From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6763 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Mon, 11 Jul 2011 08:32:28 -0400 Message-ID: References: ,<1310362598.4e1a8be6a7800@webmail.adelaide.edu.au> Reply-To: Marta Bunge 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 1310398389 15125 80.91.229.12 (11 Jul 2011 15:33:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 11 Jul 2011 15:33:09 +0000 (UTC) Cc: To: , Original-X-From: majordomo@mlist.mta.ca Mon Jul 11 17:33:05 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 1QgITo-0001Fq-Fc for gsmc-categories@m.gmane.org; Mon, 11 Jul 2011 17:33:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42056) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QgISM-0006ba-2w; Mon, 11 Jul 2011 12:31:34 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QgISL-00077K-9i for categories-list@mlist.mta.ca; Mon, 11 Jul 2011 12:31:33 -0300 In-Reply-To: <1310362598.4e1a8be6a7800@webmail.adelaide.edu.au> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6763 Archived-At: Dear David=2C =A0 =A0 =A0 Whatever is published in categories is of public domain so anyone can intervene. You have nothing to apologize for= . =A0 =A0 =A0 I am not acquainted with recent work of Dorette Pronk=2C but I read her 1995 Utrecht thesis in detail since I was a= sked to do so by her advisor. In it=2C she refers to my paper (Marta Bunge=2C "A= n application of descent to a classification theorem for toposes" =2C Math. Proc. Camb. Phil. Soc. 107 (1990) 59-79)=2C where I prove=2C in Corollary 5= .4 to the main Theorem 5.1=2C the following=2C which is=2C oin the case of groupo= ids=2C what you call the third way. It says that=2C if W is the class of isomorphisms c= lasses of weak equivalences of etale complete groupoids (ECG)=2C then W admits a calculus of right fractions=2C and the functor B from Gpd to Top induces an equivalence ECG[W-1] \iso [Top]=2C where [Top} denotes the category of Grotehndieck toposes (over a base S not necessarily Sets) and isomorphism classes of geometric morphisms. So=2C the purpose of the third way=2C in my= view=2C is to prove classification theorems. However=2C I am not au courant of more recent developments. =A0 =A0 =A0 As for the other two approaches I mentioned in my correspondence with Andre Joyal=2C their equivalence is not= that obvious. In the 1-dimensional case=2C this is done in Bunge-Pare (1979) Proposition 2.7=2C and in the 2-dimensional case it is done in Bunge-Hermid= a (2010) Theorem 4-9. =A0 =A0 =A0 Concerning size matters=2C let me observe that my construction of the stack completion (Bunge=2C Cahiers 1979) is meaningful regardless of size questions=2C that is=2C for any base topos S.= The outcome=2C however=2C of applying it to an internal category need no longer= be internal. For this reason I introduce an "axiom of stack completions" which guarantees that stack completions of internal categories be again internal=2Cand which is satisfied by any S a Grothehdieck topos. The questi= on of stating such an axiom as an additional axiom to the ones for elementary top= oses was proposed as a problem by Lawvere in his Montreal lectures in 1974.=A0 =A0 =A0 =A0 Good luck with your projects. Marta > Date: Mon=2C 11 Jul 2011 15:06:38 +0930 > From: david.roberts@adelaide.edu.au > To: joyal.andre@uqam.ca=3B martabunge@hotmail.com > CC: categories@mta.ca > Subject: stacks (was: size_question_encore) >=20 > Dear Marta=2C Andr=E9=2C and others=2C >=20 > this is perhaps a bit cheeky=2C because I am writing this in reply to Mar= ta's > email to Andr=E9=2C quoted below. To me it almost feels like reading anot= hers' mail=3B > please forgive the stretch of etiquette. >=20 > --- >=20 > 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. >=20 > However=2C I would advertise a third way=2C and that is to localise the (= or a!) > 2-category of categories internal to the base directly=2C rather than usi= ng a > model category=2C which is a tool (among other things) to localise the 1-= category > of internal categories. Dorette Pronk proved a few special cases of this = in her > 1996 paper discussing bicategorical localisations=2C namely algebraic=2C > differentiable and topological stacks=2C all of a fixed sort. >=20 > By this I mean she took a static definition of said stacks=2C rather than= working > with a variable notion of cover=2C as one finds=2C for example in algebra= ic > geometry: Artin stacks=2C Deligne-Mumford stacks=2C orbifolds etc=2C or a= s in Behrang > Noohi's 'Foundations of topological stacks'=2C where one can have a varia= ble class > of 'local fibrations'=2C which control the behaviour of the fibres of > source/target maps of a presenting groupoid. >=20 > With enough structure on the base site (say=2C existence and stability un= der > pullback of reflexive coequalisers)=2C then one can define (in roughly hi= storical > order=2C as far as I know): >=20 > 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 >=20 > 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)=2C bo= th of > which are a localisation of the same 2-category at the 'weak equivalences= '. >=20 > Without existence of reflexive coequalisers (say for example when working= in > type-theoretic foundations)=2C then one can consider ordinary (as opposed= to > 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=2C is a stack really a stack of bundles=2C or a stack of > cocycles/descent data. >=20 > 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=2C and if one has enough projectives (of the appropriate variety= )=2C then > an internal groupoid A (say) with object of objects projective satisfies >=20 > Gpd(S)(A=2CB) ~~> Gpd_W(S)(A=2CB) >=20 > for all other objects B=2C and where Gpd_W(S) denotes the 2-categorical > localisation of Gpd(S) at W. >=20 > 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)=2C there is a need to have a handle on the size of = the > hom-categories=2C to keep local smallness. One achieves this by demanding= that 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. >=20 > All the best=2C >=20 > David >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]