faithful functor



A functor F:CDF: C \to D from the category CC to the category DD is faithful if for each pair of objects x,yCx, y \in C, the function

F:C(x,y)D(F(x),F(y))F : C(x,y) \to D(F(x), F(y))

between hom sets is injective.

More abstractly, we may say a functor is faithful if it is 22-surjective – or loosely speaking, ‘surjective on equations between given morphisms’.

See also faithful morphism for a generalization to an arbitrary 2-category.



A faithful functor reflects epimorphisms and monomorphisms.

(The simple proof is spelled out for instance at epimorphism.)

Last revised on September 30, 2016 at 04:00:16. See the history of this page for a list of all contributions to it.