group completion

> Not to be confused with *[[completion of a group]]*.

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#### Algebra
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#### Group Theory
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#### Monoid theory
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#Contents#
* table of contents
{:toc}

## Idea

The [[forgetful functor]] $U$ from [[abelian groups]] to [[commutative monoids]] has a [[left adjoint]] $G$. This is called _group completion_.
A standard presentation of the group completion is the *[[Grothendieck group of a commutative monoid]]*. As such group completion plays a central role in the definition of [[K-theory]].

More generally in [[(∞,1)-category theory]] and [[higher algebra]] there is
the [[left adjoint|left]] [[adjoint (∞,1)-functor]]

$$
  K \;\colon\;  CMon_\infty(\infty Grpd) \longrightarrow AbGrp_\infty(\infty Grpd)
$$

to the inclusion of [[abelian ∞-groups]] ([[connective spectra]]) into [[commutative ∞-monoids]] in [[∞Grpd]] ([[E-∞ spaces]]). This may be called _$\infty$-group completion_.

This serves to define [[algebraic K-theory of symmetric monoidal (∞,1)-categories]].

In [[algebraic topology]] a key role was played by models of this process at least on [[H-spaces]] ([Quillen 71](#Quillen71), [May, def. 1.3](#May)). If $N$ is a [[topological monoid]], let $B N$ denotes its [[bar construction]] ("[[classifying space]]") and $\Omega B N$ the [[loop space]] of that. Then this
$$
  N \longrightarrow \Omega B N
$$
represents the group completion of $N$ ([Quillen 71, section 9](#Quillen71), [May, theorem  1.6](#May)). This crucially enters the construction of the [[K-theory of a permutative category]].

According to ([Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7](#DwyerKan80)) this indeed gives a model for the total [[derived functor]] of 1-categorical group completion.

## Related concepts

* [[Grothendieck group]]

* [[localization of a monoid]]

## References

Classical accounts include

* {#Quillen71} [[Daniel Quillen]], _On the group completion of a simplicial monoid_ [pdf](http://www.maths.ed.ac.uk/~aar/papers/quillencomp.pdf), MIT preprint 1971, Memoirs of the AMS vol 529, 1994, pp. 89-105

* {#May} [[Peter May]], _$E_\infty$-Spaces, group completions, and permutative categories_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/13.pdf))

* {#DwyerKan80} [[William Dwyer]], [[Daniel Kan]], _Simplicial localization of categories_, Journal of pure and applied algebra 17 (1980) 267-284

$\infty$-Group completion is discussed in

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

* {#Nikolaus13} [[Thomas Nikolaus]] _Algebraic K-Theory of $\infty$-Operads_ ([arXiv:1303.2198](http://arxiv.org/abs/1303.2198))

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], def. 6.1 in  _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

and specifically its monoidal properties in

* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).

[[!redirects group completions]]

[[!redirects ∞-group completion]]
[[!redirects ∞-group completions]]


group completion theorem

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#### Algebraic topology
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[homology|homological]] version of
what is called the _group completion theorem_ ([McDuff & Segal 1976](#McDuffSegal76)) relates the [[Pontrjagin ring]] of a [[topological monoid]] $A$ to that of its [[group completion]] $\Omega B A$.

## References

The original articles are

* {#BarrattPriddy72} [[Michael Barratt]], [[Stewart Priddy]], _On the homology of non-connected monoids and their associated groups_, Commentarii Mathematici Helvetici (1972) 47: 1 ([doi:10.1007/BF02566785](https://doi.org/10.1007/BF02566785))

* {#McDuffSegal76} [[Dusa McDuff]], [[Graeme Segal]], _Homology fibrations and the “group-completion” theorem_, Inventiones mathematicae, October 1976, Volume 31, Issue 3, pp 279–284 ([doi:10.1007/BF01403148](https://doi.org/10.1007/BF01403148))

* {#Quillen94} [[Daniel Quillen]], _On the group completion of a simplicial monoid_, Appendix Q in: [[Eric Friedlander]], [[Barry Mazur]], _Filtrations on the homology of algebraic varieties_, Memoir of the A.M.S., Vol. 110, no. 529 (1994) ([doi:10.1090/memo/0529](http://dx.doi.org/10.1090/memo/0529), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/quillencomp.pdf))

Alternative proof using a [[model category]] of [[bisimplicial sets]]:

* [[Ieke Moerdijk]], _Bisimplicial Sets and the Group-Completion Theorem_ In: [[Rick Jardine]], [[Victor Snaith]] (eds.) _Algebraic K-Theory: Connections with Geometry and Topology_, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 279. Springer, Dordrecht (1989) ([doi:10.1007/978-94-009-2399-7_10](https://doi.org/10.1007/978-94-009-2399-7_10))

Alternative formulation for the case of [[commutative monoid|commutative]] [[topological monoids]]:

* [[Oscar Randal-Williams]], _Group-completion, local coefficient systems and perfection_, Q. J. Math. **64** 3 (2013) 795–803 [[pdf](https://www.dpmms.cam.ac.uk/~or257/GCrem.pdf), [doi:10.1093/qmath/hat024](https://doi.org/10.1093/qmath/hat024)]

* Simon Gritschacher, _A remark on the group-completion theorem_ ([arxiv:1709.02036](https://arxiv.org/abs/1709.02036))

Generalization to [[generalized homology]] represented by [[ring spectra]] and relation to the [[Quillen plus construction]]:

* [[Thomas Nikolaus]], _The group completion theorem via localizations of ring spectra_, 2017 ([pdf](https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Group_completion.pdf))

A proof based on Nikolaus's proof was written up in

* Oscar Bendix Harr, _Group completion is a completion_, [PDF](http://web.math.ku.dk/~jg/students/oscar-bsthesis.pdf).

[[!redirects group completion theorems]]
[[!redirects group-completion theorem]]
[[!redirects group-completion theorems]]


group extension

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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _group extension_ of a [[group]] $G$ by a group $A$ is third group $\hat G$ that sits in a [[short exact sequence]], that can usefully be thought of as a [[fiber sequence]], $A \to \hat G \to G$.

## Definition
 {#Definition}

+-- {: .num_defn #GroupExtension}
###### Definition

Two consecutive [[homomorphisms]] of  [[groups]]

\[\label{shortExtension}
  A
  \overset{i}\hookrightarrow \hat G\overset{p}\to G
\]

are a **[[short exact sequence]]** if

1. $i$ is [[monomorphism]],

1. $p$ an [[epimorphism]]

1. the [[image]] of $i$ is all of the [[kernel]] of $p$: $ker(p)\simeq im(i)$.

We say that such a short exact sequence exhibits $\hat G$ as an **extension** of $G$ by $A$.

If $A \hookrightarrow \hat G$ factors through the [[center]] of $\hat G$ we say that this is a **[[central extension]]**.

=--

+-- {: .num_remark }
###### Remark

Sometimes in the literature one sees $\hat G$ called an extension "of $A$ by $G$". This is however in conflict with terms such as _[[central extension]]_, _[[extension of principal bundles]]_, etc, where the extension is always regarded _of the base, by the [[fiber]]_.  (On the other hand, our terminology conflicts with the usual meaning of "extension" in algebra.  For example, in Galois theory if $k$ is a field, then an extension of $k$ contains $k$ as a subfield.)

=--

Under the [[looping and delooping]]-equivalence, this is equivalently reformulated as follows. For $G \in$ [[Grp]] a [[group]], write $\mathbf{B}G \in $ [[Grpd]] for its [[delooping]] groupoid.

+-- {: .num_defn }
###### Definition

A sequence $A \to \hat G \to G$ is a [[short exact sequence]] of groups precisely if the [[delooping]]

$$
  \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G
$$

is a [[fiber sequence]] in the [[(2,1)-category]] [[Grpd]].

=--

This says that group extensions are special cases of the general notion discussed at _[[∞-group extension]]_. See there for more details.

+-- {: .num_defn #MorphismOfGroupExtensions}
###### Definition

A [[homomorphism]] of extensions $f : \hat G_1 \to \hat G_2$
of a given $G$ by a given $A$ is a [[group homomorphism]] of this form which fits into a [[commuting diagram]]

$$
  \array{
     && \hat G_1
    \\
    & \nearrow && \searrow
    \\
    A &&\downarrow^{\mathrlap{f}}&& G
    \\
    & \searrow && \nearrow
    \\
    && \hat G_2
  }
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

A morphism of extensions as in def. \ref{MorphismOfGroupExtensions} is necessarily an [[isomorphism]].

\[
  \label{equivExt}
  \array{
     1\to &A&\stackrel{i}\to &\hat G_1&\stackrel{p}\to &G&\to 1
     \\
    &\downarrow\mathrlap{=}&&\downarrow\mathrlap\epsilon&&\downarrow\mathrlap=&
    \\
    1\to &A&\stackrel{i'}\to &\hat G_2&\stackrel{p'}\to& G&\to 1
  }
  \,.
\]

=--

+-- {: .proof}
###### Proof

By the [[short five lemma]].

=--

## Properties

We discuss properties of group extensions in stages,

* [General properties](#PropertiesGeneral)

* [Central group extensions](#PropertiesCentralGroupExtensions)

* [Abelian group extensions](#PropertiesAbelianGroupExtensions)

* [Schreier theory for nonabelian extensions](#SchreierTheory)

### General
 {#PropertiesGeneral}

#### Fibers of extensions are normal subgroups

+-- {: .num_prop }
###### Proposition

For $A \hookrightarrow \hat G \to G$ a group extension, the inclusion $A \hookrightarrow \hat G$ is a [[normal subgroup]] inclusion.

=--

+-- {: .proof}
###### Proof

We need to check that for all $a \in A \hookrightarrow G$ and $g \in G$ the result of the [[adjoint action]] $g a g^{-1}$ formed in $\hat G$ is again in $A \stackrel{i}{\hookrightarrow} \hat G$.

Since $p : \hat G \to G$ is a group homomorphism we have that

$$
  \begin{aligned}
    p(g a g^{-1}) & = p(q) p(a) p(g^{-1})
    \\
    & = p(g) p(a) p(g)^{-1}
    \\
    & = p(g) p(g)^{-1}
    \\
    & = 1
  \end{aligned}
$$

and hence $g a g^{-1}$ is in the [[kernel]] of $p$. By the defining exactness property therefore it is in the [[image]] of $i$.

=--

#### Extensions as torsors / principal bundles
  {#Torsors}

+-- {: .num_prop}
###### Proposition

For $A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G$ a group extension, we have that $p : \hat G \to G$ is an $A$-[[torsor]] over $G$ where  the [[action]] of $A$ on $\hat G$ is defined by

$$
  \rho : A \times \hat G \stackrel{(i,Id)}{\to} \hat G \times_G \hat G \stackrel{\cdot}{\to} \hat G
  \,.
$$

=--

+-- {: .proof}
###### Proof

That $\rho$ is indeed an action _over_ $B$ in that

$$
  \array{
     A \times \hat G &&\stackrel{\rho}{\to}&& \hat G
     \\
     & {}_{\mathllap{ p \circ p_2}}\searrow && \swarrow_{\mathrlap{p}}
     \\
     &&  G
  }
$$

follows from the fact that $p$ is a group homomorphism and that $A$ is in its [[kernel]].

That $A$ is actually _equal_ to the kernel gives the principality condition

$$
  (\rho, p_2) : A\times \hat G \stackrel{\simeq}{\to} \hat G \times_G \hat G
  \,.
$$

=--

For $A$ an [[abelian group]] we may understand the $A$-torsor/$A$-[[principal bundle]] $\hat G$ as the [[delooping]] of the $\mathbf{B}A$-[[principal 2-bundle]] $\mathbf{B} \hat G \to \mathbf{BG}$ that is classified by (is the [[homotopy fiber]] of) the 2-[[cocycle]] in [[group cohomology]]  $c : \mathbf{B}G \to \mathbf{B}^2 A$ that classifies the extension.

All this is then summarized by the statement that

$$
  A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to}
  \mathbf{B}^2 A
$$

is a [[fiber sequence]] in [[∞Grpd]] (or in [[?LieGrpd]] if we have [[Lie group]] extensions, etc).

Here we may think of $\mathbf{B}\hat G$ as being the $\mathbf{B}A$-[[principal 2-bundle]] over $\mathbf{B}G$ classified by $c$. See the examples discussed at [[bundle gerbe]].

#### Split extensions and semidirect product groups
 {#SplitExtensionsAndSemidirectProductGroups}

+-- {: .num_defn #SplitExtension}
###### Definition

A group extension $A \to \hat G \stackrel{p}{\to} G$ is called **split** if there is a [[section]] [[homomorphism]] $\sigma \colon G \to \hat G$, hence a group homomorphism such that $p \circ \sigma = id_G$.

=--

+-- {: .num_remark}
###### Remark

It is important here that $\sigma$ is itself required to be a group homomorphism, not just a [[function]] on the underlying sets. The latter always exists as soon as the [[axiom of choice]] holds, since $p$ is an [[epimorphism]] by definition.

=--

+-- {: .num_prop #SplitExtensionsAreSemidirectProducts}
###### Proposition

Split extensions $\hat G$ of $G$ by $A$, def. \ref{SplitExtension}, are equivalently [[semidirect product groups]]

$$
  A \hookrightarrow \hat G \simeq A \rtimes_\rho G \to G
$$

for some [[action]] $\rho \colon G \times A  \to A$ of $G$ on $A$.

=--

This means that the underlying set is $U(A \rtimes_\rho G) = U(A) \times U(G)$ and the group operation in $A \rtimes_\rho G$ is

$$
  (a_1, g_1) \cdot (a_2, g_2)
  =
  (a_1 \cdot \rho(g_1)(a_2) , g_1 \cdot g_2)
  \,.
$$

The inclusion of $A$ is by

$$
  a \mapsto (a,e)
$$

and the projection to $G$ is by

$$
  (a,g) \mapsto g
  \,.
$$

+-- {: .proof}
###### Proof

Given a split extension $A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G$ with splitting $\sigma \colon G \to \hat G$, define an [[action]] of $G$ on $A$ by the restriction of the [[adjoint action]] $\rho_{ad}$ of $\hat G$ on itself to $A$:

$$
  \rho
    \colon
  A \times G
   \stackrel{(i,\sigma)}{\to}
 \hat G \times \hat G
   \stackrel{\rho_{ad}}{\to}
 \hat G
$$

$$
  (a,g) \mapsto \sigma(g)^{-1} \cdot a \cdot \sigma(g)
  \,.
$$

Then (...)

=--

+-- {: .num_prop}
###### Proposition

A split extension $A \to \hat G \to G$ is a [[central extension]] precisely if the [[action]] $\rho$ induced from it as in prop. \ref{SplitExtensionsAreSemidirectProducts} is trivial.

=--

+-- {: .proof}
###### Proof

For it to be a central extension the inclusion $A \to A \rtimes_\rho G$ has to land in the [[center]] of $A \rtimes_\rho G$, hence all elements $a \in A$ have to commute as elements $(a,e) \in A \rtimes_\rho G$ with all elements of $A \rtimes_\rho G$. But consider elements of the form $(e,g) \in A \rtimes_\rho G$ for all $g \in G$. Then

$$
  (a,e) \cdot (e,g) = (a, g)
$$

but

$$
  (e,g) \cdot (a,e) = (\rho(g)(a), g)
  \,.
$$

For these to be equal for all $a \in A$, $\rho(g)$ has to be the identity. Since this is to be true for all $g \in G$, the action has to be trivial.

=--

+-- {: .num_prop}
###### Remark

This means in particular that split central extensions are product groups $A \to G$. If all groups involved are [[abelian groups]], then these are equivalently the [[direct sums]] $A \oplus G$ of abelian groups. In this way the notion of split group extension reduces to that of [[split short exact sequences]] of abelian groups.

=--

+-- {: .num_prop}
###### Remark

If we have a split extension the different  splittings are given by [[derivation on a group|derivation]]s, but with possibly non-abelian values. In fact if we have $s: G\to A\rtimes G$ is a section then $s(g) = (a(g),g)$, and the multiplication in $A\rtimes G$ implies that $a: G\to A$ is a derivation. These are considered as the (possibly non-abelian) 1-cocycles of $G$ with (twisted) coefficients in $A$, as considered in, for instance, Serre's notes on [[Galois cohomology]].

=--
### Central group extensions
 {#PropertiesCentralGroupExtensions}

We discuss properties of _central_ group extensions, those where $A \hookrightarrow \hat G$ factors through the [[center]] of $\hat G$. This is a special case of the general discussion below in _[Nonabelian group extensions (Schreier theory)](#SchreierTheory)_ but is considerably less complex to write out in components.

We first discuss the

* [Classification by group cohomology](#CentralExtensionClassificationByGroupCohomology)

of central extensions in components, and then show in

* [Formulation in homotopy theory](#FormulationInHomotopyTheory)

how this follows from a more systematic abstract theory.

#### Classification by group cohomology
 {#CentralExtensionClassificationByGroupCohomology}

We discuss the classification of _central_ extensions by [[group cohomology]]. This is a special case of the more general (and more complicated) discussion below in _[Nonabelian group extensions (Schreier theory)](#SchreierTheory)_.

For $G$ a [[group]] and $A$ an [[abelian group]], write

$$
  H^2_{grp}(G,A) \;\; \in Ab
$$

for the degree-2 [[group cohomology]] of $G$ with [[coefficients]] in $A$, and write

$$
  Ext(G,A) \;\;\in Ab
$$

for the group of central extensions of $G$ by $A$.

+-- {: .num_theorem }
###### Theorem

There is a [[natural equivalence]]

$$
  Ext(G,A) \simeq H^2_{Grp}(G,A)
  \,.
$$

=--

We prove this below as prop. \ref{ExtractionAndReconstructionConsituteEquivalence}. Here we first introduce stepwise the ingredients that go into the proof.

+-- {: .num_defn #CentralExtensionAssociatedTo2Cocycle}
###### Definition
**(central extension associated to group 2-cocycle)**

For $[c] \in H^2_{Grp}(G,A)$ a [[group cohomology]] class represented by a [[cocycle]] $c \colon G \times G \to A$, define a group

$$
  G \times_c A \in Grp
$$

as follows. The underlying set is the [[cartesian product]] $U(G) \times U(A)$ of the underlying sets of $G$ and $A$. The group operation on this is given by

$$
  (g_1, a_1) \cdot (g_2, a_2)
  \coloneqq
  (g_1 \cdot g_2 ,\; a_1 + a_2 + c(g_1, g_2))
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

This defines indeed a group: the [[cocycle]] condition on $c$ gives precisely the  [[associativity]] of the product on $G \times_c A$.
Moreover, the construction extends to a [[homomorphism]] of groups

$$
  Rec : H^2_{Grp}(G,A) \to Ext(G,A)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Forming the product of three elements of $G \times_c A$ bracketed to the left is, according to def. \ref{CentralExtensionAssociatedTo2Cocycle},

$$
  \left(
    \left(g_1, a_1\right) \cdot \left(g_2, a_2\right)
  \right)
  \cdot
  \left(
    g_3, a_3
  \right)
 =
  \left(
    g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3
    +
    c(g_1, g_2) +  c\left(  g_1 g_2, g_3 \right)
  \right)
  \,.
$$

Bracketing the same three elements to the right yields

$$
  \left(g_1, a_1\right)
   \cdot
  \left(
     \left(g_2, a_2\right)
     \cdot
   \left(
    g_3, a_3
  \right)
 \right)
 =
  \left(
    g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3
    +
    c(g_2, g_3) +  c\left(  g_1 , g_2 g_3 \right)
  \right)
  \,.
$$

The difference between the two expressions is read off to be precisely

$$
  (1, (d c) (g_1, g_2, g_3))
 \,,
$$

where $d c$ denotes the group cohomology differential of $c$. Hence this vanishes precisely if $c$ is a group 2-cocycle, hence we have an associative product.

To see that it has inverses, notice that for all $(g,a)$ we have

$$
  (g,a)
  \cdot
  (g^{-1}, - a - c(g,g^{-1}))
  =
  (e, a - a - c(g,g^{-1})+ c(g,g^{-1}) )
$$

and hence inverses are given by $(g,a)^{-1} = (g^{-1}, -a - c(g,g^{-1}))$. Hence $G \times_c A$ is indeed a group.

By the discussion at [group cohomology  -- degree-2](group+cohomology#Degree2) we may assume without restriction that $c$ is a normalized cocycle, hence that $c(e,-) = c(-,e) = 0$. Using this we find that the inclusion

$$
  i \colon A \to G \times_c A
$$

given by $a \mapsto (e,a)$ is a group homomorphism. Moreover, the projection on the underlying sets evidently yields a group homomorphism $p \colon G \times_c A \to G$ given by $(g,a) \mapsto g$. The kernel of this is $A$, and hence

$$
  A \stackrel{i}{\hookrightarrow} G \times_c A \stackrel{p}{\to} G
$$

is indeed a group extension. It is a [[central extension]] again using the assumption that $c$ is normalized $c(g,e) = c(e,g) = 0$:

$$
  (g,a) \cdot (e,\tilde a) = (g, a + \tilde a + 0) = (e,\tilde a) \cdot (g,a)
  \,.
$$

Finally to see that the construction is independent of the choice of coycle $c$ representing $[c]$, let $\tilde c$ be another representative which differs by a [[coboundary]] $h \colon G \to A$ with

$$
  \tilde c (g_1,g_2) \coloneqq c(g_1,g_2) - h(g_1) - h(g_2) + h(g_1 g_2)
  \,.
$$

We claim that then we have a homomorphism of central extensions (hence an isomorphism) of the form

$$
  \array{
     A &\to& G \times_c A &\to& G
     \\
     \downarrow^{\mathrlap{id}}
      &&
     \downarrow^{\mathrlap{(id_G, p_2 -h \circ p_1)}} && \downarrow^{\mathrlap{=}}
     \\
     A &\to& G \times_{\tilde c} A &\to& G
  }
  \,.
$$

To see this we check for all elements that

$$
  \begin{aligned}
   (g_1, a_1 - h(g_1)) \cdot (g_2, a_2 - h(g_2))
   & =
   (g_1 g_2, a_1 + a_2 - h(g_1) - h(g_2) + c(g_1, g_2))
   \\
   & =
  (g_1 g_2, a_1 + a_2 + \tilde c(g_1, g_2) - h(g_1 g_2) )
  \end{aligned}
  \,.
$$

Hence the construction of $G \times_c A$ indeed defines a function $H^2_{Grp}(G,A) \to CentrExt(G,A)$.

=--

Assume the [[axiom of choice]] in the ambient [[foundations]].

+-- {: .num_defn #2CocycleExtractedFromCentralExtension}
###### Definition
**(2-cocycle extracted from central extension)**

Given a central extension $A \to \hat G \to G$ define a group 2-cocycle $c : G \times G \to A$ as follows.

Choose a [[section]] $\sigma : U(G) \to U(\hat G)$ of the underlying [[sets]] (which exists by the [[axiom of choice]] and the fact that $p : \hat G \to G$ is by definition an [[epimorphism]]). Then define $c$ by

$$
  c
  \colon
  (g_1, g_2)
    \mapsto
  \sigma(g_1) \sigma(g_2) \sigma(g_1 g_2)^{-1}
  \in A
  \,,
$$

where on the right we are using that by the section-property of $\sigma$ and the group homomorphism property of $p$

$$
  p(\sigma(g_1) \sigma(g_2) \sigma(g_1 g_2)^{-1}) = 1
$$

and hence by the exactness of the extension the argument is in $A \hookrightarrow \hat G$.

=--

Below in remark \ref{RelationBetweenComponentCocycleExtractionAndHomotopyTheory} is a discussion of how this construction arises from a more systematic discussion in [[homotopy theory]].

+-- {: .num_prop}
###### Proposition

The construction of prop. \ref{2CocycleExtractedFromCentralExtension} indeed yields a 2-cocycle in [[group cohomology]]. It extends to a morphism

$$
  Extr \colon Ext(G,A) \to H^2_{Grp}(G,A)
  \,.
$$

=--

+-- {: .proof}
###### Proof

The cocycle condition to be checked is that

$$
  c(g_1, g_2) - c(g_0 g_1, g_2) + c(g_0, g_1 g_2) - c(g_0, g_1)
  =
  1
$$

for all $g_0, g_1, g_2 \in G$. Writing this out with def. \ref{2CocycleExtractedFromCentralExtension} yields

$$
  \sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2)
  \left(\sigma(g_0 g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_0 g_1 g_2)\right)^{-1}
  \sigma(g_0)^{-1} \sigma(g_1 g_2)^{-1} \sigma(g_0 g_1 g_2)
  \left( \sigma(g_0)^{-1} \sigma(g_1)^{-1} \sigma(g_0 g_1)  \right)^{-1}
  \,.
$$

Here it is sufficient to observe that for every term also the inverse term appears.

To see that this is a well-defined map to $H^2_{grp}(G,A)$ we need to check that for $\tilde \sigma : G \to \hat G$ a different choice of section, the corresponding cocycles differ by a group coboundary $\tilde c - c = d h$. Clearly this is obtained by setting

$$
  h \colon g \mapsto \tilde \sigma(g)\sigma(g)^{-1}
  \,,
$$

where we use that the right hand side is in $A \hookrightarrow \hat G$ since because both $\sigma$ and $\tilde \sigma$ are sections of $p$, the image of the right hand under $p$ is the neutral element in $G$.

=--

+-- {: .num_prop #ExtractionAndReconstructionConsituteEquivalence}
###### Proposition

The two morphisms of  def. \ref{CentralExtensionAssociatedTo2Cocycle} and
def. \ref{2CocycleExtractedFromCentralExtension} exhibit the [[equivalence]]

$$
  H^2_{Grp}(G,A) \stackrel{\underoverset{\simeq}{Extr}{\leftarrow}}{\underset{Rec}{\to}} CentrExt(G,A)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Let $[c] \in H^2_{Grp}(G,A)$. Then by construction of $\hat G \coloneqq G \times_c A$ there is a canonical section of the underlying function of sets $U(G \times_c A) \to U(G)$ given by $(id_{U(G)}, 0) U(G) \to U(G) \times U(A)$. The cocycle induced by this section sends

$$
  \begin{aligned}
    (g_1, g_2) & \mapsto (g_1, 0) (g_2, 0) (g_1 g_2, 0)^{-1}
    \\
    & =
    (g_1, 0) (g_1, 0) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) )
    \\
    & =  (g_1 g_2, c(g_1, g_2) ) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) )
    \\
    & = (e, c(g_1, g_2) - c(g_1 g_2, (g_1 g_2)^{-1}) + c(g_1 g_2, (g_1 g_2)^{-1}))
   \\
   & = (e, c(g_1, g_2))
  \end{aligned}
  \,,
$$

which is $c(g_1, g_2) \in A \hookrightarrow G \times_c A$, and hence this recovers the 2-cocycle that we started with.

This shows that $Extr \circ Rec = id$ and in particular that $Rec$ is a [[surjection]]. It is readily seen that the [[kernel]] of $Rec$ is trivial, and so it is an equivalence.

=--

#### Formulation in homotopy theory
 {#FormulationInHomotopyTheory}

We discuss the classification of central group extensions by degree-2 [[group cohomology]] in the more abstract context of [[homotopy theory]] (via the translation discussed at _[[looping and delooping]]_), complementing the [above](CentralExtensionClassificationByGroupCohomology
) component-wise discussion.

Let

$$
  A \hookrightarrow \hat G \to G
$$

be a central group extension, def. \ref{GroupExtension}, hence with $A$ an [[abelian group]] included in the [[center]] of $G$. Then $A$ is in particular a [[normal subgroup]] and hence the homorphism

$$
  (A \to \hat G)
$$

may be regarded as a [[crossed module]] of groups. This is equivalently a [[strict 2-group]] structure on the [[groupoid]] whose objects are $\hat G$ and whose morphisms are labeled in $A$

$$
  (A \to \hat G) =
  \{
     g \stackrel{a}{\to} a \cdot g
  \}
  \,.
$$

+-- {: .num_defn }
###### Definition

Write

$$
  \mathbf{B}(A \to \hat G) \in Grpd
$$

for the [[delooping]] of this [[2-group]] to a one-object [[2-groupoid]].

The [[∞-nerve]] (or [[Duskin nerve]]) $N \mathbf{B}(A \to \hat G) \in $ [[sSet]] of this is a (3-[[coskeleton|coskeletal]]) [[Kan complex]] that realizes this as a [[truncated object of an (infinity,1)-category|2-truncated]]  [[∞-groupoid]].

$$
  \mathbf{B}(A \to \hat G) \in \infty Grpd
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

The obvious strict [[2-functor]]

$$
  \mathbf{B}(A \to \hat G) \stackrel{\simeq}{\to} \mathbf{B}H
$$

is an [[weak homotopy equivalence|equivalence]] of 2-groupoids.

=--

+-- {: .proof}
###### Proof

One way to see this is to notice that this is a [[k-surjective functor]] for all $k \in \{0,1,2,3\}$, hence a weak equivalence in the [[folk model structure]] on $\omega$-groupoids. Equivalently, under the [[nerve]] the morphism of [[simplicial sets]]

$$
  N\mathbf{B}(A \to \hat G) \to N \mathbf{B}H
$$

is an acyclic [[Kan fibration]], hence a [[weak equivalence]] in the standard [[model structure on simplicial sets]].

=--

+-- {: .num_prop }
###### Proposition

The extension $A \to \hat G \to G$ sits in a long [[homotopy fiber sequence]] in the [[(∞,1)-category]] [[∞Grpd]] of the form

$$
  A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 A
$$

which in [[Kan complexes]]/[[simplicial sets]] is [[presentable (infinity,1)-category|presented]] by  the [[zigzag]] of  [[n-functors]] between [[strict ∞-groupoid]] (sequence of [[2-anafunctors]]) of the form

$$
  \array{
     && && && && && \mathbf{B}(A \to \hat G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A
     \\
     && && && && && {}^{\mathrlap{\simeq}}\downarrow
     \\
     &&  && (A \to \hat G) &\to& \mathbf{B}A &\to& \mathbf{B}\hat G &\to&  \mathbf{B}G
     \\
     && && \downarrow^{\mathrlap{\simeq}}
     \\
     A &\to& \hat G &\to& G
  }
  \,.
$$

In particular, the induced [[connecting homomorphism]]

$$
  \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 A
$$

is the [[group cohomology]] [[cocycle]] that classifies the delooped extension as a $\mathbf{B}A$-[[principal 2-bundle]].

=--

+-- {: .proof}
###### Proof

One sees directly that the morphisms $\mathbf{B}\hat G \to \mathbf{B}G$ and  $\mathbf{B}(A \to \hat G ) \to \mathbf{B}^2 A $ as well as their loopings $\hat G \to G$ and  $(A \to \hat G) \to G$ are [[Kan fibrations]]. By the discussion at [[homotopy pullback]] this means that the set-theoretic [[fibers]] of these morphisms are models for their [[homotopy fibers]]. But the ordinary [[kernel]] of $\mathbf{B}(A \to \hat G) \to \mathbf{B}(A \to 1) = \mathbf{B}^2 A$ is manifestly $\mathbf{B} \hat G$, and so on.

=--

+-- {: .num_remark #RelationBetweenComponentCocycleExtractionAndHomotopyTheory}
###### Remark

The construction in def. \ref{2CocycleExtractedFromCentralExtension}

$$
  Rec : CentrExt(G,A) \to H^2_{Grp}(G,A)
$$

is precisely the result of moving set-theoretically through the [[zigzag]]

$$
  \array{
    \mathbf{B}(A \to \hat G) &\to& \mathbf{B}(A \to 1) = \mathbf{B}^2 A
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}G
  }
$$

from the bottom left to the top right,
and that this is well-defined on [[cohomology]] comes down to the statement that the vertical morphism is a [[weak homotopy equivalence]].

This is a nonabelian analog of the discussion at _[[mapping cone]]_ in the section _[Homology exact sequences and fiber sequences](mapping%20cone#HomologyExactSequencesAndFiberSequences)_.

=--

### Abelian group extensions
 {#PropertiesAbelianGroupExtensions}

+-- {: .num_remark}
###### Remark

For $A, G \in $ [[Ab]] $\hookrightarrow$ [[Grp]] even a [[central extension]] $\hat G$ of $G$ by $A$ is not necessarily itself an abelian group.

=--

But by prop. \ref{ExtractionAndReconstructionConsituteEquivalence} above it is so if the group 2-cocycle that classifies the extension is symmetric:

+-- {: .num_defn}
###### Definition

A 2-cocycle $c \colon G \times G \to A$ in [[group cohomology]] is **symmetric** if

$$
  \forall_{g_1, g_2 \in G} c(g_1, g_2) = c(g_2, g_1)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

A group 2-cocycle cohomologous to a symmetric group 2-cocycle is itself symmetric. Hence we may speak of symmetric group cohomology classes in degree 2.

=--

+-- {: .num_defn}
###### Definition

Write

$$
  H^2_{Grp}(G,A)_{sym} \hookrightarrow H^2_{Grp}(G,A)
$$

for the set (group) of classes of symmetric group 2-cocycles on $G$ with coefficients in $A$.

=--

+-- {: .num_defn}
###### Definition

For $G,A \in Ab \hookrightarrow Grp$, write
$Ext(G,A)$
for the subset of equivalence class of abelian group extensions of $G$ by $A$.

=--

The theory of _abelian group extensions_ in [[Ab]] is naturally and classically treated with tools of [[homological algebra]], such as the theory of [[Ext]]-functors.

For the moment see at _[[projective resolution]]_ the section

* _[Projective resolutions adapted to group cocycles](projective+resolution#ProjectiveResolutionsForGroupCocycles)_

and

* _[Derived Hom-functor / Ext-functor](projective+resolution#DerivedHomFunctor)_ .

### Nonabelian group extensions (Schreier theory)
 {#SchreierTheory}

We discuss the classification theory for the general case of nonabelian
group extensions, first in the form of

* _[Traditional Schreier theory](#SchreierTheoryTraditional)_

and then more abstractly in the language of [[homotopy theory]] in

* _[Homotopy theory perspective on Schreier theory](#SchreierTheorynPOV)_

#### Traditional description
 {#SchreierTheoryTraditional}

[[Otto Schreier]] (1926) and [[Samuel Eilenberg|Eilenberg]]-[[Saunders MacLane|Mac Lane]] (late 1940-s) developed a theory of classification of nonabelian extensions of abstract groups leading to the low dimensional [[nonabelian group cohomology]]. This is sometimes called **Schreier's theory** of nonabelian group extensions.

The traditional Schreier-Mac Lane way to obtain nonabelian group 2-cocycle
from a group extension as above starts with choosing a set-theoretic section of $p:G\to B$.

**Note.** The exposition which follows in this long "traditional" section of this entry is mainly from personal notes of Zoran Škoda from 1997.

Each element $g$ of $G$ defines an inner automorphism $\phi(g)$ of $K$
by $\phi(g)(k) = g k g^{-1}$.
The restriction $\phi|_K$ takes
(by definition) values in the subgroup $Int(K)$ of inner automorphisms
of $K$. In fact $\phi:G\to Inn(G)\subset Aut(K)$ is a homomorphism of groups.

If $g_1$ and $g_2$ are in the same left coset, that is $g_1K = g_2K$,
then there is $k \in K$, $g_1 = g_2k$, so that $\forall k' \in K$ we have
$\phi(g_1k') = \phi(g_2k k') = \phi(g_2)\phi(k k')$ and therefore
$\phi(g_1K) \subset \phi(g_2)Int(K)$. Thus we obtain a well-defined map
$\phi_* : G/K \rightarrow Aut(K)/Int(K)$.
Choose a set-theoretic section of the projection $p : G \rightarrow B$
and let
\[\psi \stackrel{def}{=} \phi \circ \sigma: B \rightarrow Aut(K).\]
**Warning.** Unlike $\phi$, the map $\psi$ is *not* a homomorphism of groups.

We attempt to reconstruct $G$ from
the knowledge of $\psi$ and $K$.
As a set, $G$ can be naturally identified with $B \times K$.
Indeed, write each element $g \in G$ as
$\sigma(b)k, b \in B, k \in K$ by setting
$b = p(g), k = \sigma(p(g))^{-1}g$.
Elements $b \in B$ and $k \in K$ in that decomposition are unique,
and we get a bijection
\begin{equation}
B\times K\ni (b,k)\mapsto\sigma(b)k \in G,
\end{equation}
whose inverse is the map $g \mapsto (p(g), \sigma(p(g))^{-1}g)$.
By means of that bijection, $B \times K$ inherits the group structure
from $G$. Let us figure out the multiplication rule on $B \times K.$
If $\sigma(b_1)k_1 = g_1$, and $\sigma(b_2)k_2 = g_2$, then,
\begin{equation}
g_1g_2 = \sigma(b_1)k_1\sigma(b_2)k_2 =
\sigma(b_1)\sigma(b_2)\sigma(b_2)^{-1}k_1\sigma(b_2)k_2.
\end{equation}
 Now $p(\sigma(b_1)\sigma(b_2)) = p(b_1b_2)$ so
$$\chi(b_1,b_2) \stackrel{def}{=}
 \sigma(b_1b_2)^{-1}\sigma(b_1)\sigma(b_2) \in K.
$$
This formula clearly
defines a function $\chi : B \times B \rightarrow K$. In this notation,
$$\array{
g_1g_2 & = & \sigma(b_1b_2)\chi(b_1,b_2)\phi(\sigma(b_2)^{-1})(k_1)k_2 \\
       & = & \sigma(b_1b_2)\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2.
}$$
and using bijection of $G$ with $B\times K$ this can be expressed in terms of elements in $B\times K$ so that
\begin{equation}\label{mrule}
(b_1,k_1)(b_2,k_2) = (b_1b_2,\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2).
\end{equation}

According to this formula, *all the information about the multiplication is encoded in functions $\chi : B \times B \rightarrow Aut(K)$
and $\psi : B \rightarrow Aut(K)$, and we may forget about $\sigma$* at this point. However, _not every pair $(\chi,\psi)$ will
give some multiplication rule on $B \times K$_.
Let $a,b,c \in B$,
and $e = e_K$ be the unity element in $K$.
Then
$$
[(a,e)(b,e)](c,e) = (a b, \chi(a,b))(c,e) =
 (a b c, \chi(a b,c) \psi(c)^{-1}(\chi(a,b))).
$$
From the other side, this has to be the same, by associativity, to
$$
(a,e)[(b,e)(c,e)] = (a,e)(bc,\chi(b,c)) = (a b c,\chi(a,b c)\chi(b,c))
$$
where we took into account that expressions like $[\psi^{-1}(b)(e)] = e$,
because $\psi(b)$ is an *automorphism* for each $b \in B$.

 Comparing the expressions above we obtain
\[\label{psi1}
 \chi(a b,c)\psi(c)^{-1}(\chi(a,b)) = \chi(a,b c)\chi(b,c),
    for all a,b,c \in B.
\]

If the pair $(\chi,\psi)$ is constructed as above, then

$$\array{
\psi(a)\psi(b)k & = & \phi(\sigma(a))\phi(\sigma(b))k \\
        & = & \sigma(a)\sigma(b)k\sigma(b)^{-1}\sigma(a)^{-1} \\
        & = & \sigma(a b)\chi(a,b)k\chi(a,b)^{-1}\sigma(a b)^{-1} \\
        & = & \psi(a b) Ad_K(\chi(a,b)) k,
}$$

where $Ad_K$ is the canonical map $K \rightarrow Int(K)$, $k\mapsto k(-)k^{-1}$.

Thus we obtain the relation
\[\label{psi2}
\psi(a)\psi(b) = \psi(a b) Ad_K(\chi(a,b))
\]

+-- {: .num_defn}
###### Definition
Let $B$ and $K$ be two groups.
Let $\chi: B \times B \rightarrow K$
and $\psi : B \rightarrow Aut(K)$ satisfy \eqref{psi1}
and \eqref{psi2}. Then we call that the family
$\{\chi(b_1,b_2)| b_1,b_2 \in B\}$
is a factor system (This term is due Schreier(1924))
or a **nonabelian group 2-cocycle with automorphisms**, and
the family $\{\psi(b) | b \in B \}$ -- a system of automorphisms
=--

A 2-cocycle $\chi$ is **counital** if $\chi(b,e) = \chi(e,b) = e$, for all $b \in B$.

If $K$ is commutative, then $\psi$ is always a homomorphism (cf. (eq:psi2)). Then $K$ is a right $B$-module through $\psi(-)^{-1}$. That justifies the sometimes used term "(right) cocycle $B$-module" for $(K,\psi,\chi)$. If $\psi$ is trivial ($\psi(b) = Id_K, \forall b \in B$)
then the cocycle condition (eq:psi1) becomes
\begin{equation}
\chi(a b,c)\chi(a,b) = \chi(a,b c)\chi(b,c).
\end{equation}

+-- {: .num_theorem}
###### Theorem
If formulas (eq:psi1) and (eq:psi2)
are both satisfied, then the formula (eq:mrule)
for multiplication of pairs defines a group multiplication on $B \times K$. That set, together
with multiplication (eq:mrule) is called the
**cocycle cross product** of $B$ and $K$ with cocycle $\chi$ and
action $\psi$. If the cocycle is trivial i.e. $\chi(\cdot,\cdot) = e_K$,
we call it the **(external) semidirect product**.
=--

+-- {: .proof}
###### Proof

We have checked above the [[associativity]] for pairs of the form $(a,e)$ etc. This was useful to find the cocycle condition correctly. Now the general associativity should be a similar calculation with general elements. Using (eq:psi1) and (eq:psi2) it can be done.

$$\array{
\psi(a)\psi(e)k & =& \psi(a)Ad_K(\chi(a,e))k \\
& = &\psi(a)\chi(a,e)k\chi(a,e)^{-1}
}$$
where we used \eqref{psi2}.

Thus $Ad_K(\chi(a,e)) = \psi(e)$ and therefore it does not
depend on $a$.

Then use (eq:psi1) with $b = c = e$ to get
$\psi(e)^{-1}(\chi(a,e)) = \chi(e,e), \forall a \in B$.

Thus $\chi(a,e)^{-1}(\chi(a,e))\chi(a,e) = \chi(e,e)$,
that is $\chi(a,e)$ does not depend on $a$.

Now we claim that the *unit* element is given by $(e, \chi(e,e)^{-1})$.
To verify that it is also a right unit we compute

$$\array{
(a,b)(e,\chi(e,e)^{-1}) & = & (a, \chi(a,e) \psi(e)^{-1}(b)\chi(e,e)^{-1}) \\
& = & (a, \chi(a,e)\chi(e,e)^{-1}b\chi(e,e)\chi(e,e)^{-1})
}$$

what is equal to $(a,b)$ by just proved statement that
$\chi(a,e)$ does not depend on $a$.

Now use (eq:psi1) with $a = b = e$ to get

\[\label{psiac}
\psi(c)^{-1}(\chi(e,e)) = \chi(e,c), \forall c \in B.
\]

Thus we can verify that $(e, \chi(e,e)^{-1})$ is a left unit too
by a calculation as follows. Namely

$$(e,\chi(e,e)^{-1})(a,b)= (a,\chi(e,a)\psi(a)^{-1}(\chi(e,e)^{-1})b)$$

by the definition of the product. Then by (eq:psiac), this equals to

$$(a,\psi(a)^{-1}(\chi(e,e))\psi(a)^{-1}(\chi(e,e)^{-1})b)$$

and, because $\psi(a)^{-1}$ is an antiautomorphism, this is finally
equal to $(a,b)$.

Now check that each element $(a,b)$ can be
factorized as $(a,e)(e,\chi(e,e)^{-1}b)$.
In order to show that $(a,b)$ has
an inverse it is then enough to
show that both $(a,e)$ and
$(e,\chi(e,e)^{-1}b)$ have inverses.

Claim: the inverse of $(a,e)$ is

$$(a^{-1},\chi(a,a)^{-1}\chi(e,e)^{-1}).$$

To this aim, we calculate

$$(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) =(e,\chi(e,e^{-1}),$$

because $\psi(a)^{-1}(e) = e$. Furthermore,

$$(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(e,e^{-1}),$$

because $\psi(a)^{-1}(e) = e$. Next,

$$(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})(a,e) =
(e,\chi(a^{-1},a)\psi(a)^{-1}(\chi(a,a^{-1}) \chi(e,e)^{-1}))$$

what equals $(e,\chi(e,e)^{-1})$.

Indeed, (eq:psi1) with $a = a, b = a^{-1}, c = a$
reads $\chi(e,a) \psi(a)^{-1}(\chi(a, a^{-1}))$ $=\chi(a,e)\chi(a^{-1},a)$.

Then apply (eq:psiac) and take inverse of both sides
to obtain

$$\psi(a)^{-1}(\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}))
= \chi(a^{-1},a)^{-1}\chi(a,e)^{-1}.$$

Then recall that $\chi(a,e)$ does not depend on $a$ and multiply by  $\chi(a^{-1},a)$ from the left.

Claim: the inverse of $(e,\chi(e,e)^{-1}k)$
is $(e,\chi(e,e)k^{-1})$.
Here the verification is symmetric ($k$ vs. $k^{-1}$) for the left and for the right inverse and immediate.
=--

Given groups $K$ and $B$ and any maps $\chi$ and $\psi$ satisfying (eq:psi1) and (eq:psi2), needed to define a cocycle cross product $B\times_\chi K$ of $K$ and $B$, one defines the map $i : K \rightarrow B \times_\chi K$ by $k \mapsto (e,\chi(e,e)^{-1}k)$.
Then $i$ is a monomorphism of groups, $i(K)$ is
a normal subroup of the cocycle cross product of $B$ and $K$,
and there is a canonical isomorphism $B \cong G/K$.
We define the set-theoretic maps $\sigma',\chi'$ and $\psi'$
as follows. $\sigma' : B \rightarrow B \times K$
is defined by $\sigma'(b) = (b, e)$ , for all $b \in B$.
Then $\chi' : B \times B \to i(K)$ is defined by
$\chi'(b_1,b_2) = \sigma'(b_1b_2)^{-1}\sigma'(b_1)\sigma'(b_2)$
and $\psi' : B \to Aut(i(K))$ is defined
    by $\psi'(b)i(k) = \sigma'(b)i(k)\sigma'(b)^{-1}$.
Using the natural identifications $i : K \cong i(K)$,
and $i_{Aut} : Aut(i(K)) \cong Aut(K)$, we have $\psi' = i_{Aut}\circ \psi$ and $\chi' = i \circ \chi$. Now

$$\array{
\chi'=i\circ\chi
&\Leftrightarrow&(b_1,e)(b_2,e)(e,\chi(e,e)^{-1}k)
=(b_1 b_2,e)(e,\chi(e,e)^{-1}\chi(b_1,b_2)k)\\
&\Leftrightarrow& (b_1b_2,\chi(b_1,b_2))(e,\chi(e,e)^{-1}k) = (b_1 b_2,\chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k)\\
&\Leftrightarrow&
\chi(b_1 b_2,e)\psi(e)^{-1}(\chi(b_1,b_2))\chi(e,e)^{-1}k = \chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k
}$$

for all $b_1,b_2 \in B$ for all $k \in K$ in all these lines.
The last line is true by (eq:psi1).

Similarly, $\psi' = i_{Aut} \circ \psi$ iff $(b,e)(e,\chi(e,e)^{-1}k) = (e,\chi(e,e)^{-1}\psi(b)k)(b,e)$ for all $b$ and $k$.

Here the LHS computes as $(b,k)$ using $\chi(b,e) = \chi(e,e)$, and the RHS is

$$(e,\psi(b)k)(b,e) = (b, \chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}\psi(b)(k)))
= (b, k)$$

by (eq:psiac).

+-- {: .num_prop }
###### Proposition

The following are equivalent

* (i) extension \eqref{shortExtension} is split

* (ii) for extension \eqref{shortExtension}
there is a subgroup $B_1 \subset G$ such that
$B_1 \cap i(K) = 1$ and $B_1i(K) = G$
($G$ is an internal semidirect product of $K$ and $B_1$).

* (iii) extension \eqref{shortExtension} is
isomorphic to an external semidirect product
of $K$ and $B$.

=--

+-- {: .proof}
###### Proof

(i) $\Rightarrow$ (ii) If the extension is split then
there is a *homomorphism* $\sigma : B \rightarrow G$
such that $p \circ \sigma = id_B$. Let $B_1 = \sigma(B)$.
By exactness of \eqref{shortExtension}), all
elements in $i(K)$ map $p$ sends to 1, and by
 $p \circ \sigma = id_B$ map $p|_{B_1}$ is injection,
therefore the only element in $i(K)$
which belongs to $B_1$ is 1.

$B_1i(K) = G$ is also obvious: e.g. for given $g \in G,$
$p(g) = p\sigma p(g)$, so that $p((\sigma p (g))^{-1}g) = 1$
what means $(\sigma p (g))^{-1}g \in {Ker}(p)$ so that
$g = (\sigma p (g))i(k)$ for some $k \in K$ by exactness.

(ii) $\Rightarrow$ (iii) Our previous elaborate discussion of
cocycle cross products makes it obvious:
choosing a section $\sigma$ which is a homomorphism
gives $\chi(a,b) = 1$, and we can construct
equivalent external semidirect product
as a cocycle cross product with trivial $\chi$.

(iii) $\Rightarrow$ (i) Equivalence of extensions preserves the
property of the corresponding short exact sequence to be split.
Every external semidirect product is as a set $K\times B$
and the product is given by formula \eqref{mrule} without a cocycle.
The map $\sigma : B \rightarrow G$,
$B \ni b \mapsto (1_K,b) \in K \times B$, splits the sequence.

=--

+-- {: .num_defn }
###### Definition

An extension \eqref{shortExtension} is Abelian
iff $K$ is Abelian. An Abelian extension \eqref{shortExtension}
is central iff it is isomorphic to a cocycle cross product
extension with all the automorphisms $\psi(b), b \in B$ trivial.
We say that the extension \eqref{shortExtension}
is Abelian iff $G$ is Abelian.

=--

Remarks. (i) Note that \eqref{psi2} implies that $\psi$
is a homomorphism if $K$ in the case of Abelian extensions (for
any choice of set-theoretic section $\sigma$.

(ii) If $G$ is Abelian then \eqref{shortExtension} is central,
but not every central extension is corresponding to an Abelian $G$.
Abelian extensions in terms of the above definition trivially
(strictly!) include both central extensions and extensions with $G$
central.
By abuse of language one sometimes says for $G$
to be an extension of $K$ what leads to strange expression that not every
Abelian extension (as extension -- in terms of the definition above) is Abelian (as a group).

#### Comparing different extensions;  2-coboundaries {#2Coboundaries}

Let us now investigate when two extensions
$G_1$ and $G_2$ of $B$ by $K$,
given by $\psi,\chi$ and $\psi',\chi'$ respectively, are equivalent, cf. diagram \eqref{equivExt}.

We know that $\epsilon|_K : i(k) \stackrel{\epsilon}\mapsto i'(k)$, for all $k \in K$.
The formula for $i$ in \luse{crossform} says that whenever we represent
an extension as a cocycle extension we have $i(k) = (e,\chi(e,e)^{-1}k).$
Thus $\epsilon(e,\chi(e,e)^{-1}k) = (e,\chi'(e,e)^{-1}k)$, for all $k \in K.$ Also recall (or recalculate) that every element $(a,k)$ in $G$ can be
factorized as $(a,e)(e,\chi(e,e)^{-1}k)$.
By the definition $\epsilon$ is a homomorphism of groups,
so $\epsilon(a,k) = \epsilon(a,e)\epsilon(e,\chi(e,e)^{-1}k)$.
Also the cosets are preserved, so $\epsilon(a,e) = (a,\lambda(a))$
where $\lambda : B \rightarrow K$ is some set-theoretic map.
Thus

$$\array{
\epsilon(a,k) & = & (a,\lambda(a))(e,\chi'(e,e)^{-1}k) \\
    & = & (a, \chi'(a,e)\psi'(e)^{-1}(\lambda(a))\chi'(e,e)^{-1}k) \\
    & = & (a, \lambda(a)k).
}$$

Now multiply more general elements in $G$:

$$
\array{
\epsilon((b_1,k_1)(b_2,k_2)) = (b_1,\lambda(b_1)k_1)(b_2,\lambda(b_2)k_2) \\
= (b_1b_2,\chi'(b_1,b_2)\psi'(b_2)^{-1}(\lambda(b_1)k_1)\lambda(b_2)k_2)
}$$

what should be the same as

\[
\epsilon((b_1b_2,\chi(b_1,b_2)\psi(b_2)^{-1}(k_1)k_2))
 = (b_1b_2,\lambda(b_1b_2)\chi(b_1b_2)\psi(b_2)^{-1}(k_1)k_2)
\]

In a special case, when $k_1 = e_K$ we have therefore

\begin{equation}
\chi(b_1,b_2) = \lambda(b_1b_2)^{-1}\chi'(b_1,b_2)
\psi'(b_2)^{-1}(\lambda(b_1))\lambda(b_2) \label{equiv1}
\end{equation}

In order to obtain a relation between $\psi'(b)(k)$
and $\psi(b)(k)$ note that
\[
\epsilon ((e,\chi(e,e)^{-1}k)(b,e)) = (e, \chi'(e,e)^{-1}k)(b,\lambda(b)).
\]
That is equivalent to any in the following chain of formulas:
$$\array{
\epsilon (b,\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k)) &=&
    (b, \psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b)) \\
\Leftrightarrow \lambda(b)\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k))
&=& \chi'(e,b)\psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b)
}$$

Then by \eqref{psiac}, it follows that
$$\array{
     \lambda(b)\chi(e,b)\chi(e,b)^{-1}\psi(b)^{-1}(k)
&=& \chi'(e,b)\chi'(e,b)^{-1}\psi'(b)^{-1}(k)\lambda(b)   \\
\Leftrightarrow \lambda(b)\psi(b)^{-1}(k))
&=& (\psi'(b)^{-1}(k))\lambda(b) \\
\Leftrightarrow (Ad_K(\lambda(b)) \circ\psi(b)^{-1})(k)
&=& \psi'(b)^{-1}(k)
}$$

Now invert the maps in $Aut(K)$ to obtain

\[
\psi'(b) = \psi(b)Ad_K(\lambda(b)^{-1}) \label{equiv2}
\]

Thus we obtain

+-- {: .num_theorem }
###### Theorem

Two extensions of a group $B$ by group $K$ with
corresponding systems $(\psi,\chi)$ and $(\psi',\chi')$ are
equivalent iff there is a [[homomorphism]]
$\lambda: B \rightarrow K$ such that
the relations \eqref{equiv1}
and \eqref{equiv2} are valid.

=--

+-- {: .proof}
###### Proof

If function $\lambda$ takes values in the center of $B$
then \eqref{equiv2} implies that $\psi' = \psi : B \rightarrow Aut(K)$
and conversely.

If instead of functions $\psi$ and $\psi'$ we consider the
respective maps into the group of external automorphisms
(cosets of automorphisms with respect
to the group of internal homomorphisms)
$[\psi], [\psi']:~B \rightarrow Aut(K)/Int(K)$,
then the equivalent extensions define the same maps.
By \eqref{psi2} these maps are actually homomorphisms (unlike e.g.$\psi$).
For a given $\psi$ if there is $\chi$ so that $(\psi,\chi)$ does define
an extension of $B$ by $K$ we say that
the extension is *associated* to (the homomorphism) $[\psi]$.
That does not mean that any given homomorphism
in $hom_{Group}(B,Aut(K)/Int(K))$
is associated to any extension, nor it means that if
a homomorphism is associated to some extension,
that every its representative in $hom_{Set}(B,Aut(K))$
is a part of a pair $(\psi,\chi)$ defining an extension.
To see that situation in more detail we start with a *given*
automorphism, which we call $\theta$ ,
and *choose* an element $\psi(a)\in\theta(a)$, the representative
of a coset in $Aut(K)/Int(K)$;
that choice should be specified for all $a \in B$.
Note that for any $\rho \in Aut(K), a \in K$ we have, by direct inspection,
$\rho Ad_K(a)\rho^{-1} = Ad_K(\rho(a))$. Thus there is a well-defined function

$$  Ad_K \circ h : B \times B \rightarrow Int(K), \,\,\,
(Ad_K\circ h)(a,b) := \psi(a b)^{-1}\psi(a)\psi(b)$$

- indeed

\[\psi(a)Ad_K(r_1)\psi(b)Ad_K(r_2) = \psi(a)\psi(b)Ad_K(\psi(b)^{-1}(r_1)r_2)\]

so choosing $\psi(a b) \in [\psi(a b)]$ is the same as choosing it in $[\psi(a)][\psi(b)]$ and guarantees that $\psi(a b)^{-1}\psi(a)\psi(b)$ is in $Int(K)$. Let us choose some $h$ so that $Ad_K \circ h$ is interpretable as a genuine composition.

$$\array{
(\psi(a)\psi(b))\psi(c) & = & \psi(a b)Ad_K(h(a,b))\psi(c) \\
    & = & \psi(a b)\psi(c)\psi(c)^{-1}Ad_K(h(a,b))\psi(c) \\
    & = & \psi(a b c)Ad_K(h(a b,c))Ad_K(\psi(c)^{-1}h(a,b)) \\
    & = & \psi(a b c)Ad_K(h(a b,c)\psi(c)^{-1}h(a,b))
}$$

what is by associativity the same as

$$\array{
\psi(a)(\psi(b)\psi(c)) & = & \psi(a)\psi(b c)Ad_K(h(b,c)) \\
    & = & \psi(a b c)Ad_K(h(a,b c))Ad_K(h(b,c)) \\
    & = & \psi(a b c)Ad_K(h(a,b c)h(b,c)).
}$$

Thus $Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) = Ad_K(h(a,b c)h(b,c)).$
Two elements of $K$ generate the same automorphism iff they
differ by a central element. Thus

\begin{equation} \label{2semicoc}
h(a b,c)\psi(c)^{-1}h(a,b) = h(a,b c)h(b,c)z(a,b,c)
\end{equation}

for a unique central element $z(a,b,c) \in Z(K).$
 The correspondence $z : (a,b,c) \mapsto z(a,b,c)$ maps
$B \times B \times B$ into $Z(K)$.

=--

+-- {: .num_prop }
###### Proposition

$z$ is an (Abelian) 3-cocycle with values
 in $_/Z(K)_{\psi^{-1}}$
($Z(K)$ understood as trivial-$\psi^{-1}$ $B$-bimodule):

\begin{equation}
z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c) = z(a,b,c d)z(a b,c,d)
\label{3coc}
\end{equation}

=--

+-- {: .proof}
###### Proof

To see this we calcuate
$$\array{
h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)]
& = h(a b c,d)[\psi(d)^{-1}h(a,b c)h(b,c)z(a,b,c)] \\
& = h(a,b c d)h(b c,d)z(a,b c,d)[\psi(d)^{-1}h(b,c)z(a,b,c)] \\
& = h(a,b c d)h(b,c d)h(c,d)z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c)
}$$
Compare
$$\array{
h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)]
& = h(a b c,d)[\psi(d)^{-1}h(a,b c)]\psi(d)^{-1}\psi(c)^{-1}h(a,b) \\
& = h(a b c,d)[\psi(d)^{-1}h(a b,c)]h(c,d)^{-1}[\psi(c d)^{-1}h(a,b)]h(c,d) \\
& = h(a b c,d)h(a b,c d)z(a b,c,d)[\psi(c d)^{-1}h(a,b)]h(c,d) \\
& = h(a,b c d)h(b,c d)h(c,d)z(a,b,c d)z(a b,c,d)
}$$

=--

+-- {: .num_prop }
###### Proposition

(i) If we choose a different $h$
such that

$$Ad_K(h(a,b)) = \psi(a b)^{-1}\psi(a)\psi(b),$$

then $z$ will change only up to a 3-coboundary $d f,$
i.e. there is a function $f : B \times B \rightarrow Z(K)$,
such that $z' = (d f)z$ where

\[ (d f)(a,b,c) = f^{-1}(b,c)f^{-1}(a,b c)f(a b,c)\psi(c)^{-1}f(a,b),\,\,\,\,\,
for all a,b,c \in B.\]

(ii) Conversely, if $z$ is
a 3-cocycle obtained from $\psi$ as above and $d f$ is a 3-coboundary,
then there is a $h'$ determining
the same inner automoprhism of $K$
such that the corresponding 3-cocycle $z'$ is equal to $(df)z$.

(iii) Let $\psi, \psi' : B \rightarrow Aut(K)$ be
two set-theoretic sections so that $[\psi] = [\psi'] = \theta : B
\rightarrow Aut(K)/Int(K)$, then (for arbitrary choice of
$h$, $h'$) the cocycles $z$ and $z'$
obtained as above differ only up to a 3-coboundary. $\|$

=--

+-- {: .proof}
###### Proof

(i) Choose two different $h',h: B \times B \rightarrow K$
such that $Ad_K(h') = Ad_K(h)$. Then $h'(a,b) = h(a,b)f(a,b)$ where
$f : B \times B \rightarrow Z(K)$ is some function with values
in center of $K$.  A direct comparison of \eqref{2semicoc} written for
$h,z$ and $h',z'$ respectively proves the assertion.

(ii) Trivial: Any $f : B \times B \rightarrow Z(K)$
such that $h' = hf$ will not change the inner automorphism.
Thus any central 3-coboundary  $df$ can be obtained by
changing a choice of $h$.

(iii) $[\psi'] = [\psi]$ implies that exists $k : B \rightarrow K,
\psi'(a) = \psi(a)Ad_K(k(a)).$ Then

$$\array{
\psi'(a b)Ad_K(h'(a,b)) & = & \psi'(a)\psi'(b) = \psi(a)Ad_K(k(a))\psi(b)Ad_K(k(b))\\
& = & \psi(a)\psi(b)Ad_K([\psi(b)^{-1}k(a)]k(b)) \\
& = & \psi(a b)Ad_K(h(a,b)[\psi(b)^{-1}k(a)]k(b)) \\
& = & \psi'(a b)Ad_K(k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b)).
}$$

Thus $ h'(a,b) = k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b),$
for appropriate choice of $h'$ - what can change $z'$ up to coboundary -
using the freedom from (i). If we want formula involving $\psi'$
instead than we use $\psi'(a) = \psi(a)Ad_K(k(a))$ to obtain
$k(a b)h'(a,b) = h(a,b)k(b)[\psi'(b)^{-1}k(a)]$.
Using that and previous identities,

$$\array{
k(a b c)h'(a b,c)\psi'(c)^{-1}h'(a,b)
&=& h(a b,c)k(c)[\psi'(c)^{-1}k(a b)]\psi'(c)^{-1}h'(a,b) \\
&=& h(a b,c)k(c)\psi'(c)^{-1}k(a b)h'(a,b) \\
&=& h(a b,c)k(c)\psi'(c)^{-1}h(a,b)k(b)[\psi'(b)^{-1}k(a)] \\
&=& h(a b,c)[\psi(c)^{-1}h(a,b)]k(c)\psi'(c)^{-1}k(b)[\psi'(b)^{-1}k(a)] \\
&=& h(a,b c)h(b,c)z(a,b,c)k(c)
    [\psi'(c)^{-1}k(b)][\psi'(c)^{-1}\psi'(b)^{-1}k(a)] \\
&=& h(a,b c)h(b,c)k(c)[\psi'(c)^{-1}k(b)]
    [\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\
&=& h(a,b c)k(b c)h'(b,c)[\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\
&=& h(a,b c)k(b c)h'(b,c)h'(b,c)^{-1}[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\
&=& h(a,b c)k(b c)[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\
&=& k(a b c)h'(a,b c)h'(b,c)z(a,b,c)
}$$

for all $a,b,c \in B$. Thus $h'(a b,c)\psi'(c)^{-1}h'(a,b) = h'(a,b c)h'(b,c)z(a,b,c)$ i.e.
our choice of $h'$ insured no change in $z$. Of course that means
that in arbitrary choice of $h'$
we do not miss more than a coboundary by (i).

=--

+-- {: .num_cor }
###### Corollary

A given homomorphism
$\theta : B \times B \rightarrow Aut(K)/Int(K)$
is associated to some extension of $B$ by $K$ iff $z$ is
a 3-coboundary.

=--

+-- {: .proof}
###### Proof

Indeed, if $\theta$ is associated to an extension,
then we know that there is an isomorphism of the extension with
a cross product given by some cocycle $\chi$ and
some automorphism $\psi$ such that $[\psi] = \theta$. But using
the identification, $\chi = h$ for that particular choice
of $\psi$, so that $z = 1$. By the proposition, every
other $z$ obtained from $\theta$ is in the same cohomology class,
thus every such $z$ is a coboundary. Conversely, if $z$ is
a coboundary, then by the proposition, we can change it to $z = 1$,
and then we have all the conditions for a cross product
extension satisfied.

=--

#### Formulation in homotopy theory
 {#SchreierTheorynPOV}

One may regard the above from the [[nPOV]] as a special case of the way cocycles in the general notion of [[cohomology]] classify their [[homotopy fibers]]. More on this is at

* [[group cohomology]]

* [[nonabelian group cohomology]].

## Examples

By the above classification theorems, all the examples at _[[group cohomology]]_ equivalently induce examples for group extensions. And indeed by definition every [[short exact sequence]] defines an extension.

But examples of fundamental importance include for instance

* the [[real numbers]] as an extension of the [[circle group]]

  $$
    \mathbb{Z} \to \mathbb{R} \to U(1)
    \,.
  $$

* the [[spin group]] as an extension of the [[special orthogonal group]]

  $$
    \mathbb{Z}_2 \to Spin \to SO
  $$

* etc.

* [[universal central extension]]

## Related concepts

* [[group cohomology]]

  * [[Galois cohomology]]

  * [[nonabelian group cohomology]], [[groupoid cohomology]]

* **group extension**, [[∞-group extension]]

   [[Ext-group]], [[central extension]], [[maximal central extension]]

  * [[Baer sum]]

  * [[solvable group]], [[nilpotent group]]

  * [[ring extension]]

* [[Lie group cohomology]]

  * [[∞-Lie groupoid cohomology]]

* [[central extension of groupoids]]

* [[Lie algebra extension]]

* [[central charge]]

## References

### General

Original articles:

* {#EilenbergMacLane42} [[Samuel Eilenberg]], [[Saunders MacLane]], *Group Extensions and Homology*, Annals of Mathematics **43** 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), [jstor:1968966](https://www.jstor.org/stable/1968966)]

* [[Samuel Eilenberg]], [[Saunders MacLane]], *Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel*,  Ann. of Math. (2)  48,  (1947). 326--341 [jstor:1969174](http://www.jstor.org/pss/1969174)

* [[Saunders MacLane]], *Cohomology theory in abstract groups*. III. Operator homomorphisms of kernels.  Ann. of Math. (2)  50,  (1949). 736--761.

* [[Lawrence Breen]], Théorie de Schreier supérieure, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 465--514 [numdam](http://www.numdam.org/item?id=ASENS_1992_4_25_5_465_0).

Textbooks accounts

* A. G. Kurosh, Theory of groups

* [[Kenneth Brown]],  _Cohomology of Groups_, Graduate Texts in Mathematics, **87**, Springer 1982 ([doi:10.1007/978-1-4684-9327-6](https://link.springer.com/book/10.1007/978-1-4684-9327-6))

Lecture notes and similar include

* [[Brian Conrad]], _Group cohomology and group extensions_ ([pdf](http://math.stanford.edu/~conrad/249BPage/handouts/gpext.pdf))

* [[Terry Tao]], _Some notes on group extensions_ ([blog](http://terrytao.wordpress.com/2010/01/23/some-notes-on-group-extensions/))

* [[Patrick Morandi]], _Group extensions and $H^3$_ ([pdf](http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf))

  _Nobabelian cohomology_ ([pdf](http://sierra.nmsu.edu/morandi/notes/nonabeliancohomology.pdf))

* Raphael Ho, _Classifications of group extensions and $H^2$_ ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Ho.pdf))

See also:

* [[R. Brown]], [[T. Porter]], _On the Schreier theory of non-abelian extensions: generalisations and computations_, Proc. Roy. Irish Acad. Sect. A, 96 (1996), 213 – 227.

* [[Manuel Bullejos]], [[Antonio M. Cegarra]],  A 3-dimensional non-abelian cohomology of groups with applications to homotopy classification of continuous maps Canad. J. Math., vol. 43, (2), 1991, 1-32.

* [[Antonio M. Cegarra]], [[Antonio R. Garzón]], A long exact sequence in non-abelian cohomology, Proc. Int. Conf. Como 1990., Lec. Notes in Math. 1488, Springer 1991.

A theory for central 2-group extensions is here:

* [[Antonio R. Garzón]] and E.M. Vitale, On the second cohomology categorical group and a Hochschild-Serre 2-exact sequence, Theory and Applications of Categories, Vol. 30 (2015), 933-984.  ([pdf](http://www.tac.mta.ca/tac/volumes/30/27/30-27.pdf))

See also references to Dedecker listed [[zoranskoda:Paul Dedecker|here]].

### Applications

A bit of discussion of some occurences of central extensions of groups in [[physics]] is in

* G. Tuynman and W. Wiegerinck, _Central extensions of physics_ ([pdf](http://math.univ-lille1.fr/~gmt/PaperFolder/CentralExtensions.pdf))

(In fact there are many more than mentioned in that introduction.)

Extensions of [[supergroups]] are discussed in

* {#Drupieski14} [[Christopher Drupieski]], _Strict polynomial superfunctors and universal extension classes for algebraic supergroups_ ([arXiv:1408.5764](http://arxiv.org/abs/1408.5764))

[[!redirects Schreier's theory]]
[[!redirects Schreier theory]]

[[!redirects extension of groups]]
[[!redirects extensions of groups]]

[[!redirects central extension of groups]]
[[!redirects central extensions of groups]]

[[!redirects group extension]]
[[!redirects group extensions]]

[[!redirects abelian group extension]]
[[!redirects abelian group extensions]]


group field theory

The _group field theory_ is a particualr kind of a quantum field theory on a group manifold. It is sometimes used as a particular formulation of a [[spin foam]] approach to 3d quantum gravity.

* wikipedia [group field theory](http://en.wikipedia.org/wiki/Group_field_theory)
* [[Laurent Freidel]], _Group Field Theory: An overview_, Int.J.Theor.Phys.44:1769-1783,2005, [arxiv/hep-th/0505016](http://arxiv.org/abs/hep-th/0505016), [doi](http://dx.doi.org/10.1007/s10773-005-8894-1)
* Daniele Oriti, _The group field theory approach to quantum gravity_, [gr-qc/0607032](http://arxiv.org/abs/gr-qc/0607032)
* Winston Fairbairn, [[Etera R. Livine]], _3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory_, Class.Quant.Grav. __24__:5277-5297, 2007, [arxiv/gr-qc/0702125](http://arxiv.org/abs/gr-qc/0702125), [doi](http://dx.doi.org/10.1088/0264-9381/24/20/021)
* Adrian Tanasa, _Multi-orientable group field theory_, [arxiv/1109.0694](http://arxiv.org/abs/1109.0694)


group functor

A _group functor_ is a [[group object]] in a functor category.

## Terminology

The terminology of group-functors is mainly used in the theory of [[algebraic group|algebraic groups]] in algebraic geometry. If $G$ is a group functor the multiplication morphism $\pi:G\times G\to G$ is called _group law on_ $G$. An example of special interest in this context is that of a [[group scheme]].

## References

* Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)


group homology

#Contents#
* table of contents
{:toc}

## Idea

Group homology is the [[homology]] dual of [[group cohomology]].

See for instance at _[group cohomology -- In terms of homological algebra](group+cohomology#InTermsOfHomologicalAlgebra)_ and replace [[Ext]] by [[Tor]].

Accordingly, the group homology of a [[discrete group]] $G$ is equivalently the [[ordinary homology]] of its [[classifying space]] $B G$ (the [[Eilenberg-MacLane space]] $K(G,1)$):

$$
  H^{grp}_\bullet(G)
  \;\simeq\;
  H_\bullet\big( B G \big)
  \,.
$$

(eg. [Brown 1982, §4.1](#Brown82)).

## References

* {#Brown82} [[Kenneth Brown]], *The Homology of a Group*,  Chapter II of:  *Cohomology of Groups*, Graduate Texts in Mathematics, **87**, Springer (1982) [[doi:10.1007/978-1-4684-9327-6](https://link.springer.com/book/10.1007/978-1-4684-9327-6)]


group homotopy

##Tentative##

If [[group cohomology]] studies morphism between [[Eilenberg-MacLane object]]s /[[Eilenberg-MacLane space|spaces]] $[K(A, n), K(B, m)]$, usually in the case $n = 1$, one might imagine **group homotopy** would study $[M(A, n), M(B, m)]$, especially for $m = 1$, where the $M(A, n)$ are [[Moore space|Moore spaces]]. (Or should we take co-Moore spaces?)

One difference is that $K(G, 1)$ is defined for nonabelian $G$, whereas $M(G, 1)$ is only defined for abelian $G$.

+-- {: .query}

[[David Corfield]]: An asymmetry in the structure and co-structure of the circle? $S^1$ is both $K(\mathbb{Z}, 1)$ and $M(\mathbb{Z}, 1)$, and supports an abelian group structure but a nonabelian [[cogroup]] structure, so first cohomology must be abelian, but not necessarily first homotopy.

[[Urs Schreiber]]: might it be more an asymmetry in the definition: since [[Moore space]] involves [[homology]] which is (in the sense described there) the _abelian_ version of [[homotopy]].

[[David Corfield]]: some people, e.g., Hatcher, talk of homotopy with coefficients as probing spaces with co-Moore spaces, based on cohomology, so I'm not sure if your point is right. The ordinary homotopy groups come from maps out of co-Moore spaces for $G = \mathbb{Z}$, i.e., the spheres. These are all cogroups, but not abelian in the case of the 1-sphere. Cohomology groups come from mapping into E-M spaces. In the case of $K(\mathbb{Z}, 1)$ we again have the circle, but this time it supports an abelian group. I guess $K(\mathbb{Z}, 1)$ comes from forming loop spaces from above, higher $n$, so will be abelian, whereas $M(\mathbb{Z}, 1)$ comes from suspending from below, and only goes abelian at the second stage.

=--

According to Baues, the groups $[M(A, n), M(B, m)]$ are not at all understood. This is clear, since for $A = B = \mathbb{Z}$, these are the homotopy groups of the spheres.

>Need to say something about homology decomposition and homotopy decomposition (see Baues, Homotopy Types. Homology decomposition fails to be natural, but Baues introduces 'boundary invariants' which are natural.


group object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _group object_ in a [[cartesian category|cartesian]] [[category]] $C$ is a [[group]] [[internalization|internal]] to $C$ (see at *[[internalization]]* for more on the general idea).

Given a non-cartesian [[monoidal category]] one can still make sense of group objects in the [[formal duality|dual]] guise of [[Hopf monoids]], see there for more and see Rem. \ref{GroupObjectsInGeneralMonoidalCategories} below.

## Definition

### In terms of internal group objects

\begin{definition}
\label{GroupObjectInCartesianCategory}
**(group object in [[cartesian monoidal category]])**
\linebreak
A **group object** or **internal group** [[internalization|internal to]] a category $\mathcal{C}$ with [[finite products]] (binary [[Cartesian products]] and a [[terminal object]] $\ast$) is

* an [[object]] $G$ in $\mathcal{C}$

* and [[morphisms]] in $\mathcal{C}$ as follows:

  * [[neutral element]]:

    $\mathrm{e} \,\colon\,  \ast \to G$

  * [[inverse elements]]:

    $(-)^{-1} \,\colon\, G\to G$

  * [[binary operation]] $m \colon G\times G \to G$,

such that the following [[commuting diagram|diagrams commute]]:

$$
\array{
  G\times G\times G
  & \stackrel{id\times m}{\longrightarrow}
  &
  G\times G
  \\
  {}^{
    \mathllap{
    m\times id
    }
  }
  \big\downarrow && \big\downarrow m
  \\
  G\times G & \stackrel{m}{\longrightarrow} & G
}
$$
(expressing the fact multiplication is associative),
$$
\array{
  G
  & \stackrel{(\mathrm{e},id)}{\longrightarrow}
  & G\times G
  \\
  {}^{\mathllap{(\id,\mathrm{e})}}
  \big\downarrow
  &\underset{\id}{\searrow}&
  \big\downarrow m
  \\
  G\times G & \underset{m}{\longrightarrow} &G
}
$$
(telling us that the [[neutral element]] is a left and right [[unit element]]), and
$$
\array{
G
  &
   \overset{
     ((-)^{-1},id)
   }
   {\longrightarrow}
   &
   G\times G
   \\
   {}^{
     \mathllap{
       (id,(-)^{-1})
     }
   }
   \big\downarrow
   &
   \underset{id}{\searrow}
   &
   \big\downarrow m
   \\
   G\times G
   &
   \stackrel{m}{\longrightarrow}
   &
   G
}
$$
(telling us that the inverse map really does take an [[inverse element|inverse]]).
\end{definition}

\begin{remark}
\label{UseOfDiagonalMorphism}
The [[associativity law]] technically factors through the [[isomorphisms]] between $(G\times G)\times G$ and $G\times (G\times G)$.

The [[pairing]] $(f,g)$ denotes $(f\times g)\circ\Delta$ where $\Delta$ is a [[diagonal morphism]].
\end{remark}

\begin{remark}
Even if $C$ doesn\'t have *all* binary products, as long as products with $G$ (and the terminal object $*$) exist, then one can still speak of a group object $G$ in $\mathcal{C}$, as above.
\end{remark}

\begin{remark}
\label{GroupObjectsInGeneralMonoidalCategories}
**(group objects in general monoidal categories: [[Hopf monoids]])**
\linebreak

Notice that the use of [[diagonal maps]] (Rem. \ref{UseOfDiagonalMorphism}) in Def. \ref{GroupObjectInCartesianCategory} precludes direct generalization of this definition of group objects to non-[[cartesian monoidal category|cartesian]] [[monoidal categories]], where such maps in general do not exist.

Hence, while the [[underlying]] [[monoid object]] may generally be defined in any [[monoidal category]], the internal formulation of existence of [[inverse elements]] requires [[extra structure]], such as that of a compatible [[comonoid object]]-[[structure]] to substitute for the missing [[diagonal maps]].

Given this, inverses may be encoded by an *[[antipode]]* map and the resulting "monoidal group objects" are known as *[[Hopf monoids]]*. These subsume and generalize *[[Hopf algebras]]*, which are widely studied, for instance in their role as [[quantum groups]].
\end{remark}

### In terms of presheaves of groups
 {#InTermsOfPresheavesOfGroups}

+-- {: .num_prop}
###### Proposition

Given a [[cartesian monoidal category]] $\mathcal{C}$, the category of internal groups in $\mathcal{C}$ (in the sense of Def. \ref{GroupObjectInCartesianCategory}) is [[equivalence of categories|equivalent]] to the [[full subcategory]] of the category of [[presheaves]] of [[groups]] $Grp^{C^{op}}$ on $C$, spanned by those presheaves whose underlying set part in $Set^{C^{op}}$ is [[representable functor|representable]].

=--

This is a special case of the general theory of _[[structures in presheaf toposes]]_.

It means that the [[forgetful functor]] from the [[functor category]] $Func\big(\mathcal{C}^{op}, Grp\big)$ to the [[presheaf category]] $Func\big(\mathcal{C}^{op}, Set\big)$ (obtained by composing with the [[forgetful functor]] [[Grp]] $\to$ [[Set]]) creates representable group objects from  representable objects.

We unwind how this works:

An object $G$ in $\mathcal{C}$ equipped with internal group structure is identified equivalently with a [[diagram]] of [[functors]] of the form

\[
  \label{PresheafWithValuesInGroups}
  \array{
    && Grp
    \\
    & \mathllap{{}^{(G,\cdot)}}\nearrow & \big\downarrow
    \\
    \mathcal{C}^{op} &\underset{y(G)}{\longrightarrow}& Set
  }
  \,,
\]

where $\mathcal{C}^{op}$ is the [[opposite category]] of $C$, [[Grp]] is the category of [[groups]] with [[group homomorphisms]] between them, and [[Set]] is the category of [[sets]] with [[maps]]/[[functions]] between them. Finally,

$$
  \array{
    y \colon & C &\xhookrightarrow{\phantom{--}}& PSh(C)
    \\
    & G &\mapsto& Hom_C(-,G)
  }
$$

is the [[Yoneda embedding]] of $\mathcal{C}$ into its [[category of presheaves]] $PSh(C) \,\coloneqq\, Func(C^{op}, Set)$, which sends each [[object]] $G$ to the [[representable presheaf]] that it represents.

Since the [[Yoneda embedding]] is [[fully faithful functor|fully faithful]], it is natural to leave it notationally implicit and to write $G(S)$ (for $S \in \mathcal{C}$) as shorthand for

$$
  G(S) \coloneqq  y(G)(S) \coloneqq  Hom_C(S,G)
  \,.
$$

(This a also referred to as "$G$ seen at stage $S$", or similar.)

Now, the lift (eq:PresheafWithValuesInGroups) of such a presheaf of sets to a presheaf of groups equips for each object $S \in \mathcal{C}$ the set $G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) $ with an ordinary group [[structure]]
$\big(G(S), \cdot_S, \athrm{e}_S\big)$, in particular with a product operation (a map of sets) of the form

$$
  \cdot_S
    \,\colon\,
  G(S) \times G(S) \longrightarrow G(S)
  \,.
$$

Moreover, since morphisms in [[Grp]] are [[group homomorphisms]], it follows
that for every morphism $f \colon S \to T$ in $C$ we get a
[[commuting diagram]] of the form

$$
  \array{
    G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S)
    \\
    \big\uparrow\mathrlap{^{G(f)\times G(f)}}
    && \big\uparrow\mathrlap{^{G(f)}}
    \\
    G(T) \times G(T) &\underset{\cdot_T}{\longrightarrow}& G(T)
    \mathrlap{\,.}
  }
$$

Taken together this means that there is a morphism

$$
  y(G \times G)  \longrightarrow y(G)
$$

of [[representable presheaves]]. By the [[Yoneda lemma]], this *uniquely*
comes from a morphism $\cdot \colon G \times G \to G$ in $\mathcal{C}$, which is the
product of the group structure on the object $G$ that we are after.

etc.

### As data structure
 {#AsDataStructure}

In the language of [[dependent type theory]] (using the notation for [[dependent pair types]] *[[dependent functions and dependent pairs -- table|here]]*) the type of group data structures is:

<img src="/nlab/files/GroupDataType-230121.jpg" width="740">

## Examples
 {#Examples}

* A group object in [[Sets]] is a [[group]].

* A group object in [[TopologicalSpaces]] is a [[topological group]].

* A group object in [[SimplicialSets]] is a [[simplicial group]].

* A group object in [[Ho(Top)]] is an [[H-group]].

* A group object in [[Diff]] is a [[Lie group]].

* A group object in [[SDiff]] is a [[supergroup|super Lie group]].

* A group object in [[Grp]] is an [[abelian group]] (using the [[Eckmann-Hilton argument]]).

* A group object in [[Ab]] is an abelian group again.

* A group object in [[Cat]] is a strict [[2-group]].

* A group object in [[Grpd]] is a strict $2$-group again.

* A group object in [[CRing]]$^{op}$ is a commutative [[Hopf algebra]].

* A group object in a [[functor category]] is a [[group functor]].

* A group object in [[schemes]] is a [[group scheme]].

* A group object in an [[opposite category]] is a [[cogroup object]].

* A group object in [[G-sets]]/[[G-spaces]] is a $G$-[[equivariant group]], namely a [[semidirect product group]].

* A group object in [[stacks]] is a [[group stack]].

## Theory

The basic results of elementary group theory apply to group objects in any category with finite products.  (Arguably, it is precisely the *elementary* results that apply in any such category.)

The theory of group objects is an example of a [[Lawvere theory]].

## Related concepts

* [[monoid]], [[monoid object]], [[monoid object in an (∞,1)-category]]

* [[group]], **group object**, [[group object in an (∞,1)-category]]

* [[group action]], [[action object]]

* [[groupoid]], [[groupoid object]], [[groupoid object in an (∞,1)-category]]

* [[infinity-groupoid]], [[infinity-groupoid object]], [[groupoid object in an (∞,1)-category]]

* [[ring]], [[ring object]]

## References
 {#References}

The general definition of internal groups seems to have first been formulated in:

* {#Grothendieck61} [[Alexander Grothendieck]], p. 104 (7 of 21) of: [[FGA]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-III.pdf))

following the general principle of [[internalization]] formulated in:

* {#Grothendieck60} [[Alexander Grothendieck]], p. 340 (3 of 23) in: [[FGA]] *Technique de descente et théorèmes d'existence en géométrie algébriques. II: Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/SB_1958-1960__5__369_0), [pdf](http://www.numdam.org/item/SB_1958-1960__5__369_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-II.pdf))

reviewed in:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]], Section 2.2 of: _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) [[MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001), [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)]

On [[internalization]], [[H-spaces]], [[monoid objects]], [[group objects]] in [[algebraic topology]]/[[homotopy theory]] and introducing the [[Eckmann-Hilton argument]]:

* [[Beno Eckmann]], [[Peter Hilton]], *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221 ([doi:10.4064/fm-50-2-207-221](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III primitive categories_,  Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))

With emphasis of the role of the [[Yoneda lemma]]:

* [[Saunders MacLane]], chapter III, section 6 in: _[[Categories for the Working Mathematician]]_, Springer (1971)

Review:

* [[John Michael Boardman]], *Algebraic objects in categories*, Chapter 7 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford (1995) [[doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)]

* [[Magnus Forrester-Barker]], *Group Objects and Internal Categories* [[arXiv:math/0212065](https://arxiv.org/abs/math/0212065)]

In the broader context of internalization via [[sketches]]:

* [[Michael Barr]], [[Charles Wells]], Section 4.1 of: _[[Toposes, Triples, and Theories]]_, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in [Theory and Applications of Categories, No. 12 (2005) pp. 1-287](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)

With focus on internalization in [[sheaf toposes]]:

* [[Saunders Mac Lane]], [[Ieke Moerdijk]], Section II.7 of: _[[Sheaves in Geometry and Logic]]_, Springer 1992 ([doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0))

[[!redirects group objects]]

[[!redirects internal group]]
[[!redirects internal groups]]
[[!redirects inner group]]
[[!redirects inner groups]]

[[!redirects abelian group object]]
[[!redirects abelian group objects]]


group of ideles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _group of ideles_ $\mathbb{I}$ is the [[group of units]] in the [[ring of adeles]] $\mathbb{A}$:

$$
  \mathbb{I} = \mathbb{A}^\times \coloneqq GL_1(\mathbb{A})
  \,.
$$

In classical [[algebraic number theory]] one embeds a [[number field]] into the [[cartesian product]] of its [[completions]] at its [[archimedean absolute values]]. This embedding is very useful in the [[proofs]] of several fundamental [[theorems]]. However, it was noticed by [[Claude Chevalley]] and [[André Weil]] that the situation was improved somewhat if the number field is embedded in the cartesian product of its [[formal completions]] at all of its [[absolute values]]. With a few additional restrictions, these objects are known as the _[[adeles]]_, and the [[group of units|units]] of this ring are called the _ideles_.

When considering the adeles and ideles, it is their [[topology]] as much as their algebraic structure that is of interest. Many important results in [[number theory]] translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the [[ideal class group]] and the [[Dirichlet unit theorem]] are equivalent to a certain quotient of the ideles being compact and discrete.

([Weston, p. 1](#Weston))

## Definition

+-- {: .num_defn }
###### Definition

The [[group of units]] of the ring of adeles $\mathbb{A}_{\mathbb{Q}}$ is called the _group of ideles_

$$
  \mathbb{I}_{\mathbb{Q}} \coloneqq GL_1(\mathbb{A}_{\mathbb{Q}}) = \mathbb{A}_{\mathbb{Q}}^\times
  \,.
$$

It is a [[topological group]] via identification with the set $\{(x, x^{-1}) \in \mathbb{A}_\mathbb{Q}^2: \; x \in \mathbb{I}_\mathbb{Q}\}$, seen as a [[subspace]] of $\mathbb{A}_\mathbb{Q}^2$.
=--

+-- {: .num_remark}
###### Remark
The topology on $\mathbb{I}_\mathbb{Q}$ is strictly finer than the subspace topology inherited from $\mathbb{A}_\mathbb{Q}$. For example, the set $\mathbb{R}^\times \times \prod_p \mathbb{Z}_p^\times$ is a neighborhood of $1$ in $\mathbb{I}_\mathbb{Q}$, but not in the subspace topology. Cf. the discussion [here](https://math.stackexchange.com/a/145452/43208). Note: multiplicative inversion is not continuous in the subspace topology.
=--

The same definition holds for the [[ring of adeles]] of any other [[global field]] $K$, here one writes

$$
  \mathbb{I}_K \coloneqq GL_1(\mathbb{A}_K)
$$

or similar. The notation $J_K$ is also common.

+-- {: .num_defn #IdeleClassGroup}
###### Definition

The [[quotient]]

$$
  K^\times \backslash \mathbb{I}_K
  =
  GL_1(K)\backslash GL_1(\mathbb{A}_K)
$$

is called the _idele class group_ of $K$.

=--

+-- {: .num_remark }
###### Remark

The idele class group, def. \ref{IdeleClassGroup}, appears prominently in the description of the [moduli space of line bundles](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence) over the [[arithmetic curve]] on which $K$ is the [[rational functions]]. From there it appears in the abelian [[Langlands correspondence]] and in the abelian case of [[Tamagawa measures]].

=--

The idele class group is a key object in _[[class field theory]]_.

## Properties

### Product formula
 {#ProductFormula}

Recall the [[p-adic norm]] ${\vert -\vert}_p$ on $\mathbb{Q}$ for $p$ a [[prime number]], given by

$$
  {\left \vert \frac{a}{b}p^\ell \right \vert}_p
  \coloneqq
  p^{-\ell}
$$

for $a,b$ coprime to $p$. The usual [[absolute value]] [[norm]] one writes

$$
  {\vert -\vert}_\infty
$$

and associates with the "[[place at infinity|prime at infinity]]". When an index runs over the set of all primes ("finite primes") union with the "prime at infinity" one usually writes it "$v$" instead of $p$.

This induces:

+-- {: .num_defn #IdeleNorm}
###### Definition

The _idele norm_

$$
  {\vert -\vert} \colon \mathbb{I}_{\mathbb{Q}} \longrightarrow \mathbb{C}^\times
$$

is the function given by

$$
  {\vert \alpha\vert} \coloneqq \underset{v}{\prod} {\vert \alpha_v\vert}_v
  \,.
$$

=--

Notice that by construction there is a [[diagonal]] map $\mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}}$.

+-- {: .num_prop #ProductFormula}
###### Proposition
**(product formula)**

The idele norm, def. \ref{IdeleNorm}, is trivial on the diagonal of $\mathbb{Q}^\times$ inside the ideles, in that

$$
  (\alpha \in \mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}})
  \;\Rightarrow\;
  {\vert \alpha\vert}
  \coloneqq
  \underset{v}{\prod} {\vert \alpha_v\vert}_v
  = 1
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

The product formula, prop. \ref{ProductFormula}, says that the idele norm descends to the idele class group, def. \ref{IdeleClassGroup}.

=--

(e.g. [Garrett 11, section 1](#Garrett11))

### Strong approximation theorem for the idele class group
 {#StrongApproximationTheorem}

+-- {: .num_prop #StrongApproximationTheorem}
###### Proposition
**(strong approximation form ideles)**

The idele class group, def. \ref{IdeleClassGroup}, may be expressed as

$$
  \mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times
  \simeq
  (0,\infty)
   \times
  \underset{p}{\prod}
  \mathbb{Z}_p^\times
  \,.
$$

=--

(e.g. [Goldfeld-Hundley 11, prop. 1.4.5 and below (2.2.7)](#GoldfeldHundley11))

This implies that the [[ring of adeles]] may be decomposed into a rational and an idele class factor as:

$$
  \begin{aligned}
    \mathbb{A}_{\mathbb{Q}}^\times
     & \simeq
    \mathbb{Q}^\times
    \times
    (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times)
    \\
    & \coloneqq
    \underset{n \in \mathbb{Q}^\times}{\cup}
    n \cdot
    (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times)
 \end{aligned}
  \,.
$$

(e.g. [Goldfeld-Hundley 11, prop. 1.4.6 and below (2.2.7)](#GoldfeldHundley11))

This decomposition is crucial in the discussion of the [[Riemann zeta function]] (see there) as an [[adelic integral]].

### Automorphic forms and Relation to Dirichlet characters

The [[automorphic forms]] of the idele group are essentially
[[Dirichlet characters]] in disguise ([Goldfeld-Hundley 11, below def. 2.1.4](#GoldfeldHundley11))

### Function field analogy
 {#FunctionFieldAnalogy}

Via the [[function field analogy]] one may understand any [[number field]] or [[function field]] $F$ as being the [[rational functions]] on an [[arithmetic curve]] $\Sigma$. Under this identification the [[ring of adeles]] $\mathbb{A}_F$ of $F$ has the interpretation of being the [[ring of functions]] on all punctured [[formal disks]] around all points of $\Sigma$, such that only finitely many of them do not extend to the given point. ([Frenkel 05, section 3.2](#Frenkel05)).

This means for instance that the [[general linear group]] $GL_n(\mathbb{A}_F)$ with [[coefficients]] in the [[ring of adeles]] has the interpretation as being the [[Cech cohomology|Cech cocycles]] for [[algebraic vector bundles]] of [[rank]] $n$ on an [[algebraic curve]] with respect to any [[cover]] of that curve by the complement of a finite number of points together with the [[formal disks]] around these points. Here for $n = 1$ then $GL_1(\mathbb{A}_F)$ is the group of ideles.

This is part of a standard construction of the [[moduli stack of bundles]] on algebraic curves, see at _[Moduli space of bundles and the Langlands correspondence](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.

[[!include function field analogy -- table]]

## Related concepts

* [[differential cohesion and idelic structure]]

## References

Basics are recalled in

* _Adeles_ [pdf](http://wiki.epfl.ch/gant/documents/lecture2-cib2011.pdf)

* Pete Clark, _Adeles and Ideles_ ([pdf](http://math.uga.edu/~pete/8410Chapter6.pdf))

* Erwin Dassen , _Adeles & Ideles_ ([pdf](http://www.math.leidenuniv.nl/~astolk/monday/notes/dassen-adeles-ideles.pdf))

* {#Weston} Tom Weston, _The idelic approach to number theory_ ([pdf](http://www.math.umass.edu/~weston/oldpapers/idele.pdf))

* {#GoldfeldHundley11} {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

* {#Garrett11} [[Paul Garrett]], _Iwasawa-Tate on ζ-functions and L-functions_, 2011 ([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf)
[[!redirects Poisson formula]]

Discussion in the context of the [[geometric Langlands correspondence]] is in

* {#Frenkel05} [[Edward Frenkel]], section 3.2 of _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

[[!redirects idele]]
[[!redirects ideles]]

[[!redirects idele class group]]
[[!redirects idele class groups]]

[[!redirects idele group]]
[[!redirects idele groups]]

[[!redirects idèle group]]
[[!redirects group of idèles]]


group of Lie type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Groups of Lie type

## Definition

A __group of Lie type__ $G(k)$ is a (not necessarily [[finite group|finite]]) [[group]] of [[rational point]]s of a [[reductive group|reductive]] linear [[algebraic group]] $G$ with values in the [[field]] $k$.

+-- {: .un_remark}
###### Warning

Not to be confused with the notion of _[[Lie group]]_.
=--

[[!redirects group of Lie type]]
[[!redirects groups of Lie type]]


group of order 2

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

There is, up to [[isomorphism]], a unique simple [[group]] of [[order of a group|order]] 2:

it has two elements $(1,\sigma)$, where $\sigma \cdot \sigma = 1$.

This is usually denoted $\mathbb{Z}_2$ or $\mathbb{Z}/2\mathbb{Z}$, because it is the [[cokernel]] (the [[quotient]] by the [[image]] of) the  homomorphism

$$
  \cdot 2 : \mathbb{Z} \to \mathbb{Z}
$$

on the additive group of [[integers]]. As such $\mathbb{Z}_2$ is the special case of a [[cyclic group]] $\mathbb{Z}_p$ for $p = 2$ and hence also often denoted $C_2$.

## Properties

### ADE-Classification

In the [[ADE-classification]] of [[finite subgroups of SU(2)]], the group of order 2 is the smallest non-trivial group, and the smallest in the A-series:

[[!include ADE -- table]]

## Related concepts

* [[prime field]]

* [[Bockstein homomorphism]]

* [[Stiefel-Whitney class]]

* [[Steenrod square]]

[[!redirects group of order two]]

[[!redirects Z/2]]
[[!redirects Z_2]]

[[!redirects cyclic group of order two]]
[[!redirects cyclic group of order 2]]

[[!redirects cyclic group with 2 element]]

[[!redirects finite group of order 2]]
[[!redirects finite cyclic group of order 2]]

[[!redirects finite group of order two]]
[[!redirects finite cyclic group of order two]]


group of units

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### Group Theory
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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition
 {#Definition}

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###### Definition

For $R$ a [[ring]], its _group of units_, denoted $R^\times$ or $GL_1(R)$, is the [[group]] whose elements are the elements of $R$ that are invertible under the product, and whose group operation is the multiplication in $R$.

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###### Remark

$GL_1(R)$ is an [[affine variety]] (in fact an affine [[algebraic group]]) over $R$, namely $\{(x, y) \in R^2: x y = 1\}$.

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This leads us to the following alternative perspective:

+-- {: .num_defn}
###### Definition

In a [[category]] with [[finite limits]], with $R$ a [[ring object]] therein, the _group of units_ of $R$ is the [[equalizer]] of the two maps $m, c_1: R \times R \to R$, where $m$ is the ring multiplication and $c_1$ is the [[constant map]] with value the multiplicative identity.
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Cf. Example \ref{idele} below.

## Properties

### Relation to the multiplicative group

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###### Proposition

The group of units of $R$ is equivalently the collection of [[morphisms]]
from $Spec R$ into the [[group of units]] $\mathbb{G}_m$

$$
  GL_1(R) = R^\times \simeq Hom(Spec R, \mathbb{G}_m)
  \,.
$$

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### Relation to the group ring

+-- {: .num_remark}
###### Remark

There is an [[adjunction]]

$$
  (R[-]\dashv (-)^\times)
  \colon
  Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp
$$

between the [[category]] of [[associative algebras]] over $R$ and that of [[groups]], where $R[-]$ forms the [[group algebra]] over $R$ and where $(-)^\times$ assigns to an $R$-algebra its group of units.

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## Examples

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###### Example

The multiplicative group of the [[ring]] of [[integers]] modulo $n$ is the  _[[multiplicative group of integers modulo n]]_.

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+-- {: .num_example #idele}
###### Example

The group of units of the [[ring of adeles]] $\mathbb{A}$ is the [[group of ideles]]. The topology on the idele group $\mathbb{I}$ arises by considering $\mathbb{I}$ as an affine variety in $\mathbb{A}^2$ as above, and giving it the subspace topology. This is _not_ the subspace topology induced by the inclusion $\mathbb{I} \hookrightarrow\mathbb{A}$ into the ring of adeles.

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+-- {: .num_example}
###### Example
The group of units of the $p$-adic integers $\mathbb{Z}_p$ fits in an [[exact sequence]]

$$1 \to 1 + p \mathbb{Z}_p \hookrightarrow \mathbb{Z}_p^\times \to (\mathbb{Z}/(p))^\times \to 1$$

where the [[quotient]] is isomorphic to the [[cyclic group]] $\mathbb{Z}/(p-1)$ (see [[root of unity]]) and the [[kernel]] is, at least when $p \gt 2$, isomorphic to the *additive group* $\mathbb{Z}_p$. Explicitly, for such $p$ the formal [[exponential map]] $\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$ converges when $x \in p \mathbb{Z}_p$ and maps $p \mathbb{Z}_p$ isomorphically onto the multiplicative group $1 + p \mathbb{Z}_p$. The formal logarithm $\log(x) = \sum_{n \geq 1} \frac{(-1)^{n-1} (x - 1)^n}{n}$ is also [[convergence|convergent]] for $x \in 1 + p \mathbb{Z}_p$ and provides the [[inverse]].

By [[Henselian ring|Hensel's lemma]], the group of units $\mathbb{Z}_p^\times$ has $(p-1)^{th}$ [[roots of unity]] and therefore the exact sequence above [[split exact sequence|splits]]. This splitting descends to the quotient ring $\mathbb{Z}/(p^n)$ and its group of units, giving an isomorphism $GL_1(\mathbb{Z}/(p^n)) \cong \mathbb{Z}/(p^{n-1}) \oplus \mathbb{Z}/(p-1)$.
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## Related concepts

* **group of units**/[[multiplicative group]], [[Picard group]], [[Brauer group]]

* [[multiplicative submonoid of cancellative elements]]

* [[∞-group of units]]

* [[multiplicative group]]

* [[exponential sequence]], [[Kummer sequence]]

## References

For instance:

* [[Anthony Knapp]], p. 143 in: *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]

[[!redirects groups of units]]