From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2728 Path: news.gmane.org!not-for-mail From: "Steve Lack" Newsgroups: gmane.science.mathematics.categories Subject: RE: Pullback & coproduct of toposes Date: Tue, 22 Jun 2004 09:41:13 +1000 Message-ID: References: <059095AA-BEE2-11D8-A8D3-000A279156EE@cs.man.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018856 5505 80.91.229.2 (29 Apr 2009 15:27:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:36 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Tue Jun 29 13:30:48 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 29 Jun 2004 13:30:48 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BfLRB-0003d4-00 for categories-list@mta.ca; Tue, 29 Jun 2004 13:26:25 -0300 In-Reply-To: <059095AA-BEE2-11D8-A8D3-000A279156EE@cs.man.ac.uk> Importance: Normal Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:2728 Archived-At: >-----Original Message----- >From: cat-dist@mta.ca [mailto:cat-dist@mta.ca]On Behalf Of Barney Hilken >Sent: Wednesday, 16 June 2004 1:38 AM >To: categories@mta.ca >Subject: categories: Pullback & coproduct of toposes > > >I don't have access to a decent library at the moment, and I can't >afford a copy of the Elephant myself, so can anyone let me know: > >In the category of toposes and geometric morphisms, under what >conditions is coproduct stable under pullback? > Answer: always. Let f:E-->S+S' be a morphism of toposes. Identify S+S' with the product of the categories S and S'. Then in S+S' the terminal object (1,1) is a coproduct (1,0)+(0,1). Now apply the inverse image functor f* to obtain a decomposition 1=X_1+X_2 of the terminal object 1 in E. By extensivity of E, then, the category E is equivalent to the product E/X_1 x E/X_2; in other words, the topos E is the coproduct of the toposes E/X_1 and E/X_2. (Where E/X_1 and E/X_2 are of course the pullbacks along f of the injections S-->S+S' and S'-->S+S'.) This argument is contained in Marta Bunge & Stephen Lack, Van Kampen theorems for toposes, Adv. Math. 179:291-317, 2003. where it is seen as part of the fact that the 2-category of toposes is extensive. Steve Lack.