In this 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. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being an infinity-groupoid. The combinatorics of n-morphisms are moreover 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 recall how the Morse cochain complex of a Morse function on a smooth compact manifold, can be endowed with an A-infinity-algebra structure by counting moduli spaces of perturbed Morse gradient trees. Given two Morse functions, I will finally explain how to realize this higher algebra of A-infinity algebras in Morse theory, by constructing n-morphisms between their respective Morse cochain complexes.