Introduction to Category Theory/Functors

Functor

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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

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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.

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