From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6468 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-category Date: Thu, 13 Jan 2011 15:02:52 -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 1295032070 6512 80.91.229.12 (14 Jan 2011 19:07:50 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 14 Jan 2011 19:07:50 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Fri Jan 14 20:07:45 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 1Pdozx-0008JS-0M for gsmc-categories@m.gmane.org; Fri, 14 Jan 2011 20:07:45 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:57758) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PdozX-0000Rs-Rq; Fri, 14 Jan 2011 15:07:19 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PdozN-0007Xd-Ms for categories-list@mlist.mta.ca; Fri, 14 Jan 2011 15:07:10 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6468 Archived-At: Dear Jean, 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. An anafunctor is really a simple thing: a morphism in the bicategory of fractions obtained from Cat by inverting the functors which are fully faithful and essentially surjective. It can be represented by a span A <-- F --> B whose left leg is fully faithful and surjective on objects. One intuition is that the objects of F over a\in A are different "ways to compute a value" of the anafunctor at a. Different "ways to compute a value" may give different values, but they will be canonically isomorphic. For example, let P --> 2 be a fibration, with fibers B and A. Then there is (without AC) an anafunctor A --> B, where the objects of F are the cartesian arrows of P over the nonidentity arrow of 2, and the projections assign to such an arrow its domain and codomain. More generally, if Cat_ana denotes the bicategory of categories and anafunctors, then from any fibration P --> C we can construct (without AC) a pseudofunctor C^{op} --> Cat_ana. Moreover, if we allow morphisms between fibrations to be anafunctors as well, then the bicategory of fibrations over C is biequivalent to the bicategory of pseudofunctors C^{op} --> Cat_ana. (This should not be read as saying anything more than it says; in particular I would not claim that fibrations are always "the same as" indexed categories even from this viewpoint. For fixed C, they form equivalent bicategories, which makes them sufficiently "the same" for some purposes, but, as you have pointed out, not for other purposes.) Similarly, regarding "internalization," any ordinary (non-cloven) fibration does give rise to an internal fibration in the bicategory Cat_ana. The same is true for internal fibrations and anafunctors in a topos (the relevant "non-cartesian" parts of the internal logic of the topos E having been incorporated into the definition of Cat_ana(E)). Unfortunately, since Cat_ana is only a bicategory, not a strict 2-category, we do not get the strict notion of internal fibration, but the weaker version as defined by Street, in which cartesian liftings exist only up to isomorphism. I think this is a nice example of when one may be "forced" to use Street fibrations rather than Grothendieck ones (never claiming, of course, that there is anything necessarily "wrong" with Grothendieck fibrations when they suffice). For example, if p: P --> C is a (Grothendieck) fibration, f: A --> C and g: A --> P are functors and m: f --> pg is a natural transformation, then we can define an anafunctor A <-- H --> P in which the objects of H are pairs (a,n), where a is an object of A and n: x --> g(a) is a cartesian arrow in P with p(n) = m_a. The functor H --> A is surjective on objects because p is a fibration. Then the composite anafunctor ph: A --> C is naturally isomorphic to f, and there is a natural transformation from h to g which lies over m (modulo this isomorphism) and which is cartesian in Cat_ana(A,P) over Cat_ana(A,C). One can generalize to the case when f, g, and p are also anafunctors and p is a Street fibration (suitably interpreted for an anafunctor). In general, it seems to me that there are two overall approaches to doing category theory without AC (including with internal categories in a topos): 1) Embrace anafunctors as "the right kind of morphism between categories" in the absence of AC. As I mentioned above, many familiar facts about category theory which normally use AC remain true without it, if all notions are replaced by their corresponding "ana-" versions. Of course, this approach has the disadvantage that anafunctors are more complicated than ordinary functors, and form a bicategory rather than a strict 2-category; thus one may be forced into using other weaker notions like Street fibrations, bilimits, etc. 2) Insist on using only ordinary functors, so that we can work with the strict 2-category Cat, which is simpler and stricter than Cat_ana. However, many theorems which are true under AC now become false. In addition to the properties of fibrations as above, one also has to distinguish between "having limits" in the sense of "every diagram has a limit" versus the sense of "there is a function assigning a limit to every diagram." Personally, while there is nothing intrinsically wrong with (2), I think (1) gives a more satisfactory theory. It also has connections to applications outside of category theory. For instance, anafunctors between internal categories in a topos are more or less equivalent to morphisms between their stack completions, and in various parts of mathematics internal categories, and notions equivalent to anafunctors, are frequently used as representatives of stacks (Lie groupoids, Hopf algebroids, moduli stacks, etc.). So it is not just a philosophical reason to prefer (1). However, I respect that others may disagree, and I'd be interested in hearing about mathematical reasons to prefer (2). Regards, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]