Bénabou-Roubaud theorem

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## Idea

The theorem ([Bénabou-Roubaud 70](#BenabouRoubaud70)) identifies the [[Eilenberg-Moore category]] of algebras over the [[monad]] induced from a [[base change]] [[adjoint triple]] for some [[bifibration]] satisfying the [[Beck-Chevalley condition]] with the [[category of descent data]] along this morphism. This is the basis for the monadic reformulation of [[descent]] theory: [[monadic descent]].

## Context

A [[functor]] $P : F\to A$ is a [[Grothendieck opfibration]] if $P^{op}:F^{op}\to A^{op}$ is a [[Grothendieck fibration]], and a functor $P:F\to A$ is a __[[bifibration]]__ if $P$ is both a Grothendieck opfibration and fibration (no additional compatibility asked!). Thus we can talk about [[cartesian morphism|cartesian]] and [[cocartesian morphism|cocartesian]] arrows in $F$.

Given a bifibration  $P : F\to A$, automatically for any morphism $a:A_1\to A_0$ in $A$ the "inverse image" (or "pullback" or "restriction") functor $a^*:F(A_0)\to F(A_1)$ is [[right adjoint]] to the "pushforward" functor $a_!:F(A_1)\to F(A_0)$; with unit $\eta^a : Id_{F(A_1)} \to a^* a_!$ and counit $\epsilon^a : a_! a^* \to Id_{F(A_0)}$.

(Note that in [[topos theory]] and [[algebraic geometry]], functors $a^*$ called "[[inverse images]]" usually have *right* adjoints $a_*$.  This situation can be reconciled with the setup of bifibrations either by taking fiberwise opposites, so that left and right adjoints are switched, or by taking opposites of both the base and total categories, so that the direct and inverse images are switched.  However, there are also many bifibrations arising in other contexts in which $a^*$ has both a left adjoint $a_!$ *and* a right adjoint $a_*$, although the latter cannot then be described cleanly in fibrational terms.)

The [[adjunction]] $a_!\dashv a^*$ generates a [[monad]] $\mathbf{T}^a=(T^a,\mu^a,\eta^a)$ in the usual way: the functor is $T^a = a^* a_!\colon  F(A_1)\to F(A_1)$, the multiplication is $\mu^a = a^* \epsilon^a a_!\colon  T^a \circ T^a \to T^a$, and the unit is just the unit of the adjunction.  Denote by $F^a$ the [[Eilenberg–Moore category]] $F(A_1)^{\mathbf{T}^a}$ of modules (algebras) over the monad $\mathbf{T}^a$, with canonical forgetful functor $U^{\mathbf{T}} \colon  F^a \to F(A_1)$ and canonical [[comparison functor]] $\Phi^a \colon F(A_0) \to F^a$.

Now we assume that $A$ has [[pullbacks]], and that $P$ satisfies what is nowadays called the **[[Beck-Chevalley property]]**, namely that for each commutative square

$$
  \array{
    N_0 & \stackrel{\chi'}\leftarrow & N_1
     \\
     {}^{\mathllap{k_0}}\downarrow
     &&
     \downarrow^{\mathrlap{k_1}}
     \\
     M_0 &\stackrel{\chi}\leftarrow & M_1
  }
$$

in $F$ such that its image in $A$ is a [[pullback]] square, if $\chi$ and $\chi'$ are cartesian and $k_0$ is cocartesian then $k_1$ is cocartesian.

An equivalent way to state the condition is that for any pullback square

$$
  \array{
    x & \overset{a}{\to} & y
    \\
    {}^{\mathllap{c}} \downarrow
    &&
   \downarrow^{\mathrlap{d}}
   \\
   z & \underset{b}{\to} & w
  }
$$

in $A$, the canonical transformation $c_! a^* \to b^* d_!$ is an isomorphism.   In the Bénabou--Roubaud paper this is called the *Chevalley property* and said to make $P$ into a *Chevalley functor*.

Denote by $A_2 :=A_1\times_{A_0}A_1$ the pullback of $a$ along itself, with the canonical projections $a_1,a_2\colon A_2\to A_1$.  Now consider the lift of the cartesian square defining $A_2$ to $F$ in such a way that $a_1$ is lifted to a cartesian arrow, $a_2$ to a cocartesian arrow, and $a$ to a cocartesian arrow.  Then by the universality there is a lift of $a$ completing the square, and by the Beck--Chevalley property it is cartesian. Together with the isomorphism given by adjunction this gives a morphism

$$
K^a: Hom_{F(A_2)}(a_1^*(M_1),a_2^*(M_1))\to Hom_{F(A_1)}(T^a(M_1),M_1).
$$

One checks that an invertible morphism $\phi\colon a_1^*(M_1)\to a_2^*(M_1)$ satisfies the cocycle equation
(making it into a [[descent]] datum) iff $K^a(\phi)$ is an action of $\mathbf{T}^a$ on $M_1$, and similarly for the unitality axiom.

## The theorem

Denote by $Desc(a)$ the category of [[descent]] data for the fibration $P$ along the morphism $a$; it comes with canonical functors

$$
  \Psi^a\colon F(A_0)\to Desc(a)
$$

and

$$
  U^a\colon Desc(a)\to F(A_1)
  \,.
$$

The __Bénabou--Roubaud theorem__ asserts that this induces an [[equivalence of categories]] between $Desc(a)$ and $F^a$.

In addition, this equivalence satisfies some naturality properties, including that it commutes appropriately with the canonical functors to the fibers $F(A_0)$ and $F(A_1)$. Combining this theorem with Beck's [[monadicity theorem]], it becomes a practical tool for establishing a descent property in bifibrations, with variants in some other setups (to be covered later).

There are several characterizations of a [[Beck-Chevalley property]] for bifibrations:

+-- {: .un_prop}
###### Proposition
(Duško Pavlović, in Category theory Como 1990, LNM 1488, Springer 1991)

Let $p: F\to B$ be a bifibration, $Q = (f,g,s,t)$ a square in $B$ such that $f\circ g = s\circ t$, and $\Theta = (\phi, \gamma, \sigma, \theta)$ a square in $F$ such that $\phi\circ\gamma=\sigma\circ\theta$, with $p(\phi)=f$, $p(\gamma)=g$, $p(\sigma)=s$ and $p(\theta)=t$. The following conditions are equivalent:

a) if $\theta$ and $\phi$ are cartesian and if $\sigma$ is cocartesian *then* $\gamma$ must be cocartesian;

b) if $\sigma$ and $\gamma$ are cocartesian and if $\theta$ is cartesian *then* $\phi$ must be cartesian;

c) if $\theta$ is cartesian and if $\sigma$ is cocartesian *then* $\phi$ is cartesian iff $\gamma$ is cocartesian.

If some inverse image functors $f^*$ and $t^*$ and some direct image functors $g_!$ and $s_!$ are chosen,
then every square $\Theta$ over $Q$ satisfies conditions (a-c) iff there is a canonical natural isomorphism

d) $f^*\circ s_! \cong g_! \circ t^*$.
=--

## Related concepts

* [[Beck-Chevalley condition]]

* [[monadicity theorem]]

## References

The original article:

* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270**  (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]],   English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]

See also:

* {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], _Facets of Descent I_, Applied Categorical Structures 1994, Volume 2, Issue 3, pp 245-281

* {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], _Facets of Descent II_, Applied Categorical Structures September 1997, Volume 5, Issue 3, pp 229-248

* X. Guo, thesis, p. 58 of _Monadicity, purity and descent equivalence_, [pdf](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp02/NQ59136.pdf)

There has been some historical discussion on this in the category list; Zoran's response is [here](http://mathlight.wordpress.com/2010/01/07/becks-theorem-vs-benabou-roubaud).

The following reference presents a proof for a (generalization) of the Bénabou-Roubaud Theorem (pages 435 and 436):

* [[Fernando Lucatelli Nunes]], *Pseudo-Kan extensions and descent theory*, TAC [33-15] (http://www.tac.mta.ca/tac/volumes/33/15/33-15abs.html)

[[!redirects Benabou-Roubaud theorem]]
[[!redirects Benabou–Roubaud theorem]]
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[[!redirects Bénabou–Roubaud theorem]]
[[!redirects Bénabou--Roubaud theorem]]

[[!redirects Bénabou-Roubaud's theorem]]


Bézout domain

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## Definition

A **Bézout domain** is an [[integral domain]] that is also a [[Bézout ring]].

### In constructive mathematics

In constructive mathematics, there are different types of integral domains, yielding different types of Bézout domains: the gcd function and Bézout coefficient functions are no longer valued in $R \setminus \{0\}$ in one variable, but in $\{x \in R \vert x \neq 0\}$, $\{x \in R \vert x \# 0\}$, or some other definition, depending on what the base [[integral domain]] ends up being (classical, Heyting, discrete, residue, et cetera).

## Properties

Every [[GCD domain]] of dimension at most 1 is a Bézout domain.

### In classical mathematics

In [[classical mathematics]], every [[Bézout domain]] $R$ that satisfies one of the following conditions is a [[principal ideal domain]]

1. $R$ is a [[unique factorization domain]]

2. $R$ is a [[Noetherian ring]]

3. $R$ is an [[atomic domain]]

4. $R$ satisfies the ascending chain condition on principal ideals.

### In constructive mathematics

In [[constructive mathematics]], Bézout domains are usually better behaved because many important rings may fail to be [[principal ideal domains]].

For instance, the ring of integers is a principal ideal domain if and only if the [[excluded middle|law of excluded middle]] holds: In one direction, the usual proofs rely on being able to [[decidable proposition|decide]] whether any particular integer belongs to the ideal or not.

For the converse, let $\varphi$ be an arbitrary proposition. Consider the ideal $\{ x \in \mathbb{Z} | (x = 0) \vee \varphi \}$. By assumption, it is generated by some number $n$. Since the integers are [[decidable equality|discrete]], it holds that $n = 0$ or $n \neq 0$. In the first case $\neg\varphi$ holds, in the second $\varphi$.

However, this ideal cannot be proved to be finitely generated either.  If an ideal is generated by $n_1, \ldots, n_k$, then we may form their [[gcd]] one step at a time, which we can do algorithmically.  Therefore, $\mathbb{Z}$ remains a Bézout domain.

As a result, in constructive mathematics not every [[Bézout domain]] that is a [[unique factorization domain]] is a [[principal ideal domain]], as $\mathbb{Z}$ is both a Bézout domain and a unique factorization domain, but is not a principal ideal domain.

Some authors have tried to define a principal ideal domain as a [[Noetherian ring|Noetherian]] Bézout  domain, but it is unknown if this still coincides in constructive mathematics with the definition of principal ideal domain as a [[integral domain]] whose [[ideals]] are all [[principal ideals]].

## Examples

* Every [[field]] $K$ is a Bézout domain where for all elements $a \in K$ and $b \in K$, $\gamma(a, b) = a \cdot b$, $\beta_1(a, b) = 1$, and $\beta_2(a, b) = 1$

* The ring of integers $\mathbb{Z}$

* For any [[discrete field]] $K$, the [[polynomial ring]] $K[x]$ on one generator is a Bézout domain.

* The ring of [[entire function|entire]] [[holomorphic functions]] on the [[complex plane]]

### Non-examples

* For any Bézout [[unique factorization domain]] $R$, the polynomial ring $R[x]$ on one generator is a [[unique factorization domain]] which is not a Bézout domain. The ideal generated by $2$ and $x$ is not a [[principal ideal]], as $1$ is the greatest common divisor of $2$ and $x$, but $1$ is not in the ideal generated by $2$ and $x$.

* In particular, any integral domain extension $\mathbb{R}[x, y]$ of the [[rational numbers]] by a [[transcendental number]] $x \in \mathbb{R}$is not a Bézout domain.

* Similarly, for any [[field]] $K$, the polynomial ring $R[x, y]$ on two generators is a [[unique factorization domain]] which is not a Bézout domain. The ideal generated by $x$ and $y$ is not a [[principal ideal]], as $1$ is the greatest common divisor of $x$ and $y$, but $1$ is not in the ideal generated by $x$ and $y$.

* In particular, any integral domain extension $\mathbb{Q}[x, y]$ of the [[rational numbers]] by two [[transcendental numbers]] $x \in \mathbb{R}$ and $y \in \mathbb{R}$ which are [[linearly independent]] as [[basis]] [[rational vector space|$\mathbb{Q}$-vectors]] is not a Bézout domain.

## See also

* [[Bézout ring]]

* [[GCD domain]]

* [[Euclidean domain]]

* [[principal ideal domain]]

* [[function field analogy]]

## References

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

See also:

* Wikipedia, _[Bézout domain](http://en.wikipedia.org/wiki/Bézout_domain)

[[!redirects Bézout domains]]

[[!redirects Bezout domain]]
[[!redirects Bezout domains]]


Bézout monoid

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## Definition

A [[monoid]] (or [[semigroup]]) $M$ is a __left Bézout monoid__ if all [[finitely generated]] left [[ideal of a monoid|ideals]] of $M$ are left [[principal ideal of a monoid|principal ideals]] in $M$. Similarily, $M$ is a __right Bézout monoid__ if all [[finitely generated]] right [[ideal of a monoid|ideals]] of $M$ are right [[principal ideal of a monoid|principal ideals]]. Finally, $M$ is a __Bézout monoid__ if it is both a left Bézout monoid and a right Bézout monoid.

## Related concepts

* [[ideal of a monoid]]

* [[principal ideal of a monoid]]

* [[principal ideal monoid]]


Bézout ring

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## Definition ##

There are multiple definitions of a Bézout ring:

### In commutative rings ###

\begin{definition}
A [[commutative ring]] $R$ is a **Bézout ring** if for every element $a \in R$ and $b \in R$, there exists elements $x \in R$, $y \in R$ called **Bézout coefficients** and $g \in R$ called a **common divisor**, such that $a \cdot x + b \cdot y = g$, $g \vert a$ and $g \vert b$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **Bézout ring** if it has functions $\beta_0:R \times R \to R$, $\beta_1:R \times R \to R$, $\gamma:R \times R \to R$ such that for every element $a \in R$ and $b \in R$, $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gamma(a, b)$, $\gamma(a, b) \vert a$ and $\gamma(a, b) \vert b$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **Bézout ring** if it has functions $\beta_0:R \times R \to R$, $\beta_1:R \times R \to R$, $\gamma:R \times R \to R$, $q_0:R \times R \to R$, $q_1:R \times R \to R$ such that for every element $a \in R$ and $b \in R$, $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gamma(a, b)$, $q_0(a, b) \cdot \gamma(a, b) = a$ and $q_1(a, b) \cdot \gamma(a, b) = b$.
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **Bézout ring** if every ideal with 2 generators is a [[principal ideal]]:
\end{definition}

\begin{definition}
A [[commutative ring]] $R$ is a **Bézout ring** if every finitely generated ideal is a [[principal ideal]].
\end{definition}

If the commutative ring is a [[GCD ring]] and the common divisor is the [[greatest common divisor]], then the Bézout ring condition $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gcd(a, b)$ is called the **Bézout identity**. There might be multiple Bézout coefficient functions for each Bézout ring.

The third definition implies that Bézout rings are [[algebraic theory|algebraic]].

## See also ##

* [[commutative ring]]

* [[GCD ring]]

* [[Bézout domain]]

* [[Euclidean ring]]

## References

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

* Wikipedia, _[Bézout's identity](http://en.wikipedia.org/wiki/Bézout's_identity)

[[!redirects Bézout rings]]

[[!redirects Bezout ring]]
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Bézout's theorem

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## Idea

Bézout's theorem states sufficient conditions (transversality) such that the [[intersection product]] of two [[algebraic curves]] inside an ambient [[algebraic surface]] gives literally the number of points at which the curves intersect.

This is the origin of _[[intersection theory]]_, see there for more.

The refinement of Bézout's theorem for non-transversally intersecting varieties is _[[Serre's intersection formula]]_

## References

* Wikipedia, _[Bézout's_theorem](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem)_

[[!redirects Bézout theorem]]

[[!redirects Bezout's theorem]]
[[!redirects Bezout theorem]]


C(n)-extendible cardinal

# $C(n)$-extendible cardinals
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## Idea

A $C(n)$-extendible cardinal is a type of [[large cardinal]] which generalizes [[extendible cardinals]].  They come up in formulating refined versions of [[Vopěnka's principle]].

## Definition

We work in [[ZFC]].  Let $C(n)$ denote the class of infinite [[cardinal numbers]] $\kappa$ such that $V_\kappa$ (see [[von Neumann hierarchy]]) is a $\Sigma_n$-elementary submodel of the universe $V$, i.e. such that the embedding preserves and reflects the truth of $\Sigma_n$-[[Lévy hierarchy|formulas]].  In particular:

* $\kappa$ is $C(0)$ if and only if it is infinite, and

* $\kappa$ is $C(1)$ if and only if it is uncountable and $V_\kappa$ coincides with the *hereditarily $\kappa$-sized sets* $H(\kappa)$, i.e. those whose [[transitive closure]] has cardinality $\lt\kappa$.

+-- {: .un_defn}
###### Definition
If $\kappa\lt\lambda$ and both are in $C(n)$, we say $\kappa$ is **$\lambda$-$C(n)$-extendible** if there is an [[elementary embedding]] $j:V_\lambda \to V_\mu$ with critical point $\kappa$, such that $\mu\in C(n)$, $j(\kappa)\gt \lambda$, and $j(\kappa)\in C(n)$.

We say $\kappa\in C(n)$ is **$C(n)$-extendible** if it is $\lambda$-$C(n)$-extendible for all $\lambda\in C(n)$ with $\lambda\gt\kappa$.
=--

In particular:

* $\kappa$ is $C(0)$-extendible if and only if it is [[extendible cardinal|extendible]].

## References

* Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, [[Jiri Rosicky]] _Definable orthogonality classes in accessible categories are small_, [arXiv](http://arxiv.org/abs/1101.2792)
{#BagariaCasacubertaMathiasRosicky}

[[!redirects C(n)-extendible cardinals]]


C* algebraic deformation quantization

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## Idea
 {#Idea}

What is called the *strict* or *$C^\ast$-algebraic* form of  [[deformation quantization]] (sometimes just: *strict quantization*) is an attempt to formalize [[quantization]] of [[phase spaces]] or more generally of [[Poisson manifolds]] by _[[continuous field of C*-algebras|continuously deforming]]_, in a precise sense, their [[commutative algebra|commutative]] [[algebras of functions]] ([[algebras of observables]]) to non-commutative [[C*-algebras]] whose [[commutators]] are, to "first order" in a suitable sense, determined by the given [[Poisson bracket]].

This is in contrast to _[[formal deformation quantization]]_, where one asks not for [[C*-algebras]] but just for [[formal power series algebras]]. Where formal deformation quantization is _[[perturbative quantum field theory|perturbative]]_ quantization ([[perturbation]] in [[Planck's constant]], see [Collini (2016)](perturbative+algebraic+quantum+field+theory#Collini16)), strict deformation quantization is meant to reflect [[non-perturbative quantum field theory|non-perturbative]] quantization.

In general, by *[[deformation quantization]]* one means notions of [[quantization]] of [[Poisson manifolds]] $\big(X, \{-,-\}\big)$ in terms of [[sequences]] of [[noncommutative algebra|non-commutative]] [[algebras]] $A_{\hbar}$ parameterized by specific admissible (formal) values of [[Planck's constant]] $\hbar$ including $\hbar = 0$, where $A_0 = C^\infty(X)$ is the ordinary commutative [[algebra of functions]] on the [[underlying]] [[smooth manifold]]. The idea is that $A_{\hbar}$ is the [[algebra of observables]] of the corresponding [[quantum system]] at that value of $\hbar$, arising from "deforming" the commutative product of $A_0$ in a way that increases with $\hbar$ and is [[infinitesimal|infinitesimally]] controlled by the given [[Poisson bracket]] $\{-,-\}$.

Beware that the term "[[deformation quantization]]" is often taken by default to refer to the historically first notion of [[formal deformation quantization]], where $\hbar$ is just a formal [[variable]] (i.e. an [[infinitesimal]], but not an actual [[number]]) and the [[underlying]] [[vector space]] of all algebras in question is that of [[formal power series]] in $\hbar$.

One might naively imagine that the [[formal power series]] appearing in [[formal deformation quantization]] have a finite [[radius of convergence]] $\epsilon \in \mathbb{R}_+$ thus yielding actual (non-formal) deformation quantizations for $\hbar \lt \epsilon$, but in practice this happens rarely (see the first references [below](#ConvergingFormalDeformationQuantization)). Indeed, [[geometric quantization]] makes manifest that [[prequantization]] conditions typically force admissible values of [[Planck's constant|$\hbar$]] to form a [[discrete space|discrete]] [[subspace]] of $\mathbb{R}_+$ with only an [[accumulation point]] at $\hbar = 0$.

Therefore, in *strict* or $C^\ast$-algebraic deformation quantization the parameter [[Planck's constant|$\hbar$]] is typically allowed to take discrete [[positive number|positive]] [[real numbers|real values]] with an [[accumulation point]] at $\hbar = 0$, and where to each such value is associated an actual [[C*-algebra]]-[[algebra of observables|of observables]]. There are a variety of similar but different proposals for what exactly this should mean in detail, see [Hawkins (2008a), Section 2](#Hawkins08a) for overview and references.

In its focus on [[algebras of observables]] the notion of deformation quantization is roughly dual to *[[geometric quantization]]*, which primarily constructs the [[spaces of quantum states]]. In special sitations both notions are compatible, but in general there is a large amount of ambiguity in [[quantization]], between but also within the different approaches.

Typically the $C^\ast$-algebraic deformation takes the quantum algebra to be a suitable [[convolution algebra]] of suitably [[polarization|polarized]] sections over a [[Lie groupoid]] that [[Lie integration|Lie integrates]] a [[Poisson Lie algebroid]] which encodes the original [[Poisson bracket]] to be quantized [[Hawkins (2008b)](#Hawkins08b)], see at _[[geometric quantization of symplectic groupoids]]_.

While there are good examples of strict $C^\ast$-algebraic deformation quantization for toy examples such as low spacetime dimension (notably [[quantum mechanics]]) to date no examples of interacting field theories in spacetime dimension $\geq 4$ have a known non-perturbative quantization. (For the case of [[Yang-Mills theory]]/[[QCD]] the construction of its non-perturbative quantization is one of the open "Millennium Problems" listed by the Clay Mathematics Institute, see at _[[quantization of Yang-Mills theory]]_.)

## Properties

### Relation to formal deformation quantization

Under favorable circumstances, one can form from a strict $C^\ast$-algebraic deformation
quantization given by a [[continuous field of C*-algebras]] over a subset of the interval the "differentiation"
as $\hbar \in [0,1]$ tends to 0, such that this reproduces a [[formal deformation quantization]].

Conversely, a natural intuition might be that given a [[formal deformation quantization]] then the subalgebra of [[convergence|converging]] [[power series]] inside all [[formal power series]] has a completion to a [[C*-algebra]] which constitutes a strict deformation quantization.

While this seems natural, the only actual example where this is understood to date
seems to be the simple case of the standard Poisson structure on $\mathbb{R}^{2n}$
with its [[Weyl algebra]] [[star product]]. (See [this MO discussion](http://mathoverflow.net/a/141410/381)).

## Examples

* [[non-perturbative quantum field theory]], [[non-perturbative effect]]

* [[deformation quantization of the 2-sphere]]

## Related concepts

* [[geometric quantization of symplectic groupoids]]

* [[non-perturbative quantum field theory]]

[[!include infinitesimal and local - table]]

## References

### Converging formal deformation quantization
 {#ConvergingFormalDeformationQuantization}

On [[convergence]] of [[formal power series]] in [[formal deformation quantization]]:

* M. Bordemann, M. Brischle, C. Emmrich, [[Stefan Waldmann]], _Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for $\mathbb{C}P^n$_ ([arXiv:q-alg/9512019](http://arxiv.org/abs/q-alg/9512019))

* H. Omori, Y. Maeda, N. Miyazaki, A. Yoshida, _Deformation quantization of Fréchet-Poisson algebras of Heisenberg type_ 2001 ([pdf](http://www.math.keio.ac.jp/library/research/report/2001/01002.pdf))

Textbook account:

* [[Stefan Waldmann]], _Poisson-Geometrie und Deformationsquantisierung_, Springer (2007) [[doi:10.1007/978-3-540-72518-3](https://doi.org/10.1007/978-3-540-72518-3)]

### Strict deformation quantization proper

The notion of strict $C^\ast$-algebraic deformation quantization was introduced in:

* {#Rieffel89} [[Marc A. Rieffel]], *Deformation quantization of Heisenberg manifolds*, Communications in Mathematical Physics **122** (1989) 531–562 [[doi:10.1007/BF01256492](https://doi.org/10.1007/BF01256492)]

  > (motivated by [[noncommutative tori]])

* {#Rieffel90} [[Marc Rieffel]], _Deformation quantization and operator algebras_, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411-423, Proc. Sympos. Pure Math. __51__, Part 1, Amer. Math. Soc. (1990) [[pdf](http://math.berkeley.edu/~rieffel/papers/deformation.pdf), [[Rieffel-DefQuantization.pdf:file]] [MR91h:46120](http://www.ams.org/mathscinet-getitem?mr=1077400)]

* [[Marc Rieffel]], _Deformation quantization for actions of $\mathbb{R}^d$_, Mem. Amer. Math. Soc. __106__ 506 (1993) [[ams:memo-106-506](https://bookstore.ams.org/memo-106-506), [MR94d:46072](http://www.ams.org/mathscinet-getitem?mr=1184061)]

* [[Marc Rieffel]], _Quantization and $C^\ast$-algebras_, Contemporary Mathematics **167** (1994) [[pdf](http://math.berkeley.edu/~rieffel/papers/quantization.pdf), [[Rieffel-CStarQuantization.pdf:file]]]

See also:

* Andrzej Sitarz, *Rieffel's deformation quantization and isospectral deformations*, International Journal of Theoretical Physics **40** (2001) 1693 [[arXiv:math/0102075](https://arxiv.org/abs/math/0102075), [doi:10.1023/A:1011956229254](https://doi.org/10.1023/A:1011956229254)]

Specifically on [[Lie group|Lie]]-[[group algebras]] as strict deformation quantization of [[Lie-Poisson structures]]:

* [[Marc A. Rieffel]], *Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures*, American Journal of Mathematics **112** 4 (1990) 657-685 [[doi:10.2307/2374874](https://doi.org/10.2307/2374874), [jstor:2374874](https://www.jstor.org/stable/2374874)]

and generalization to [[groupoid algebras]] of [[Lie groupoids]] [[Lie integration|integrating]] given [[Lie algebroids]]:

* [[Nicolaas P. Landsman]], Ex. 2 in: *Lie Groupoid $C^\ast$-Algebras and Weyl Quantization*, Communications in Mathematical Physics **206** (1999) 367–381 [[doi:10.1007/s002200050709](https://doi.org/10.1007/s002200050709)]

* [[Nicolaas P. Landsman]], B. Ramazan, Ex. 11.1 in: *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) 159-192 [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]

see also:

* [[Klaas Landsman]], *Strict deformation quantization of a particle in external gravitational and Yang-Mills fields*, Journal of Geometry and Physics __12__ 2 (1993) 93-132 [doi:10.1016/0393-0440(93)90010-C]

Review:

* [[Marc Rieffel]], _Questions on quantization_, Contemp. Math. **228** (1998) 315-326 [[arXiv:quant-ph/9712009](http://arxiv.org/abs/quant-ph/9712009)]

* [[Nicolaas P. Landsman]], *Classical and quantum representation theory* [[arXiv:hep-th/9411172](https://arxiv.org/abs/hep-th/9411172)]

Comparative review of notions of strict deformation quantization:

* {#Hawkins08a} [[Eli Hawkins]], Section 2 of: *An Obstruction to Quantization of the Sphere*, Communications in Mathematical Physics **283** (2008) 675–699  [[arXiv:0706.2946](http://arxiv.org/abs/0706.2946), [doi:10.1007/s00220-008-0517-2](https://doi.org/10.1007/s00220-008-0517-2)]

Discussion of strict deformation quantization in terms of [[geometric quantization of symplectic groupoids]] via [[polarization|polarized]] twisted [[groupoid convolution algebras]] is in

* {#Hawkins08b} [[Eli Hawkins]], *A Groupoid Approach to Quantization*, J. Symplectic Geom. **6** 1  (2008) 61-125 [[arXiv:math/0612363](https://arxiv.org/abs/math/0612363), [euclid](https://projecteuclid.org/journals/journal-of-symplectic-geometry/volume-6/issue-1/A-groupoid-approach-to-quantization/jsg/1215032733.full)]

For the special case of [[Moyal deformation quantization]] [[Hawkins (2008b), section 6.2](#Hawkins08b)] this construction had been suggested without proof in

* [[Alan Weinstein]], in P. Donato et al. (eds.) _Symplectic Geometry and Mathematical Physics,  (Birkhäuser, Basel, 1991); p. 446.

and a detailed proof was given in

* {#GraciaBondiaVarilly94} [[José Gracia-Bondia]], [[Joseph Varilly]], _From geometric quantization to Moyal quantization_, J.Math.Phys. 36 (1995) 2691-2701 ([arXiv:hep-th/9406170](https://arxiv.org/abs/hep-th/9406170))

see also

* {#BordemannMeinrenkenSchlichenmaier93} Martin Bordemann, [[Eckhard Meinrenken]], Martin Schlichenmaier, _Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$_,  	Commun.Math.Phys. 165 (1994) 281-296 ([arXiv:hep-th/9309134](http://arxiv.org/abs/hep-th/9309134))

On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

[[!redirects C-star algebraic deformation quantizations]]

[[!redirects strict deformation quantization]]
[[!redirects strict deformation quantizations]]

[[!redirects C-star algebraic deformation quantization]]
[[!redirects C*-algebraic deformation quantization]]

[[!redirects strict C*-algebraic deformation quantization]]

[[!redirects strict algebraic deformation quantization]]
[[!redirects strict algebraic deformation quantizations]]

[[!redirects continuous deformation quantization]]
[[!redirects continuous deformation quantizations]]

[[!redirects Rieffel quantization]]
[[!redirects Rieffel quantizations]]


C*Alg

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#### Operator algebra
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#Contents#
* table of contents
{:toc}

## Definition

The [[category]] whose [[objects]] are [[C-star algebras]] and whose [[morphisms]] are $C^\ast$-[[homomorphisms]].

## Properties

The universal additive split-exact and homotopy-invariant [[localization]] of $C^\ast$ at the $C^\ast$-algebra of [[compact operators]] is [[KK-theory]]. See there for more.

## Related concepts

* [[Gelfand duality]]

* [[noncommutative topology]]

* [[homotopical structure on C*-algebras]], [[model structure on operator algebras]]

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

* [[AQFT]]

category: category


C-field gauge algebra -- references

#### Supergravity C-Field gauge algebra
 {#ReferencesCFieldGaugeAlgebra}

Identifying the super-graded gauge algebra of the [[supergravity C-field|C-field]] in [[D=11 supergravity]] (with non-trivial [[super Lie algebra|super Lie bracket]] $[v_3, v_3] = -v_6$):

* {#CremmerJuliaLuPope} [[Eugene Cremmer]], [[Bernard Julia]], H. Lu, [[Christopher Pope]], Equation (2.6) of *Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities*, Nucl.Phys. B **535** (1998) 242-292 \[doi:10.1016/S0550-3213(98)00552-5, [arXiv:hep-th/9806106](https://arxiv.org/abs/hep-th/9806106)\]

* I.V. Lavrinenko, H. Lu, [[Christopher N. Pope]], [[Kellogg S. Stelle]], (3.4) in: *Superdualities, Brane Tensions and Massive IIA/IIB Duality*, Nucl. Phys. B **555** (1999) 201-227 [[arXiv:hep-th/9903057](https://arxiv.org/abs/hep-th/9903057), doi:10.1016/S0550-3213(99)00307-7]

* [[Jussi Kalkkinen]], [[Kellogg S. Stelle]], (75) of: *Large Gauge Transformations in M-theory*, J. Geom. Phys. **48** (2003) 100-132 [[arXiv:hep-th/0212081](https://arxiv.org/abs/hep-th/0212081), doi:10.1016/S0393-0440(03)00027-5]

* [[Igor A. Bandos]], [[Alexei J. Nurmagambetov]], [[Dmitri P. Sorokin]], (86) in: *Various Faces of Type IIA Supergravity*, Nucl.Phys. B **676** (2004) 189-228 [[arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153), [doi:10.1016/j.nuclphysb.2003.10.036](https://doi.org/10.1016/j.nuclphysb.2003.10.036)]

Identification as an [[L-infinity algebra|$L_\infty$-algebra]] (a [[dg-Lie algebra]], in this case):

* [[Hisham Sati]], (4.9) in: *Geometric and topological structures related to M-branes*, in *Superstrings, Geometry, Topology, and $C^\ast$-algebras*, Proc. Symp. Pure Math. **81** (2010) 181-236 [[arXiv:1001.5020](http://arXiv.org/abs/1001.5020), [ams:pspum/081](http://www.ams.org/books/pspum/081)]

and identificatoin with the rational [[Whitehead L-infinity algebra|Whitehead $L_\infty$-algebra]] (the [[rational homotopy theory|rational Quillen model]] ) of the [[4-sphere]] (cf. *[[schreiber:Hypothesis H]]*):

* [[Hisham Sati]], [[Alexander Voronov]], (13) in: *Mysterious Triality and M-Theory* [[arXiv:2212.13968](https://arxiv.org/abs/2212.13968)]

* [[Hisham Sati]], [[Urs Schreiber]], (22) in: *[[schreiber:Flux Quantization on Phase Space]]* [[arXiv:2312.12517](https://arxiv.org/abs/2312.12517)]


C-field tadpole cancellation

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#### String theory
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[[!include string theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[M-theory]] [[KK-compactification|compactified]] on 8-dimensional [[compact topological space|compact]] [[fibers]] $X^{(8)}$ (see _[[M-theory on 8-manifolds]]_) [[tadpole cancellation]] for the [[supergravity C-field]] has been argued ([Sethi-Vafa-Witten 96](#SethiVafaWitten96), [Becker-Becker 96](#BeckerBecker96), [Dasgupta-Mukhi 97](#DasguptaMukhi97)) to be the condition

$$
  N_{M2}
  \;+\;
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;=\;
  \underset{
    I_8(X^8)
  }{
    \underbrace{
      \tfrac{1}{48}\big( p_2 - (\tfrac{1}{2}p_1)^2  \big)[X^{8}]
    }
  }
  \;\;\;\;
  \overset{!}{\in}
  \mathbb{Z}
  \,,
$$

where

1. $N_{M2}$ is the net number of [[M2-branes]] in the spacetime (whose [[worldvolume]] appears as points in $X^{(8)}$);

1. $G_4$ is the [[field strength]]/flux of the [[supergravity C-field]]

1. $p_1$ is the [[first Pontryagin class]] and $p_2$ the [[second Pontryagin class]] combining to [[I8]], all regarded here in [[rational homotopy theory]].

If $X^{8}$ has

* [[Spin(7)-structure]] (hence in particular if it is a [[Calabi-Yau manifold]], which has $SU(4) = $ [[Spin(6)]]-structure)

or

* [[Sp(2).Sp(1)-structure]]

then

$$
  \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2  \big)
  \;=\;
  \chi
$$

is the [[Euler class]] (see [this Prop.](Spin7-manifold#CharacteristicClassesForSpinStructure) and [this Prop.](quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure), respectively), hence in these cases the condition is equivalently

\[
  \label{DasguptaMukhiConditionInTermsOfEuler}
  N_{M2}
  \;=\;
  -
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;+\;
  \tfrac{1}{24}\chi[X^8]
  \;\;\;\;
  \in
  \mathbb{Z}
  \,,
\]

where $\chi[X]$ is the [[Euler characteristic]] of $X$.

## Properties

### Integrality on $K3 \times K3$
 {#IntegralityOnK3TimesK3}

That (eq:DasguptaMukhiConditionInTermsOfEuler) should be an [[integer]] is a highly non-trivial condition on the manifold $X^8$.

One case where this is satisfied is for $X^8$ being the [[product space]] of [[K3]] with itself.

To see this, one needs the [[shifted C-field flux quantization]]-condition

\[
  \label{ShiftedCFieldFluxQuantization}
  [\tilde G_4]
  \;\in\;
  H^4(X^8, \mathbb{Z})
  \to
  H^4(X^8, \mathbb{R})
\]

for

\[
  \label{ShiftedCFieldFLux}
  \tilde G_4
  \;\coloneqq\;
  G_4 + \tfrac{1}{4}p_1(\nabla_{T X^8})
\]

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

With the [[shifted C-field flux quantization]] (eq:ShiftedCFieldFLux), the Dasgupta-Mukhi-expression (eq:DasguptaMukhiConditionInTermsOfEuler) for the number of [[M2-branes]] is indeed an [[integer]] on the [[product space]] $X^8 = K3 \times K3$ of [[K3]] with itself:

\[
  \tfrac{1}{24}\chi[K3 \times K3]
  -
  \tfrac{1}{2} \big( G_4[K3 \times K3]\big)^2
  \;\in\;
  \mathbb{Z}
  \,,
\]

=--

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

After replacing $G_4$ by $\widetilde G_4$ (eq:ShiftedCFieldFLux) the expression becomes

\[
  \label{ExpressingDasgupteMukhInTildeG4}
  \begin{aligned}
    &
    \tfrac{1}{24}\chi
    -
    \tfrac{1}{2} \big( G_4\big)^2
    \\
    & =
    \tfrac{1}{24}\chi
    -
    \tfrac{1}{2}\big( \tfrac{1}{4} p_1 \big)^2
    -
    \underset{
      \in \mathbb{Z}
    }{
    \underbrace{
      \tfrac{1}{2}
      \widetilde G_4 \cdot \big( \widetilde G_4 - \tfrac{1}{2}p_1\big)
    }
    }
    \,,
  \end{aligned}
\]

where the summand over the brace is an integral class (by [this Corollary](Wu+class#DivisibilityOfCupSquare)), because $K3$ is a [[spin manifold]] so that $\tfrac{1}{2}p_1$ is the [[Wu class]] $\nu_4$ (by [this Prop.](Wu+class#InTermsOfPontryagin)).

Hence it is now sufficient to show that the first two summands on the right of (eq:ExpressingDasgupteMukhInTildeG4) are both [[integers]], when evaluated on $K3 \times K3$.

But we have

1. $\chi[K3\times K3] \;=\; 24^2$ (by [this Prop.](K3+surface#EulerCharacteristicOfK3TimesK3))

1. $ p_1[K3 \times K3] \;=\; - 2 \times 48$ (by [this Prop.](K3+surface#FirstPontryaginClassOfK3TimesK3))

Hence

$$
  \begin{aligned}
    \tfrac{1}{24}\chi[K3 \times K3]
    & = \tfrac{1}{24} 24^2
    \\
    & = 24 \;\in\; \mathbb{Z}
  \end{aligned}
$$

and

$$
  \begin{aligned}
    \tfrac{1}{32} (p_1[K3 \times K3])^2
    & =
    \tfrac{1}{32} \times
    (
      \underset{
        \mathclap{
          = 3 \times 32
        }
      }
      {
        \underbrace{ 2 \times 48}
      }
    )^2
    \\
    & =
    9 \times 32
    \;\in\;
    \mathbb{Z}
  \end{aligned}
$$

=--

## Related concepts

* [[RR-field tadpole cancellation]] in [[type II string theory]]

* [[shifted C-field flux quantization]] in [[M-theory]]

* [[24 branes transverse to K3]] in [[F-theory]] and [[HET-theory]]

## References
 {#References}

The C-field tadpole cancellation condition is claimed in

* {#SethiVafaWitten96} [[Savdeep Sethi]], [[Cumrun Vafa]], [[Edward Witten]], p. 2 of: _Constraints on Low-Dimensional String Compactifications_, Nucl. Phys. B **480** (1996) 213-224 [[arXiv:hep-th/9606122](https://arxiv.org/abs/hep-th/9606122), doi:10.1016/S0550-3213(96)00483-X]

referring for proof to the computation in

* [[Cumrun Vafa]], [[Edward Witten]], p. 11 (12 of 14) of: _A One-Loop Test Of String Duality_, Nucl. Phys. B **480** (1996) 213-224 [[arxiv:hep-th/9505053](https://arxiv.org/abs/hep-th/9505053), doi:10.1016/0550-3213(95)00280-6]

A comment is also in

* [[Edward Witten]], p. 3 of _On Flux Quantization In M-Theory And The Effective Action_, J. Geom. Phys. **22**  (1997) 1-13 [[arXiv:hep-th/9609122](https://arxiv.org/abs/hep-th/9609122), doi:10.1016/S0393-0440(96)00042-3]

Another condition appears in

* {#BeckerBecker96} [[Katrin Becker]], [[Melanie Becker]], around (2.58) in: *M-Theory on Eight-Manifolds*, Nucl. Phys. B **477** (1996) 155-167 [[arXiv:hep-th/9605053](https://arxiv.org/abs/hep-th/9605053), doi:10.1016/0550-3213(96)00367-7]

The formulas of [Sethi-Vafa-Witten 96](#SethiVafaWitten96) and [Becker-Becker 96](#BeckerBecker96) have been plugged together in

* {#DasguptaMukhi97} Keshav Dasgupta, [[Sunil Mukhi]], equation (1) in: *A Note on Low-Dimensional String Compactifications*, Phys. Lett. B **398** (1997) 285-290 [[arXiv:hep-th/9612188](https://arxiv.org/abs/hep-th/9612188), doi:10.1016/S0370-2693(97)00216-5]

Further discussion:

* [[Sergei Gukov]], [[Cumrun Vafa]], [[Edward Witten]], _CFT's From Calabi-Yau Four-folds_, Nucl. Phys. B584:69-108, 2000; Erratum-ibid.B608:477-478, 2001 ([arXiv:hep-th/9906070](https://arxiv.org/abs/hep-th/9906070))

* Keshav Dasgupta, Govindan Rajesh, [[Savdeep Sethi]], _M Theory, Orientifolds and G-Flux_, JHEP 9908 (1999) 023 ([arXiv:hep-th/9908088](https://arxiv.org/abs/hep-th/9908088))

Lecture notes:

* [[Timo Weigand]], p. 86 of _TASI Lectures on F-theory_ ([arXiv:1806.01854](https://arxiv.org/abs/1806.01854))

Application in [[dualities in string theory]]:

* {#Sethi07} [[Savdeep Sethi]], around (2) in _A Note on Heterotic Dualities via M-theory_, Phys. Lett. B659:385-387, 2008 ([arXiv:0707.0295](https://arxiv.org/abs/0707.0295))

Application in [[string phenomenology]]

* [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, equation (9) in _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

Discussion via [[Hypothesis H]]:

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_,
Comm. Math. Phys. **377**  (2020) 1961-2025 ([doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2), [arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

surveyed in:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!redirects C-field tadpole cancellation condition]]
[[!redirects tadpole cancellation condition for the supergravity C-field]]


C-infinity topology

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=--
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#### Mapping space
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[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Whitney $C^k$ topology_ is a [[topology]] on the set $C^\infty(X,Y)$ of [[smooth function]]s between two [[smooth manifold]]s, induced from the natural topology of the order-$k$ [[jet spaces]] of maps from $X$ to $Y$.

Often, and espcially for $k = \infty$ this is just called the _$C^k$-topology_ .

## Related concepts

* [[topology of mapping spaces]]

* [[differential topology of mapping spaces]]

* [[compact-open topology]]

## References

* Kriegl, Michor, ...

(...)

[[!redirects C-k topology]]
[[!redirects C-k topologies]]

[[!redirects C-infinity topologies]]

[[!redirects Whitney topology]]
[[!redirects Whitney topologies]]

[[!redirects Whitney C-k topology]]
[[!redirects Whitney C-k topologies]]

[[!redirects Whitney C-infinity topology]]
[[!redirects Whitney C-infinity topologies]]


c-principle

## Idea

A variant of [[h-principle]] with [[cobordisms]] thrown in.

## Related concepts

* [[h-principle]]

* [[Oka principle]]

## References

* [[Michael Freedman]], _Controlled Mather-Thurston theorems_, [arXiv:2006.00374](https://arxiv.org/abs/2006.00374).


c-Reedy category

# c-Reedy categories

* contents
{: toc}

## Idea

A *c-Reedy category* is a generalization of a [[Reedy category]], even more general than a [[generalized Reedy category]], in which the level morphisms need not even be invertible.

## Definition

A **c-Reedy category** is a category $C$ equipped with an ordinal-valued degree function on its objects, and subcategories $\overset{\leftrightarrow}{C}$, $\overset{\to}{C}$, and $\overset{\leftarrow}{C}$ containing all the objects, such that

* $\overset{\leftrightarrow}{C} \subseteq \overset{\to}{C}\cap \overset{\leftarrow}{C}$.
* Every morphism in $\overset{\leftrightarrow}{C}$ is level (i.e. its domain and codomain have the same degree).
* Every morphism in $\overset{\to}{C}\setminus\overset{\leftrightarrow}{C}$ strictly raises degree, and every morphism in $\overset{\leftarrow}{C}\setminus\overset{\leftrightarrow}{C}$ strictly lowers degree.
* Every morphism $f$ factors as $\overset{\to}{f} \overset{\leftarrow}{f}$, where $\overset{\to}{f} \in \overset{\to}{C}$ and $\overset{\leftarrow}{f}\in\overset{\leftarrow}{C}$, and the category of such factorizations with connecting maps in $\overset{\leftrightarrow}{C}$ is connected.
* For any $x$ and any degree $\delta\lt\deg(x)$, the functor $\overset{\leftarrow}{C}(x,-):\overset{\leftrightarrow}{C}_{=\delta} \to \Set$ is a coproduct of retracts of representables, where $\overset{\leftrightarrow}{C}_{=\delta}$ denotes the full subcategory of $\overset{\leftrightarrow}{C}$ on objects of degree $\delta$.

## References

* [[Mike Shulman]], *Reedy categories and their generalizations*, [blog](https://golem.ph.utexas.edu/category/2015/06/what_is_a_reedy_category.html), [arXiv](http://arxiv.org/abs/1507.01065)

[[!redirects c-Reedy categories]]


C-star dynamical system

[[!redirects C-star-system]]
[[!redirects C-star-system]]

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=--
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=--

# Contents
* table of contents
{:toc}

## Idea

A __$C^*$-dynamical system__, or only $C^*$-system
is a [[C-star-algebra]] continuously [[action|acted]] upon by
a [[group]] of $*$-[[automorphisms]]. In [[quantum mechanics]] as well as in [[AQFT]] the [[observables]] of the theory are [[self-adjoint operator]]s of (a [[local net]] of) [[C-star-algebra]]s, in this context the [[global gauge group]] of the theory is the maximal group of [[unitary operators]] that leave all observables invariant, the algebra and the gauge group form a $C^*$-system.

## Definition

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

A **$C^*$-system** $(\mathcal{A}, \alpha_G)$ consists of a $C^*$-algebra $\mathcal{A}$, a [[locally compact space|locally compact]] group $G$ and a [[continuous map|continuous]] [[homomorphism]] $\alpha$ of $G$ into the [[group]] $aut(\mathcal{A})$  of $*$-[[automorphisms]] of $\mathcal{A}$ equipped with the [[topology]] of pointwise convergence.

=--

If the algebra is a $*$-[[star-algebra|algebra]] only, then some authors call it a __$*$-system__.

Sometimes the continuity condition is dropped entirely or replaced by some weaker assumption, therefore one should always check what -- if any -- continuity assumption an author makes.

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

The **fixed point algebra** of a $C^*$-system  $(\mathcal{A}, \alpha_G)$ is $\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G  \}$. If the fixed point algebra is trivial then $\alpha_G$ **acts ergodically**.

=--

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

A [[state]] $\rho$ of the algebra $\mathcal{A}$ is an **invariant state** if
$$
\rho (A) = \rho(\alpha_g A) \; \forall A \in \mathcal{A}, \; \forall g \in G.
$$

=--

## Properties ##

+-- {: .un_lemma }
###### Lemma

The set of invariant states is convex, weak-$*$ closed and weak-$*$ compact.
(see [[operator topology]]).

=--

## References ##

* [[Hellmut Baumgärtel]], Manfred Wollenberg, _Causal nets of operator algebras._ Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete))

* chapter 6 in Gerd Petersen, _Pullback and pushout constructions in $C^\ast$-algebra theory_, J. Funct. Analysis __167__, 243--344 (1999) [pdf](http://www.math.ru.nl/~mueger/ped2.pdf)

category: operator algebras
[[!redirects C-star-system]]
[[!redirects C-star-systems]]
[[!redirects C-star system]]
[[!redirects C-star systems]]
[[!redirects C-star dynamical systems]]
[[!redirects invariant state]]


C-star-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Noncommutative geometry
+-- {: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definitions

### Abstract $C^\ast$-algebras

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

A $C^*$-algebra is a [[Banach algebra]] $(A, {\|-\|})$ over a [[topological field]] $K$ (often the field $K \coloneqq \mathbb{C}$ of [[complex numbers]]) equipped with an [[anti-involution]] $(-)^\ast$ compatible with [[complex conjugation]] if appropriate (that is: a Banach [[star-algebra]]) that satisfies the __$C^*$-identity__
$$
  {\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|}
$$
or equivalently the __$B^*$-identity__
$$
  {\|{A^* A}\|} = {\|{A}\|^2}
  \,.
$$

A [[homomorphism]] of $C^\ast$-algebras is a map that preserves all this structure. For this it is sufficient for it to be a [[star-algebra]] homomorphisms.

$C^\ast$-algebras with these homomorphisms form a [[category]] [[C*Alg]].

=--

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

Often one sees the definition without the clause (which should be in the definition of Banach $*$-algebra) that the involution is an [[isometry]] (so that ${\|A^*\|} = {\|A\|}$, which is key for the equivalence of the $B^*$ and $C^*$ identities).  This follows easily from the $B^*$-identity, while it follows from the $C^*$-identity after some difficulty.
=--

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

There are different concepts for the [[tensor product]] of $C^*$-algebras, see for example at _[[spatial tensor product]]_.

=--

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

$C^*$-algebras equipped with the [[action]] of a [[group]] by [[automorphisms]] of the algebra are called _[[C-star-systems]]_ .

=--
### Concrete $C^\ast$-algebras and $C^\ast$-representations

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

Given a [[complex numbers|complex]] [[Hilbert space]] $H$, a __[[concrete structure|concrete]] $C^*$-algebra__ on $H$ is a $*$-[[subalgebra]] of the algebra of [[bounded operators]] on $H$ that is [[closed subspace|closed]] in the [[norm topology]].

=--

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

A __[[representation of a C-star algebra|representation]]__ of a $C^*$-algebra $A$ on a [[Hilbert space]] $H$ is a $*$-homomorphism from $A$ to the algebra of [[bounded operators]] on $H$.

=--

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

It is immediate that concrete $C^*$-algebras correspond precisely to [[faithful representations]] of abstract $C^*$-algebras.  It is an important theorem that *every* $C^*$-algebra has a faithful representation; that is, every abstract $C^*$-algebra is [[isomorphism|isomorphic]] to a concrete $C^*$-algebra.

=--

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

The original definition of the term '$C^*$-algebra' was in fact the concrete notion; the 'C' stood for 'closed'.  Furthermore, the original term for the abstract notion was '$B^*$-algebra' (where the 'B' stood for 'Banach').  However, we now usually interpret '$C^*$-algebra' abstractly.  (Compare '$W^*$-algebra' and '[[von Neumann algebra]]'.)

=--

### In $\dagger$-compact categories {#DaggerFormulation}

The notion of $C^*$-algebra can be abstracted to the general context of [[symmetric monoidal †-categories]], which serves to illuminate their role in _[[quantum mechanics in terms of †-compact categories]]_.

For a discussion of this in the finite-dimensional case see for instance ([Vicary](#Vicary)).

## Properties
 {#Properties}

### Category theoretic properties

$C^*$-algebras are [[monadic]] over sets.
More precisely, the forgetful functor $\mathbf{C^*Alg}\to\mathbf{Set}$ that assigns to each algebra the set of points in its unit ball is monadic. See [Pelletier & Rosicky (1993)](#PelletierRosicky93).

See also *[[operator algebras]]*.

### Partial order and positive elements
 {#PartialOrderAndPositiveElements}

The [[self-adjoint elements]] in a $C^\ast$-algebra $\mathcal{A}$

$$
  Herm(\mathcal{A})
  \;\coloneqq\;
  \big\{
    A \,\in\, \mathcal{A}
    \,\big\vert\,
    A^\ast \,=\, A
  \big\}
$$

form a [[partial order|partially ordered]] [[complex vector space]] by declaring an element $A$ to be "larger" than some $B$ if the difference is a [[normal operator]]

$$
  A \geq B
  \;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;
  \underset{C \in \mathcal{A} }{\exists}
  \;\;
  A - B \,=\, C^\ast C
  \,.
$$

In particular, the positive elements are exactly the [[normal operators]]

$$
  A \geq 0
  \;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;
  \underset{C \in \mathcal{A} }{\exists}
  \;\;
  A \,=\, C^\ast C
  \,.
$$

(It is (only) this [[partial order]] on the [[underlying]] [[complex vector space]] of $\mathcal{A}$ that determines which [[linear functions]] $\mathcal{A} \to \mathbb{C}$ count as *[[states on a C-star algebra|states]]*.)

E.g. [Murphy (1990) §2.2](#Murphy90), [Blackadar (2006) §II.3.1](#Blackadar06)

Discussion in the context of [[algebraic quantum field theory]]: [Bratteli & Robinson (1979) §2.2.2](AQFT#BratteliRobinson79), [Fredenhagen (2003) p. 6](AQFT#Fredenhagen03).

### Gelfand-Naimark theorem

The _[[Gelfand-Naimark theorem]]_ says that every [[C*-algebra]] is [[isomorphism|isomorphic]] to a $C^\ast$-algebra of [[bounded linear operators]] on a [[Hilbert space]].  In other words, every abstract $C^*$-algebra may be made into a concrete $C^*$-algebra.

### Gelfand-Naimark-Segal construction

The [[Gelfand-Naimark-Segal construction]] ([[GNS construction]]) establishes a correspondence between cyclic $*$-[[representation]]s of $C^*$-[[C*-algebra|algebras]] and certain linear functionals (usually called _[[state on an operator algebra|states]]_) on those same $C^*$-algebras.  The correspondence comes about from an explicit construction of the [[star-representation|*-representation]] from one of the [[linear functionals|linear functionals]] (states).

### Gelfand duality

[[Gelfand duality]] says that every ([[unital algebra|unital]]) _[[commutative C-star-algebra|commutative]]_ $C^*$-algebra over the [[complex numbers]] is that of complex-valued [[continuous functions]] from some [[compactum|compact Hausdorff topological space]]: there is an [[equivalence of categories]] $C^* CAlg \simeq $ [[Top]]${}_{cpt}$.

Accordingly one may think of the study of non-commutative $C^\ast$-algebras as _[[non-commutative topology]]_.

### General
 {#General}

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

For $A$ and $B$ two $C^\ast$-algebras and $f : A \to B$
a [[star-algebra]] [[homomorphism]] the set-theoretic [[image]]
$f(A) \subset B$ is a $C^\ast$-subalgebra of $B$, hence is also the
[[image]] of $f$ in $C^\ast Alg$.

=--

This is ([KadisonRingrose, theorem 4.1.9](#KadisonRingrose)).

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

There is a [[functor]]

$$
  \mathcal{C} : C^\ast Alg \to Poset
$$

to the [[category]] [[Poset]] of [[posets]], which sends each $A \in C^\ast Alg$
to its [[poset of commutative subalgebras]] $\mathcal{C}(A)$
and sends each morphism $f : A \to B$ to the [[functor]]
$\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)$ which
sends a [[commutative C-star-algebra|commutative]] subalgebra $C \subset A$ to $f(C) \subset B$.

=--

### Construction as groupoid convolution algebras

Many $C^\ast$-algebras arise as [[groupoid algebras]] of [[Lie groupoids]]. See at _[groupoid algebra - References - For smooth geometry](category+algebra#ReferencesForSmoothGeometry)_

### Homotopy theory

There is [[homotopy theory]] of $C^\ast$-algebras, being a non-commutative generalization of that of [[Top]].
(e.g. [Uuye 12](#Uuye)). For more see at _[[homotopical structure on C*-algebras]]_.

## Examples

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

Any algebra $M_n(A)$ of [[matrices]] with [[coefficients]] in a $C^\ast$-algebra is again a $C^\ast$-algebra. In particular $M_n(\mathbb{C})$ is a $C^\ast$-algebra for all $n \in \mathbb{N}$.

=--

\begin{example}
\label{ContinuousFunctionsVanishingAtInfinity}
For $A$ a $C^\ast$-algebra and for $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]], the set of [[continuous functions]] $X \to A$ which [[vanish at infinity]] is again a $C^\ast$-algebra by extending all operations pointwise.
(This algebra is [[unital algebra|unital]] precisely if $A$ is and if $X$ is a [[compact topological space]].)

This algebra is denoted

$$
  C_0(X,A) \in C^\ast Alg
  \,.
$$

If $A = \mathbb{C}$ then one usually just writes

$$
  C_0(X) \coloneqq C_0(X, \mathbb{C})
  \,.
$$

This are the $C^\ast$-algebras to which the [[Gelfand duality]] theorem applies and which are the default [[algebras of observables]] in [[classical physics]] (for $X$ a [[phase space]], cf. eg. [Landsman (2017), §3](#Landsman17)).
\end{example}

\begin{remark}\label{CompactlySupportedContinuousFunctions}
The subalgebra $C_{00}(X) \subset C_0(X)$ of [[compact support|compactly supported]] among all [[vanishing at infinity]]-functions (Exp. \ref{ContinuousFunctionsVanishingAtInfinity}) is not in general itself a $C^\ast$-algebra, but is a very well-behaved [[ideal]] inside $C_0(X)$, cf. [Amini (2004)](#Amini04).
\end{remark}

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

A [[uniformly hyperfinite algebra]] is in particular a $C^\ast$-algebra, by definition.

=--

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

A [[von Neumann algebra]] is in particular a $C^\ast$-algebra, by definition.

=--

## Related concepts

* [[commutative C-star-algebra]], [[Gelfand duality]], [[noncommutative topology]]

* [[separable C*-algebra]], [[homogeneous C*-algebra]]

* [[nuclear C*-algebra]]

* [[unitisation of C*-algebras]]

* [[von Neumann algebra]], [[enveloping von Neumann algebra]]

* [[JB-algebra]], [[JLB-algebra]]

* [[dense subalgebra]], [[F-star-algebra]]

* [[multiplier algebra]]

* [[graded C*-algebra]]

* [[tensor product of C*-algebras]]

* [[crossed product C*-algebra]]

* [[Cuntz algebra]]

* [[Poincaré duality C*-algebra]]

* [[Hilbert C*-module]], [[Hilbert C*-bimodule]], [[amplimorphism]]

* [[C*-category]]

* [[C*-coalgebra]], [[Hopf C*-algebra]]

* [[continuous field of C*-algebras]]

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

  * [[asymptotic C*-homomorphism]], [[l.m.c.-C*-algebra]]

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

* [[AQFT]]

  * [[state on an operator algebra]]

## References

Monographs:

* {#Dixmier77} [[Jacques Dixmier]], *$C^\ast$-algebras*, North Holland (1977)

* {#KadisonRingrose} [[Richard V. Kadison]], [[John R. Ringrose]], _Fundamentals of the theory of operator algebras_, chapter 4 in: Vol I *Elementary Theory*, Graduate Studies in Mathematics **15**, AMS 1997 ([ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0888.46039&format=complete))

* {#Murphy90} [[Gerard Murphy]], _$C^\ast$-algebras and Operator Theory_, Academic Press (1990) [[doi:10.1016/C2009-0-22289-6](https://doi.org/10.1016/C2009-0-22289-6)]

* [[Ian Putnam]], *Lecture notes on $C^\ast$-algebras* (2019) [[pdf](https://www.math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf), [[Putnam-CStarAlgebras.pdf:file]]]

* {#Blackadar06} [[Bruce Blackadar]], *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

An exposition that explicitly gives [[Gelfand duality]] as an [[equivalence of categories]] and introduces all the notions of [[category theory]] necessary for this statement is in

* Ivo Dell'Ambrogio, _Categories of $C^\ast$-algebras_ ([pdf](http://www.math.ethz.ch/u/ambrogio/exercise_C_-algebras.pdf))

See also:

* Wikipedia, _[$C^\ast$-algebra](https://en.wikipedia.org/wiki/C*-algebra)_

For [[operator algebra]]-theory see there and see

* [[Stanisław Woronowicz]], _Unbounded elements affiliated with $C^\ast$-algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991)

* [[Stanisław Woronowicz]], K. Napiórkowski, _[[Operator theory in the C*-algebra framework]]_, Reports on Mathematical Physics Volume 31, Issue 3, June 1992, Pages 353–371 ([publisher](http://www.sciencedirect.com/science/article/pii/003448779290025V), [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/OP.pdf))

On category-theoretic properties:

* {#PelletierRosicky93} J Wick Pelletier,  J Rosicky, *On the equational theory of $C^*$-algebras*, Algebra Universalis **30** (1993) 275-284

A characterizations of injections of commutative sub-$C^*$-algebras -- hence of the [[poset of commutative subalgebras]] of a $C^*$-algebra -- is in

* [[Chris Heunen]], _Characterizations of categories of commutative $C^*$-algebras_ ([arXiv:1106.5942](http://arxiv.org/abs/1106.5942))

General properties of the [[category]] of $C^\ast$-algebras are discussed in

* [[Ralf Meyer]], _Categorical aspects of bivariant K-theory_, ([arXiv:math/0702145](http://arxiv.org/abs/math/0702145))
 {#Meyer}

Specifically [[pullback]] and [[pushout]] of $C^\ast$-algebras is discussed in

* Gerd Petersen, _Pullback and pushout constructions in $C^\ast$-algebra theory_ ([pdf](http://www.math.ru.nl/~mueger/ped2.pdf))

See also

* {#Amini04} [[Massoud Amini]], *Locally Compact Pro-$C^\ast$-Algebras*, Canadian Journal of Mathematics **56** 1 (2004) 3-22  [[arXiv:math/0205253](https://arxiv.org/abs/math/0205253), [doi:10.4153/CJM-2004-001-6](https://doi.org/10.4153/CJM-2004-001-6)]

The [[homotopy theory]] of $C^\ast$-algebras (a [[category of fibrant objects]]-structure on $C^\ast Alg$) is discussed in

* Otgonbayar Uuye, _Homotopy theory for $C^\ast$-algebras_ ([arXiv:1011.2926](http://arxiv.org/abs/1011.2926))
 {#Uuye}

For more along such lines see the references at _[[KK-theory]]_ and _[[E-theory]]_.

Discussion of $C^\ast$-algebras as [[algebras of observables]] in [[quantum physics]]/[[quantum probability theory]]:

* {#Landsman17} [[Klaas Landsman]], *Foundations of quantum theory -- From classical concepts to Operator algebras*, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf)]

[[!redirects C-star-algebra]]
[[!redirects C-star-algebras]]
[[!redirects C-star algebra]]
[[!redirects C-star algebras]]
[[!redirects C*-algebra]]
[[!redirects C* algebra]]
[[!redirects C*-algebras]]
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[[!redirects C-* algebra]]
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[[!redirects C-* algebras]]
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[[!redirects B-*-algebras]]
[[!redirects B-* algebras]]

[[!redirects C*-identity]]
[[!redirects B*-identity]]


C-star-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A C\*-category can be thought of as a [[horizontal categorification]] of a C\*-[[C-star-algebra|algebra]]. Equivalently,  a C\*-algebra $A$ is thought of as a pointed one-object C\*-category $\mathbf{B}A$ (the [[delooping]] of $A$).  Accordingly, a more systematic name for C\*-categories would be __C\*-[[algebroids]]__.

## Definition

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

A (unital) __C\*-category__ is a \*-[[star-category|category]] [[enriched category|enriched]] in the category [[Ban]] of [[Banach spaces]] such that:

1. Every arrow $a \in Hom(x,y)$ satisfies the C\*-identity ${\|a^* a\|} = {\|a\|}^2$.

2. Composition satisfies ${\|{b a}\|} \leq {\|b\|} {\|a\|}$ for all composable pairs of arrows $a$ and $b$. (That is, we give $Ban$ the [[projective tensor product]].)

3. For every arrow $a \in Hom(x,y)$ there exists an arrow $b \in Hom(x,x)$ such that $a^\ast a = b^ \ast b$.
=--

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

Condition (3) above is equivalent to requiring that every arrow of the form $x^* x$ is positive in the sense of C\*-algebras. Unlike C\*-algebras, this does not follow automatically, as can be seen by considering the category with two objects $x,y$ with all morphism sets a copy of $\mathbb{C}$ and with involution defined on $a \in Hom(x,y)$ by $a^* = \overline{a}$ if $x=y$ and $a^* = -\overline{a}$ otherwise.
=--

+-- {: .num_remark}
###### Remark
A __C\*-category__ can be defined analogously to unital C\*-categories, using enriched [[nonunital categories]] instead of (unital) [[enriched categories]].
=--

## Examples

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

The $C^\ast$-[[representation category]] of a weak [[Hopf C-star-algebra|Hopf]] $C^\ast$-algebra (see there for details) is naturally a [[rigid monoidal category|rigid monoidal]] $C^\ast$-category.
=--

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

The category $Hilb$ of Hilbert spaces and bounded linear maps is a C\*-category.
=--

## Representation Theory

C\*-algebras can be represented as algebras of bounded linear operators on some choice of Hilbert space, using the [[Gelfand-Naimark-Segal construction|G.N.S. construction]]. C\*-categories have an analogue of the G.N.S. construction that allows them to represented on the category $Hilb$ of [[Hilbert space|Hilbert spaces]] and bounded linear maps.

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

For any (small) C\*-category $\mathcal{C}$ there exists a faithful \*-functor $\rho \colon \mathcal{C} \to Hilb$.
=--

## Related concepts

* [[GNS construction]]

* [[star-category]], [[dagger-category]]

* [[W-star category]]

* [[unitary fusion category]]

* [[spaceoid]]

* [[KK-theory]]

## References

With emphasis on the special case of [[W-star category|$W^\ast$-categories]]:

* {#GhezLinaRoberts85} P. Ghez, R. Lima, [[John E. Roberts]], Prop. 1.9 in: *$W^\ast$-categories*,  Pacific J. Math. **120** 1 (1985) 79-109 [[euclid:pjm/1102703884](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-120/issue-1/Wast-categories/pjm/1102703884.full)]

[[!redirects C-star-category]]
[[!redirects C-star-categories]]
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[[!redirects C*-categories]]
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[[!redirects C-star categories]]

[[!redirects C-star-algebroid]]
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[[!redirects C*-algebroid]]
[[!redirects C*-algebroids]]
[[!redirects C-star algebroid]]
[[!redirects C-star algebroids]]


C-star-coalgebra

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

# $C^*$-coalgebras
* table of contents
{: toc}

## Idea

$C^*$-coalgebras are like $C^*$-[[C-star-algebra|algebras]], but [[coalgebras]].  Their [[dual vector space|duals]] are $W^*$-[[W-star-algebra|algebras]].

## Definition

Let $A$ be a [[Banach *-coalgebra]] over the [[ground field]] $K$.  Let $\lambda\colon A \to K$ be a [[bounded linear functional]] on $A$, and let $\lambda^*$ be the composite
$$ A \overset{*}\to A \overset{\lambda}\to K \overset{\bar{}}\to K $$
(where $\bar{}$ is [[complex conjugation]], trivial if $K$ is [[real field|real]]); although some of the maps in this composite may be only [[antilinear map|antilinear]], the composite $\lambda^*$ is linear (over all of $K$).  Now consider the composite
\[ (\lambda {\displaystyle\hat{\otimes}_\pi} \lambda^*) \circ \Delta\colon A \to K {\displaystyle\hat{\otimes}_\pi} K \cong K ;\label{Cstarfunctional} \]
since $\Delta$ and $*$ are short, the norm of this functional is at most ${\|\lambda\|^2}$.

$A$ is a __$C^*$-coalgebra__ if the norm of the map in \eqref{Cstarfunctional} is exactly ${\|\lambda\|^2}$, for every bounded linear functional $\lambda$.

Although there is an asymmetry in \eqref{Cstarfunctional} (in the relative placement of $\lambda$ and $\lambda^*$), if we start with $\lambda^*$ instead of $\lambda$, we see that the [[universal quantification|universally quantified]] definition of $C^*$-coalgebra is symmetric.

## Justification

If we take the formal dual of everything in the definition above, then $A$ becomes a [[Banach *-algebra]] and $\lambda\colon K \to A$ becomes (multiplication of scalars by) an element $x$ of $A$.  The formal dual of the composite \eqref{Cstarfunctional} is (multiplication of scalars by) the element $x^* x$.  The requirement that the norm of this be exactly the square of the norm of $x$ is the $B^*$-identity that defines a $C^*$-[[C-star-algebra|algebra]].

So the definition of $C^*$-coalgebra dualises *everything* in the definition of $C^*$-algebra, down to using [[coelement]]s (in this case linear functionals) instead of [[elements]].

## Dual $W^*$-algebras

In general, the [[dual in a closed category|dual]] of a [[coalgebra]] is an [[algebra]], in any [[closed monoidal category]].  In particular, the [[dual vector space|dual]] of a [[Banach coalgebra]] is a [[Banach algebra]].  The involution $*$ gets along with this just fine; the dual of a [[Banach *-coalgebra]] is a [[Banach *-algebra]].  Finally, the linear functional in \eqref{Cstarfunctional} becomes the element in the $B^*$-identity for the dual, so the dual of a $C^*$-coalgebra is a $C^*$-[[C-star-algebra|algebra]].

But we have more!  If $A$ is a $C^*$-coalgebra, then (as we\'ve just seen) $A^*$ is a $C^*$-algebra; but since $A^*$ has a [[predual]] $A$, this means that $A^*$ is actually a $W^*$-[[W-star-algebra|algebra]] as well.

+-- {: .standout}
Which $W^*$-algebras arise in this way?
=--

## Examples (and non-examples)

The [[sequence space]] $l^1$ is a $C^*$-coalgebra, whose dual $W^*$-algebra is the sequence space $l^\infty$.  (For details of the comultiplication on $l^1$, see the examples in [[Banach coalgebra]].)

Although $l^\infty$ is a Banach coalgebra (under 'coconvolution'), it is *not* a $C^*$-coalgebra (at least not under coconvolution).

+-- {: .query}
(YC) Moreover, the TVS-isomorphism class of the predual of a $W^*$-algebra is very restricted (let alone its isomorphism class in Bang). In particular $\ell^\infty$ doesn't have a snowball's chance in Texas of being a $C^*$-coalgebra under any kind of choice of comultiplication, because it's the wrong kind of Banach space.)
=--

Although the dual of the [[Lebesgue space]] $L^1$ (on the [[real line]] $\mathbb{R}$ with [[Lebesgue measure]]) is the $W^*$-algebra $L^\infty$, $L^1$ is *not* a $C^*$-coalgebra, nor even a Banach coalgebra (at least not in the obvious way).  Essentially, this is because the [[diagonal subset|diagonal]] in $\mathbb{R}^2$ has [[null set|measure zero]] (so $\Delta$ takes an element of $L^1(\mathbb{R})$, interpreted as an [[absolutely continuous measure]] on $\mathbb{R}$, to a measure on $\mathbb{R} \times \mathbb{R}$ that is *not* absolutely continuous and so cannot be reinterpreted as an element of $L^1(\mathbb{R} \times \mathbb{R}) \cong L^1(\mathbb{R}) {\displaystyle\hat{\otimes}_\pi} L^1(\mathbb{R})$).

[[!redirects C-star coalgebra]]
[[!redirects C-star coalgebras]]
[[!redirects C-star-coalgebra]]
[[!redirects C-star-coalgebras]]
[[!redirects C* coalgebra]]
[[!redirects C* coalgebras]]
[[!redirects C*-coalgebra]]
[[!redirects C*-coalgebras]]
[[!redirects C-star cogebra]]
[[!redirects C-star cogebras]]
[[!redirects C-star-cogebra]]
[[!redirects C-star-cogebras]]
[[!redirects C* cogebra]]
[[!redirects C* cogebras]]
[[!redirects C*-cogebra]]
[[!redirects C*-cogebras]]


C-star-correspondence

+-- {: .rightHandSide}
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

# $C^*$-correspondences
* table of contents
{: toc}

## Idea

Roughly speaking, a $C^*$-[[C-star-correspondence|correspondence]] between two $C^*$-[[C*-algebra|algebras]] $A$ and $B$ is a generalised $C^*$-[[C*-algebra|homomorphism]] from $A$ to $B$.

## Definition {#Definition}

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

Let $A$ and $B$ be $C^*$-[[C*-algebra|algebras]]. An $A,B$-correspondence is a pair $(E,\varphi)$ consisting of a right Hilbert $B$-[[Hilbert C*-module|module]] $E$ and a non-degerate $C^*$-homomorphism $\varphi \colon A\to \mathcal{L}_B(E)$; _i.e_., the image $\varphi(A)E$ is dense in $E$ (as Hilbert $B$-modules).

We write

$$
A\stackrel{(E,\varphi)}{\to} B,
$$
or just $A\stackrel{E}{\to}B$, and call this a $C^*$-[[C-star-correspondence|correspondence]].
=--

## Properties

### Composition of correspondences

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

Let $A\stackrel{(E,\varphi)}{\longrightarrow}B$ and $B\stackrel{(F,\psi)}{\longrightarrow}C$ be $C^*$-[[C-star-correspondence|correspondences]]. Then the [[internal tensor product]] $E\otimes_{\psi}F$ is a Hilbert right $C$-[[Hilbert C*-module|module]]. The composition

$$

A \stackrel{(E,\varphi)\circ(F,\psi)}{\longrightarrow}  C

$$

is defined by the the pair $(E\otimes_{\psi}F,\varphi\otimes \psi)$, where

$$
\begin{array}{lccc}
\varphi\otimes\psi \colon & A & \to & \mathcal{L}_C(E\otimes_{\psi}F) \\
& a & \mapsto & \varphi(a)(\cdot)\otimes_{\psi}(\cdot)
\end{array}
$$
=--

### Unitary intertwiners

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

Let $A\stackrel{(E,\varphi)}{\longrightarrow}B$ and $A\stackrel{(F,\psi)}{\longrightarrow}B$ be $C^*$-[[C-star-correspondence|correspondences]]. A _unitary intertwiner_
$$
u\colon E\Rightarrow F
$$
is a unitary $u\in \mathcal{L}_B(E,F)$ such that for all $a\in A$ the following diagram commutes

$$
\array{E & \stackrel{\varphi(a)}{\longrightarrow} & E\\
 ^u\downarrow && \downarrow^u\\
F & \stackrel{\psi(a)}{\longrightarrow} & F
 }
$$
=--

## The $2$-category $\mathfrak{Corr}$

...

## Related concepts

* [[Hilbert bimodule]]

## References

* [[Alcides Buss]], [[Chenchang Zhu]], [[Ralf Meyer]], _A higher category approach to twisted actions on $C^*$-algebras,_ Proceedings of the Edinburgh Mathematical Society, **56** (2013), pp 387-426, doi:[10.1017/S0013091512000259](
https://doi.org/10.1017/S0013091512000259), arXiv:[0908.0455](http://arxiv.org/abs/0908.0455)

Relation to [[KK-theory]]:

* [[El-kaioum Moutuou]], _Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory_, Journal of K-theory, **13** (2014) pp83-113, doi:[10.1017/is013010018jkt244](
https://doi.org/10.1017/is013010018jkt244), arXiv:[1305.2495](https://arxiv.org/abs/1305.2495)

[[!redirects C*-correspondence]]
[[!redirects C*-correspondences]]
[[!redirects C-star-correspondence]]
[[!redirects C-star-correspondences]]


C-star-system

[[!redirects C-star-system]]
[[!redirects C-star-system]]

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

# Contents
* table of contents
{:toc}

## Idea

A __$C^*$-dynamical system__, or only $C^*$-system
is a [[C-star-algebra]] together with an
[[action]] of a [[group]] of [[automorphisms]]. In [[quantum mechanics]] as well as in [[AQFT]] the [[observables]] of the theory are [[self-adjoint operator]]s of (a [[local net]] of) [[C-star-algebra]]s, in this context the [[global gauge group]] of the theory is the maximal group of [[unitary operators]] that leave all observables invariant, the algebra and the gauge group form a $C^*$-system.

## Definition

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

A **$C^*$-system** $(\mathcal{A}, \alpha_G)$ consists of a $C^*$-algebra $\mathcal{A}$, a [[locally compact space|locally compact]] group $G$ and a [[continuous map|continuous]] [[homomorphism]] $\alpha$ of $G$ into the [[group]] $aut(\mathcal{A})$ of $*$-[[automorphisms]] of $\mathcal{A}$ equipped with the [[topology]] of pointwise convergence.

=--

If the algebra is a $*$-[[star-algebra|algebra]] only, then some authors call it a __$*$-system__.

Sometimes the continuity condition is dropped entirely or replaced by some weaker assumption, therefore one should always check what -- if any -- continuity assumption an author makes.

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

The **fixed point algebra** of a $C^*$-system  $(\mathcal{A}, \alpha_G)$ is $\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G  \}$. If the fixed point algebra is trivial then $\alpha_G$ **acts ergodically**.

=--

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

A [[state]] $\rho$ of the algebra $\mathcal{A}$ is an **invariant state** if
$$
\rho (A) = \rho(\alpha_g A) \; \forall A \in \mathcal{A}, \; \forall g \in G.
$$

=--

## Properties ##

+-- {: .un_lemma }
###### Lemma

The set of invariant states is convex, weak-$*$ closed and weak-$*$ compact.
(see [[operator topology]]).

=--

## References ##

* [[Hellmut Baumgärtel]], Manfred Wollenberg: _Causal nets of operator algebras._ Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete))

* chapter 6 in Gerd Petersen, _Pullback and pushout constructions in $C^\ast$-algebra theory_, J. Funct. Analysis __167__, 243--344 (1999) [pdf](http://www.math.ru.nl/~mueger/ped2.pdf)

category: operator algebras
[[!redirects C-star-system]]
[[!redirects C-star-systems]]
[[!redirects C-star system]]
[[!redirects C-star systems]]
[[!redirects C-star dynamical systems]]
[[!redirects invariant state]]


c-theorem

+-- {: .rightHandSide}
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### Context
#### Quantum field theory
+-- {: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# The Zamolodchikov $c$-theorem
* table of contents
{: toc}

## Idea

The fixed point of a [[renormalization]]-group (RG) transform applied to a [[field theory]] should, intuitively, be described by a theory which is scale-invariant.  The **Zamolodchikov $c$-theorem** makes this precise:  for any renormalizable 2D field theory, there exists a function $c(g)$ of its coupling constants $g = (g_1,g_2,\ldots)$ which

* decreases monotonically with successive RG transforms, and
* at the fixed point of the RG, is equal to the [[Virasoro algebra|central charge]] of the [[conformal field theory]] which describes that critical point.

[Cardy (1996)](#Cardy1996) observes,

: The $c$-theorem has the interpretation that renormalization group flows go 'downhill'.  In particular, it rules out the existence (for systems satisfying reflection positivity) of limit cycles and other esoteric behaviour in renormalization group flows.  It also severely restricts the possible fixed points to which unstable directions at a given fixed point may flow. ... An appealing physical interpretation of the $[c(g)]$-function is as a kind of entropy of information about the critical system.  Under renormalization, information is lost about the short distance behaviour of the correlation functions.  However, this cannot be taken too literally---for example, even at infinite temperature a block spin transformation results in loss of information about the microstates of the system, yet no renormalization group flow takes place.  Presumably, a more complete interpretation along these lines needs to account for the fact that the central charge is sensitive to only the effectively gapless degrees of freedom.

Attempts to find analogues for the 2D $c$-theorem in higher [[dimensions]] have involved versions of [[holographic principle|holographic duality]] such as the [[AdS/CFT correspondence]].  See [Myers and Sinha (2010)](#Myers2010), [(2011)](#Myers2011).

## Proof

We follow the original proof given by [Zamolodchikov (1986)](#Zamolodchikov).

We start with a field theory whose action functional is an integral of a local [[Lagrangian]] density:

$$ S = \int \mathcal{L}(g,a,x) d x, $$

where $a$ is the ultraviolet cutoff of the theory.  The basic assumption of RG is that there exists a single-parameter family of transformations on $Q$, the space of coupling constants,

$$ R_t: Q \rightarrow Q,$$

which has the property that a field theory with the action $S(R_t g, e^t a)$ is equivalent to the original theory with action $S(g, a)$.  Here, _equivalence_ means that the [[correlation function]]s of the two theories agree, as long as we consider scales larger than the renormalized UV cutoff, $x \gg e^t a$.  The RG flow is described by the [[beta function]]s:

$$  d g^i = \beta^i(g) \,d t. $$

At the fixed points, $\beta^i(g) = 0$.  (The beta functions in RG theory should not be confused with the [[Euler beta function]]).  Here, we are using the term in the same sense as when people say, "The beta function in [[QCD]] is negative."

The local energy-momentum tensor $T_{\mu\nu}(x) = T_{\nu\mu}(x)$ satisfies the conservation equation $\partial_\mu T_{\mu\nu} = 0$.  Define the complex coordinates

$$ (z, \bar{z}) = (x^1 + i x^2, x^1 - i x^2) $$

and define $T = T_{zz}$, $\Theta = T_{z\bar{z}}$.  The local scalar fields are given by

$$\Phi_i(g,x) = \frac{\partial}{\partial g^i} \mathcal{L}(g,a,x).$$

If the field theory is renormalizable, we can expand the field $\Theta$ in the basis given by the $\Phi_i$:

$$\Theta = \beta^i(g) \Phi_i.$$

Take a convenient distance scale $x_0$, significantly larger than the UV cutoff of the theory.  This arbitrary value is the "normalization point".  Define

$$ C(g) = 2z^4 \left.\langle T(x) T(0) \rangle\right|_{x^2 = x_0^2} $$

$$ H_i(g) = z^2 x^2 \left.\langle T(x) \Phi_i(0) \rangle\right|_{x^2 = x_0^2} $$

$$ G_{i j}(g) = x^4 \left.\langle \Phi_i(x) \Phi_j(0) \rangle\right|_{x^2 = x_0^2} $$

The positivity condition of the field theory implies that the symmetric [[matrix]] $G_{i j}(g)$ is positive definite, and that we can use it as the [[metric space|metric]] in the space of coupling constants.

The **Callan--Symanzik equation** describes how the $n$-point [[correlation function]]s vary under the RG transform.  For our purposes, the scaling matrix $\gamma$ which appears in the Callan-Symanzik equation is given by

$$ \gamma_i^{\,j} (g) \Phi_j = \left(\frac{a}{2} \frac{\partial}{\partial a} - \beta^k \frac{\partial}{\partial g^k}\right) \Phi_i = (\partial_i \beta^j) \Phi_j.$$

We combine our expansion of the field $\Theta$ with the conservation condition $\partial_\mu T_{\mu\nu} = 0$ and the Callan--Symanzik equation to get

$$ \frac{1}{2} \beta^i\partial_i C = -3\beta^i H_i + \beta^i\beta^k\partial_k H_i + \beta^k(\partial_k \beta^i) H_i, $$

and

$$ \beta^k \partial_k H_i + (\partial_i \beta^k)H_k - H_i = -2\beta^k G_{i k} + \beta^j \beta^k G_{i j} + \beta^j (\partial_i \beta^k) G_{j k} + \beta^j (\partial_j \beta^k) G_{i k}, $$

where we have written $\partial_i$ for the derivative in the coupling-constant space, $\partial/\partial g^i$.

For the function

$$c(g) = C(g) + 4\beta^i H_i = 6\beta^i \beta^j G_{i j},$$

we have that

$$\beta^i \partial_i c(g) = -12 \beta^i \beta^j G_{i j}.$$

This verifies that

$$ \frac{d c}{d t} = \beta^i(g) \partial_i c(g) \leq 0, $$

establishing that the function $c(g)$ decreases monotonically as we apply the RG transform.

Consider a fixed point $g_*$, and choose a coordinate system centered on that point, so that the metric becomes

$$ G_{i j}(g) = \delta_{i j} + \mathcal{O}(g^2). $$

Near the fixed point, we can calculate $c(g)$ using [[perturbation theory]].  This calculation results in

$$ c(g) = \tilde{c}(g_*) - 3(2 - d_i) g^i g^i + 2C_{i j k} g^i g^j g^k + \mathcal{O}(g^4). $$

We have written $d_i$ for the [[anomalous dimension]]s of the vectors $\Phi_i(g_*, x)$, which at the fixed point are conformal fields.  It then follows that the 2D field theory at the fixed point has conformal symmetry, with an infinite set of generators

$$\{L_n | n = 0, \pm 1, \pm 2, \ldots \}$$

which satisfy the [[Virasoro algebra]]

$$ [L_n, L_m] = (n - m)L_{n+m} + \frac{\tilde{c}}{12}(n^3 - n) \delta_{n+m,0}. $$

The fixed-point value of the $c$-function becomes the central charge of the Virasoro algebra.  The central charge is the numerical coefficient in the correlation function

$$\langle T(z) T(0) \rangle = z^{-4} \tilde{c}(g_*) / 2.$$

Near the fixed point, we can also compute the beta functions in perturbation theory.

$$ \beta^i(g) = \frac{2 - d_i}{2} g^i - \frac{1}{2} C_{i j k} g^j g^k + \mathcal{O}(g^3).$$

Note that to maintain consistency of indices across both sides of this equation, we do not sum over $i$ in the first term.  If the perturbations are "soft", satisfying $|2 - d_i| \ll 1$, then the coefficients $C_{i j k}$ form the structure constants of a [[CFT]] operator algebra.  Raising the indices on $G_{i j}$ by $G^{i k}G_{k j} = \delta^i_{\,j}$, we can write

$$\beta^i(g) = -\frac{1}{12} G^{i j}(g) \partial_j c(g).$$

## References

* A. B. Zamolodchikov (1986), _"Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory_, JETP Lett. **43**: 730--732. ([pdf](http://www.jetpletters.ac.ru/ps/1413/article_21504.pdf))
  {#Zamolodchikov}
* J. Cardy (1996), _Scaling and Renormalization in Statistical Physics._ Cambridge Lecture Notes in Physics.
  {#Cardy1996}
* J. Cardy (2010), _The Ubiquitous 'c': From the Stefan--Boltzmann Law to Quantum Information_, J.Stat.Mech. **2010**: P10004, ([arXiv:1008.2331](http://arxiv.org/abs/1008.2331))
  {#Cardy2010}
* R. C. Myers and A. Sinha (2010), _Seeing a $c$-theorem with holography_, Physical Review D **82**: 046006, ([arXiv:1006.1263](http://arxiv.org/abs/1006.1263))
  {#Myers2010}
* R. C. Myers and A. Sinha (2011), _Holographic $c$-theorems in arbitrary dimensions_, Journal of High Energy Physics **2011**, 1: 1--53, ([arXiv:1011.5819](http://arxiv.org/abs/1011.5819))
  {#Myers2011}
* Z. Komargodski and A. Schwimmer (2011), _On renormalization group flows in four dimensions_, ([arXiv:1107.3987](http://arxiv.org/abs/1107.3987))
  {#Komargodski2011}
* Wikipedia, [Callan--Symanzik equation](http://en.wikipedia.org/wiki/Callan%E2%80%93Symanzik_equation)


C. A. Cremonini

## Selected writings

On [[integration over supermanifolds]] and [[picture changing operators]] in view of [[super Chern-Simons theory]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Pictures from Super Chern-Simons Theory_ ([arXiv:1907.07152](https://arxiv.org/abs/1907.07152))

* [[Carlo A. Cremonini]], *The Geometry of Picture Changing Operators* [[arXiv:2305.02828](https://arxiv.org/abs/2305.02828)]

On super-[[Wilson lines]] via [[integration over supermanifolds]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], S. Penati, _Supersymmetric Wilson Loops via Integral Forms_ ([arXiv:2003.01729](https://arxiv.org/abs/2003.01729))

On [[Lie algebra cohomology]] of [[super Lie algebras]] (see also the [[Green-Schwarz action functional|brane scan]]) in relation to [[integration over supermanifolds|integrable forms]] of coset [[supermanifolds]]:

* [[Roberto Catenacci]], [[C. A. Cremonini]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces_ ([arXiv:2012.05246](https://arxiv.org/abs/2012.05246))

* [[C. A. Cremonini]], [[Pietro Antonio Grassi]], *Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms* ([arXiv:2106.11786](https://arxiv.org/abs/2106.11786))

On [[self-dual field theory]] via [[supergeometry]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Self-Dual Forms in Supergeometry I: The Chiral Boson_ ([arXiv:2012.10243](https://arxiv.org/abs/2012.10243))

On the [[self-dual higher gauge field]] on the [[M5-brane]] lifted to [[superspace]] in the [[D'Auria-Fré formulation of supergravity]]:

* {#ACDGMNRT22} [[Laura Andrianopoli]], [[C. A. Cremonini]], [[Riccardo D'Auria]], [[Pietro A. Grassi]], R. Matrecano, R. Noris, [[Lucrezia Ravera]], [[Mario Trigiante]], *M5-brane in the superspace approach*, Phys. Rev. D **106** 2 (2022) 026010 [[arXiv:2206.06388](https://arxiv.org/abs/2206.06388), [doi:10.1103/PhysRevD.106.026010](https://doi.org/10.1103/PhysRevD.106.026010)]

[[!redirects Carlo A. Cremonini]]
[[!redirects Carlo Alberto Cremonini]]

category:people


C. A. R. Hoare

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tony_Hoare)

## Selected writings

Introducing [[Hoare logic]]:

* {#Hoare69} [[C. A. R. Hoare]], *An axiomatic basis for computer programming*, Communications of the ACM **12** 10 (1969) 576–580 [[doi:10.1145/363235.363259](https://doi.org/10.1145/363235.363259)]

  also:

  in: *Programming Methodology. Texts and Monographs in Computer Science*, Springer (1978) 89-100 [[doi:10.1007/978-1-4612-6315-9_9](https://doi.org/10.1007/978-1-4612-6315-9_9)]

category: people

[[!redirects Tony Hoare]]
[[!redirects Charles Antony Richard Hoare]]
[[!redirects Charles A. R. Hoare]]


C. L. Mallows

## Selected writings

Introducing the [[Mallows kernel]]:

* [[C. L. Mallows]], *Non-Null Ranking Models. I*,  Biometrika Vol. 44, No. 1/2 (Jun., 1957), pp. 114-130 ([jstor:2333244](https://www.jstor.org/stable/2333244))

category: people


C. P. Boyer > history

> see *[[Charles P. Boyer]]*


C. R. F. Maunder

* [Wikipedia entry](https://de.wikipedia.org/wiki/Richard_Maunder) (deutsch)

## Selected writings

On the [[Atiyah-Hirzebruch spectral sequence]] for [[Whitehead-generalized cohomology theories]]:

* {#Maunder63} [[C. R. F. Maunder]], *The spectral sequence of an extraordinary cohomology theory*, Mathematical Proceedings of the Cambridge Philosophical Society **59** 3 (1963) 567- 574 $[$[doi:10.1017/S0305004100037245](https://doi.org/10.1017/S0305004100037245)$]$

On [[algebraic topology]]

* {#Maunder70} [[C. R. F. Maunder]], *Algebraic Topology*, Cambridge University Press, Cambridge (1970, 1980) $[$[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf)$]$

category: people

[[!redirects Richard Maunder]]


C. S. Seshadri

[[!redirects C. Seshadri]]

* [Wikipedia entry](http://en.wikipedia.org/wiki/C._S._Seshadri)

## related $n$Lab entries

* [[Narasimhan-Seshadri theorem]]

* [[stable vector bundle]]

category: people


C. T. J. (Kit) Dodson

[[!redirects C. T. J. Dodson]]
* [Webpage](http://www.maths.manchester.ac.uk/~kd/homepage/dodson.html)

category:people


C6-field

+-- {: .rightHandSide}
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[higher U(1)-gauge field]] which is the [[electric-magnetic duality|EM-dual]] of the [[C3-field]] in [[11-dimensional supergravity]]. Modeled by a [[circle n-bundle with connection|circle 6-bundle with connection]]. Couples to the [[M5-brane]].

## Related concepts

[[!include table of branes]]