Category Theory/Functors

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This is the Functors chapter of Category Theory.

Definition[edit]

A functor is a morphism between categories. Given categories  \mathcal{B} and  \mathcal{C} , a functor  T: \mathcal{C} \to \mathcal{B} has domain  \mathcal{C} and codomain  \mathcal{B} , and consists of two suitably related functions:

  • The object function  T , which assigns to each object  c in  \mathcal{C} , an object  Tc in  \mathcal{B} .
  • The arrow function (also  T ), which assigns to each arrow  f: c \to c' in  \mathcal{C} , an arrow  Tf: Tc \to Tc' in  \mathcal{B} , such that it satisfies  T(1_c)=1_{Tc} and  T(g \circ f)=Tg \circ Tf where  g \circ f is defined.

Examples[edit]

  • The power set functor is a functor  \mathcal{P}:\textbf{Set} \to \mathbf{Set} . Its object function assigns to every set  X , its power set  \mathcal{P}X and its arrow function assigns to each map  f:X \to Y , the map  \mathcal{P}f: \mathcal{P}X \to \mathcal{P}Y .
  • The inclusion functor  \mathcal{I}:\mathcal{S} \to \mathcal{C} sends every object in a subcategory  \mathcal{S} to itself (in  \mathcal{C} ).

Types of functors[edit]

  • A functor  T: \mathcal{C} \to \mathcal{B} is an isomorphism of categories if it is a bijection on both objects and arrows.
  • A functor  T: \mathcal{C} \to \mathcal{B} is called full if, for every pair of objects  c,c' in  \mathcal{C} and every arrow  g:Tc \to Tc' in  \mathcal{B} , there exists an arrow  f:c \to c' in  \mathcal{C} with  g=Tf . In other words,  T is surjective on arrows given objects  c,c' .
  • A functor  T: \mathcal{C} \to \mathcal{B} is called faithful if, for every pair of objects  c,c' in  \mathcal{C} and every pair of parallel arrows  f_1,f_2:c \to c' in  \mathcal{C} , the equality  Tf_1 = Tf_2:Tc \to Tc' implies that  f_1=f_2 . In other words,  T is injective on arrows given objects  c,c' . The inclusion functor is faithful.
  • A functor  T:\mathcal{C} \to \mathcal{B} is called forgetful if it "forgets" some or all aspects of the structure of  \mathcal{C} .