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.
The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'.
However, it is expected that every topological space will have a homotopy
-group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group
One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces
since they are the fundamental building blocks for constructing higher groups, and homotopy types in general.
through a simplicial construction,[1] and it behaves functorially.
This construction gives an equivalence between groups and 1-groups.
Note that some authors write
2-groups can be described using crossed modules and their classifying spaces.
, where the fibration sequence is now coming from a map
Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 3-types of groups.
with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower.
, the existence of universal covers gives us a homotopy type
which fits into a fibration sequence giving a homotopy
These can be understood explicitly using the previous model of 2-groups, shifted up by degree (called delooping).
fits into a Postnikov tower with associated Serre fibration giving where the
, giving a cohomology class in
The previous construction gives the general idea of how to consider higher groups in general.
with the latter bunch being abelian, we can consider the associated homotopy type
, making it easier to construct the rest of the homotopy type using the Postnikov tower.
-group is a higher group, or simple space, with trivial
This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoids[3]pg 295 since the groupoid structure models the homotopy quotient
This gives a Postnikov tower where the first non-trivial map
In principle[4] this cohomology group should be computable using the previous fibration
with the Serre spectral sequence with the correct coefficients, namely
-group, would require several spectral sequence computations, at worst
there exists[5] canonical homomorphisms giving a technique for relating n-groups constructed from a complex manifold
Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space
, giving higher cohomology groups