Operads and Operadic Algebras in Homotopy Theory

TL;DR

This expository chapter provides an overview of operads in homotopy theory, covering foundational definitions, structural properties, model structures, and applications to loop spaces and homotopy types.

Contribution

It offers a comprehensive exposition of the basic topics in the homotopy theory of operadic algebras, including new insights into applications and comparison techniques.

Findings

Operads and their algebras are fundamental in homotopy theory.

Model structures facilitate the study of algebra categories over operads.

Applications include $n$-fold loop spaces and algebraic models of homotopy types.

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).