From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8612 Path: news.gmane.org!not-for-mail From: Zhen Lin Low Newsgroups: gmane.science.mathematics.categories Subject: Re: Are Joyal--Tierney fibrations exponentiable? Date: Sat, 2 May 2015 20:32:54 +0100 Message-ID: References: <20150502085012.GA6806@mathematik.tu-darmstadt.de> Reply-To: Zhen Lin Low NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1430619692 12018 80.91.229.3 (3 May 2015 02:21:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 3 May 2015 02:21:32 +0000 (UTC) Cc: categories list , maw@mawarren.net To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Sun May 03 04:21:19 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YojWw-0002Wr-Rt for gsmc-categories@m.gmane.org; Sun, 03 May 2015 04:21:19 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39321) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YojVl-0000gz-3H; Sat, 02 May 2015 23:20:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YojVl-0001XP-O0 for categories-list@mlist.mta.ca; Sat, 02 May 2015 23:20:05 -0300 In-Reply-To: <20150502085012.GA6806@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8612 Archived-At: Dear Thomas, Thank you for your reply. Michael Warren also pointed out to me (in a private reply) that one can construct dependent products for split fibrations of internal groupoids. Unfortunately, this doesn't quite answer my question. As far as I know, Joyal--Tierney fibrations are cloven (or rather, cleavable), but they do not have to split. After all, in the case where the Grothendieck topos we start with is Set, the Joyal--Tierney model structure coincides with the standard model structure on Grpd, in which the fibrations really are just isofibrations in the usual sense. (I assume the axiom of choice here; the theory of cofibrantly generated model categories breaks down otherwise.) Incidentally, since you bring up universes, perhaps I should explain why I am focusing on Joyal--Tierney fibrations: I am wondering about the strength of propositional resizing. As you say, the groupoid interpretation makes sense constructively; but it is not hard to see that propositional resizing in the groupoid interpretation implies a weak form of the axiom of choice (namely, WISC) in the ambient set theory. This essentially boils down to the difference between weak equivalences (= fully faithful and essentially surjective on objects) and equivalences. The difference disappears when one restricts to Joyal--Tierney fibrations: this is a special case of the so-called "Whitehead theorem" in model category theory. -- Zhen Lin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]