A duality between monads and monadic morphisms

Abstract

We establish a duality between monads and monadic morphisms in any $(\infty,2)$-category and characterize monadic morphisms in a wide class of examples. This duality unifies several dualities between algebraic structures and their representations, and provides a general mechanism for transferring structure from a monad to its $\infty$-category of algebras. This transfer of structure yields uniform constructions of tensor products for algebras over lax symmetric monoidal and oplax symmetric monoidal monads, extending classical tensor products for modules and operadic algebras. Using this framework, we construct a relative tensor product for algebras over lax monoidal monads, a tensor product for algebras over Hopf $\infty$-operads and equip the $\infty$-category of operadic algebras with canonical enrichment.