nLab faithful functor

Definition

In 1-category theory

A functor $F \colon \mathcal{C} \to \mathcal{D}$ from a category $\mathcal{C}$ to a category $\mathcal{D}$ is called faithful, if for each pair of objects $x, y \in \mathcal{C}$, its function $F_{x,y}$ on hom-sets is injective:

$$\array{ \mathcal{C}(x,y) &\xhookrightarrow{\;\; F_{x,y} \;\;}& \mathcal{D}(F(x), F(y)) \\ (x \overset{\phi}{\to} y) &\mapsto& \big( F(x)\overset{F(\phi)}{\to} F(y) \big) \,. }$$

More abstractly, we may say that 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).

Warning

This generalization is about extending to morphisms in general (∞,1)-categories the fact that in $\infty{}Grpd$, $0$-truncated morphisms give a reasonable notion of faithful functor.

In particular, the notion of a “faithful morphism in the (∞,1)-category of (∞,1)-categories” does not give the right notion of a “faithful functor between (∞,1)-categories”.

Properties

Proposition

A faithful functor reflects epimorphisms and monomorphisms.

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

Related concepts

• embedding of categories

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor

• 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

  • homotopy final functor, final (∞,1)-functor