What profunctors lack compared to arrows is the ability to compose them. If we add composition, will we get an arrow? MONOIDS This is exactly the question tackled in section 6 of “Notions of Computation as Monoids,” which unpacks a result from the (rather dense) “Categorical semantics for arrows”. “Notions” is a great paper because … Read more
In this code, they key piece is the delay 0 arrow in the rec block. To see how it works, it helps to think of values as varying over time and time as chopped into slices. I think of the slices as ‘days’. The rec block explains how each day’s computation works. It’s organised by … Read more
A free functor is left adjoint to a forgetful functor. For the adjunction you need to have the isomorphism (natural in x and y): (Free y :~> x) <-> (y :~> Forget x) In what category should this be? The forgetful functor forgets the Arrow instance, so it goes from the category of Arrow instances … Read more
Every applicative yields an arrow and every arrow yields an applicative, but they are not equivalent. If you have an arrow arr and a morphism arr a b it does not follow that you can generate a morphism arr o (a \to b) that replicates its functionality. Thus if you round trip through applicative you … Read more
Please take a look in http://www.cs.yale.edu/homes/hudak/CS429F04/AFPLectureNotes.pdf, which explains how Arrows work in FRP. 2-tuples are used in defining Arrows because it’s needed to represent an arrowized function taking 2 arguments. In FRP, constants and variables are often represented as arrows which ignores its “input”, e.g. twelve, eleven :: Arrow f => f p Int twelve … Read more
There are two excellent papers by Lindley, Wadler & Yallop (discussed at LTU here). The most important thing to understand is that there are more things which are arrows than there are things which are monads. Conversely, monads are strictly more powerful than arrows (the second paper above specifies precisely in which fashion). In particular, … Read more
I don’t know a tutorial, but I think it’s easiest to understand arrows if you look at some concrete examples. The biggest problem I had learning how to use arrows was that none of the tutorials or examples actually show how to use arrows, just how to compose them. So, with that in mind, here’s … Read more