# Category theory/Functor

## Contents

## Definition of a Functor

Given that and are categories, a *functor* is a pair of mappings (the subscripts are generally omitted in practice).

### Axioms

- If in , then in
- If in and in , then
- For all objects in ,

### Examples of functors

- , the functor giving the free monoid over a set
- , 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 , there is an
**identity functor**(again, the subscript is usually ommitted) given by the rule for all objects and arrows . - If and , then , 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 , there exists a category whose arrows are all functors between categories in . 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)
```