Commutative graded monads

TL;DR

This paper introduces the concept of commutative graded monads and proves that their Kleisli algebras naturally inherit a canonical monoidal structure, extending known results for monoidal monads.

Contribution

It defines commutative graded monads and provides a strict two-categorical proof of their Kleisli algebras' monoidal structure, generalizing previous monad results.

Findings

Kleisli algebras for commutative graded monads have a canonical monoidal structure

The result generalizes the monoidal structure for monoidal monads to graded cases

Provides a strict two-categorical proof of the main theorem

Abstract

It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for such monads equally carry a canonical monoidal structure which reduce to the monoidal monad case when the commutative graded monad in question is graded over the trivial monoidal category.