A witness that (a ~ s, b ~ t).

Note: Composition with an Equality is index-preserving.

A Simple Equality.

Isomorphism families can be composed with another Lens using (.) and id.

Note: Composition with an Iso is index- and measure- preserving.

 type Iso' = Simple Iso

A Prism l is a 0-or-1 target Traversal that can also be turned around with re to obtain a Getter in the opposite direction.

There are two laws that a Prism should satisfy:

First, if I re or review a value with a Prism and then preview or use (^?), I will get it back:

 preview l (review l b) ≡ Just b

Second, if you can extract a value a using a Prism l from a value s, then the value s is completely described my l and a:

If preview l s ≡ Just a then review l a ≡ s

These two laws imply that the Traversal laws hold for every Prism and that we traverse at most 1 element:

 lengthOf l x <= 1

It may help to think of this as a Iso that can be partial in one direction.

Every Prism is a valid Traversal.

Every Iso is a valid Prism.

For example, you might have a Prism' Integer Natural allows you to always go from a Natural to an Integer, and provide you with tools to check if an Integer is a Natural and/or to edit one if it is.

 nat :: Prism' Integer Natural
 nat = prism toInteger $ \ i ->
    if i < 0
    then Left i
    else Right (fromInteger i)

Now we can ask if an Integer is a Natural.

>>> 5^?nat
Just 5

>>> (-5)^?nat
Nothing

We can update the ones that are:

>>> (-3,4) & both.nat *~ 2
(-3,8)

And we can then convert from a Natural to an Integer.

>>> 5 ^. re nat -- :: Natural
5

Similarly we can use a Prism to traverse the Left half of an Either:

>>> Left "hello" & _Left %~ length
Left 5

or to construct an Either:

>>> 5^.re _Left
Left 5

such that if you query it with the Prism, you will get your original input back.

>>> 5^.re _Left ^? _Left
Just 5

Another interesting way to think of a Prism is as the categorical dual of a Lens -- a co-Lens, so to speak. This is what permits the construction of outside.

Note: Composition with a Prism is index-preserving.

A Simple Prism.

A Lens is actually a lens family as described in http://comonad.com/reader/2012/mirrored-lenses/.

With great power comes great responsibility and a Lens is subject to the three common sense Lens laws:

1) You get back what you put in:

 view l (set l b a)  ≡ b

2) Putting back what you got doesn't change anything:

 set l (view l a) a  ≡ a

3) Setting twice is the same as setting once:

 set l c (set l b a) ≡ set l c a

These laws are strong enough that the 4 type parameters of a Lens cannot vary fully independently. For more on how they interact, read the "Why is it a Lens Family?" section of http://comonad.com/reader/2012/mirrored-lenses/.

Every Lens can be used directly as a Setter or Traversal.

You can also use a Lens for Getting as if it were a Fold or Getter.

Since every Lens is a valid Traversal, the Traversal laws are required of any Lens you create:

 l pure ≡ pure
 fmap (l f) . l g ≡ getCompose . l (Compose . fmap f . g)

 type Lens s t a b = forall f. Functor f => LensLike f s t a b

 type Lens' = Simple Lens

A Traversal can be used directly as a Setter or a Fold (but not as a Lens) and provides the ability to both read and update multiple fields, subject to some relatively weak Traversal laws.

These have also been known as multilenses, but they have the signature and spirit of

 traverse :: Traversable f => Traversal (f a) (f b) a b

and the more evocative name suggests their application.

Most of the time the Traversal you will want to use is just traverse, but you can also pass any Lens or Iso as a Traversal, and composition of a Traversal (or Lens or Iso) with a Traversal (or Lens or Iso) using (.) forms a valid Traversal.

The laws for a Traversal t follow from the laws for Traversable as stated in "The Essence of the Iterator Pattern".

 t pure ≡ pure
 fmap (t f) . t g ≡ getCompose . t (Compose . fmap f . g)

One consequence of this requirement is that a Traversal needs to leave the same number of elements as a candidate for subsequent Traversal that it started with. Another testament to the strength of these laws is that the caveat expressed in section 5.5 of the "Essence of the Iterator Pattern" about exotic Traversable instances that traverse the same entry multiple times was actually already ruled out by the second law in that same paper!

 type Traversal' = Simple Traversal

The only LensLike law that can apply to a Setter l is that

 set l y (set l x a) ≡ set l y a

You can't view a Setter in general, so the other two laws are irrelevant.

However, two Functor laws apply to a Setter:

 over l id ≡ id
 over l f . over l g ≡ over l (f . g)

These can be stated more directly:

 l pure ≡ pure
 l f . untainted . l g ≡ l (f . untainted . g)

You can compose a Setter with a Lens or a Traversal using (.) from the Prelude and the result is always only a Setter and nothing more.

>>> over traverse f [a,b,c,d]
[f a,f b,f c,f d]

>>> over _1 f (a,b)
(f a,b)

>>> over (traverse._1) f [(a,b),(c,d)]
[(f a,b),(f c,d)]

>>> over both f (a,b)
(f a,f b)

>>> over (traverse.both) f [(a,b),(c,d)]
[(f a,f b),(f c,f d)]

A Setter' is just a Setter that doesn't change the types.

These are particularly common when talking about monomorphic containers. e.g.

 sets Data.Text.map :: Setter' Text Char

 type Setter' = Setter'

A Getter describes how to retrieve a single value in a way that can be composed with other LensLike constructions.

Unlike a Lens a Getter is read-only. Since a Getter cannot be used to write back there are no Lens laws that can be applied to it. In fact, it is isomorphic to an arbitrary function from (a -> s).

Moreover, a Getter can be used directly as a Fold, since it just ignores the Applicative.

A Fold describes how to retrieve multiple values in a way that can be composed with other LensLike constructions.

A Fold s a provides a structure with operations very similar to those of the Foldable typeclass, see foldMapOf and the other Fold combinators.

By convention, if there exists a foo method that expects a Foldable (f a), then there should be a fooOf method that takes a Fold s a and a value of type s.

A Getter is a legal Fold that just ignores the supplied Monoid.

Unlike a Traversal a Fold is read-only. Since a Fold cannot be used to write back there are no Lens laws that apply.

An Action is a Getter enriched with access to a Monad for side-effects.

Every Getter can be used as an Action.

You can compose an Action with another Action using (.) from the Prelude.

A MonadicFold is a Fold enriched with access to a Monad for side-effects.

Every Fold can be used as a MonadicFold, that simply ignores the access to the Monad.

You can compose a MonadicFold with another MonadicFold using (.) from the Prelude.

Every IndexedLens is a valid Lens and a valid IndexedTraversal.

 type IndexedLens' i = Simple (IndexedLens i)

Every IndexedTraversal is a valid Traversal or IndexedFold.

The Indexed constraint is used to allow an IndexedTraversal to be used directly as a Traversal.

The Traversal laws are still required to hold.

 type IndexedTraversal' i = Simple (IndexedTraversal i)

Every IndexedSetter is a valid Setter.

The Setter laws are still required to hold.

 type IndexedSetter' i = Simple (IndexedSetter i)

Every IndexedGetter is a valid IndexedFold and can be used for Getting like a Getter.

Every IndexedFold is a valid Fold and can be used for Getting.

An IndexedAction is an IndexedGetter enriched with access to a Monad for side-effects.

Every Getter can be used as an Action.

You can compose an Action with another Action using (.) from the Prelude.

An IndexedMonadicFold is an IndexedFold enriched with access to a Monad for side-effects.

Every IndexedFold can be used as an IndexedMonadicFold, that simply ignores the access to the Monad.

You can compose an IndexedMonadicFold with another IndexedMonadicFold using (.) from the Prelude.

An IndexPreservingLens leaves any index it is composed with alone.

 type IndexPreservingLens' = Simple IndexPreservingLens

An IndexPreservingLens leaves any index it is composed with alone.

 type IndexPreservingTraversal' = Simple IndexPreservingTraversal

An IndexPreservingSetter can be composed with a IndexedSetter, IndexedTraversal or IndexedLens and leaves the index intact, yielding an IndexedSetter.

 type IndexedPreservingSetter' i = Simple IndexedPreservingSetter

An IndexPreservingGetter can be used as a Getter, but when composed with an IndexedTraversal, IndexedFold, or IndexedLens yields an IndexedFold, IndexedFold or IndexedGetter respectively.

An IndexPreservingFold can be used as a Fold, but when composed with an IndexedTraversal, IndexedFold, or IndexedLens yields an IndexedFold respectively.

An IndexPreservingAction can be used as a Action, but when composed with an IndexedTraversal, IndexedFold, or IndexedLens yields an IndexedMonadicFold, IndexedMonadicFold or IndexedAction respectively.

An IndexPreservingFold can be used as a Fold, but when composed with an IndexedTraversal, IndexedFold, or IndexedLens yields an IndexedFold respectively.

A Simple Lens, Simple Traversal, ... can be used instead of a Lens,Traversal, ... whenever the type variables don't change upon setting a value.

 _imagPart :: Simple Lens (Complex a) a
 traversed :: Simple (IndexedTraversal Int) [a] a

Note: To use this alias in your own code with LensLike f or Setter, you may have to turn on LiberalTypeSynonyms.

This is commonly abbreviated as a "prime" marker, e.g. Lens' = Simple Lens.

Many combinators that accept a Lens can also accept a Traversal in limited situations.

They do so by specializing the type of Functor that they require of the caller.

If a function accepts a LensLike f s t a b for some Functor f, then they may be passed a Lens.

Further, if f is an Applicative, they may also be passed a Traversal.

 type LensLike' f = Simple (LensLike f)

This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.

This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.

 type Over' p f = Simple (Over p f)

Convenient alias for constructing indexed lenses and their ilk.

Convenient alias for constructing simple indexed lenses and their ilk.

 type LensLike f s t a b = Overloading (->) (->) f s t a b

 type Overloading' p q f s a = Simple (Overloading p q f) s a

 type LensLike f s t a b = Overloaded (->) f s t a b

 type Overloaded' p q f s a = Simple (Overloaded p q f) s a