Relating Idioms, Arrows and Monads from Monoidal Adjunctions

TL;DR

This paper explores the deep categorical relationships between monads, arrows, and idioms using monoidal categories and adjunctions, providing a unified theoretical framework for understanding these computational notions.

Contribution

It introduces a categorical model linking monads, arrows, and idioms via monoidal adjunctions, extending prior characterizations with a unified formal approach.

Findings

Categorical connections between monads, arrows, and idioms established.

Development of a categorical version of Lindley, Yallop, and Wadler's characterization.

Enhanced understanding of the structural relationships in computational effects.

Abstract

We revisit once again the connection between three notions of computation: monads, arrows and idioms (also called applicative functors). We employ monoidal categories of finitary functors and profunctors on finite sets as models of these notions of computation, and develop the connections between them through adjunctions. As a result, we obtain a categorical version of Lindley, Yallop and Wadler's characterisation of monads and idioms as arrows satisfying an isomorphism.