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 $A_{\infty }$-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 $C$.

arXiv commentPaper significantly expanded with two new sections