n-group (category theory)

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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]\displaystyle{ 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 2-groups under the moniker 'gr-category'.

The general definition of [math]\displaystyle{ n }[/math]-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy [math]\displaystyle{ n }[/math]-group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group [math]\displaystyle{ \pi_n }[/math], or the entire Postnikov tower for [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ G }[/math] can be turned into an Eilenberg-Maclane space [math]\displaystyle{ K(G,1) }[/math] through a simplicial construction,[1] and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write [math]\displaystyle{ K(G,1) }[/math] as [math]\displaystyle{ BG }[/math], and for an abelian group [math]\displaystyle{ A }[/math], [math]\displaystyle{ K(A,n) }[/math] is written as [math]\displaystyle{ B^nA }[/math].

2-groups

Main pages: 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 [math]\displaystyle{ (\pi_1,\pi_2, t,\omega) }[/math] where [math]\displaystyle{ \pi_1,\pi_2 }[/math] are groups with [math]\displaystyle{ \pi_2 }[/math] abelian,

[math]\displaystyle{ t:\pi_1 \to \text{Aut}(\pi_2) }[/math]

a group morphism, and [math]\displaystyle{ \omega \in H^3(B\pi_1,\pi_2) }[/math] a cohomology class. These groups can be encoded as homotopy [math]\displaystyle{ 2 }[/math]-types [math]\displaystyle{ X }[/math] with [math]\displaystyle{ \pi_1(X) = \pi_1 }[/math] and [math]\displaystyle{ \pi_2(X) = \pi_2 }[/math], with the action coming from the action of [math]\displaystyle{ \pi_1(X) }[/math] on higher homotopy groups, and [math]\displaystyle{ \omega }[/math] coming from the Postnikov tower since there is a fibration

[math]\displaystyle{ B^2\pi_2 \to X \to B\pi_1 }[/math]

coming from a map [math]\displaystyle{ 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]\displaystyle{ \pi_1, e, \ldots, e, \pi_n }[/math], where the fibration sequence is now

[math]\displaystyle{ B^n\pi_n \to X \to B\pi_1 }[/math]

coming from a map [math]\displaystyle{ B\pi_1 \to B^{n+1}\pi_n }[/math] whose homotopy class is an element of [math]\displaystyle{ 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 3-types of groups.[2] Essential, these are given by a triple of groups [math]\displaystyle{ (\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 3-group as a homotopy 3-type [math]\displaystyle{ X }[/math], the existence of universal covers gives us a homotopy type [math]\displaystyle{ \hat{X} \to X }[/math] which fits into a fibration sequence

[math]\displaystyle{ \hat{X} \to X \to B\pi_1 }[/math]

giving a homotopy [math]\displaystyle{ \hat{X} }[/math] type with [math]\displaystyle{ \pi_1 }[/math] trivial on which [math]\displaystyle{ \pi_1 }[/math] acts on. These can be understood explicitly using the previous model of [math]\displaystyle{ 2 }[/math]-groups, shifted up by degree (called delooping). Explicitly, [math]\displaystyle{ \hat{X} }[/math] fits into a postnikov tower with associated Serre fibration

[math]\displaystyle{ B^{3}\pi_3 \to \hat{X} \to B^2\pi_2 }[/math]

giving where the [math]\displaystyle{ B^3\pi_3 }[/math]-bundle [math]\displaystyle{ \hat{X} \to B^2\pi_2 }[/math] comes from a map [math]\displaystyle{ B^2\pi_2 \to B^4\pi_3 }[/math], giving a cohomology class in [math]\displaystyle{ H^4(B^2\pi_2, \pi_3) }[/math]. Then, [math]\displaystyle{ X }[/math] can be reconstructed using a homotopy quotient [math]\displaystyle{ \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 n group with groups [math]\displaystyle{ \pi_1,\pi_2,\ldots, \pi_n }[/math] with the latter bunch being abelian, we can consider the associated homotopy type [math]\displaystyle{ X }[/math] and first consider the universal cover [math]\displaystyle{ \hat{X} \to X }[/math]. Then, this is a space with trivial [math]\displaystyle{ \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]\displaystyle{ \hat{X} // \pi_1 }[/math] gives a reconstruction of [math]\displaystyle{ X }[/math], showing the data of an [math]\displaystyle{ n }[/math]-group is a higher group, or Simple space, with trivial [math]\displaystyle{ \pi_1 }[/math] such that a group [math]\displaystyle{ 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[3]pg 295 since the groupoid structure models the homotopy quotient [math]\displaystyle{ -// \pi_1 }[/math].

Going through the construction of a 4-group [math]\displaystyle{ X }[/math] is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume [math]\displaystyle{ \pi_1 = e }[/math] is trivial, so the non-trivial groups are [math]\displaystyle{ \pi_2,\pi_3,\pi_4 }[/math]. This gives a postnikov tower

[math]\displaystyle{ X \to X_3 \to B^2\pi_2 \to * }[/math]

where the first non-trivial map [math]\displaystyle{ X_3 \to B^2\pi_2 }[/math] is a fibration with fiber [math]\displaystyle{ B^3\pi_3 }[/math]. Again, this is classified by a cohomology class in [math]\displaystyle{ H^4(B^2\pi_2, \pi_3) }[/math]. Now, to construct [math]\displaystyle{ X }[/math] from [math]\displaystyle{ X_3 }[/math], there is an associated fibration

[math]\displaystyle{ B^4\pi_4 \to X \to X_3 }[/math]

given by a homotopy class [math]\displaystyle{ [X_3, B^5\pi_4] \cong H^5(X_3,\pi_4) }[/math]. In principle[4] this cohomology group should be computable using the previous fibration [math]\displaystyle{ B^3\pi_3 \to X_3 \to B^2\pi_2 }[/math] with the Serre spectral sequence with the correct coefficients, namely [math]\displaystyle{ \pi_4 }[/math]. Doing this recursively, say for a [math]\displaystyle{ 5 }[/math]-group, would require several spectral sequence computations, at worse [math]\displaystyle{ n! }[/math] many spectral sequence computations for an [math]\displaystyle{ n }[/math]-group.

n-groups from sheaf cohomology

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

[math]\displaystyle{ \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 n-groups constructed from a complex manifold [math]\displaystyle{ X }[/math] and sheaf cohomology on [math]\displaystyle{ X }[/math]. This is particularly applicable for complex tori.

See also

References

  1. ↑ "On Eilenberg-Maclane Spaces". http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf. 
  2. ↑ Conduché, Daniel (1984-12-01). "Modules croisés généralisés de longueur 2" (in en). 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. https://www.worldcat.org/oclc/534951159. 
  4. ↑ "Integral cohomology of finite Postnikov towers". http://doc.rero.ch/record/482/files/Clement_these.pdf. 
  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. https://www.worldcat.org/oclc/851380558. 

Algebraic models for homotopy n-types

Cohomology of higher groups

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]\displaystyle{ X }[/math] with values in a higher group [math]\displaystyle{ \mathbb{G}_\bullet }[/math], giving higher cohomology groups [math]\displaystyle{ \mathbb{H}^*(X,\mathbb{G}_\bullet) }[/math]. If we are considering [math]\displaystyle{ X }[/math] 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. 
  • Debremaeker, Raymond (2017). Cohomology with values in a sheaf of crossed groups over a site. 



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