Full and faithful functors
for every pair of objects X and Y in C. The functor F is said to be
- faithful if FX,Y is injective
- full if FX,Y is surjective
- fully faithful (= full and faithful) if FX,Y is bijective
for each X and Y in C.
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 : X → Y 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 : C → D is a full and faithful functor and then .
- The forgetful functor U : Grp → Set 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 Ab → Grp is fully faithful, since each abelian group maps to a unique group, and any group homomorphism between abelian groups is preserved in Grp.
- Mac Lane (1971), p. 15
- Jacobson (2009), p. 22
- Mac Lane (1971), p. 14