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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]