nLab
n-group

Definition

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

If it is deloopable, an $n$-group $G$ is the hom-object $G={\mathrm{Aut}}_{BG}(*)$ of an n-groupoid $BG$ with a single object $*$.

If $BG$ is a strict n-groupoid, then the corresponding $n$-group is called strict. 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.

Examples

• circle n-group

• string 2-group

• fivebrane 6-group

Related concepts

See also

• group

• 2-group

  • strict 2-group
  • crossed module

• cat-n-group

• k-tuply groupal n-groupoid .

• ∞-group

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