The standard model structure on Cat is canonical

When I previously posted about a model structure on Cat, I called the usual (categorical equivalence, injective-on-objects, isofibration) model structure the “standard model structure”. The nLab calls it the canonical model structure on Cat, but I dislike that name because it always seems to suggest that there is some mechanical procedure for constructing it. As it turns out – there is!

Recall that Rezk’s classifying diagram for a (small) category 𝒞 is the bisimplicial set (= simplicial simplicial set) defined by N(𝒞)n, m = Fun([n] × I[m], 𝒞), where [n] is the standard n-simplex in Cat and I is the groupoid completion functor (= left adjoint of the inclusion GrpdCat). Rezk [2001] has shown that N : CatssSet is fully faithful and is homotopically conservative in the sense of sending categorical equivalences to degreewise weak homotopy equivalences and reflecting degreewise weak homotopy equivalences as categorical equivalences. As is usual with presheaf categories, N has a left adjoint, namely the colimit-preserving functor τ1 : ssSetCat that sends the bisimplicial set representing [n, m] to the category [n] × I[m].

I claim that the standard model structure on Cat is the model structure obtained by transferring the projective model structure on ssSet (i.e. degreewise weak homotopy equivalences and degreewise Kan fibrations) along the left adjoint τ1 : ssSetCat. We have already remarked that N preserves and reflects weak equivalences, so it is enough to show that N also preserves and reflects fibrations. This is fairly straightforward.


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