From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6477 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 19:32:59 +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 1295353405 21407 80.91.229.12 (18 Jan 2011 12:23:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 18 Jan 2011 12:23:25 +0000 (UTC) To: Michal Przybylek , Categories Original-X-From: majordomo@mlist.mta.ca Tue Jan 18 13:23:21 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 1PfAal-0002pe-Na for gsmc-categories@m.gmane.org; Tue, 18 Jan 2011 13:23:20 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:55702) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PfAad-0000zU-8X; Tue, 18 Jan 2011 08:23:11 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PfAaZ-0007Pg-1k for categories-list@mlist.mta.ca; Tue, 18 Jan 2011 08:23:07 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6477 Archived-At: I was I think a bit hasty in my last post. I thought it was possible to separate the use of Choice in the metalogic and in Set, but I can't see how to stop Choice 'filtering down'. However if we work not with categories of sets but more general categories, I can get a more definite answer. Let S be a site (with a subcanonical singleton pretopology) so that the bicategory Cat_ana(S) is defined, and also assume that coequalisers of reflexive pairs exist in S. Theorem: If Cat_ana(S) is equivalent to a 2-category Cat(C) for some category C with finite products and coequalisers of reflexive pairs, then covers are split in S. Proof: In what follows an _equivalence_ of bicategories is defined to be a 2-functor (weak or strict) which is essentially surjective and locally fully faithful and essentially surjective. If one has Choice in the metalogic, then one can find a 2-functor which is an inverse up to a isotransformation etc. Definition: Let B be a bicategory. An object x in B is called a _discrete object_ if B(w,x) is equivalent to a set for all objects w. Let do(B) denote the full sub-bicategory on the discrete objects. For any object a in a category C there is a discrete object disc(a) in Cat(C), and disc:C --> Cat(C) is a functor. There is also a 2-functor Cat(S) --> Cat_ana(S) for a site S (with subcanonical singleton pretopology), which is the identity on objects. Discrete objects in Cat(S) are precisely discrete objects in Cat_ana(S). Lemma: If B = Cat(C) for some category C with finite products and coequalisers of reflexive pairs of arrows, then disc:C --> do(Cat(C)) is an equivalence. Lemma: if B = Cat_ana(S) for some site S with coequalisers of reflexive pairs of arrows, then disc:S --> do(Cat_ana(S)) is an equivalence. Lemma: Let F:B --> B' be an 2-functor. Then there is a 2-functor do(B) --> do(B') (i.e. discrete objects are mapped to discrete objects). If F is an equivalence then it reflects discrete objects. Corollary: If F:B --> B' is an equivalence there is an equivalence do(B) --> do(B') given by restriction of F. So if we have an equivalence Cat(C) --> Cat_ana(S) and both C and S satisfy the conditions of the first two lemmas, we have a co-span of equivalences C --> do(Cat_ana(S)) <-- S Thus if one doesn't mind inverting equivalences as defined here, we have an equivalence S --> C of categories. Lemma: Given an equivalence of categories S --> C there is an equivalence Cat(S) --> Cat(C). Thus we have an equivalence Cat(S) --> Cat_ana(S). But this implies that the appropriate version of internal Choice holds in S. # Going back to Michal's question, this would imply that in the topos S all regular epimorphisms split, which is of course not always true. David On 17 January 2011 09:21, David Roberts wrote: > 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]