Homotopy Unital A∞-morphisms with Several Entries

We show that morphisms from n homotopy unital A∞-algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A ∞, whose algebras are homotopy unital A∞-algebras. The operad A∞ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A ∞-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. The present work is a sequel to [Lyu15]. We use freely notations and notions from the previous article. There polymodule cooperads were defined and an example of A∞-polymodule cooperad F was given. Here we describe three more examples of polymodule cooperads: an A∞-polymodule cooperad F, an A hu ∞-polymodule cooperad F hu and A ∞-polymodule cooperad F. Here A∞ (resp. A hu ∞) is an operad of conventional (resp. homotopy unital) A∞-algebras, and A∞, A hu ∞ are their signless versions. Also F (resp. F ) is a signless version of F (resp. F). We develop the idea of “isomorphism” of operads and polymodule cooperads changing degrees. Operads A∞ and A∞, A hu ∞ and A hu ∞, and polymodule cooperads F and F, F hu and F are “isomorphic” in this sense. Both categories of dg-operads and of polymodule dg-cooperads have a model structure. It is known that the dg-operad A∞ (resp. A hu ∞) is a cofibrant resolution of the dg-operad As (resp. As1 ) of non-unital (resp. unital) associative dg-algebras. We show that the polymodule cooperad F (resp. F) is a cofibrant resolution of the polymodule cooperad responsible for morphisms and composition in the multicategory of non-unital (resp. unital) associative dg-algebras. Polymodule cooperads F , F (resp. F , F) are means to represent morphisms and their composition in multicategories of conventional (resp. homotopy unital) A∞-algebras or A∞-algebras. The composition is recovered via convolution of polymodule cooperad and a lax Cat-multifunctor Hom built from dg-modules. Verification that changing degrees does not lead out of polymodule cooperads is straightforward but lengthy.