In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of under the moniker 'gr-category'.

-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group.

Examples

Eilenberg-MacLane spaces

One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and , and for an abelian group.

2-groups

The definition and many properties of 2-groups are already known. can be described using crossed modules abelian,

a group homomorphism a cohomology class. These groups can be encoded as homotopy coming from the Postnikov tower since there is a fibration

, where the fibration sequence is now

.

3-groups

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy of groups. with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this as a homotopy which fits into a fibration sequence

acts on. These can be understood explicitly using the previous model of fits into a Postnikov tower with associated Serre fibration

.

n-groups

The previous construction gives the general idea of how to consider higher groups in general. For an , showing the data of an is a higher group, or simple space acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoidspg 295.

. This gives a Postnikov tower

, there is an associated fibration

. In principle with the Serre spectral sequence. Doing this recursively, say for a many spectral sequence computations for an .

n-groups from sheaf cohomology

For a complex manifold with universal cover, and a sheaf there exists canonical homomorphisms

giving a technique for relating . This is particularly applicable for complex tori.

See also

• ∞-groupoid

• Crossed module

• Homotopy hypothesis

• Abelian 2-group

References

Hoàng Xuân Sính, Gr-catégories, PhD thesis, (1973)

Algebraic models for homotopy n-types

• - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

Cohomology of higher groups

Cohomology of higher groups over a site

as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.