nLab
faithful functor

Contents

Definition

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.

Revised on January 16, 2016 09:53:49 by Anonymous Coward (213.192.19.34)