Monads need not be endofunctors

TL;DR

This paper generalizes monads to relative monads, which operate between different categories, and demonstrates their properties, constructions, and connections to other categorical concepts like arrows and monoids.

Contribution

It introduces the concept of relative monads, extending the theory of monads to functors between different categories, with applications to various structures.

Findings

Relative monads generalize traditional monads.

Kleisli and Eilenberg-Moore constructions extend to relative monads.

Relative monads form monoids in functor categories and relate to monads and arrows.

Abstract

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads.