RE : size_question_encore

[Marta Bunge <martabunge@hotmail.com>]

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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