Diagonals and A-Infinity Tensor Products

Extending work of Saneblidze–Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of 𝐴 ∞ -algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of 𝐴 ∞ -algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.