N-group (category theory) explained

N-group (category theory) should not 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

may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of under the moniker 'gr-category'.

The general definition of

n

-group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group

\pin

, or the entire Postnikov tower for

n=infty

.

Examples

Eilenberg-Maclane spaces

One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces

K(A,n)

since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group

G

can be turned into an Eilenberg-Maclane space

K(G,1)

through a simplicial construction,[1] and it behaves functorially. This construction gives an equivalence between groups and . Note that some authors write

K(G,1)

as

BG

, and for an abelian group

A

,

K(A,n)

is written as

BnA

.

2-groups

See main article: articles, Double groupoid and 2-group.

The definition and many properties of 2-groups are already known. can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple

(\pi1,\pi2,t,\omega)

where

\pi1,\pi2

are groups with

\pi2

abelian,

t:\pi1\to\operatorname{Aut}\pi2

a group homomorphism, and

\omega\in

3(B\pi
H
1,\pi

2)

a cohomology class. These groups can be encoded as homotopy

X

with

\pi1X=\pi1

and

\pi2X=\pi2

, with the action coming from the action of

\pi1X

on higher homotopy groups, and

\omega

coming from the Postnikov tower since there is a fibration
2\pi
B
2

\toX\toB\pi1

coming from a map

B\pi1\to

3\pi
B
2
. Note that this idea can be used to construct other higher groups with group data having trivial middle groups

\pi1,e,\ldots,e,\pin

, where the fibration sequence is now
n\pi
B
n

\toX\toB\pi1

coming from a map

B\pi1\toBn+1\pin

whose homotopy class is an element of

Hn+1(B\pi1,\pin)

.

3-groups

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy of groups.[2] Essentially, these are given by a triple of groups

(\pi1,\pi2,\pi3)

with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this as a homotopy

X

, the existence of universal covers gives us a homotopy type

\hat{X}\toX

which fits into a fibration sequence

\hat{X}\toX\toB\pi1

giving a homotopy

\hat{X}

type with

\pi1

trivial on which

\pi1

acts on. These can be understood explicitly using the previous model of, shifted up by degree (called delooping). Explicitly,

\hat{X}

fits into a Postnikov tower with associated Serre fibration

B3\pi3\to\hat{X}\to

2\pi
B
2
giving where the
3\pi
B
3
-bundle

\hat{X}\to

2\pi
B
2
comes from a map
2\pi
B
2

\to

4\pi
B
3
, giving a cohomology class in

H4(B

2\pi
2,

\pi3)

. Then,

X

can be reconstructed using a homotopy quotient

\hat{X}//\pi1\simeqX

.

n-groups

The previous construction gives the general idea of how to consider higher groups in general. For an with groups

\pi1,\pi2,\ldots,\pin

with the latter bunch being abelian, we can consider the associated homotopy type

X

and first consider the universal cover

\hat{X}\toX

. Then, this is a space with trivial

\pi1(\hat{X})=0

, making it easier to construct the rest of the homotopy type using the Postnikov tower. Then, the homotopy quotient

\hat{X}//\pi1

gives a reconstruction of

X

, showing the data of an is a higher group, or simple space, with trivial

\pi1

such that a group

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

-//\pi1

.

Going through the construction of a 4-group

X

is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume

\pi1=e

is trivial, so the non-trivial groups are

\pi2,\pi3,\pi4

. This gives a Postnikov tower

X\toX3\to

2\pi
B
2

\to*

where the first non-trivial map

X3\to

2\pi
B
2
is a fibration with fiber
3\pi
B
3
. Again, this is classified by a cohomology class in

H4(B

2\pi
2,

\pi3)

. Now, to construct

X

from

X3

, there is an associated fibration
4\pi
B
4

\toX\toX3

given by a homotopy class

[X3,

5\pi
B
4]

\cong

5(X
H
3,\pi

4)

. In principle[4] this cohomology group should be computable using the previous fibration
3\pi
B
3

\toX3\to

2\pi
B
2
with the Serre spectral sequence with the correct coefficients, namely

\pi4

. Doing this recursively, say for a, would require several spectral sequence computations, at worst

n!

many spectral sequence computations for an .

n-groups from sheaf cohomology

X

with universal cover

\pi:\tilde{X}\toX

, and a sheaf of abelian groups

l{F}

on

X

, for every

n\geq0

there exists[5] canonical homomorphisms
n(\pi
\phi
1

X,H0(\tilde{X},\pi*l{F}))\toHn(X,l{F})

giving a technique for relating constructed from a complex manifold

X

and sheaf cohomology on

X

. This is particularly applicable for complex tori.

See also

References

  1. Web site: On Eilenberg-Maclane Spaces. live. https://web.archive.org/web/20201028001827/http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf. 28 Oct 2020.
  2. Conduché. Daniel. 1984-12-01. Modules croisés généralisés de longueur 2. Journal of Pure and Applied Algebra. en. 34. 2. 155–178. 10.1016/0022-4049(84)90034-3. 0022-4049.
  3. Book: Goerss, Paul Gregory.. Simplicial homotopy theory. 2009. Birkhäuser Verlag. Jardine, J. F., 1951-. 978-3-0346-0189-4. Basel. 534951159.
  4. Web site: Integral cohomology of finite Postnikov towers. live. https://web.archive.org/web/20200825141534/http://doc.rero.ch/record/482/files/Clement_these.pdf. 25 Aug 2020.
  5. Book: Birkenhake, Christina. Complex Abelian Varieties. 2004. Springer Berlin Heidelberg. Herbert Lange. 978-3-662-06307-1. Second, augmented. Berlin, Heidelberg. 573–574. 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

X

with values in a higher group

G\bullet

, giving higher cohomology groups

H*(X,G\bullet)

. If we are considering

X

as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.