RE: size_question_encore

["André Joyal" <joyal.andre@uqam.ca>]

dear Marta,

I apologise, I had forgoten our conversation!
My memory was never good, and it is getting worst.

You wrote:

  >No, I am not thinking of the analogue of Steve Lack's model  
structure since, strictly speaking,
  >it has nothing to do with stacks. Comments to that effect (with  
which Steve agrees) are included
  >in the Bunge-Hermida paper. It was actually a surprise to discover  
that after trying to do what
  >you suggest and failing.

I disagree with your conclusion. I looked at your paper with Hermida.
We are not talking about the same model structure. The fibrations in  
2Cat(S) defined
by Steve Lack (your definition 7.1) are too weak when the topos S does  
not satify the axiom of choice.
Equivalently, his generating set of trivial cofibrations is too small.

Nobody has read my paper with Myles
<A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial  
groupoids, JPAA, Vol 89, 1993>.

Best,
André



-------- Message d'origine--------
De: Marta Bunge [mailto:martabunge@hotmail.com]
Date: sam. 09/07/2011 13:41
À: Joyal, André; edubuc@dm.uba.ar; categories@mta.ca
Objet : RE: RE : RE : categories: size_question_encore


Dear Andre,
You were indeed aware of my work and that with Pare on stacks since  
you are one of the few we thank for useful conversations! There were  
two ways to define stacks and one of them was your suggestion. One  
could say that one is expressed directly in terms of descent and the  
other in terms of weak equivalences. As it turns out, both are needed  
in my construction of the stack completion and similarly in the 2- 
dimensional case.
As for which method is preferable, I do not know. Whether one  
constructs stack completions for categories in a Grothendieck topos  
using the carving out from presheaf toposes (my method), or by means  
of a model structure (yours), one has to resort to the existence of a  
generating family to keep them small.
No, I am not thinking of the analogue of Steve Lack's model structure  
since, strictly speaking, it has nothing to do with stacks. Comments  
to that effect (with which Steve agrees) are included in the Bunge- 
Hermida paper. It was actually a surprise to discover that after  
trying to do what you suggest and failing. I attach my paper with  
Hermida in this connection. Section 3 makes clear what happens with  
Lack's model structure in dimension 1, and Section 7 considers the 2- 
dimensional analogue, also not suitable to get the 2-stack completion.
I really meant an extension of the Joyal-Tierney model structure.  
Thanks for pointing out Moerdijk's work, and your old one with  
Tierney. I will eventually look into those.
No need to respond to this.
Best regards, and many thanks,Marta



  > Subject: RE : RE : categories: size_question_encore
  > Date: Sat, 9 Jul 2011 12:18:45 -0400
  > From: joyal.andre@uqam.ca
  > To: martabunge@hotmail.com; edubuc@dm.uba.ar; categories@mta.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
  >

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