nLab
faithful functor

Contents

Contents

Definition

In 1-category theory

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’.

In higher category theory

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

And see 0-truncated morphism for generalization to (∞,1)-categories (see there).

Properties

Proposition

A faithful functor reflects epimorphisms and monomorphisms.

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

Last revised on September 9, 2018 at 07:33:17. See the history of this page for a list of all contributions to it.