Operads and Operadic Algebras in Homotopy Theory

Abstract

This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads, the definition of algebras over operads, structural aspects of categories of algebras over operads, model structures on algebra categories, and comparison of algebra categories when changing operad or underlying category. In addition, it includes two applications of the theory: The original application to $n$-fold loop spaces, and an application to algebraic models of homotopy types (chosen purely on the basis of author bias).