Full and faithful functors

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In category theory, a faithful functor (resp. a full functor) is a functor that is injective (resp. surjective) when restricted to each set of morphisms that have a given source and target.

Formal definitions[edit]

Explicitly, let C and D be (locally small) categories and let F : CD be a functor from C to D. The functor F induces a function

F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))

for every pair of objects X and Y in C. The functor F is said to be

for each X and Y in C.

Properties[edit]

A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : XY and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : CD is a full and faithful functor and F(X)\cong F(Y) then X \cong Y.

Examples[edit]

  • The forgetful functor U : GrpSet is faithful as each group maps to a unique set and the group homomorphism are a subset of the functions. This functor is not full as there are functions between groups which are not group homomorphisms. A category with a faithful functor to Set is (by definition) a concrete category; in general, that forgetful functor is not full.
  • The inclusion functor AbGrp is fully faithful, since each abelian group maps to a unique group, and any group homomorphism between abelian groups is preserved in Ab.

See also[edit]

Notes[edit]

  1. ^ Mac Lane (1971), p. 15
  2. ^ a b Jacobson (2009), p. 22
  3. ^ Mac Lane (1971), p. 14

References[edit]