nLab
faithful functor

Definition

In 1-category theory

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

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

between hom sets is injective.

More abstractly, we may say a functor is faithful if it is $2$-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.)

Related concepts

• essentially surjective functor

• full functor

• full and faithful functor

• equivalence of categories