Hi Corey, On Dec 14, 2007 8:44 PM, Corey O'Connor <[EMAIL PROTECTED]> wrote: > The reason I find all this odd is because I'm not sure how the type > class Functor relates to the category theory concept of a functor. How > does declaring a type constructor to be an instance of the Functor > class relate to a functor? Is the type constructor considered a > functor?
Recall the definition of functor. From Wikipedia: "A functor F from C to D is a mapping that * associates to each object X in C an object F(X) in D, * associates to each morphism f:X -> Y in C a morphism F(f):F(X) -> F(Y) in D such that the following two properties hold: * F(idX) = idF(X) for every object X in C * F(g . f) = F(g) . F(f) for all morphisms f:X -> Y and g:Y -> Z." http://en.wikipedia.org/wiki/Functor We consider C = D = the category of types. Note that any type constructor is a mapping from types to types -- thus it associates to each object (type) X an object (type) F(X). Declaring a type constructor to be an instance of Functor means that you have to provide 'fmap :: (a -> b) -> (f a -> f b)" -- that is, a mapping that associates to each morphism (function) "fn :: a -> b" a morphism "fmap fn :: f a -> f b". Making sure that the two laws are fulfilled is the responsibility of the programmer writing the instance of Functor. (I.e., you're not supposed to do this: instance Functor Val where fmap f (Val x) = Val (x+1).) Hope this helps with seeing the correspondence? :-) - Benja _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe