From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5888 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: Isomorphisms of categories Date: Sun, 30 May 2010 12:55:36 -0500 Message-ID: <4C02A698.9090706@math.uchicago.edu> References: <4C02A580.2000606@math.upenn.edu> Reply-To: Peter May NNTP-Posting-Host: lo.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 1275350028 14824 80.91.229.12 (31 May 2010 23:53:48 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 31 May 2010 23:53:48 +0000 (UTC) Cc: categories@mta.ca To: jds@math.upenn.edu Original-X-From: categories@mta.ca Tue Jun 01 01:53:46 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 1OJEni-000408-Ab for gsmc-categories@m.gmane.org; Tue, 01 Jun 2010 01:53:46 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OJEIS-0006zt-AH for categories-list@mta.ca; Mon, 31 May 2010 20:21:28 -0300 In-Reply-To: <4C02A580.2000606@math.upenn.edu> Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5888 Archived-At: On 5/30/10 12:50 PM, jim stasheff wrote: > Peter May wrote: >> DeTeXing an exercise I routinely assign, here is >> an example of an isomorphism of categories that is >> not `accidental' in Peter Johnstone's sense and is >> always used in practice as an isomorphism and not >> merely an equivalence. >> >> >> The fundamental theorem of Galois theory: >> >> Let G = Gal(E/F) be the Galois group of a finite >> Galois extension E/F. Define an isomorphism of >> categories between the category of intermediate >> fields F\subset K\subset E and field maps >> K >--> L that fix F pointwise and the category >> of orbits G/H and G-maps between them. >> >> > and an isomprhic category of coverings spaces such that... > > jim No-no, that is in fact the very next exercise: Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]