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, formula_1 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 formula_1-group is a matter of ongoing research. However, it is expected that every topological space will have a "homotopy formula_1-group" at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group formula_4, or the entire Postnikov tower for formula_5.
Examples.
Eilenberg-Maclane spaces.
One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces formula_6 since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group formula_7 can be turned into an Eilenberg-Maclane space formula_8 through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write formula_8 as formula_10, and for an abelian group formula_11, formula_6 is written as formula_13.
2-groups.
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 formula_14 where formula_15 are groups with formula_16 abelian,
formula_17
a group homomorphism, and formula_18 a cohomology class. These groups can be encoded as homotopy formula_19-types formula_20 with formula_21 and formula_22, with the action coming from the action of formula_23 on higher homotopy groups, and formula_24 coming from the Postnikov tower since there is a fibration
formula_25
coming from a map formula_26. Note that this idea can be used to construct other higher groups with group data having trivial middle groups formula_27, where the fibration sequence is now
formula_28
coming from a map formula_29 whose homotopy class is an element of formula_30.
3-groups.
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. Essentially, these are given by a triple of groups formula_31 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 formula_20, the existence of universal covers gives us a homotopy type formula_33 which fits into a fibration sequence
formula_34
giving a homotopy formula_35 type with formula_36 trivial on which formula_36 acts on. These can be understood explicitly using the previous model of 2-groups, shifted up by degree (called delooping). Explicitly, formula_35 fits into a Postnikov tower with associated Serre fibration
formula_39
giving where the formula_40-bundle formula_41 comes from a map formula_42, giving a cohomology class in formula_43. Then, formula_20 can be reconstructed using a homotopy quotient formula_45.
"n"-groups.
The previous construction gives the general idea of how to consider higher groups in general. For an "n"-group with groups formula_46 with the latter bunch being abelian, we can consider the associated homotopy type formula_20 and first consider the universal cover formula_33. Then, this is a space with trivial formula_49, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient formula_50 gives a reconstruction of formula_20, showing the data of an formula_1-group is a higher group, or simple space, with trivial formula_36 such that a group formula_7 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 formula_55.
Going through the construction of a 4-group formula_20 is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume formula_57 is trivial, so the non-trivial groups are formula_58. This gives a Postnikov tower
formula_59
where the first non-trivial map formula_60 is a fibration with fiber formula_40. Again, this is classified by a cohomology class in formula_43. Now, to construct formula_20 from formula_64, there is an associated fibration
formula_65
given by a homotopy class formula_66. In principle this cohomology group should be computable using the previous fibration formula_67 with the Serre spectral sequence with the correct coefficients, namely formula_68. Doing this recursively, say for a formula_69-group, would require several spectral sequence computations, at worst formula_70 many spectral sequence computations for an formula_1-group.
"n"-groups from sheaf cohomology.
For a complex manifold formula_20 with universal cover formula_73, and a sheaf of abelian groups formula_74 on formula_20, for every formula_76 there exists canonical homomorphisms
formula_77
giving a technique for relating "n"-groups constructed from a complex manifold formula_20 and sheaf cohomology on formula_20. This is particularly applicable for complex tori.
References.
Cohomology of higher groups over a site.
Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space formula_20 with values in a higher group formula_81, giving higher cohomology groups formula_82. If we are considering formula_20 as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.