#
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.

http://www.tac.mta.ca/tac/volumes/30/46/30-46.pdf

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