Algebra+Homotopy=Operad

TL;DR

This survey introduces operads as algebraic tools for encoding higher homotopies and explores their broad applications across algebra, geometry, topology, and physics.

Contribution

It provides an elementary introduction to operads and demonstrates their universal role in various mathematical and physical theories.

Findings

Operads encode higher homotopies effectively.

Applications span algebra, geometry, topology, and physics.

Accessible to students with basic category theory knowledge.

Abstract

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.)