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.

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