From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9354 Path: news.gmane.org!.POSTED!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Cartesian morphism ~~> fibration Date: Thu, 21 Sep 2017 07:41:42 +0930 Message-ID: References: <20170920174906.GE8154@mathematik.tu-darmstadt.de> Reply-To: David Roberts NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1506003203 22685 195.159.176.226 (21 Sep 2017 14:13:23 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 21 Sep 2017 14:13:23 +0000 (UTC) Cc: "categories@mta.ca list" To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Thu Sep 21 16:13:10 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 1dv2E0-00059x-UD for gsmc-categories@m.gmane.org; Thu, 21 Sep 2017 16:13:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43850) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dv2F0-0000PZ-KU; Thu, 21 Sep 2017 11:14:10 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dv2D6-0001TG-81 for categories-list@mlist.mta.ca; Thu, 21 Sep 2017 11:12:12 -0300 In-Reply-To: <20170920174906.GE8154@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9354 Archived-At: Dear Thomas, Thanks for that example (and to someone else who, off-line, gave me the example where B is trivial). Here's the version for categories fibred in groupoids https://stacks.math.columbia.edu/tag/06N7 So I guess this extends your example where the codomain is a discrete fibration, merely having to replace the domain by an equivalent category. This makes the original cartesian functor a Street fibration, I believe. This is all in the context of stacks, and in particular algebraic or other presentable sacks, which is what I'm looking at, though in greater generality than the Stacks Project. David On 21 Sep. 2017 3:19 am, "Thomas Streicher" < streicher@mathematik.tu-darmstadt.de> wrote: >> I'm trying to find a reference for the following result, if indeed it is >> true. > > I think the claim is wrong in general. Let P : X->B be a fibration of > categories with a terminal object, i.e. P has a right adjoint right > inverse One. Then One : Id_B -> P is a cartesian functor though > itself not a fibration in general (e.g. B = 1 and X the ordinal 2 then > One picks 1 from 2 which has empty fibre over 0). > > However, if P is a fibration and Q is a discrete fibration and F is a > functor with QF = P then F is a fibration iff F is a cartesian functor > from P to Q. > > Thomas > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]