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N-group (category theory)

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Source updated April 5, 2026

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 $n=\infty$ .

01Examples

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 ^{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 $5$ -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.

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Sources and credits

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N-group (category theory)

01Examples

Eilenberg-MacLane spaces

2-groups

3-groups

n-groups

n-groups from sheaf cohomology

Sources and credits

Related topics

∞-groupoid

Crossed module

Homotopy hypothesis