The Morita model structure on Cat

Dreaming of mathematics

There is a well-known model structure on Cat where the weak equivalences are the categorical equivalences, i.e. the functors that are fully faithful and essentially surjective on objects, the cofibrations are the functors that are injective on objects, and the fibrations are the isofibrations, i.e. the functors that lift isomorphisms. Let us say that a functor f : 𝒞 → 𝒟 is a Morita equivalence if the induced functor f* : [𝒟op, Set] → [𝒞op, Set] is a categorical equivalence. Clearly, every categorical equivalence is a Morita equivalence. Does the left Bousfield localisation of Cat with respect to Morita equivalences exist?

The standard model structure on Cat is combinatorial and simplicial, and all objects are cofibrant, so the model structure is also left proper. Thus, we may apply Smith’s theorem on the existence of left Bousfield localisations. It is a straightforward exercise to…

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