Re: Fibrations in a 2-Category

[Marta Bunge <marta.bunge@mcgill.ca>]

Dear all,
>>
The discussion about the equivalence between the bicategories of anafunctors and of representable distributors, due entirely to Benabou as mentioned  below, has an additional interpretation in terms of stack completions which may be of interest to some of you.  In what follows, S denotes an elementary topos in the sense of Lawvere and Tierney. 
>>
The first thing I noticed is that a  "representable distributor" from C to D  (in the sense of Benabou) is none other than a functor from  C to D*, where D* is the (intrinsic) stack completion of D.  This follows from the characterization of (intrinsic) stack completions given in my paper "Stack completions and Morita equivalence for categories in a topos", Cahiers de Top. Geom Diff. XX-4 (1979) 404-436. In plain terms, my characterization says that, for any locally internal fibration A over a topos S,  the stack completion A* of A is obtained as the middle term in the factorization of the Yoneda embedding of A into the category of S-valued presheaves on A through its `weakly-essential image'. In the case of a category D in a topos S, its stack completion is that of the fibration [D] over S,   called the "externalization" of D in my paper with Bob Pare, "Stacks and equivalences of indexed categories", Cahiers de Top. Geo. Diff XX-4 (1979) 373-399. 
> >
In principle then, the theory of anafunctors introduced by Michael Makkai  (and with which I am not really acquainted) could be recast more simply and to advantage in these terms, exploiting for this purpose the universal property of stack completions. For categories in S, this says that an anafunctor from C to D is simply a functor from [C] to [D]*. Just as in the case of anafunctors, for a general topos S,  the fibration [D]* over S for a category D in S need not be equivalent to (the externalization [D*]  of) an internal category D* in S.  
>>
However, for a Grothendieck topos S, this is the case on account of the  existence of a generating set.  An alternative construction of the stack  completion of a category in S is that of Andre Joyal and Myles Tierney, "Strong stacks and classifying spaces" in Category Theory, Proceedings of  Como 1990, LNM 1488, Springer, 213-236, 1991, by means of a Quillen model structure on Cat(S) whose weak equivalences are the weak equivalence functors.  This construction applies for instance to a Grothendieck topos S. In particular, for S Grothendieck topos, there is then an equivalence between the bicategory of anafunctors in Cat(S) and the Kleisli bicategory of the stack completion 2-monad on Cat(S).  
>>

Marta Bunge

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Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
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> Date: Mon, 31 Jan 2011 14:33:20 -0800
> Subject: Re: categories: Re: Fibrations in a 2-Category
> From: mshulman@ucsd.edu
> To: categories@mta.ca
> CC: marta.bunge@mcgill.ca; jean.benabou@wanadoo.fr; droberts@maths.adelaide.edu.au
> 
> Dear all,
> 
> In case there is any confusion, let me clarify that I have never
> claimed, myself, to have first invented/discovered the equivalence
> between the bicategories of anafunctors and of representable
> distributors.  I thought of it as a "folklore" sort of fact, hence why
> I did not attribute it, but Jean has pointed out that his email of 22
> Jan appears to be the first time anyone has written it down.
> 
> Best,
> Mike
  		 	   		  

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