A lens, prism or a traversal.

The pr and ls are used to track review and get capabilities, leave them being type variables. The consumers of lens will try to unify either of these with Y type (= ()).

The s, t, a and b are "input object type", "output object type", "input part type" and "output part type", accordingly.

This means that you can get a from s, and given a function a -> b make a t out of s.

Later in this text I will use term "final" for "out" types and "initial" for "in" types.

The lens is "not a prism".

The prism is "not a lens".

The traversal is neither "lens" or "prism".

The isomorphism is both "lens" and "prism".

Lens that cannot change type of the object.

Prism that cannot change type of the object.

Traversal that cannot change type of the object.

Iso that cannot change type of the object.

A lens constructor.

Parameters are: getter and setter.

If the lens can change type of an object, the setter must return the object of a new type.

A prism constructor.

Parameters are: reconstructor and a splitter.

Reconstructor takes a final value and constructs a final object.

The splitter turns initial object either to final object directly (if initial object is of wrong variant), or spits out a.

A traversal constructor.

Parameters are: toList and a mapper.

We need toList, because there is no Foldable typeclass in Elm.

The mapper is a "mapSomething" function update s.

An isomorphism constructor.

An identity optic.

Optical composition.

Retrieve the only element using a lens.

Use prism to reconstruct.

Check if s is related to a prism.

Note: You can change Y to pr here and it will still compile. I restricted it to prisms, because semantics for traversals is kinda be questionable.

Retrieve up to one element using any optic.

Retrieve all elements using any optic.

Update over any optic.

Assign into any optic.