Canonical Gradings of Monads

TL;DR

This paper introduces a canonical way to grade monoids in monoidal categories relative to a class of morphisms, with applications to graded monads and computational effects, providing a unified framework for understanding algebraic structures.

Contribution

It defines a canonical grading of monoids in monoidal categories relative to a class of morphisms, especially when that class forms a factorization system, and applies this to graded monads and algebraic operations.

Findings

Canonical gradings exist for monoids under certain conditions.

Characterization of canonical gradings for various monads.

Application to models of computational effects.

Abstract

We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.