From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6586 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibrations in a 2-Category Date: Mon, 14 Mar 2011 17:57:04 -0400 Message-ID: References: <20110122220701.C8B538626@mailscan1.ncs.mcgill.ca>,,,<20110131223321.3F49B57D7@mailscan2.ncs.mcgill.ca> 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 1300191224 25501 80.91.229.12 (15 Mar 2011 12:13:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 15 Mar 2011 12:13:44 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Tue Mar 15 13:13:40 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 1PzT87-0001MC-Cb for gsmc-categories@m.gmane.org; Tue, 15 Mar 2011 13:13:39 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42194) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PzT82-0001gV-IP; Tue, 15 Mar 2011 09:13:34 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PzT7v-0000pO-94 for categories-list@mlist.mta.ca; Tue, 15 Mar 2011 09:13:27 -0300 In-Reply-To: <20110131223321.3F49B57D7@mailscan2.ncs.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6586 Archived-At: Dear all=2C >> The discussion about the equivalence between the bicategories of anafunctor= s and of representable distributors=2C due entirely to Benabou as mentioned= below=2C has an additional interpretation in terms of stack completions wh= ich may be of interest to some of you. =A0In what follows=2C S denotes an e= lementary topos in the sense of Lawvere and Tierney.=A0 >> The first thing I noticed is that a =A0"representable distributor" from C t= o D =A0(in the sense of Benabou) is none other than a functor from =A0C to = D*=2C where D* is the (intrinsic) stack completion of D. =A0This follows fr= om the characterization of (intrinsic) stack completions given in my paper = "Stack completions and Morita equivalence for categories in a topos"=2C Cah= iers de Top. Geom Diff. XX-4 (1979) 404-436. In plain terms=2C my character= ization says that=2C for any locally internal fibration A over a topos S=2C= the stack completion A* of A is obtained as the middle term in the factori= zation of the Yoneda embedding of A into the category of S-valued presheave= s on A through its `weakly-essential image'.=A0In the case of a category D = in a topos S=2C its stack completion is that of the fibration [D] over S=2C= =A0called the "externalization" of D in my paper with Bob Pare=2C "Stacks = and equivalences of indexed categories"=2C Cahiers de Top. Geo. Diff XX-4 (= 1979) 373-399.=A0 >=A0> In principle then=2C the theory of anafunctors introduced by Michael Makkai= (and with which I am not really acquainted) could be recast more simply an= d to advantage in these terms=2C exploiting for this purpose the universal = property of stack completions. For categories in S=2C this says that an ana= functor from C to D is simply a functor from [C] to [D]*. Just as in the ca= se of anafunctors=2C for a general topos S=2C=A0=A0the fibration [D]* over = S for a category D in S need not be equivalent to (the externalization [D*]= of) an internal category D* in S.=A0=A0 >> However=2C for a Grothendieck topos S=2C this is the case on account of the= existence of a generating set. =A0An alternative construction of the stack= completion of a category in S is that of Andre Joyal and Myles Tierney=2C = "Strong stacks and classifying spaces" in Category Theory=2C Proceedings of= Como 1990=2C LNM 1488=2C Springer=2C 213-236=2C 1991=2C by means of a Quil= len model structure on Cat(S) whose weak equivalences are the weak equivale= nce functors. =A0This construction applies for instance to a Grothendieck t= opos S.=A0In particular=2C for S Grothendieck topos=2C there is then an equ= ivalence between the bicategory of anafunctors in Cat(S) and the Kleisli bi= category of the stack completion 2-monad on Cat(S). =A0 >> Marta Bunge ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics=20 McGill UniversityBurnside Hall=2C Office 1005 805 Sherbrooke St. West Montreal=2C QC=2C Canada H3A 2K6 Office: (514) 398-3810/3800 =A0 Home: (514) 935-3618 marta.bunge@mcgill.ca=20 http://www.math.mcgill.ca/~bunge/ ************************************************ > Date: Mon=2C 31 Jan 2011 14:33:20 -0800 > Subject: Re: categories: Re: Fibrations in a 2-Category > From: mshulman@ucsd.edu > To: categories@mta.ca > CC: marta.bunge@mcgill.ca=3B jean.benabou@wanadoo.fr=3B droberts@maths.ad= elaide.edu.au >=20 > Dear all=2C >=20 > In case there is any confusion=2C let me clarify that I have never > claimed=2C myself=2C to have first invented/discovered the equivalence > between the bicategories of anafunctors and of representable > distributors. I thought of it as a "folklore" sort of fact=2C hence why > I did not attribute it=2C but Jean has pointed out that his email of 22 > Jan appears to be the first time anyone has written it down. >=20 > Best=2C > Mike = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]