From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5505 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: small2 Date: Sun, 10 Jan 2010 15:08:54 -0500 Message-ID: Reply-To: Marta Bunge NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1263167969 32757 80.91.229.12 (10 Jan 2010 23:59:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 10 Jan 2010 23:59:29 +0000 (UTC) To: Original-X-From: categories@mta.ca Mon Jan 11 00:59:21 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NU7gm-0004qh-Kr for gsmc-categories@m.gmane.org; Mon, 11 Jan 2010 00:59:20 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NU7Rd-0004zv-LY for categories-list@mta.ca; Sun, 10 Jan 2010 19:43:41 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5505 Archived-At: This is in reply to a posting by Bob Pare.=A0 Dear Bob:=A0 You wrote:=A0 =93As usual it helps put things in relief to generalize them.=A0Consider ca= tegory theory in a world based on a topos S. A small category is a=A0catego= ry object in S. A large category like S or Group(S) is an indexed=A0categor= y=2C given by a pseudo-functor S^op -> Cat=2C or a fibration over S if=A0yo= u prefer. A small category C gives a large one by homming. But something is= =A0lost in the process.=94=A0 That is precisely how I always thought=A0of small versus large relative to = a topos S (the universe of discourse).=A0Further=2C a large category A (fib= ered over S) "is" small if and only=A0if the corresponding pseudo-functor A= : S^op--> Cat is representable.=A0As you=A0say=2C a small category C (inter= nal to the topos S) may always be considered as a=A0large one (via its exte= rnalization [C]: S^op ->Cat).=A0 These considerations came up in my and=A0our joint work on stacks (Cahiers= =2C 1979)=2C and more recently in my work with=A0Claudio Hermida on 2-stack= s. The notions of a stack and of a 2-stack are taken =A0to be intrinsic to = a topos S=2C that is=2C relative to the topology of its=A0(regular) epimorp= hisms=2C as introduced by Lawvere 1974.=A0 Dimension 1. A small category C (internal to a topos=A0S) always has a stac= k completion when regarded as a large category via its=A0externalization [C= ]. The stack completion of C is given by yon: [C] -> [C]*=3D LocRep(S^(C^op= ))=2C a weak equivalence functor. Applied to a groupoid G=2C this=A0gives t= he classification theorem for G-torsors (Diaconescu 1975).=A0An axiom=A0of = stack completions (ASC) in its rough form says that S satisfies it if for= =A0ever small category C in S=2C the fibration LocRep(S^{C^op}) is represen= table by=A0a category C* so that [C]* and [C*] are equivalent as fibrations= . As shown by=A0Joyal and Tierney (1991) by means of Quillen model structur= es=2C but also by a=A0general argument involving the existence of a set of = generators=2C (Duskin 1980)=2C=A0Grothendieck toposes satisfy (ASC).=A0 Dimension 2. The 2-dimensional analogue of the above=A0set-up was discussed= in my lecture at CT 2008 (joint work with Claudio Hermida).=A0Our main res= ult is that=2C for a topos S satisfying (ASC)=2C any 2-category 1-stack=A0C= in S=2C regarded as a 2-fibration=2C has a 2-stack completion=2C to wit yo= n: [C]=A0-> [C]*=3DLocRep(Stack^(C^op))=2C a weak 2-equivalence 2-functor. = Applied to a=A02-gerbe G and suitably interpreted=2C this gives a classific= ation theorem for=A0G-2-torsors.=A0The validity of an appropriately formula= ted (ASC)^2 for a=A0Grothendieck topos S (see slides for my lecture at CT 2= 008) is true by a=A0general argument involving the existence of a set of ge= nerators. What is still=A0missing=2C however=2C is a construction of a smal= l 2-category 1-stack C*representing the 2-fibration [C]*=3D LocRep(Stack^(C= ^op)) in the case of a=A0Grothendieck topos S. The Quillen model structure = on 2-Cat given by Lack 2002=A0is not suitable for this purpose.=20 Dimension n. Analogue results in higher=A0dimensions are less tractable but= a pattern emerges from the passage from=A0dimension 1 to dimension 2. =A0 Remark (concerning a previous posting of yours): Although using the S-index= ed=A0versions of fibrations over S is useful=2C just as presentations of gr= oups are=A0useful=2C the entire discussion of stacks can be carried out at = the level of=A0fibrations (ditto 2-fibrations). I fail to understand what i= s all the fuss=A0about the use of S-indexed categories if taken in that spi= rit. Certainly not deserving=A0ridicule! =A0You also wrote:=A0 =93Well this is my second posting in a=A0week bringing my life-time total= =A0to three! Seems a good place to stop.=94 =A0To me=2C that is a poor reason to give.=A0This forum could certainly pro= fit from your interventions=2C but I understand=A0your qualms=2C as I mysel= f quite often abstain from an urge to intervene.=A0 =A0Happy New Year to you and everyone=2C=A0Marta ************************************************ 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/ ************************************************ = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]