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