From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9191 Path: news.gmane.org!.POSTED!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: when is Fam (E) a topos? Date: Sat, 22 Apr 2017 17:35:43 +0100 (BST) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: blaine.gmane.org 1492962123 13299 195.159.176.226 (23 Apr 2017 15:42:03 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 23 Apr 2017 15:42:03 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Sun Apr 23 17:41:59 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 1d2Je9-0003HH-JK for gsmc-categories@m.gmane.org; Sun, 23 Apr 2017 17:41:57 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43371) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1d2Jdt-0006IT-GB; Sun, 23 Apr 2017 12:41:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1d2Jd7-00025M-DR for categories-list@mlist.mta.ca; Sun, 23 Apr 2017 12:40:53 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9191 Archived-At: I was surprised by Thomas's previous post, because I knew that if E has set-indexed copowers then Fam(E) can be identified with the glueing of Delta, and is thus a topos. (I haven't seen Pieter Hofstra's thesis, so I wasn't aware that he had made a different claim.) In fact set-indexed copowers in E (a slightly weaker condition than cocompleteness, cf. A2.1.7 in the Elephant) is necessary as well as sufficient for Fam(E) to be a topos. Here's a proof: Let me write objects of Fam(E) in the form (I, (A_i | i \in I)) where I is a set and the A_i are objects of E. Noting that the forgetful functor sending (I,(A_i)) to I is represented by the object (1,(0)) where 0 is the initial object of E, it's easy to see that if Fam(E) is cartesian closed then objects of the form (1,(A)) form an exponential ideal, i.e. any exponential (1,(A))^(I,(B_i)) is of the form (1,(C)). In particular, if the exponential (1,(A))^(I,(1 | i \in I)) exists, it is of the form (1,(C)) where C is an I-fold power of A in E. So E has arbitrary set-indexed powers; but E^op is monadic over E, so it also has set-indexed powers, i.e. E has set-indexed copowers. Peter Johnstone On Fri, 21 Apr 2017, Thomas Streicher wrote: >> 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/ ]