RE: size_question_encore

["André Joyal" <joyal.andre@uqam.ca>]

Dear Marta,

I thank you for your message and for drawing my attention to your work.
I apologise for not having refered to it.

  >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.

I guess you are thinking of having the analog of Steve Lack's model  
structure
but for the category of 2-categories internal to a Grothendieck topos S.
That is a good question. I am not aware that this has been done (but  
my knowledge of the litterature is lacunary).
You may also want to establish the analog of Moerdijk's model  
structure for the category of internal 2-groupoids.
I am confident that these model structure exists.
They should be closely related to a model structure on internal  
simplicial groupoids
<A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial  
groupoids, JPAA, Vol 89, 1993>.
And also related to the model structure on simplicial sheaves,  
described in my letter
to Grothendieck in 1984, but unfortunately not formally published.

Best regards,
Andre




-------- Message d'origine--------
De: Marta Bunge [mailto:martabunge@hotmail.com]
Date: ven. 08/07/2011 09:00
À: Joyal, André; edubuc@dm.uba.ar; categories@mta.ca
Objet : RE : categories: size_question_encore


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.
  >

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