n-group (category theory)

Not to be confused with p-group.

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

Main articles: Double groupoid and 2-group

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.

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)". Archived from the original on 2022-08-27.
• Baez, John C.; Lauda, Aaron D. (2003). "Higher-Dimensional Algebra V: 2-Groups". arXiv:math/0307200v3.
• 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. arXiv:0708.1741.
• "Classification of weak 3-groups". MathOverflow.
• 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. S2CID 123554728.

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. arXiv:math/9812035. Bibcode:1999MPCPS.127..497B. doi:10.1017/S030500419900393X. S2CID 17663055.
• Arvasi, Z.; Ulualan, E. (2006). "On algebraic models for homotopy 3-types" (PDF). Journal of Homotopy and Related Structures. 1: 1–27. arXiv:math/0602180.
• Brown, Ronald (1992). "Computing homotopy types using crossed n-cubes of groups". Adams Memorial Symposium on Algebraic Topology. pp. 187–210. arXiv:math/0109091. doi:10.1017/CBO9780511526305.014. ISBN 9780521420747. S2CID 2750149.
• Joyal, André; Kock, Joachim (2007). "Weak units and homotopy 3-types". Categories in Algebra, Geometry and Mathematical Physics. Contemporary Mathematics. Vol. 431. pp. 257–276. doi:10.1090/conm/431/08277. ISBN 9780821839706. S2CID 13931985.
• Algebraic models for homotopy n-types at the 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. Bibcode:1946PNAS...32..277E. doi:10.1073/pnas.32.11.277. PMC 1078947. PMID 16588731.
• Thomas, Sebastian (2009). "The third cohomology group classifies crossed module extensions". arXiv:0911.2861 [math.KT].
• Thomas, Sebastian (January 2010). "On the second cohomology group of a simplicial group". Homology, Homotopy and Applications. 12 (2): 167–210. arXiv:0911.2864. doi:10.4310/HHA.2010.v12.n2.a6. S2CID 55449228.
• Noohi, Behrang (2011). "Group cohomology with coefficients in a crossed module". Journal of the Institute of Mathematics of Jussieu. 10 (2): 359–404. arXiv:0902.0161. doi:10.1017/S1474748010000186. S2CID 7835760.

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". 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].