Full and Faithful Functors

Full And Faithful Functors

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

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

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

• faithful if F_X,Y is injective
• full if F_X,Y is surjective
• fully faithful if F_X,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.