Control.Category.Structures

This module defines some basic structures in a category in a more fine-grained way then Control.Arrow.

Unfortunately names in this module clash with Control.Arrow.

Documentation

class Category cat => Products cat where

A category with finite products.

Minimal complete definition

(***) | first, second

Methods

first :: cat a b -> cat (a, c) (b, c)

Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.

first a is equal to a *** id.

second :: cat a b -> cat (c, a) (c, b)

A mirror image of first.

second a is equal to id *** a.

(***) :: cat a b -> cat c d -> cat (a, c) (b, d) infixl 3

A product of two arrows. Split the input between the two argument arrows and combine their output.

Instances

Products (->)

Products ReifiedSemiIso'

Monad m => Products (Kleisli m)

Products cat => Products (Dual cat)

Products cat => Products (ReaderCT env cat)

class Category cat => Coproducts cat where

A category with finite coproducts.

Minimal complete definition

(+++) | left, right

Methods

left :: cat a b -> cat (Either a c) (Either b c)

Feed marked inputs through the argument arrow, passing the rest through unchanged to the output.

left a is equal to a +++ id.

right :: cat a b -> cat (Either c a) (Either c b)

A mirror image of left.

right a is equal to id +++ a.

(+++) :: cat a b -> cat c d -> cat (Either a c) (Either b d) infixl 2

A coproduct of two arrows. Split the input between the two argument arrows, retagging and merging their outputs.

Instances

Coproducts (->)

Coproducts ReifiedSemiIso'

Monad m => Coproducts (Kleisli m)

Coproducts cat => Coproducts (Dual cat)

Coproducts cat => Coproducts (ReaderCT env cat)

class Category cat => CatPlus cat where

A category cat is a CatPlus when cat a b is a monoid for all a, b.

Methods

cempty :: cat a b

The identity of /+/.

(/+/) :: cat a b -> cat a b -> cat a b infixl 3

An associative operation on arrows.

Instances

CatPlus ReifiedSemiIso'

MonadPlus m => CatPlus (Kleisli m)

CatPlus cat => CatPlus (Dual cat)

CatPlus cat => CatPlus (ReaderCT env cat)

class CatTrans t where

A category transformer.

Methods

clift :: Category cat => cat a b -> t cat a b

Lift an arrow from the base category.

Instances

CatTrans (ReaderCT env)