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I appreciate orbifold taking the time to reply but I couldn't understand his answer at all, even after perfectly understanding what others have said. There are

I appreciate orbifold taking the time to reply but I couldn't understand his answer at all, even after perfectly understanding what others have said. There are 3 sentences in that paragraph the second of which is very large and difficult to parse and missing an end parenthesis somewhere. Why do you get a composition of functions by applying a functor to individual functions? It seems like a functor is just something you can map over. When a functor is mapped over the end result is another functor. Not a function nor a composition of functions.

Dewie 12y ago
> Why do you get a composition of functions by applying a functor to individual functions? It seems like a functor is just something you can map over. When a functor is mapped over the end result is another functor. Not a function nor a composition of functions. A function is a Functor. ;) If you have a function f and a function g, fmap f g = h, and h is a function. A more common way to write this is function composition of f and g, which can be written: f . g. The dot (.) is function composition. Which means that fmap f g = f . g, which means that fmap = (.). > When a functor is mapped over the end result is another functor. Not a function nor a composition of functions. But they was describing the case when the Functor in question is a function.
orbifold 12y ago
I was unsure what notation to use, so I used english instead. In Haskell a Functor f is characterized by a higher order function fmap :: (a -> b) -> (f a -> f b), which is required to satisfy the laws fmap (g . f) = (fmap g) . (fmap f) and fmap id = id where (.) :: (b -> c) -> (a -> b) -> (a -> c) is the function composition operator and id :: a -> a the identity function. Those are the laws the last two sentences try to phrase in english. In mathematics a functor F between categories C and D maps objects in C to objects in D and any morphism f: X -> Y in C to a morphism F f : F X -> F Y in D, in such a way that for morphisms f : X -> Y and g : Y -> Z in C one has F (g . f) = F g . F f and F id_X = id_{F X}. So you see Haskell and math notation are almost identical, although you can express the laws only as compiler rules in Haskell.
dllthomas 12y ago
I think it gets worlds better with diagrams. I'll try to revisit this and provide some, when I've a bit more time.

Dewie 12y ago

> Why do you get a composition of functions by applying a functor to individual functions? It seems like a functor is just something you can map over. When a functor is mapped over the end result is another functor. Not a function nor a composition of functions. A function is a Functor. ;) If you have a function f and a function g, fmap f g = h, and h is a function. A more common way to write this is function composition of f and g, which can be written: f . g. The dot (.) is function composition. Which means that fmap f g = f . g, which means that fmap = (.). > When a functor is mapped over the end result is another functor. Not a function nor a composition of functions. But they was describing the case when the Functor in question is a function.

orbifold 12y ago

I was unsure what notation to use, so I used english instead. In Haskell a Functor f is characterized by a higher order function fmap :: (a -> b) -> (f a -> f b), which is required to satisfy the laws fmap (g . f) = (fmap g) . (fmap f) and fmap id = id where (.) :: (b -> c) -> (a -> b) -> (a -> c) is the function composition operator and id :: a -> a the identity function. Those are the laws the last two sentences try to phrase in english. In mathematics a functor F between categories C and D maps objects in C to objects in D and any morphism f: X -> Y in C to a morphism F f : F X -> F Y in D, in such a way that for morphisms f : X -> Y and g : Y -> Z in C one has F (g . f) = F g . F f and F id_X = id_{F X}. So you see Haskell and math notation are almost identical, although you can express the laws only as compiler rules in Haskell.

dllthomas 12y ago

I think it gets worlds better with diagrams. I'll try to revisit this and provide some, when I've a bit more time.