#
nLab

empty category

### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Definition

The **empty category** is the category with no objects, and hence no morphisms, i.e. the category whose class of objects is the empty set.

This is the initial object in Cat.

This is a groupoid, so we may call it the **empty groupoid**. One may similarly speak of the **empty n-category**, the **empty infinity-groupoid**, etc etc etc.

## Properties

The empty category is discrete, hence may be identified with a set: the empty set. This set is a subsingleton, so we may also identify it with a truth value: the false one.

The empty category is initial in Cat (as well as Grpd, ∞Grpd, Set, etc).