From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9188 Path: news.gmane.org!.POSTED!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: when is Fam (E) a topos? Date: Wed, 19 Apr 2017 11:23:04 +0200 Message-ID: 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 1492644383 28933 195.159.176.226 (19 Apr 2017 23:26:23 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 19 Apr 2017 23:26:23 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Apr 20 01:26:14 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 1d0yzF-0007IW-Dl for gsmc-categories@m.gmane.org; Thu, 20 Apr 2017 01:26:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42260) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1d0yxr-0003il-Is; Wed, 19 Apr 2017 20:24:47 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1d0yx8-0006Ax-8X for categories-list@mlist.mta.ca; Wed, 19 Apr 2017 20:24:02 -0300 Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9188 Archived-At: A couple of days ago I made the wrong claim that > If BB is a topos and P : XX -> BB is a fibration then P is a fibration > of toposes iff XX is a topos and P is a logical functor. The following shows how wrong this claim is. 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. Thus, for the motivating examples of fibred toposes the total category is a topos only in the trivial case! Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]