From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9418 Path: news.gmane.org!.POSTED!not-for-mail From: =?UTF-8?Q?Andr=c3=a9e_Ehresmann?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Some questions about different notions of "theory" Date: Thu, 9 Nov 2017 12:26:15 +0100 Message-ID: References: Reply-To: =?UTF-8?Q?Andr=c3=a9e_Ehresmann?= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit X-Trace: blaine.gmane.org 1510260338 13389 195.159.176.226 (9 Nov 2017 20:45:38 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 9 Nov 2017 20:45:38 +0000 (UTC) To: Mike Stay , categories Original-X-From: majordomo@mlist.mta.ca Thu Nov 09 21:45:34 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eCthe-0003IS-61 for gsmc-categories@m.gmane.org; Thu, 09 Nov 2017 21:45:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59386) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eCtiK-0007a4-PB; Thu, 09 Nov 2017 16:46:16 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eCtgt-0002ba-K3 for categories-list@mlist.mta.ca; Thu, 09 Nov 2017 16:44:47 -0400 Content-Language: en-GB Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9418 Archived-At: In answer to Mike: "3) nLab says a sketch is a small category T equipped with subsets (L,C) of its limit cones and colimit cocones. A model of a sketch is a Set-valued functor preserving the specified limits and colimits. Is preserving limits and colimits like a ring homomorphism? Preserving both limits and colimits sounds like it ought to involve profunctors, but maybe I'm level slipping." Sketches were introduced by Charles Ehresmann in 1966 (and developed by us and our research students) in several papers, among which I'll cite: (i) The main Charles' paper "ESQUISSES ET TYPES DES STRUCTURES ALG??BRIQUES", Bul. Inst. Pol. Iasi, Tome XIV-1-2, 1968 (in French), reprinted in http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_IV_1.pdf on p. 19-32 (with several Comments in English I added on p. 329-332); (ii) Our joint paper "Categories of sketched structures", Cahiers Top.GDC XIII-2, 1972 http://www.numdam.org/article/CTGDC_1972__13_2_104_0.pdf I have been surprised to see the definition given in nLab of a sketch as a category with some limit-cones and co-limit-cones. For us, a 'sketch' is a category S (or even a graph) with some distinguished cones and co-cones (but not necessarily (co-)limit-cones). And in the above citation we construct the 'prototype' of the sketch which is constructed by forcing the (co-)cones to become (co-)limit-cones. Indeed, our idea was that a sketch describes a 'smaller' presentation of a generalized algebraic structure (for instance of categories themselves), and from it we deduced its prototype and also its type (analogue of the theory for Lawvere algebraic structures). The 'realisation' (or 'model') of a sketch S in a category C is then a functor from S to C which transforms the distinguished (co-)cones into (co-)limit-cones. There is no need of profunctors here. Let us note that C is not required to admit all (co-)limits, just the images of the distinguished (co-)cones must become (co-)limit-cones. An initial 1959 example (at the basis of Charles' definition) was the notion of a 'differentiable category' which is a model of the sketch of categories into the category of differentiable maps (which admits limits but only specific co-limits). Sincerely Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]