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 $n=\mathrm{\infty }$.

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 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,

$t:{\pi }_1\to \text{Aut}({\pi }_2)$

a group morphism, 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

${\pi }_1,e,\dots ,e,{\pi }_n$

, 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 3-types of groups. Essential, 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 ${\pi }_1,{\pi }_2,\dots ,{\pi }_n$ 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 $n$-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 worse

$n!$

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

${ℱ}$

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 n-groups 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)
  • "Thesis of Hoàng Xuân Sính (Gr-catégories)". https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html. 
• Baez, John C.; Lauda, Aaron D. (2003). Higher-Dimensional Algebra V: 2-Groups. Bibcode: 2003math......7200B. 
• Roberts, David Michael; Schreiber, Urs (2008). "The inner automorphism 3-group of a strict 2-group". Journal of Homotopy and Related Structures 3: 193–244. 
• "Classification of weak 3-groups". https://mathoverflow.net/questions/271877/classification-of-weak-3-groups. 
• Jardine, J. F. (January 2001). "Stacks and the homotopy theory of simplicial sheaves". Homology, Homotopy and Applications 3 (2): 361–384. doi:10.4310/HHA.2001.v3.n2.a5. 

Algebraic models for homotopy n-types

• Blanc, David (1999). "Algebraic invariants for homotopy types". Mathematical Proceedings of the Cambridge Philosophical Society 127 (3): 497–523. doi:10.1017/S030500419900393X. Bibcode: 1999MPCPS.127..497B. 
• Arvasi, Z.; Ulualan, E. (2006). "On algebraic models for homotopy 3-types". Journal of Homotopy and Related Structures 1: 1–27. http://www.emis.de/journals/JHRS/volumes/2006/n1a1/v1n1a1.pdf. 
• Brown, Ronald (1992). "Computing homotopy types using crossed n-cubes of groups". Adams Memorial Symposium on Algebraic Topology. pp. 187–210. doi:10.1017/CBO9780511526305.014. ISBN 9780521420747. 
• Joyal, André; Kock, Joachim (2007). "Weak units and homotopy 3-types". Categories in Algebra, Geometry and Mathematical Physics. Contemporary Mathematics. 431. pp. 257–276. doi:10.1090/conm/431/08277. ISBN 9780821839706. 
• Algebraic models for homotopy n-types in nLab - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

Cohomology of higher groups

• Eilenberg, Samuel; MacLane, Saunders (1946). "Determination of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants". Proceedings of the National Academy of Sciences 32 (11): 277–280. doi:10.1073/pnas.32.11.277. PMID 16588731. Bibcode: 1946PNAS...32..277E. 
• Thomas, Sebastian (2009). The third cohomology group classifies crossed module extensions. 
• Thomas, Sebastian (January 2010). "On the second cohomology group of a simplicial group". Homology, Homotopy and Applications 12 (2): 167–210. doi:10.4310/HHA.2010.v12.n2.a6. 
• Noohi, Behrang (2011). "Group cohomology with coefficients in a crossed module". Journal of the Institute of Mathematics of Jussieu 10 (2): 359–404. doi:10.1017/S1474748010000186. 

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

• Jibladze, Mamuka; Pirashvili, Teimuraz (2011). Cohomology with coefficients in stacks of Picard categories. 
• Debremaeker, Raymond (2017). Cohomology with values in a sheaf of crossed groups over a site.