From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/408 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: pushouts in toposes Date: Sun, 29 Jun 1997 11:36:43 -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 1241016949 25628 80.91.229.2 (29 Apr 2009 14:55:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:49 +0000 (UTC) To: categories Original-X-From: cat-dist Sun Jun 29 11:37:13 1997 Original-Received: by mailserv.mta.ca; id AA26707; Sun, 29 Jun 1997 11:36:43 -0300 Original-Lines: 27 Xref: news.gmane.org gmane.science.mathematics.categories:408 Archived-At: Date: Sun, 22 Jun 1997 11:54:47 -0300 From: RJ Wood A belated, somewhat tangential comment, on the distributivity condition A v /\Bi = /\(A v Bi) that Peter mentioned in his posts. The subobject classifier, Omega, satisfies this condition (internally) if and only if the topos is boolean. See the proof of Theorem 10 in Constructive Complete Distributivity II, Math Proc Cam Phil Soc, (1991) 110, 245-249, by Rosebrugh and Wood, which shows that if Omega^op is Heyting then Omega is Boolean. This result was discovered independently by Richard Squire in his thesis. It has always struck me as somewhat surprising but in the Rosebrugh/ Wood proof it is an immediate consequence of the corollary of the following result which I believe is due to Benabou and which seems to be not well known: LEMMA If f <= id:Omega--->Omega then f(u) = u/\f(true). COROLLARY If f <= id:Omega--->Omega and f(true) = true then f = id. (If Omega^op is Heyting, apply the corollary to --, where - is the negation for Omega^op.) RJ