More recently (Bunge-Hermida, MakkaiFest, 2011), we have carried out
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
Dear Marta, I thank you for your message and for drawing my attention to your work. I apologise for not having refered to it. the 2-analogue of the 1-dimensional 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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