# nLab full sub-2-category

Contents

### Context

#### 2-Category theory

2-category theory

# Contents

## Definition

A 2-functor $F : C \to D$ exhibits the 2-category $C$ as a full sub-2-category of $D$ if for all objects $c_1,c_2 \in C$ the component functor $F_{c_1, c_2}$ is an equivalence of categories

$F_{c_1, c_2} : C(c_1,c_2) \stackrel{\simeq}{\to} D(F(c_1), F(c_2)) \,,$

hence if $F$ is a 2-fully-faithful 2-functor.

## Properties

If $C$ and $D$ are ordinary categories regarded as 2-categories, a full sub 2-category $F : C \hookrightarrow D$ is equivalently a full subcategory of $D$.

If $D$ is a (2,1)-category a full sub-2-category is equivalently a full sub-(∞,1)-category.

Last revised on May 27, 2020 at 12:37:25. See the history of this page for a list of all contributions to it.