From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6261 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 13:07:29 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285898412 7295 80.91.229.12 (1 Oct 2010 02:00:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:00:12 +0000 (UTC) Cc: Thomas Streicher , categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:00:10 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1Uuv-0006bu-8M for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:00:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52501) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1Uu9-0005cy-2B; Thu, 30 Sep 2010 22:59:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1Uu6-0001ea-BQ for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 22:59:18 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6261 Archived-At: > It is true that if one has a Grothendieck fibration between categories > in the ordinary sense, then it only becomes an internal fibration in > the naively defined 2-category Cat if we have the axiom of choice, > since the latter amounts to saying that we can simultaneously choose > cartesian liftings for any families of objects of E and morphisms of B > we might want to pull them back along. =A0(I don't think any "global > choice" is necessary, since the definition doesn't require us to make > such a choice simultaneously for every possible family--only that for > any particular family, we -could- make such a choice.) Though we can always make such a simultaneous choice as soon as we have it in one particular case. Given the Grothendieck fibration p: E->B in Cat, and letting X denote the comma object (B,p), it is enough to choose a lifting for the X-indexed family of morphisms of B corresponding to the projection X --> B^2 at the X-indexed family of objects of E corresponding to the projection X --> E; for then to give a Y-indexed family of morphisms in B and a Y-indexed family of objects of E over their codomains is to give a morphism Y -> X, and so the chosen lifting for the latter induces a chosen lifting for the former. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]