nLab
full functor

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

$$F:\mathrm{hom}(x,y)\to \mathrm{hom}(F(x),F(y))$$F : hom(x,y) \to hom(F(x), F(y))

is onto.

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.

Discussion

Mathieu says: I agree that, for functors, there is no reason to say “fully faithful” rather than “full and faithful”. But for arrows in a 2-category (like in the new version of the entry on subcategories), there are reasons. I quote myself (from my thesis): «Remark: we say fully faithful and not full and faithful, because the condition that, for all $X:C$, $C(X,f)$ be full is not equivalent in $Grpd$ to $f$ being full. Moreover, in $Grpd$, this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a $Grpd$-category which, in $Grpd$ and $Symm2Grp$ (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for $Cat$, for example.

Mike says: Do you have a reason to care about full functors which are not also faithful? I’ve never seen a very compelling one. (Maybe I should just read your thesis…) I agree that “full morphism” (in the representable sense) is not really a useful/correct concept in a general 2-category, and that therefore “full and faithful” is not entirely appropriate, so I usually use “ff” in that context. I’ve changed the entry above a bit to reflect your comment; is it satisfactory now? Maybe all this should actually go at full and faithful functor (and/or fully faithful functor)?