Faithful functor
A functor which is "injective on Hom-sets" . Explicitly, a functor is called faithful if, given any two morphisms
in
with the same domain and codomain, the equation
implies
. The name derives from the representation theory of groups: a permutation (respectively,
-linear) representation of a group
is faithful if and only if it is faithful when considered as a functor
(respectively
). A faithful functor reflects monomorphisms (that is,
monic implies
monic) and epimorphisms; hence if the domain category
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) |
Faithful functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Faithful_functor&oldid=17844