Notions of Computation as Monoids

TL;DR

This paper unifies various notions of computation—monads, applicative functors, and arrows—by modeling them as monoids within a monoidal category, enabling new insights and constructions across these concepts.

Contribution

It introduces a unified monoidal category framework for monads, applicative functors, and arrows, revealing their structural similarities and enabling generalized constructions.

Findings

Unified framework for computation notions as monoids

Derived useful constructions like free monoids and Cayley representations

Clarified relationships between different computational abstractions

Abstract

There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as monoids in a monoidal category. We demonstrate that at this level of abstraction one can obtain useful results which can be instantiated to the different notions of computation. In particular, we show how free constructions and Cayley representations for monoids translate into useful constructions for monads, applicative functors, and arrows. Moreover, the uniform presentation of all three notions helps in the analysis of the relation between them.