n-group (category theory)

Not to be confused with p-group.

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. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the monniker 'gr-category'.

The general definition of n-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy n-group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π_n, or the entire Postnikov tower for n = ∞.

The definition and many properties of 2-groups are already known. A 1-group is simply a group, and the only 0-group is trivial. 2-groups can be described using crossed modules.

References

Hoàng Xuân Sính, Gr-catégories, PhD thesis, (1973)
John C. Baez and Aaron D. Lauda, Higher-Dimensional Algebra V: 2-Groups, Theory and Applications of Categories 12 (2004), 423–491.
David Michael Roberts and Urs Schreiber, The inner automorphism 3-group of a strict 2-group, Journal of Homotopy and Related Structures, vol. 3(1) (2008), pp.193–245.

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