October 1993 - Categories - categories.org.au

Name for this concept?
by William H. Rowan 02 Nov '93

02 Nov '93

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. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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KZ-terminology
by pavlovic＠triples.Math.McGill.CA 28 Oct '93

28 Oct '93

In a paper recently announced on this network ("The Chu construction and the cofree models of classical linear logic", available from triples.math.mcgill.ca), I remarked that the Chu construction was a KZ-codoctrine. The term KZ-doctrine was coined, I believe, by Ross Street in his "Fibrations in bicategories" (Cahiers XXI-2) --- to abbreviate the term Kock-Z\"oberlein-doctrine. Now I have been warned from several sides that this abbreviation awakes in German-speaking people associations to concentration camps. I was going to shrug my shoulders and say that people called Dick usually don't pay any attention to the association their name may awake (especially in children), but then I remembered that a guy I know, whose name happens to be Adolf, had a serious fight with his father about this. History seems to be more serious than anatomy. So what do you say: Should we change the terminology or not? Regards to all, Dusko Pavlovic ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Localization
by Marek Golasinski 28 Oct '93

28 Oct '93

Dear Colleagues, It is well known that one of possible model for the homotopy category there is the category Cat of small categories. Does anybody know a construction of a P-localization (P is a set of primes) using the categorical structure only? Many thanks in advance for your reply. With my best wishes, Marek G. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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FYI
by James Stasheff 26 Oct '93

26 Oct '93

------------------------------------------------------------------------------ send mail only to hep-th(a)xxx.lanl.gov, do not reply to no-reply(a)xxx.lanl.gov . rigorously identical files also available from hep-th(a)babbage.sissa.it send complaints regarding untexable papers directly to submitter. use a single `get' to request multiple papers, `list macros' for available macro packages, and `help' for a list of available commands and other info. ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ received from Mon Oct 25 00:06:08 MDT 1993 to Tue Oct 26 00:06:04 MDT 1993 ------------------------------------------------------------------------------ \\

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None
by Dr John W. Barrett 26 Oct '93

26 Oct '93

Spherical Categories, by John W. Barrett and Bruce W. Westbury, 16 pages, amstex \\ This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple. \\ (hep-th/9310164 , 100kb) %-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%- ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Universal Algebra v. Category Theory
by es＠math.mcgill.ca 24 Oct '93

24 Oct '93

Here are some remarks in connection with the Universal Algebra versus Category Theory debate initiated by Saunders MacLane. (i) There is yet another way to describe a (multisorted) algebraic theory, alias clone, namely as a Cartesian multicategory, that is, a Gentzen style deductive system with appropriate equations between deductions. Deductions f:A_1...A_n ---> B should be viewed as operations, in the spirit of the formulas as types paradigm. (ii) Why anyone would deny that the subalgebras of an algebra form a complete lattice is beyond me. The subobject lattice and congruence lattice are both subsumed in the lattice of subcongruences, which I found myself in my paper "Goursat's theorem and the Zassenhaus lemma", Can. J. Math. 10(1957), 45-56. Subcongruences have recently been resurrected as partial equivalence relations; they can of course be defined in any category. (iii) If one studies algebras on graphs instead of sets, as pioneered by Burroni, one may view many structured categories, even toposes, as algebras. (iv) To avoid problems with empty types, e.g. lack of transitivity, one should declare variables both in equations and deductions. Phil Scott and I did so in our book of 1986, by placing a subscript X={x_1,...x_n} on the equality and entailment symbols. Unfortunately, some other books did not follow this practice. Jim Lambek ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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my new address
by Marek Zawadowski 14 Oct '93

14 Oct '93

Marek Zawadowski Instytut Matematyki Uniwersytet Warszawaski 00-913 Warszawa ul. S.Banacha 2 Poland e-mail: zawado(a)mimuw.edu.pl Please send all correspondence there, Marek Zawadowski ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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chu is cofree
by pavlovic＠triples.Math.McGill.CA 14 Oct '93

14 Oct '93

A paper on THE CHU CONSTRUCTION AND COFREE MODELS OF CLASSICAL LINEAR LOGIC by Dusko Pavlovic is available by anonymous FTP from triples.math.mcgill.ca. The files chu-{A4,US}.{dvi,ps}.Z are the directory pub/pavlovic. Each of them contains the whole paper, but in various formats. This version supersedes the previous ones, that I had sent to some people. Abstract We describe a couniversal property of the Chu construction. It induces a comonad on a category of autonomous categories. *-Autonomous categories are exactly the coalgebras for this comonad. The models of full classical linear logic (with the exponentials) are then obtained as the coalgebras on the models of intuitionistic linear logic --- this time for a comonad derived from the Chu construction. In view of the computational interpretations of linear logic, these results suggest an interesting connection of the functional and the concurrent programming. For this reason, some effort has been spent to make all the constructions effective. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Re: universal algebra and partial algebra
by sjv＠doc.ic.ac.uk 14 Oct '93

14 Oct '93

Ronnie Brown asks - >Is it naive to ask that universal algebra theory should also include >partial algebras, and in particular groupoids, categories, multiple >groupoids, multiple categories (of various kinds)? It seems that where the methods of universal algebra generalize well to cover partial algebras is in _essentially_ algebraic theories, i.e. those in which there is a hierarchy of operators such that the domain of definition of each is defined by a conjunction of equations involving more primitive operators. This certainly covers categories and groupoids - anyone who knows Phillip Higgins' book can see how smoothly universal algebra works for them. As for multiple groupoids and multiple categories, I can't say because I'm not familiar with them. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Re: On UA-CAT Workshop at MSRI - Empty Carriers
by John C. Mitchell 11 Oct '93

11 Oct '93

I'm glad to see this discussion. In the book that I am (still) writing, I went to some length to do all the universal algebra with explicit lists of variables. At times, I have wondered whether this was syntactic overkill. However, it makes for clean completeness theorems and existence of initial algebras. In addition, it meshes well with later chapters on typed lambda calculi, where contexts (lists of variables and their types) are needed. One minor question, while I am thinking of it. I used "sort" in the chapter on algebra, and "type" in the chapters on lambda calculi, etc. What do the UA-literate think about using "type" uniformly throughout? Would it be offensive to simplify terminology this way? BTW, there is not much math in the chapter on UA, just soundness, completeness, homomorphisms, initiality, and a number of CS examples (stacks, queues, trees, ...), ending with a brief overview of rewrite systems. John Mitchell ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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