From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7317 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: The Idea of Structure as Data and Conditions Date: Tue, 29 May 2012 23:26:17 -0700 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1338370884 10916 80.91.229.3 (30 May 2012 09:41:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 30 May 2012 09:41:24 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed May 30 11:41:23 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SZfP8-0001ZA-QC for gsmc-categories@m.gmane.org; Wed, 30 May 2012 11:41:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45575) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SZfO0-0000ht-Bp; Wed, 30 May 2012 06:40:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SZfNz-0005OG-QH for categories-list@mlist.mta.ca; Wed, 30 May 2012 06:40:11 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7317 Archived-At: On 5/27/2012 11:00 PM, FEJ Linton wrote: > Virtually no one ever wants to restrict attention to functions that respect > (preserve or reflect) membership (other than "preserve" between ordinals). Wouldn't that depend on the context in which membership arises? Certainly group homomorphisms aren't expected to respect membership of group elements in groups, but neither are they expected to respect the composition of group homomorphisms equipping the category Grp. The latter kind of respect is accorded categories by functors between them. By the same token the former kind is accorded elementary models of set theory by elementary homomorphisms between them, the appropriate counterpart of functors in that context. Elementary homomorphisms preserve elementary structure, which in the case of models of set theory has membership as a basic part. ZFC structure is very different (at the bottom few layers) from the algebraic-in-Grph structure of the category Set, which has function composition as a basic part. For those who prefer algebra to logic, Joyal, Moerdijk and Awodey offer Algebraic Set Theory, AST, as a middle ground here. This replaces membership by (set-sized) unions and singleton a |--> {a}. Homomorphisms then have their usual algebraic meaning, which is arguably less fiddly than for elementary maps. Union allows the subset relation to be defined as X <= Y iff X U Y = Y, from which one can then define membership X e Y as {X} <= Y. Both relations are preserved by the homomorphisms of AST. For a crash course see Awodey's http://www.andrew.cmu.edu/user/awodey/preprints/astIntroFinal.pdf ZFC, Set, and AST differ only at the bottom few layers, above which foundational variations are tied to more fundamental issues involving Choice vs. Determinacy etc. One might compare the differences at the bottom with the wave-particle dichotomy in quantum mechanics or the event-state and time-information dichotomies in concurrency that I spoke on at Physics & Computation 1992 and 1994, see http://boole.stanford.edu/pub/ph94.pdf and the earlier (1992) http://boole.stanford.edu/pub/ql.pdf Or at least that's how it all looks to this outsider. Happy to be corrected on details I've got wrong. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]