From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5890 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: isomorphisms Date: Sun, 30 May 2010 21:51:24 -0500 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1275350065 14918 80.91.229.12 (31 May 2010 23:54:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 31 May 2010 23:54:25 +0000 (UTC) To: Categories list Original-X-From: categories@mta.ca Tue Jun 01 01:54:24 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 1OJEoH-0004MY-M5 for gsmc-categories@m.gmane.org; Tue, 01 Jun 2010 01:54:21 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJEKJ-00072x-HA for categories-list@mta.ca; Mon, 31 May 2010 20:23:23 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5890 Archived-At: It seems to me that there is an important distinction here that is not being emphasized. Isomorphisms of categories can be *technically* quite useful. Knowing that a given equivalence of categories is an isomorphism, rather than merely an equivalence, can certainly make things much simpler, or even make things possible that we didn't know how to do before. Many examples of this sort have been mentioned. Another that should be added to the list is the theory of strict 2-categories, 2-limits, 2-adjoints, and so on, all of which is defined using ordinary enriched category theory over Cat, and hence involves many isomorphisms of hom-categories. However, I find that in most or all of these examples, one is not actually interested in the fact of an isomorphism of categories for its own sake. There is no "real meaning" in the fact that two categories are isomorphic, rather than equivalent; generally it's a technical accident of how we chose to define them. It may be a very *convenient* technical accident, but it is an accident nonetheless. If we chose our definitions differently, or worked in a different foundational system (such as one where the notion of "isomorphism of categories" cannot even be defined), some or all of our isomorphisms of categories might change to become only equivalences, but I don't believe that any of the real content of category theory would go away; it would just become harder to prove. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]