September 1990 - Categories - categories.org.au

CT91
by Tom Fox 18 Sep '90

18 Sep '90

CATEGORY THEORY 1991 - Clarification of dates The conference will be held JUNE 23-30, 1991, as first announced. Subject: An answer, after 11 weeks absence From: "Fred E.J. Linton" <FLINTON(a)Wesleyan.bitnet> To: inhb(a)musicb.mcgill.ca Mike, On the very day I arrived in Iceland for the Jonsson symposium, you asked me: This sounds like something you would have done. If B is a category with a triple T and if K is the Kleisli category, then the category of T-algebras can be identified as the full subcategory of Func(K\op,Set) consisting of all functors R:K\op --> Set such that R o F\op: B\op --> Set is representable. Can you give me a reference? If I had SLN 80 around, I would expect to find it there, but it is easier to ask you. Now that I'm back, I can answer: yep, that's what I did, more or less: but not quite as you say -- that is, not literally the full subcategory you say, but rather the pairs consisting of such functors as one entry and a matching representing object as the other. Done first in the La Jolla volume, but just over Sets , then in SLN 80 in the very first article (An outline of functorial semantics, pp. 7-52), where I wrote A where you write B , and finally, for the "relative category" setting, in a preprint published by the Banach Center in Warsaw in 1974 entitled Relative Functorial Semantics, III: Triples vs. Theories (three whole pages), where I used S (suggesting Sets ) as the notation for the closed, or monoidal, or more generally just multilinear base category relatiove to which all the relative category theory was to be done. OK? -- Fred

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by Michael Barr 12 Sep '90

12 Sep '90

Dear Bob: Since Paul Johnson put his question on the full net, let me answer it there. 1. An equational theory is sketchable iff the theory has rank. This follows from theorems of Pare and Makkai and is sort of obvious anyway. 2. Yes, provided the base is accessible. 3. You had better ask Makkai or Pare. But my guess is the following: If you look at all the categories that are lambda-accessible, then there will be canonical sketches built from the lambda-accessible models. But there are theories of arbitrarily high arities whose models are, for example, the category of sets. Sounds paradoxical? Hint: the underlying functor is not the usual, but is the functor represented by a big (but still small) set. Michael

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by PJOHNSON＠Wesleyan.bitnet 10 Sep '90

10 Sep '90

I would like to learn some fundamental facts (that don't seem to be visible in Makkai and Pare', although I may be overlooking them) about jus how sketchable is a generalization of equational: 1. It seems clear that the models of an equational theory (a category algebraic over Sets) is sketchable. I'm thinking of a product sketch on (the opposite of) the Kleisli category. But is such a category accessible, or only in the case the theory has rank (a cardinal bound on the arities of the operations) ? 2. Are model categories for equational theories on an arbitrary base category again sketchable over Sets via the same construction suggested above? 3. Is there such a thing as a canonical sketch, a kind of complete syntactic description of a sketchable theory? And if so, is there an adunction between syntax and semantics?

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by Michael Barr 08 Sep '90

08 Sep '90

Category Theory, 1991 23--30 July, McGill University, Montreal Groupe Interuniversitaire en Etudes Categoriques Request for financial support --------------------------------------------------------------------- We will have a limited amount of funds to support people who cannot attend without such support. It will be MUCH easier to fund people who require only housing and food than people who also require transportation. In particular, people from eastern Europe and the Soviet Union will be expected to try to arrange transit on their own national airline. Other people are well advised to try the Canadian airline Nationair which seems to have the cheapest fares we have seen. Because of present uncertainties, it may be difficult to get firm prices, but you will have to try to get an estimate if you wish to ask us for travel funds. People who do not request travel funds will have a much better chance of being funded for food and housing. Owing to the exigiencies of the funding organization, we must have your request, including a title for your talk, as soon as possible and no later than October 5. If you receive this by Email you will not get it by post. Please reply the same way. If you receive it by post and can reply by Email, please do so, giving the same information. We will then need an abstract by March 15, 1991 and will be attempting to announce the amounts we can provide by the middle of April. We cannot extend support to people who are not giving talks. --------------------------------------------------------------------- Name: Address: Email: Telephone: Fax: --------------------------------------------------------------------- Title of talk: --------------------------------------------------------------------- Estimated amount required for travel: __Canadian __US $ --------------------------------------------------------------------- Signed:

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by unknown＠categories.mta.ca 01 Sep '90

01 Sep '90

--------------------------------------------------------------------- Please return this form to: Category Theory, 1991 Department of Math & Stat, McGill University 805 Sherbrooke St. W Montreal, Quebec Canada H3A 2K6 Email: mt16(a)mcgilla.bitnet or mt16(a)musica.mcgill.ca

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