# Full and faithful functors

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In category theory, a faithful functor (resp. a full functor) is a functor that is injective (resp. 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 each group maps to a unique set and the group homomorphism are a subset of the functions. 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.
• The inclusion functor AbGrp is fully faithful, since each abelian group maps to a unique group, and any group homomorphism between abelian groups is preserved in Grp.

## Notes

1. ^ Mac Lane (1971), p. 15
2. ^ a b Jacobson (2009), p. 22
3. ^ Mac Lane (1971), p. 14