A projection on data, which only knows how to execute a strict
left-fold.
It is a monad and a monoid, and is very useful for efficiently
aggregating the projections on data intended for left-folding, since
its concatenation (
<>) has complexity of
O(1).
The intuition for this abstraction can be derived from lists.
Let's consider the
foldl' function for lists:
foldl' :: (b -> a -> b) -> b -> [a] -> b
If we rearrange its parameters we get
foldl' :: [a] -> (b -> a -> b) -> b -> b
Which in Haskell is essentially the same as
foldl' :: [a] -> (forall b. (b -> a -> b) -> b -> b)
We can isolate that part into an abstraction:
newtype Unfoldl a = Unfoldl (forall b. (b -> a -> b) -> b -> b)
Then we get to this simple morphism:
list :: [a] -> Unfoldl a
list list = Unfoldl (\ step init -> foldl' step init list)
We can do the same with say
Data.Text.Text:
text :: Text -> Unfoldl Char
text text = Unfoldl (\ step init -> Data.Text.foldl' step init text)
And then we can use those both to concatenate with just an
O(1) cost:
abcdef :: Unfoldl Char
abcdef = list ['a', 'b', 'c'] <> text "def"
Please notice that up until this moment no actual data materialization
has happened and hence no traversals have appeared. All that we've
done is just composed a function, which only specifies which parts of
data structures to traverse to perform a left-fold. Only at the moment
where the actual folding will happen will we actually traverse the
source data. E.g., using the "fold" function:
abcdefLength :: Int
abcdefLength = fold Control.Foldl.length abcdef