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 {\displaystyle 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 moniker 'gr-category'. The general definition of n {\displaystyle n} -group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy n {\displaystyle n} -group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π n {\displaystyle \pi _{n}} , or the entire Postnikov tower for n = ∞ {\displaystyle n=\infty } .
Examples
Eilenberg-MacLane spaces One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces K ( A , n ) {\displaystyle K(A,n)} since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group G {\displaystyle G} can be turned into an Eilenberg-MacLane space K ( G , 1 ) {\displaystyle K(G,1)} through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write K ( G , 1 ) {\displaystyle K(G,1)} as B G {\displaystyle BG} , and for an abelian group A {\displaystyle A} , K ( A , n ) {\displaystyle K(A,n)} is written as B n A {\displaystyle B^{n}A} .
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 ( π 1 , π 2 , t , ω ) {\displaystyle (\pi _{1},\pi _{2},t,\omega )} where π 1 , π 2 {\displaystyle \pi _{1},\pi _{2}} are groups with π 2 {\displaystyle \pi _{2}} abelian,
t : π 1 → Aut π 2 {\displaystyle t:\pi _{1}\to \operatorname {Aut} \pi _{2}}
a group homomorphism, and ω ∈ H 3 ( B π 1 , π 2 ) {\displaystyle \omega \in H^{3}(B\pi _{1},\pi _{2})} a cohomology class. These groups can be encoded as homotopy 2 {\displaystyle 2} -types X {\displaystyle X} with π 1 X = π 1 {\displaystyle \pi _{1}X=\pi _{1}} and π 2 X = π 2 {\displaystyle \pi _{2}X=\pi _{2}} , with the action coming from the action of π 1 X {\displaystyle \pi _{1}X} on higher homotopy groups, and ω {\displaystyle \omega } coming from the Postnikov tower since there is a fibration
B 2 π 2 → X → B π 1 {\displaystyle B^{2}\pi _{2}\to X\to B\pi _{1}}
coming from a map B π 1 → B 3 π 2 {\displaystyle B\pi _{1}\to B^{3}\pi _{2}} . Note that this idea can be used to construct other higher groups with group data having trivial middle groups π 1 , e , … , e , π n {\displaystyle \pi _{1},e,\ldots ,e,\pi _{n}} , where the fibration sequence is now
B n π n → X → B π 1 {\displaystyle B^{n}\pi _{n}\to X\to B\pi _{1}}
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