Dear Andre, Sounds good, except that I do not quite see what the status of such an axiom is if added to ET + NNO. I do not doubt that any GT satisfies it. An alternative, which I had in mind and on which spoke about it in my 1 hour lecture at Calais 2008, is to add (ASC)^n for each n>0 to ET + NNO. How to state (ASC)^n in elementary terms relies on Lemma 8.2 of Bunge-Hermida, of which I have sketched a proof by induction. It says that for every epimorphism e from J to I in E, the induced n-functor (F_e)^n from the n-kernel of e to the discrete n-category on I is a weak n-equivalence n-functor (as defined in Def. 8.1). Intended definition: An n-category C in E is an n-stack if for every epimorphism e, C inverts (F_e) in the sense of n-equivalence. This is an elementary axiom for each n >0, and one could add as many as one needed for a specific purpose. In constructing the 2-stack completion, I only needed the n=1 case. So, for 3-stack completions, n = 1 and n=2 would suffice. Etc. Beyond that I cannot envisage any uses of n-stacks. But then, I am not a higher-order category person. In any case, this is getting interesting. Best regards, Marta
From: joyal.andre@uqam.ca To: martabunge@hotmail.com Subject: A new axiom? Date: Wed, 13 Jul 2011 10:32:30 -0400 CC: mshulman@ucsd.edu; david.roberts@adelaide.edu.au; categories@mta.ca
Dear Marta and all,
The category of simplicial objects in a Grothendieck topos admits a model structure in which the weak equivalences are the local weak homotopy equivalences and the cofibrations are the monomorphisms (I have described the model structure in my 1984 letter to Grothendieck). A higher stack can be defined to be a simplicial object which is globally homotopy equivalent to a fibrant object.
The notion of internal simplicial object can be defined in any elementary topos with natural number object. The local weak homotopy equivalences between simplicial objects can be defined internally.
It seems reasonable to introduce a new axiom for an elementary topos E (with natural number object). It may be called the Model Structure Axiom:
The MSA axiom: "The category of simplicial objects in E admits a model structure in which the weak equivalences are the local weak homotopy equivalences and the cofibrations are the monomorphisms"
A nice thing about this axiom is that it implies the existence of n- stack completion for every n. It also implies the existence of infinity-stack completion.
Best, André
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