Algebra + homotopy = operad

Summary

“If I could only understand the beautiful consequences following from the concise proposition d² = 0.”

—Henri Cartan

This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in algebra, geometry, topology, and mathematical physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.)

Introduction

Galois explained to us that operations acting on the solutions of algebraic equations are mathematical objects as well. The notion of an operad was created in order to have a well defined mathematical object which encodes “operations”. Its name is a portemanteau word, coming from the contraction of the words “operations” and “monad”, because an operad can be defined as a monad encoding operations. The introduction of this notion was prompted in the 60’s, by the necessity of working with higher operations made up of higher homotopies appearing in algebraic topology.

Algebra is the study of algebraic structures with respect to isomorphisms. Given two isomorphic vector spaces and one algebra structure on one of them, one can always define, by means of transfer, an algebra structure on the other space such that these two algebra structures become isomorphic.

Homotopical algebra is the study of algebraic structures with respect to quasiisomorphisms, i.e. morphisms of chain complexes which induce isomorphisms in homology only.