May 1993 - Categories - categories.org.au

2-category algebras?
by baez＠ucrmath.ucr.edu 02 Jun '93

02 Jun '93

We all know about group algebras, and this leads naturally to the idea of a category algebra: form a vector space having as a basis the morphisms in a given category, with the obvious product (taking the product of two morphisms to be zero if the head of one isn't the tail of the other). I used an example of this in my paper "Quantum Gravity and the Algebra of Tangles," but I have never seen a reference to the general concept, and I would appreciate one. What I am now interested in, however, is generalizing this notion to 2-category algebras. This might be related to recent algebraic work of Ruth Lawrence, which I haven't actually seen. Any leads? Regards, John Baez

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The group ring of a small category
by MTHFWL＠ubvms.cc.buffalo.edu 01 Jun '93

01 Jun '93

Dear John Baez, see my abstract in the American Math Society Notices for 1963 Meeting Number 601, paper number 37, page 280. /A longer exposition including applications to network design, cohomology etc. was written then and I can make you a copy if you are interested. Even in 30 years not all the conclusions have been drawn. My colleague S.D. Schack also independently discovered the construction and has used it extensively in deformation theory. An important variant which applies to free categories and others was discovered by Pierre Leroux and used to construct the Moebius function of a category. Greetings, F.W. Lawvere +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Zariski Categories
by Allan Adler 30 May '93

30 May '93

I'm looking at the book Categories of Commutative Algebras, by Yves Diers. Before plunging into the details, I'm wondering about a couple of things: (1) Does Grothendieck's theorem about faithfully flat descent work in the setting he develops? (2) What he calls algebraic spaces seem more like "bunches of varieties". The terminology "algebraic spaces" as used by Artin means something else. So does the notion of algebraic space in the sense of Artin have a place in this generalization? Allan Adler ara(a)altdorf.ai.mit.edu +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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linear topology, hopf algebras and *-autonomous categories
by blute＠triples.Math.McGill.CA 29 May '93

29 May '93

The following paper is available by anonymous ftp from the site triples.math.mcgill.ca. It is in the file pub/blute in dvi and ps format(compressed). Anyone who would like the paper and cannot access it this way can of course contact me. Regards, Rick Blute LINEAR TOPOLOGY, HOPF ALGEBRAS AND *-AUTONOMOUS CATEGORIES Richard F. Blute McGill University Abstract The goal of this paper is to construct models of variants of linear logic in categories of vector spaces. We begin by recalling work of Barr, in which vector spaces are augmented with linear topologies to obtain new examples of symmetric monoidal closed (autonomous) categories. The advantage of these categories is that they have large subcategories of reflexive objects which are also autonomous, in fact *-autonomous. Thus, we obtain models of (a fragment of) classical linear logic. These models have an advantage over other models arising from linear algebra in that they do not identify the two multiplicative connectives, tensor and par. Furthermore, the models will be complete and cocomplete. We then extend Barr's work to vector spaces with additional structure, in particular to representations of Hopf algebras. As special cases, we examine cocommutative Hopf algebras, and quasitriangular Hopf algebras, also known as quantum groups. In the quasitriangular case, the representations actually form a braided *-autonomous category, first defined by the author. They thus give a model of a braided variant of linear logic, also defined by the author. This is the first example of such a model which does not identify the two multiplicative connectives. Hopf algebras, and more generally bialgebras, have been of interest recently as a possible framework for the study of concurrency, in that they encode the paradigm of interleaving processes, also known as fair merge, as well as other structural operations on concurrent processes. This was first observed by D. Benson. As a result of the work in this paper, such bialgebras will yield models of a fragment of linear logic. While these models do not equate the multiplicative connectives they do validate the MIX rule, thus giving models of a slightly larger theory, first studied by Fleury and Retore. Also, if additional restrictions are placed on the Hopf algebra, we obtain models of cyclic linear logic, defined by Yetter. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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coequalisers & visit
by Paul Taylor 28 May '93

28 May '93

Thanks for sending me exactly the kind of counterexample I need for my foundations book. It will do very nicely as an exercise in the section on factorisation systems. I shall have to think a bit more about the one involving presentations of categories which Art Stone and others mentioned. I am coming to Canada for LiCS in Montreal next month. My flight plan is to arrive in Toronto on Saturday 12 June and return from Montreal (via (!!) Toronto) after LiCS. I wonder whether the people at York would like to hear a seminar from me in return for some local accommodation for a few days? Several subjects are on offer: stable domain/category theory (see the papers in my ftp archive, which I think soem of you already have), intuitionistic ordinals, synthetic domain theory, as you please. Paul Taylor +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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coequalisers again
by Paul Taylor 26 May '93

26 May '93

The more I think about coequalisers the less I understand them. DOES ANYONE KNOW of a category in which REGULAR EPIS DO NOT COMPOSE? (Maybe the opposite of the category of commutative rings?) Here is a summary of the results of the previous discussion: I asked what a REGULAR EPI is. I asked this because the observation that all monos in a topos are regular is often made immediately after introducing the subobject classifier, by considering the characteristic map and constantly true. This is not a co-equivalence relation, but regular epis are usually defined in terms of equivalence relations. However if the kernel pair of a map exists, it coequalises this if it coequalises anything, so the definitions agree. You can't do much logic in a category without pullbacks, but Mike Barr and Peter Freyd gave me generalisations which are applicable without them. Categories of algebras were the original examples of REGULAR CATEGORIES. In these there is a factorisation of homomorphisms into regular epis (which are charactersed as surjections) and monos. This is stable under pullback, as is the class of regular epis, and also the class of diagrams which are kernels and coequalisers. Does a regular category have all coequalisers? The definition in the literature is ambiguous, but I am inclined to agree with that in "Categories, Allegories" by Peter Freyd and Andre Scedrov (North Holland 1992), namely that they do not. Although categories of algebras have all coequalisers, and regular epis are stable, coequaliser diagrams need not be. Here is a counterexample in the category of groups given to me by Peter Freyd. 2 --------> 2 2 >-----> 2 2 ---->> 2 ---->> 1 | | | --------> | | | | -- | | | -- | | | -- | | | | | | | V | | V 2 --------> V V 2 >-----> 2 A -------------->> 3 --------> 4 Here 2^2 is the four-element normal subgroup of the group of even permuations on four symbols. The parallel pair consists of the inclusion and the map which interchanges (12)(34) with (13)(24). The equalisers and coequalisers are illustrated and the squares are pullbacks in the obvious way. There are several reasons why I am interested in all of this. One of them is the interpretation of WHILE PROGRAMS using coequalisers of functional relations, which must be stable under pullback. I would like to be able to embed such a category in a topos. There is an obvious Grothendieck topology, and the inclusion sends the coequalising maps to epis, but why should it preserve the coequaliser DIAGRAMS? (For a draft of a paper on while programs, see /theory/papers/Taylor/while-1993.dvi at theory.doc.ic.ac.uk) The other reason is an application to SYNTHETIC DOMAIN THEORY. See my paper in LICS 1991 (also /theory/papers/Taylor/lics.dvi) for the background. Eugenio Moggi asked for a good class of monos (see /theory/papers/Moggi/ELT.dvi) and I was trying to show that what I called "extremal" monos in my LiCS paper coincided with those which arise as equalisers of parallel pairs into powers of Sigma. Somewhat to my surprise, I discovered yesterday that in any category with kernel pairs and coequalisers of them (NOT NECESSARILY STABLE), a map is mono iff it is orthogonal to all regular epis where "orthogonal" means that the universal property for factorisation systems is satisfied. Moreover if regular epis compose, (reg.epi)-(mono) is a factorisation Hence the question above. Stability of regular epis is sufficient but not necessary to make them compose. For SDT I am specifically interested in "Sigma-epis" instead of monos (maps which internally satisfy the cancellation property for parallel pairs into the object Sigma, rather than all objects). For reasons concerned with Eugenio Moggi's Evaluation Logic, we want Sigma-regular monos to be "topological subspaces", ie that maps to Sigma extend along the mono. This is also sufficient to make regular monos compose. Paul Taylor +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Coequalizers
by Walter Tholen 21 May '93

21 May '93

Regarding Paul Taylor's question, here are some useful references that contain answers: G.M. Kelly, Monomorphisms, epimorphisms, and pullbacks, J. Austral. Math Soc. 9 (1969) 124-142. (Mentions an example of a complete,cocomplete, wellpowered, cowellpowered additive category due to Isbell in which reg. epis do not compose. The following articles deal with long chains of regular epis; this is a selection only:) J. MacDonald and A. Stone,The tower and regular decomposition , Cahiers TGDC 23 (1983) 197-213 R. Bo"rger and W. Tholen,Total categories and solid functors, Can. J. Math 42(1990) 213-229 (see sections 4 and5) J. Adamek and W. Tholen, Total categories with generators, J. of Alg. 133 (1990) 63-78 (see section 1) R. Bo"rger, Making factorizations compositive, Comm. Math. Univ. Carolin. (1991) 749-759 (see section 3). The last article mentions one remaining problem on the composition- cancellation behaviour of regular epis in a category (with kernel pairs and their coequalizers, say): consider the properties (A) and (B): (A) if e is split epi and d is regular epi, then the composite ed of first d and then e is regular epi, (B) if a composite ed is regular epi, then e is regular epi one always has that if (A) holds in our category, then also (B) holds. But what about the converse? (Note that neither property holds true in general, already Kelly's paper contains a counter-example.) Regards, Walter Tholen +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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an answer to Paul Taylor's question
by Art Stone 21 May '93

21 May '93

Paul Taylor asks for a category in which REGULAR EPIS DO NOT COMPOSE. The short answer is |Cat . (Where |x denotes boldface-x, look at the canonical presentation, which has length two, of |3 in terms of free categories over sets, i.e., coproducts of copies of |2. The intervening algebras in this presentation are free categories over graphs.) In slightly greater generality, how do we produce examples of categories in which there are chains of length n of regular epis -- whose composites cannot be written as composites of fewer than n regular epis (for each n > 1) ? When I worried about this several decades ago, Peter Freyd gave an answer, in his essentially equationally defined categories [e.g., in the early pages of his Aspects of Topoi] -- defined by equations on a set of partial as well as entire operations -- in which the operations can be ordered so that for each partial operation its domain is defined by equations in earlier operations. |Cat is an example when n = 2. This leads into the subject of adjoint towers [Applegate-Tierney, LNM 137; Adamek-Herrlich-Rosicky c.1987] To get a tower of height n over |Sets^m (for any m ) there must be n batches of progressively "higher" operations, each requiring the last of the preceding batches, in the definitions of its domains. For the |Cat example, the intervening category is directed |Graphs. The base category is respectively |Sets or |Sets^2 , when we use the usual one-sorted or two-sorted definition of "category". And if we define "category" in terms of (commuting) "triangle", in place of "morphism", then the tower degenerates -- after the pattern of the main theorem of Gabriel-Ulmer [LNM 195 and LNM 221]. Now, if |Cat is interesting in this way, what does it mean that |Cat is a topos wrt categories enriched over it [as remarked by Ross Street several months ago here on the catnet], and what modifications are we going to be compelled to make when we attempt to carry topos-style thinking from |Sets over to |Cat? Talk about WHILE programs suggests talk about imperative languages. Perhaps more important, even, than their dealing with STATE is the way imperative languages can be explicit about extending or shrinking storage (using pointers). Manes and Moggi have shown us how monads can be used to model STATE and given us a category theoretic translation of the distinction between call-by-value and call-by-name [begin with Moggi's paper on theory.doc.ic.ac.uk]. Is there also a 2-category theoretic translation of the distinction between call-by-value and call-by-reference, that depends on the difference between the two kinds of composition we have in 2-categories? Here I doubt if the base 2-category can be |Cat. But |Cat (x) |Cat remains a possibility, where (x) denotes Gray's assymetric tensor product [LNM 391 and in the Heller-Tierney (ed.) Eilenberg festschrift]. Art Stone +++++++++++++++++++++++++++++++++++++++++++

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None
by barr＠triples.Math.McGill.CA 20 May '93

20 May '93

Ragular epis that do not compose: In Cat, which is complete and cocomplete and is even tripleable over a category (graphs) tripleable over sets, regular epis do not compose. Let 2 be the category with two objects and one non-identity arrow from one to the other. Let N be the category with one object and a natural numbers of maps. Let N_2 be the category with one object and 2 arrows, one an identity and the other idempotent. The obvious map from N to N_2 is obviously a regular epi and so is the map from 2 to N that identifies the objects, but the composite is not. Michael ++++++++++++++++++ From: pavlovic(a)triples.Math.McGill.CA (Dusko Pavlovic) > DOES ANYONE KNOW of a category in which REGULAR EPIS DO NOT COMPOSE? > (Maybe the opposite of the category of commutative rings?) I don't suppose that this is what you mean, but isn't already something like --0--> --a--> --f--> --1--> --b--> a0=a1, fa=fb providing an answer to the question as it stands? - Dusko ++++++++++++++++++++++++++++++++++++++++++++++++

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None
by G.M. Kelly 20 May '93

20 May '93

Paul Taylor should look at my very old paper "Monomorphisms, epimorphisms, and pull-backs", J.Austral.Math.Soc. 9 (1969), 124-142. There he will find counter-examples as well as theorems. Max Kelly. +++++++++++++++++++++++++++++++++++

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