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Hask

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Revision as of 04:50, 6 September 2012

Hask is the category of Haskell types and functions.

The objects of Hask are Haskell types, and the morphisms from objects
A
to
B
are Haskell functions of type
A -> B
. The identity morphism for object
A
is
id :: A
, and the composition of morphisms
f
and
g
is
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 () = ()
This might be a problem, because
undef1 . id = undef2
. In order to make Hask a category, we define two functions
f
and
g
as the same morphism if
f x = g x
for all
x
. Thus
undef1
and
undef2
are different values, but the same morphism in Hask.

2 Hask is not Cartesian closed

Actual Hask does not have sums, products, or an initial object, and
()
is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
Why Hask isn't as nice as you'd thought.
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
Platonic 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. Instances of Functor and Monad really are endofunctors and monads.

4 Links

Retrieved from "http://www.haskell.org/haskellwiki/Hask"

Categories: Mathematics | Theoretical foundations