# Full and faithful functors

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* : *C* → *D* be a functor from *C* to *D*. The functor *F* induces a function

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

**faithful**if*F*_{X,Y}is injective^{[1]}^{[2]}**full**if*F*_{X,Y}is surjective^{[2]}^{[3]}**fully faithful**(= full and faithful) if*F*_{X,Y}is bijective

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* : *X* → *Y* 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* : *C* → *D* is a full and faithful functor and then .

## Examples

- The forgetful functor
*U*:**Grp**→**Set**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
**Ab**→**Grp**is fully faithful, since each abelian group maps to a unique group, and any group homomorphism between abelian groups is preserved in**Grp**.

## See also

## Notes

## References

- Mac Lane, Saunders (September 1998).
*Categories for the Working Mathematician*(second ed.). Springer. ISBN 0-387-98403-8. - Jacobson, Nathan (2009).
*Basic algebra*.**2**(2nd ed.). Dover. ISBN 978-0-486-47187-7.