nLab
full and faithful functor

Contents

Definition

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.

Properties

Example

A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits.

This is evident from inspection of the defining universal property.

Higher categorical generalizations

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)