What Makes a Strong Monad?

TL;DR

This paper reviews and extends the theory of strong monads, providing new characterizations and conditions for their existence and uniqueness, with implications for semantics of effectful languages.

Contribution

It introduces new equivalences and conditions for strong monads and functors, and explores strength from multiple categorical perspectives.

Findings

Presented equivalent characterizations of strong functors and monads.

Provided conditions for existence and uniqueness of strengths.

Analyzed strength from actions, enrichment, and powering perspectives.

Abstract

Strong monads are important for several applications, in particular, in the denotational semantics of effectful languages, where strength is needed to sequence computations that have free variables. Strength is non-trivial: it can be difficult to determine whether a monad has any strength at all, and monads can be strong in multiple ways. We therefore review some of the most important known facts about strength and prove some new ones. In particular, we present a number of equivalent characterizations of strong functor and strong monad, and give some conditions that guarantee existence or uniqueness of strengths. We look at strength from three different perspectives: actions of a monoidal category V, enrichment over V, and powering over V. We are primarily motivated by semantics of effects, but the results are also useful in other contexts.