4 - Presheaves: Externally

Summary

We study the analogues of Grothendieck fibrations with discrete fibres in the setting of ∞-categories, which is one of the ways to describe presheaves with values in the category of ∞-groupoids: these are the right fibrations. We construct Joyal’s contravariant model category structures, whose fibrant objects precisely are these right fibrations. There is also a dual notion of left fibrations, which are the fibrant objects of the covariant model structures. We prove an external version of the Yoneda lemma in this context: we compute the fibres of right fibrations as certain mapping spaces. We introduce final functors as well as final objects, and extend Grothendieck’s theory of smooth functors and of proper functors to ∞-categories. We prove Grothendieck’s base change formulas (which are instances of the Beck-Chevalley property), and explain how to deduce from these (generalisations of) Quillen’s theorem A and theorem B.