package preface

A Semigroup is a type t which provides a binary associative operation combine which lets you combine any two values of t into one.

A Monoid is a type t which provides a binary associative operation combine and a neutral element (neutral). In other words, a Monoid is a Semigroup with a neutral element.

A Functor represents a type that can be mapped over. So we can go from 'a t to 'b t using a function from 'a to 'b. Mapping preserve the structure of the input.

Alt is a Functor which is a kind of Semigroup over a parametrized type. In other word, Alt is a Functor with a combine operation.

An Applicative is a functor with lifting and sequencing capabilities. Applicative is more general (and by extension weaker) than a Monad. An Applicative is also a Functor.

Alternative is a kind of Monoid on Applicative. An Alternative is formally an Applicative with neutral and combine. So an Alternative is also an Applicative (and an Alt which is also a Functor).

A Selective (applicative functor) allows to declare effects statically and select which execute dynamically. It is an algebraic structure between Applicative and Monad. A Selective is also an Applicative.

A Monad allow to sequences operations that are dependent from one to another, in contrast to Applicative, which executes a series of independent actions.

Monad_plus is a kind of Monoid on Monad. A Monad_plus is formally a Monad with neutral and combine. So a Monad_plus is also a Monad.

A Comonad is the dual of the Monad.

A Foldable is a data structure wich can be fold. In other word, reduced to a summary value one element at a time

A Traversable is a data structure that can be traversed from left to right, performing an action on each element.

Contravariant is a "Contravariant functor". In other word, Contravariant is the dual of a Functor.

Divisible is a "Contravariant Applicative Functor", in other word, Divisible is the dual of an Applicative

A Bifunctor is a type constructor that takes two type arguments and is a Functor (Covariant) in both arguments.

A Profunctor is a type constructor that takes two type arguments and is a Contravariant Functor as first argument and a covariant Functor as second argument.

Strong is a Profunctor working on product types.

Choice is a Profunctor working on sum types (via Either).

Closed is a Profunctor working on exponential types (function).

A Category is a generalization of function composition to morphisms.

An Arrow is an abstract view of computation sitting between Applicative and Monad. Arrow is built on the top of Category and Strong. So an Arrow is also a Category.

An Arrow with a neutral element. So Arrow_zero is also an Arrow.

An Arrow with a combine function. So Arrow_alt is also an Arrow. Arrow_alt is a kind of Alt in the arrow hierarchy.

An Arrow_plus is the conjonction of an Arrow_zero and Arrow_alt. An Arrow_plus is a kind of Monoid in the arrow hierarchy. And it also an Arrow_alt and an Arrow_zero (which is also a Arrow).

An Arrow_choice is an Arrow with conditional capabilities. It is a kind of Selective in the arrow hierarchy. And an Arrow_choice is also Arrow.

An Arrow_apply is an Arrow with application capabilities. An Arrow_apply is also Arrow.

A Reader monad parametrized over an inner Monad and an environment. Reader is a monad transformer.

A Writer monad parametrized over an inner Monad and an tape which is a Monoid. Writer is a monad transformer.

A State monad parametrized over an inner Monad and a state (which is an arbitrary type). State is a monad transformer.

A Store comonad parametrized over an inner Comonad and a store (which is an arbitrary type). Store is a comonad transformer. Store is Costate (the dual of State).

An Env comonad parametrized over an inner Comonad and an env (which is an arbitrary type). Env is a comonad transformer. Env is Coreader (the dual of Reader).

A Traced comonad parametrized over an inner Comonad and a tape (which is an arbitrary type). Traced is a comonad transformer. Traced is Cowriter (the dual of Writer).

A Free applicative allows you to build an Preface_specs.Applicative from a given Preface_specs.Functor.

A Free selective allows you to build a rigid Preface_specs.Selective from a given Preface_specs.Functor.

A Free monad allows you to build a Preface_specs.Monad from a given Preface_specs.Functor.

A Freer monad allows you to build a Preface_specs.Monad from an arbitrary type (with one type parameter). It offers the same capabilities as a Preface_specs.Free_monad but benefits from a lighter execution cost.

Signatures representing types. Mainly used as a functor parameter