A full and faithful functor is a functor which is both full and faithful. That is, a functor $F\colon C \to D$ from a category $C$ to a category $D$ is called full and faithful if for each pair of objects $x, y \in C$, the function
between hom sets is bijective. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also full subcategory.
Together with bijective-on-objects functors, fully faithful functors form an orthogonal factorization system on $Cat$; see bo-ff factorization system. More invariantly, pair them with essentially surjective functors to get a bicategorial factorization system.
A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits, also all isomorphisms (is a conservative functor). This is evident from inspection of the defining universal property.
For (∞,1)-categories the corresponding notion of fully faithful functor is described at fully faithful (∞,1)-functor. This is part of a bigger pattern at work here which is indicated at stuff, structure, property and k-surjective functor.
Inside a 2-category there is a “representable” notion of ff morphism.
There is also a notion of fully faithful functor in enriched category theory, which in general is stronger than being an ff morphism in the 2-category of enriched categories. But it can be expressed internally in any proarrow equipment.
Last revised on November 24, 2017 at 09:42:19. See the history of this page for a list of all contributions to it.