# nLab 3-group

group theory

### Cohomology and Extensions

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Definition

A 3-group is equivalently

1. a 2-groupoid $G$ equipped with the structure of a loop space object of a connected 3-groupoid $\mathbf{B}G$ (its delooping);

2. a monoidal 2-category in which every object has an weak inverse under the tensor product.

## Properties

### Presentation by crossed complexes

Some classes of 3-groups are modeled by 2-crossed modules or crossed squares.

Revised on November 1, 2012 18:10:54 by Urs Schreiber (131.174.41.102)