A∞-morphisms with Several Entries

We show that morphisms from n A∞-algebras to a single one are maps over an operad module with n+ 1 commuting actions of the operad A∞, whose algebras are conventional A∞-algebras. The composition of A∞-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. Under these notions lie two examples of Cat-operads: that of graded modules and of complexes. It is well-known that operads play a prominent part in the study of A∞-algebras. In particular, A∞-algebras in the conventional sense [Sta63] are algebras over the dg-operad A∞, a resolution (a cofibrant replacement) of the dg-operad As of associative non-unital dg-algebras. Here and elsewhere in this article an operad is a non-symmetric operad unless it is called symmetric. More generally, there is an approach to homotopy algebras over an operad O as algebras over cofibrant dg-resolution P of this operad [Mar00]. The question arises about morphisms of homotopy algebras, the class of morphisms of P-algebras being too narrow. A possible solution [Lyu11] is to consider a bimodule F over P and to define homotopy morphisms as “maps” (analogue of “algebras”) over F , a cofibrant dg-resolution of the bimodule describing O-algebra morphisms. Composition of homotopy morphisms arises as convolution of a coalgebra structure on F and the algebra hom. Practically the same notion called co-ring over an operad was a starting point of research by Hess, Parent and Scott [HPS05]. In the present article we study multicategory of homotopy algebras (when there is one, e.g. of A∞-algebras). Let V be a bicomplete closed symmetric monoidal category. We are mostly interested in the category of complexes V = dg. A related choice is the category of graded modules V = gr. We use also the closed category of essentially small categories. For any V-multicategory C and any object X of C we can produce a V-operad End X, (End X)(n) = C(X, . . . , X } {{ } n ;X) = C(X;X). Similarly given objects X1, . . . , Xn, Y of a symmetric V-multicategory C we can consider the operad V-polymodule (the n ∧ 1-operad module) P = hom(X1, . . . , Xn;Y ). By definition, P = (P(j))j∈Nn , where P(j) = C ( ( i Xi) n i=1;Y ) (the argument Xi is repeated j i times). The collection P carries commuting left actions of operads End Xi, i ∈ n, and right action of End Y . Formalising properties of these actions we write down the definition of an n ∧ 1-operad module over operads A1, . . . , An, B. When a collection (Fn)n>0 of n ∧ 1-operad modules over an operad A is given we may consider a multicategory-to-be whose objects are A-algeReceived by the editors 2011-02-25 and, in revised form, 2015-10-27. Transmitted by James Stasheff. Published on 2015-11-05. 2010 Mathematics Subject Classification: 18D50, 18D05, 18G35.