Siddharth Bhat

27 May 2021

â€¢

5 min read

Today, we're going to be taking a look at representable functors in Haskell. But first, some incantations:

```
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE InstanceSigs #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE DataKinds #-}
```

We recall that a natural transformation between two functors is of the type:

```
type Nat f g = forall x. f x -> g x
```

Intuitively, a natural transformation between two functors is a function between functors.
for any `f x`

, we have a recipe on how to convert it into `g x`

. An example is
the natural transformation from `Maybe`

to` List`

:

```
-- maybe2list :: Nat Maybe List
maybe2list :: forall a. Maybe a -> List a
maybe2list Nothing = []
maybe2list (Just a) = [a]
```

Also recall that the function type `a -> b`

is also called as `Hom(a, b)`

or `Hom a b`

.
To stay consistent with category theory terminology, we're going to create a type
synonym:

```
type Hom a b = a -> b
```

Now, a functor `f`

is saif to be *representable* if it is the same to `Hom a`

for some
`a`

. Let's meditate on this a little bit. So, we have:

- A functor
`f`

- An object
`a`

, such that`f`

is the same as`Hom a`

- A way to convert
`f`

into`Hom a`

. That is, we have a natural transformation from`f`

to`Hom a`

. This is the same as asking for`Nat f (Hom a)`

, or`forall x. f x -> Hom a x`

, which is the same as`forall x. f x -> (a -> x)`

. - We also want a way to convert
`Hom a`

into`f`

, since`f`

is "the same" as`Hom a`

. This asks for a`Nat (Hom a) f`

, or a`forall x. Hom a x -> f x`

or`forall x. (a -> x) -> f x`

.

Intuitively, this is telling us that there's some object `a`

such that `f x`

is always equal
to `a -> x`

for any choice of `x`

. So we get an equivalence beween some "data" `f x`

, and
a function `a -> x`

. How do we encode this in haskell? What's an example? Let's see!

```
-- Functor f is representable iff isomorphic to SOME hom functor
-- f: C -> Set -- set to be isomorphic to hom functor
-- âˆƒdâˆˆC, f ~= Hom(d, -)
-- d is called as the "representing object" of functor f
class Functor f => Representable f where
type family RepresentingObj f :: *
-- fwd :: Nat f (Hom d)
-- fwd :: forall x. f x -> (Hom d) x
fwd :: forall x. f x -> (RepresentingObj f -> x)
bwd :: forall x. (RepresentingObj f -> x) -> f x -- memoization
```

The idea is encoded as shown above, where we say that a functor `f`

is `Representable`

,
if (1) it has a `RepresentingObj f`

which represents it, and (2) there are functions
to convert `f x`

into `(RepresentingObj f -> x)`

.

Consider the data type of infinite streams:

```
-- | always infinite stream
data Stream a = SCons a (Stream a)
```

for example, the infinite stream of zeros is an inhabitant of this:

```
zeros :: Stream Int
zeros = SCons 0 zeros
```

as is the stream of all natural numbers:

```
nats :: Stream Int
nats = let go n = SCons n (go (n+1)) in go 0
```

This is a functor, similar to how list is a functor:

```
instance Functor Stream where
fmap f (SCons x xs) = SCons (f x) (fmap f xs)
```

More interestingly, it is a *representable functor*,
since we can think of a `Stream a`

as a function `(Integer -> a)`

,
as we can index a stream at an arbitrary integer. Similarly, given
an `(Integer -> a)`

, we can build a `Stream a`

from it by memoizing
the function `(Integer -> a)`

into the data structure `Stream a`

:

```
instance Representable Stream where
type RepresentingObj Stream = Integer
-- fwd:: Stream a -> (Integer -> a)
fwd (SCons x _) 0 = x
fwd (SCons _ xs) n = fwd xs (n-1)
-- bwd:: (Integer -> a) -> Stream a
bwd int2x = go 0 where
go n = SCons (int2x n) (go (n+1))
```

Clearly, these are invertible, and thus we witness the equality between `Stream a`

and `Integer -> a`

Lists are an example of a type that is *not* a representable functor,
because different lists can contain a different numbers of elements. so,
for example, we can't use `Integer`

as the representing object (ie, we can't
write all lists as functions `Natural -> a`

, because the empty list
contains *no* elements, so it would have to be a function `Empty -> a`

,
while the list containing one element has a *single* element, so it
would be the function `() -> a`

, the list containing two elements
has *two* elements, which would be the function `Bool -> a`

, and so on.

Thus, it's impossible for us to find a representing object for a list!

Since we saw that lists aren't representable, because we can query "outside their domain", the question naturally becomes, can we build a data structure such that the query remains within the domain? The answer is yes, with much trickery!

First, we'll need an encoding of natural numbers as `0`

and a successor
function (intuitively, `+1`

), where we represent a number `n`

as adding `1`

`n`

times to `0`

:

```
data NAT = ZERO | SUCC NAT deriving(Eq)
```

so, zero will be `ZERO`

, one will be `SUCC ZERO`

, two will be `SUCC (SUCC ZERO)`

,
and so on. We implement `countNAT`

to convert naturals to integers:

```
countNAT :: NAT -> Int
countNAT ZERO = 0
countNAT (SUCC n) = countNAT n + 1
instance Show NAT where show n = show (countNAT n)
```

Next, we create a type `FinSet (n :: NAT) :: *`

, where `FinSet n`

contains EXACTLY `n`

members. Yes, that statement has a lot to unpack:

The type `FinSet n`

has `n :: NAT`

. So we can ask for `FinSet ZERO`

, `FinSet (SUCC ZERO)`

, `FinSet (SUCC (SUCC ZERO))`

and so on. Notice that in reality,
`ZERO`

, `SUCC ZERO`

and so on are *values* in Haskell. However, we are
creating a type `FinSet n`

which depends on *values*. To enable this, we use
the `DataKinds`

extension. Such types which can dependent on values fall under
a class of type systems known as dependent types.

Let's first see the definition, and then try and gain some intuition of how this works:

```
data FinSet (n :: NAT) where
Intros :: FinSet (SUCC n)
Lift :: FinSet n -> FinSet (SUCC n)
```

Why does this work? I claim that `FinSet n`

only has `n`

numbers. Well, let's see.
Can I ever crete a member of `FinSet ZERO`

? No, both the constructors `Intros`

and `Lift`

create `FinSet (Succ _)`

for some `_`

, so it's impossible to get my hands
on a `FinSet ZERO`

. What about `FinSet (SUCC ZERO)`

[aka `FinSet 1`

]? Yes, there's
one way to get such an element:

```
Intros :: FinSet (SUCC ZERO)
```

and that's the only way to get a member of `FinSet (SUCC ZERO)`

!
What about `FinSet (SUCC (SUCC ZERO))`

? We have two inhabitants:

```
FinSet (SUCC (SUCC ZERO))
- Intros :: FinSet (SUCC (SUCC ZERO))
- Lift (Intros :: FinSet (SUCC ZERO)) :: FinSet (SUCC (SUCC ZERO))
```

and so on. Here's the `Show`

instance for a `FinSet`

, where we think of `Intros`

as `1`

, and `Lift`

as incrementing the number.

```
countFinSet :: FinSet n -> Int
countFinSet Intros = 1
countFinSet (Lift x) = 1 + countFinSet x
instance Show (FinSet n) where
show x = show (countFinSet x)
```

We can use the same mechanism to encode `Vector n a`

, a vector
with exactly `n`

elements!

```
data Vector (n :: NAT) (a :: *) where
VNil :: Vector ZERO a
VCons :: a -> Vector n a -> Vector (SUCC n) a
instance Functor (Vector n) where
fmap _ VNil = VNil
fmap f (VCons a as) = VCons (f a) (fmap f as)
```

```
instance Representable (Vector n) where
type RepresentingObj (Vector n) = FinSet n
-- fwd :: forall x. f x -> (RepresentingObj f -> x)
fwd :: forall x. Vector n x -> ((FinSet n) -> x)
fwd (VCons x xs) Intros = x
fwd (VCons x xs) (Lift n) = fwd xs n
bwd :: forall x. ((FinSet n) -> x) -> Vector n x
bwd finset2x = undefined
```

Unfortunately, I couldn't quite figure out how to write `bwd`

in Haskell, thus
I leave it as an exercise for the reader :)

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