From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6748 Path: news.gmane.org!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: RE : size_question_encore Date: Fri, 8 Jul 2011 09:00:09 -0400 Message-ID: References: ,<9076_1310082720_4E16469F_9076_34_1_E1QeyJ6-00024q-CT@mlist.mta.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 1310171055 4539 80.91.229.12 (9 Jul 2011 00:24:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Jul 2011 00:24:15 +0000 (UTC) To: , , Original-X-From: majordomo@mlist.mta.ca Sat Jul 09 02:24:07 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 1QfLL4-0004Ez-7Z for gsmc-categories@m.gmane.org; Sat, 09 Jul 2011 02:24:06 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35792) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QfLJU-0005dD-MG; Fri, 08 Jul 2011 21:22:28 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QfLJT-0006rf-VE for categories-list@mlist.mta.ca; Fri, 08 Jul 2011 21:22:27 -0300 In-Reply-To: <9076_1310082720_4E16469F_9076_34_1_E1QeyJ6-00024q-CT@mlist.mta.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6748 Archived-At: Dear Andre=2CI welcome your suggestion of involving stacks in order to test= universality when the base topos S does not have Choice. I have been explo= iting this implicitly but systematically several times since my own constru= ction of the stack completion of a category object C in any Grothendieck to= pos S (Cahiers=2C 1979). For instance=2C I have used it crucially in my pap= er on Galois groupoids and covering morphisms (Fields=2C 2004)=2C not only = in distinguishing between Galois groupoids from fundamental groupoids=2C bu= t also for a neat way of (well) defining the fundamental groupoid topos of = a Grothendieck topos as the limit of a filtered 1-system of discrete groupo= ids=2C obtained from the naturally arising bifiltered 2-system of such by t= aking stack completions. This relates to the last remark you make in your p= osting. Concerning S_fin=2C it does not matter if=2C in constructing the ob= ject classifier=2C one uses its stack completion instead=2C since S is a st= ack (Bunge-Pare=2C Cahiers=2C 1979). In my opinion=2C stacks should be the = staple food of category theory without Choice. For instance=2C an anafuncto= r (Makkai's terminology) from C to D is precisely a functor from C to the s= tack completion of D. More recently (Bunge-Hermida=2C MakkaiFest=2C 2011)= =2C we have carried out the 2-analogue of the 1-dimensional case along the = same lines of the 1979 papers=2C by constructing the 2-stack completion of = a 2-gerbe in "exactly the same way". Concerning this=2C I have a question f= or you. Is there a model structure on 2-Cat(S) (or 2-Gerbes(S))=2C for S a = Grothedieck topos=2C whose weak equivalences are the weak 2-equivalence 2-f= unctors=2C and whose fibrant objects are precisely the (strong) 2-stacks? A= lthough not needed for our work=2C the question came up naturally after you= r paper with Myles Tierney. We could find no such construction in the liter= ature. With best regards=2C Marta > Subject: categories: RE : categories: size_question_encore > Date: Wed=2C 6 Jul 2011 21:23:36 -0400 > From: joyal.andre@uqam.ca > To: edubuc@dm.uba.ar=3B categories@mta.ca >=20 > Dear Eduardo=2C >=20 > I would like to join the discussion on the category of finite sets. >=20 > As you know=2C the natural number object in a topos can be given many cha= racterisations. > For example=2C it can be defined to be the free monoid on one generator. = Etc >=20 > Clearly the internal category S_f of finite set in the topos Set has many= equivalent descriptions. > For example=2C it is a a category with finite coproducts freely generated= by one object u. > This means that for every category with finite coproducts C and every obj= ect c of C=2C=20 > there is a finite coproducts preserving functor F:S_f--->C=20 > together with an isomorphism a:F(u)->c=2C and moreover that the > pair (F=2Ca) is unique up to unique isomorphism of pairs. > It folows from this description that the category > of finite sets is well defined up to an equivalence of categories=2C > with an equivalence which is unique up to unique isomorphism. >=20 > The situation is more complicated if we work in a general > Grothendieck topos instead of the topos of sets. > The problem arises from the fact that in a Grothendieck topos=2C a local = equivalence > may not be a global equivalence=20 > A "global" equivalence between internal categories > is defined to be an equivalence in the 2-category of internal categories = of this topos.=20 > A "local"equivalence is defined to be a functor > which is essentially surjective and fully faithful. > Every internal category C has a stack completion C--->C' > which is locally equivalent to C. > A local equivalence induces a global equivalences after stack completion. >=20 > Let me remark here that the stack completion can be obtained by using a Q= uillen model structure > introduced by Tierney and myself two decades ago. > More precisely=2C the category of small categories (internal to a Grothen= dieck topos) admits a model structure > in which the weak equivalences are the local equivalences=2C > and the cofibrations are the functors monic on objects. > An internal category is a stack iff it is globally > equivalent to a fibrant objects of this model structure. >=20 >=20 > I propose using stacks for testing the universality of categorical constr= uctions in a topos. > For example=2C in order to say that the category S_f of finite > sets in a topos is freely generated by one object u=2C we may say > that for every stack with finite coproducts C and every (globally defined= )=20 > object c of C=2C there is a finite coproduct preserving functor F:S_f--->= C=20 > together with an isomorphism a:F(u)->c=2C and moreover that the > pair (F=2Ca) is unique up to unique isomorphism of pairs. > The category of finite sets so defined is not unique=2C > but its stack completion is unique up to global equivalence. >=20 > Finally=2C let me observe that the local equivalences=20 > between the categories of finite sets are the 1-cells > of a 2-category which is 2-filtered. It is thus a 2-ind object > of the 2-category of internal categories. >=20 > I hope my observations can be useful. >=20 > Best regards=2C > Andr=E9 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]