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
dear Marta, I apologise, I had forgoten our conversation! My memory was never good, and it is getting worst. You wrote: that after trying to do what
you suggest and failing.
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
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 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]