From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6258 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Wed, 29 Sep 2010 14:25:37 -0700 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 1285802736 9985 80.91.229.12 (29 Sep 2010 23:25:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Sep 2010 23:25:36 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Thu Sep 30 01:25:35 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P161k-0006LW-Oq for gsmc-categories@m.gmane.org; Thu, 30 Sep 2010 01:25:33 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46647) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P160p-0003Zj-LX; Wed, 29 Sep 2010 20:24:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P160j-0001l3-I7 for categories-list@mlist.mta.ca; Wed, 29 Sep 2010 20:24:29 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6258 Archived-At: On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher wrote: > The notion of Grothendieck fibration is a property > of functors and not an additional structure. However, the notion of fibration > in a(n abstract) 2-category can be formulated only postulating a certain kind > of structure which, however, is unique up to canonical isomorphism. But this > amounts to defining Grothendieck fibrations in terms of cleavages (which > certainly are all canonically isomorphic). But choosing cleavages amounts > to accepting very strong choice principles which is maybe no real problem > but at least aesthetically moderately pleasing. I'm not sure what you mean here. The notion of fibration in a 2-category can be defined as a property if you like: a morphism E --> B in a 2-category K is a fibration if all the induced functors K(X,E) --> K(X,B) are fibrations and all commutative squares induced by morphisms X --> X' are morphisms of fibrations (preserve cartesian arrows). This is equivalent to giving some structure on E --> B, but that structure is unique up to unique isomorphism when it exists. (One can argue that both ordinary fibrations and fibrations in a 2-category are actually "property-like structures," or "properties that are not necessarily preserved by morphisms," since their forgetful functors are pseudomonic but not full. But that applies equally to both.) 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. (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.) However, in the absence of the axiom of choice, the naive definition of "functor" is not very well-behaved; it's better to use "anafunctors," or equivalently to invert the weak equivalences (fully-faithful and essentially surjective functors) in the naively defined 2-category Cat. In the resulting bicategory, I think any ordinary Grothendieck fibration will indeed be an internal fibration, without any need for choice. (Of course, "internal fibration" should probably now be interpreted in the looser sense of Street, as appropriate when working in a non-strict 2-category.) Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]