{{distinguish|p-group}} {{DISPLAYTITLE:''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, <math>n</math> may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of {{nowrap|2-groups}} under the moniker 'gr-category'.
The general definition of <math>n</math>-group is a matter of ongoing research. However, it is expected that every topological space will have a ''homotopy {{nowrap|<math>n</math>-group}}'' at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group <math>\pi_n</math>, or the entire Postnikov tower for <math>n=\infty</math>.
== Examples ==
=== Eilenberg-MacLane spaces === One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces <math>K(A,n)</math> since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group <math>G</math> can be turned into an Eilenberg-MacLane space <math>K(G,1)</math> through a simplicial construction,<ref>{{Cite web|last=|first=|date=|title=On Eilenberg-Maclane Spaces|url=http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf|url-status=live|archive-url=https://web.archive.org/web/20201028001827/http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf|archive-date=28 Oct 2020|access-date=|website=}}</ref> and it behaves functorially. This construction gives an equivalence between groups and {{nowrap|1-groups}}. Note that some authors write <math>K(G,1)</math> as <math>BG</math>, and for an abelian group <math>A</math>, <math>K(A,n)</math> is written as <math>B^nA</math>.
=== 2-groups === {{Main articles|Double groupoid|2-group}}
The definition and many properties of 2-groups are already known. {{nowrap|2-groups}} can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple <math>(\pi_1,\pi_2, t,\omega)</math> where <math>\pi_1,\pi_2</math> are groups with <math>\pi_2</math> abelian, :<math>t:\pi_1 \to \operatorname{Aut} \pi_2</math> a group homomorphism, and <math>\omega \in H^3(B\pi_1,\pi_2)</math> a cohomology class. These groups can be encoded as homotopy {{nowrap|<math>2</math>-types}} <math>X</math> with <math>\pi_1 X = \pi_1</math> and <math>\pi_2 X = \pi_2</math>, with the action coming from the action of <math>\pi_1 X</math> on higher homotopy groups, and <math>\omega</math> coming from the Postnikov tower since there is a fibration :<math>B^2\pi_2 \to X \to B\pi_1</math> coming from a map <math>B\pi_1 \to B^3\pi_2</math>. Note that this idea can be used to construct other higher groups with group data having trivial middle groups <math>\pi_1, e, \ldots, e, \pi_n</math>, where the fibration sequence is now :<math>B^n\pi_n \to X \to B\pi_1</math> coming from a map <math>B\pi_1 \to B^{n+1}\pi_n</math> whose homotopy class is an element of <math>H^{n+1}(B\pi_1, \pi_n)</math>.
=== 3-groups === Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy {{nowrap|3-types}} of groups.<ref>{{Cite journal|last=Conduché|first=Daniel|date=1984-12-01|title=Modules croisés généralisés de longueur 2|journal=Journal of Pure and Applied Algebra|language=en|volume=34|issue=2|pages=155–178|doi=10.1016/0022-4049(84)90034-3|issn=0022-4049|doi-access=}}</ref> Essentially, these are given by a triple of groups <math>(\pi_1,\pi_2,\pi_3)</math> with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this {{nowrap|3-group}} as a homotopy {{nowrap|3-type}} <math>X</math>, the existence of universal covers gives us a homotopy type <math>\hat{X} \to X</math> which fits into a fibration sequence :<math>\hat{X} \to X \to B\pi_1</math> giving a homotopy <math>\hat{X}</math> type with <math>\pi_1</math> trivial on which <math>\pi_1</math> acts on. These can be understood explicitly using the previous model of {{nowrap|2-groups}}, shifted up by degree (called delooping). Explicitly, <math>\hat{X}</math> fits into a Postnikov tower with associated Serre fibration :<math>B^{3}\pi_3 \to \hat{X} \to B^2\pi_2</math> giving where the <math>B^3\pi_3</math>-bundle <math>\hat{X} \to B^2\pi_2</math> comes from a map <math>B^2\pi_2 \to B^4\pi_3</math>, giving a cohomology class in <math>H^4(B^2\pi_2, \pi_3)</math>. Then, <math>X</math> can be reconstructed using a homotopy quotient <math>\hat{X}//\pi_1 \simeq X</math>.
=== ''n''-groups === The previous construction gives the general idea of how to consider higher groups in general. For an {{nowrap|''n''-group}} with groups <math>\pi_1,\pi_2,\ldots,\pi_n</math> with the latter bunch being abelian, we can consider the associated homotopy type <math>X</math> and first consider the universal cover <math>\hat{X} \to X</math>. Then, this is a space with trivial <math>\pi_1(\hat{X}) = 0</math>, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient <math>\hat{X} // \pi_1</math> gives a reconstruction of <math>X</math>, showing the data of an {{nowrap|<math>n</math>-group}} is a higher group, or simple space, with trivial <math>\pi_1</math> such that a group <math>G</math> acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoids<ref>{{Cite book|last=Goerss, Paul Gregory.|title=Simplicial homotopy theory|date=2009|publisher=Birkhäuser Verlag|others=Jardine, J. F., 1951-|isbn=978-3-0346-0189-4|location=Basel|oclc=534951159}}</ref><sup>pg 295</sup> since the groupoid structure models the homotopy quotient <math>-// \pi_1</math>.
Going through the construction of a 4-group <math>X</math> is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume <math>\pi_1 = e</math> is trivial, so the non-trivial groups are <math>\pi_2,\pi_3,\pi_4</math>. This gives a Postnikov tower :<math>X \to X_3 \to B^2\pi_2 \to *</math> where the first non-trivial map <math>X_3 \to B^2\pi_2</math> is a fibration with fiber <math>B^3\pi_3</math>. Again, this is classified by a cohomology class in <math>H^4(B^2\pi_2, \pi_3)</math>. Now, to construct <math>X</math> from <math>X_3</math>, there is an associated fibration :<math>B^4\pi_4 \to X \to X_3</math> given by a homotopy class <math>[X_3, B^5\pi_4] \cong H^5(X_3,\pi_4)</math>. In principle<ref>{{Cite web|last=|first=|date=|title=Integral cohomology of finite Postnikov towers|url=http://doc.rero.ch/record/482/files/Clement_these.pdf|url-status=live|archive-url=https://web.archive.org/web/20200825141534/http://doc.rero.ch/record/482/files/Clement_these.pdf|archive-date=25 Aug 2020|access-date=|website=}}</ref> this cohomology group should be computable using the previous fibration <math>B^3\pi_3 \to X_3 \to B^2\pi_2 </math> with the Serre spectral sequence with the correct coefficients, namely <math>\pi_4</math>. Doing this recursively, say for a {{nowrap|<math>5</math>-group}}, would require several spectral sequence computations, at worst <math>n!</math> many spectral sequence computations for an {{nowrap|<math>n</math>-group}}.
==== ''n''-groups from sheaf cohomology ==== For a complex manifold <math>X</math> with universal cover <math>\pi:\tilde{X}\to X</math>, and a sheaf of abelian groups <math>\mathcal{F}</math> on <math>X</math>, for every <math>n \geq 0</math> there exists<ref>{{Cite book|last=Birkenhake|first=Christina|title=Complex Abelian Varieties|date=2004|publisher=Springer Berlin Heidelberg|others=Herbert Lange|isbn=978-3-662-06307-1|edition=Second, augmented|location=Berlin, Heidelberg|pages=573–574|oclc=851380558}}</ref> canonical homomorphisms :<math>\phi_n:H^n(\pi_1 X, H^0(\tilde{X}, \pi^*\mathcal{F})) \to H^n(X, \mathcal{F})</math> giving a technique for relating {{nowrap|''n''-groups}} constructed from a complex manifold <math>X</math> and sheaf cohomology on <math>X</math>. This is particularly applicable for complex tori.
== See also ==
* ∞-groupoid * Crossed module * Homotopy hypothesis * Abelian 2-group
== References == <references /> * Hoàng Xuân Sính, [http://www.math.rwth-aachen.de/~kuenzer/sinh.html Gr-catégories], PhD thesis, (1973) **{{cite web|title=Thesis of Hoàng Xuân Sính (Gr-catégories)|url=https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html|archive-url=https://web.archive.org/web/20220827214622/https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html |archive-date=2022-08-27}} * {{cite arXiv |eprint=math/0307200v3|last1=Baez |first1=John C. |last2=Lauda |first2=Aaron D. |title=Higher-Dimensional Algebra V: 2-Groups |year=2003 }} * {{cite journal |last1=Roberts |first1=David Michael |last2=Schreiber |first2=Urs |title=The inner automorphism 3-group of a strict 2-group |journal=Journal of Homotopy and Related Structures |date=2008 |volume=3 |pages=193–244|arxiv=0708.1741}} *{{cite web |title=Classification of weak 3-groups |url=https://mathoverflow.net/questions/271877/classification-of-weak-3-groups |website=MathOverflow}} *{{cite journal |title=Stacks and the homotopy theory of simplicial sheaves |journal=Homology, Homotopy and Applications |date=January 2001 |volume=3 |issue=2 |pages=361–384 |last1=Jardine |first1=J. F. |doi=10.4310/HHA.2001.v3.n2.a5 |s2cid=123554728 |doi-access=free }}
=== Algebraic models for homotopy ''n''-types ===
*{{cite journal |doi=10.1017/S030500419900393X|title=Algebraic invariants for homotopy types |year=1999 |last1=Blanc |first1=David |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=127 |issue=3 |pages=497–523 |arxiv=math/9812035 |bibcode=1999MPCPS.127..497B |s2cid=17663055 }} *{{cite journal |last1=Arvasi |first1=Z. |last2=Ulualan |first2=E. |title=On algebraic models for homotopy 3-types |journal=Journal of Homotopy and Related Structures |date=2006 |volume=1 |pages=1–27 |arxiv=math/0602180|bibcode=2006math......2180A |url=http://www.emis.de/journals/JHRS/volumes/2006/n1a1/v1n1a1.pdf}} * {{cite book |doi=10.1017/CBO9780511526305.014|chapter=Computing homotopy types using crossed n-cubes of groups |title=Adams Memorial Symposium on Algebraic Topology |year=1992 |last1=Brown |first1=Ronald |pages=187–210 |arxiv=math/0109091 |isbn=9780521420747 |s2cid=2750149 }} * {{cite book |doi=10.1090/conm/431/08277|chapter=Weak units and homotopy 3-types |title=Categories in Algebra, Geometry and Mathematical Physics |series=Contemporary Mathematics |year=2007 |last1=Joyal |first1=André |author-link1=André Joyal |last2=Kock |first2=Joachim |volume=431 |pages=257–276 |isbn=9780821839706 |s2cid=13931985 }} *{{nlab|id=algebraic+models+for+homotopy+n-types|title=Algebraic models for homotopy n-types}} - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes
=== Cohomology of higher groups ===
*{{cite journal |doi=10.1073/pnas.32.11.277|title=Determination of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants |year=1946 |last1=Eilenberg |first1=Samuel |author-link1=Samuel Eilenberg |last2=MacLane |first2=Saunders |author-link2=Saunders Mac Lane |journal=Proceedings of the National Academy of Sciences |volume=32 |issue=11 |pages=277–280 |pmid=16588731 |pmc=1078947 |bibcode=1946PNAS...32..277E |doi-access=free }} *{{cite arXiv |eprint=0911.2861|last1=Thomas |first1=Sebastian |title=The third cohomology group classifies crossed module extensions |year=2009 |class=math.KT }} * {{cite journal |title=On the second cohomology group of a simplicial group |journal=Homology, Homotopy and Applications |date=January 2010 |volume=12 |issue=2 |pages=167–210 |last1=Thomas |first1=Sebastian |doi=10.4310/HHA.2010.v12.n2.a6 |s2cid=55449228 |doi-access=free |arxiv=0911.2864 }} *{{cite journal |doi=10.1017/S1474748010000186 |title=Group cohomology with coefficients in a crossed module |year=2011 |last1=Noohi |first1=Behrang |journal=Journal of the Institute of Mathematics of Jussieu |volume=10 |issue=2 |pages=359–404 |arxiv=0902.0161 |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 <math>X</math> with values in a higher group <math>\mathbb{G}_\bullet</math>, giving higher cohomology groups <math>\mathbb{H}^*(X, \mathbb{G}_\bullet)</math>. If we are considering <math>X</math> as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups. *{{cite arXiv |eprint=1101.2918|last1=Jibladze |first1=Mamuka |last2=Pirashvili |first2=Teimuraz |title=Cohomology with coefficients in stacks of Picard categories |year=2011 |class=math.AT }} *{{cite arXiv |eprint=1702.02128 |last1=Debremaeker |first1=Raymond |title=Cohomology with values in a sheaf of crossed groups over a site |year=2017 |class=math.AG }} {{Category theory}}
Category:Group theory Category:Higher category theory Category:Homotopy theory