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, $n$ may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of Template:Nowrap under the moniker 'gr-category'.

The general definition of $n$-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy Template:Nowrap at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group ${\pi }_n$, or the entire Postnikov tower for $n=\mathrm{\infty }$.

Examples

Eilenberg-Maclane spaces

One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces $K(A,n)$ since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group $G$ can be turned into an Eilenberg-Maclane space $K(G,1)$ through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and Template:Nowrap. Note that some authors write $K(G,1)$ as $BG$, and for an abelian group $A$, $K(A,n)$ is written as $B^{n}A$.

2-groups

The definition and many properties of 2-groups are already known. Template:Nowrap can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple $({\pi }_1,{\pi }_2,t,\omega )$ where ${\pi }_1,{\pi }_2$ are groups with ${\pi }_2$ abelian,

$t:{\pi }_1\to \mathrm{Aut}{\pi }_2$

a group homomorphism, and $\omega \in H^3(B{\pi }_1,{\pi }_2)$ a cohomology class. These groups can be encoded as homotopy Template:Nowrap $X$ with ${\pi }_{1}X={\pi }_1$ and ${\pi }_{2}X={\pi }_2$, with the action coming from the action of ${\pi }_{1}X$ on higher homotopy groups, and $\omega$ coming from the Postnikov tower since there is a fibration

$B^2{\pi }_2\to X\to B{\pi }_1$

coming from a map $B{\pi }_1\to B^3{\pi }_2$. Note that this idea can be used to construct other higher groups with group data having trivial middle groups ${\pi }_1,e,\dots ,e,{\pi }_n$, where the fibration sequence is now

$B^n{\pi }_n\to X\to B{\pi }_1$

coming from a map $B{\pi }_1\to B^{n+1}{\pi }_n$ whose homotopy class is an element of $H^{n+1}(B{\pi }_1,{\pi }_n)$.

3-groups

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy Template:Nowrap of groups. Essentially, these are given by a triple of groups $({\pi }_1,{\pi }_2,{\pi }_3)$ with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this Template:Nowrap as a homotopy Template:Nowrap $X$, the existence of universal covers gives us a homotopy type $\hat{X}\to X$ which fits into a fibration sequence

$\hat{X}\to X\to B{\pi }_1$

giving a homotopy $\hat{X}$ type with ${\pi }_1$ trivial on which ${\pi }_1$ acts on. These can be understood explicitly using the previous model of Template:Nowrap, shifted up by degree (called delooping). Explicitly, $\hat{X}$ fits into a Postnikov tower with associated Serre fibration

$B^3{\pi }_3\to \hat{X}\to B^2{\pi }_2$

giving where the $B^3{\pi }_3$-bundle $\hat{X}\to B^2{\pi }_2$ comes from a map $B^2{\pi }_2\to B^4{\pi }_3$, giving a cohomology class in $H^4(B^2{\pi }_2,{\pi }_3)$. Then, $X$ can be reconstructed using a homotopy quotient $\hat{X}//{\pi }_1\simeq X$.

n-groups

The previous construction gives the general idea of how to consider higher groups in general. For an Template:Nowrap with groups ${\pi }_1,{\pi }_2,\dots ,{\pi }_n$ with the latter bunch being abelian, we can consider the associated homotopy type $X$ and first consider the universal cover $\hat{X}\to X$. Then, this is a space with trivial ${\pi }_1(\hat{X})=0$, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient $\hat{X}//{\pi }_1$ gives a reconstruction of $X$, showing the data of an Template:Nowrap is a higher group, or simple space, with trivial ${\pi }_1$ such that a group $G$ acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoids^{pg 295} since the groupoid structure models the homotopy quotient $-//{\pi }_1$.

Going through the construction of a 4-group $X$ is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume ${\pi }_1=e$ is trivial, so the non-trivial groups are ${\pi }_2,{\pi }_3,{\pi }_4$. This gives a Postnikov tower

$X\to X_3\to B^2{\pi }_2\to *$

where the first non-trivial map $X_3\to B^2{\pi }_2$ is a fibration with fiber $B^3{\pi }_3$. Again, this is classified by a cohomology class in $H^4(B^2{\pi }_2,{\pi }_3)$. Now, to construct $X$ from $X_3$, there is an associated fibration

$B^4{\pi }_4\to X\to X_3$

given by a homotopy class $[X_3,B^5{\pi }_4]\cong H^5(X_3,{\pi }_4)$. In principle this cohomology group should be computable using the previous fibration $B^3{\pi }_3\to X_3\to B^2{\pi }_2$ with the Serre spectral sequence with the correct coefficients, namely ${\pi }_4$. Doing this recursively, say for a Template:Nowrap, would require several spectral sequence computations, at worst $n!$ many spectral sequence computations for an Template:Nowrap.

n-groups from sheaf cohomology

For a complex manifold $X$ with universal cover $\pi :\tilde{X}\to X$, and a sheaf of abelian groups $ℱ$ on $X$, for every $n\ge 0$ there exists canonical homomorphisms

${\varphi }_n:H^n({\pi }_{1}X,H^0(\tilde{X},{\pi }^{*}ℱ))\to H^n(X,ℱ)$

giving a technique for relating Template:Nowrap constructed from a complex manifold $X$ and sheaf cohomology on $X$. This is particularly applicable for complex tori.

See also

• ∞-groupoid
• Crossed module
• Homotopy hypothesis
• Abelian 2-group

References

• Hoàng Xuân Sính, Gr-catégories, PhD thesis, (1973)
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Algebraic models for homotopy n-types

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• Template:Nlab - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

Cohomology of higher groups

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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 $X$ with values in a higher group ${𝔾}_{\bullet }$, giving higher cohomology groups ${ℍ}^{*}(X,{𝔾}_{\bullet })$. If we are considering $X$ as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.

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