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Hask

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Hask is the name usually given to the category having Haskell types as objects and Haskell functions between them as morphisms.

A type-constructor that is an instance of the Functor class is an endofunctor on Hask.

A solution approach to the issue of partiality making many of the identities required by categorical constructions not literally true in Haskell:


1 The seq problem

The right identity law fails in Hask if we distinguish values which can be distinguished by
seq
, since:
id . undefined = \x -> id (undefined x) = \x -> undefined x
should be equal to
undefined
, but can be distinguished from it using
seq
: ghci>
(undefined :: Int -> Int) `seq` ()
   * Exception: Prelude.undefined
ghci>
(id . undefined :: Int -> Int) `seq` ()
   ()

2 The limits problem

Even in the absence of seq, bottoms cause datatypes to not actually be instances of the expected categorical constructions. For instance, using some intuition from the category of sets, one might expect the following:

data Void    -- no elements ; initial object
data () = () -- terminal object
 
data (a, b) = (a, b)               -- product
data Either a b = Left a | Right b -- coproduct

However, Void actually does contain an element, bottom, so for each x :: T, const x is a different function Void -> T, meaning Void isn't initial (it's actually terminal).

Similarly, const undefined and const () are two distinct functions into (). Consider:

t :: () -> Int
t () = 5
 
t . const ()        = \x -> 5
t . const undefined = \x -> undefined

So, () is not terminal.

Similar issues occur with (co)products. Categorically:

\p -> (fst p, snd p) = id
 
\s -> case s of Left x -> p (Left x) ; Right y -> p (Right y) = p

but in Haskell

id undefined = undefined /= (undefined, undefined) = (fst undefined, snd undefined)
 
const 5 undefined = 5
  /= undefined = case undefined of 
                   Left x  -> const 5 (Left x)
                   Right y -> const 5 (Right y)

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