Introduction to Category Theory/Functors

From Wikiversity
Jump to navigation Jump to search

Functor[edit | edit source]

A structure-preserving map between categories is called functor. A (covariant) functor F from category to category satisfies

  • sends objects of to objects of .
  • sends arrows of to arrows of .
  • If is an arrow from to in , then is an arrow from to in .
  • sends identity arrows to identity arrows: .
  • preserves compositions: .

A contravariant functor reverses arrows:

  • If is an arrow from to in , then is an arrow from to in .
  • preserves compositions: .

Natural Transformations[edit | edit source]

If F and G are covariant functors between the categories and , then a natural transformation from F to G associates to every object X in a morphism in called the component of at X, such that for every morphism in we have . This equation can conveniently be expressed by the commutative diagram

diagram defining natural transformations

If both F and G are contravariant, the horizontal arrows in this diagram are reversed. If is a natural transformation from F to G, we also write . This is also expressed by saying the family of morphisms is natural in X.

If, for every object X in C, the morphism is an isomorphism in , then is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.

Natural transformations are usually far more natural than the definition above.

Related resources[edit | edit source]