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

TL;DR

This paper develops a categorical framework using monads and comonads to study representation categories of $A__$-algebras and coalgebras, exploring their relations and introducing $A__$-contramodules.

Contribution

It introduces a unified categorical approach to $A__$-algebras and coalgebras, relating their modules and comodules via rational pairings and defining $A__$-contramodules.

Findings

Established a formalism relating $A__$-modules and comodules.

Connected $A__$-comodules and modules through rational pairings.

Introduced the concept of $A__$-contramodules.

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 $C$.