Notions of computation as monoids

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.

From monoids to near-semirings

“…Hughes uses it to optimise list concatenation [9] and Voigtländer [22] uses the codensity monad transformer to optimise monadic computations. Rivas and Jaskelioff [17] show that these two optimisations are instances of the Cayley representation for monoids in a generalised setting, and extend it to applicative functors. Our work extends this representation to include the additional operators present in non-deterministic computations by moving from generalised monoids to generalised near-semirings.…”

“…The monoid of endomorphisms over a set X is (X → X, •, id), where • is function composition and id is the identity function. Every monoid has an embedding into the monoid of endomorphisms over its carrier set, a result usually known as Cayley's theorem for monoids [17]. Proof.…”

“…Rivas and Jaskelioff [17] show that the free and Cayley constructions carry over from monoids to generalised monoids. Here we investigate whether the same is true for generalised nearsemirings.…”

“…As first indicated by Day [4] and later by Rivas and Jaskelioff [17], there are different equivalent representations of the Day convolution. We pick here the one that directly determines the familiar Applicative type class.…”

“…The free monad is just the free monoid in that category and the codensity transformation arises as the Cayley representation of that monoid [17]. Moreover, useful generality is gained by this approach, as not only monads are monoids, but also applicative functors and arrows.…”