full and faithful functor



A full and faithful functor is a functor which is both full and faithful. That is, a functor F:CDF\colon C \to D from a category CC to a category DD is called full and faithful if for each pair of objects x,yCx, y \in C, the function

F:C(x,y)D(F(x),F(y))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.



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 July 13, 2017 15:14:59 by Anonymous (