Full and faithful functors

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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 with 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, and two morphisms f : X → Y and f′ : X′ → Y′ 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.

[edit] Examples

• The forgetful functor U : Grp → Set is faithful but is not injective on objects or on morphisms. 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.

• Let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function. Then F is full, but is not surjective on objects or on morphisms.

• The forgetful functor Ab → Grp is fully faithful. It is injective on both objects and morphisms, but is surjective on neither.

[edit] See also

• full subcategory
• equivalence of categories