Hask

Hask refers to a category with types as objects and functions between them as morphisms. However, its use is ambiguous. Sometimes it refers to Haskell (actual Hask), and sometimes it refers to some subset of Haskell where no values are bottom and all functions terminate (platonic Hask). The reason for this is that platonic Hask has lots of nice properties that actual Hask does not, and is thus easier to reason in. There is a faithful functor from platonic Hask to actual Hask allowing programmers to think in the former to write code in the latter.

Hask

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
Definition	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
Platonic candidate	u1 r = case r of {}	u1 _ = ()	u1 (Left a) = f a u1 (Right b) = g b	u1 r = (f r,g r)
Example failure condition	r ~ ()	r ~ ()	r ~ () f _ = () g _ = ()	r ~ () f _ = undefined 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
Result	FAIL	FAIL	FAIL	FAIL

Platonic Hask

Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottoms. 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.

Links

• Makoto Hamana: What is the category for Haskell?
• Nils A. Danielsson, John Hughes, Patrik Jansson, and Jeremy Gibbons. Fast and loose reasoning is morally correct.