nonabelian group cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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This entry largely discusses [[Schreier theory]] of nonabelian [[group extensions]] -- but from the [[nPOV]].

#Contents#
* table of contents
{:toc}

## Idea and Definition

As [[group cohomology]] of a group $G$ is the cohomology of its [[delooping]] $\mathbf{B}G$, so _nonabelian group cohomology_ is the corresponding [[nonabelian cohomology]].

By the [[category theory|general abstract]] definition of [[cohomology]], the _[[abelian group|abelian]]_ [[group cohomology]] in degree $n \in \mathbb{N}$ of a [[group]] $G$ with coefficients in an abelian group $K$ is the set of
[[equivalence classes]] of [[morphisms]]

$$
  H^n(G,K) = \{ \mathbf{B}G  \to \mathbf{B}^n K \}_\sim
$$

in the [[(∞,1)-category]] [[∞Grpd]], from the [[delooping]] $\mathbf{B}G$ of $G$ to the $n$-fold delooping $\mathbf{B}^n K$ of $K$.

However, if the group $K$ is not abelian, then its $n$-fold delooping does not exist for $n \geq 2$, so accordingly the above does not give a prescription for cohomology of $G$ with coefficients in a nonabelian group $K$ in degree greater than 1 (and in degree 1 group cohomology is not very interesting).

But for nonabelian $K$ there are higher groupoids that _approximate_ the non-existing higher deloopings. Nonabelian group cohomology is the [[cohomology]] of $\mathbf{B}G$ with coefficients in such approximations.

More precisely, notice that for $n=2$ and $K$ abelian, the $n$-fold [[delooping]] $\mathbf{B}^2 K$ is the strict [[2-groupoid]] whose corresponding [[crossed complex]] is

$$
  [\mathbf{B}^2 K]
  =
  \left(
    K \to {*} \stackrel{\to}{\to} {*}
  \right)
  \,.
$$

But for every [[group]] $K$ there is also its [[2-group|automorphism 2-group]] $AUT(K)$. Its delooping corresponds to the [[crossed complex]]

$$
  [\mathbf{B} AUT(K)]
  =
  \left(
    K \stackrel{\delta = Ad}{\to} Aut(K) \stackrel{\to}{\to}
    {*}
  \right)
  \,,
$$

where the boundary map $\delta$ is the one that sends an element $k \in K$ to the automorphism $Ad(k) : k' \mapsto k k' k^{-1}$.

So this looks much like $\mathbf{B}^2 K$ (when that exists) only that it has more elements in degree 1.

Accordingly, what is called nonabelian group cohomology of $G$ with coefficients in $K$ is the set of equivalence classes of morphisms

$$
  H^2_{nonab}(G,K) := \{ \mathbf{B}G \to \mathbf{B}AUT(K) \}_\sim
  \,.
$$

Notice that when $K$ has nontrivial [[automorphism]]s, this differs in general from the ordinary degree 2 abelian group cohomology even if $K$ is abelian.

It is a familiar fact that abelian group cohomology classifies (shifted) central [[group extension]]s. This is really nothing but the statement that to a morphism $\mathbf{B}G \to \mathbf{B}^n K$ we may associate its [[fibration sequence]]

$$
  \array{
    \mathbf{B}^{n-1} K& \to&\mathbf{B}\hat G &\to& {*}
    \\
    \downarrow &&\downarrow && \downarrow
    \\
    {*}& \to& \mathbf{B}G &\to& \mathbf{B}^n K
  }
$$

(where both squares are [[homotopy pullback]] squares). In particular, for $n = 2$ we get ordinary central extensions

$$
  \mathbf{B}K \to \mathbf{B}\hat G \to \mathbf{B}B
  \,.
$$

which may be looped to yield exact sequences of morphisms of groups

$$
  K \to \hat G \to B
  \,.
$$

In [[Schreier theory]] one notices that similarly nonabelian group cohomology in degree 2 classifies nonabelian [[group extension]]s, i.e. sequences

$$
  K \to \hat G \to G
  \,.
$$

As we shall discuss below, by following the [[category theory|abstract nonsense]] as described above, nonabelian degree 2 cocycles really classify something slightly richer, namely exact sequences of groupoids

$$
  Aut(K) \sslash K
    \to
  Aut(K) \sslash \hat G
   \to
  {*}\sslash G
  \,,
$$

where the double slashes denote [[homotopy quotients]] represented as [[action groupoids]] (and so ${*}\sslash G = \mathbf{B}G$ is the [[delooping groupoid]]).

In the existing literature -- which does not usually present the picture quite in the way we are doing here -- nonabelian group cohomology is rarely considered beyond degree 2. But the picture does straightforwardly generalize. For instance degree 3 nonabelian cohomology of $G$ with coefficients in $K$ may be taken to be the [[cohomology]] of $\mathbf{B}G$ with coefficients in the 3-groupoid $\mathbf{B}AUT(AUT(K))$.

$$
  H^3_{nonab}(G,K) = \{\mathbf{B}G \to \mathbf{B}AUT(AUT(K))\}_\sim
  \,.
$$

And so on.

## Details

We work out in detail what nonabelian group cocycles, such as morphisms

$$
  \mathbf{B}G \to \mathbf{B}AUT(K)
$$

correspond to in terms of classical group data, using the relation between [[strict 2-groups]] and [[crossed module]]s that is spelled out in detail at [strict 2-group -- in terms of crossed modules](strict+2-group#InTermsOfCrossedModules).

For making the translation we follow the **convention LB** there.

### Degree 2 cocycles

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

Degree 2 cocycles of nonabelian group cohomology on $G$ with coefficients in $K$ are given by the following data:

* a map $\psi : G \to Aut(K)$;

* a map $\chi : G \times G \to K$

* subject to the constraint that for all $g_1, g_2 \in G$ we have

  $$
    \psi(g_1 g_2)
    =
    Ad(\chi(g_1, g_2)) \psi(g_2) \psi(g_1)
    \,.
  $$

* and subject to the cocycle condition that for all $g_1, g_2, g_3 \in G$ we have

  $$
   \chi(g_1 g_2, g_3) \psi(g_3)(\xi(g_1,g_2))
   =
   \chi(g_1, g_2 g_3) \chi(g_2, g_3)
  $$

=--

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

Use the identification of $\mathbf{B}AUT(K)$ with its [[crossed module]] $(A \stackrel{Ad}{\to} Aut(K))$ in the _convention L B_ as described at [strict 2-group -- in terms of crossed modules](strict+2-group#InTermsOfCrossedModules) to translate the relevant diagrams -- which are of the sort spelled out in great detail at [[group cohomology]]: the first three items of the above describe the maps

$$
  (\psi, \chi) :
  \left(
    \array{
       && \bullet
       \\
       & {}^{\mathllap{g_1}}\nearrow &
       \Downarrow^{\mathrlap{=}}&
       \searrow^{\mathrlap{g_2}}
       \\
       \bullet &&\stackrel{g_1 g_2}{\to} && \bullet
    }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
    \array{
       && \bullet
       \\
       & {}^{\mathllap{\psi(g_1)}}\nearrow &
       \Downarrow^{\mathrlap{\chi(g_1,g_2)}}&
       \searrow^{\mathrlap{\psi(g_2)}}
       \\
       \bullet &&\stackrel{\psi(g_1 g_2)}{\to} && \bullet
    }
  \right)
  \,.
$$

The cocycle condition is the fact that this assignment has to make all tetrahedras commute (since there are only trivial [[k-morphism]]s with $k \geq 3$ in $\mathbf{B}AUT(K)$):

$$
  \array{
    \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet
    \\
    \uparrow
    &
      \Downarrow{}^{\mathrlap{\chi(g_1, g_2)}}
    &&& \downarrow
    \\
    {}^{\mathllap{\psi(g_1)}}\uparrow
    &&{}^{\mathllap{\psi(g_1 g_2)}}\nearrow&&
    \downarrow^{\mathrlap{\psi(g_3)}}
    \\
    \uparrow &&&
     {}^{\mathllap{\chi(g_1 g_2, g_2)}}\Downarrow
   & \downarrow
    \\
    \bullet &&\stackrel{\psi(g_3)}{\to}&&
    \bullet
  }
  \;\;\;\;\;\;\;\;
  =
  \;\;\;\;\;\;\;\;
  \array{
    \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet
    \\
    \uparrow
    &&&
      {}^{\mathllap{\chi(g_2, g_3)}}
    \Downarrow
& \downarrow
    \\
    {}^{\mathllap{\psi(g_1)}}\uparrow
    &&\searrow^{\mathrlap{\psi(g_2 g_3)}}&&
    \downarrow^{\mathrlap{\psi(g_3)}}
    \\
    \uparrow &
     \Downarrow{}^{\mathrlap{\chi(g_1 , g_2 g_3)}}
    &&& \downarrow
    \\
    \bullet &&\stackrel{\psi(g_3)}{\to}&&
    \bullet
  }
$$

=--

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

Precisely the same kind of "twisted" cocycles appear as the cocycles of nonabelian [[gerbe]]s and [[principal 2-bundle]]s: for a $K$-gerbe these are cocycles with coefficients in $\mathbf{B}AUT(K)$ but on a domain that is the [[discrete groupoid]] given by the given base space.

=--

### Extensions classified by degree 2-cocycles

The following statement is classically the central statement of [[Schreier theory]]. We state and prove it in the abstract nonsense context of general [[cohomology]], where the things classified by a cocycle are nothing but its [[homotopy fiber]]s.

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

Cohomology classes of nonabelian 2-cocycles $(\psi, \chi) :  \mathbf{B}G \to \mathbf{B}AUT(K)$ are in bijection  with equivalence classes of extensions

$$
  K \to \hat G \to G
$$

=--

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

In fact, we claim a bit more: we claim that the [[fibration sequence]] to the left defined by the cocycle $(\psi, \chi) : \mathbf{B}G \to \mathbf{B}AUT(K)$ is

$$
  \cdots
  \to
  Aut(K)
  \to
  Aut(K) \sslash K
  \to
  Aut(K) \sslash \hat G
  \to
  \mathbf{B}G
  \stackrel{(\psi,\xi)}{\to}
  \mathbf{B}AUT(K)
  \,,
$$

where

$$
  \hat G \coloneqq K \times_{(\psi,\chi)} G
$$

is the twisted product of $K$ with $G$, using the maps $\chi$ and $\psi$, i.e. the group whose underlying set is the cartesian [[product]] $K \times G$ with multiplication given by

$$
  (k_1, g_1) (k_2, g_2) =
  \left(
    \chi(g_1,g_2) \psi(g_2)(k_1) k_2
    \;\;  , \;\;
    g_1 g_2
  \right)
  \,.
$$

To see this, we compute the [[homotopy pullback]]

$$
  \array{
    Aut(K)\sslash\hat G & \to & {*}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}&
    \mathbf{B}AUT(K)
  }
$$

as the ordinary [[pullback]]

$$
  \array{
    Aut(K)\sslash\hat G & \to & \mathbf{E}AUT(K)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}&
    \mathbf{B}AUT(K)
  }
$$

as described at [[generalized universal bundle]]. ($\mathbf{E}AUT(K)$ is the universal $AUT(K)$-[[principal 2-bundle]]).

Recall from the discussion there that a morphism in $\mathbf{AUT}(K)$ is a triangle

$$
  \array{
     && \bullet
     \\
     & {}^{\mathllap{\alpha}}\swarrow
     &{}^\mathrlap{k}\swArrow&
     \searrow^{\mathrlap{\beta}}
     \\
     \bullet
     &&\stackrel{\gamma}{\to}&&
     \bullet
  }
$$

in $\mathbf{B}AUT(K)$, and composition of morphisms is pasting of these triangles along their vertical edges. 2-morphisms in $\mathbf{E}AUT(K)$ are given by paper-cup pasting diagrams of such triangles in $\mathbf{B}AUT(K)$

Accordingly, the [[pullback]] $\mathbf{B}G \times_{(\psi,\xi)} \mathbf{E}AUT(K)$ has

* objects are elements of $Aut(K)$ (this is the bit not seen in the classical picture of [[Schreier theory]], as that doesn't know about [[groupoid]]s);

* morphisms are pairs

  $$
    (k,g)
    \;\;\;
    :=
    \left(
    \array{
      && \bullet
      \\
      & {}^{\mathllap{\alpha}}\swarrow
      &{}^\mathrlap{k}\swArrow&
      \searrow^{\mathrlap{\beta}}
      &&&&& \in \mathbf{AUT(K)}
      \\
      \bullet
      &&\stackrel{\psi(g)}{\to}&&
      \bullet
      \\
      \\
      \bullet
      &&\stackrel{g}{\to}&&
      \bullet
      &&&& \in \mathbf{B}G
    }
    \right)
  $$

* 2-morphisms (thought of as 2-[[simplex]]es) take two
  such triangles $(k_1, g_1)$ and $(k_2, g_2)$ to the pair $(k', g_1, g_2)$, where $k'$ is given by the pasting diagram

  $$
    \array{
      && \bullet
      \\
      &\swarrow& \downarrow & \searrow
      \\
      \downarrow &\Downarrow^{\mathrlap{k_1}}&
      \bullet &{}^{\mathllap{k_2}}\Downarrow& \downarrow
      \\
      \downarrow
      & \nearrow &\Downarrow^{\mathrlap{\chi(g_1, g_2)}}
      & \searrow & \downarrow
      \\
      \bullet && \stackrel{}{\to}
      &&
      \bullet
    }
    \,.
  $$

Translating these diagrams into forumas using the _convention LB_ as described at [strict 2-group -- in terms of crossed modules](strict+2-group#InTermsOfCrossedModules) yields the given formulas.

=--

### Homotopies between 2-cocycles {#Homotopy2Cocycle}

Given two 2-cocycles

$$
  (\psi_1, \chi_1), (\psi_2, \chi_2)
  :
  \mathbf{B}G
  \to
  \mathbf{B}AUT(K)
$$

a homotopy (coboundary) between them is a [[transfor|transformation]]

$$
  \lambda : (\psi_1, \chi_1) \Rightarrow (\psi_2, \chi_2)
  \,.
$$

Its components

$$
  \lambda
  :
  (\bullet \stackrel{g}{\to} \bullet)
  \;\;
  \mapsto
  \;\;
  \left(
   \array{
     \bullet &\stackrel{\psi_1(g)}{\to}& \bullet
     \\
     {}^{\mathllap{\lambda(\bullet)}}
     \downarrow &{}^{\mathllap{\lambda(g)}}\swArrow&
     \downarrow^{\mathrlap{\lambda(\bullet)}}
     \\
     \bullet &\stackrel{\psi_2(g)}{\to}& \bullet
   }
  \right)
$$

are given in terms of group elements by

* $\lambda(\bullet) \in Aut(K)$

* $\{\lambda(g) \in K | g \in G\}$

such that

\[
  \lambda(\bullet) \psi_1(g)
  =
  Ad(\lambda(g)) \psi_2(g) \lambda(\bullet)
  \,.
\]

The naturality condition on this datat is that
for all $g_1, g_2 \in G$ we have

$$
  \array{
    && \bullet
    \\
    & {}^{\mathllap{\psi_1(g_1)}}\nearrow
    &\Downarrow^{\chi_1(g_1,g_2)}&
    \searrow^{\mathrlap{\psi_1(g_2)}}
    \\
    \bullet &&\stackrel{\psi_1(g_2 g_1)}{\to}&& \bullet
    \\
    {}^{\mathllap{\lambda(\bullet)}}\downarrow &&
    {}^{\mathllap{\lambda(g_1, g_2)}}\swArrow
    &&
    \downarrow^{\mathrlap{\lambda(\bullet)}}
    \\
    \bullet
    &&\stackrel{\psi_2(g_2 g_1)}{\to}&&
    \bullet
  }
  \;\;\;\;\;\;\;\;
  =
  \;\;\;\;\;\;\;\;
  \array{
    & {}^{\mathllap{\psi_1(g_1)}}\nearrow
    &\downarrow& \searrow^{\mathrlap{\psi_1(g_1)}}
    \\
    \downarrow
    &{}^{\lambda(g_2)}\swArrow
     &
     \downarrow^{\mathrlap{\lambda(\bullet)}}
    &{}^{\lambda(g_2)}\swArrow&
    \downarrow
    \\
    {}^{\mathllap{\lambda(\bullet)}}\downarrow
     &
     {}^{\mathllap{\psi_2(g_1)}}
     \nearrow &
     \Downarrow^{\chi_2(g_1,g_2)}
     & \searrow^{\mathrlap{\psi_2(g_2)}}
     &
     \downarrow^{\mathrlap{\lambda(\bullet)}}
    \\
    \bullet &&\stackrel{\psi_2(g2 g_1)}{\to}&&
    \bullet
  }
$$

In terms of the _conventionl LB_ at [strict 2-group -- in terms of crossed modules](strict+2-group#InTermsOfCrossedModules), this is equivalent to the equation

\[
  \label{homotopy}
  \lambda(g_2 g_1) \;
  \rho(\lambda(\bullet))(\chi_1(g_1,g_2))
  =
  \chi_2(g_1, g_2)
  \;
  \rho(\psi_1(g_2))(\lambda(g_2))
  \;
  \lambda(g_2)
  \,.
\]

Compare this to the discussion of [2-coboundaries of extensions](group+extension#2Coboundaries) at [[group extension]].

## Nonabelian Lie algebra cohomology

When the groups in question are [[Lie group]]s, there is an [[infinitesimal object|infinitesimal]] version of nonabelian group cohomology:

* [[nonabelian Lie algebra cohomology]]

See there for details.

## Related concepts

* [[non-abelian cohomology]]

* [[group cohomology]]

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

* [[group extension]]

* [[Lie group cohomology]]

  * [[∞-Lie groupoid cohomology]]

## References

Non-abelian group cohomology in degree 1, as the set of [crossed-conjugation](crossed+homomorphism#AdjointActionOnCrossedHomomorphisms)-[[equivalence classes]] of [[crossed homomorphisms]]:

* [[Philippe Gille]], [[Tamás Szamuely]], Def. 2.3.2 in: *Central Simple Algebras and Galois Cohomology*, Cambridge University Press  2006 ([doi:10.1017/CBO9780511607219](https://doi.org/10.1017/CBO9780511607219), [pdf](http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf))

* {#Milne17} [[James Milne]], Sec. 3.k (27.a of the [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf)) in: *Algebraic Groups*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

* [[Groupprops]], *[First cohomology set with coefficients in a non-abelian group](https://groupprops.subwiki.org/wiki/First_cohomology_set_with_coefficients_in_a_non-abelian_group)*

[[!redirects non-abelian group cohomology]]


nonabelian groupoid cohomology

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

Nonabelian [[groupoid cohomology]] is the [[nonabelian cohomology]] of [[groupoid]]s.

This generalizes [[nonabelian group cohomology]], which is the special case of nonabelian group cohomology for [[groupoid]]s that are [[delooping]]s $\mathbf{B}G$ of groups $G$.

## Examples

### Atiyah Lie groupoid

The [[Atiyah Lie groupoid]] $At(P) \to \Pi(X)$ of a [[principal bundle]] $P \to X$ is the extension of $\Pi(X)$ classified by the nonabelian groupoid cocycle $\Pi(X) \to \mathbf{B} AUT(G)$ coming from a [[connection on a bundle|connection]] on $P$.


nonabelian Hodge theory

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
=--
=--

> **under construction**

#Contents#
* table of contents
{:toc}

## Idea

Nonabelian Hodge theory generalizes aspects of [[Hodge theory]] from abelian cohomology ([[abelian sheaf cohomology]]) to [[nonabelian cohomology]].

## Nonabelian Hodge theorems

### Nonabelian harmonic sections

Notice or recall (for instance from [[generalized universal bundle]] and [[action groupoid]]) the following equivalent description of [[section]]s of [[associated bundle]]s:

for $G$ a [[group]] with [[action]] $\rho$ on an object $V$ witnessed by the [[action groupoid]] sequence

$$
  V \to V//G \to \mathbf{B}G
$$

the $\rho$-[[associated bundle]] $E \to X$ to a $G$-[[principal bundle]] $P \to X$ classified by an [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} Y  \to \mathbf{B}G$ is the [[pullback]]

$$
  \array{
    E &\to& V//G
    \\
    \downarrow && \downarrow
    \\
    Y &\to& \mathbf{B}G
  }
  \,.
$$

Since this is a [[pullback]] diagram by definition, a glance at a pasting diagram of the form

$$
  \array{
    && E &\to& V//G
    \\
    & \nearrow & \downarrow && \downarrow
    \\
    Y &\stackrel{=}{\to}& Y &\to& \mathbf{B}G
  }
$$

shows that [[section]]s

$$
  \array{
     && E
     \\
     & {}^{\sigma}\nearrow & \downarrow
     \\
     Y &\stackrel{=}{\to}& Y
  }
$$

are in bijection with maps $Y \to V//G$ that make

$$
  \array{
    Y &\to& V//G
    \\
    \downarrow^= && \downarrow
    \\
    Y &\to& \mathbf{B}G
  }
$$

commute.

In the special case that $X$ is a connected [[manifold]] and $G$ a discrete group we can without restriction take $Y = \hat X//\pi_1(X)$ be the [[action groupoid]] of the  [[universal cover]] by the [[fundamental group|homotopy group]], so that the classifying map $Y \to \mathbf{B}G$ is the same as a [[group]] homomorphism

$$
  \rho : \pi_1(X) \to G
  \,.
$$

In that case the above says that a section of the associated bundle is a $\rho$-equivariant map

$$
  \phi : \hat X \to V
  \,.
$$

This is the way these sections are formulated usually in the literature. The above description has the advantage that it works more generally in [[nonabelian cohomology]] for [[principal bundle]]s generalized to [[principal ∞-bundle]]s.

Next consider furthermore the special case that $V = G/K$ is the [[coset]] [[homogeneous space]] of $G$ quotiented by a subgroup $K$. Then if $G$ is a [[Lie group]] or [[algebraic group]] consider moreover a choice of $G$-invariant metric on the quotient $G/K$. Also consider a [[Riemannian manifold]] structure on $X$.

Then

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

The **energy** of a [[section]] $\sigma$ of an associated $G/K$-bundle as above is the real number

$$
  E(\phi) := \int_X |d \phi|^2
  \,.
$$

=--

Here

* $\phi$ is the $\rho$-equivariant map describing the section as above,

* the [[norm]] is taken with respect to the chocen invariant [[metric]] on $G/K$

* and the [[integral]] is taken with respect to the [[Riemannian metric]] on $X$.

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

Such a $\phi$ is called **harmonic** if it is a [[critical point]] of $E(-)$.

=--

+-- {: .num_theorem}
###### Theorem
**(Corlette, generalizing Eells-Sampson)**

If $\rho : \pi_1(X) \to G$  is a representation with

* $G$ a [[reductive group|reductive]] [[algebraic group]]

* $K$ is a [[maximal compact subgroup]]
* $\rho(\pi_1(X))$ is

  * Zariski-dense in $G$

  * or its Zariski-closure is itself reductive

then there exists a _harmonic_ section $\phi$ in the above sense.

=--

This is due to ([Corlett 88](#Corlette88)).
A version of the proof is reproduced in [Simpson 96, p. 8](#Simpson96)

### Kähler case: Equivalence between Local systems and Higgs bundles
 {#LocalSystemsAndHiggsBundles}

The _nonabelian Hodge theorem_ due to ([Simpson 92](#Simpson92)) establishes, for $X$ a [[compact topological space|compact]] [[Kähler manifold]], an [[equivalence]]
between (irreducible) [[flat vector bundles]] on $X$ and (stable) [[Higgs bundles]] with vanishing [[first Chern class]].

#### Relation to the abelian Hodge theorem

The sense in which the nonabelian Hodge theorem of ([Simpson 92](#Simpson92)) generalizes the abelian [[Hodge theorem]] is the following ([Simpson 92, Introduction](#Simpson92)).

The abelian [[cohomology group]] $H^1(X,\mathbb{C}_{disc})$ classifies [[flat vector bundle|flat]] [[complex line bundles]] whose underlying [[line bundle]] is trivial, hence closed [[differential 1-forms]] modulo 0-forms. The abelian [[Hodge theorem]] gives for this hence the decomposition

$$
  H^1(X,\mathbb{C}_{disc})
  \simeq
  H^1(X, \mathcal{O}_X) \oplus H^0(X, \Omega^1_X)
  \,.
$$

It is this kind of relation which is generalized by the nonabelian Hodge theorem. Here one starts instead with the [[nonabelian cohomology]] set $H^1(X, GL_n(\mathbb{C})_{disc})$ which classifies [[flat vector bundle|flat]] [[rank]]-$n$ [[vector bundles]] on $X$, for $n \in \mathbb{N}$. The equivalence to [[Higgs bundles]] gives now a decomposition of these structures into a [[holomorphic vector bundle]] classified by $H^1(X, GL_n(\mathcal{O}_X))$ and a differential 1-form with values in endomorphisms of that, subject to some conditions.

#### Statement

A quick review of the theorem in ([Simpson 92](#Simpson92)) is for instance in ([Raboso 15, section 1.2](#Raboso15)). An elegant abstract reformulation in terms of [[differential cohesion]]/[[D-geometry]], following ([Simpson 96](#Simpson96)) is in ([Raboso 15, section 3.3](#Raboso15)):

Analogous to how the [[de Rham stack]] $\int_{inf} X = X_{dR}$ of $X$ is the ([[homotopy quotient|homotopy]]) [[quotient]] of $X$ by the first order [[infinitesimal neighbourhood]] of the [[diagonal]] in $X \times X$, so there is a space ([[stack]]) $X_{Dol}$ which is the formal completion of the 0-section of the [[tangent bundle]] of $X$ ([Simpson 96](#Simpson96)).

Now a [[flat vector bundle]] on $X$ is essentially just a [[vector bundle]] on the [[de Rham stack]] $X_{dR}$, and a [[Higgs bundle]] is essentially just a [[vector bundle]] on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear [[algebraic group]] over $\mathbb{C}$)

$$
  \mathbf{H}(X_{dR}, \mathbf{B}G)
  \simeq
  \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0}
  \,,
$$

where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing [[first Chern class]] (see [Raboso 15, Theorem 3.3](#Raboso15)).

#### Generalizations to twisted bundles

A generalization of the nonabelian Hodge theorem of ([Simpson 92](#Simpson92)) to [[twisted bundles]] in discussed in ([Raboso 15](#Raboso15)).

## Relation to geometric Langlands

Nonabelian Hodge theory is closely related to the [[geometric Langlands correspondence]].

## Related concepts

* [[Hodge structure]]

* [[noncommutative Hodge theory]]

## References

* wikipedia [nonabelian Hodge correspondence](https://en.wikipedia.org/wiki/Nonabelian_Hodge_correspondence)

Lecture notes on nonabelian Hodge theory include:

* [[Ron Donagi]], [[Tony Pantev]], _Lectures on the geometric Langlands
conjecture and non-abelian Hodge theory_, 2009 ([pdf](http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf))

* [[Alberto García Raboso]], [[Steven Rayan]], _Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure_,  Fields Inst. Monogr. 34 (2015), 131--171 ([arXiv](https://arxiv.org/abs/1406.1693)) ([Springer](http://link.springer.com/chapter/10.1007/978-1-4939-2830-9_5))

Corlette's nonabelian Hodge theorem can be found in:

* {#Corlette88} K. Corlette, _Flat $G$-bundles with canonical metric_, J. Diff Geometry 28 (1988)

Works by [[Carlos Simpson]] on nonabelian Hodge theory include:

* {#Simpson92} [[Carlos Simpson]], _Higgs bundles and local systems_, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 75, 5{95. MR 1179076 (94d:32027) ([numdam](http://www.numdam.org/item?id=PMIHES_1992__75__5_0))

* {#Simpson96} [[Carlos Simpson]], _The Hodge filtration on nonabelian cohomology_, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR
1492538 (99g:14028) ([arXiv:9604005](http://arxiv.org/abs/alg-geom/9604005))

* [[Carlos Simpson]], _Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology_ ([arXiv:9712020](http://arxiv.org/abs/alg-geom/9712020))

* [[Carlos Simpson]], _Algebraic aspects of higher nonabelian Hodge theory_ ([arXiv:9902067](http://arxiv.org/abs/math/9902067))

* [[Carlos Simpson]], [[Tony Pantev]], [[Ludmil Katzarkov]], _Nonabelian mixed Hodge structures_ ([arXiv](http://arxiv.org/abs/math/0006213))

The nonabelian Hodge theorem of ([Simpson 92](#Simpson92)) is generalized to [[twisted bundles]] in:

* {#Raboso15} [[Alberto García Raboso]], _A twisted nonabelian Hodge correspondence_, PhD thesis 2014 ([arXiv:1501.05872](http://arxiv.org/abs/1501.05872), [pdf slides](http://www.math.toronto.edu/agraboso/files/TwistedNAHT_Talk_Handout.pdf))

Gothen's paper starts with a survey on nonabelian Hodge correspondence,

* Peter B. Gothen, _Higgs bundles and the real symplectic group_, [arXiv:1102.4175](https://arxiv.org/abs/1102.4175)

* C. C. Liu, S. Rayan, Y. Tanaka, _The Kapustin--Witten equations and nonabelian Hodge theory_ Eur. J. Math. __8__ (Suppl 1) 23--41 (2022) [doi](https://doi.org/10.1007/s40879-022-00538-4)

[[!redirects nonabelian Hodge theorem]]
[[!redirects nonabelian Hodge theorems]]
[[!redirects nonabelian Hodge correspondence]]


nonabelian homological algebra

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

__Nonabelian homological algebra__ studies the generalizations of elements of [[homological algebra]] to ambient [[categories]] which are not [[abelian categories]].

For example, the [[5-lemma]], [[9-lemma]], the [[snake lemma]] are some of the "elements" which can be generalized to a wide class of categories. Typical classes of categories suitable for some elements of homological algebra are [[protomodular categories]], homological categories, and [[semi-abelian categories]].

## Terminology and relations to other fields

Though they may be shown to be related the subject of [[nonabelian cohomology]] is a bit different from nonabelian homological algebra.

The nonabelian groups were historical motivation for much of the subject. While abelian groups are the template of an object in an abelian category, large part of work was directed toward understanding of what is the right generalization of concept of a group, what lead to a definition of a [[semi-abelian category]].

## Contributors

Main contributors are [[Dominique Bourn]], [[George Janelidze]], [[Francis Borceux]]. Their direction of work is largely influenced by motivations from universal algebra.

A recent independent development is the work of [[Alexander Rosenberg]] listed below in references. It features also the nonadditive triangulated categories and a new version of algebraic K-theory as byproducts. In unfinished part of the work further generalization of the homological algebra in the fibered categories is considered.

## References

* [[Francis Borceux]], [[Dominique Bourn]], *[[Mal'cev, protomodular, homological and semi-abelian categories]]*, Mathematics and Its Applications __566__,  Kluwer 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

* [[George Janelidze|G. Z. Janelidze]], _On [[satellites]] in arbitrary categories_, Bull. Georgian Acad. Sci. __82__ (1976), no. 3, 529-532, in Russian, with a reprint translated in English at [arXiv:0809.1504](http://front.math.ucdavis.edu/0809.1504).

* [[Alexander Rosenberg]], _[Homological algebra of noncommutative 'spaces' I](http://www.mpim-bonn.mpg.de/preprints/send?bid=3623)_ , 199 pages, preprint Max Planck, Bonn: MPIM2008-91.


nonabelian Lie algebra cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents
{:toc}

## Idea

Abstractly, nonabelian Lie algebra cohomology is the restriction of the general notion of [[∞-Lie algebra cohomology]] to [[cocycle]]s of the form $\mathfrak{g} \to der \mathfrak{h}$, where $\mathfrak{g}$ and $\mathfrak{h}$ are ordinary [[Lie algebra]]s and $der(-)$ denotes the Lie algebra of [[derivation]]s.

Traditionally abelian [[Lie algebra cohomology]] is conceived as the cohomology of the [[Chevalley-Eilenberg complex]] of a Lie algebra and some nonabelian generalizations of this model have been given in the literature. We show below how these definitions are the [[nonabelian cohomology]] special cases of the general abstract definition of [[∞-Lie algebra cohomology]].

The coefficients are not now in a Lie algebra module (which is viewed here as an abelian Lie algebra with action of another Lie algebra), but an arbitrary Lie algebra with something that is action of another Lie algebra up to an inner automorphism.

For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of [[group extension|factor systems]].

Below, in the section [Abstract definition](#AbstractDefinition) we discuss how a nonabelian Lie algebra cocylce is a morphism

$$
  (\psi,\chi) : \mathfrak{g} \to Der(\mathfrak{k})
$$

of [[L-∞-algebra]]s to the [[strict Lie 2-algebra]] of derivations of $\mathfrak{k}$.

A generalization (indeed a [[horizontal categorification]]) is [[nonabelian Lie algebroid cohomology]].

## Nonabelian 2-cocycles

Let $F$ be a field. **Lie algebra factor system** (or a **nonabelian 2-cocycle**) on a $F$-Lie algebra $\mathfrak{b}$ with coefficients in $F$-Lie algebra $\mathfrak{k}$ is a pair $(\chi,\psi)$ where $\chi: \mathfrak{b}\wedge \mathfrak{b}\to\mathfrak{k}$ and $\psi:\mathfrak{b}\to Der(\mathfrak{k})$ are $F$-linear maps satisfying

$$
  \begin{aligned}
& \chi([b_1,b_2]\wedge b_3)-\chi(b_1\wedge [b_2,b_3])+\chi(b_2\wedge[b_1,b_3])
\\& = \psi(b_3)(\chi(b_1\wedge b_2))-\psi(b_1)(\chi(b_2\wedge b_3))+\psi(b_2)(\chi(b_1\wedge b_3))
\end{aligned}
$$

for all $b_1,b_2,b_3\in B$ and

$$
[\psi(a),\psi(b)]=\psi([a,b])+ad_{\mathfrak{k}}(\chi(a\wedge b))
$$

where $a,b\in B$ and $ad_{\mathfrak{k}}:\mathfrak{k}\to Int(\mathfrak{k})$ is the canonical map into inner automorphisms $k\mapsto [k,]$.

## Schreier's theory for Lie algebras

[[Otto Schreier]] (1926) and Eilenberg-Mac Lane (late 1940-s) developed a theory of nonabelian [[group extension|extensions of abstract groups]] leading to the low dimensional nonabelian group cohomology. For Lie algebras, the theory can be developed in the same manner. One tries to classify [[Lie algebra extension|extensions of Lie algebras]]

$$ 0\to \mathfrak{k}
\overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0$$

**Theorem.** To every Lie algebra extension as above, and a choice of $F$-linear section $\sigma:\mathfrak{b}\to\mathfrak{g}$ of $p$, one can assign a nonabelian 2-cocycle (factor system) on $\mathfrak{b}$ with values in $\mathfrak{k}$ as follows: set

$$\chi(b_1\wedge b_2):=-\sigma([b_1,b_2])+[\sigma(b_1),\sigma(b_2)]$$

and define $\phi:\mathfrak{g}\to Der(\mathfrak{k})$ by $\phi(g)(k):=[g,k]$. Then set $\psi:=\phi\circ\sigma$. Then $(\chi,\psi)$ is a nonabelian 2-cocycle on $\mathfrak{b}$ with values in $\mathfrak{k}$.

**Theorem. (cocycle crossed product of Lie algebras)** Let $(\chi,\psi)$ be a factor system as above.
Then define a $F$-linear bracket on the $F$-vector space $\mathfrak{b}\oplus\mathfrak{k}$ by

$$
[(b_1,k_1),(b_2,k_2)] = ([b_1,b_2],\chi(b_1\wedge b_2)+\psi(b_1)(k_2)-\psi(b_2)(k_1)+[k_1,k_2])
$$

Then

(i) $[,]$ is a antisymmetric and satisfies the Jacobi identity, i.e. $\mathfrak{g}:=(\mathfrak{b}\oplus\mathfrak{k},[,])$ is an $F$-Lie algebra.

(ii) $k\mapsto (0,k)$ defines an embedding $i:\mathfrak{k}\to\mathfrak{g}$ of Lie algebras and $(b,k)\mapsto b$ is a surjective homomorphism of Lie algebra $p:\mathfrak{g}\to\mathfrak{b}$ whose kernel is the Lie ideal $i(\mathfrak{k})=0\oplus\mathfrak{k}\subset\mathfrak{g}$. This way $0\to\mathfrak{k}\overset{i}\to\mathfrak{g}\overset{p}\to\mathfrak{b}\to 0$ is an extension of the base Lie algebra $\mathfrak{b}$ by the kernel Lie algebra $\mathfrak{k}$.

(iii) If the 2-cocycle is obtained from a Lie algebra extension $0\to \mathfrak{k}\overset{i_0}\to \mathfrak{g}_0\overset{p_0}\to\mathfrak{b}\to 0$ and an arbitrary $F$-linear section $\sigma_0$ of $p_0$, then the map $can_\sigma:\mathfrak{g}_0\to\mathfrak{g}$ given by $g\mapsto (p(g),-\sigma(p(g))+g)$ is well-defined and a Lie algebra isomorphism such that $can_\sigma\circ i_0=i$, $p_0=p\circ can_\sigma$, hence the two extensions are isomorphic.

In addition to the problem of extensions, nonabelian 2-cocycles appear in a more general problem of liftings of Lie algebras.

## Abstract definition {#AbstractDefinition}

We claim that the above definition of nonabelian Lie algebra cocycles may be understood naturally in terms of the general notion of [[cohomology]] and in particular is the image of the story of [[nonabelian group cohomology]] under Lie differentiation:

+-- {: .standout}

The following observation is not in the literature.

=--

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

Let $\infty Lie$ be the [[(∞,1)-category]] of [[L-∞-algebra]]s.
Let $\mathfrak{g}, \mathfrak{k}$ be [[Lie algebra]]s. Then the
degree 2 nonabelian Lie algebra cohomology of $\mathfrak{g}$ with coefficients
in $\mathfrak{k}$ is

$$
  H^2_{nonab}(\mathfrak{g}, \mathfrak{k})
  \simeq
  \pi_0 \infty Lie(\mathfrak{g}, Der(\mathfrak{k}))
  \,,
$$

where $Der(\mathfrak{k})$ is the [[strict Lie 2-algebra]] of derivations
on $\mathfrak{k}$.

More in detail:

* nonabelian degree 2 Lie algebra cocycles $(\psi,\xi)$ are in natural bijections with
  morphisms

  $$
    \mathfrak{g} \to Der(\mathfrak{k})
  $$

* coboundaries $\eta$ between cocycles $(\psi_1,\xi_1)$ and $(\psi_2,\xi_2)$
  correspond to homotopies between these

  $$
    \array{
      & \nearrow\searrow^{\mathrlap{(\psi_1,\chi_1)}}
      \\
      \mathfrak{g}
      &\Downarrow^{\eta}&
      Der(\mathfrak{k})
      \\
      & \searrow\nearrow^{\mathrlap{(\psi_2,\chi_2)}}
    }
  $$

  and this correspondence is precise if we take the homotopy to be
  induced from the "standard [[cylinder object]]", described below.

=--

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

Checking this is a straightforward matter of unwinding the definitions
of morphisms of $L_\infty$-algebras.

Which is what we indicate.

We model $\infty Lie$ as usual a subcategory of [[dg-algebra]]s of
[[semifree dga]]s, by representing each $L_\infty$-algebra $\mathfrak{g}$
by its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$.

For the [[Lie algebra]] $\mathfrak{g}$ itself with Lie bracket
$[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ this is the
[[semifree dga]]

$$
  CE(\mathfrak{g}) =
  (\wedge^\bullet \mathfrak{g}^* , \; d = [-,-]^* )
  \,,
$$

where the differential is on generators the dual of the Lie bracket,
$[-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* $
extended as a graded derivation to all of $\wedge^\bullet \mathfrak{g}^*$.

For any [[strict Lie 2-algebra]] coming from a [[differential crossed module]]
$(\mathfrak{h}_1 \stackrel{\delta}{\to} \mathfrak{h}_1)$ with action
$\rho : \mathfrak{h}_1 \to der(\mathfrak{h}_2)$ -- that we think of
in the following as equivalently a linear map
$\rho : \mathfrak{h}_1 \otimes \mathfrak{h}_2 \to \mathfrak{h}_2$ --
the [[Chevalley-Eilenberg algebra]] is

$$
  CE(\mathfrak{h}_2 \stackrel{\delta}{\to} \mathfrak{h}_1)
  =
  \left(
    \wedge^\bullet ( \mathfrak{h}_1^* \oplus \mathfrak{h}_2^* )
    ,
    \;
    d_{\delta}
  \right)
$$

with $\mathfrak{h}_1^*$ in degree 1 and $\mathfrak{h}_2^*$ in degree 2, and with the
differential given on degree 1 generators by

$$
  d_\delta |_{\mathfrak{h}_1^*} = [-,-]_{\mathfrak{h}_1}^* + \delta^*
  : \mathfrak{h}_1^* \to
    \mathfrak{h}_1^* \wedge \mathfrak{h}_1^*
    \oplus
    \mathfrak{h}_2^*
$$

and on degree 2 generators by

$$
  d_\delta |_{\mathfrak{h}_2^*} = \rho^*
  :
  \mathfrak{h}_2^* \to \mathfrak{h}_1^* \otimes \mathfrak{h}_2^*
  \,.
$$

The case of the derivation [[strict Lie 2-algebra]] of a Lie algebra
$\mathfrak{k}$ is the special case of this for

$$
  Der(\mathfrak{k}) = (\mathfrak{k} \stackrel{ad}{\to} der(\mathfrak{k}))
  \,.
$$

Now a morphism

$$
  (\psi, \chi) : \mathfrak{g} \to Der(\mathfrak{k})
$$

of $\infty$-Lie algebras is given by a morphism

$$
  CE(\mathfrak{g}) \leftarrow CE(Der(\mathfrak{k})) :
  (\psi^*, \chi^*)
$$

of [[dg-algebra]]s.

Morphisms of [[dg-algebra]]s are given by morphisms of the underlying
graded algebras, subject to the respect for the differentials. Morphisms
of the underlying graded [[Grassmann algebra]]s are given by
grading preserving linear maps on the spaces of generators.

So the underlying maps

$$
  \wedge^\bullet \mathfrak{g}^*
   \leftarrow
  \wedge^\bullet (\mathfrak{h}_1^* \oplus \mathfrak{h}_2^*)
  :
  (\psi^* , \chi^*)
$$

come from linear maps

$$
  \mathfrak{g}^* \leftarrow \mathfrak{h}_1^* : \psi^*
$$

and

$$
  \mathfrak{g}^* \wedge \mathfrak{g}^* \leftarrow \mathfrak{h}_2^* : \chi^*
$$

i.e. form linear maps

$$
  \psi : \mathfrak{g} \to der(\mathfrak{k})
$$

and

$$
  \chi : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{k}
  \,.
$$

This is the underlying data of the nonabelian 2-cocycle. Now the
respect for the differentials on the Chevalley-Eilenberg algebras will
give the cocycle condition:

let $\omega \in \mathfrak{h}_2^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)$
be any degree 2 element, then respect for the differential implies that

$$
  \array{
    \omega(\chi([-,-],-))
    =
    \omega(\rho(\psi(-)(\chi(-,-))))
    &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega(\rho(-)(-)))
    \\
    \uparrow^{d_\mathfrak{g} = [-,-]^*} && \uparrow^{\mathrlap{d_{\delta}}}
    \\
    \omega(\xi(-,-))
    &\stackrel{(\psi^*, \chi^*)}{\leftarrow}&
    \omega
  }
  \,.
$$

Since this has to hold for all $\omega$, we get the first part of the cocycle condition:

$$
  \chi([-,-],-) = \rho(\psi(-)\chi(-,-))
$$

(both sides here regarded as elements of a graded Grassmann algebra as indicated above,
so with all antisymmetrization on the arguments implicit).

Similarly, for $\lambda \in \mathfrak{h}_1^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)$
be any degree 1 element, then respect for the differential implies that

$$
  \array{
     \lambda(\psi([-,-]))
     =
     \lambda([\psi(-), \psi(-)])
     +
     \lambda(ad(\chi(-,-)))
     &\stackrel{}{\leftarrow}&  \lambda([-,-]_{\mathfrak{h}_1}) + \lambda(ad(-))
     \\
     \uparrow && \uparrow
    \\
    \lambda(\psi(-))
    &\stackrel{(\psi^* , \chi^*)}{\leftarrow}&
    \lambda
  }
  \,.
$$

Again, this has to hold for all $\lambda$, so we have the auxiliary condition on the
cocycle

$$
  \psi([-,-])
  =
  [\psi(-),\psi(-)]
  +
  ad(\xi(-,-))
  \,.
$$

This shows that morphisms $\mathfrak{g} \to Der(\mathfrak{k})$ are
in bijection to the nonabelian cocycles.

It remains to show that the homotopies map to coboundaries. For that
we may take in $\infty Lie$ the standard [[cylinder object]] of some
$CE(\mathfrak{g})$ to be

$$
  CE(\mathfrak{g})\otimes CE(\mathfrak{g})
  \leftarrow
  C^\bullet(\Delta^1)\otimes CE(\mathfrak{g})
  \leftarrow
  CE(\mathfrak{g})
  \,,
$$

where $C^\bullet(\Delta^1)$ is the [[semifree dga]] of cochains on the
cellular 1-[[simplex]], i.e.

$$
  C^\bullet(\Delta^1) =
  (\wedge^\bullet (\langle a,b \rangle \oplus \langle c \rangle)
  , d a = - d b = d c
  )
  \,,
$$

with $a,b$ generators in degree 0 and $c$ in degree 1. Using this,
write out the data implied by a morphism $\eta$ that is a [[left homotopy]]

$$
  \array{
    CE(\mathfrak{g})
    \\
    \downarrow & \nwarrow^{\mathrlap{(\psi_1^*, \chi_1^*)}}
    \\
    C^\bullet(\Delta^1)\otimes CE(\mathfrak{g})
    &\stackrel{\eta}{\leftarrow}&
    CE(Der(\mathfrak{k}))
    \\
    \uparrow & \swarrow_{\mathrlap{(\psi_2^*, \chi_2^*)}}
    \\
    CE(\mathfrak{g})
  }
$$

along the above lines.

Notice that in $dgAlg^{op}$ every object is cofibrant, so that this is indeed a left homotopy. See [[∞-Lie algebra cohomology]] for more on this.

=--

## References

On original source is

* G. Hochschild, _Lie algebra kernels and cohomology_, Amer. J. Math. __76__, n.3 (1954) 698--716.

The notation above is from personal notes of Z. Škoda (1997). A systematic theory has been many times partly rediscovered from soon after the Eilenberg--Mac Lane work on group extension, among first by Hochschild and then by many others till nowdays. Here is a recent online account emphasising parallels with [[differential geometry]]:

*  [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, Extensions of Lie algebras ([math.DG/0005042](http://arxiv.org/abs/math/0005042))

A more conceptual picture is in a work of [[Danny Stevenson]] which extends also to its [[categorification]], extensions of [[Lie 2-algebra]]s. See

* [[Danny Stevenson]], Lie 2-algebras and the geometry of gerbes, Unni Namboodiri Lectures 2006 [slides](http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf)

There is also

* N. Inassaridze, E. Khmaladze, and M. Ladra, _Non-abelian Cohomology and Extensions of Lie Algebras_ Journal of Lie Theory,   Volume 18 (2008) 413–432 ([pdf](http://www.rmi.acnet.ge/~khmal/PAPERS/LieTheory2008.pdf))


nonabelian Lie algebroid cohomology

[[nonabelian cohomology|Nonabelian]] [[Lie algebroid]] [[cohomology]] is a [[horizontal categorification]] of [[nonabelian Lie algebra cohomology]].

* Olivier Brahic, Extensions of Lie Brackets, [arXiv:0810.1462](http://arXiv.org/abs/0810.1462)

treats extensions even over different bases.


nonabelian Poincaré duality

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Ran space]]

* [[factorization algebra]], [[topological chiral homology]]

* [[Weil conjecture on Tamagawa numbers]]

## References

### General

* [[Jacob Lurie]], _Nonabelian Poincaré duality_ February 2014 ([pdf](http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf))

### Application to Tanagawa numbers

The announced proof by [[Jacob Lurie]] and [[Dennis Gaitsgory]] via nonabelian Poincaré duality of the [[Weil conjecture on Tamagawa numbers]] was announced in

* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincaré Duality_, talk at  [FRG Chern-Simons workshop](http://people.mpim-bonn.mpg.de/teichner/Older/FRG-3.html), Jan. 15-17, 2011

and details are at

* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincare Duality (282y)_, lecture notes, 2014 ([web](http://www.math.harvard.edu/~lurie/282y.html))

### Application to stable splitting of mapping spaces

Generalization via nonabelian Poincaré duality of [[stable splitting of mapping spaces]] from [[codomains]] which are $n$-fold [[suspensions]] to general [[n-connective spaces]]:

* Lauren Bandklayder, _Stable splitting of mapping spaces via nonabelian Poincaré duality_ ([arxiv:1705.03090](https://arxiv.org/abs/1705.03090))

[[!redirects nonabelian Poincare duality]]


nonabelian sheaf cohomology

_Nonabelian sheaf cohomology_ is a notion in the context of [[nonabelian cohomology]]:

Given an [[higher category theory|infinity-category]]-valued (pseudo)presheaf $\mathbf{A} : Spaces^{op} \to \infty-Cat$, its _cohomology_ $H(-,\mathbf{A})$ is the $\infty$-category valued (pseudo)presheaf which assigns to each space the directed limit over all [[descent|descent data]]:
$$
  H(X, \mathbf{A}) := colim_{Y^\bullet \to X}
  Desc(Y^\bullet, \mathbf{A})
  \,.
$$
Here the colimit is over all [[hypercover|hypercovers]] of $X$.

#Remarks#

* [[duality|Dual]] to nonabelian sheaf cohomology is [[nonabelian cosheaf homotopy]].

* A presheaf whose value on each space is [[equivalence|equivalent]] to its descent $\infty$-category for any cover of that space is an [[infinity-stack]].


nonabelian Stokes theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
+-- {: .toc .clickDown tabindex="0"}
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _nonabelian Stokes theorem_ (e.g. [Schreiber-Waldorf 11, theorem 3.4](#SchreiberWaldorf11)) is a generalization of the [[Stokes theorem]] to [[Lie algebra valued differential 1-forms]] and with [[integration of differential forms]] refined to [[parallel transport]].

## Statement

If $\hat A \in \Omega^1(D^2, \mathfrak{g})$ is a [[Lie algebra valued 1-form]] on the 2-[[disk]] then the [[parallel transport]] $\mathcal{P} \exp(\int_{S^1} A)$ of its restriction $A \in \Omega^1(S^1, \mathfrak{g})$ to the [[boundary]] [[circle]], hence its [[holonomy]] (for a fixed choice of base point) is equal to a certain kind of adjusted 2-dimensional integral of its [[curvature]] 2-form $F_A$ over $D^2$.

In particular if $F_{\hat A} = 0$ then the [[holonomy]] of $A$ is trivial.

## Properties

### Relation to higher parallel transport

In terms of the notion of [[connection on a 2-bundle]] the nonabelian Stokes theorem says that if $A \in \Omega^1(X, \mathfrak{g})$ is a [[Lie algebra valued 1-form]], then $(F_A, A)$ is a [[Lie 2-algebra valued 2-form]] with values in the [[inner derivation Lie 2-algebra]] $inn(\mathfrak{g})$ of $\mathfrak{g}$ whose [[curvature]] 3-form $H = \mathbf{d}_A F_A$ vanishes (which is the [[Bianchi identity]] for $F_A$) and its [[higher parallel transport]] exists. The [[2-functor|2-functorial]] source-target matching condition in this higher parallel transport is the statement of the nonabelian Stokes theorem.

### Relation to Lie integration

For $F_A = 0$ the nonabelian Stokes theorem may be regarded as proving that the [[Lie integration]] of $\mathfrak{g}$ by "the path method" (see at [[Lie integration]]) is indeed the [[simply connected topological space|simply connected]] [[Lie group]] corresponding to $\mathfrak{g}$ by [[Lie theory]].

## References

* Robert L. Karp, Freydoon Mansouri, Jung S. Rno, *Product Integral Formalism and Non-Abelian Stokes Theorem*, J. Math. Phys. **40** (1999) 6033-6043 [[arXiv:hep-th/9910173](https://arxiv.org/abs/hep-th/9910173), [doi:10.1063/1.533068](https://doi.org/10.1063/1.533068)]

* R. L. Karp, F. Mansouri, J. S. Rno, *Product Integral Representations of Wilson Lines and Wilson Loops, and Non-Abelian Stokes Theorem*, Turk. J. Phys. **24** (2000) 365-384 [[arXiv:hep-th/9903221](https://arxiv.org/abs/hep-th/9903221), [journal page](https://search.trdizin.gov.tr/tr/yayin/detay/67090)]

* Boguslaw Broda, *Non-Abelian Stokes theorem in action*, in *Modern Nonlinear Optics* Part 2, Wiley (2001) 429-468 [[arXiv:math-ph/0012035](https://arxiv.org/abs/math-ph/0012035), [ISBN:978-0-471-46612-3](https://www.wiley.com/en-ae/Modern+Nonlinear+Optics,+Volume+119,+Part+2,+2nd+Edition-p-9780471466123)]

In the context of [[higher parallel transport]] in [[principal 2-bundles with connection]]:

* {#SchreiberWaldorf11} [[Urs Schreiber]], [[Konrad Waldorf]], Theorem 3.4 in: *Smooth Functors vs. Differential Forms*, Homology, Homotopy Appl. **13** 1 (2011) 143-203 [[arXiv:0802.0663](http://arxiv.org/abs/0802.0663), [euclid:hha/1311953350](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-1/Smooth-functors-vs-differential-forms/hha/1311953350.full)]

See also:

* Seramika Ariwahjoedi, Freddy Permana Zen, *Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions*, Symmetry **2023** 15 11 (2000) [[doi:10.3390/sym15112000]( https://doi.org/10.3390/sym15112000)]


nonassociative algebra

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## Definition

Let $k$ be a [[commutative unital ring]], usually a [[field]] (but conceivably even a commutative [[rig]]).

A __nonassociative $k$-algebra__ is a $k$-[[module]] $V$ equipped with a bilinear product $V\otimes V\to V$.

This product is typically neither associative nor unital, although it can be (an example of the [[red herring principle]]).

Mathematicians working in the field of nonassociative algebras often say simply 'algebra' meaning a nonassociative algebra.

## Examples

Some interesting subclasses are [[Lie algebra]], [[Jordan algebra]], [[Leibniz algebra]], [[alternative algebra]], [[associative unital algebra]], [[composition algebra]]...

The [[octonions]] are a (slightly) non-associative real [[normed division algebra]].

## Related concepts

* [[nonassociative ring]]

* [[noncommutative algebra]], [[nonunital algebra]]

## References

* Richard D. Schafer, _Introduction to Non-Associative Algebras_, Dover, New York, 1995. ([pdf](http://www.gutenberg.org/files/25156/25156-pdf.pdf))

[[!redirects nonassociative algebra]]
[[!redirects nonassociative algebras]]
[[!redirects non-associative algebra]]
[[!redirects non-associative algebras]]

[[!redirects not-necessarily associative algebra]]
[[!redirects not-necessarily associative algebras]]


nonassociative group

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## Idea

A non-associative [[group]], or an [[invertible quasigroup|invertible]] [[loop]]. Nonassociative is used in the sense of not-necessarily associative, in the same sense that a [[nonassociative algebra]] is not-necessarily associative.

## Definition

A __nonassociative group__ or __invertible loop__ is a [[loop]] $(G,\backslash,/,1)$ with a unary operation called the __inverse__ $(-)^{-1}:G \to G$ such that

* $a^{-1} \cdot a = 1$
* $a \cdot a^{-1} = 1$

for all $a \in G$.

### Without division

A __nonassociative group__ or __invertible loop__ is a [[unital magma]] $(G,(-)\cdot(-):G\times G\to G),1:G)$ with a unary operation called the __inverse__ $(-)^{-1}:G \to G$ such that

* $a^{-1} \cdot a = 1$
* $a \cdot a^{-1} = 1$
* $(a \cdot b^{-1}) \cdot b = a$
* $(a \cdot b) \cdot b^{-1} = a$
* $b \cdot (b^{-1} \cdot a) = a$
* $b^{-1} \cdot (b \cdot a) = a$

for all $a,b \in G$.

## Properties

Every non-associative group is a [[loop (algebra)|loop]] with a two-sided [[inverse]].

## Examples

* Every group is a nonassociative group.

## Related concepts

* [[group]] (associative version)

* [[loop]] (non-invertible version)

* [[invertible quasigroup]] (non-unital version)

* [[commutative loop]] (commutative version)

* [[centipede mathematics]]

[[!redirects nonassociative group]]
[[!redirects nonassociative groups]]

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[[!redirects invertible unital magmas]]


nonassociative ring

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## Idea

The concept of _nonassociative ring_ is like that of [[ring]] but without the requirement of [[associativity]].

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###### Remark on terminology

The term "nonassociative ring" may be regarded as an example of the "[[red herring principle]]", as a nonassociative ring is not in general a ring in the modern sense of the word.

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## Definition

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###### Definition

A __nonassociative ring__ is a [[set]] $R$ with operations of addition and multiplication, such that:

*  $R$ is a [[unital magma]] under multiplication;
*  $R$ is an [[abelian group]] under addition;
*  multiplication distributes over addition.

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More sophisticatedly, just as a ring is a [[monoid object]] in [[Ab]], so

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###### Definition

A _nonassociative ring_ is a [[unital magma]] object in [[Ab]].

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###### Remark

A non-associative ring may well have associativiy, i.e. it may be in the image of the [[forgetful functor]] from associative rings to nonassociative rings. But if so, then this element is still not part of the defining data and in particular a [[homomorphism]] of non-associative rings need not to preserve associativity.

=--

## Properties

For any nonassociative commutative ring $R$, a non-associative commutative ring extension (in the sense of [[field extension]]) of $R$ is just a $R$-[[nonassociative algebra]].

## Examples

Examples include [[Lie rings]] and [[nonassociative algebras]] such as [[alternative algebras]].

## Related concepts

* [[nonunital ring]]

* [[noncommutative ring]]

* [[unital magma object]]

[[!redirects nonassociative rings]]

[[!redirects non-associative ring]]
[[!redirects non-associative rings]]


Noncommutative Algebra

This page is a link to Lurie's paper, _Noncommutative Algebra_.

* [[Jacob Lurie]], _Noncommutative Algebra_ ([arXiv](http://arxiv.org/abs/math/0702299))

on [[higher algebra]] and in particular on [[A-infinity ring]]s and [[A-infinity algebra]]. This article is now subsumed as a part of monograph [[Higher Algebra]].

category: reference


noncommutative algebra

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## Idea

An [[algebra]] which is not a [[commutative algebra]].

## Related concepts

* [[associative algebra]], [[nonassociative algebra]]

* [[nonunital algebra]]

* [[deformation quantization]]

* [[algebra of quantum observables]]

* [[noncommutative geometry]]

  * [[noncommutative algebraic geometry]]

  * [[noncommutative derived algebraic geometry]]

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noncommutative algebraic geometry

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## Idea

__Noncommutative algebraic geometry__  is the study of  'spaces' represented or defined in terms of algebras, or categories. Commutative [[algebraic geometry]], restricts attention to spaces whose local description is via _commutative_ [[ring]]s and [[algebra]]s, while noncommutative algebraic geometry allows for more general local (or affine) models. The categories are viewed as categories of [[quasicoherent sheaf|quasicoherent modules]] on noncommutative locally affine space, and by affine one can think of many algebraic models, e.g. $A_\infty$-algebras; the algebra and its category of modules are in the two descriptions viewed as representing the same space (Morita equivalence should not change the space). Categories are needed as the global "algebra of functions" is usually an incomplete global description while it may be  good enough locally. The categories are typically [[abelian category|abelian]], [[triangulated category|triangulated]], [[dg-category|dg-categories]] or [[A-infinity-category|A-infinity-catgories]].

Noncommutative algebraic geometry extends the [[algebraic geometry|algebraic geometric]] methods ([[local ring]]s, intersection theory etc.) to study noncommutative algebras, and conversely, uses noncommutative algebras in the study of commutative [[algebraic variety|algebraic varieties]] ([[Brauer group]]s, noncommutative desingularizations, [[stack]]s etc.). But some features, phenomena and methods do not have commutative analogues.

## Relation to other kinds of geometry

### Relation to ordinary algebraic geometry
 {#RelationToOrdinaryAlgebraicGeometry}

The direct "naive" generalization of Grothendieck-style [[algebraic geometry]] via [[sheaves]] on a [[site]] ([[Zariski site]], [[etale site]] etc.) of [[commutative rings]]-[[opposite category|op]] to non-commutative rings does not work, for reasons discussed in some detail in ([Reyes 12](#Reyes12)). This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of [[monoidal categories]] "of ([[quasicoherent sheaf|quasicoherent]]) [[abelian sheaves]]" ("[[2-rings]]"). See at _[[2-algebraic geometry]]_ for more.

### Relation to noncommutative geometry a la Connes

Noncommutative algebraic geometry may be considered a subfield of general [[noncommutative geometry]]. Among prominent other subfields, the most influential is the direction lead by [[Alain Connes]].
In Connes' noncommutative geometry the algebras in question are [[operator algebras]] viewed as algebras of continuous, smooth or measurable functions or more general continuous global sections. There it is customary and sufficient to describe a space globally by a single algebra: smooth and continuous context allow for partition of unity so there is no need for sheaves and gluing local descriptions, to define the very spaces.

## History

The subject is older than mainstream [[noncommutative geometry]] and it is very much implicit in Pierre Gabriel's thesis "Des catégories Abéliennes" (Bull. Soc. Math. France 1962). The [[Gabriel spectrum]] (consisting of isomorphism classes of injective indecomposables) amounts to a reconstruction of a [[noetherian scheme|noetherian]] [[quasiseparated scheme|quasiseparated]] [[quasicompact scheme|quasicompact]] [[scheme]] from its category of [[quasicoherent sheaf|quasicoherent]] $\mathcal{O}$-[[module]]s (now generalized to all schemes, as the [[Gabriel-Rosenberg theorem]]). He looked at [[localization]] functors, and corresponding subcategories ([[reflective subcategory|reflective]] and coreflective), and proved some theorems on localization of [[quasicoherent sheaf|quasicoherent sheaves]]. At the same time, [[Alexander Grothendieck|Grothendieck]] taught us that _to study a space it is enough to have a [[category of sheaves]] -- a [[Grothendieck topos]] -- on this would-be space_.

Together with two Auslander-Goldman papers " _Maximal orders_ " and " _The Brauer group of a commutative ring_ " -- both in Trans. AMS 1960 -- Gabriel's paper is a precursor of a more articulated development starting with Mike Artin's " _On Azumaya algebras and finte dimensional representations of rings_ " (J. Alg. 1969). Artin's 1969 paper characterizes [[Azumaya algebra]]s via their identities and paved the way to generalize [[algebraic geometry]] to [[ring]]s satisfying polynomial identities, via their [[representation]]s and invariant theory. Subsequent major contributions were made by Claudio Procesi with his book " _Rings with polynomial identities_ " (Marcel Dekker 1973) and his paper " _The invariant theory of $n\times n$ matrices_" (Adv. Math. 1976) and by Mike Artin and Bill Schelter's " _A version of Zariski's main theorem for PI rings_ " (Amer. J. Math. 1979). The link with [[representation theory]], using seminal work by Paul Cohn and [[George Bergman]], was clarified by Claus Ringel " _The simple Artinian spectrum of a finite dimensional algebra_ " (in Dekker Lect. Notes 51, 1979) and culminated in Aidan Schofield's book " _Representations of rings over skew fields_ " (1985). These developments have been influential for recent work in '[[formal scheme|formal]]' noncommutative geometry, such as [[Maxim Kontsevich]]'s "_Formal (non)commutative symplectic geometry" (1993), [[Joachim Cuntz]] and [[Daniel Quillen]]'s "Algebra extensions and nonsingularity" (JAMS 1995) and Kontsevich-Rosenberg's "Noncommutative smooth spaces" (cf. bibl. and [[quasi-free algebra]]).

In the 70ties, [[localization]] theory was popular among ring theorists, and, several people tried to generalize [[structure sheaf|structure sheaves]] to noncommutative algebras. Gabriel-like localizations via alternative formalism of "kernel functors" (Goldman 1969) led to the books by Jonathan Golan " _Localization of noncommutative rings_ " (Marcel Dekker 1975) and [[Fred Van Oystaeyen]] " _Prime spectra in noncommutative algebra_ " (Springer LNM 444, 1975), and making the link with the Artin-Procesi approach, the book

* [[Fred Van Oystaeyen]], [[Alain Verschoren]], _Non-commutative algebraic geometry_, Springer LNM 887, 1981.

Localizations were viewed as analogues of open sets. However, the [[lattice of torsion theories]] is not [[distributive lattice|distributive]], what presented major difficulties at the time. Around 1995, a major [[descent]] theorem was proved by van Oystaeyen and Willaert and independently in a work of A. Rosenberg in Stockholm (1988). It is sort of generalized [[sheaf]] condition for quasicoherent modules, with respect to [[cover]]s by Gabriel localizations. This fact is now seen almost obvious; namely it is the comonadic descent for a [[comonad]] induced by a conservative family of flat localizationshaving right adjoint, which in the Gabriel localization clearly satisfy the conditions for Beck's [[comonadicity theorem]].

Cohn's universal localizations were used by Paul Cohn in " _The affine scheme of a general ring_ " (in Springer LNM 753, 1979) and later by Claus Ringel and Aidan Schofield as mentioned before. In England, Goldie's localization theory led to the theory of 'cliques' and 'clans' of prime ideals (Jategaonkar, Hajarnavis and others) which may in retrospect be viewed as influential to present local description via [[quiver]]s and [[A-infinity-category|A-infinity structures]].

Around 1980, the Leningrad school of [[integrable system]]s (Sklyanin, Fadeev, Takhtajan) created examples of [[quantum groups]], and soon after that were joined by Drinfel'd, Novikov, Jimbo, and independently, Manin and Woronowicz; these in turn give good examples in __affine noncommutative geometry__ (spaces represented by a single nongraded ring), moreover the actions of group-like objects are considered starting the subject of [[equivariant noncommutative algebraic geometry]].

In the early 80ties maximal orders were revisited as noncommutative versions of normal [[variety|varieties]]. In particular their local structure was investigated. Extending prior work by Mark Ramras, Mike Artin succeeded in describing the etale local structure of maximal orders over surfaces. In trying to generalize this to higher dimensions, one was quickly led to impose strong homological properties on the orders (and later, more general algebras). This led to the investigation of homological homogeneous rings (work by Ken Brown and others), Artin-Schelter regular algebras and, finally, Auslander-regular rings.

In this vein, around 1989, the [[noncommutative projective geometry]]  was developed by Artin, Zhang and collaborators, where Serre's theorem on $Proj$ of a graded ring is taken as a definition in the noncommutative case: [[quasicoherent sheaf|quasicoherent sheaves]] over a $Proj$ of a graded ring is the category of graded modules modulo the modules of finite length. They assume strong regularity properties on graded rings, limiting to the geometry which is very close to commutative projective geometry. Using [[Ore localization]], [[F. van Oystaeyen]] has defined another version of noncommutative projective geometry, based on his notion of a [[schematic algebra]].

A huge project started, trying to classify Auslander-regular algebras, graded-connected and generated in degree one (that is, noncommutative analogons of projective n-space) for small dimensions n. In dimension 3, the initial work was done by Artin and Schelter, culminating in the influential papers (ATV1), (ATV2) by Mike Artin, John Tate and [[Michel Van den Bergh]] (see bibl.). In this classification a class of algebras re-appeared which were initially discovered by Odeskii and Feigin, the so called Sklyanin algebras. Ring-theoretic investigations of the Sklyanin algebras were done by Tate and Van den Bergh (Noetherianity for all dimensions n) and in dimension 4 a geometric study via 'point' and 'line' modules was carried out by Thierry Levasseur, Toby Stafford and Paul Smith. Further classifications in dimension 3 (dropping the degree one generator condition) and 4 were done by these people and their students. In the ungraded case, maximal orders with excellent homological properties were recently used in connection with quotient-singularities and their desingularizations as in the noncommutative crepant resolution by Michel Van den Bergh (Abel symposium 2002).

## Approaches

### Spectrum of an abelian category

When one is serious about the point of view that every space is represented by some [[abelian category]] and that every [[abelian category]] represents a (noncommutative) space this way, one wants to find the _spectrum_ of an abelian category in the way that one forms the [[spectrum (geometry)]] of a [[ring]]. In particular since in the noncommutative case the classical spectra from ring theory (like primitive spectrum and Gabriel's spectrum are usually too small to represent noncommutative spaces)

Motivated by this, [[A. L. Rosenberg]] has developed many spectral constructions extending from the left spectrum of a ring to various spectra of [[abelian category|abelian categories]], [[triangulated category|triangulated categories]] and [[exact category|right exact categories]] (in his sense) and created a categorical version of a notion of (quasicompact relative) [[noncommutative scheme]] $X$ over a base category $C$ (viewed as category of sheaves on $Spec k$). Many other variants of spectra of abelian categories can be constructed; there is a general pattern though, cf. [[spectral cookbook]].

### Sheaves of sets on a $Q$-category

A space in the sense of a Grothendieck school is a sheaf $F$ of sets on the category of affine schemes $Aff$ equipped with some subcanonical Grothendieck topology and which is locally affine, i.e. there is a cover of $F$ (in the sense of induced Grothendieck topology on the category of presheaves) by representables. The noncommutative analogue of $Aff$ is the opposite to the category of noncommutative unital rings $NAff = Ring^{op}$. The problem is that the natural candidates for covers in $NAff$ are not making classes which satisfy the axioms for [[Grothendieck topology]], namely, the crucial stability under pullback fails. However $A$ can be equipped with a structure of a [[Q-category]], and then one can consider the sheaves of sets on a Q-category and define in an analogous way when they are locally affine. Similar approach works for noncommutative stacks.

### Derived noncommutative geometry
  {#DerivedNonCommutativeGeometry}

[[derived noncommutative geometry|Derived noncommutative geometry]] is a subject related to the commutative [[derived algebraic geometry]] of [[Carlos Simpson|C. Simpson]]'s school. Cyclic and [[Hochschild homology]] play a major role.

Here one thinks of [[dg-categories]], [[A-infinity categories]], [[Spectra]]-[[enriched categories]] and ultimately of [[stable (∞,1)-categories]] as [[formal duals]] to non-commutative derived spaces:

Any [[locally presentable (∞,1)-category|locally presentable]] [[stable (∞,1)-category]] may be regarded as a stable [[(∞,1)-topos]] (the "stable Giraud theorem", see [there](stable+infinity-category#StabGiraud)).
Therefore, thinking of a non-commutative space as the [[formal dual]] to a  [[triangulated category]] or rather to a [[stable (∞,1)-category]] is directly analogous to the way  [[topos theory]] and in particular [[higher topos theory]] characterizes generalized spaces as formal duals of [[toposes]].

More in detail, every [[stable (∞,1)-category]] with a set of generators is [[equivalence of (infinity,1)-categories|equivalently]] the $\infty$-category of [[(∞,1)-modules]] over an [[A-∞ algebra]] (or rather $A_\infty$-algebroid, in general, due to [Schwede-Shipley 01](stable+infinity-category#SchwedeShipley)) and in this sense manifestly the formal dual to a non-commutative derived variety (see [there](stable+infinity-category#AsCategoriesOfModules) for more).

The notion of [[noncommutative motive]] as localizations of stable $\infty$-categories (see [there](noncommutative+motive#AsUniversalAditiveInvariant)), due to [Blumberg-Gepner-Tabuada 10](noncommutative+motive#BlumbergGepnerTabuada10), directly ties into this.

The most nontrivial result seem to be a noncommutative
version of the [[degeneration of Hodge-to-de-Rham spectral sequence]], conjectured by Kontsevich and, in one version, [proved](http://front.math.ucdavis.edu/0611.5623) by [[D. Kaledin]].

#### Homological mirror symmetry

[Homological mirror symmetry](http://en.wikipedia.org/wiki/Homological_mirror_symmetry) by [[Maxim Kontsevich]] is a source of many *examples* of spaces of [[derived algebraic geometry]], which are represented by [[triangulated category|triangulated categories]] or their [[dg-category|dg-]] or $A_\infty$-[[enhanced triangulated category|enhancements]]; this has been enhanced by the insight in the structure of [[derived category|derived categories]] of [[coherent sheaf|coherent sheaves]] and their deformations like Landau-Ginzburg models, by [Beilinson](http://en.wikipedia.org/wiki/Alexander_Beilinson)(1977/8), Kapranov (1985-), Orlov, Bondal, van den Bergh, Polishchuk, Bridgeland and others.

### Other motivations

Another big source of examples is deformation quantization and related topics involving usage of $A_\infty$ and $L_\infty$-algebras. Understanding of the role of quivers in noncommutative geometry has also being stimulated by independent and original works of Lieven le Bruyn and [[Arfinn Laudal]]. Laudal was also  motivated by deformation theory.

## Related concepts

[[!include Isbell duality - table]]

$\,$

[[!include noncommutative motives - table]]

## References

* the entries on the subject include: [[noncommutative scheme]], [[sheaf on a noncommutative space]], [[noncommutative projective geometry]], [[localization]], [[schematic algebra]], [[Q-category]], [[quasicoherent sheaf]], [[spectrum of an abelian category]], [[spectral cookbook]], [[quantum group]], [[noncommutative principal bundle]], [[quantum homogeneous space]], [[equivariant noncommutative geometry]]
* list of pages with the tag [noncommutative geometry](/nlab/list/noncommutative+geometry)
* related subjects include: [[noncommutative geometry]], [[Hopf algebra]], [[gebra]], [[bialgebroid]], [[quantization]], [[deformation theory]]

* M. Artin, J. J. Zhang, _Noncommutative projective schemes_, Adv. Math. __109__ (1994), no. 2, 228--287, [doi](http://dx.doi.org/10.1006/aima.1994.1087)

* A. Bondal, M. van den Bergh, _Generators and representability of functors in commutative and noncommutative geometry_, Moscow MJ 2003

* M. Kontsevich, A. Rosenberg, _Noncommutative smooth spaces_,  The Gelfand Mathematical Seminars, 1996--1999,  85--108, Birkhäuser 2000; [arXiv:math/9812158](http://arxiv.org/abs/math/9812158)

* [[M. Kontsevich]], A. Rosenberg, _Noncommutative spaces_, preprint MPI-2004-35 ([dvi](http://www.mpim-bonn.mpg.de/preblob/2303),[ps](http://www.mpim-bonn.mpg.de/preblob/2331)), _Noncommutative spaces and flat descent_, MPI-2004-36 [dvi](http://www.mpim-bonn.mpg.de/preblob/2304),[ps](http://www.mpim-bonn.mpg.de/preblob/2332), _Noncommutative stacks_, MPI-2004-37 [dvi](http://www.mpim-bonn.mpg.de/preblob/2305),[ps](http://www.mpim-bonn.mpg.de/preblob/2333)

* [[A. L. Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125, [MR99h:14002](http://www.ams.org/mathscinet-getitem?mr=1622759), [doi](http://dx.doi.org/10.1023/A:1000479824211)

* O. A. Laudal, _Noncommutative algebraic geometry_, Rev. Mat. Iberoamericana __19__, n. 2 (2003), 509--580; ([euclid](http://projecteuclid.org/euclid.rmi/1063050166)).

* (ATV2) M. Artin, J. Tate, [[M. Van den Bergh]], _Modules over regular algebras of dimension $3$_, Invent. Math. __106__ (1991), no. 2, 335–388, [MR93e:16055](http://www.ams.org/mathscinet-getitem?mr=1128218), [doi](http://dx.doi.org/10.1007/BF01243916)

* (ATV1) Michael Artin, John Tate, [[Michel Van den Bergh]], _Some algebras associated to automorphisms of elliptic curves_, The [[Grothendieck Festschrift]], Vol. I, 33–85, Progr. Math. __86__, Birkhäuser 1990.

* the n-geometry cafe, _the NAG canon_ ([blog](http://www.noncommutative.org/index.php/the-nag-canon.html))

* [[F. van Oystaeyen]], _Algebraic geometry for associative algebras_, Marcel Dekker 2000. vi+287 pp.

* S. Mahanta, _On some approaches towards non-commutative algebraic geometry_, [math.QA/0501166](http://arxiv.org/abs/math/0501166)

* [[L. Katzarkov]], [[Maxim Kontsevich|M. Kontsevich]], T. Pantev, _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)

* [[D. Kaledin]], _Tokyo lectures "Homological methods in non-commutative geometry"_, [pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex)

* [[Zoran Skoda]], _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770); preprint MPIM2009-3.

* MathOverflow, [Theories of Noncommutative Geometry](http://mathoverflow.net/questions/10512/theories-
of-noncommutative-geometry)

* {#Reyes12} Reyes, _Sheaves that fail to represent matrix rings_ ([arXiv:1211.4005](http://arxiv.org/abs/1211.4005))

Discussion of noncommutative [[formal geometry]]
of infinitesimal neighborhood of commutative schemes within noncommutative
ambient schemes is in

* {#Kapranov98} [[Mikhail Kapranov]], _Noncommutative geometry based on commutator expansions_ ([arXiv:9802041](http://arxiv.org/abs/math/9802041))

For more on this see at _[[Kapranov's noncommutative geometry]]_.
See also the references at _[[derived noncommutative geometry]]_

category: algebraic geometry, noncommutative geometry
[[!redirects non-commutative algebraic geometry]]


noncommutative associated bundle

# Noncommutative associated bundles
* table of contents
{: toc}

## Distinguished from other noncommutative bundles

Here we discuss associating a bundle with fiber $F$ to a [[noncommutative principal bundle]] with structure "group" (e.g. Hopf algebra or generalization). There are also other kinds of [[noncommutative vector bundle]]s (e.g. locally free sheaves on nc spaces; duals of projective modules etc.).

## Affine case

The simplest case of the [[noncommutative principal bundle]] is the case of Hopf-Galois extension.

* the total and base space are affine, i.e. represented by noncommutative algebras $P$ and $U$. There is a monomorphism of algebras
$U\hookrightarrow P$ (which is viewed as a comorphism of the projection among the spectra)

* $P$ is equipped with a right coaction $P\to P\otimes H$ of a fixed Hopf algebra $H$, which is an algebra map (we say that $P$ is a right $H$-comodule algebra)

* Hopf-Galois condition is satisfied: $U\hookrightarrow P$ is a $H$-[[Hopf-Galois extension]] (typically required to be faithfully flat).

The space of __global sections of the associated vector bundle__ with  $V$ as a fiber is then simply the [[cotensor product]] with the vector space $P\Box^H V$, for which we need a left $H$-comodule structure $V\to H\otimes V$. This construction however does not give the total space of the associated bundle as a noncommutative space. In commutative algebraic case to get the total algebra we need to replace vector space by the symmetric algebra of the dual, and then do the cotensor product. This can be repeated in the noncommutative case to get some algebra, however this does not reproduce the same space of sections as the cotensor product with the underlying vector space. Instead one can take the tensor algebra of the dual of the vector space as the noncommutative analogue of the symmetric algebra. After taking the cotensor product, one gets much bigger total algebra. However, it does not reproduce the quantum examples. Sometimes one should take some sort of a q-symmetric algebra, like generalizations of quantum Manin plane, what gives a consistent result just in some special cases.

## Gluing via localizations

One can glue Hopf-Galois extensions along [[noncommutative localization]]s to more global objects. For example, both the total space and the base space can be represented by [[noncommutative scheme]]s. An early example with nonaffine base space is described in Zoran Škoda, "Localizations for construction of quantum coset spaces" (most proofs given elsewhere). This is an application of using [[descent in noncommutative algebraic geometry]]. To get to the sheaves of sections of associated bundles, one defines the sections as a cotensor product *locally* on affine pieces and then glues. A simple case is treated in Zoran Škoda, "Coherent states for Hopf algebras".

## Literature

### Affine case

* [[S. Majid]], [[T. Brzeziński]], _Quantum group gauge theory on quantum spaces_, Commun. Math. Phys. __157__, 591--638 (1993); Erratum 167, 235 (1995) [MR94g:58015](http://www.ams.org/mathscinet-getitem?mr=1243712), [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104254023)

### Nonaffine case

* Zoran Škoda, _Localizations for construction of quantum coset spaces_, Banach Center Publications __61__, 265–298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090)

* Zoran Škoda, _Coherent states for Hopf algebras_, Letters in Mathematical Physics __81__, N.1, pp. 1-17, July 2007, [pdf](http://www.irb.hr/korisnici/zskoda/skodacohstates.pdf), (earlier different arXiv version: math.QA/0303357)

[[!redirects noncommutative associated bundle]]
[[!redirects noncommutative associated bundles]]
[[!redirects non-commutative associated bundle]]
[[!redirects non-commutative associated bundles]]


noncommutative black hole

__Noncommutative black hole__ is a black hole solution of gravity theory on a noncommutative space-time.

* Thorsten Ohl, Alexander Schenkel, _Cosmological and black hole spacetimes in twisted noncommutative gravity_, JHEP 10, 052 (2009) [doi](http://dx.doi.org/10.1088/1126-6708/2009/10/052), [arxiv/0906.2730](http://arxiv.org/abs/0906.2730), _Symmetry reduction in twisted noncommutative gravity with applications to cosmology and black holes_, JHEP 0901, 084 (2009) [doi](http://dx.doi.org/10.1088/1126-6708/2009/01/084), [arxiv/0810.4885](http://arxiv.org/abs/0810.4885)

* J. C. L´opez-Dom´ınguez, O. Obreg´on, C. Ram´ırez and
M. Sabido, _Noncommutative black holes_, [pdf](http://iopscience.iop.org/1742-6596/91/1/012010/pdf/1742-6596_91_1_012010.pdf)

* Myung, Kim, Park, _Thermodynamics and evaporation of the
noncommutative black hole_ [pdf](http://iopscience.iop.org/1126-6708/2007/02/012/pdf/1126-6708_2007_02_012.pdf)

* [[Shahn Majid]], _Scaling limit of the noncommutative black hole_, [arxiv/1009.5406](http://arxiv.org/abs/1009.5406)
[[!redirects noncommutative black holes]]


noncommutative differential calculus

# Noncommutative differential calculus
* table of contents
{: toc}

## Disambiguation

There are several notions of noncommutative differential calculus. See related pages [[regular differential operator in noncommutative geometry]],
[[bicovariant differential calculus]], [[differential bimodule]], [[differential monad]], [[cyclic homology]]...

This entry is about the version of "Cartan calculus" for associative algebras introduced by [[Boris Tsygan]], [[Dmitri Tamarkin]] and [[Ryszard Nest]], and studied also by Dolgushev, Kowalzig and others.

## Definition

Let $k$ be a unital commutative ground ring. A
[[Gerstenhaber algebra]] $(V^\bullet,\cup)$
over $k$ is a
$\mathbf{N}$-graded-commutative algebra
with a graded Lie structure on $V[1]$
satisfying the Leibniz rule
(an analogue of a Poisson bracket in dg-world)

$$
\{ c, a\cup b\} = \{c, a\}\cup b +
(-1)^{p q} a \cup \{ c, b\}
$$

A [[Gerstenhaber module]] $(\Omega_\bullet,\cap)$ is a
graded module over a Gerstenhaber algebra $(V^\bullet,\cup)$
$$
\cap : V\otimes \Omega\to\Omega,\,\,\,\,\,\,\,
V^p\otimes\Omega_n\ni a\otimes x \mapsto a\cap x
\in\Omega_{n-p}
$$
with a graded Lie algebra action of $V[1]$,
$$
\mathcal{L} : V[1]\otimes\Omega\to\Omega,\,\,\,\,\,\,
V^{p+1}\otimes\Omega_n\ni a\otimes x \mapsto \mathcal{L}_a(x)
\in\Omega_{n-p},
$$
satisfying the mixed Leibniz rule
$$
b \cap \mathcal{L}_a(x) = \{ b, a\}\cap x +
(-1)^{p q} \mathcal{L}_a(b\cap x).
$$

A Gerstenhaber module $\Omega$ over a Gerstenhaber
algebra $V$ is a __Batalin-Vilkovisky module__
if it is equipped with a $k$-linear differential
of degree +1,
$$
B : \Omega_\bullet\to\Omega_{\bullet+1},\,\,\,\,\,\,B^2 = 0,
$$
such that $\mathcal{L}_\alpha$ for $a\in V^p$ and $B$ satisfy the generalization of the [[Cartan's homotopy formula]]
$$
\mathcal{L}_a(x) = B(a\cap x) - (-1)^p a\cap B(x)
$$

A __Tsygan-Tamarkin-Nest noncommutative (differential) calculus__ is a pair $(V, \Omega)$
of a Gerstenhaber algebra $V$
and a Batalin-Vilkovisky $V$-module $\Omega$.

See also the case of [[Batalin-Vilkovisky algebra]].

## Related concepts

* [[differential calculus]]

## References

* [[Dmitri Tamarkin]], [[Boris Tsygan]], _Noncommutative differential calculus, homotopy BV algebras and formality conjectures_, Metods of Functional Analysis and topology, 1, 2001 arXiv:[math.KT/0002116](https://arxiv.org/abs/math/0002116)

* D. Tamarkin, B. Tsygan, _The ring of differential operators on forms in noncommutative calculus_, Graphs
and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc. 2005, pp. 105--131  [doi](https://doi.org/10.1090/pspum/073/2131013) [MR2131013](https://mathscinet.ams.org/mathscinet-getitem?mr=2131013)

* R. Nest, B. Tsygan, _On the cohomology ring of an algebra_, Advances in geometry, Progr. Math. __172__, Birkhauser 1999, pp. 337--370 [arxiv:math.QA/9803132](http://arxiv.org/abs/math/9803132)

* V. Dolgushev, D. Tamarkin, B. Tsygan, _Noncommutative calculus and the Gauss-Manin connection_, in: Higher structures in geometry and physics, 139--158, Progr. Math., 287, Birkhäuser/Springer, New York, 2011 , [arXiv:0902.2202](http://arxiv.org/abs/0902.2202)

* K. Behrend, B. Fantechi, _Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections_,
[pdf](https://www.math.ubc.ca/~behrend/GeBaViLa.pdf)

* Pedro Tamaroff, _The Tamarkin-Tsygan calculus of an algebra a la Stasheff_, Homology, Homotopy and Applications __23__:2 (2021) 257--282 [arXiv:1907.08888](https://arxiv.org/abs/1907.08888) [doi](https://doi.org/10.4310/HHA.2021.v23.n2.a14)

Many examples of such noncommutative differential calculi are constructed using [[Hopf algebroid]]s
(Schauenburg's version) in works

* Niels Kowalzig, _Gerstenhaber and Batalin-Vilkovisky structures on modules over operads_, [arxiv:1312.1642](https://arxiv.org/abs/1312.1642);
_Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids_ [arXiv:1305.2992](https://arxiv.org/abs/1305.2992)

* [[Niels Kowalzig]], [[Ulrich Kraehmer]], _Batalin-Vilkovisky structures on Ext and Tor_,   Journal für die reine und angewandte Mathematik 697 (2014)
[doi](https://doi.org/10.1515/crelle-2012-0086) [arXiv:1203.4984](https://arxiv.org/abs/1203.4984)

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[[!redirects noncommutative differential calculus]]
[[!redirects noncommutative differential calculi]]
[[!redirects Batalin-Vilkovisky module]]
[[!redirects Batalin–Vilkovisky module]]
[[!redirects Batalin--Vilkovisky module]]


noncommutative divisor

See page 3 in

* [[Paul Seidel]], _Fukaya $A_\infty$ structures associated to Lefschetz fibrations. II_, [arXiv:1404.1352](http://arxiv.org/abs/1404.1352).


noncommutative Fourier transform

This entry is about the noncommutative Fourier transform in the sense of [[Mikhail Kapranov]].

There is a different notion of a Fourier transform in the context of Hopf algebras which is sometimes also called the noncommutative Fourier transform, but in $n$Lab it will be described in an entry under the other common name, [[quantum group Fourier transform]] (studied by [[Shahn Majid]] and others).

## Motivation

In physics there is a notion of large N limit of gauge theories; N here denotes the size of matrices, e.g. related to the gauge SO(N) group. For large N remarkable simplifications happen to the behaviour of correlation functions and the gauge theory is very much like string theory.

In some other formal considerations one also sees that if one discretizes the space of loops in some space and approximates them with discrete walks and the steps in different dimensional directions are labeled by different noncommutative variables then one has a rough approximation of loops by words. Thus the walks on free groups and spaces of loops are somehow comparable. There is a conjectural noncommutative Fourier transform which should make this precise. Some examples of this type of theory are already constructed.

## Literature

* lecture at msri 2000 _Noncommutative neighborhoods and noncommutative Fourier transform_: [link](http://msri.org/publications/ln/msri/2000/interact/kapranov/1/index.html)

* M. Kapranov, _Noncommutative geometry and path integrals_, [math.QA/0612411](http://arxiv.org/abs/math/0612411)


Noncommutative Geometry

+-- {: .rightHandSide}
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#### Geometry
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=--
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=--

This entry collects some links related to the textbook

* [[Alain Connes]],

  **Noncommutative geometry**

  Academic Press, San Diego, CA, 1994

  [ISBN:9780080571751](https://www.elsevier.com/books/noncommutative-geometry/connes/978-0-08-057175-1)

  [pdf](http://www.alainconnes.org/docs/book94bigpdf.pdf)

on [[noncommutative geometry]] in the [[operator algebra]]-theoretic and [[spectral triple|spectral]] sense.

## Related entries

* [[noncommutative topology]]

* [[C*-algebra]]

* [[groupoid convolution algebra]]

* [[noncommutative topology of quasiperiodicity]]

* [[Hilbert module]], [[KK-theory]]

* [[spectral triple]]

category: reference


noncommutative geometry

+-- {: .rightHandSide}
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+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Noncommutative geometry
+--{: .hide}
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=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Spaces can be characterized by their [[algebra|algebras]] of functions (see at [[Isbell duality - table]]).

Using this [[duality]] between [[space and quantity]] one can define generalized spaces in terms of generalizations of their algebras of functions.

The idea of noncommutative geometry is to encode everything about the geometry of a space algebraically and then allow all commutative function algebras to be generalized to possibly non-commutative algebras.

More generally, __noncommutative geometry__ means replacing the space by some structure carried by an entity (or a collection of entities) living on that would-be space. The entity may be for example a function, vector bundle, coherent sheaf, a complex of sheaves and KK-theory class. Objects organize into [[associative algebras]], [[operator algebras]],
[[categories]], higher categories ($k$-linear or not) and so on; and sometimes such a collection represents a space.  [[reconstruction|Reconstruction]] theorems are theorems on construction of a genuine (say topological) "underlying" space of such an entity or collection.  Spectral theories are procedures (sometimes functors, often not) which recover some form of underlying space called [[spectrum (geometry)|spectrum]], often just partially or under strong assumptions on (the data determining) the noncommutative space.

Under the process of forming [[groupoid convolution algebras]] a good bit of commutative but [[higher geometry]] translates into noncommutative geometry. This is for instance the origin of the role of noncommutative geometry in [[twisted K-theory]].

## Definitions

### Connes' noncommutative geometry

A particular and most prominent realization of the program of noncommutative geometry has been lead by [[Alain Connes]]. This is really "[[spectral geometry|spectral]]" and possibly non-commutative [[Riemannian geometry]], where the metric structure on a possibly noncommutative "[[algebra of functions]]" is encoded by a structure called a _[[spectral triple]]_ (see there for more).

The central ingredients in Connes' noncommutative geometry are

* the idea to characterize a (noncommutative) space by a [[C-star algebra]] $A$, to be thought of as the $C^*$-[[algebra of functins|algebra of global functions]] on that space; this approach has been occasionally considered earlier e.g. in the book of Semadeni on Banach algebras.

* There is a refined, quantized differential calculus where the differential is given by a commutator formula involving a Fredholm operator; the setup in which this is taken place involves cyclic cocycles discovered by Tsygan and Connes.

* the noncommutative analog of the structure of a [[Riemannian manifold]] with a spin structure in terms of generalized [[Dirac operator]]s $D$ acting on a representation space of the algebra $A$. Metric information on the space is then encoded in the spectrum of $D$.

For that reason [[Alain Connes|Connes]]' noncommutative manifolds are well described as **spectral geometry**. The Dirac operator is, of course, very much related to the quantized differential calculus of Connes.

* Noncommutative measure spaces are represented by noncommutative von Neumann algebras.

In noncommutative geometry various homotopical and (co)homological invariants were introduced by large amount of improvisation, similar to the beginnings of algebraic topology, but more recently there are few systematic approaches to homotopy theory emerging. See [[model structure on operator algebras]].

## Related concepts

* [[noncommutative topology]], [[noncommutative measure theory]], [[noncommutative probability theory]]

* [[groupoid algebra]]

* [[noncommutative torus]], [[fuzzy sphere]]

* [[noncommutative topology of quasiperiodicity]]

* [[noncommutative scheme]]

* [[non-commutative analytic space]]

* [[QFT on non-commutative spacetime]]

* [[noncommutative motive]]

* [[Riemann hypothesis and physics]]

* [[D-brane geometry]]

* [[noncommutative algebraic geometry]]

* [[derived noncommutative geometry]]

[[!include Isbell duality - table]]

## References

### General

* [[Alain Connes]], _[[Noncommutative Geometry]]_, Academic Press, San Diego, CA, 1994 ([ISBN:9780080571751](https://www.elsevier.com/books/noncommutative-geometry/connes/978-0-08-057175-1), [pdf](http://www.alainconnes.org/docs/book94bigpdf.pdf))

* [[José Gracia-Bondia]], [[Joseph Varilly]], Héctor Figueroa, _Elements of noncommutative geometry_, Birkhäuser 2001. xviii+685 pp. ([doi:10.1007/978-1-4612-0005-5](https://link.springer.com/book/10.1007/978-1-4612-0005-5), [gBooks](http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false))

Exposition:

* {#Lupercio20} [[Ernesto Lupercio]], _Non-commutative Geometry Indomitable_ ([arXiv:2008.08529](https://arxiv.org/abs/2008.08529))

See also:

* {#Connes95} [[Alain Connes]], _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](http://www.alainconnes.org/docs/reality.pdf))

* [[Alain Connes]], _A walk in the non-commutative garden_ ([arXiv:math/0601054](http://arxiv.org/abs/math/0601054))

With a view towards [[motives in physics]]:

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_

### Very early sources and schools

There are many sources of noncommutative spaces, e.g. [[quantization]] in [[physics]] (Snyder studied an interesting noncommutative space in the late 1940s). It has been often noticed, since Gel'fand--Neimark's work (see [[Gelfand spectrum]]), that many geometric notions for commutative [[Banach algebra]]s are interesting in the noncommutative case; among early enthusiasts one could mention [[Irving Segal]]; Semadeni's monograph on Banach spaces of continuous maps also predates the sudden expansion of the field in the late 1970s when [[Alain Connes]] ([web site](http://www.alainconnes.org/)) brought about a whole revolution in mathematics using the framework of noncommutative geometry based on [[operator algebra]]s, and boosted by the discovery of cyclic (co)homology by Connes and Boris Tsygan and its connection to [[K-theory]].

In [[ring theory]], there were many efforts at building [[spectrum (geometry)|spectra]] of noncommutative rings or [[abelian category|abelian categories]] (the spectrum of indecomposable injectives of Gabriel, the affine spectrum of P. M. Cohn
and so on) and stronger results in special cases like P. I. rings (polynomial identities rings) by M. Artin, F. van Oystaeyen and others in the 1970s. Golan and van Oystaeyen took noncommutative localization seriously in the mid 1970s with very promising results; [[Alexander Rosenberg|A. Rosenberg]] found in 1982 the so-called [[left spectrum]] of a noncommutative ring, straightforwardly later generalized to a spectrum of an [[abelian category]]. All these efforts belong to an early phase of [[noncommutative algebraic geometry]]. In the late 1980s, the study of [[derived categories]] of (quasi)coherent sheaves complemented that effort ([[Mikhail Kapranov|Kapranov]], Bondal, Orlov, ...). Though the effort was rather disconnected (from ring-theoretic mainstream noncommutative algebraic geometry) at the beginning, it can be now more easily judged as closely related; this is also the time of birth of the influential school of noncommutative [[projective geometry]] (Artin, Smith, Zhang, ...).

The quantization program and the study of [[integrable systems]] brought about a number of interesting examples (in the early phase by G. Kac, Sklyanin, Fadeev, Drinfel'd, Woronowicz, Jimbo, [[Yuri Manin|Manin]], Reshetikin, Lusztig, Majid and others) with group-like flavour, the study of [[quantum groups]] which were studied in a number of formalisms from operator algebraic to algebro-geometric and purely categorical. [[action|Actions]] are very important in noncommutative geometry, and are some of the main examples in Connes' school like group(oid) $C^*$-algebras, crossed product operator algebras and the study of functions on orbifolds and foliations. They also play an important role in [[equivariant noncommutative algebraic geometry]]; cf. the central notions like [[Hopf–Galois extension]] and entwining structure.

Noncommutative formal geometry, concerned with objects like infinitesimal neighborhoods of subvariaties, power series in noncommutative variables and so on, has been appearing more sporadically than the operator algebraic and algebro-geometric frameworks. Among pioneering works, there are [[Kontsevich]]'s (formal) noncommutative symplectic geometry (1992) and [[Kapranov's noncommutative geometry]] "based on commutator expansions". Noncommutative [[analytic geometry]] is even now only vaguely outlined in existing works.

[[!redirects noncommutative geometries]]

[[!redirects non-commutative geometry]]
[[!redirects non-commutative geometries]]

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[[!redirects noncommutative spaces]]

[[!redirects non-commutative space]]
[[!redirects non-commutative spaces]]

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noncommutative geometry - contents

**[[noncommutative geometry]]**

([[geometry]] $\leftarrow$ [[Isbell duality]] $\to$ [[algebra]])

## Topology

* [[noncommutative topology]]

* [[C*-algebra]], [[Gelfand duality]]

## Smooth and Riemannian geometry

* [[dense subalgebra]]

* [[spectral triple]]

* [[2-spectral triple]]

## Algebraic geometry

* [[noncommutative algebraic geometry]]

## Homotopy theory

* [[noncommutative stable homotopy theory]]

  * [[model structure on operator algebras]]

  * [[homotopical structure on C*-algebras]]

* [[KK-theory]], [[E-theory]]

## Relation to physics

* [[deformation quantization]]

* [[algebra of observables]], [[local net of observables]]

* [[Bohr topos]]

[Edit this sidebar](/nlab/edit/noncommutative+geometry+-+contents)


This entry collects material related to the book

* [[Alain Connes]], [[Matilde Marcolli]],

  _Noncommutative Geometry, Quantum Fields and Motives_,

  Vol. 55, AMS Bookstore, 2008

  [pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf)

on [[noncommutative geometry]], [[quantum field theory]] and [[motives in physics]].

#Contents#
* table of contents
{:toc}

## Chapter 1. Quantum fields, noncommutative spaces, and motives

### 1. Introduction

### 2. Basics of perturbative QFT

* [[perturbative quantum field theory]]

### 3. Feynman diagrams

* [[Feynman diagram]]

#### 4. Dimensional regularization

* [[dimensional regularization]]

#### 5. The graph by graph method of Bogoliubov-Parasiuk-Hepp-Zimmermann

* [[BPHZ renormalization]]

#### 6. The Connes-Kreimer theory of perturbative renormalization

* [[Hopf algebra renormalization]]

#### 7. Renormalization and the Riemann-Hilbert correspondence

* [[Riemann-Hilbert correspondence]]

#### 8. Motives in a nutshell

* [[motive]], [[pure motive]]

#### 9. The Standard model of elementary particle physics

* [[standard model of particle physics]]

#### 10. The framework of (metric) noncommutative geometry

* [[noncommutative geometry]]

* [[spectral triple]], [[spectral geometry]]

#### 11. The spectral action principle

* [[spectral action]]

#### 12. Noncommutative geometry and the Standard Model

#### 13. The finite noncommutative geometry

#### 14. The product geometry

#### 15. Bosons as inner fluctuations

* [[Higgs boson]]

#### 16. The spectral action and the Standard Model Lagrangian

* [[spectral action]]

#### 17. The Standard Model Lagrangian from the spectral action

#### 18. Functional integral

#### 19. Dimensional regularization and noncommutative geometry

* [[dimensional regularization]]

## Chapter 2. The Riemann zeta function and noncommutative geometry

#### 1. Introduction

#### 2. Counting primes and zeta function

* [[prime number]]

* [[zeta function]]

* [[Riemann zeta function]]

#### 3. Classical and quantum mechancis of zeta

## Chapter 3. Quantum statistical mechanics and Galois symmetries

## Chapter 4. Endomotives, thermodynamics, and the Weil explicit formula

category: reference