  We construct an A-infinity structure on the tensor product of two A-infinity
algebras by using the simplicial decomposition of the Stasheff polytope. The
key point is the construction of an operad AA-infinity based on the simplicial
Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and
the operad A-infinity is a retract by deformation of it. We compare these
constructions with analogous constructions due to Saneblidze-Umble and
Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff
polytope.
