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We've spent quite a bit of time talking about categories, and special entities in them - morphisms and objects, and special kinds of them, and properties we can find.
And one of the main messages visible so far is that as soon as we have an algebraic structure, and homomorphisms, this forms a category. More importantly, many algebraic structures, and algebraic theories, can be captured by studying the structure of the category they form.
So obviously, in order to understand Category Theory, one key will be to understand homomorphisms between categories.
1.1 Homomorphisms of categories
A category is a graph, so a homomorphism of a category should be a homomorphism of a graph that respect the extra structure. Thus, we are led to the definition:
Definition A functor from a category C to a category D is a graph homomorphism F0,F1 between the underlying graphs such that for every object :
- F1(gf) = F1(g)F1(f)
Note: We shall consistently use F in place of F0 and F1. The context should be able to tell you whether you are mapping an object or a morphism at any given moment.
1.1.1 Examples and non-examples
- Monoid homomorphisms
- Monotone functions between posets
- Pick a basis for every vectorspace, send and to the matrix representing that morphism in the chosen bases.
1.2 Interpreting functors in Haskell
One example of particular interest to us is the category Hask. A functor in Hask is something that takes a type, and returns a new type. Not only that, we also require that it takes arrows and return new arrows. So let's pick all this apart for a minute or two.Taking a type and returning a type means that you are really building a polymorphic type class: you have a family of types parametrized by some type variable. For each type
The rules we expect a Functor to obey seem obvious: translating from the categorical intuition we arrive at the rules
- andfmap id = id
- fmap (g . f) = fmap g . fmap f
data Boring a = Boring instance Functor Boring where fmap f = const Boring
data List a = Nil | Cons a (List a) instance Functor List where fmap f Nil = Nil fmap f (Cons x lst) = Cons (f x) (fmap f lst) data Maybe a = Nothing | Just a instance Functor Maybe where fmap f Nothing = Nothing fmap f (Just x) = Just (f x) data Either b a = Left b | Right a instance Functor (Either b) where fmap f (Left x) = Left x fmap f (Right y) = Right (f y) data LeafTree a = Leaf a | Node [LeafTree a] instance Functor LeafTree where fmap f (Node subtrees) = Node (map (fmap f) subtrees) fmap f (Leaf x) = Leaf (f x) data NodeTree a = Leaf | Node a [NodeTree a] instance Functor NodeTree where fmap f Leaf = Leaf fmap f (Node x subtrees) = Node (f x) (map (fmap f) subtrees)
2 Natural transformations
3 The category of categories
- For now, I wanna introduce functors as morphisms of categories, then introduce the category of categories, and the functor categories, and then talk about functors as containers and the HAskell way of dealing with them.