From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3747 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Beck-Chevalley for presheaves on groupoids? Date: Wed, 2 May 2007 10:55:07 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019498 10123 80.91.229.2 (29 Apr 2009 15:38:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:38:18 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Fri May 4 15:30:29 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 04 May 2007 15:30:29 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Hk2O2-0004W8-Am for categories-list@mta.ca; Fri, 04 May 2007 15:20:10 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:3747 Archived-At: Hi - One of you must know the answer to this! Suppose we have a weak pullback (= pseudo-pullback) square of groupoids: G -> H | | v v K -> L Suppose we take presheaves on all four. We can get a square hom(G^{op},Set) -> hom(H^{op},Set) ^ ^ | | hom(K^{op},Set) -> hom(L^{op},Set) where the arrows pointing forward - in the same direction as the original arrows - are defined using pushforward, and the arrows pointing backward are defined using pullback. Does this square commute up to natural isomorphism? Do you know a reference somewhere? Some side remarks: 1) This seems related to the "Beck-Chevalley condition". 2) It may work for categories as well as groupoids, but I happen to need it only for groupoids. 3) I really need it with the category Vect replacing Set, so if you know a general result for any sufficiently nice category playing the role of Set here, that would be wonderful. Best, jb