nLab n-group

Contents

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.