n-group (category theory)

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}}

coming from a map B π 1 → B n + 1 π n {\displaystyle B\pi _{1}\to B^{n+1}\pi _{n}} whose homotopy class is an element of H n + 1 ( B π 1 , π n ) {\displaystyle H^{n+1}(B\pi _{1},\pi _{n})}.

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 ( π 1 , π 2 , π 3 ) {\displaystyle (\pi _{1},\pi _{2},\pi _{3})} 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 X {\displaystyle X}, the existence of universal covers gives us a homotopy type X ^ → X {\displaystyle {\hat {X}}\to X} which fits into a fibration sequence

X ^ → X → B π 1 {\displaystyle {\hat {X}}\to X\to B\pi _{1}}

giving a homotopy X ^ {\displaystyle {\hat {X}}} type with π 1 {\displaystyle \pi _{1}} trivial on which π 1 {\displaystyle \pi _{1}} acts on. These can be understood explicitly using the previous model of 2-groups, shifted up by degree (called delooping). Explicitly, X ^ {\displaystyle {\hat {X}}} fits into a Postnikov tower with associated Serre fibration

B 3 π 3 → X ^ → B 2 π 2 {\displaystyle B^{3}\pi _{3}\to {\hat {X}}\to B^{2}\pi _{2}}

giving where the B 3 π 3 {\displaystyle B^{3}\pi _{3}}-bundle X ^ → B 2 π 2 {\displaystyle {\hat {X}}\to B^{2}\pi _{2}} comes from a map B 2 π 2 → B 4 π 3 {\displaystyle B^{2}\pi _{2}\to B^{4}\pi _{3}}, giving a cohomology class in H 4 ( B 2 π 2 , π 3 ) {\displaystyle H^{4}(B^{2}\pi _{2},\pi _{3})}. Then, X {\displaystyle X} can be reconstructed using a homotopy quotient X ^ / / π 1 ≃ X {\displaystyle {\hat {X}}//\pi _{1}\simeq X}.

n -groups

The previous construction gives the general idea of how to consider higher groups in general. For an n-group with groups π 1 , π 2 , … , π n {\displaystyle \pi _{1},\pi _{2},\ldots ,\pi _{n}} with the latter bunch being abelian, we can consider the associated homotopy type X {\displaystyle X} and first consider the universal cover X ^ → X {\displaystyle {\hat {X}}\to X}. Then, this is a space with trivial π 1 ( X ^ ) = 0 {\displaystyle \pi _{1}({\hat {X}})=0}, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient X ^ / / π 1 {\displaystyle {\hat {X}}//\pi _{1}} gives a reconstruction of X {\displaystyle X}, showing the data of an n {\displaystyle n}-group is a higher group, or simple space, with trivial π 1 {\displaystyle \pi _{1}} such that a group G {\displaystyle G} 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 − / / π 1 {\displaystyle -//\pi _{1}}.

Going through the construction of a 4-group X {\displaystyle X} is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume π 1 = e {\displaystyle \pi _{1}=e} is trivial, so the non-trivial groups are π 2 , π 3 , π 4 {\displaystyle \pi _{2},\pi _{3},\pi _{4}}. This gives a Postnikov tower

X → X 3 → B 2 π 2 → ∗ {\displaystyle X\to X_{3}\to B^{2}\pi _{2}\to *}

where the first non-trivial map X 3 → B 2 π 2 {\displaystyle X_{3}\to B^{2}\pi _{2}} is a fibration with fiber B 3 π 3 {\displaystyle B^{3}\pi _{3}}. Again, this is classified by a cohomology class in H 4 ( B 2 π 2 , π 3 ) {\displaystyle H^{4}(B^{2}\pi _{2},\pi _{3})}. Now, to construct X {\displaystyle X} from X 3 {\displaystyle X_{3}}, there is an associated fibration

B 4 π 4 → X → X 3 {\displaystyle B^{4}\pi _{4}\to X\to X_{3}}

given by a homotopy class [ X 3 , B 5 π 4 ] ≅ H 5 ( X 3 , π 4 ) {\displaystyle [X_{3},B^{5}\pi _{4}]\cong H^{5}(X_{3},\pi _{4})}. In principle this cohomology group should be computable using the previous fibration B 3 π 3 → X 3 → B 2 π 2 {\displaystyle B^{3}\pi _{3}\to X_{3}\to B^{2}\pi _{2}} with the Serre spectral sequence with the correct coefficients, namely π 4 {\displaystyle \pi _{4}}. Doing this recursively, say for a 5 {\displaystyle 5}-group, would require several spectral sequence computations, at worst n ! {\displaystyle n!} many spectral sequence computations for an n {\displaystyle n}-group.

n -groups from sheaf cohomology

For a complex manifold X {\displaystyle X} with universal cover π : X ~ → X {\displaystyle \pi:{\tilde {X}}\to X}, and a sheaf of abelian groups F {\displaystyle {\mathcal {F}}} on X {\displaystyle X}, for every n ≥ 0 {\displaystyle n\geq 0} there exists canonical homomorphisms

ϕ n : H n ( π 1 X , H 0 ( X ~ , π ∗ F ) ) → H n ( X , F ) {\displaystyle \phi _{n}:H^{n}(\pi _{1}X,H^{0}({\tilde {X}},\pi ^{*}{\mathcal {F}}))\to H^{n}(X,{\mathcal {F}})}

giving a technique for relating n-groups constructed from a complex manifold X {\displaystyle X} and sheaf cohomology on X {\displaystyle X}. This is particularly applicable for complex tori.

n-group (category theory)

Examples

Eilenberg-MacLane spaces

2-groups

3-groups

n -groups

n -groups from sheaf cohomology

See also

Algebraic models for homotopy n -types

Cohomology of higher groups

Cohomology of higher groups over a site