In mathematics, an ngroup, or ndimensional higher group, is a special kind of ncategory that generalises the concept of group to higherdimensional algebra. Here, may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an indepth study of 2groups under the moniker 'grcategory'.
The general definition of group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group , or the entire Postnikov tower for .
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Transcription
Examples
EilenbergMaclane spaces
One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group can be turned into an EilenbergMaclane space through a simplicial construction,^{[1]} and it behaves functorially. This construction gives an equivalence between groups and 1groups. Note that some authors write as , and for an abelian group , is written as .
2groups
The definition and many properties of 2groups are already known. 2groups can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple where are groups with abelian,
a group morphism, and a cohomology class. These groups can be encoded as homotopy types with and , with the action coming from the action of on higher homotopy groups, and coming from the Postnikov tower since there is a fibration
coming from a map . Note that this idea can be used to construct other higher groups with group data having trivial middle groups , where the fibration sequence is now
coming from a map whose homotopy class is an element of .
3groups
Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 3types of groups.^{[2]} Essential, these are given by a triple of groups with only the first group being nonabelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this 3group as a homotopy 3type , the existence of universal covers gives us a homotopy type which fits into a fibration sequence
giving a homotopy type with trivial on which acts on. These can be understood explicitly using the previous model of groups, shifted up by degree (called delooping). Explicitly, fits into a postnikov tower with associated Serre fibration
giving where the bundle comes from a map , giving a cohomology class in . Then, can be reconstructed using a homotopy quotient .
ngroups
The previous construction gives the general idea of how to consider higher groups in general. For an n group with groups with the latter bunch being abelian, we can consider the associated homotopy type and first consider the universal cover . Then, this is a space with trivial , making it easier to construct the rest of the homotopy type using the postnikov tower. Then, the homotopy quotient gives a reconstruction of , showing the data of an group is a higher group, or Simple space, with trivial such that a group 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 .
Going through the construction of a 4group is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume is trivial, so the nontrivial groups are . This gives a postnikov tower
where the first nontrivial map is a fibration with fiber . Again, this is classified by a cohomology class in . Now, to construct from , there is an associated fibration
given by a homotopy class . In principal^{[4]} this cohomology group should be computable using the previous fibration with the Serre spectral sequence with the correct coefficients, namely . Doing this recursively, say for a group, would require several spectral sequence computations, at worse many spectral sequence computations for an group.
ngroups from sheaf cohomology
For a complex manifold with universal cover , and a sheaf of abelian groups on , for every there exists^{[5]} canonical homomorphisms
giving a technique for relating ngroups constructed from a complex manifold and sheaf cohomology on . This is particularly applicable for complex tori.
See also
References
 ^ "On EilenbergMaclane Spaces" (PDF). Archived (PDF) from the original on 28 Oct 2020.
 ^ Conduché, Daniel (19841201). "Modules croisés généralisés de longueur 2". Journal of Pure and Applied Algebra. 34 (2): 155–178. doi:10.1016/00224049(84)900343. ISSN 00224049.
 ^ Goerss, Paul Gregory. (2009). Simplicial homotopy theory. Jardine, J. F., 1951. Basel: Birkhäuser Verlag. ISBN 9783034601894. OCLC 534951159.
 ^ "Integral cohomology of finite Postnikov towers" (PDF). Archived (PDF) from the original on 25 Aug 2020.
 ^ Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 573–574. ISBN 9783662063071. OCLC 851380558.
 Hoàng Xuân Sính, Grcatégories, PhD thesis, (1973)
 "Thesis of Hoàng Xuân Sính (Grcatégories)". Archived from the original on 20220827.
 Baez, John C.; Lauda, Aaron D. (2003). "HigherDimensional Algebra V: 2Groups". arXiv:math/0307200v3. Bibcode:2003math......7200B.
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(help)  Roberts, David Michael; Schreiber, Urs (2008). "The inner automorphism 3group of a strict 2group". Journal of Homotopy and Related Structures. 3: 193–244. arXiv:0708.1741.
 "Classification of weak 3groups". 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 ntypes
 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 3types" (PDF). Journal of Homotopy and Related Structures. 1: 1–27. arXiv:math/0602180.
 Brown, Ronald (1992). "Computing homotopy types using crossed ncubes 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 3types". 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 ntypes at the nLab  musings by Tim porter discussing the pitfalls of modelling homotopy ntypes with ncubes
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.
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(help)  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. 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 with values in a higher group , giving higher cohomology groups . If we are considering 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.
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(help)  Debremaeker, Raymond (2017). "Cohomology with values in a sheaf of crossed groups over a site". arXiv:1702.02128.
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