The formal theory of monoidal monads

TL;DR

This paper provides a formal 3-categorical explanation for the natural monoidal structures on categories of Kleisli and Eilenberg-Moore algebras for (lax and oplax) monoidal monads, revealing a deep structural isomorphism.

Contribution

It introduces a purely formal, 3-categorical framework explaining the monoidal structures on algebra categories for monoidal monads, generalizing to various contexts.

Findings

Establishes an isomorphism between lax monoidal monads and monoidal objects with oplax morphisms.

Shows the natural monoidal structures on Kleisli and Eilenberg-Moore categories.

Highlights the broad applicability of the phenomenon across different categorical settings.

Abstract

We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. As we explain at the end of the paper a similar phenomenon occurs in many other situations.