RE: RE: size_question_encore

RE: RE: size_question_encore

[André Joyal]

Dear Marta,

You wrote:

  >What we wanted (though we did not need it) was a model structure  
similar to the Joyal-Terney but in dimension 2,
  >and of course it would be different from this one. I am sure too  
that it exists.

I am glad we agree.

Best, -Andre

-------- Message d'origine--------
De: Marta Bunge [mailto:martabunge@hotmail.com]
Date: dim. 10/07/2011 14:26
À: Joyal, André; categories@mta.ca
Objet : RE: categories: RE: size_question_encore


Dear Andre,
An addendum to my previous is in order. In Section 7 of Bunge-Hermida  
we actually discuss Lack's model structure on 2-Cat(S). We did this  
only to show that it is not suitable to get 2-stack completions, in  
case someone would think of this idea. The (ELP) is too restrictive  
for that. This may have led to your misunderstanding when we actually  
agree. What we wanted (though we did not need it) was a model  
structure similar to the Joyal-Terney but in dimension 2, and of  
course it would be different from this one. I am sure too that it  
exists.
Best,Marta



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RE : size_question_encore

[Marta Bunge]

Dear Andre,I welcome your suggestion of involving stacks in order to test  universality when the base topos S does not have Choice. I have been exploiting this implicitly but systematically several times since my own construction of the stack completion of a category object C in any Grothendieck topos S (Cahiers, 1979). For instance, I have used it crucially in my paper on Galois groupoids and covering morphisms (Fields, 2004), not only in distinguishing between Galois groupoids from fundamental groupoids, but 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 groupoids, obtained from the naturally arising bifiltered 2-system of such by taking stack completions. This relates to the last remark you make in your posting. Concerning S_fin, it does not matter if, in constructing the object classifier, one uses its stack completion instead, since S is a stack (Bunge-Pare, Cahiers, 1979). In my opinion, stacks should be the staple food of category theory without Choice. For instance, an anafunctor (Makkai's terminology) from C to D is precisely a functor from C to the stack completion of D. More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out the 2-analogue of the 1-dimensional case along the same lines of the 1979 papers, by constructing the 2-stack completion of a 2-gerbe in "exactly the same way". Concerning this, I have a question for you. Is there a model structure on 2-Cat(S) (or 2-Gerbes(S)), for S a Grothedieck topos, whose weak equivalences are the weak 2-equivalence 2-functors, and whose fibrant objects are precisely the (strong) 2-stacks? Although not needed for our work, the question came up naturally after your paper with Myles Tierney. We could find no such construction in the literature.  With best regards, Marta

> Subject: categories: RE : categories: size_question_encore
> Date: Wed, 6 Jul 2011 21:23:36 -0400
> From: joyal.andre@uqam.ca
> To: edubuc@dm.uba.ar; categories@mta.ca
> 
> Dear Eduardo,
> 
> I would like to join the discussion on the category of finite sets.
> 
> As you know, the natural number object in a topos can be given many characterisations.
> For example, it can be defined to be the free monoid on one generator. Etc
> 
> Clearly the internal category S_f of finite set in the topos Set has many  equivalent descriptions.
> For example, 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 object c of C, 
> there is a finite coproducts preserving functor F:S_f--->C 
> together with an isomorphism a:F(u)->c, and moreover that the
> pair (F,a) 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,
> with an equivalence which is unique up to unique isomorphism.
> 
> 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, a local equivalence
> may not be a global equivalence 
> A "global" equivalence between internal categories
> is defined to be an equivalence in the 2-category of internal categories of this topos. 
> 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.
> 
> Let me remark here that the stack completion can be obtained by using a Quillen model structure
> introduced by Tierney and myself two decades ago.
> More precisely, the category of small categories (internal to a Grothendieck topos) admits a model structure
> in which the weak equivalences are the local equivalences,
> 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.
> 
> 
> I propose using stacks for testing the universality of categorical constructions in a topos.
> For example, in order to say that the category S_f of finite
> sets in a topos is freely generated by one object u, we may say
> that for every stack with finite coproducts C and every (globally defined) 
> object c of C, there is a finite coproduct preserving functor F:S_f--->C 
> together with an isomorphism a:F(u)->c, and moreover that the
> pair (F,a) is unique up to unique isomorphism of pairs.
> The category of finite sets so defined is not unique,
> but its stack completion is unique up to global equivalence.
> 
> Finally, let me observe that the local equivalences 
> 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.
> 
> I hope my observations can be useful.
> 
> Best regards,
> André
> 

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