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).

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.

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

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