nLab n-group

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.

Examples

• A $1$-group is simply a group; see also 2-group and 3-group.

• For $n \lt 1$, there is a single $n$-group, the point.

• For arbitrary $n$, there is a circle n-group.

• In string theory, we have the string 2-group and the fivebrane 6-group.

Related concepts

See also

• group

• 2-group

  • strict 2-group

  • crossed module

• 3-group

• 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 (doi:10.1016/S0022-4049(98)00143-1)