In this talk, I will introduce the notion of n-morphisms between two $A_\infty$- algebras. These higher morphisms are such that 0-morphisms correspond to $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies. I will then prove that the set of higher morphisms between two $A_\infty$-algebras provide a satisfactory framework to study the higher algebra of $A_\infty$-algebras : this set defines in fact a simplicial set, which has the property of being a Kan complex.
I will then show how the combinatorics of n-morphisms between $A_\infty$-algebras are encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra. They are constructed from the standard simplices and multiplihedra, by lifting the Alexander-Whitney map to the level of simplices. The combinatorics arising in this context are moreover conveniently described in terms of overlapping partitions.
If time permits, I will finally explain how to realize this higher algebra of $A_\infty$-algebras in Morse theory, by counting the points of 0-dimensional moduli spaces of perturbed Morse gradient trees.
