From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9465 Path: news.gmane.org!.POSTED!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: How analogous are categorial and material set theories? Date: Fri, 8 Dec 2017 17:15:36 -0800 Message-ID: References: Reply-To: Vaughan Pratt 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 1512923265 14171 195.159.176.226 (10 Dec 2017 16:27:45 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 10 Dec 2017 16:27:45 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun Dec 10 17:27:40 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 1eO4S4-0003PK-5M for gsmc-categories@m.gmane.org; Sun, 10 Dec 2017 17:27:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40983) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eO4SW-0002IJ-Pr; Sun, 10 Dec 2017 12:28:08 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eO4R3-0005YA-A5 for categories-list@mlist.mta.ca; Sun, 10 Dec 2017 12:26:37 -0400 In-Reply-To: Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9465 Archived-At: I prefer to think of what Steve presumably has in mind here as an equational theory where composition is 2-ary where 2 is not 1+1 but rather o--->o. One difference between equational logic and first order logic is that only the former has well-defined homomorphisms (not sure if Gerald Sacks would have agreed).?? Just as group theory (in Steve's sense) has homomorphisms, so does category theory in that sense have functors. I'm not sure how one argues that CT has natural transformations however.?? They seem to enter as part of the metatheory, which as usually presented seems to be pretty set theoretic in its outlook. How do NT's look in an HOTT account of CT? The language of the Big Bang Theory is pretty family-oriented, except for equality which seems somewhat controversial.?? But I digress. Vaughan On 12/07/17 10:49 PM, Steve Vickers wrote: > Dear Patrik, > > The theory of categories is a first order theory, so what exactly are you denying here? > > Steve. > >> On 7 Dec 2017, at 18:58, peklund@cs.umu.se wrote: >> >> Is Category Theory a Theory? I think not. At least not in a logical >> sense. >> > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]