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Thomason model structure

Idea

By regarding a simplicial set as an object in the standard model structure on simplicial sets, one effectively identifies it (up to weak equivalence) with that ∞-groupoid that it presents under Kan fibrant replacement.

If the original simplicial set is the nerve of a category, the corresponding Kan fibrant replacement is something like the $\mathrm{\infty }$-groupoidification of that category.

This way each ordinary category models an ∞-groupoid. The Thomason model category structure on Cat exhibits this: in this model category a morphism between two categories is a weak equivalence, precisely if it induces a weak equivalence of the corresponding $\mathrm{\infty }$-groupoids.

It turns out the Thomason model structure on Cat is Quillen equivalent to the standard model structure on simplicial sets.

This is remarkable, in that it says that every homotopy type, i. e. every weak equivalence class of $\mathrm{\infty }$-groupoids, is obtained by $\mathrm{\infty }$-groupoidifying just categories.

In fact, every cofibrant object in this structure is a poset. Since every object in a model category is weakly equivalent to a cofibrant one, this means that even the nerves of just posets are sufficient to model all homotopy types.

This is a rather curious aspect of the Thomason model on Cat: it does not really have anything intrisically to do with categories, but rather uses these as a way to present ∞-groupoids. It particular, it does not see the equivalences of categories. There is a different model structure on Cat in which weak equivalences are the “true” weak equivalences of categories (not of anything constructed from them). This is called the folk model structure on Cat.

References

The original reference is

• R. W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21, no. 3 (1980), pp. 305–324. MR0591388 (82b:18005) numdam scan

A correction to this article was made in

• Denis-Charles Cisinski, Les morphisme de Dwyer ne sont pas stables par rétractes, Cahiers Topologie et Géom. Différrentielle Catégoriques, 40 no. 3 (1999), pp. 227–231. (Numdam)

There it was clarified that every cofibrant object in the Thomason model structure is a poset.

A useful review of the Thomason model structure and a generalization of the model structure to n-fold categories is given in

• Thomas M. Fiore, Simona Paoli, A Thomason Model Structure on the Category of Small n-fold Categories (arXivhomason Model Structure on the Category of Small n-fold Categories))