# 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.