# nLab fully faithful morphism

### Context

#### 2-category theory

2-category theory

# Contents

## Definition

Let $K$ be a 2-category.

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

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

## 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 $V$-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)