From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6476 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-category Date: Mon, 17 Jan 2011 09:21:41 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1295353364 21256 80.91.229.12 (18 Jan 2011 12:22:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 18 Jan 2011 12:22:44 +0000 (UTC) Cc: Categories To: Michal Przybylek Original-X-From: majordomo@mlist.mta.ca Tue Jan 18 13:22:40 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PfAa6-0002Ve-JJ for gsmc-categories@m.gmane.org; Tue, 18 Jan 2011 13:22:38 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:55694) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PfAZj-0000vk-HO; Tue, 18 Jan 2011 08:22:15 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PfAZe-0007Ok-UO for categories-list@mlist.mta.ca; Tue, 18 Jan 2011 08:22:11 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6476 Archived-At: Hi Michal, it is not *always* false. Consider the topoi Set and Set_choice, where the first is the category of sets without choice and the second is with choice. Then the bicategory of categories, anafunctors and transformations in Set is equivalent (assuming choice in the metalogic) to the 2-category of categories, functors and natural transformations in Set_choice. This is (essentially) shown by Makkai in his original anafunctors paper. However, I doubt that it is always true (only a hunch). Also, one does not need a topos as an ambient category in which to define anafunctors, only a site where the Grothendieck pretopology is subcanonical and singleton (single maps as covering families). The topos case is when you take the regular pretopology. And although you did not ask for a reference, here's one: http://arxiv.org/abs/1101.2363 which builds on internal anafunctors introduced here http://arxiv.org/abs/math.CT/0410328 and Makkai's original paper is available in parts from here: http://www.math.mcgill.ca/makkai/anafun/ David On 15 January 2011 09:14, Michal Przybylek wrote: > On Fri, Jan 14, 2011 at 12:02 AM, Michael Shulman wrote: > >> One way to deal with the difficulty you mention is by using >> "anafunctors," which were introduced by Makkai precisely in order to >> avoid the use of AC in category theory. > > [...] > > Interesting. But before I ask for references on ``anafunctors'' I > would like to know the following - is it false that for any (say) > topos T there exists a category C whose 2-category of internal > categories, functors, and natural transformations is (weakly) > equivalent to the bicategory Cat_ana(T)? > > > Best, > MRP [For admin and other information see: http://www.mta.ca/~cat-dist/ ]