category-extras-0.53.6: Various modules and constructs inspired by category theoryContentsIndex
Control.Category.Cartesian
Portabilitynon-portable (class-associated types)
Stabilityexperimental
MaintainerEdward Kmett <ekmett@gmail.com>
Contents
Pre-(Co)Cartesian categories
(Co)Cartesian categories
Description
Synopsis
module Control.Category.Associative
module Control.Category.Monoidal
class (Associative k p, Coassociative k p, Braided k p) => PreCartesian k p | k -> p where
fst :: k (p a b) a
snd :: k (p a b) b
diag :: k a (p a a)
(&&&) :: k a b -> k a c -> k a (p b c)
bimapPreCartesian :: PreCartesian k p => k a c -> k b d -> k (p a b) (p c d)
braidPreCartesian :: PreCartesian k p => k (p a b) (p b a)
associatePreCartesian :: PreCartesian k p => k (p (p a b) c) (p a (p b c))
coassociatePreCartesian :: PreCartesian k p => k (p a (p b c)) (p (p a b) c)
class (Associative k s, Coassociative k s, Braided k s) => PreCoCartesian k s | k -> s where
inl :: k a (s a b)
inr :: k b (s a b)
codiag :: k (s a a) a
(|||) :: k a c -> k b c -> k (s a b) c
bimapPreCoCartesian :: PreCoCartesian k s => k a c -> k b d -> k (s a b) (s c d)
braidPreCoCartesian :: PreCoCartesian k s => k (s a b) (s b a)
associatePreCoCartesian :: PreCoCartesian k s => k (s (s a b) c) (s a (s b c))
coassociatePreCoCartesian :: PreCoCartesian k s => k (s a (s b c)) (s (s a b) c)
class (Monoidal k p i, PreCartesian k p) => Cartesian k p i | k -> p i
class (Comonoidal k s i, PreCoCartesian k s) => CoCartesian k s i | k -> s i
Documentation
module Control.Category.Associative
module Control.Category.Monoidal
Pre-(Co)Cartesian categories
class (Associative k p, Coassociative k p, Braided k p) => PreCartesian k p | k -> p where

NB: This is weaker than traditional category with products! That is Cartesian, below. The problem is (->) lacks an initial object, since every type is inhabited in Haskell. Consequently its coproduct is merely a semigroup, not a monoid as it has no identity, and since we want to be able to describe its dual category, which has this non-traditional form being built over a category with an associative bifunctor rather than as a monoidal category for the product monoid.

Minimum definition: 

 fst, snd, diag 
 fst, snd, (&&&)
Methods
fst :: k (p a b) a
snd :: k (p a b) b
diag :: k a (p a a)
(&&&) :: k a b -> k a c -> k a (p b c)
bimapPreCartesian :: PreCartesian k p => k a c -> k b d -> k (p a b) (p c d)
free construction of Bifunctor for the product Bifunctor Prod k if (&&&) is known
braidPreCartesian :: PreCartesian k p => k (p a b) (p b a)
free construction of Braided for the product Bifunctor Prod k
associatePreCartesian :: PreCartesian k p => k (p (p a b) c) (p a (p b c))
free construction of Associative for the product Bifunctor Prod k
coassociatePreCartesian :: PreCartesian k p => k (p a (p b c)) (p (p a b) c)
free construction of Coassociative for the product Bifunctor Prod k
class (Associative k s, Coassociative k s, Braided k s) => PreCoCartesian k s | k -> s where
Methods
inl :: k a (s a b)
inr :: k b (s a b)
codiag :: k (s a a) a
(|||) :: k a c -> k b c -> k (s a b) c
bimapPreCoCartesian :: PreCoCartesian k s => k a c -> k b d -> k (s a b) (s c d)
free construction of Bifunctor for the coproduct Bifunctor Sum k if (|||) is known
braidPreCoCartesian :: PreCoCartesian k s => k (s a b) (s b a)
free construction of Braided for the coproduct Bifunctor Sum k
associatePreCoCartesian :: PreCoCartesian k s => k (s (s a b) c) (s a (s b c))
free construction of Associative for the coproduct Bifunctor Sum k
coassociatePreCoCartesian :: PreCoCartesian k s => k (s a (s b c)) (s (s a b) c)
free construction of Coassociative for the coproduct Bifunctor Sum k
(Co)Cartesian categories
class (Monoidal k p i, PreCartesian k p) => Cartesian k p i | k -> p i
class (Comonoidal k s i, PreCoCartesian k s) => CoCartesian k s i | k -> s i
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