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Showing content from https://wiki.haskell.org/Category_theory/Functor below:

Category theory/Functor - HaskellWiki

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

1. If f : A → B in 𝒜 , then F ( f ) : F ( A ) → F ( B ) in ℬ
2. If f : B → C in 𝒜 and g : A → B in 𝒜 , then F ( f ) ∘ F ( g ) = F ( f ∘ g )
3. For all objects A in 𝒜 , i d F ( A ) = F ( i d A )

• F r e e : S e t → M o n , the functor giving the free monoid over a set
• F r e e : S e t → 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

• 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 → 𝒞 and G : 𝒜 → ℬ , then F ∘ G : 𝒜 → 𝒞 , with composition defined component-wise.

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

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.

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)

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

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