Full and faithful functors

In category theory, a faithful or full functor is a functor which is injective or surjective when restricted to each set of morphisms with a given source and target.

Related Topics:
Category theory - Functor - Injective - Surjective - Morphism

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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

Related Topics:
Locally small - Categories

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:F_{X,Y}colonmathrm{Hom}_{mathcal C}(X,Y) ightarrowmathrm{Hom}_{mathcal D}(FX,FY)

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

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• 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 in C.

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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.

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Full and faithful functors: Examples ►