full and faithful functor

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

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

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.

This is evident from inspection of the defining universal property.

There is a bigger pattern at work here which is indicated at stuff, structure, property and k-surjective functor.

For (∞,1)-categories the corresponding notion of fully faithful functor is described at

Revised on July 13, 2017 15:14:59
by Anonymous
(166.70.31.28)