Koszul duality in algebraic topology

Abstract

The most prevalent examples of Koszul duality of operads are the self-duality of the associative operad and the duality between the Lie and commutative operads. At the level of algebras and coalgebras, the former duality was first noticed as such by Moore, as announced in his ICM talk at Nice (Moore in Actes du Congres International des Mathematiciens, Tome 1. Gauthier-Villars, Paris, pp. 335–339, 1971). This particular duality has typically been called Moore duality, and some prefer to call the general phenomenon Koszul–Moore duality. The second duality at the level of algebras was realized in the seminal work of Quillen on rational homotopy theory (Quillen in Ann Math 90(2):205–295, 1969). Our aim in these notes based on our talk at the Luminy workshop on Operads in 2009 is to try to provide some historical, topological context for these two classical algebraic dualities. We first review the original cobar and bar constructions used to study loop spaces and classifying spaces, emphasizing the less-familiar geometry of the cobar construction. Then, after some elementary topology, we state duality between bar and cobar complexes in that setting. Before explaining Quillen’s work, we also share some other ideas—calculations of Cartan–Serre and Milnor–Moore and philosophy of Eckmann–Hilton—which may have influenced him. After stating Quillen’s duality, we share some recent work which relates these constructions to geometry through Hopf invariants and in particular linking phenomena.