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A short course on \(\infty \)-categories. (English) Zbl 1473.55014

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 549-617 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: This chapter discusses non-technical account of some ideas in the theory of \(\infty\)-categories, as originally introduced by Boardman-Vogt in their study of homotopy-invariant algebraic structures. \(\infty\)-categories have applications in many areas of pure mathematics. The theory of \(\infty\)-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 \(\infty\)-categories, a class of \(\infty\)-categories having very good formal properties. The theory of presentable \(\infty\)-categories has two precursors; locally presentable categories in the classical context as well as combinatorial model categories in the homotopical framework. Stable \(\infty\)-categories are an enhancement of triangulated categories. The chapter discusses the stabilization of nice \(\infty\)-categories which is obtained by passing to internal spectrum objects.
For the entire collection see [Zbl 1468.55001].

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U40 Topological categories, foundations of homotopy theory
18N99 Higher categories and homotopical algebra

Citations:

Zbl 1468.55001
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