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
- <math>F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(FX,FY)</math>
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 if FX,Y is bijective
for each X and Y in 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.
Examples
The forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms. This functor is not full as there are functions between groups which are not group homomorphisms. More generally, for any concrete category the forgetful functor to Set is faithful (but usually 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 neither surjective on objects or morphisms.
The forgetful functor Ab → Grp is fully faithful. However, it is neither surjective on objects or morphisms.