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

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • essentially surjective functor

    • surjective on objects functor

    • bijective on objects functor

  • essentially surjective and full functor

  • equivalence of categories

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories