes, (and if and when there is any discussion forthcoming!) Note my new e-mail address: rowan@crl.com. There is a place on this system where I will be able to put files of common interest. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Question about discrete opfibrations Date: Wed, 3 Nov 93 17:28:07 +0100 From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens) Suppose D is a finite category. Does there exist a unique discrete opfibration (dof) p:D->C such that C has a minimal number of objects? (With "unique", I mean that any two dofs p:D->C and p':D->C', where C and C' have the minimal number of objects, are isomorphic in the sense that there exists an isomorphism i:C->C' such that p' = ip) If so, is there an efficient algorithm to compute this discrete opfibration? If not so, can you give a counter-example? Thanks in advance for any replies, Frank Piessens Katholieke Universiteit Leuven. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Names of these concepts Date: Wed, 3 Nov 93 16:52:56 +1100 From: kelly_m@maths.su.oz.au (Max Kelly) Yes, of course, I misread the question in haste, thinking it referred to a class of morphisms closed under left & right composition with any morphisms. Sorry - Max Kelly. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Classes of arrows closed under isomorphism Date: Thu, 4 Nov 1993 08:05:28 -0800 From: "William H. Rowan" Many thanks for all the replies to my question about what to call a class of arrows closed under composition with isomorphisms. I now think a class of arrows "closed under isomorphism" or "isomorphism closed" is a good terminology. As for what good is it, it is true that if X is closed under isomorphism, the image FX under a functor F is not necessarily closed under isomorphism, but certainly FX generates such an isomorphism-closed class. And, if F and F' are naturally isomorphic, then FX and F'X generate the same isomorphism-closed class. This is very elementary, but I like it. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: European Colloquium Category Theory -ECCT- Note from moderator: The following was posted on the Usenet Newsgroup sci.math.research yesterday. It is presumably of interest to subscribers to this list. ++++++++++++++++++++++++++++++++++++++++++++ From: damphous@univ-tours.fr Date: Thu, 4 Nov 93 14:56:40 GMT EUROPEAN COLLOQUIUM on CATEGORY THEORY in Tours (France) ---- ECCT ---- FIRST ANNOUNCEMENT (November 4th 1993) A European Colloquium on Category Theory (ECCT) is planned in TOURS (France) from July 22nd to July 29th, the week before the International Congress of Mathematics in Z\"urich. Mathematicians present in Europe at that time are most welcome to pre-register now. Having an estimate of the participation is important for us at this stage of the organiza- tion To pre-register, you must proceed as follows: Register on the ECCT list: send an e-mail to ECCT-request@univ-tours.fr, with the following line in the body of the message (do not put a subject in this e-mail). SUBSCRIBE When pre-registering, you will automatically be sent all pertaining information as the organization progresses. For any further inform- ation, contact Pierre Damphousse or Ren\'e Guitart at : damphous@univ-tours.fr or guitart@univ-tours.fr ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: e.e.c.t. and c.a.e.n. 94 Date: Wed, 10 Nov 93 18:08:44 +0100 From: ageron@univ-caen.fr (Pierre Ageron) Two events in France next summer might be of interest to category theorists: - the EUROPEAN COLLOQUIUM ON CATEGORY THEORY (E.E.C.T.) (all areas of category theory) in Tours, July 22-29 organizers: Pierre Damphousse and Rene Guitart to pre-register: send the one-word message "SUBSCRIBE" to EECT-request@univ-tours.fr (no subject) - the workshop CATEGORIES, ALGEBRES, ESQUISSES ET NATURALITES (C.A.E.N. 94) (main themes: sketches, categories with structure, applications to logic and algorithmics) in Caen (Normandy), September 27-30 organizer: Pierre Ageron to get more information: send a message to ageron@univ-caen.fr ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: category theory & linear optimization Date: Wed, 10 Nov 93 17:03:46 PST From: Michael J. Healy (206) 865-3123 I apologize in advance if this is not relevant or is already familiar, but there is work in using categorical methods for software synthesis to synthesize optimization algorithms and codes. One reference for this is D. R. Smith and M. R. Lowry (1990). Algorithm Theories and Design Tactics, Science of Computer Programming 14, North-Holland, pp. 305-321. Regards, Mike Healy ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Why nothing beyond natural transformations? Date: Thu, 11 Nov 93 11:18:42 GMT From: jrk@sys.uea.ac.uk (Richard Kennaway) I have a rather basic but vague question. Categories, functors, and natural transformations seem to form the first three elements of a series which could be continued indefinitely. Why is it that these three suffice, and that further members of the sequence are almost never required? Similarly, there are (1-)categories and 2-categories, and further members of this sequence can be defined, yet they are rarely needed. I have once seen 4-categories referred to, but only once. Is there some intuitive explanation for this? -- ____ Richard Kennaway __\_ / School of Information Systems Internet: jrk@sys.uea.ac.uk \ X/ University of East Anglia uucp: ...mcsun!ukc!uea-sys!jrk \/ Norwich NR4 7TJ, U.K. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Higher-dimensional categories Date: Thu, 11 Nov 93 14:25:17 PST From: baez@ucrmath.ucr.edu (john baez) I'm very interested in the work of Louis Crane, Dan Freed and others on how extending TQFTs to higher dimensions or higher codimensions can be nicely phrased in the language of n-categories. Luckily James Dolan is here at UCR now and is educating me in such matters. He and I are beginning to struggle towards a nice concept of "weak n-categories," or even better "weak omega-categories," or perhaps even better, "weak Z-categories" (in some sense a homotopical analog of chain complexes). I am interested in these things for doing physics, but I dimly realize that lots of people have struggled with these concepts, and I want to get a better grasp of what the key achievements and problems in this subject are. I have read Kapranov and Voevodsky's massive preprint on Braided Monoidal 2-categories, 2-vector spaces and Zamolodchikov tetrahedra equations, and soon I should receive a copy of the new paper on coherence in 3-categories by Gordon, Powell, and Street. What are the other main things I should find out about? How come all you categorists haven't yet invented a notion of "weak Z-categories," in which there are n-morphisms for all integers n, and all relations between n-morphisms are expressed in terms of (n+1)-morphisms? Regards, John Baez ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Why nothing beyond natural transformations? Date: Fri, 12 Nov 1993 08:16:25 -0500 From: James Stasheff the intuition behind those lacks may be lack of intuition (or good examples) on our part but the times they are achanging jim ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Question about discrete opfibrations Date: Fri, 12 Nov 93 13:22:14 SET From: Reinhard.Boerger@FernUni-Hagen.de Let C be a category with 2 objects A,B and 2 morphisms u,v:A->B. let D be the binary copower of C, i.e. the product of C with the 2-object discrete category and let F:D->C be the codiagonal funjctor mapping both copies of C identically to C. Let G:C->D be the functor mpping one coy of C identically and interchanging u and v in the other copy. Then F and G are opfibations, which are not isomorphic under D (though they are isomorphic over C). Greetings Reinhard Boerger ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Higher-dimensional categories (2 posts) Date: Fri, 12 Nov 93 07:34:09 EST From: barr@triples.Math.McGill.CA (Michael Barr) John Baez and others interested in this question might begin by checking out the work of Rainer Vogt. Once when I was in Aarhus, he was telling me that a weak infinity category was a simplicial set that satisfied the Kan condition for interior faces only. That is, for any collection of n n-simplexes x^0,...,x^{i-1},x^{i+1},...x^n with 0 < i < n, such that d^jx^k = d^{k-1}x^j, whenever j < k and none of j, k, k-1 was i, there is an (n+1)-simplex x such that d^jx=x^j for j >< i. Applied in dimension 2, this gives a weak composite. In dimension 2 there is only one interior face and if this condition applied uniquely, you would have a category with d^1x being the composite of x^0 and x^2. Michael Barr ++++++++++++++++++++++++++++++++++++++++++ From: "Allen Knutson" Date: Fri, 12 Nov 93 9:14:40 EST Have you looked at Quillen's "Homotopical Algebra"? I didn't much myself, but it sounds potentially relevant, and Quillen's a mighty smart guy. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: job ad From: Richard Wood Date: Wed, 3 Nov 1993 16:42:56 -0400 Dalhousie University Department of Mathematics, Statistics and Computing Science Halifax, Nova Scotia Canada B3H 3J5 phone (902) 494-2572 FAX (902) 494-5130 Applications are invited at the Assistant Profesor level for two anticipated tenure track positions in the Division of Mathematics. Duties include teaching at the Graduate and Undergraduate levels and maintenance of a strong research programme. Selection is based on demonstration of promise of excellence in research and teaching. To apply, please send a curriculum vitae, selected reprints, and three letters of reference to Dr. R.P. Gupta, Chair Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, Nova Scotia, Canada B3H 3J5 phone: (902) 494-2572 FAX: (902) 494-5130 The deadline date for applications is January 15, 1994. Dalhousie University is an Employment Equity/Affirmative Action Employer. The University encourages applications from qualified women, aboriginal peoples, visible minorities and persons with disabilities. In accordance with Canadian Immigration requirements, this advertisement is directed to Canadian Citizens and Permanent Residents. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Why nothing beyond natural transformations? Date: Sun, 14 Nov 93 15:18:58 +1100 From: street@macadam.mpce.mq.edu.au >Date: Thu, 11 Nov 93 11:18:42 GMT >From: jrk@sys.uea.ac.uk (Richard Kennaway) > >I have a rather basic but vague question. > >Categories, functors, and natural transformations seem to form the first >three elements of a series which could be continued indefinitely. Why is >it that these three suffice, and that further members of the sequence are >almost never required? They don't, and they are. Lo, before the forties, everyone thought they knew what "natural" was and thought not of defining it. Then came the definition [E-ML] in terms of "categories". Categories themselves were seen as unnatural and unnecessary for many years; and still are by some. As you have indicated, "2-categories" are now seen as quite natural by many. To some of us, n-categories are as natural as n-simplexes and n-cubes. >Similarly, there are (1-)categories and 2-categories, and further members >of this sequence can be defined, yet they are rarely needed. I have once >seen 4-categories referred to, but only once. > >Is there some intuitive explanation for this? There may be a mathematical reason for feeling there is a barrier at n = 3. It is easy to define n-categories and there are interesting examples of these. But structural examples often form something less; let's call them weak n-categories: the r-th composition is only associative up to coherent (r-1)-equivalence etc. You may have met weak 2-categories which are Benabou's "bicategories". Now every bicategory is equivalent (in the approp sense) to a 2-category. Yet, not all weak 3-categories (Gordon-Power-St have called them tricategories) are equiv to 3-categories. Fear of the unknown is natural; at the 3-level and above we are forced into more unfamiliar and exotic phenomena. Exotic, yes, but not unnatural. This is where cubes and braids lurk, and who would call them unnatural? Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Higher-dimensional categories Date: Sun, 14 Nov 1993 07:24:24 -0500 (EST) From: GLENN@CUA.EDU I've been working for several months on an approach to higher dimensional composition and the accompanying higher dim category theory (functors, natural transformations, limits, etc...). (It may be related to Vogt's work, cited in Barr's reply, but I'm unfamiliar with that work.) I'd be happy to send a "work-in-progress" report (in Latex format) giving details on what I've discovered thus far. The "dimension 1" cases are ordinary categories. In the dimension n case: objects are (n-1)-simplexes, maps are n-simplexes and composites of maps are (n+1)-simplexes of a simplicial set C. As Barr described in his response, the partial binary operation which defines composition in an ordinary category, associates a map (= 1-simplex) to a pair of maps whose source and target match appropriately. In the n-dim case there is a partial (n+1)-ary operation which, given n+1 n-simplexes whose faces match appropriately, produces their "composite", another n-simplex whose faces match the given ones according to the simplicial identities. One has to specify *which* of the faces of the n+1 simplex is the composite of the others. It can be the face opposite vertex i for any i = 0,..., n+1. The most efficient way to specify all this is by saying that the standard function from C_{n+1} ( = the n+1 simplexes of C) to the set of open i-boxes of C of dimension n+1 is an isomorphism. Examples (though special cases) of such structures are already known: n-dimensional hypergroupoids. These include Eilenberg-MacLane spaces and also arise from the singular complex of any topological space. Rreferences to hypergroupoids include my 1982 paper ("Realization of Cohomology Classes in Arbitrary Exact Categories", Jour. of Pure and Appl. Alg. 25 (1982) 33-105) and papers by Jack Duskin. There seems to be a reasonable "category theory" for this kind of higher-dimensional composition. For example, functors C --> C' and natural transformations of such functors occur as simplexes in the function complex C'^C (of dimensions n-1 and n respectively). If C and C' are higher dim categories in the sense I described above, then so is C'^C. Paul Glenn Dept. of Mathematics Catholic University of America Washington, DC 20064 Internet: glenn@cua.edu Phone: 202 319 5221 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Higher-dimensional categories Date: Mon, 15 Nov 93 10:11:01 +1100 From: street@macadam.mpce.mq.edu.au I am very interested in seeing the paper of Paul Glenn, but in the meantime, perhaps relevant is Conjecture 5.3 of my "The algebra of oriented simplexes" JPAA 49 (1987) 283-335. This is a slightly more explicit form of a conjecture ( pre 1978 ) of John E. Roberts who motivated my work. Dominic Verity has proved this conjecture (it was announced in Bangor early this year and at the MSRI Conference last July). Verity's preprint on this will be available very soon. This work characterizes n-categories as simplicial sets with certain elements at each dimension distinguished (and called "hollow" in loc cit, but now we call them "thin" in accord with the Welsh School). The kinds of operations Barr and Glenn hint at are expressed as UNIQUE horn filler conditions. Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Three questions: on equalizers, e-mail to Manes, arrows as axioms Date: Wed, 17 Nov 1993 06:23:37 +0100 From: France.Dacar@IJS.si Greetings. These days I'm going through a little categories-refresher course. Among my reading matter are the "Categories, Functors..." by Arbib and Manes, and Manes' "Algebraic Theories". My three questions are related to these. Q1: In "Algebraic Theories", there are several propositions (say 4.22, 4.23, 4.24) which include the following assumption: "Let (K,E,M) be a regular category _such that every equalizer is in in M_." This mystifies me, because an equalizer is _always_ an M-monic, for any image-factorization system (E,M). In any category, if an equalizer f factors as f = ge (composition to the left...) with e epi, then e is an isomorphism. This is quite trivial; since I cannot imagine that this could have been overlooked for as long as image-factorization systems were around up to the Manes' book, I am starting to doubt my ability to spot a mistake in my reasoning... Q2: How can I reach Manes or Arbib? They're not on the structures mailing list. I have collected quite an errata on their book which they still might find useful. Q3: This last question is about Universal Algebra vs. Monads and friends. My knowledge of the relevant literature is rather narrow, so this question has been probably adressed somewhere; I'd like the relevant pointers. As a kind of self-imposed exercise I have tried to fill in the gap I felt there exists between the old good techniquies of UA and that of high-flying categorial treatment involving monads and such. I developed a little "elementary theory of UA in categories" which uses arrows as axioms, in the following sense. Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. Then, playing around with conditions imposed on the category C and on the axioms A, I can, step by step, reconstruct most of the "standard" results of UA, up to and beyond the Birkhoff's theorem. All along the reasoning stays close to that in AU, just translated into the language of arrows. At certain point a monad emerges, and then I can make the jump upwards to more rarefied regions. For me this treatment by arrows as axioms provided the "missing link". On the road up to the final ascent to monads there are plenty of points where one can go off in some other direction and still get some benefit by transplanting part of UA techniques "in abstract" to an un-algebraic matter. One can also generalize: there are other conditions on an object expressible by a single arrow, one can allow restricted boolean combinations of one-arrow conditions as basic statements (say Horn implications), and so on. If this was already done, where can I look it up? If not, is it worth publishing, and where? ------------------------------------------------------------------------ France Dacar Email: france.dacar@ijs.si Computer Science Department Phone: +386 61 1-259-199 / 768 Jozef Stefan Institute Fax: +386 61 1-258-058 Jamova 39, 61000 Ljubljana, Slovenia ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: higher dimensional groupoids Date: Wed, 17 Nov 1993 17:48:37 +0000 From: mas010@bangor.ac.uk (Prof R. Brown) Higher dimensional groupoids One reason to see how higher dimensional categories exist and are useful is to look at the groupoid case. Higher dimensional groupoids form a useful higher dimensional version of groups and the fundamental group. The old result that "a set with two compatible group structures is just an abelian group" (basically, 1932, with Cech's description of higher homotopy groups) was for long regarded as an obstruction to such a programme. In fact, sets with 2 compatible groupoid structures model 2-dimensional homotopy theory, and so on for n compatible groupoid structures. This makes them of course very complicated, even in dimension 2. This fact is interesting in itself. A clear problem was to define a higher homotopy groupoid. One solution was by the writer and PJ Higgins, as may be found in joint work in (Proc London Math Soc 1978, for dimension 2, JPAA 1981 for all higher dimensions). This gives what we called the fundamental omega-groupoid of a filtered space. Later, Loday found a more complex and more general model, the fundamental cat^n-group of an n-cube of spaces. The existence of these functors is non-trivial. My purpose in considering higher dimensional groupoids since about 1966 was the possibility of higher dimensional van Kampen Theorems, modelled on the groupoid version. The references are the above papers with Higgins, and also with J-L Loday (Topology, 1987, Proc London Math Soc 1987), with his more complicated algebraic model. These theorems and models now allow for specific computations of some homotopy invariants and even some homotopy types (my paper in the Adams Memorial Colloquium). The intuition is that one needs algebraic structures which can model the geometric notion of subdivision, and which have structure in a range of dimensions, to carry the information about how bits of a space fit together. Multiple groupoids (or categories) seem well placed for that.The intuition is related to old ideas in topology of "What is a cycle?". A cycle should be some kind of composition of the little pieces. How should one accomplish this, algebraically? One has to move away from a linear notation and an always defined composition. Taking the free abelian group on the little bits seems like a cop-out (but of course, it has its uses!). One aspect of the theory is the equivalence of various views of a given structure. This is referred to by Ross Street in his communication, with regard to infinity-categories. So we have a set of equivalences between crossed complexes, omega-groupoids (RB-PJH, JPAA 1981), infinity-categories (CTGDC 1981), cubical T-complexes (CTGDC 1981), simplicial T-complexes (Ashley's thesis of 1978, published in Diss Math 155, 1988), polyhedral T-complexes (Jones, 1984, published as Diss Math 156, 1988). Here T-stands for "thin". These canonical structures are a generalisation of identities which appear only from dimension 2. The idea appears in old axioms for groups; with Keith Dakin in Bangor in 1975; and independently with Roberts in Australia at the same time, with the name "hollow". Thin elements play a crucial role in computations in and manipulations with these objects. Intuitively, thin elements have commutative boundary, and commutative boundaries have thin fillers. Thin elements also define compositions: all faces but one of a thin element "compose" to give the remaining face. For more on the philosophy of this, see Jones' thesis, which deals with compositions of general faces. Even more complex is the equivalence between cat^n-groups and crossed n-cubes of groups (Ellis-Steiner, JPAA, 1987). These equivalences allow a translation from a linear notation to a "higher dimensional" notation. Verity's equivalence in the category case mentioned by Ross Street, is in the same spirit as, and generalises, the equivalence between simplicial T-complexes and infinity-groupoids (go through crossed complexes, combining RB-PJH with Ashley). These things work. I just realised that one can do many sums with results of the first paper of RB-PJH, which for example allows the computation not only of \pi_2(BH \cup CBG) when G is a subgroup of H, for specific G,H, but also of the first k-invariant, i.e. the 2-type, of this space. These ideas are all modelled on groupoids, as is natural for the homotopy applications and the Generalised Van Kampen Theorems. A categorical version has not been done in full generality (polyhedrally). One does not expect the same type of application. There are relations with cohomology and knot theory which need more investigation. A lot of these ideas were discused in the visit of Ross and Dominic Verity to Bangor in May/June, 1993. Ronnie Brown ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Three questions: on equalizers, e-mail to Manes, arrows as axioms Date: Wed, 17 Nov 1993 18:46:40 -0800 From: "William H. Rowan" Concerning axiomatization of universal algebra using arrows, usually we consider that an identity in universal algebra is given categorically by giving a pair of elements of a free algebra F (or, a pair of homomorphisms from a free algebra on one element into F) and then an algebra X satisfies iff all homomorphisms F->X coequalize . I think this is explained sketchily in MacLane, CWM. This is subsumed by your framework because we only need to form the coequalizer h:F->G of and then take {h} as your class A of arrows. Perhaps your framework has greater expressive power than the usual one in universal algebra, in which case it might be worth publishing, but I can offer no strong opinion to that effect, and indeed you will probably have to figure this out yourself. You also need to consider that perhaps someone else has an equivalent or even more general framework which is all worked out, such as sketches or something. Perhaps some other people will have other comments. Bill Rowan +++++++++++++++++++ Date: Thu, 18 Nov 93 10:07:16 CST From: strecker@math.ksu.edu (George Strecker) Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. I believe that this was first done by Banaschewski and Herrlich Houston Math. J. (1977) 149-171. See also Section 22 of the book "Abstract & Concrete Categories" [Wiley Interscience]. George Strecker ++++++++++++++++++++++ From: koslowj@math.ksu.edu (Juergen Koslowski) Date: Thu, 18 Nov 93 9:20:56 CST Concerning question 1 of France Dacar: > Q1: In "Algebraic Theories", there are several propositions > (say 4.22, 4.23, 4.24) which include the following assumption: > "Let (K,E,M) be a regular category _such that every equalizer is in > in M_." This mystifies me, because an equalizer is _always_ an > M-monic, for any image-factorization system (E,M). In any > category, if an equalizer f factors as f = ge (composition to the > left...) with e epi, then e is an isomorphism. This is quite > trivial; since I cannot imagine that this could have been > overlooked for as long as image-factorization systems were around > up to the Manes' book, I am starting to doubt my ability to spot > a mistake in my reasoning... > I don't have my copy of Manes' AT handy, but it seems to me that the assumption that E consists of epis is not warranted. After all, E = all morphisms and M = all isos constitute an image-factorization system, and not all equalizers are isos. Regards, J"urgen -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++ Date: Thu, 18 Nov 93 11:20:58 -0500 From: cfw2@po.CWRU.Edu (Charles F. Wells) The work described in Point 3 of France Dacar's recent message looks very much like a special case of work of Andreka and Nemeti and Guitart and Lair. The basic references are given below in BibTeX input form, which is (I fondly believe) self-explanatory: @ARTICLE{ author = "Andreka, H. and I. Nemeti", title = "Formulas and ultraproducts in categories", journal = "Beit. zur Alg. und Geom.", volume = 8, year = 1979 } @ARTICLE{ author = "Guitart, Ren\'e and Christian Lair", title = "Calcul Syntaxique des Mod\`eles et Calcul des Formules Internes", journal = diagrammes, year = "1980", volume = "4", } I do not have the Andreka and Nemeti article. These articles are also relevant: @ARTICLE{ author = "Guitart, Ren\'e and Christian Lair", title = "Limites et Co-limites pour Repre\-senter les For\-mu\-les", journal = diagrammes, year = "1982", volume = "7", } @ARTICLE{ author = "Guitart, Ren\'e", title = "On the Geometry of Computations (I)", journal = cahiers, year = "1986", volume = "27", pages = "107--136", } -- 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: Arrows as Axioms Date: Fri, 19 Nov 93 18:56:34 +0100 From: momathie@mathp7.jussieu.fr (Monique Mathieu) Ce dont parle F.Dacar est largement etudie dans : 1. H. Andreka et I. Nemeti , "Generalization of the Concept of Variety and Quasi-Variety to Partial Algebras through Category Theory" Diss. Math. CCIV , Warszawa , 1983. On peut en trouver une "version modernisee" dans : 2. F. Cury ,"Completion et Completude selon H.Andreka. et I.Nemeti. Diagrammes 29 , Paris 1992. Le cas particulier de "il existe un unique" se touve dans : 3. L. Coppey, Theories Algebriques et Extensions de Pre-faisceaux" Cahier de Top. et de Geometrie Diff. , Vol. XIII - 1 et XIII - 4 , Paris 1972 , 4. L. Coppey et C. Lair , "AMEN, Algebricite - Monadicite - Esquissabilite et Non-algebricite , Diagrammes 13 , Paris 1985. Anterieurement, H. Andreka, I. Nemeti et I. Sain avaient meme etendu la satisfaction (par un objet) d'un ensemble A de fleches en la satisfaction d'un ensemble de familles projectives de fleches. Voir, par exemple : 5. A. Andreka et I. Nemeti , "A General Axiomatizability Theorem formulated in terms of Cone Injective Subcategories" , Coll. Math. Soc. , J. Bolyai 29 , Univ. Alg. Esztergon (Hungary) 1977. 6. I. Nemeti et I. Sain , "Cone Implicational Subcategories and some Birkhoff-Type Theorems" Coll. Math. Soc. J. Bolyai 29 , Univ. Alg. Esztergon (Hungary) 1977. 7. H. Andreka et I. Nemeti , "Los Lemma holds in every Category" Studia Scientiarum Mathematicarum Hungarica 13 , 1978. 8. H. Andreka et I. Nemeti , "Injectivity in Categories to represent all First Order Formulas" , Demonstratio Math. Vol XII - 3 , 1979. L'aspect completement categorique des choses consiste a satisfaire des ensembles de cones projectifs (d'indexations non necessairement discretes). Ceci a ete traite originellement (a la suite des travaux de H. Andreka, I. Nemeti et I. Sain) tout d'abord par : 9. R. Guitart et C. Lair , "Calcul Syntaxique des Modeles et Calcul des Formules Internes" , Diagrammes 4 , Paris 1980. Le lien systematique avec les categories de modeles d'esquisses (i.e. de theories du premier ordre) est largement explicite par : 10. C. Lair , "Categories Qualifiables et Categories Esquissables" Diagrammes 17 , Paris 1987. Ces derniers temps, certains reprennent les idees qui se trouvaient dans 1. pour, vraisemblablement, aboutir dans quelques annees a celles qui se trouvent dans 10?? 11. J. Adamek et J. Rosicky , "Locally Presentable and Accessible Categories" Livre a paraitre. 12. H. Makkai , "Generalized Sketches as a Framework for Completeness Theorem" Pre-print , 1993. Monique Mathieu, Universite Paris 7, Denis Diderot. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: normed categories Date: Sun, 21 Nov 93 07:26:51 CST From: koslowj@math.ksu.edu (Juergen Koslowski) In "METRIC SPACES, GENERALIZED LOGIC, AND CLOSED CATEGORIES" (Rend. del Sem. Mat. e Fis. di Milano 43 (1973)) Bill Lawvere suggests the notion of a "normed category" as a category enriched in a suitable (symmetric monoidal) closed category S(R). (Here R denotes the interval [0,\infty], ordered by >= and with + as tensor and truncated - as internal Hom.) Has anybody worked out the details? Presumably, the objects of S(R) are to be subsets of R. This is suggested by the claim that there ought to be a closed functor inf from S(R) to R that induces the passage from normed categories to metric spaces (and by the use of the same symbol "S" that is used to denote the category of sets). As with any commutative monoid, the power-set of R under inclusion carries a symmetric monoidal closed structure: A + B = { a+b | a\in A and b\in B } (point-wise) defines the tensor and C - A = { x\in R | A + {x} contained in C } (not point-wise!) defines the internal Hom. {0} is the unit object. Clearly, inf turns into a strong functor as required. But how am I to interpret "the fundamental property of a normed category" (top of p. 140), namely (*) |f| + |g| >= |fg|? This inequality seems to apply to ordinary (= Set-enriched) categories X that carry an extra structure, namely a function |-| from Mor(X) to R. Presumably, this function also ought to satisfy (**) 0 >= |id_Y| for any X-object Y. E.g., one could define |f| to be 0 for every isomorphism of X, and 1 for every other morphism. In general this does NOT yield an S(R)-enriched category in the sense defined above. In fact, the definition above only seems to support the equality |f| + |g| = |fg|. Can further morphisms be added to S(R) without destroying the monoidal closed structure? Yes, if A +{q} is contained in B one can interpret q as a morphism from A to B. (The original inclusions arise for q = 0.) All these new morphisms f satisfy x <= f(x). But in view of the preceding paragraph one would prefer to have morphisms g that satisfy x >= g(x). Since this latter condition does not allow any new morphisms with domain {0}, the monoidal closed structure cannot be maintained. I must be missing something obvious. E.g., the definition of S(R) above doesn't use the symmetric monoidal closedness of R itself at all. Could someone please point me in the right direction? Thank you! J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) aberne@dhvrrzn1.uni-hannover.d400.de ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: normed categories Date: Sun, 21 Nov 1993 10:42:22 -0500 (EST) From: MTHFWL@ubvms.cc.buffalo.edu No,it is not the power set of a closed category in which the normed categories are enriched, but rather the portion of the presheaf category (closed via Brian Day) consisting of coproducts of representables, for which there is an alternative description of the closed structure. Some details of this construction were published in a paper by Betti and Galluzzi. It is also mentioned in the introduction to Springer Lecture Notes 274 that real numbers are merely the poset reflection of the category of dynamical systems. -Bill ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Arrows as Axioms From: Jiri Rosicky Date: Mon, 22 Nov 1993 14:15:41 +0100 (MET) One more reference added to the list of Monique Mathieu: 11a. J.Adamek, J.Rosicky: On injectivity in locally presentable categories, Trans. Amer. Math. Soc. 336 (1993), 785-804 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Sydney ftp site: current contents Date: Wed, 24 Nov 1993 18:12:58 +1100 From: walters_b@maths.su.oz.au (Bob Walters) SYDNEY CATEGORIES COMBINATORICS and COMPUTER SCIENCE CCC Material Available by Anonymous FTP from maths.su.oz.au 129.78.68.2 ======================================================================== This file is README in the sydcat directory of maths.su.oz.au, 129.78.68.2, accessible by anonymous ftp. The sydcat directory is for FTP distribution of recent publications, programs, seminar listings and other material of the Sydney Category Theory, Combinatorics, and Computer Science Group. ======================================================================== This group consists of staff and students at the University of Sydney Macquarie University University of New South Wales University of Technology Sydney Sydney, Australia, including the following: Murray Adelman murray@macadam.mpce.mq.edu.au Brian Day Lee Flax flax@macadam.mpce.mq.edu.au Robbie Gates gates_r@maths.su.oz.au Amitavo Islam islam_a@maths.su.oz.au Barry Jay cbj@socs.uts.edu.au Mike Johnson mike@macadam.mpce.mq.edu.au Giulio Katis katis_p@maths.su.oz.au Max Kelly kelly_m@maths.su.oz.au Mark Leeming leeming_m@maths.su.oz.au Stephen Ma ma_s@maths.su.oz.au Wafaa Khalil khalil_w@maths.su.oz.au Wesley Phoa wes@cs.unsw.oz.au Usha Sridhar sridhar_u@maths.su.oz.au Ross Street street@macadam.mpce.mq.edu.au Sun Shu-Hao sun_s@maths.su.oz.au Dominic Verity domv@macadam.mpce.mq.edu.au Henry Weld weld_h@maths.su.oz.au Bob Walters walters_b@maths.su.oz.au Karl Wehrhahn wehrhahn_k@maths.su.oz.au =======================Available material=============================== The following files and directories are available: ADDRESS lists are in sydcat/addresses BIBLIOGRAPHY files are in sydcat/bibliography BOOK descriptions are in sydcat/books PAPERS:dvi files of papers of author are in sydcat/papers/author CONFERENCE details are in sydcat/conference SEMINARS - some details are in sydcat/seminars SOFTWARE - programs are in sydcat/software Here is the current list of contents: addresses/catcurrent The Category Theory address list maintained by Max Kelly and Michael Johnson Updated regularly addresses/structdir Vaughan Pratt's email address list Updated October 1993 Release 3.0 Master copy: Boole.Stanford.EDU:~ftp/pub/struct.dir Maintainer: Vaughan Pratt, pratt@cs.stanford.edu bibliography A directory for bibliographies: contains some Bibtex files of sydcat publications bibliography/reports List of Research Reports of the Pure Mathematics Department, University of Sydney, Australia Updated regularly. books/hibi Contents and bibliographic details of Takayuki Hibi "Algebraic combinatorics on convex polytopes" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia books/walters Information about, corrections to, and bibliographic details of R.F.C. Walters "Categories and Computer Science" published by Carslaw Press in Australia and by Cambridge University Press elsewhere Contents and bibliographic details of RFC Walters "Number theory an introduction" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia books/wehrhahn Contents and bibliographic details of KH Wehrhahn "Combinatorics: an introduction" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia conferences Details of conferences papers/jay A directory containing papers of Barry Jay Title: Compositional Characterization of Observable Program Properties Authors: B. Steffen,C. Barry Jay, M. Mendler Process: Aarhus TR and accepted by Journal Title: Coherence in Category Theory and the Church-Rosser Property Author:CBJ Process: published in Studia Logica Title: Modelling reduction in confluent categories Author: CBJ Process: Published in the Durham Proceedings Title: Fixpoint and Loop Constructions as Colimits Author: CBJ Process: Como Proceedings Title:Long $\beta\eta$ normal forms and confluence (revised) Author: CBJ Process: static cf virtues.tex Title: Extending properties to categories of partial maps Author: CBJ Process : static Title: Partial Functions, Ordered Categories, Limits and Cartesian Closure Author: CBJ Process: proceedings of the HOW Title: Tail recursion through universal invariants Author: CBJ Process: to appear in TCS papers/johnson M.S.Johnson Linear term rewriting systems are higher dimensional string rewriting systems, {\em Proc of the Institute for Mathematics and its Applications\/} (1991), 101--110. M.S.Johnson (with G.P.~Monro and C.N.G.~Dampney) A mathematical foundation for ERA, {\em Proc of the Institute for Mathematics and its Applications\/} (1991), 77--84. M.S.Johnson (with R.F.C.~Walters) Category theoretic modelling of digital circuits and systems, {\em Proc of the Pan-Commonwealth Conference on Mathematical Modelling} {\em in Circuit Design}, Commonwealth Science Council (1992), 199--213. M.S.Johnson (with R.F.C.~Walters) Algebra objects and algebra families for finite limit theories, {\em Journal of Pure and Applied Algebra} {\bf 83} (1992) 283--293. M.S.Johnson (with S.-H. Sun) Remarks on representations of universal algebras by sheaves of quotient algebras, M.S.Johnson {\em Proceedings of the Canadian Mathematical Society\/} {\bf 13} (1992) 299--307. M.S.Johnson (with C.N.G.~Dampney and P.~Deuble) Taming large complex information systems, M.S.Johnson {\em Proceedings of Complex Systems} '92, IOS Press, Amsterdam, 210--222. M.S.Johnson (with R.~Buckland) An application of logic programming in pure mathematics, to appear in {\em Proceedings of ACSC}, 1993. M.S.Johnson (with C.N.G.~Dampney) Category theory and information systems engineering, {\em Proceedings of AMAST93}, Unviersity of Twente, Holland, 95--103. M.S.Johnson On the value of commutative diagrams in information modelling, to appear in {\em Springer Workshops in Computing}, 16 pages. papers/kelly G.M. Kelly, Stephen Lack: Finite-product-preserving functor, Kan extensions, and strongly-finitary 2-monads G.M. Kelly, Stephen Lack, R.F.C. Walters, Coinverters in categories with structure galaus.dvi G.Janelidze and G.M.Kelly, Galois theory and a general notion of central extension. (A4 version) galam.dvi G.Janelidze and G.M.Kelly, Galois theory and a general notion of central extension. (American quarto version) janmark.dvi G.Janelidze and L.Ma'rki, Radicals of ringa and pullbacks. papers/phoa Wesley Phoa bohm.ps "From term models to domains" --describes a category of `synthetic domains' for the closed term model in which terms with the same Bohm tree are identified Wesley Phoa graph.ps "Building domains from graph models" --describes a category of `synthetic domains' in the realizability topos arising from the r.e. graph model of the lambda-calculus Wesley Phoa pcf.ps "A note on PCF and the untyped lambda-calculus" --proof of computational adequacy of an untyped translation of call-by-name PCF Wesley Phoa poly.ps "A simple categorical semantics for first-order polymorphism" --describes how any cartesian closed category can be used to model ML polymorphism, using the notion of `polynomial category' M.P. Fourman and Wesley Phoa sml.ps "A proposed categorical semantics for Pure ML" (with M. P. Fourman, LFCS) --sketch of a semantics for SML using synthetic domain theory; focuses on the Modules system Wesley Phoa subtypes.ps "Using fibrations to understand subtypes" --informal account of categorical models for subtyping and bounded quantification Wesley Phoa synth.ps "Effective domains and intrinsic structure" --describes a category of `synthetic domains' in the effective topos Wesley Phoa tech.ps Replacing fibs.ps, topoi.ps, eff.ps, these notes provide an introduction to (some aspects of) a) fibrations and polymorphic lambda calculus b) constructive logic, categorical logic and topos theory c) Kleene realizability; PERs and omega-sets d) the effective topos; modest sets and how they model polymorphism They assume some basic knowledge of category theory, logic and typed lambda calculus. No familiarity with indexed categories or with categorical logic or topos theory is required. The notes do not attempt to be comprehensive, but simply try to give a reasonably relaxed account of the material. They are about 150pp including the index and appendixes. There are plenty of exercises. papers/sun Shu-Hao Sun, RFC Walters, Representations of modules and cauchy completeness Shu-Hao Sun, Adjunction of the associated sheaf functor of non-commutative rings Shu-Hao Sun, Non-commutative Deligne formula Shu-Hao Sun, Biregular rings and their duality Shu-Hao Sun, Non-commutative quasi-coherent sheaves Shu-Hao Sun, Equivalence of algebraic and geometric local cohmology Shu-Hao Sun, Structure sheaves for non-commutative rings Shu-Hao Sun, Duality on compact prime ringed spaces Shu-Hao, Generalized Grothendieck topologies papers/walters Aurelio Carboni, Stephen Lack, R. F. C. Walters, An introduction to extensive and distributive categories, Jun92. S. Carmody, R.F.C. Walters, The Todd-Coxeter Procedure and Left Kan Extension, Mar91 S. Carmody, R.F.C. Walters, Computing quotients of actions of a free category, Mar91. M. S. Johnson, R.F.C. Walters, Algebra Families, Apr92 P. Katis, N. Sabadini, RFC Walters, On discrete dynamical systems and concurrency, Dec93 G.M. Kelly, Stephen Lack, R.F.C. Walters, Coinverters in categories with structure, Jun92 Wafaa Khalil, R.F.C. Walters, An imperative language based on distributive categories II Apr92 Wafaa Khalil, R.F.C. Walters, Functional processors and operations on them in extensive categories, May93 Wafaa Khalil, Eric Wagner, R.F.C. Walters, Fix-point semantics for programs in distributive categories, May93 Mark Leeming, R.F.C. Walters, Computing left kan extensions using the Todd-Coxeter procedure N. Sabadini, S. Vigna, RFC Walters An automata-theoretic approach to concurrency through distributive categories: on morphisms, Jan93 N. Sabadini, R.F.C. Walters On functions and processors: an automata-theoretic approach to concurrency through distributive categories, Nov92 N. Sabadini, H. Weld, R.F.C. Walters, Distributive automata and asynchronous circuits, May93 N. Sabadini, S. Vigna, RFC Walters A notion of refinement for automata, Jan93 Shu-Hao Sun, RFC Walters, Representations of modules and cauchy completeness, Mar93 R.F.C. Walters, An imperative language based on distributive categories, 91 seminars/sydcat sydcat.tex is a listing of seminars given at the Sydney Category Seminar Not being currently maintained. seminars/cics cics.tex is a listing of seminars given at the Sydney Categories in Computer Science Seminar. software/kan_1.0 kan (vers 1.0) (Sean Carmody, Craig Reilly, Bob Walters) An implementation of the algorithm developed in 1990 by Carmody & Walters to compute (finite) left Kan extensions is now operational (programmed by Reilly and Carmody). There are also some sample input files as well as a file called KAN.info which gives further details of the program and one called README which describes the sample input files. If you experiment with the program, we would be very interested to hear any comments or suggestions (especially with regards any bugs which you -- hopefully won't -- find). Future versions of the program will include a more standardised i/o format, and will renumber the elements of the output sets will be of the form {1,2,..,n} (a set which may currently be given as {1,3,7} would become {1,2,3}). Sean Carmody. email: carmody_s@maths.su.oz.au 21 May 1991 software/kan_2.0 A later version of kan, but Sean left Sydney for Cambridge before it was properly documented. software/buckland Representing pasting schemes in Prolog A program written by Richard Buckland, with Michael Johnson. For details see programs/buckland/readme Further details email t-richbu@microsoft.com mike@macadam.mpce.mq.edu.au -- Bob Walters Department of Pure Mathematics, University of Sydney, NSW 2006, Australia Internet: walters_b@maths.su.oz.au Phone: +61 2 692 2966 FAX: +61 2 692 4534 ==========================Instructions====================================== FTP LOGIN. Give the following commands. ftp maths.su.oz.au Login: anonymous (if you don't have an account on maths) Paswd: yoursurname (though any string will work) bin (if you are retrieving a .dvi file) prompt off (if you want no ? prompts from mget) cd sydcat (change directory to _public/sydcat ls -lt (see what's there, most recent first) mget filename-1 ... filename-n (e.g. mget catcurrent.Z) quit (exit from FTP) DVI. If you wish to print paper, calg say, retrieve calg.dvi and associated .eps and .sty files from the subdirectory calg (cd calg first). You must first give the bin command to ftp since .dvi files are not text files. You will then need a dvi to postscript converter which will include the .eps files. Print the resulting postscript file on your host. PROBLEMS. If you have problems in either retrieving or compiling papers, please contact Bob Walters. NOTE. Please note that the IP satellite link between Australia and the rest of the world is saturated most of the time. Large file transfers to non-Australian sites should be spaced out, and should preferably take place between the hours 2300 and 0800 local Eastern Australian time (the local time appears in the ftpd banner at connection). ============================================================================ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Initial model of higher sketches Date: Fri, 26 Nov 93 19:00:04 +0900 From: Hiroyuki Miyoshi Does anyone know results about the existance of initial model (in Cat or other categories) of 2-sketch or V-sketch with indexed limits, and that of other equivalent (or more general) formulations (e.g. higher-dimensional sketch of Power and Wells)? I think of results analogous to the existance of the (family of) initial model in Set of FL or FLS sketch. Any relevant informations and references are much welcome. Thanks in advance. ---- Hiroyuki Miyoshi Department of Mathematics, Faculty of Science and Technology, Keio Univ. OFFICE: c/o Nobuo Saito, Faculty of Environmental Information, Keio Univ. 5322 Endoh, Fujisawa 252, JAPAN EMAIL: miyoshi@slab.sfc.keio.ac.jp ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Initial model of higher sketches Date: Fri, 26 Nov 93 20:39:37 EST From: otto@triples.Math.McGill.CA (James Otto) >Does anyone know results about the existance of initial model (in Cat >or other categories) of 2-sketch or V-sketch with indexed limits, and >that of other equivalent (or more general) formulations (e.g. >higher-dimensional sketch of Power and Wells)? > >I think of results analogous to the existance of the (family of) >initial model in Set of FL or FLS sketch. Any relevant informations >and references are much welcome. > >Thanks in advance. ... >Hiroyuki Miyoshi ... >EMAIL: miyoshi@slab.sfc.keio.ac.jp i'll overly busy until the end of '93, but here's some thoughts. you can probably describe (e.g. see my thesis, summer '94) the modeling of your V-sketchs in appropriate V-cats as set models of an FL sketch, or, more humanely, of an essentially algebraic presentation. then the initial Herbrand model of that gives you an initial V-cat model. actually there is some fiddling about approximating pseudo maps by strict maps, e.g. approximating non locally finitely presentable cats by locally finitely presentable ones. e.g. see the Australians on flexible limits. also M. Makkai, here at McGill, has a view of sketches in which weak initial models are seen as injective hulls. bon soir, J. Otto ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Initial model of higher sketches Date: Mon, 29 Nov 93 14:34:35 +1100 From: kelly_m@maths.su.oz.au (Max Kelly) Miyoshi might be interested in my article "Structures defined by finite limits in the enriched context I ", Cahiers de Topologie et Ge'om. Diff. 23(1982), 3-42. Max Kelly. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: (A\otimes B)(C\otimes D)=(AC)\otimes (BD) Date: Mon, 29 Nov 93 09:52:49 EST From: stiller@blaze.cs.jhu.edu Is there a categorical analogue of the following well-known matrix identity: If A, B, C, D are matrices over a field and juxtaposition denotes matrix multiplication and \otimes denotes Kronecker product, then (A\otimes B)(C\otimes D)=(AC)\otimes (BD) whenever the dimensions are consistent? Note: I am a category theory novice, but the proof of this identity uses so few of the field axioms, and is so useful, (and curiously looks like the interchange of horizontal and vertical compositions of natural transformations which I presume is coincidental) that I was just curious if an analogue of this identity is true in more general categories than matrices over rings. -- Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University. stiller@cs.jhu.edu. "Tertan I am, but what is Tertan? Of this time, of that place, of some parentage, what does it matter?" ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD) Date: Mon, 29 Nov 93 15:56:21 PST From: baez@ucrmath.ucr.edu (john baez) Lewis Stiller writes: Is there a categorical analogue of the following well-known matrix identity: If A, B, C, D are matrices over a field and juxtaposition denotes matrix multiplication and \otimes denotes Kronecker product, then (A\otimes B)(C\otimes D)=(AC)\otimes (BD) whenever the dimensions are consistent? Note: I am a category theory novice, but the proof of this identity uses so few of the field axioms, and is so useful, (and curiously looks like the interchange of horizontal and vertical compositions of natural transformations which I presume is coincidental) that I was just curious if an analogue of this identity is true in more general categories than matrices over rings. ---------- I am just learning what some of the people on this group invented, so only the enthusiasm of a novice can justify my attempt to explain this. Briefly, this "exchange identity" (or maybe it's called "interchange") is indeed symptomatic of a very general phenomenon in category theory, and the analogy with horizontal and vertical compositions is NOT coincidental. A 2-category is, roughly, a category in which homsets are categories and composition is a (bi)functor. One always has an exchange identity in such a structure. The category of categories is a 2-category, indeed the primordial one. A monoidal category (a category with tensor products satisfying nice axioms) can be viewed as a 2-category in such a way that your exchange identity becomes a special case. Another nice example is the 2-category in which objects are maps from a point into a topological space, morphisms are maps from a unit interval, and morphisms-between-morphisms (so-called 2-morphisms) are maps from a square. Well, here associativity does not hold "on the nose" but only "up to reparametrization" so we are touching upon the great puzzle of "weak n-categories". In any event, this situation provides a very nice geometrical interpretation of the exchange identity. jb ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD) Date: Tue, 30 Nov 93 15:11:36 +1100 From: street@macadam.mpce.mq.edu.au >Is there a categorical analogue of the following well-known matrix >identity: If A, B, C, D are matrices over a field and juxtaposition >denotes matrix multiplication and \otimes denotes Kronecker product, >then > >(A\otimes B)(C\otimes D)=(AC)\otimes (BD) > There is a (skeletal) category Mat whose objects are finite ordinals, whose arrows n --> m are mxn matrices, and whose composition is matrix multiplication. Mat becomes a monoidal category with Kronecker product as its tensor product. Your equation expresses the functoriality of this tensor product Mat x Mat ---> Mat which is also strictly associative. The category Vect of finite dimensional vector spaces is equivalent to Mat and ordinary tensor product of vector spaces. Every monoidal category is equivalent to a strictly associative one (coherence theorem), and Mat is a concrete way of doing this for Vect. Category theorists call your equation "the middle-four-interchange-law" since it involves interchanging the middle two of a string of four terms. It is the axiom which makes 2-categories interesting. I made much use of all this, including Kronecker product, in my Myhill Lectures at Buffalo, 20-23 April 1993 [available as Macquarie University Math Report 93-130 (June 1993), submitted for publication]. Sincerely, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD) Date: Tue, 30 Nov 93 14:21:42 EST From: barr@triples.Math.McGill.CA (Michael Barr) I'd like to modify it. I got too clever, forgetting that commuting with sums doesn't need any additional naturality. > Date: Mon, 29 Nov 93 09:52:49 EST > From: stiller@blaze.cs.jhu.edu > > Is there a categorical analogue of the following well-known matrix > identity: If A, B, C, D are matrices over a field and juxtaposition > denotes matrix multiplication and \otimes denotes Kronecker product, > then > > (A\otimes B)(C\otimes D)=(AC)\otimes (BD) > > whenever the dimensions are consistent? > > Note: I am a category theory novice, but the proof of this identity > uses so few of the field axioms, and is so useful, (and curiously > looks like the interchange of horizontal and vertical compositions of > natural transformations which I presume is coincidental) that I was > just curious if an analogue of this identity is true in more general > categories than matrices over rings. > -- > Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University. > stiller@cs.jhu.edu. "Tertan I am, but what is Tertan? Of this time, of > that place, of some parentage, what does it matter?" > > > The stated identity is obviously true in any additive category with a monoidal structure, provided the monoidal structure commutes with the finite sums. This is guaranteed if the monoidal structure is part of a closed monoidal structure. Just a consequence of the functoriality of the finite sums. Michael