ABSTRACT

This chapter discusses non-technical account of some ideas in the theory of ∞-categories, as originally introduced by Boardman–Vogt in their study of homotopy-invariant algebraic structures. ∞-categories have applications in many areas of pure mathematics. The theory of ∞-categories should really be thought of as homotopy coherent category theory. In particular, the basic notion of a functor is to model the idea of having a homotopy coherent diagram. Homotopy coherent category theory has quite some history and references include. The chapter discusses a short introduction to presentable ∞-categories, a class of ∞-categories having very good formal properties. The theory of presentable ∞-categories has two precursors; locally presentable categories in the classical context as well as combinatorial model categories in the homotopical framework. Stable ∞-categories are an enhancement of triangulated categories. The chapter discusses the stabilization of nice ∞-categories which is obtained by passing to internal spectrum objects.