full and faithful functor

A **full and faithful functor** is a functor which is both full and faithful. “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 $Cat$. 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 September 30, 2016 03:59:09
by Urs Schreiber
(89.15.237.91)