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, 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 moniker 'gr-category'.
The general definition of -group is a matter of ongoing research. 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 , or the entire Postnikov tower for .
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. For instance, every group can be turned into an Eilenberg-Maclane space through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write as , and for an abelian group , is written as .
The definition and many properties of 2-groups are already known. 2-groups can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple where are groups with abelian,
a group morphism, and a cohomology class. These groups can be encoded as homotopy -types with and , with the action coming from the action of on higher homotopy groups, and coming from the Postnikov tower since there is a fibration
coming from a map . Note that this idea can be used to construct other higher groups with group data having trivial middle groups , where the fibration sequence is now
coming from a map whose homotopy class is an element of .
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. Essential, these are given by a triple of groups with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this 3-group as a homotopy 3-type , the existence of universal covers gives us a homotopy type which fits into a fibration sequence
giving a homotopy type with trivial on which acts on. These can be understood explicitly using the previous model of -groups, shifted up by degree (called delooping). Explicitly, fits into a postnikov tower with associated Serre fibration
giving where the -bundle comes from a map , giving a cohomology class in . Then, can be reconstructed using a homotopy quotient .
The previous construction gives the general idea of how to consider higher groups in general. For an n group with groups with the latter bunch being abelian, we can consider the associated homotopy type and first consider the universal cover . Then, this is a space with trivial , making it easier to construct the rest of the homotopy type using the postnikov tower. Then, the homotopy quotient gives a reconstruction of , showing the data of an -group is a higher group, or Simple space, with trivial such that a group 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 since the groupoid structure models the homotopy quotient .
Going through the construction of a 4-group is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume is trivial, so the non-trivial groups are . This gives a postnikov tower
where the first non-trivial map is a fibration with fiber . Again, this is classified by a cohomology class in . Now, to construct from , there is an associated fibration
given by a homotopy class . In principal this cohomology group should be computable using the previous fibration with the Serre spectral sequence with the correct coefficients, namely . Doing this recursively, say for a -group, would require several spectral sequence computations, at worse many spectral sequence computations for an -group.
giving a technique for relating n-groups constructed from a complex manifold and sheaf cohomology on . This is particularly applicable for complex tori.
Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space with values in a higher group , giving higher cohomology groups . If we are considering as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.