Higher algebra of A-infinity algebras in Morse theory

In this short talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.

Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.