From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6770 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE: stacks (was: size_question_encore) Date: Tue, 12 Jul 2011 08:30:30 -0400 Message-ID: References: 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 1310479360 11570 80.91.229.12 (12 Jul 2011 14:02:40 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 12 Jul 2011 14:02:40 +0000 (UTC) Cc: David Roberts , , To: Mike Shulman Original-X-From: majordomo@mlist.mta.ca Tue Jul 12 16:02:35 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QgdXm-0003YD-U6 for gsmc-categories@m.gmane.org; Tue, 12 Jul 2011 16:02:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46993) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QgdVU-00021l-9E; Tue, 12 Jul 2011 11:00:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QgdVT-0006K8-Gh for categories-list@mlist.mta.ca; Tue, 12 Jul 2011 11:00:11 -0300 Importance: Normal In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6770 Archived-At: Dear Michael=2C I remind you that it was Benabou who observed (and proved) the equivalence = between=A0anafunctors and representable distributors=2C reproduced in the n= Lab as "folklore"=2C from which I concluded (without any knowledge of anafu= nctors) that the stack completion of a category C in S represents anafuncto= rs with target C. I am glad that Makkai is now aware of this fact=2C which = gives a universal flavor to his subject=2C whatever "morally" means.=A0I kn= ow nothing about the "axiom of cardinal selection".=A0 As for there being an example of an elementaru topos which does not satisfy= the "axiom of stack compleitons"=2C Joyal gave one lomg ago and Lawvere me= ntioned it in his 1974 Montreal lectures. Take a group G with a proper clas= s of subgroups having a small index in G. The topos [G=2C Sets] is an examp= le.=A0Also=2C so far as I know=2C it is not yet known (Hyland=2C Robinson= =2C and Rosolini=2C"The discrete objects in the effective topos"=2C Proc. L= ondon Math. Soc. (3) 60 (1990=2C 1-36)) whether the full internal subcatego= ry Q on the subquotients of N in Eff (the effective topos) has an internal = stack completion. The stack completion is identified as Orth(Delta 2)=2C fa= milies of discrete objects.=A0 =A0Regards=2CMarta =A0 > Date: Mon=2C 11 Jul 2011 18:20:42 -0700 > Subject: Re: categories: RE: stacks (was: size_question_encore) > From: mshulman@ucsd.edu > To: martabunge@hotmail.com > CC: david.roberts@adelaide.edu.au=3B joyal.andre@uqam.ca=3B categories@mt= a.ca >=20 > Is the "axiom of stack completions" related to the "axiom of small > cardinality selection" used by Makkai to prove that the bicategory of > anafunctors is cartesian closed? I think I recall a remark in > Makkai's paper to the effect that the stack completion of a category C > is at least morally the same as the category Ana(1=2CC) of "ana-objects" > of C. >=20 > Are there known examples of elementary toposes which violate the axiom > of stack completions? >=20 > On Mon=2C Jul 11=2C 2011 at 5:32 AM=2C Marta Bunge wrote: >> 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 topo= s S. =A0The >> outcome=2C however=2C of applying it to an internal category need no lo= nger =A0be >> 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 qu= estion of >> stating such an axiom as an additional axiom to the ones for elementary= toposes >> was proposed as a problem by Lawvere in his Montreal lectures in 1974. = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]