# Faithful functor

A functor which is "injective on Hom-sets" . Explicitly, a functor $ F : \mathfrak C \rightarrow \mathfrak D $
is called faithful if, given any two morphisms $ \alpha , \beta : A \rightarrow B $
in $ \mathfrak C $
with the same domain and codomain, the equation $ F \alpha = F \beta $
implies $ \alpha = \beta $.
The name derives from the representation theory of groups: a permutation (respectively, $ R $-
linear) representation of a group $ G $
is faithful if and only if it is faithful when considered as a functor $ G \rightarrow \mathop{\rm Set} $(
respectively $ G \rightarrow \mathop{\rm Mod} _ {R} $).
A faithful functor reflects monomorphisms (that is, $ F \alpha $
monic implies $ \alpha $
monic) and epimorphisms; hence if the domain category $ \mathfrak C $
is balanced (i.e. has the property that any morphism which is both monic and epic is an isomorphism) then it also reflects isomorphisms. A functor with the latter property is generally called conservative; however, some authors include this condition in the definition of faithfulness.

In Russian literature there seems to be some confusion between the terms "faithful functor" and "exact functor" , see also Exact functor.

#### References

[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |

**How to Cite This Entry:**

Faithful functor.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Faithful_functor&oldid=51478