nLab full functor

Definition

A functor $F\colon C \to D$ from a category $C$ to a category $D$ is called full if for each pair of objects $x, y \in C$, the function

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

between hom sets is surjective.

More abstractly, we may say a functor is full if it is 1-surjective – or, in simple terms, ‘surjective on morphisms between given objects’. (Note that a functor may be full without being surjective on morphisms, overall, since $F$ is allowed to not hit morphisms between objects that are not in the image of $F$.)

Fullness is most important for functors which are also faithful, and full and faithful functors are often called fully faithful. For ordinary functors this may sound odd, because there is no real sense in which “full” modifies “faithful.” However, in some contexts (such as for morphisms in a general 2-category), there is a good notion of “full-and-faithful” or “fully faithful,” but the right notion of “full” alone is not so clear. “Fully faithful” is also sometimes abbreviated to “ff”; see also bo-ff factorization system.

A subcategory is called a full subcategory if its inclusion functor (which is automatically faithful) is also full, and any full and faithful functor exhibits an equivalence of its domain with a full subcategory of its codomain.

Examples

Proposition

Given a pair of adjoint functors

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} { \phantom{AA} \bot \phantom{AA} } \mathcal{D}$$

• the right adjoint $R$ is a full functor precisely if the component of the counit over every object $x$ is a split monomorphism $L R x \stackrel{}{\to} x$;

• the left adjoint $L$ is a full functor precisely if the component of the unit over every object $x$ is a split epimorphism $x \to R L x$.

For proof see this Prop at adjoint functor.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • 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

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense 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