monocle

A Fold can be seen as a Getter with many targets or a weaker PTraversal which cannot modify its target.

Fold is on the top of the Optic hierarchy which means that Getter, PTraversal, POptional, PLens, PPrism and PIso are valid Fold

Extension methods for Fold

A Getter can be seen as a glorified get method between a type S and a type A.

A Getter is also a valid Fold

Extension methods for Fold

Extension methods for monomorphic Iso

Extension methods for monomorphic Lens

Extension methods for monomorphic Optional

Iso is a type alias for PIso where S = A and T = B:

type Iso[S, A] = PIso[S, S, A, A]

An Iso defines an isomorphism between a type S and A:

 get
--------------------> S A <-------------------- reverseGet 

A PIso allows to lift a function f: A => B to S => T and a function g: T => S to B => A

 g S T S
<-------- T
| ↑ | ↑
| | | | get | | reverseGet reverse.reverseGet | | reverse.get
| | | | ↓ f | ↓ | A --------> B A B 

A PIso is also a valid Getter, Fold, PLens, PPrism, POptional, PTraversal and PSetter

A PLens can be seen as a pair of functions:

• get: S => A i.e. from an S, we can extract an A
• set: (B, S) => T i.e. if we replace an A by a B in an S, we obtain a T

A PLens could also be defined as a weaker PIso where replace requires an additional parameter than reverseGet.

PLens stands for Polymorphic Lens as it replace and modify methods change a type A to B and S to T. Lens is a type alias for PLens restricted to monomorphic updates:

type Lens[S, A] = PLens[S, S, A, A]

A PLens is also a valid Getter, Fold, POptional, PTraversal and PSetter

Typically a PLens or Lens can be defined between a Product (e.g. case class, tuple, HList) and one of its component.

A POptional can be seen as a pair of functions:

• getOrModify: S => Either[T, A]
• replace : (B, S) => T

A POptional could also be defined as a weaker PLens and weaker PPrism

POptional stands for Polymorphic Optional as it replace and modify methods change a type A to B and S to T. Optional is a type alias for POptional restricted to monomorphic updates:

type Optional[S, A] = POptional[S, S, A, A]

A PPrism can be seen as a pair of functions:

• getOrModify: S => Either[T, A]
• reverseGet : B => T

A PPrism could also be defined as a weaker PIso where get can fail.

Typically a PPrism or Prism encodes the relation between a Sum or CoProduct type (e.g. sealed trait) and one of its element.

PPrism stands for Polymorphic Prism as it replace and modify methods change a type A to B and S to T. Prism is a type alias for PPrism where the type of target cannot be modified:

type Prism[S, A] = PPrism[S, S, A, A]

A PPrism is also a valid Fold, POptional, PTraversal and PSetter

A PSetter is a generalisation of Functor map:

• map: (A => B) => F[A] => F[B]
• modify: (A => B) => S => T

PSetter stands for Polymorphic Setter as it replace and modify methods change a type A to B and S to T. Setter is a type alias for PSetter restricted to monomorphic updates:

type Setter[S, A] = PSetter[S, S, A, A]

PTraversal, POptional, PPrism, PLens and PIso are valid PSetter

A PTraversal can be seen as a POptional generalised to 0 to n targets where n can be infinite.

PTraversal stands for Polymorphic Traversal as it replace and modify methods change a type A to B and S to T. Traversal is a type alias for PTraversal restricted to monomorphic updates:

type Traversal[S, A] = PTraversal[S, S, A, A]

Extension methods for monomorphic Setter

Extension methods for monomorphic Traversal