From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6269 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Fri, 1 Oct 2010 14:36:53 +0200 Message-ID: Reply-To: Thomas Streicher 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 1286025156 29160 80.91.229.12 (2 Oct 2010 13:12:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 2 Oct 2010 13:12:36 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sat Oct 02 15:12:35 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P21tD-0000jE-0A for gsmc-categories@m.gmane.org; Sat, 02 Oct 2010 15:12:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53837) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P21s6-0003iG-6g; Sat, 02 Oct 2010 10:11:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P21s1-0006x4-GB for categories-list@mlist.mta.ca; Sat, 02 Oct 2010 10:11:21 -0300 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6269 Archived-At: Dear Michael, > Does this clarify what I meant? Yes, indeed that has been very helpful.=20 > Perhaps "not well-behaved" was a poor choice of words since it implies > a value judgement, but I think it is correct to say that in the > absence of choice, category theory becomes very unfamiliar unless we > replace functors with anafunctors. For instance, if we insist on > using only functors, then a category with finite products does not > necessarily become a monoidal category, as there is no "product" > functor from A=D7A to A. Also, without choice the 2-category Cat using > functors is not even a regular 2-category, let alone a 2-topos. Since > the defining characteristics of Set include that it is a well-pointed > 1-topos, it seems unlikely to me that one will be able to get much of > anywhere with a version of Cat that is not a 2-topos. Well, but then we can work with categories with a chosen structure, e.g. chosen products; that's what is recommended in the Elephant and it looks = to me as close to practice; it's very unlikely to have a nonconstructive pro= of of existence of products. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]