From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4995 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Topology on cohomology groups Date: Sat, 20 Jun 2009 06:32:46 -0400 (EDT) Message-ID: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1245608236 7930 80.91.229.12 (21 Jun 2009 18:17:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 21 Jun 2009 18:17:16 +0000 (UTC) To: Steve Vickers , Original-X-From: categories@mta.ca Sun Jun 21 20:17:14 2009 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 4.50) id 1MIRbJ-0004kB-Se for gsmc-categories@m.gmane.org; Sun, 21 Jun 2009 20:17:10 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIR2y-0006WD-5p for categories-list@mta.ca; Sun, 21 Jun 2009 14:41:40 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4995 Archived-At: Very tentatively, I have a memory that Lefschetz in the 30s invented linearly compact vector spaces (and proved that the "category" of them was dual to the "category" of discrete vector space (this was before categories) for the purpose of making cohomology more closely dual to homology. Michael On Fri, 19 Jun 2009, Steve Vickers wrote: > A cohomology group can easily be an infinite power of the coefficient > group. But such a group has a natural non-discrete topology, namely the > compact-open (which in this case is also the product topology). > > Are there approaches to cohomology that, as part of the process, also > supply topologies on the cohomology groups? > > [I'm trying to understand the topos-theoretic account of cohomology as > in Johnstone's "Topos Theory". But it looks heavily dependent on having > a classical base topos, since it uses the classical proof of sufficiency > of injectives (together with the existence of Barr covers) to deduce the > same property internally in any Grothendieck topos. For a more fully > constructive theory I wonder if one needs to take better care of the > topologies.] > > Steve Vickers. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]