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.
Keywords: A∞-algebras, A∞-coalgebras, monads, comonads
2020 MSC: 18C15, 18C20, 18G70
Theory and Applications of Categories, Vol. 45, 2026, No. 48, pp 2064-2114.
Published 2026-09-23.
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