Protocols

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A Functor provides a type with the ability to transform its value type into another type, while preserving its structure.

Using the encoding for HKTs in Bow, in the type Kind<F, A>, A is the value type and F represents the structure of the type. An instance of Functor for F allows to transform A into another type, while maintaining F unchanged.

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An Alternative is an Applicative with MonoidK capabilities.

An Applicative Functor is a Functor that also provides functionality to lift pure expressions, and sequence computations and combine their results.

Instances of this typeclass must obey the following laws:

1. Identity

   ap(pure(id), v) == v
   

2. Composition

   ap(ap(ap(pure(compose), u), v), w) == compose(u, compose(v, w))
   

3. Homomorphism

   ap(pure(f), pure(x)) == pure(f(x))
   

4. Interchange

   ap(fa, pure(b)) == ap(pure({ x in x(a) }), fa)
   

An ApplicativeError is an Applicative that has capabilities to raise and handle error values.

It has an associated type E to represent the error type that the implementing instance is able to handle.

A Bimonad has the same capabilities as Monad and Comonad.

A Comonad is the dual of a Monad. It provides capabilities to compose functions that extract values from their context.

Implementations of this instance must obey the following laws:

extract(duplicate(fa)) == fa
map(fa, f) == coflatMap(fa, { a in f(extract(a)) }
coflatMap(fa, extract) == fa
extract(coflatMap(fa, f)) == f(fa)

A Contravariant Functor is the dual of a Covariant Functor, usually referred to as just Functor. Whereas an intuition behind Covariant Functors is that they can be seen as containing or producing values, Contravariant Functors can be seen as consuming values.

EquatableK provides capabilities to check equality of values at the kind level.

Foldable describes types that have the ability to be folded to a summary value.

A FunctorFilter provides the same capabilities as a Functor, while filtering out elements simultaneously.

An Invariant Functor provides a type the ability to transform its value type into another type. An instance of Functor or Contravariant are Invariant Functors as well.

A Monad provides functionality to sequence operations that are dependent from one another.

Instances of Monad must obey the following rules:

flatMap(pure(a), f) == f(a)
flatMap(fa, pure) == fa
flatMap(fa) { a in flatMap(f(a), g) } == flatMap(flatMap(fa, f), g)

Also, instances of Monad derive a default implementation for Applicative.ap as:

ap(ff, fa) == flapMap(ff, { f in map(fa, f) }

A MonadCombine has the capabilities of MonadFilter and Alternative together.

A MonadError has the same capabilities as a Monad and an ApplicativeError.

A MonadFilter is a Monad with FunctorFilter capabilities. It also provides functionality to create a value that represents an empty element in the context implementing an instance of this typeclass.

Implementing this typeclass automatically derives an implementation for FunctorFilter.mapFilter.

A MonadReader is a Monad with capabilities to read values from a shared environment.

A MonadState is a Monad that maintains a state.

A MonadWriter is a Monad with the ability to produce a stream of data in addition to the computed values.

A monoid is an algebraic structure that has the same properties as a semigroup, with an empty element.

A MonoidK is a SemigroupK that also has an empty element.

NonEmptyReducible defines a Reducible in terms of another type that has an instance of Foldable and a method that is able to split a structure of values in a first value and the rest of the values in the structure.

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The Selective typeclass represents Selective Applicative Functors, described in this paper. Selective Applicative Functors enrich Applicative Functors by providing an operation that composes two effectful computations where the second depends on the first.

A semigroup is an algebraic structure that consists of a set of values and an associative operation.

SemigroupK is a Semigroup that operates on kinds with one type parameter.

Traverse provides a type with the ability to traverse a structure with an effect.

TraverseFilter represents array-like structures that can be traversed and filtered as a single combined operation. It provides the same capabilities as Traverse and FunctorFilter together.