Arrows, like Monads, are Monoids

Abstract

Monads are by now well-established as programming construct in functional languages. Recently, the notion of "Arrow" was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing that they correspond to monoids in suitable subcategories of bifunctors C op × C → C. This shows that, at a suitable level of abstraction, arrows are like monads -which are monoids in categories of functors C → C. Freyd categories have been introduced by Power and Robinson to model computational effects, well before Hughes' Arrows appeared. It is often claimed (informally) that Arrows are simply Freyd categories. We shall make this claim precise by showing how monoids in categories of bifunctors exactly correspond to Freyd categories.

Monads need not be endofunctors

“…The untyped λ-calculus syntax as has been identified as a monoid in [Fin, Set] by Fiore et al [14]. Heunen and Jacobs [18] have shown that arrows on C are actually monoids in the category [C op × C, Set] of endoprofunctors; Jacobs et al have proved the Freyd construction of [30] is, in a good sense, the Kleisli construction for arrows. Spivey [31] has studied a generalization of monads, which differs from ours, but is similar in spirit and related (see Conclusion).…”

Monads Need Not Be Endofunctors

“…The result of Heunen and Jacobs [18] about arrows being monoids follows as an instance of a generality about relative monads. Heunen and Jacobs [18] considered the special case of arrows and showed an arrow to be a monoid in [J op × J, Set] (the category of endoprofunctors on J) as a monoidal category, which is, of course, an equivalent statement.…”

The Essence of Dataflow Programming

“…Functional reactive programming is of course the same as dataflow programming, except that it is done by functional programmers rather than the traditional dataflow languages community. The exact relationship between Hughes's arrows and Power and Robinson's symmetric premonoidal categories has been established recently by Jacobs and colleagues [21,22].…”