n-group in nLab

Contents

Contents

Definition

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

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

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

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

Examples

• A 11-group is simply a group; see also 2-group and 3-group.

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

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

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

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)