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

## Examples

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### Eilenberg-Maclane spaces

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One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces K ( A , n ) {\displaystyle K(A,n)} since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group  G {\displaystyle G} can be turned into an Eilenberg-Maclane space  K ( G , 1 ) {\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  K ( G , 1 ) {\displaystyle K(G,1)} as  B G {\displaystyle BG} , and for an abelian group A {\displaystyle A} ,  K ( A , n ) {\displaystyle K(A,n)} is written as  B n A {\displaystyle B^{n}A} .

### 2-groups

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

t : π 1 -> Aut ⁡ π 2 {\displaystyle t:\pi _{1}\to \operatorname {Aut} \pi _{2}}

a group homomorphism, and  ω ∈ H 3 ( B π 1 , π 2 ) {\displaystyle \omega \in H^{3}(B\pi _{1},\pi _{2})} a cohomology class. These groups can be encoded as homotopy  2 {\displaystyle 2} -types X {\displaystyle X} with  π 1 X = π 1 {\displaystyle \pi _{1}X=\pi _{1}} and  π 2 X = π 2 {\displaystyle \pi _{2}X=\pi _{2}} , with the action coming from the action of  π 1 X {\displaystyle \pi _{1}X} on higher homotopy groups, and  ω {\displaystyle \omega } coming from the Postnikov tower since there is a fibration

B 2 π 2 -> X -> B π 1 {\displaystyle B^{2}\pi _{2}\to X\to B\pi _{1}}

coming from a map  B π 1 -> B 3 π 2 {\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  π 1 , e , … , e , π n {\displaystyle \pi _{1},e,\ldots ,e,\pi _{n}} , where the fibration sequence is now

B n π n -> X -> B π 1 {\displaystyle B^{n}\pi _{n}\to X\to B\pi _{1}}

coming from a map  B π 1 -> B n + 1 π n {\displaystyle B\pi _{1}\to B^{n+1}\pi _{n}} whose homotopy class is an element of  H n + 1 ( B π 1 , π n ) {\displaystyle H^{n+1}(B\pi _{1},\pi _{n})} .

### 3-groups

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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  ( π 1 , π 2 , π 3 ) {\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 X {\displaystyle X} , the existence of universal covers gives us a homotopy type  X ^ -> X {\displaystyle {\hat {X}}\to X} which fits into a fibration sequence

X ^ -> X -> B π 1 {\displaystyle {\hat {X}}\to X\to B\pi _{1}}

giving a homotopy  X ^ {\displaystyle {\hat {X}}} type with  π 1 {\displaystyle \pi _{1}} trivial on which  π 1 {\displaystyle \pi _{1}} acts on. These can be understood explicitly using the previous model of 2-groups, shifted up by degree (called delooping). Explicitly,  X ^ {\displaystyle {\hat {X}}} fits into a Postnikov tower with associated Serre fibration

B 3 π 3 -> X ^ -> B 2 π 2 {\displaystyle B^{3}\pi _{3}\to {\hat {X}}\to B^{2}\pi _{2}}

giving where the  B 3 π 3 {\displaystyle B^{3}\pi _{3}} -bundle  X ^ -> B 2 π 2 {\displaystyle {\hat {X}}\to B^{2}\pi _{2}} comes from a map  B 2 π 2 -> B 4 π 3 {\displaystyle B^{2}\pi _{2}\to B^{4}\pi _{3}} , giving a cohomology class in  H 4 ( B 2 π 2 , π 3 ) {\displaystyle H^{4}(B^{2}\pi _{2},\pi _{3})} . Then,  X {\displaystyle X} can be reconstructed using a homotopy quotient  X ^ / / π 1 ≃ X {\displaystyle {\hat {X}}//\pi _{1}\simeq X} .

### n-groups

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The previous construction gives the general idea of how to consider higher groups in general. For an n-group with groups  π 1 , π 2 , … , π n {\displaystyle \pi _{1},\pi _{2},\ldots ,\pi _{n}} with the latter bunch being abelian, we can consider the associated homotopy type  X {\displaystyle X} and first consider the universal cover  X ^ -> X {\displaystyle {\hat {X}}\to X} . Then, this is a space with trivial  π 1 ( X ^ ) = 0 {\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  X ^ / / π 1 {\displaystyle {\hat {X}}//\pi _{1}} gives a reconstruction of  X {\displaystyle X} , showing the data of an  n {\displaystyle n} -group is a higher group, or simple space, with trivial  π 1 {\displaystyle \pi _{1}} such that a group  G {\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  − / / π 1 {\displaystyle -//\pi _{1}} .

Going through the construction of a 4-group  X {\displaystyle X} is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume  π 1 = e {\displaystyle \pi _{1}=e} is trivial, so the non-trivial groups are  π 2 , π 3 , π 4 {\displaystyle \pi _{2},\pi _{3},\pi _{4}} . This gives a Postnikov tower

X -> X 3 -> B 2 π 2 -> ∗ {\displaystyle X\to X_{3}\to B^{2}\pi _{2}\to *}

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

B 4 π 4 -> X -> X 3 {\displaystyle B^{4}\pi _{4}\to X\to X_{3}}

given by a homotopy class  [ X 3 , B 5 π 4 ] ≅ H 5 ( X 3 , π 4 ) {\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  B 3 π 3 -> X 3 -> B 2 π 2 {\displaystyle B^{3}\pi _{3}\to X_{3}\to B^{2}\pi _{2}} with the Serre spectral sequence with the correct coefficients, namely  π 4 {\displaystyle \pi _{4}} . Doing this recursively, say for a  5 {\displaystyle 5} -group, would require several spectral sequence computations, at worst  n ! {\displaystyle n!} many spectral sequence computations for an  n {\displaystyle n} -group.

#### n-groups from sheaf cohomology

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

ϕ n : H n ( π 1 X , H 0 ( X ~ , π ∗ F ) ) -> H n ( X , F ) {\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  X {\displaystyle X} and sheaf cohomology on  X {\displaystyle X} . This is particularly applicable for complex tori.

## See also

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

## References

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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

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* 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

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* 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

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Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space  X {\displaystyle X} with values in a higher group  G ∙ {\displaystyle \mathbb {G} _{\bullet }} , giving higher cohomology groups  H ∗ ( X , G ∙ ) {\displaystyle \mathbb {H} ^{*}(X,\mathbb {G} _{\bullet })} . If we are considering  X {\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].

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