From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9190 Path: news.gmane.org!.POSTED!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: when is Fam (E) a topos? Date: Fri, 21 Apr 2017 11:01:11 +0200 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: blaine.gmane.org 1492774137 14524 195.159.176.226 (21 Apr 2017 11:28:57 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 21 Apr 2017 11:28:57 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Apr 21 13:28:53 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1d1Wk8-0003hQ-QV for gsmc-categories@m.gmane.org; Fri, 21 Apr 2017 13:28:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42699) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1d1WkI-0005ec-2G; Fri, 21 Apr 2017 08:29:02 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1d1WjX-0006Xp-NI for categories-list@mlist.mta.ca; Fri, 21 Apr 2017 08:28:15 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9190 Archived-At: > Let E be a topos then Fam(E) -> Set is certainly a fibered topos > but by Th.6.2.3 of Pieter Hofstra's Thesis Fam(E) is a topos iff E is > an atomic category (in the sense of Johnstone's 1977 book on Topos Theory, > exercise 12 on p. 257). But in atomic categories all morphisms are epic > and thus Fam(E) is a topos only if E is trivial. Alas, there is a flaw in Pieter's Th.6.2.3 (which certainly is not crucial for the main results of his otherwise very nice Thesis). Actually, it can be seen quite easily: if E is a cocomplete topos then Fam(E) is equivalent to the glueing of Delta : Set -> E which is known to be a topos. So it seems to be open to characterize those toposes E for which Fam(E) is a topos. In particular, I don't know the answer for E the free topos (with nno) or a realizability topos. In the latter case we know that glueing of Nabla (right adjoint to Gamma) is a topos but it's different from Fam(E). I'd be grateful about any suggestions even for these particular cases! Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]