7 - Homotopical Algebra

Summary

This final chapter revisits homotopical algebra from scratch, with the tools of ∞-category theory. This is all about localisations of ∞-categories and their interactions with (co)limits. We first define localisation by a universal property and explain why this exists. A fundamental tool here is the calculus of fractions, which we introduce and study with the greatest level of generality, but always with the aim of reaching results and properties which can be used in practice. In particular, we explain why the presence of nice classes of fibrations or of cofibrations, such as in model categories, do indeed provide examples. We never use any explicit model of localisations, but rather use the theory of Kan extensions. In particular, we construct derived functors as Kan extensions. We prove coherence results expressing how the formation of ∞-categories of functors can be compatible with localisations. We show that the ∞-category of ∞-groupoids is indeed the localisation of the category of simplicial sets by weak homotopy equivalences (thus proving a conjecture of Nichols-Barrer), and explain the basics of the theory of presentable ∞-categories.