From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6481 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-category Date: Tue, 18 Jan 2011 15:45:42 -0800 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1295533420 8938 80.91.229.12 (20 Jan 2011 14:23:40 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 20 Jan 2011 14:23:40 +0000 (UTC) Cc: Michal Przybylek , Categories To: David Roberts Original-X-From: majordomo@mlist.mta.ca Thu Jan 20 15:23:35 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 1PfvQF-0004PO-6X for gsmc-categories@m.gmane.org; Thu, 20 Jan 2011 15:23:35 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:35697) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PfvQ4-0001H1-KQ; Thu, 20 Jan 2011 10:23:24 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PfvPz-0000ja-1z for categories-list@mlist.mta.ca; Thu, 20 Jan 2011 10:23:19 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6481 Archived-At: On Mon, Jan 17, 2011 at 1:02 AM, David Roberts wrote: > 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. I don't think this argument quite works, but I think one can show something almost as good, namely that if S is a topos, C has finite limits, and Cat_ana(S) is equivalent to Cat(C), then S satisfies the *internal* axiom of choice. > 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. This isn't quite right; a discrete object in Cat(C) is essentially an internal equivalence relation in C, and so do(Cat(C)) is the category of equivalence relations and functors between them (not morphisms between their quotients). This is a full subcategory of the free exact completion C_{ex/lex}, which contains the free regular completion C_{reg/lex} (the category of kernels). We can say that disc:C --> do(Cat(C)) is fully faithful, and its essential image consists of the projective objects in do(Cat(C)) (assuming that C has finite limits). In particular, do(Cat(C)) has enough projectives. > 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. This I believe if S is exact, such as a topos. The point is that an effective equivalence relation in S, regarded as an internal category in S, admits a surjective weak equivalence to its quotient object, regarded as a discrete internal category. Thus, the two become equivalent in Cat_ana(S), though not in general in Cat(S). > 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. An equivalence of bicategories certainly preserves and reflects discrete objects, which is all that matters for this proof. (But a general functor of bicategories need not preserve discrete objects.) > So if we have an equivalence Cat(C) --> Cat_ana(S) ... then we can conclude that S, being equivalent to do(Cat_ana(S)) and hence to do(Cat(C)), has enough projectives, namely C. Already this is a nontrivial restriction on a topos (or set-theoretic axiom), although it can hold in the absence of IAC. However, we can say more. First, if we identify C with the subcategory of projectives in S, then the equivalence functor Cat(C) --> Cat_ana(S) must be, up to equivalence, the inclusion which regards internal categories in C as internal categories in S, and internal functors as internal anafunctors. For being an equivalence, it in particular preserves lax codescent objects; but every internal category is a lax codescent object formed of discrete internal categories, and the functor C --> Cat(C) --> Cat_ana(S) is what we used to identify C with the projective objects of S. Therefore, since this functor is an equivalence, every internal category in S must be equivalent, in Cat_ana(S), to an internal category in C, i.e. an internal category in S formed of projective objects. Now for any object A of S, we have an internal category 1+A+1 \rightrightarrows 1+1 with "two objects" and A as the object-of-morphisms from one to the other (and only identity arrows otherwise). If this category is equivalent in Cat_ana(S) to one composed of projective objects, then we must have a surjective weak equivalence to it from such a category, which is equivalent to giving a well-supported projective object P such that PxPxA is projective. Thus, any object A is "locally projective", which is sufficient for IAC. (If we are talking about set theoretical foundations, rather than working in a topos, we could then pick an element p of P, which exists since it is well-supported. Then since the projectives are closed under finite limits, the fiber of PxPxA over (p,p), namely A, would be projective, and hence AC holds.) I also think it's worth mentioning that if S merely has enough projectives, then we can identify Cat_ana(S) (up to equivalence of bicategories) with a full sub-2-category of Cat(S), consisting of those internal categories whose object-of-objects is projective (but with no condition on the object-of-morphisms). In fact, such categories are the cofibrant objects in a model structure on Cat(S), in which everything is fibrant and whose weak equivalences are the internally fully-faithful and essentially-surjective functors. Thus, this is a particular case of the fact that morphisms in the homotopy (2-)category of a model category are represented by maps from a cofibrant replacement to a fibrant replacement. (When S is a Grothendieck topos, there is also a model structure on Cat(S) with those weak equivalences in which every object is cofibrant, and in which the fibrant objects are stacks. I believe this was proven by Joyal and Tierney in their paper "Strong stacks and classifying spaces".) The set-theoretic axiom that "there exist enough projective sets" is a weak form of choice called the "presentation axiom" or "COSHEP" ("the Category Of Sets Has Enough Projectives"). It implies dependent choice and some other weak forms of choice, and tends to hold in models arising from type theory. So if one is willing to accept that axiom in lieu of full AC, or one is working in a topos that has enough projectives (such as, notably, the effective topos), then one can avoid talking about anafunctors by restricting to internal categories with projective object-of-objects. I don't know whether there is a dual set-theoretic "axiom of small stack completions". Regards, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]