faithful functor



In 1-category theory

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

𝒞(x,y) F x,y 𝒟(F(x),F(y)) (xϕy) (F(x)F(ϕ)F(y)). \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 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).


This generalization is about extending to morphisms in general (∞,1)-categories the fact that in Grpd\infty{}Grpd, 00-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”.



A faithful functor reflects epimorphisms and monomorphisms.

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

basic properties of…