Hask
From HaskellWiki
(Difference between revisions)
(→"Platonic" Hask) |
(id is a function :)) |
||
| (One intermediate revision not shown.) | |||
| Line 1: | Line 1: | ||
'''Hask''' is the [[Category theory|category]] of Haskell types and functions. | '''Hask''' is the [[Category theory|category]] of Haskell types and functions. | ||
| - | The objects of '''Hask''' are Haskell types, and the morphisms from objects <hask>A</hask> to <hask>B</hask> are Haskell functions of type <hask>A -> B</hask>. The identity morphism for object <hask>A</hask> is <hask>id :: A</hask>, and the composition of morphisms <hask>f</hask> and <hask>g</hask> is <hask>f . g = \x -> f (g x)</hask>. | + | The objects of '''Hask''' are Haskell types, and the morphisms from objects <hask>A</hask> to <hask>B</hask> are Haskell functions of type <hask>A -> B</hask>. The identity morphism for object <hask>A</hask> is <hask>id :: A -> A</hask>, and the composition of morphisms <hask>f</hask> and <hask>g</hask> is <hask>f . g = \x -> f (g x)</hask>. |
== Is '''Hask''' even a category? == | == Is '''Hask''' even a category? == | ||
| Line 80: | Line 80: | ||
<br /><hask>sndP . u = g</hask> | <br /><hask>sndP . u = g</hask> | ||
|- | |- | ||
| - | ! scope="row" | | + | ! scope="row" | Candidate |
| <hask>u1 r = case r of {}</hask> | | <hask>u1 r = case r of {}</hask> | ||
| <hask>u1 _ = ()</hask> | | <hask>u1 _ = ()</hask> | ||
Current revision
Hask is the category of Haskell types and functions.
The objects of Hask are Haskell types, and the morphisms from objectsA
B
A -> B
A
id :: A -> A
f
g
f . g = \x -> f (g x)
Contents |
1 Is Hask even a category?
Consider:
undef1 = undefined :: a -> b undef2 = \_ -> undefined
Note that these are not the same value:
seq undef1 () = undefined seq undef2 () = ()
undef1 . id = undef2
f
g
f x = g x
x
undef1
undef2
2 Hask is not Cartesian closed
Actual Hask does not have sums, products, or an initial object, and()
| Initial Object | Terminal Object | Sum | Product | Product | |
|---|---|---|---|---|---|
| Type | data Empty | data () = () | data Either a b = Left a | Right b | data (a,b) = (,) { fst :: a, snd :: b} | data P a b = P {fstP :: !a, sndP :: !b} |
| Requirement | There is a unique function
u :: Empty -> r | There is a unique function
u :: r -> () | For any functions
f :: a -> r g :: b -> r there is a unique function u :: Either a b -> r such that: u . Left = f u . Right = g | For any functions
f :: r -> a g :: r -> b there is a unique function u :: r -> (a,b) such that: fst . u = f snd . u = g | For any functions
f :: r -> a g :: r -> b there is a unique function u :: r -> P a b such that: fstP . u = f sndP . u = g |
| Candidate | u1 r = case r of {} | u1 _ = () | u1 (Left a) = f a u1 (Right b) = g b | u1 r = (f r,g r) | u1 r = P (f r) (g r) |
| Example failure condition | r ~ () | r ~ () | r ~ () f _ = () g _ = () | r ~ () f _ = undefined g _ = undefined | r ~ () f _ = () g _ = undefined |
| Alternative u | u2 _ = () | u2 _ = undefined | u2 _ = () | u2 _ = undefined | |
| Difference | u1 undefined = undefined u2 undefined = () | u1 _ = () u2 _ = undefined | u1 undefined = undefined u2 undefined = () | u1 _ = (undefined,undefined) u2 _ = undefined | f _ = () (fstP . u1) _ = undefined |
| Result | FAIL | FAIL | FAIL | FAIL | FAIL |
3 "Platonic" Hask
Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottom values. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads.
