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| Hask is the largest category of types in Haskell but you can find more well behaved categories. See also the [[http://hackage.haskell.org/package/category-extras|category-extras]] package. | Hask is the largest category of types in Haskell but you can find more well behaved categories. See also the [[http://hackage.haskell.org/package/category-extras|category-extras]] package. | ||
| - | Another deficit is that when it comes to passing functions as arguments, Haskell sees more than just Hask morphisms. A nicer definition of the largest category of Haskell would be if arrows weren't Haskell functions identified by function extensionally ($\forall x. f(x)=g(x)\implies f=g$), but rather identified if equivalent when passed to other Haskell functions as arguments ($\forall h. h(f)=h(g)\implies f=g$). However, this doesn't work: With the definition of concatenation given above, the category laws imply that '(undefined :: Int -> Char).id' must be '\x ->(undefined :: Int -> Char) x'. But there are functions which can detect the non-extensive property of 'undefined :: Int -> Char' being the 'undefined' term for the type 'Int -> Char'. The function 'seq' (which is implemented to make enforcement of strict evaluation possible) will return a different result when passed the extensionally equal functions 'undefined :: Int -> Char' and '\x ->(undefined :: Int -> Char) x'. | + | Another deficit is that when it comes to passing functions as arguments, Haskell sees more than just Hask morphisms. A nicer definition of the largest category of Haskell would be if arrows weren't Haskell functions identified by function extensionally |
| + | ($\forall x. f(x)=g(x)\implies f=g$), | ||
| + | but rather identified if equivalent when passed to other Haskell functions as arguments | ||
| + | ($\forall h. h(f)=h(g)\implies f=g$, see indiscernibility of identicals ). | ||
| + | However, this doesn't work: With the definition of concatenation given above, the category laws imply that '(undefined :: Int -> Char).id' must be '\x ->(undefined :: Int -> Char) x'. But there are functions which can detect the non-extensive property of 'undefined :: Int -> Char' being the 'undefined' term for the type 'Int -> Char'. The function 'seq' (which is implemented to make enforcement of strict evaluation possible) will return a different result when passed the extensionally equal functions 'undefined :: Int -> Char' and '\x ->(undefined :: Int -> Char) x'. | ||
| === Functors === | === Functors === | ||
