From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3485 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Artin glueing for quasitoposes Date: Tue, 31 Oct 2006 18:07:26 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019335 8969 80.91.229.2 (29 Apr 2009 15:35:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:35:35 +0000 (UTC) To: Categories mailing list Original-X-From: rrosebru@mta.ca Tue Oct 31 19:19:17 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 31 Oct 2006 19:19:17 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Gf2lD-0000fL-EG for categories-list@mta.ca; Tue, 31 Oct 2006 19:11:11 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 48 Original-Lines: 29 Xref: news.gmane.org gmane.science.mathematics.categories:3485 Archived-At: Given a subobject of 1 in a topos, it's well known that one can `split' the topos into complementary open and closed subtoposes, and reconstruct the topos (up to equivalence) by applying Artin glueing to these two subtoposes. Hands up, all those of you who thought that the same thing works for quasitoposes ... I thought so! It isn't true, as I have just discovered. Certainly, given a strong subobject U >--> 1 in a quasitopos E, one can construct the `closed complement' of the open subquasitopos E/U, in exactly the same way as one does for a topos: let's denote it by C(U). It's also true that one has a `fringe functor' from E/U to C(U), and that one gets a comparison functor from E to the quasitopos obtained by glueing along this functor (again, the glueing construction works for quasitoposes just as it does for toposes). But, for this comparison to be an equivalence, one needs to know that the inverse image functors E --> E/U and E --> C(U) are (not just jointly faithful, but) jointly isomorphism-reflecting. And that can fail: I have a counterexample in a slice of the quasitopos of Frechet spaces (Elephant, A2.6.4(c). Has anyone noticed this failure before? If so, has anyone actually written it up? Peter Johnstone