An introduction to higher categorical algebra

Abstract

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the derived $\infty$-category of a commutative ring, before turning to our main example, the $\infty$-category of spectra. We then go on to consider ring spectra and their $\infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and deformation theory.