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 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 '' 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 . Note that some authors write $K(G,1)$ as $BG$, and for an abelian group $A$, $K(A,n)$ is written as $B^nA$.

2-groups

The definition and many properties of 2-groups are already known. 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,
:$t:\pi_1 \to \operatorname \pi_2$
a group homomorphism, and $\omega \in H^3(B\pi_1,\pi_2)$ a cohomology class. These groups can be encoded as homotopy $X$ 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 $B\pi_1 \to B^\pi_n$ whose homotopy class is an element of $H^(B\pi_1, \pi_n)$.

3-groups

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 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 as a homotopy $X$, the existence of universal covers gives us a homotopy type $\hat \to X$ which fits into a fibration sequence
:$\hat \to X \to B\pi_1$
giving a homotopy $\hat$ type with $\pi_1$ trivial on which $\pi_1$ acts on. These can be understood explicitly using the previous model of , shifted up by degree (called delooping). Explicitly, $\hat$ fits into a Postnikov tower with associated Serre fibration
:$B^\pi_3 \to \hat \to B^2\pi_2$
giving where the $B^3\pi_3$-bundle $\hat \to B^2\pi_2$ 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 $\hat//\pi_1 \simeq X$.

''n''-groups

The previous construction gives the general idea of how to consider higher groups in general. For an 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 $\hat \to X$. Then, this is a space with trivial $\pi_1(\hat) = 0$, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient $\hat // \pi_1$ gives a reconstruction of $X$, showing the data of an 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 _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 , would require several spectral sequence computations, at worst $n!$ many spectral sequence computations for an .

''n''-groups from sheaf cohomology

For a complex manifold $X$ with universal cover $\pi:\tilde\to X$, and a sheaf of abelian groups $\mathcal$ on $X$, for every $n \geq 0$ there exists canonical homomorphisms
:$\phi_n:H^n(\pi_1 X, H^0(\tilde, \pi^*\mathcal)) \to H^n(X, \mathcal)$
giving a technique for relating 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)
**
*
*
*
*

Algebraic models for homotopy ''n''-types

*
*
*
*
* - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

Cohomology of higher groups

*
*
*
*

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 $\mathbb_\bullet$, giving higher cohomology groups $\mathbb^*(X, \mathbb_\bullet)$. If we are considering $X$ as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.
*
*
{{Category theory

Group theory
Higher category theory
Homotopy theory