A new axiom?

["André Joyal" <joyal.andre@uqam.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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