# Homotopy unital $A_\infty$-morphisms with several entries

## Volodymyr Lyubashenko

We show that morphisms from n homotopy unital $A_\infty$-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad $A_\infty^\hu$, whose algebras are homotopy unital $A_\infty$-algebras. The operad $A_\infty$ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of $A_\infty^\hu$\n-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures.

Keywords: $A_\infty$-algebra, $A_\infty$-morphism, multicategory, multifunctor, operad, operad module, polymodule cooperad

2010 MSC: 18D50, 18D05, 18G35

Theory and Applications of Categories, Vol. 30, 2015, No. 46, pp 1552-1623.

Published 2015-11-05.

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