# nLab n-group

group theory

### Cohomology and Extensions

#### Higher category theory

higher category theory

# Contents

## Definition

An $n$-group is a group object internal to $(n-1)$-groupoids.

If it is deloopable, an $n$-group $G$ is the hom-object $G = Aut_{\mathbf{B}G}({*})$ of an n-groupoid $\mathbf{B}G$ with a single object ${*}$.

If $\mathbf{B}G$ is a strict n-groupoid, then the corresponding $n$-group is called a strict $n$-group. Strict $n$-groups are equivalent to crossed complexes of groups, of length $n$.

Under the homotopy hypothesis $n$-groups correspond to pointed connected homotopy n-types.

## References

The homotopy theory of k-tuply groupal n-groupoids is discussed in

• A.R. Garzón, J.G. Miranda, Serre homotopy theory in subcategories of simplicial groups Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123

Revised on February 25, 2013 21:54:04 by Tim Porter (95.147.236.96)