The Morita model structure on Cat

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 verify that left Bousfield localisation with respect to the inclusion of the free idempotent into the free split idempotent gives the desired model structure: the local objects are the Cauchy-complete categories, so the local equivalences are the Morita equivalences. (That Morita equivalences are local equivalences is easy; for the converse, consider a small category of sufficiently large sets.)

But one could (in principle) also establish the existence of the Morita model structure on Cat by hand: the fibrations are the isofibrations that also lift splittings of idempotents, i.e. the isofibrations with the right lifting property with respect to the inclusion mentioned above. Thus the Morita model structure on Cat is generated by the same cofibrations as the standard model structure on Cat and just one extra trivial cofibration.

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