From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/401 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in toposes Date: Tue, 17 Jun 1997 23:51:29 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016944 25592 80.91.229.2 (29 Apr 2009 14:55:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:44 +0000 (UTC) To: categories Original-X-From: cat-dist Tue Jun 17 23:51:51 1997 Original-Received: by mailserv.mta.ca; id AA26346; Tue, 17 Jun 1997 23:51:29 -0300 Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:401 Archived-At: Date: Tue, 17 Jun 1997 10:32:21 -0400 (EDT) From: Peter Freyd Cesc Rossello asks about pushouts in topoi. In particular, assume that for each i, Ai ---> Bi | | A ---> B f is a pushout (same f each i) and that the vertical maps are monic (henceforth to be treated notationally as inclusion maps). Let A0 be the intersection of the Ai's and B0 the intersection of the Bi's. Then is it the case that A0 ---> B0 | | A ---> B is also a pushout? Yes for finite families, no for arbitrary families. The case for finite families is an straightforward consequence of the representation theorem for pre-topoi and the fact that such representations preserve pushouts of monics. (See 1.636 and 1.65 in Categories, Allegories.) For the failure in the infinite-family case specialize to the case that f:A --> B is also an inclusion map. The result, if true, would translate to: A v /\Bi = /\(A v Bi). Take sheaves on any non-discrete T1-space, X, for a counterexample. Let B be the terminal sheaf (i.e. X itself), A the complement of some non-isolated point and {Bi} the family of all other complements of one-element sets.