A duality between monads and monadic morphisms

TL;DR

This paper develops a duality between monads and monadic morphisms in higher categories, enabling the transfer of algebraic structures and the construction of tensor products for various algebraic objects.

Contribution

It introduces a duality framework in $( abla,2)$-categories that unifies algebraic dualities and extends tensor product constructions to complex algebraic structures.

Findings

Unified duality between monads and monadic morphisms in higher categories

Construction of tensor products for algebras over monads and operads

Enrichment of the $ abla$-category of operadic algebras

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.