Abstract

The aim of this short course is to give a non-technical account of some ideas in the theory of ∞-categories (a.k.a. quasi-categories, inner Kan complexes, weak Kan complexes, Boardman complexes, or quategories), as originally introduced by Boardman–Vogt [23, p. 102] in their study of homotopy-invariant algebraic structures. Recently, ∞-categories have been studied intensively by Joyal [67, 68, 69], Lurie [82, 79, 80, 81], and others (this includes the Riehl–Verity program which was started in [97]). ∞-categories have applications in many areas of pure mathematics (and some of them are taken up in various chapters of this Handbook). In this chapter we are more modest. We simply try to emphasize the philosophy and some of the main ideas of ∞-category theory, and we sketch the lines along which the theory is developed. In particular, this means that there is no claim of originality.