# Haskell's strength: generalising with lenses

This post is about the `strength`

function, Lenses,
and *strong functors*.
Specifically, it’s about how we can generalise
`strength`

using lenses to work on any product type, not just tuples.

If you like, you can skip straight to the good bit where we derive a generalised strength.

For our purposes, `strength`

is a function which “everts” a tuple
containing some functor, `(a, f b)`

, turning it inside out to result
in a functor of a tuple `f (a, b)`

.
We can write this function as follows:

```
strength :: Functor f => (a, f b) -> f (a, b)
= fmap (\b -> (a, b)) fb strength (a, fb)
```

Note that the choice of `f b`

being the second item in the tuple is
fairly arbitrary, and we could just as easily have put it as the first.
Our choice is important to make code cleaner later in the article.

In Category Theory, a functor allowing this operation is said to be
*strong*.
A more formal definition of a strong functors can be found in Edward
Kmett’s excellent article
“Deriving strength from laziness”,
which also notes that all Haskell `Functor`

s are strong.

## So what?

This might seem a bit pointless, but turns out to be useful in a number
of cases. For example, take MapReduce.
a MapReduce program defines two functions,
`map`

, and `reduce`

.
Haskell types for these functions might look like this:

```
mrMap :: ka -> va -> [(kb, vb)]
mrReduce :: kb -> [vb] -> [output]
```

where `ka`

denotes “key type a”, equivalent to `in_key`

from the
MapReduce slides, and “vb” denotes “value b”, equivalent to
`intermediate_value`

.
Let’s suppose we want to write the “identity” reduce function. That is,
a function which yields the output of the `mrMap`

function unchanged.

One such implementation is simply

```
mrReduce' :: kb -> [vb] -> [(kb, vb)]
= map (\v -> (k, v)) vs mrReduce' k vs
```

But it’s already looking very familiar. We can write that in terms of
`strength`

like this:

```
mrReduce :: kb -> [vb] -> [(kb, vb)]
= strength (k, vs)
mrReduce k vs
-- Or even more simply...
= curry strength mrReducePF
```

## Another example: JSON objects to key/value pairs

In case you’re not particularly convinced, here’s another example.

Let’s say we’re dealing with a JSON object, and we want to parse it into a list of keys and values. An example object might look like this:

```
{
"foo": 3,
"bar": 1,
"baz": 4
}
```

And we want to end up with something like this:

```
result :: [(Text, Int)]
= [("foo", 3), ("bar", 1), ("baz", 4)] result
```

We’ll use the excellent `Data.Aeson`

library,
which provides a function
`parseJSON :: FromJSON a => Value -> Parser a`

which we can use to encode and decode to and from text.

We can use this, combined with `strength`

, to convert an Object- a
`HashMap Text Value`

- into a list of key/value pairs.

First, we’ll tackle converting a single key/value pair:

```
parsePair :: FromJSON v => (Text, Value) -> Parser (Text, v)
= strength . over _2 parseJSON parsePair
```

Now we can simply apply it to each of the object’s properties and combine the monadic results using mapM:

```
parseKVP :: FromJSON v => Value -> Parser [(Text, v)]
Object ps) = mapM parsePair $ toList ps
parseKVP (= mzero parseKVP _
```

## What if it’s not a pair?

We’ve seen that our examples worked out particularly well for us, but only because we picked an implementation of strength matching our use-cases. How would we deal with nested Functors as the first item? For that matter, what about 3- and 4-tuples? In fact, what about records?

What we want is to be able to use `strength`

on *any* product type,
including records, with some easy way to specify which field is the
functor we want to evert.

## Lenses to the rescue

Fortunately, the lens package is here to help. This gives us
a nice way of passing around getters and setters, and crucially lets
us do *polymorphic updates*. That means that when modifying a field, we
can also change its type
(and therefore also the type of the ‘parent’ record).
We’ve seen this already in the definition of `parsePair`

, where the
second item in a tuple was modified “in place”, while changing its type
from `Value`

to `Parser v`

.

## The essence of strength

Clearly, our new function won’t be able to use pattern matching. We’ll rewrite the original function without pattern matching first.

```
strength' :: Functor f => (a, f b) -> f (a, b)
= fmap (g x) (f x)
strength' x where f x = snd x
= \b -> (fst x, b) g x
```

Note how it’s of the form `fmap (g x) (f x)`

, where:

`f x`

extracts the functor from the record`g x`

is a function putting values of type`b`

into a tuple

We can look at f and g as getters and setters of sorts, which is exactly what lenses do best. Now we can rewrite it like this:

```
strength'' :: Functor f => (a, f b) -> f (a, b)
= fmap (g x) (f x)
strength'' x where f = view _2
= \b -> set _2 b x g x
```

## The good bit

From here it’s pretty clear how to proceed: we just need to replace the hard-coded lens with one passed in as an argument. Note that you’ll need the RankNTypes extension for this to work (i’m not sure why!)

```
strong :: Functor f => Lens s t (f a) a -> s -> f t
= fmap (set l ?? x) (view l x) strong l x
```

Note we’ve replaced the lambda expression using `??`

, an infix version
of `flip`

which you can read as a placeholder for a missing argument.

Finally, to explain the type signature. The type `Lens s t a b`

can be
read as
“A lens into a record of type `s`

to a field of type ‘a’. The field can
be modified to be of type ‘b’, resulting in a record of type ‘t’.”

What we’ve specified is a `Lens s t (f a) a`

, which is a lens on a
record `s`

, containing a field of type `f a`

. The “target” record is of
type `t`

, and has a field of type `a`

- the type of elements of the
functor. Our `strong`

function now returns an `f t`

, the wrapped record.

Finally, some examples of usage:

```
Just "hi", 1) == Just ("hi", 1)
strong _1 ("key", [1..3]) == [("key", 1), ("key", 2), "key", 3)]
strong _2 (
data KVP k v = KVP
_key :: k
{ _val :: v
,
}'KVP
makeLenses '
KVP "key" [1..3]) == [KVP { _key = "Hi", _val = 1 } ... ] strong val (
```

Pretty cool!

## Further down the rabbit hole

This isn’t the end of the story, however. If we examine the type of
`strong`

more closely, we find something interesting.
The `Lens`

type is defined thusly:

`type Lens s t a b = Functor f => (a -> f b) -> s -> f t`

If we replace these types with those from our earlier definition
of `strong`

, we find that

`Lens s t (f a) a == Functor f => (f a -> f a) -> s -> f t`

Which means that given a lens of this type, we can apply it to the identity function, and obtain the very same function:

```
strong' :: Lens s t (f a) a -> s -> f t
= l id strong' l
```

As it turns out, this function is already *in* Control.Lens, as the
`sequenceAOf`

function, which goes even further to work
on `LensLike`

lenses which don’t even require the Functor constraint
for `f`

.