Re: Simplicial versus (cubical with connections)

["Jonathan CHICHE 齊正航" <chichejonathan@gmail.com>]

There is another way to state that the cube category with connections  
behaves "as well as" the simplex category. Both are strict test  
categories (as defined by Grothendieck in "Pursuing Stacks"). See  
http://www.math.jussieu.fr/~maltsin/ps/cubique.pdf. Without  
connections, the cube category is a test category, but not a strict  
one, so that the product in the cube category does not reflect the  
product of homotopy types. This issue vanishes if connections are  
allowed. Grothendieck explicitly wrote in "Pursuing Stacks" that he  
believed that, homotopically speaking, any strict test category was  
"as good as" the simplex category. For instance, he conjectured there  
that an analog of the Dold-Kan correspondence (which he called Dold- 
Puppe) holds for every strict test category. (As regards the  
existence of a Quillen model structure the cofibrations of which are  
monomorphisms on the presheaf category, and so on, see the  
introduction to Astérisque 301 by Maltsiniotis and Astérisque 308 by  
Cisinski.)

Best regards,

Jonathan Chiche

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