Koszul duality in algebraic topology - an historical perspective

TL;DR

This paper provides a historical overview of Koszul duality in algebraic topology, focusing on classical constructions and recent advances linking homotopy invariants to duality between algebraic structures.

Contribution

It offers an historical perspective on Koszul duality, highlighting classical constructions and recent developments connecting homotopy invariants with dual algebraic structures.

Findings

Survey of classical bar and cobar constructions

Explanation of Koszul duality of operads

Recent work linking homotopy invariants to algebraic duality

Abstract

We survey the topology which led to the original bar and cobar constructions, for both associative algebras and coalgebras and for Lie algebras and commutative coalgebras. These constructions are often viewed as part of the larger theory of Koszul duality of operads, so this survey is meant to offer an historical perspective on the most prominent cases of that theory. We also explain recent work which shows that Hopf/linking invariants for homotopy are at the heart of the duality between commutative algebras and Lie coalgebras.