Functors supporting a zip operation that takes the union of non-uniform shapes.

If your functor is actually a functor from Kleisli Maybe to Hask (so it supports maybeMap :: (a -> Maybe b) -> f a -> f b), then an Align instance is making your functor lax monoidal w.r.t. the cartesian monoidal structure on Kleisli Maybe, because These is the cartesian product in that category (a -> Maybe (These b c) ~ (a -> Maybe b, a -> Maybe c)). This insight is due to rwbarton.

Minimal definition: either align or alignWith.

Laws:

join align = fmap (join These)
align (f <$> x) (g <$> y) = bimap f g <$> align x y
alignWith f a b = f <$> align a b
align x (align y z) = fmap assoc (align (align x y) z)

Note: join f x = f x x

And an addition property if f is Foldable, which tries to enforce align-feel: neither values are duplicated nor lost.

toList x = toListOf (folded . here) (align x y)
         = mapMaybe justHere (toList (align x y))

A unit of align.

Laws:

(`align` nil) = fmap This
(nil `align`) = fmap That

Align two structures and combine with mappend.

See salign. malign will be deprecated after Semigroup becomes a super class of Monoid

Align two structures and combine with <>.

Since: 0.7.3

Align two structures as in zip, but filling in blanks with Nothing.

Align two structures as in zipWith, but filling in blanks with Nothing.

Left-padded zip.

Left-padded zipWith.

Right-padded zip.

Right-padded zipWith.

Alignable functors supporting an "inverse" to align: splitting a union shape into its component parts.

Minimal definition: nothing; a default definition is provided, but it may not have the desired definition for all functors. See the source for more information.

Laws:

unalign nil                 = (nil,           nil)
unalign (This        <$> x) = (Just    <$> x, Nothing <$  x)
unalign (That        <$> y) = (Nothing <$  y, Just    <$> y)
unalign (join These  <$> x) = (Just    <$> x, Just    <$> x)
unalign ((x `These`) <$> y) = (Just x  <$  y, Just    <$> y)
unalign ((`These` y) <$> x) = (Just    <$> x, Just y  <$  x)

Foldable functors supporting traversal through an alignable functor.

Minimal definition: crosswalk or sequenceL.

Laws:

crosswalk (const nil) = const nil
crosswalk f = sequenceL . fmap f

Bifoldable bifunctors supporting traversal through an alignable functor.

Minimal definition: bicrosswalk or bisequenceL.

Laws:

bicrosswalk (const empty) (const empty) = const empty
bicrosswalk f g = bisequenceL . bimap f g