In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms corresponds to A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. I will then prove that the set of higher morphisms between two A-infinity algebras provide a satisfactory framework to study the higher algebra of A-infinity algebras : this set defines in fact a simplicial set, which has the property of being a Kan complex whose homotopy groups can be explicitly computed.
If time permits, I will finally show how the combinatorics of n-morphisms between A-infinity 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.