Homotopy Operadic Algebras

Abstract

When a chain complex is equipped with some compatible algebraic structure, its homology inherits this algebraic structure. The purpose of this chapter is to show that there is some hidden algebraic structure behind the scene. More precisely if the chain complex contains a smaller chain complex, which is a deformation retract, then there is a finer algebraic structure on this small complex. Moreover, the small complex with this new algebraic structure is homotopy equivalent to the starting data. The operadic framework with the Koszul duality theory enables us to state explicitly this transfer of structure result. In this chapter, we give four equivalent definitions for the notion of homotopy -algebra. We also introduce the notion of ∞-morphisms.