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Concept
N-group (category theory)
Summary
In mathematics, an n-group, or n-dimensional higher group, is a special kind of 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 .
Double groupoid and 2-group
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 fibrationcoming 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 nowcoming 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.
Official source
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