RE: A new axiom?

[Marta Bunge <martabunge@hotmail.com>]

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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