On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras

Abstract

In this paper, we use the language of monads, comonads and Eilenberg-Moore categories to describe a categorical framework for A∞-algebras and A∞-coalgebras, as well as A∞-modules and A∞-comodules over them respectively. The resulting formalism leads us to investigate relations between representation categories of A∞algebras and A∞-coalgebras. In particular, we relate A∞-comodules and A∞-modules by considering a rational pairing between an A∞-coalgebra C and an A∞-algebra A. The categorical framework also motivates us to introduce A∞-contramodules over an A∞-coalgebra C.