Menu
From functors that are not applicatives:
A type constructor which is a Functor but not an Applicative. A simple example is a pair:
instance Functor ((,) r) where fmap f (x,y) = (x, f y)But there is no way how to define its
Applicativeinstance without imposing additional restrictions onr. In particular, there is no way how to definepure :: a -> (r, a)for an arbitraryr.
Here, pure fails to be definable for all types at once; however, for any concrete type T, one can make ((,) T) an applicative.
Question: Is there an example of a concrete functor (i.e., no type variables involved) that is a functor but not an applicative?
836
Applicative functors aren't explicitly modelled in F#, but they're easy enough to add if you need them. F# doesn't have typeclasses, so implementing applicative functors tend to be more on a case-by-case basis.
If we want to make a type constructor an instance of Functor, it has to have a kind of * -> *, which means that it has to take exactly one concrete type as a type parameter. For example, Maybe can be made an instance because it takes one type parameter to produce a concrete type, like Maybe Int or Maybe String.
As a variation, an applicative functor maps objects in an 'elevated world' using functions from the same 'elevated world'. In Haskell, an applicative functor is defined like this:
So what's a functor F: BG → BH F: B G → B H? It's precisely a group homomorphism from G G to H H! So in this example, a functor is just a function (which happens to be compatible with the group structure).
I don't have 50 reputation to comment here, so I'll try to do it as an answer:
however, for any concrete type T, one can make ((,) T) an applicative.
...
There's a theorem in mathematics that any collection with at least 2 elements can be made into a monoid. So for any concrete type T, it could in principle be made a member of Monoid, and then could in principle be made Applicative. What's wrong with this reasoning?
What about the tuple from the uninhabited type? (,) Void
It is a Functor,right?
Could you derive Applicative for it? How would pure be implemented?
169
There is nice example in reactive-banana library.
It features Event a types, which represents a single simultaneous event in time (think, an impulse), and Behavior a type, which represents a value that is available in any moment (for instance, emitting a value from the last event).
Behavior is an Applicative, because you can combine two of them - they have a value in any point of time.
Event, though, is a Functor only, because you can't combine them. Given two Events you can't be sure they will happen simultaneosly.
26
If you love us? You can donate to us via Paypal or buy me a coffee so we can maintain and grow! Thank you!
Donate Us With
