From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6762 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: stacks (was: size_question_encore) Date: Mon, 11 Jul 2011 15:06:38 +0930 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 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310398305 14512 80.91.229.12 (11 Jul 2011 15:31:45 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 11 Jul 2011 15:31:45 +0000 (UTC) Cc: categories To: =?iso-8859-1?b?QW5kcuk=?= Joyal , martabunge@hotmail.com Original-X-From: majordomo@mlist.mta.ca Mon Jul 11 17:31:40 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QgISR-0000Pc-AJ for gsmc-categories@m.gmane.org; Mon, 11 Jul 2011 17:31:39 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42026) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QgIQw-0006OO-4Q; Mon, 11 Jul 2011 12:30:06 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QgIQv-000732-8J for categories-list@mlist.mta.ca; Mon, 11 Jul 2011 12:30:05 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6762 Archived-At: Dear Marta, Andr=E9, and others, this is perhaps a bit cheeky, because I am writing this in reply to Marta= 's email to Andr=E9, quoted below. To me it almost feels like reading anothe= rs' mail; please forgive the stretch of etiquette. --- Marta raised an interesting point that stacks can be described in (at lea= st) two ways: via a model structure and via descent. The former implicitly (in th= e case of topoi: take all epis) or explicitly needs a pretopology on the base ca= tegory in question. This is to express the notion of essential surjectivity. However, I would advertise a third way, and that is to localise the (or a= !) 2-category of categories internal to the base directly, rather than using= a model category, which is a tool (among other things) to localise the 1-ca= tegory of internal categories. Dorette Pronk proved a few special cases of this = in her 1996 paper discussing bicategorical localisations, namely algebraic, differentiable and topological stacks, all of a fixed sort. By this I mean she took a static definition of said stacks, rather than w= orking with a variable notion of cover, as one finds, for example in algebraic geometry: Artin stacks, Deligne-Mumford stacks, orbifolds etc, or as in B= ehrang Noohi's 'Foundations of topological stacks', where one can have a variabl= e class of 'local fibrations', which control the behaviour of the fibres of source/target maps of a presenting groupoid. With enough structure on the base site (say, existence and stability unde= r pullback of reflexive coequalisers), then one can define (in roughly hist= orical order, as far as I know): representable internal distributors/profunctors =3D meriedric morphisms (generalising Pradines) =3D Hilsum-Skandalis morphisms =3D (internal) saturated anafunctors =3D (incorrectly) Morita morphisms =3D right principal bibundles/bitorsors and then (it is at morally true that) the 2-category of stacks of groupoi= ds is equivalent to the bicategory with objects internal groupoids and 1-arrows= the above maps (which have gathered an interesting collection of names), both= of which are a localisation of the same 2-category at the 'weak equivalences= '. Without existence of reflexive coequalisers (say for example when working= in type-theoretic foundations), then one can consider ordinary (as opposed t= o saturated) anafunctors. Whether these also present the 2-category of stac= ks is a (currently stalled!) project of mine. The question is a vast generalisati= on of this: without the 'clutching' construction associating to a Cech cocyle a= actual principal bundle, is a stack really a stack of bundles, or a stack of cocycles/descent data. The link to the other two approaches mentioned by Marta is not too obscur= e: the class of weak equivalences in the 2-categorical and 1-categorical approac= hes are the same, and if one has enough projectives (of the appropriate variety),= then an internal groupoid A (say) with object of objects projective satisfies Gpd(S)(A,B) ~~> Gpd_W(S)(A,B) for all other objects B, and where Gpd_W(S) denotes the 2-categorical localisation of Gpd(S) at W. One more point: Marta mentioned the need to have a generating family. Whi= le in the above approach one keeps the same objects (the internal categories/groupoids), there is a need to have a handle on the size of th= e hom-categories, to keep local smallness. One achieves this by demanding t= hat for every object of the base site there is a *set* of covers for that object = cofinal in all covers for that object. Then the hom-categories for the localised 2-category are essentially small. All the best, David Quoting Andr=E9 Joyal : > dear Marta, > > I apologise, I had forgoten our conversation! > My memory was never good, and it is getting worst. > > You wrote: > > >No, I am not thinking of the analogue of Steve Lack's model > structure since, strictly speaking, > >it has nothing to do with stacks. Comments to that effect (with > which Steve agrees) are included > >in the Bunge-Hermida paper. It was actually a surprise to discover > that after trying to do what > >you suggest and failing. > > I disagree with your conclusion. I looked at your paper with Hermida. > We are not talking about the same model structure. The fibrations in > 2Cat(S) defined > by Steve Lack (your definition 7.1) are too weak when the topos S does > not satify the axiom of choice. > Equivalently, his generating set of trivial cofibrations is too small. > > Nobody has read my paper with Myles > groupoids, JPAA, Vol 89, 1993>. > > Best, > Andr=E9 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]