# 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 2-groups 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 $n$-group 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 $n=\infty$.

## 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 1-groups. 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. 2-groups 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,
    $t:\pi _{1}\to \operatorname {Aut} \pi _{2}$
a group homomorphism, and $\omega \in H^{3}(B\pi _{1},\pi _{2})$ a cohomology class. These groups can be encoded as homotopy $2$-types $X$ 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 $\pi _{1},e,\ldots ,e,\pi _{n}$, 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 3-types 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 3-group as a homotopy 3-type $X$, 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 2-groups, 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 n-group with groups $\pi _{1},\pi _{2},\ldots ,\pi _{n}$ 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 $n$-group 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$^{pg295}$ 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 $5$-group, would require several spectral sequence computations, at worst $n!$ many spectral sequence computations for an $n$-group.

#### n-groups from sheaf cohomology

For a complex manifold $X$ with universal cover $\pi :{\tilde {X}}\to X$, and a sheaf of abelian groups ${\mathcal {F}}$ on $X$, for every $n\geq 0$ there exists canonical homomorphisms
    $\phi _{n}:H^{n}(\pi _{1}X,H^{0}({\tilde {X}},\pi ^{*}{\mathcal {F}}))\to H^{n}(X,{\mathcal {F}})$
giving a technique for relating n-groups constructed from a complex manifold $X$ and sheaf cohomology on $X$. This is particularly applicable for complex tori.