3 - The Homotopy Theory of ∞-Categories

Summary

This chapter defines the homotopy theory of ∞-categories: the goal is to have a robust notion of `invariance under equivalences of ∞-categories’. Both because it has a special place in the tradition of homotopy theory, and because it plays a central role in the theory of ∞-categories, we first construct the Kan-Quillen model category structure on the category of simplicial sets, which determines the classical homotopy theory of Kan complexes. We then proceed to the construction of the Joyal model structure, which determines the homotopy theory of ∞-categories. Related tasks consist in proving that ∞-categories exactly are fibrant objects of the Joyal model structure, identifying the classes of Kan complexes and of ∞-groupoids, constructing ∞-categories of functors, and proving that a natural transformation is invertible if and only if it is object-wise invertible. We also construct Serre’s long exact sequence associated to a Kan fibration, and prove the related result that equivalences of ∞-categories precisely are fully faithful and essentially surjective functors.