# nLab fully faithful morphism

Contents

### Context

#### 2-category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# 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

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

It is said to be co-fully faithful or corepresentably fully faithful if for all objects $X \in K$ , the functor

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

is a full and faithful functor. Note that the shortened name “co-fully faithful” can be misleading, as a functor is corepresentably fully faithful if and only if it is representably fully faithful as a 1-cell in $Cat^{op}$ (rather than $Cat^{co}$.

## 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. In general, fully faithful morphisms should be defined with respect to a proarrow equipment (or some other context for formal category theory): in particular, this recovers $V$-fully-faithfulness. A 1-cell $f : A \to B$ in a proarrow equipment is fully faithful if the unit of the adjunction $\eta \colon 1 \to f^* f_*$ is invertible.

## Examples

In the 2-category Cat the full and faithful morphisms are precisely the full and faithful functors; and the corepresentably fully faithful morphisms are precisely the absolutely dense functors.

Last revised on April 22, 2023 at 16:10:13. See the history of this page for a list of all contributions to it.