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, ${\displaystyle n}$ 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 ${\displaystyle n}$-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy ${\displaystyle n}$-group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group ${\displaystyle \pi _{n}}$, or the entire Postnikov tower for ${\displaystyle n=\infty }$.

Examples

Eilenberg-Maclane spaces

One of the principal examples of higher groups come from the homotopy types of Eilenberg–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 behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write ${\displaystyle K(G,1)}$ as ${\displaystyle BG}$, and for an abelian group ${\displaystyle A}$, ${\displaystyle K(A,n)}$ is written as ${\displaystyle B^{n}A}$.

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 ${\displaystyle (\pi _{1},\pi _{2},t,\omega )}$ where ${\displaystyle \pi _{1},\pi _{2}}$ are groups with ${\displaystyle \pi _{2}}$ abelian,

${\displaystyle t:\pi _{1}\to \operatorname {Aut} \pi _{2}}$

a group homomorphism, and ${\displaystyle \omega \in H^{3}(B\pi _{1},\pi _{2})}$ a cohomology 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 the Postnikov 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 ${\displaystyle \pi _{1},e,\ldots ,e,\pi _{n}}$, 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 ${\displaystyle H^{n+1}(B\pi _{1},\pi _{n})}$.

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.^[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 this 3-group as a homotopy 3-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 of 2-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 ${\displaystyle {\hat {X}}//\pi _{1}\simeq X}$.

n-groups

The previous construction gives the general idea of how to consider higher groups in general. For an n-group with groups ${\displaystyle \pi _{1},\pi _{2},\ldots ,\pi _{n}}$ 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, or simple 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 by simplicial groups, but simplicial 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

${\displaystyle X\to X_{3}\to B^{2}\pi _{2}\to *}$

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.

n-groups from sheaf cohomology

For a complex manifold ${\displaystyle X}$ with universal cover ${\displaystyle \pi :{\tilde {X}}\to X}$, and a sheaf of abelian groups ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$, for every ${\displaystyle n\geq 0}$ there exists^[5] canonical homomorphisms

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

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

See also

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

References

1. "On Eilenberg-Maclane Spaces" (PDF). Archived (PDF) from the original on 28 Oct 2020.
2. Conduché, Daniel (1984-12-01). "Modules croisés généralisés de longueur 2". Journal of Pure and Applied Algebra. 34 (2): 155–178. doi:10.1016/0022-4049(84)90034-3. ISSN 0022-4049.
3. Goerss, Paul Gregory. (2009). Simplicial homotopy theory. Jardine, J. F., 1951-. Basel: Birkhäuser Verlag. ISBN 978-3-0346-0189-4. OCLC 534951159.
4. "Integral cohomology of finite Postnikov towers" (PDF). Archived (PDF) from the original on 25 Aug 2020.
5. Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 573–574. ISBN 978-3-662-06307-1. OCLC 851380558.

• 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. Bibcode:2006math......2180A.
• 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 ${\displaystyle X}$ with values in a higher group ${\displaystyle \mathbb {G} _{\bullet }}$, giving higher cohomology groups ${\displaystyle \mathbb {H} ^{*}(X,\mathbb {G} _{\bullet })}$. If we are considering ${\displaystyle X}$ 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].