Inmathematics, ann-group, orn-dimensional higher group, is a special kind ofn-category that generalises the concept ofgroup tohigher-dimensional algebra. Here,${\displaystyle n}$ may be anynatural number orinfinity. The thesis ofAlexander Grothendieck's studentHoàng Xuân Sính was an in-depth study of2-groups under the moniker 'gr-category'.

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

One of the principal examples of higher groups come from the homotopy types ofEilenberg–MacLane spaces${\displaystyle K(A,n)}$ since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group${\displaystyle G}$ can be turned into an Eilenberg-Maclane space${\displaystyle K(G,1)}$ through a simplicial construction,^[1] and it behavesfunctorially. This construction gives an equivalence between groups and1-groups. Note that some authors write${\displaystyle K(G,1)}$ as${\displaystyle BG}$, and for anabelian group${\displaystyle A}$,${\displaystyle K(A,n)}$ is written as${\displaystyle B^{n}A}$.

The definition and many properties of2-groups are already known.2-groups can be described usingcrossed modules and their classifying spaces. Essentially, these are given by a quadruple${\displaystyle ({\pi }_1,{\pi }_2,t,\omega )}$ where${\displaystyle {\pi }_1,{\pi }_2}$ are groups with${\displaystyle {\pi }_2}$ abelian,

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

agroup homomorphism, and${\displaystyle \omega \in H^3(B{\pi }_1,{\pi }_2)}$ acohomology class. These groups can be encoded as homotopy${\displaystyle 2}$-types${\displaystyle X}$ with${\displaystyle {\pi }_{1}X={\pi }_1}$ and${\displaystyle {\pi }_{2}X={\pi }_2}$, with the action coming from the action of${\displaystyle {\pi }_{1}X}$ on higher homotopy groups, and${\displaystyle \omega }$ coming from thePostnikov tower since there is a fibration

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

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

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

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

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy3-types of groups.^[2] Essentially, these are given by a triple of groups${\displaystyle ({\pi }_1,{\pi }_2,{\pi }_3)}$ with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this3-group as a homotopy3-type${\displaystyle X}$, the existence of universal covers gives us a homotopy type${\displaystyle \hat{X}\to X}$ which fits into a fibration sequence

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

giving a homotopy${\displaystyle \hat{X}}$ type with${\displaystyle {\pi }_1}$ trivial on which${\displaystyle {\pi }_1}$ acts on. These can be understood explicitly using the previous model of2-groups, shifted up by degree (called delooping). Explicitly,${\displaystyle \hat{X}}$ fits into a Postnikov tower with associated Serre fibration

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

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

The previous construction gives the general idea of how to consider higher groups in general. For ann-group with groups${\displaystyle {\pi }_1,{\pi }_2,\dots ,{\pi }_n}$ with the latter bunch being abelian, we can consider the associated homotopy type${\displaystyle X}$ and first consider the universal cover${\displaystyle \hat{X}\to X}$. Then, this is a space with trivial${\displaystyle {\pi }_1(\hat{X})=0}$, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient${\displaystyle \hat{X}//{\pi }_1}$ gives a reconstruction of${\displaystyle X}$, showing the data of an${\displaystyle n}$-group is a higher group, orsimple space, with trivial${\displaystyle {\pi }_1}$ such that a group${\displaystyle G}$ acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized bysimplicial groups, butsimplicial groupoids^{[3]pg 295} since the groupoid structure models the homotopy quotient${\displaystyle -//{\pi }_1}$.

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

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

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

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

given by a homotopy class${\displaystyle [X_3,B^5{\pi }_4]\cong H^5(X_3,{\pi }_4)}$. In principle^[4] this cohomology group should be computable using the previous fibration${\displaystyle B^3{\pi }_3\to X_3\to B^2{\pi }_2}$ with the Serre spectral sequence with the correct coefficients, namely${\displaystyle {\pi }_4}$. Doing this recursively, say for a${\displaystyle 5}$-group, would require several spectral sequence computations, at worst${\displaystyle n!}$ many spectral sequence computations for an${\displaystyle n}$-group.

For acomplex manifold${\displaystyle X}$ withuniversal cover${\displaystyle \pi :\tilde{X}\to X}$, and asheaf of abelian groups${\displaystyle \mathcal{F}}$ on${\displaystyle X}$, for every${\displaystyle n\ge 0}$ there exists^[5] canonicalhomomorphisms

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

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

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

• Hoàng Xuân Sính,Gr-catégories, PhD thesis, (1973)
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Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space${\displaystyle X}$ with values in a higher group${\displaystyle {\mathbb{G}}_{\bullet }}$, giving higher cohomology groups${\displaystyle {\mathbb{H}}^{\ast }(X,{\mathbb{G}}_{\bullet })}$. If we are considering${\displaystyle X}$ as a homotopy type and assuming thehomotopy hypothesis, then these are the same cohomology groups.

• Jibladze, Mamuka; Pirashvili, Teimuraz (2011). "Cohomology with coefficients in stacks of Picard categories".arXiv:1101.2918 [math.AT].
• Debremaeker, Raymond (2017). "Cohomology with values in a sheaf of crossed groups over a site".arXiv:1702.02128 [math.AG].

• Group theory
• Higher category theory
• Homotopy theory