Faithful functor

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

In other words, a functor F : C → D is faithful if the maps

$F_{X,Y}:\mathrm{Mor}_{\mathcal C}(X,Y)\rightarrow\mathrm{Mor}_{\mathcal D}(FX,FY)$

are injective for every pair of objects X and Y in C.

Note that 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.

For example, the forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms.

See also:

full functor
subcategory

Faithful functor

See also: Faithful functor, Category theory, Forgetful functor, Full functor, Functor, Injective, Morphism, Subcategory