From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8491 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Category Theory for the Sciences Date: Sun, 01 Feb 2015 14:10:27 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1422834072 10906 80.91.229.3 (1 Feb 2015 23:41:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 1 Feb 2015 23:41:12 +0000 (UTC) To: Categories mailing list Original-X-From: majordomo@mlist.mta.ca Mon Feb 02 00:41:12 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YI48d-0003Gy-HH for gsmc-categories@m.gmane.org; Mon, 02 Feb 2015 00:41:11 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:35528) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YI486-0002tt-L1; Sun, 01 Feb 2015 19:40:38 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YI487-0006Ui-5c for categories-list@mlist.mta.ca; Sun, 01 Feb 2015 19:40:39 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8491 Archived-At: On 1/30/2015 11:03 AM, Fred E.J. Linton wrote: > Just see how late, and how lackadaisically, Yoneda's Lemma enters into [Spivak's book]. As those who heard my CT'2011 talk (which grew out of my CT'2004 talk on communes) may recall, I'm in favour of exploiting the Yoneda Lemma in the way automobile manufacturers exploit the internal combustion engine: not as something whose mechanism is to be understood but merely as a means of propulsion controlled by the accelerator pedal. Such an approach could potentially make it accessible to more than just physicists, in particular to a wide range of workers in the social sciences. To that end, define a Sigma-category (C,Sigma) to be any category C equipped with a distinguished set Sigma of objects of C. In the obvious (to this audience) way, this determines a multisorted unary theory T, namely T = J' as the opposite of the full subcategory J of C with ob(J) = Sigma. In T, the objects represent the sorts and the morphisms the operations of the theory. In C, every object represents some model of T and every morphism represents some homomorphism of those models, not necessarily faithfully (a homomorphism may have more than one representative). What I find particularly appealing about this presentation of multisorted unary theories and (some of) their models and homomorphisms is that it extends so straightforwardly to Sigma-Pi-categories (C,Sigma,Pi). Here Pi is a second subset of ob(C) dual to Sigma in the sense that (a) whereas Sigma consists of the *sorts* of T, Pi consists of its *properties*; and (b) whereas the *elements* of a model M are the morphisms from Sigma to M, with a: s --> M being an element of sort s, the *states* of M are the morphisms from M to Pi, with x: M --> p being a state for property p. [Two asides: 1. There is a nice alliteration pun here between the duality of sorts and properties and that between sums and products. 2. Up to equivalence there is an obvious notion of maximal Sigma-Pi-category subject to leaving elements and states invariant. With that notion, the following special cases arise: (i) for Pi empty: the presheaf category Set^T; (ii) for Sigma = {I}, Pi = {_|_}, rigid in the sense that |C(I,I)| = |C(_|_,_|_)| = 1: the (ordinary) Chu category Chu(Set, C(I, _|_)); and (iii) for Sigma = Pi: what Bill Lawvere has called the Isbell envelope E(J) (J as above).] One social science that could find Sigma-Pi-categories useful is philosophy, which could find in them a single mathematical home for all three of the problems of (a) Cartesian dualism; (b) extensionality of properties (as the sets of states of a model); and (c) logical consistency of qualia (as morphisms from Sigma to Pi). (Slightly) more on this application in Section 3.3 of Fundamenta Informaticae 103 (2010) 203?218 at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.395.2995&rep=rep1&type=pdf Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]