nLab
fully faithful morphism

Contents

Definition

Let KK be a 2-category.

A morphism f:ABf:A\to B in KK is called (representably) fully-faithful (or sometimes just ff) if for all objects XKX \in K , the functor

K(X,A)K(X,B)K(X,A) \to K(X,B)

is full and faithful.

Remarks

Variations

This is not always the “right” notion of fully-faithfulness in a 2-category. In particular, in enriched category theory this definition does not recapture the correct notion of enriched fully-faithfulness. It is possible, however, to characterize VV-fully-faithful functors 2-categorically; see codiscrete cofibration.

Examples

In the 2-category Cat the full and faithful morphisms are precisely the full and faithful functors.

Revised on February 2, 2011 10:43:42 by Urs Schreiber (82.113.99.3)