From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5879 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology Date: Sat, 29 May 2010 14:47:16 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1275231462 3346 80.91.229.12 (30 May 2010 14:57:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 30 May 2010 14:57:42 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun May 30 16:57:40 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OIjxI-0006AY-8i for gsmc-categories@m.gmane.org; Sun, 30 May 2010 16:57:36 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIjSg-0006Le-O5 for categories-list@mta.ca; Sun, 30 May 2010 11:25:58 -0300 Content-Disposition: inline In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5879 Archived-At: Colin McLarty wrote in part: >As to articulating a way to avoid ever using identity of objects and >identity of categories, John Baez writes [snip] >Has anyone yet offered a first-order (or ML-typetheoretic) >axiomatization of mathematics along Makkai's lines? I don't know very much about what's been done along Makkai's lines; I also would like to see a specific (if not final) set of axioms. But category theory has been done in Martin-L=F6of type theory: http://www.cs.st-andrews.ac.uk/~rd/publications/CTMLTT.pdf It has also been done in the type-theoretic proof assistant Coq: http://coq.inria.fr/distrib/v8.2/contribs-20090527/ConCaT.html In both of these, there *is* a notion of equality (or better, identity) at all types, hence a notion of identity of objects of any given category= , allowing one to define isomorphism of categories, etc. However, this logicians' identity does not match mathematicians' equality= ; the easiest way to see this is that there are no quotient types. (This means that already to do set theory, let alone category theory, you must define a set to be a type equipped with an equivalence relation. Such a thing is also called "setoid", depending on which author you read.= ) You an also use Mike Shulman's SEAR, in the variant without identity. http://ncatlab.org/nlab/show/SEAR http://ncatlab.org/nlab/show/SEAR#eqfree This looks much more like the ordinary language of mathematics. (Actually, one could modify ETCS in a similar way, although it would no longer deserve to be called "ETCS".) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]