# Full and faithful functors

(Redirected from Faithful functor)

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

## Formal definitions

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

${\displaystyle F_{X,Y}\colon \mathrm {Hom} _{\mathcal {C}}(X,Y)\rightarrow \mathrm {Hom} _{\mathcal {D}}(F(X),F(Y))}$

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

for each X and Y in C.

## Properties

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 : XY 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.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : CD is a full and faithful functor and ${\displaystyle F(X)\cong F(Y)}$ then ${\displaystyle X\cong Y}$.

## Examples

• The forgetful functor U : GrpSet is faithful as two group homomorphisms with the same domains and codomains are equal if they are given by the same functions on the underlying sets. This functor is not full as there are functions between groups that 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.
• The inclusion functor AbGrp is fully faithful, since Ab is by definition the full subcategory of Grp induced by the abelian groups.