Category theory/Functor

Definition of a Functor

Given that $A$ and $B$ are categories, a functor $F : A \to B$ is a pair of mappings $(F_{o b j e c t s} : O b (A) \to O b (B), F_{a r r o w s} : A r (A) \to A r (B))$ (the subscripts are generally omitted in practice).

Axioms

If $f : A \to B$ in $A$ , then $F (f) : F (A) \to F (B)$ in $B$
If $f : B \to C$ in $A$ and $g : A \to B$ in $A$ , then $F (f) \circ F (g) = F (f \circ g)$
For all objects $A$ in $A$ , $i d_{F (A)} = F (i d_{A})$

Examples of functors

$F r e e : S e t \to M o n$ , the functor giving the free monoid over a set
$F r e e : S e t \to G r p$ , the functor giving the free group over a set
Every monotone function is a functor, when the underlying partial orders are viewed as categories
Every monoid homomorphism is a functor, when the underlying monoids are viewed as categories

Functor operations

For all categories $C$ , there is an identity functor $I d_{C}$ (again, the subscript is usually ommitted) given by the rule $F (a) = a$ for all objects and arrows $a$ .
If $F : B \to C$ and $G : A \to B$ , then $F \circ G : A \to C$ , with composition defined component-wise.

These operations will be important in the definition of a monad.

The category Cat

The existence of identity and composition functors implies that, for any well-defined collection of categories $E$ , there exists a category ${C a t}_{E}$ whose arrows are all functors between categories in $E$ . Since no category can include itself as an object, there can be no category of all categories, but it is common and useful to designate a category small when the collection of objects is a set, and define Cat to be the category whose objects are all small categories and whose arrows are all functors on small categories.

Functors in Haskell

Properly speaking, a functor in the category Haskell is a pair of a set-theoretic function on Haskell types and a set-theoretic function on Haskell functions satisfying the axioms. However, Haskell being a functional language, Haskellers are only interested in functors where both the object and arrow mappings can be defined and named in Haskell; this effectively restricts them to functors where the object map is a Haskell data constructor and the arrow map is a polymorphic function, the same constraints imposed by the class Functor:

class Functor f where
  fmap :: (a -> b) -> (f a -> f b)

External links

Wikipedia's Functor article
nlab's functor and endofunctor articles
Typeclassopedia Functor article