Difference between revisions of "Category theory/Functor"
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=== The category Cat === |
=== The category Cat === |
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− | The existence of identity and composition functors implies that, for any well-defined collection of categories <math>E</math>, there exists a category <math>\mathrm{Cat}_E</math> whose arrows are all functors between categories in <math>E</math>. Since no category can include itself as an object, there can be no category of all categories, but it is |
+ | The existence of identity and composition functors implies that, for any well-defined collection of categories <math>E</math>, there exists a category <math>\mathrm{Cat}_E</math> whose arrows are all functors between categories in <math>E</math>. 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 === |
=== Functors in Haskell === |
Revision as of 07:23, 17 May 2009
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)