From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6024 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Homomorphisms that are pullbacks Date: Mon, 02 Aug 2010 09:20:55 -0400 Message-ID: Reply-To: "Fred E.J. Linton" 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 1280788438 24045 80.91.229.12 (2 Aug 2010 22:33:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 2 Aug 2010 22:33:58 +0000 (UTC) To: Tom Leinster , Original-X-From: majordomo@mlist.mta.ca Tue Aug 03 00:33:57 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Og3Zz-0001xm-3s for gsmc-categories@m.gmane.org; Tue, 03 Aug 2010 00:33:55 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50912) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Og3YT-00033S-8I; Mon, 02 Aug 2010 19:32:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Og3YK-0002Xi-7p for categories-list@mlist.mta.ca; Mon, 02 Aug 2010 19:32:12 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6024 Archived-At: On Mon, 02 Aug 2010 08:36:21 AM EDT, Tom Leinster as= ked, > Let T be a monad on some category. A map of T-algebras is a commutativ= e > square > = > TA ----> TB > | | > | | > V V > A -----> B. > = > When is this square a pullback? One very simple instance, in Banach spaces (with norm-non-increasing linear mappings), with T the double-dualization monad: if B is reflexive,= = then such a square is a pullback iff A is reflexive, too. The superficial similarity with the finite/discrete case of Tom's = local homeomorphism remark in the instance of compact Hausdorff spaces = may, with luck, be more than just coincidental ... (the case I mean is that if B is a finite discrete space, then the square (in KT_2 spaces, = with T the Stone-Cech compactification monad) is a pullback iff A is = finite discrete as well). Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]