Hi Christopher,
a data type can be an instance of Category only if it has kind * -> * -> *.
It must have 2 type parameters so that you could have types like 'cat a a'.
Some simple examples:
import Prelude hiding (id, (.))
import Control.Category
import Data.Monoid
-- See https://en.wikipedia.org/wiki/Opposite_category
newtype Op c a b = Op (c b a)
instance Category c => Category (Op c) where
id = Op id
(Op x) . (Op y) = Op (y . x)
-- A category whose morphisms are bijections between types.
data Iso a b = Iso (a -> b) (b -> a)
instance Category Iso where
id = Iso id id
(Iso f1 g1) . (Iso f2 g2) = Iso (f1 . f2) (g2 . g1)
-- A product of two categories forms a new category:
data ProductCat c d a b = ProductCat (c a b) (d a b)
instance (Category c, Category d) => Category (ProductCat c d) where
id = ProductCat id id
(ProductCat f g) . (ProductCat f' g') = ProductCat (f . f') (g . g')
-- A category constructed from a monoid. It
-- ignores the types. Any morphism in this category
-- is simply an element of the given monoid.
newtype MonoidCat m a b = MonoidCat m
instance (Monoid m) => Category (MonoidCat m) where
id = MonoidCat mempty
MonoidCat x . MonoidCat y = MonoidCat (x `mappend` y)
Many interesting categories can be constructed from various monads using
Kleisli. For example, Kleisli Maybe is the category of partial functions.
Best regards,
Petr
2012/12/20 Christopher Howard
I've perhaps been trying everyones patiences with my noobish CT questions, but if you'll bear with me a little longer: I happened to notice that there is in fact a Category class in Haskell base, in Control.Category:
quote: -------- class Category cat where
A class for categories. id and (.) must form a monoid.
Methods
id :: cat a a
the identity morphism
(.) :: cat b c -> cat a b -> cat a c
morphism composition --------
However, the documentation lists only two instances of Category, functions (->) and Kleisli Monad. For instruction purposes, could someone show me an example or two of how to make instances of this class, perhaps for a few of the common types? My initial thoughts were something like so:
code: -------- instance Category Integer where
id = 1
(.) = (*)
-- and
instance Category [a] where
id = [] (.) = (++) -------
But these lead to kind mis-matches.
-- frigidcode.com
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