| Portability | non-portable (class-associated types) |
|---|---|
| Stability | experimental |
| Maintainer | Edward Kmett <ekmett@gmail.com> |
Control.Category.Cartesian
Description
- module Control.Category.Associative
- module Control.Category.Monoidal
- class (Associative k p, Coassociative k p, Braided k p) => PreCartesian k p | k -> p where
- 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
- 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 whereSource
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
Instances
bimapPreCartesian :: PreCartesian k p => k a c -> k b d -> k (p a b) (p c d)Source
braidPreCartesian :: PreCartesian k p => k (p a b) (p b a)Source
associatePreCartesian :: PreCartesian k p => k (p (p a b) c) (p a (p b c))Source
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)Source
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 whereSource
Methods
Instances
bimapPreCoCartesian :: PreCoCartesian k s => k a c -> k b d -> k (s a b) (s c d)Source
braidPreCoCartesian :: PreCoCartesian k s => k (s a b) (s b a)Source
associatePreCoCartesian :: PreCoCartesian k s => k (s (s a b) c) (s a (s b c))Source
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)Source
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 iSource
Instances
| (Monoidal k p i, PreCartesian k p) => Cartesian k p i |
class (Comonoidal k s i, PreCoCartesian k s) => CoCartesian k s i | k -> s iSource
Instances
| (Comonoidal k s i, PreCoCartesian k s) => CoCartesian k s i |