Azumaya algebra

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## Idea

An *Azumaya algebra* over a [[commutative unital ring]] $R$ is an [[associative algebra|algebra]] over $R$ that has an inverse up to [[Morita equivalence]].  That is, $A$ is an Azumaya algebra if there is an $R$-algebra $B$ such that $B \otimes_R A$ is Morita equivalent to $R$, which is the unit for the tensor product of $R$-algebras.   Thus, Morita equivalence classes of Azumaya algebras over $R$ form a group, which is called the [[Brauer group]] of $R$.

## Definition

### Traditional

In what follows, $R$ is a [[commutative unital ring]] and algebras over $R$ are assumed associative and unital but not necessarily commmutative.  An __Azumaya algebra__ over $R$ is an algebra $A$ over $R$ obeying any of the following equivalent conditions:

* There exists an $R$-algebra $B$ such that the [[tensor product]] of $R$-algebras $B \otimes_R A$ is [[Morita equivalence|Morita equivalent]] to $R$.

* The $R$-algebra $A^{\mathrm{op}} \otimes_R A$ is [[Morita equivalence|Morita equivalent]] to $R$, where $A^{\mathrm{op}}$ is the opposite algebra of $A$.

* The [[center]] of $A$ is $R$, and $A$ is a [[separable algebra]].

* As a left $R$-[[module]], $A$ finitely generated, faithful and [[projective object|projective]], and the canonical morphism $A\otimes_R A^{op}\to End_R(A)$ is an [[isomorphism]].

When $R$ is a field, an Azumaya algebra is the same as a [[central simple algebra]] over $R$.

For any commutative ring $R$ there is a [[monoidal bicategory]] with

* algebras over $R$ as objects,
* bimodules as morphisms,
* bimodule homomorphisms as 2-morphisms.

Given any monoidal bicategory we can take its [[core]]: that is, the sub-monoidal bicategory where we only keep objects invertible up to equivalence, morphisms invertible up to 2-isomorphism, and invertible 2-morphisms.  This core is a [[3-group]], sometimes called the [[Picard 3-group]], and it has [[Azumaya algebras]] over $R$ as its objects.

### Over a scheme

More generally, [[Grothendieck]] defines an __Azumaya algebra__ over a [[scheme]] $X$ as a [[sheaf]] $\mathcal{A}$ of $\mathcal{O}_X$-algebras such that for each point $x\in X$, the corresponding [[stalk]] $\mathcal{A}_x$ is an Azumaya $\mathcal{O}_{X,x}$-algebra.

The [[Brauer group]] $Br(X)$ classifies Azumaya algebras over $X$ up to a suitably defined equivalence relation: $\mathcal{A}\sim\mathcal{B}$ if $\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{B}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F})$ for some locally free sheaves of $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{F}$ of finite rank.  The group operation of $Br(X)$ is induced by the tensor product. The Brauer group can be reexpressed in terms of second [[nonabelian cohomology]]; indeed a sheaf of Azumaya algebras over $X$ determines an $\mathcal{O}_X^*$-[[gerbe]] (or $U(1)$-gerbe in the [[manifold]] context).

[[Karoubi K-theory]] involves an element in a Brauer group and in the original Karoubi--Donovan paper is related to a twisting with a "local system" which involves Azumaya algebras.

### In terms of (derived) étale cohomology
 {#InTermsOfEtaleCohomology}

For $R$ a [[ring]] and $H^n_{et}(-,-)$ the [[etale cohomology]], $\mathbb{G}_m$ the [[multiplicative group]] of the [[affine line]]; then

* $H^0_{et}(R, \mathbb{G}_m) = R^\times$ ([[group of units]])

* $H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ ([[Picard group]]: iso classes of invertible $R$-modules)

* $H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)$ ([[Brauer group]] Morita classes of Azumaya $R$-algebras)

More generally, this works for $R$ a (connective) [[E-infinity ring]]
(the following is due to [[Benjamin Antieau]] and [[David Gepner]]).

Let $GL_1(R)$ be its [[infinity-group of units]].
If $R$ is [[connective spectrum|connective]], then the first [[Postnikov tower|Postikov stage]] of the [[Picard group|Picard]] [[infinity-groupoid]]

$$
  Pic(R) \coloneqq Mod(R)^\times
$$

is

$$
  \array{
   \mathbf{B}_{et} GL_1(-) &\to& Pic(-)
   \\
   && \downarrow
   \\
   && \mathbb{Z}
  }
  \,,
$$

where the top morphism is the inclusion of locally free $R$-modules.

so $H^1_{et}(R, GL_1)$ is not equal to $\pi_0 Pic(R)$, but it is off only by $H^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}$.

Let $Mod_R$ be the [[(infinity,1)-category]] of $R$-[[module spectra|modules]].

There is a notion of $Mod_R$-[[enriched (infinity,1)-category]], of "$R$-linear $(\infty,1)$-categories".

$Cat_R \coloneqq Mod_R$-modiles in [[presentable (infinity,1)-categories]].

Forming module $(\infty,1)$-categories is then an [[(infinity,1)-functor]]

$$
  Alg_R \stackrel{Mod}{\to} Cat_R
$$

Write $Cat'_R \hookrightarrow Car_R$ for the image of $Mod$.  Then define the [[Brauer group|Brauer]] [[infinity-group]] to be

$$
  Br(R) \coloneqq (Cat'_R)^\times
$$

One shows (Antieau-Gepner) that this is exactly the Azumaya $R$-algebras modulo Morita equivalence.

**Theorem** (B. Antieau, D. Gepner)

1. For $R$ a connective $E_\infty$ ring, any Azumaya $R$-algebra $A$ is étale locally trivial: there is an [[etale topology|etale cover]] $R \to S$ such that $A \wedge_R S \stackrel{Morita \simeq}{\to} S$.

   (Think of this as saying that an Azumaya $R$-algebra is étale-locally a Matric algebra, hence Morita-trivial: a "bundle of compact operators" presenting a (torsion) $GL_1(R)$-2-bundle).

1. $Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the [[etale cohomology]].

**Corollary**

1. $Br$ is [[connected object in an (infinity,1)-topos|connected]]. Hence $Br \simeq \mathbf{B}_{et} \Omega Br $.

1. $\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$

[[Postnikov tower]] for $GL_1(R)$:

$$
  for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n
$$

hence for $R \to S$ étale

$$
  \pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S
$$

This is a [[quasi-coherent sheaf]] on $\pi_0 R$ of the form $\tilde N$ (quasicoherent sheaf associated with a module), for $N$ an $\pi_0 R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves

$$
  H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0
$$

For every [[(infinity,1)-sheaf]] $G$ of [[infinity-groups]], there is a [[spectral sequence]]

$$
  H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R)
$$

(the second argument on the left denotes the $qth$ Postnikov stage). From this one gets the following.

* $\tilde \pi_0 Br \simeq *$

* $\tilde \pi_1 Br \simeq \mathbb{Z}$;

* $\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m$

* $\tilde \pi_n Br$ is quasicoherent for $n \gt 2$.

there is an [[exact sequence]]

$$
  0 \to
  H_{et}^2(\pi_0 R, \mathbb{G}_m)
  \to \pi_0 Br(R)
  \to H_{et}^1(\pi_0 R, \mathbb{Z})
  \to 0
$$

(notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$)

this is [[split exact sequence|split exact]] and so computes $\pi_0 Br(R)$ for connective $R$.

Now some more on the case that $R$ is not connective.

Suppose there exists $R \stackrel{\phi}{\to} S$ which is a faithful
[[Galois extension]] for $G$ a [[finite group]].

**Examples**

1. (real into complex [[K-theory spectrum]]) $KO \to KU$ (this is $\mathbb{Z}_2$)

1. [[tmf]] $\to tmf(3)$

Give $R \to S$, have a [[fiber sequence]]

$$
  Gl_1(R/S) \stackrel{fib}{\to} GL_1(R) \to GL_1(S)
  \to
  Pic(R/S) \stackrel{fib}{\to} Pic(R) \to Pic(S)
  \to
  Br(R/S) \stackrel{fib}{\to} Br(R) \to Br(S) \to \cdots
$$

**Theorem** (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R \stackrel{\simeq}{\to} S^{hG}$ ([[homotopy fixed points]])

1. $Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$

1. $Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$

it follows that there is a homotopy fixed points spectral sequence

$$
  H^p(G, \pi_\bullet \Sigma^n GL_1(S))
  \Rightarrow
  \pi_{-n} GL_1(S)
$$

**Conjecture** The spectral sequence gives an Azumaya $KO$-algebra $Q$ which is a nontrivial element in $Br(KO)$ but becomes trivial in $Br(KU)$.

### Azumaya categories

Borceux and Vitale have noted that the [[monoidal bicategory]] of  $R$-algebras, bimodules and bimodule morphisms can be generalized in the context of [[enriched category]] theory, leading to a concept of "Azumaya category".  An Azumaya algebra over the commutative ring $R$ is then a one-object Azumaya category enriched over $R Mod$.

More precisely, they consider an arbitrary [[Benabou cosmos]] $V$, meaning a [[complete]] and [[cocomplete]] [[symmetric monoidal]] [[closed category|closed]] category.   This gives a monoidal bicategory $V Mod$ with

* $V$-enriched categories as objects,
* $V$-enriched [[profunctors]] as morphisms, and
* $V$-natural transformations between $V$-enriched profunctors as 2-morphisms.

The [[core]] of this monoidal bicategory is a 3-group, and they call the objects of the core **Azumaya categories**.

* [[Francis Borceux]] and [[Enrico Vitale]], Azumaya categories, *Applied Categorical Structures* **10** (2002), 449-467. ([pdf](https://perso.uclouvain.be/enrico.vitale/Azumaya.pdf))

## Related concepts

* [[group of units]], [[Picard group]], [[Brauer group]], [[Picard 3-group]]

## References

* [G. Cortiñas](http://mate.dm.uba.ar/~gcorti), [[Charles Weibel]], _Homology of Azumaya algebras_, Proc. AMS __121__, 1, pp. 1994 ([jstor](http://www.jstor.org/stable/2160364))

* [[John Duskin]], _The Azumaya complex of a commutative ring_, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) [doi:10.1007/BFb0081352](http://dx.doi.org/10.1007/BFb0081352)

* {#GrothendieckBrauer} [[Alexander Grothendieck]], _Le groupe de Brauer I, II, III_, in Dix exposes sur la cohomologie des schemas (I: Algèbres d'Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)

* [[Max Karoubi]], [[Peter Donovan]], _Graded Brauer groups and $K$-theory with local coefficients_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1970__38_/PMIHES_1970__38__5_0/PMIHES_1970__38__5_0.pdf))

* M-A. Knus, M. Ojanguren, _Théorie de la descente et algèbres d'Azumaya_, Lec. Notes in Math. __389__, Springer 1974, [doi:10.1007/BFb0057799](http://dx.doi.org/doi:10.1007/BFb0057799), MR0417149

* J. Milne, _Étale cohomology_, Princeton Univ. Press

* [[Ross Street]], _Descent_, Oberwolfach preprint (sec. 6, Brower groups) [pdf](http://www.math.mq.edu.au/~street/Descent.pdf); _Some combinatorial aspects of descent theory_, Applied categorical structures __12__ (2004) 537-576, [math.CT/0303175](http://arxiv.org/abs/math/0303175) (sec. 12, Brower groups)

* [[Enrico Vitale]], _A Picard-Brauer exact sequence of categorical groups_, [pdf](http://www.math.ucl.ac.be/membres/vitale/cat-gruppi2.pdf)

* Ana-L. Agore, [[Stefan Caenepeel]], Gigel Militaru,  _Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings_, Appl. Categor. Struct. 22, 29--42 (2014) [doi](https://doi.org/10.1007/s10485-012-9294-3)

The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in the "[[bigger Brauer group]]" $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) Azumaya algebras is due to

* {#Toen10} [[Bertrand Toën]], _Derived Azumaya algebras and generators for twisted derived categories_ ([arXiv:1002.2599](http://arxiv.org/abs/1002.2599))

The comparison of the Artin's theorem on characterization of Azumaya algebras
and Tomiyama-Takesaki's theorem on $n$-[[homogeneous C*-algebra]]s is in chapter 9 of

* Edward Formanek, _Noncommutative invariant theory_, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 [doi](http://dx.doi.org/10.1090/conm/043)

See also

* Wikipedia, [Azumaya algebra](http://en.wikipedia.org/wiki/Azumaya_algebra)

* Urs Schreiber, [Picard and Brauer 2-groups](http://golem.ph.utexas.edu/string/archives/000786.html), String Theory Coffee Table, 2006.

* John Baez, [The Brauer 3-group](https://golem.ph.utexas.edu/category/2020/05/the_brauer_3group.html), $n$-Category Café, 2020.

[[!redirects Azumaya algebras]]
[[!redirects central separable algebra]]


B meson

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## Idea

In the [[standard model of particle physics]] a _B-meson_ is a [[meson]] which is a [[bound state]] of a [[bottom quark]] and any other [[quark]] from the first two [[generations of fermions]] ([[up quark]] or [[down quark]] or [[strange quark]] or [[charm quark]]).

Hence the class of B-mesons goes along side that of [[K-mesons]] (light mnesons with a [[strange quark]]) and [[D-mesons]] (light meson with a [[charm quark]]).

[[!include flavors of fundamental particles -- table]]

Current [[experiments]] studying B-meson [[physics]] are the \

* [[LHCb experiment]]

* [[Belle experiment]]

* [[BaBar experiment]]

All these are increasingly seeing [[flavour anomalies]] (see there) in $B$-meson decays at high [[statistical significance]], pointing to physics beyond the [[standard model of particle physics]].

## Related concepts

* [[upsilon-meson]]

* [[K-meson]], [[D-meson]]

* [[pion]], [[omega meson]], [[rho meson]]

* [[XYZ meson]]

* B-meson [[experiments]]

  * [[LHCb experiment]]

  * [[Belle collaboration]]

  * [[BaBar experiment]]

* B-[[baryon]]: [[Lambda baryon]]

* observations

  * [[flavour anomaly]]

  * [[V_cb puzzle]]

  * [[pentaquark]]

## References

### General

* [[Patrick Koppenburg]], [[Zdenek Dolezal]], [[Maria Smizanska]], _Rare decays of b hadrons_, Scholarpedia, 11(6):32643 ([doi:10.4249/scholarpedia.32643](http://dx.doi.org/10.4249/scholarpedia.32643))

See also

* Wikipedia, _B meson_

### Semileptonic decay

[[semileptonic decay]]:

Review:

* R. Casalbuoni, A. Deandrea, N.Di Bartolomeo, F. Feruglio, R. Gatto, G. Nardulli, _Phenomenology of Heavy Meson Chiral Lagrangians_, Phys. Rept. 281:145-238, 1997 ([arXiv:hep-ph/9605342](https://arxiv.org/abs/hep-ph/9605342))

* Jochen Dingfelder, Thomas Mannel, _Leptonic and semileptonic decays of $B$ mesons_, Rev. Mod. Phys. 88 (2016) 3, 035008 ([spire:1488176](https://inspirehep.net/literature/1488176), [doi:10.1103/RevModPhys.88.035008](https://doi.org/10.1103/RevModPhys.88.035008))

See also

* J. P. Lees et al. (BABAR Collaboration), _Measurement of the $B^+ \to \omega \ell^+ \nu$ branching fraction with semileptonically tagged B mesons_, Phys. Rev. D 88, 072006 (2013) ([doi:10.1103/PhysRevD.88.072006](https://doi.org/10.1103/PhysRevD.88.072006))

* Xian-Wei Kang, Tao Luo, Yi Zhang, Ling-Yun Dai, Chao Wang, _Semileptonic $B$ and $B_s$ decays involving scalar and axial-vector mesons_, Eur. Phys. J. C 78, 909 (2018) ([doi:10.1140/epjc/s10052-018-6385-9](https://doi.org/10.1140/epjc/s10052-018-6385-9))

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

### In holographic QCD

Discussion in [[holographic light front QCD]]:

* Mohammad Ahmady, _Holographic light-front QCD in B meson phenomenology_ (arXiv:2001.00266)

[[!redirects B mesons]]

[[!redirects B-meson]]
[[!redirects B-mesons]]


B-bordism

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## Idea

A _$B$-bordism_ is a [[cobordism]] $W$ equipped with extra "topological [[structure]]" with respect to _[[(B,f)-structure]]_, i.e.  in the form of a lift of the classifying map $W \to B O$ of its [[tangent bundle]]/[[stable normal bundle]] through some [[fibration]] $B \to B O$ over the [[classifying space]] of the [[orthogonal group]].

Commonly considered are lifts through the [[Whitehead tower]] of the [[orthogonal group]]), corresponding, in this order, to cobordisms with

* [[spin structure]] - [[spin bordism]]

* [[string structure]] - [[string bordism]]

* [[fivebrane structure]] - [[fivebrane bordism]]

etc.

## Examples

[[!include flavours of cobordism cohomology theories -- table]]

## References

* [[Manifold Atlas]], _[B-Bordism](http://www.map.mpim-bonn.mpg.de/B-Bordism)_

[[!redirects B-bordisms]]


B-brane

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## Idea

The [[branes]]  of the [[B-model]] [[topological string]], hence a [[TQFT]]-shadow of the [[D-branes]] of the [[superstring]]. Form the [[derived category]] of [[coherent sheaves]] on the [[target space|target]] [[spacetime]].

## Related concepts

* [[A-brane]], [[homological mirror symmetry]]

* [[2d TQFT]], [[TCFT]]

* [[Kapustin-Witten TQFT]]

* [[Bridgeland stability condition]]

* [[quiver gauge theory]]

[[!include table of branes]]

## References

A classical review is

* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))

Further surveys of the literature include

* [[Kevin H. Lin]], MO comment on _[Matrix factorization and physics](http://mathoverflow.net/a/9748/381), 2009

Formulation of the [[chain complexes]] of [[holomorphic vector bundles]] on the B-branes via [[Lie infinity-algebroid representation]] (see there) of the holomorphic [[tangent Lie algebroid]] is discussed in

* {#Bergman08} [[Aaron Bergman]], _Topological D-branes from Descent_ ([arXiv:0808.0168](http://arxiv.org/abs/0808.0168))

The definition of a [[triangulated category]] of B-branes for the [[Landau-Ginzburg model]] via [[matrix factorization]] was proposed by [[Maxim Kontsevich]] and is written out in

* [[Anton Kapustin]], Yi Li, _D-Branes in Landau-Ginzburg Models and Algebraic Geometry_ ([arXiv:hep-th/0210296](http://arxiv.org/abs/hep-th/0210296))

* {#Orlov} [[Dmitri Orlov]], _Triangulated categories of singularities and D-branes in Landau-Ginzburg models_, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248 ([arXiv:math/0302304](http://arxiv.org/abs/math/0302304))

* [[Dmitri Orlov]]_Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503–531, Progr. Math., 270, Birkhäuser Boston,
Inc., Boston, MA, 2009 ([arXiv:math.ag/0503632](http://arxiv.org/abs/math.ag/0503632))

[[!redirects B-branes]]


B-model

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## Idea

Witten introduced two topological twists of a supersymmetric nonlinear [[sigma model]], which is a certain N=2 superconformal field theory attached to a compact [[Calabi-Yau variety]] $X$. One of them is the _B-model_ [[topological string]]; it is a 2-dimensional [[topological conformal field theory|topological N=1 superconformal field theory]]. In Kontsevich's version, instead of SCFT with Hilbert space, one assembles all the needed data in terms of [[Calabi-Yau category|Calabi-Yau A-infinity-category]] which is the A-infinity-category of coherent sheaves on the underlying variety. In fact only the structure of a derived category is sufficient (and usually quoted): it is now known (by the results of [[Dmitri Orlov]] and [[Valery Lunts]]) that under mild assumptions on the variety, a derived category of coherent sheaves has a unique [[enhanced triangulated category|enhancement]].

The B-model arose in considerations of [[string theory|superstring]]-propagation on Calabi--Yau varieties: it may be motivated by considering the [[vertex operator algebra]] of the 2d[[CFT|SCFT]] given by the N=2 supersymmetric nonlinear [[sigma-model]] with target $X$ and then changing the fields so that one of the super-[[Virasoro algebra|Virasoro]] generators squares to 0. The resulting "topologically twisted" algebra may then be read as being the [[BRST complex]] of a [[TCFT]].

One can also define a B-model for [[Landau-Ginzburg model|Landau?Ginzburg models]]. The category of [[D-brane|D-branes]] for the string --  the [[B-branes]] -- is given by the category of [[matrix factorization|matrix factorizations]] (this was proposed by Kontsevich and elaborated by Kapustin-Li; see also papers of Orlov). For the genus 0 closed string theory, see the work of Saito.

By [[homological mirror symmetry]], the B-model is dual to the  [[A-model]].

## Properties

### Second quantization / effective background field theory

The [[second quantization]] [[effective field theory]] defined by the [[perturbation series]] of the B-model is supposed to be "Kodaira-Spencer gravity" / "BVOC theory" in 6d ([BCOV 93](#BCOV93), [Costello-Li 12](#CostelloLi12), [Costello-Li 15](#CostelloLi15)).

For more on this see at _[TCFT -- Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals)_.

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[sigma-model]]

  * [[AKSZ sigma-model]]

    * [[Poisson sigma-model]]

      * [[A-model]], **B-model**

      * [[half-twisted model]]

      * [[twistor string theory]]

    * [[Courant sigma-model]]

      * [[Chern-Simons theory]]

  * [[topological membrane]]

* [[topologically twisted D=4 super Yang-Mills theory]]

* [[Landau-Ginzburg model]]

[[!include gauge theory from AdS-CFT -- table]]

## References
 {#References}

### General

The B-model was first conceived in

* [[Edward Witten]], _Topological sigma models_, Commun. Math. Phys. __118__ (1988) 411--449, [euclid](http://projecteuclid.org/euclid.cmp/1104162092), [MR90b:81080](http://www.ams.org/mathscinet-getitem?mr=0958805)

An early review is in

* [[Edward Witten]]. _Mirror manifolds and topological field theory_, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056)).

The motivation from the point of view of [[string theory]] with an eye towards the appearance of the Calabi-Yau categories is reviewed for instance in

* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))

A summary of these two reviews is in

* H. Lee, _Review of topological field theory and homological mirror symmetry_ ([pdf](http://people.maths.ox.ac.uk/leeh/files/CYMSmini.pdf))

That the B-model [[Lagrangian]] arises in [[AKSZ theory]] by [[gauge fixing]] the [[Poisson sigma-model]] was observed in

* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, around page 28 in _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997

For survey of the literature see also

* [[Kevin H. Lin]], MO comment on _[Matrix factorization and physics](http://mathoverflow.net/a/9748/381), 2009

The B-model on [[genus]]-0 [[cobordism]]s had been constructed in

* S. Barannikov, [[Maxim Kontsevich]], _Frobenius manifolds and formality of Lie algebras of polyvector fields_ ,  Internat. Math. Res. Notices  1998,  no. 4, 201--215; [math.QA/9710032](http://arxiv.org/abs/alg-geom/9710032) [doi](http://dx.doi.org/10.1155/S1073792898000166)

The construction of the B-model as a [[TCFT]] on [[cobordisms]] of arbitrary [[genus]] was given in

* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([math.QA/0509264](http://arxiv.org/abs/math/0509264))

See also the MathOverflow discussion: [higher-genus-closed-string-b-model](http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model)

### Second quantization to Kodeira-Spencer gravity
 {#ReferencesBCOV}

The [[second quantization]] [[effective field theory|effective]] field theory defined by the B-model [[perturbation series]] ("Kodeira-Spencer gravity"/"BCOV theory") is discussed in

Discussion of how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]] is in

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

* {#CostelloLi12} [[Kevin Costello]], [[Si Li]], _Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model_ ([arXiv:1201.4501](http://arxiv.org/abs/1201.4501))

* [[Si Li]], _BCOV theory on the elliptic curve and higher genus mirror symmetry_ ([arXiv:1112.4063](http://arxiv.org/abs/1112.4063))

* [[Si Li]], _Variation of Hodge structures, Frobenius manifolds and Gauge theory_ ([arXiv:1303.2782](http://arxiv.org/abs/1303.2782))

* {#CostelloLi15} [[Kevin Costello]], Si Li, _Quantization of open-closed BCOV theory, I_ ([arXiv:1505.06703](http://arxiv.org/abs/1505.06703))

### Computation via topological recursion

Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of  a surface|genus]] proofs of [[mirror symmetry]] is due to

* {#BouchardKlemmMarinoPasquetti09} [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], _Remodeling the B-model_, Commun.Math.Phys.287:117-178, 2009 ([arXiv:0709.1453](https://arxiv.org/abs/0709.1453))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string I: Deriving the matrix model_ ([arXiv:1003.1737](https://arxiv.org/abs/1003.1737))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string II: The spectral curve and mirror geometry_ ([arXiv:1007.2194](https://arxiv.org/abs/1007.2194))

* {#EynardOrantin12} [[Bertrand Eynard]], [[Nicolas Orantin]], _Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture_ ([arXiv:1205.1103](https://arxiv.org/abs/1205.1103))

* {#FangLiuZong13} Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, _All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds_ ([arXiv:1310.4818](https://arxiv.org/abs/1310.4818))

[[!redirects Kodeira-Spencer gravity]]
[[!redirects BCOV theory]]

[[!redirects B-models]]


B. A. Blackadar > history


B. Andrei Bernevig

* [webpage](https://phy.princeton.edu/people/bogdan-bernevig)

## Selected writings

Predicting a [[quantum spin Hall effect]] in [[topological insulators]] based on mercury telluride (HgTe quantum wells):

* [[Andrei Bernevig]], [[Shou-Cheng Zhang]], _Quantum Spin Hall Effect_, Phys. Rev. Lett. 96, 106802 (2006) ([arXiv:cond-mat/0504147](https://arxiv.org/abs/cond-mat/0504147), [doi:10.1103/PhysRevLett.96.106802](https://doi.org/10.1103/PhysRevLett.96.106802))

* [[Andrei Bernevig]], [[Taylor Hughes]], [[Shou-Cheng Zhang]], _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science 15 December 2006: __314__, n. 5806, pp. 1757-1761 ([doi:10.1126/science.1133734](http://dx.doi.org/10.1126/science.1133734), [arXiv:cond-mat/0611399](https://arxiv.org/abs/cond-mat/0611399))

On higher order [[topological insulators]] (with protected corner-modes beyond the edge-modes):

* [[Frank Schindler]], Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, [[B. Andrei Bernevig]], [[Titus Neupert]], *Higher-Order Topological Insulators*, Science Advances 01  **4** 6 (2018) eaat0346 [[arXiv:1708.03636](https://arxiv.org/abs/1708.03636), [doi:10.1126/sciadv.aat0346](https://doi.org/10.1126/sciadv.aat0346)]

category: people

[[!redirects Andrei Bernevig]]
[[!redirects Bogdan Andrei Bernevig]]


B1-homotopy theory

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* table of contents
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## Idea

_$\mathbb{B}^1$-homotopy theory_ is the study of the [[localization]] of an [[(∞,1)-sheaf (∞,1)-category]] over a [[site]] of [[analytic spaces]] at morphisms $\mathbb{B}^1 \to *$, where $\mathbb{B}^1 \simeq Spm(k\{t\})$ is the _Tate ball_.

## Related concepts

* [[A1-homotopy theory]]

* [[complex analytic ∞-groupoid]]

## References

* {#Ayoub06} [[Joseph Ayoub]], _Motives of rigid varieties and
the motivic nearby functor_, talk notes 2006 ([pdf](http://user.math.uzh.ch/ayoub/Other-PDF/Venise_Hodge.pdf))

* [[Joseph Ayoub]], _Motifs des variétés analytiques rigides_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/MotVarRig.pdf))

* [[Joseph Ayoub]], Floian Ivorra and Julien Sebag, _Motives of rigid analytic tubes and nearby motivic sheaves_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/Motive-of-Tube.pdf))


b1-meson

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## Idea

One of the [[light meson|light]] [[vector mesons]], with [[Wigner classification]] being a [[Lorentz group]] [[bivector]]
([DHR 11](#DHR11), [DHR 12](#DHR12)) and [[isospin]] [[vector]].

## Related concepts

* [[h1-meson]]

* [[B-field]]

## References

* J. C. R. Bloch, Yu. L. Kalinovsky, C. D. Roberts, S. M. Schmidt, _Describing $a_1$ and $b_1$ decays_, Phys. Rev. D60:111502, 1999 ([arXiv:nucl-th/9906038](https://arxiv.org/abs/nucl-th/9906038))

Discussion within [[holographic QCD]]:

* {#DHR11} Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Completing the framework of AdS/QCD: $h_1$/$b_1$ mesons and excited $\omega$/$\rho$'s_, JHEP 05(2011)107 ([arXiv:1101.3315](https://arxiv.org/abs/1101.3315))

* {#DHR12}  Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Successes and Failures of a More Comprehensive Hard Wall AdS/QCD_, JHEP 04(2013)104 ([arXiv:1210.6351](https://arxiv.org/abs/1210.6351))

[[!redirects b1-mesons]]


B6-field

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## Idea

The [[higher U(1)-gauge field]] which is the [[electric-magnetic duality|EM dual]] to the [[B2-field]], for instance in [[dual heterotic string theory]]. Modeled by [[circle n-bundle with connection|circle 6-bundle with connection]]. Couples to the [[NS5-brane]].

## Related concepts

[[!include table of branes]]

[[!redirects B6-fields]]


Baas-Sullivan theory

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##Idea

The Baas-Sullivan construction allows [[generalized cohomology theories]] to be seen as a [[cobordism theory]] in which the cobordisms may have singularities.

## References

Some of the original work is contained in these papers:

* Nils Andreas Baas, Bordism theories with singularities, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970), Vol. I, pp. 1–16. Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970.

* Nils Andreas Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302.

* Nils Andreas Baas, On formal groups and singularities in complex cobordism theory, Math. Scand. 33 (1973), 303–313


Babak Haghighat

* [personal page](https://ymsc-strings.github.io/people/babak.html)

* [YMSC institute page](https://ymsc.tsinghua.edu.cn/en/info/1032/1214.htm)

* [YMSC research group page](https://ymsc-strings.github.io/people-faculty.html)

* [BIMSA page](https://www.bimsa.cn/newsinfo/581931.html)

* [InSpire page](https://inspirehep.net/authors/1057034)

* [GoogleScholar page](https://scholar.google.com/citations?user=N1SYs84AAAAJ&hl=en)

## Selected writings

On the [[M5-brane elliptic genus]]:

* Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 [[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)]

On [[M-strings]]:

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))

On [[E-strings]]:

* {#HLV14} [[Babak Haghighat]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _Fusing E-string to heterotic string: $E + E \to H$_, Phys. Rev. D 90, 126012 (2014) ([arXiv:1406.0850](https://arxiv.org/abs/1406.0850))

and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and "[[M3]]"-[[defect branes]] inside [[M5-branes]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[Riemann surfaces]] [[4d superconformal gauge field theory]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Shuwei Liu]], [[Marcus Sperling]], _4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$_,  J. High Energ. Phys. **2020** 152 (2020) [[arXiv:1907.00536](https://arxiv.org/abs/1907.00536), doi:10.1007/JHEP01(2020)152]

On [[braid group representations]] for [[su(2)-anyon]]-[[anyon statistics|statistics]] from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] of bundles of [[conformal blocks]] over [[configuration spaces of points]]:

* [[Xia Gu]], [[Babak Haghighat]], [[Yihua Liu]], *Ising- and Fibonacci-Anyons from KZ-equations*,  J. High Energ. Phys. **2022** 15 (2022) [[arXiv:2112.07195](https://arxiv.org/abs/2112.07195), doi:10.1007/JHEP09(2022)015]

On [[QFT with defects|defects]] in the [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on [[4-manifolds]]:

* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), doi:10.1007/JHEP11(2023)208]

On [[conformal blocks]] for [[Liouville theory]]:

* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]

* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]

category: people


BaBar experiment

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#### Experiments
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## Idea

A [[B-meson]]-[[experiment]].

Together with the [[LHCb experiment]] and the [[Belle experiment]], BaBar is responsible for detecting the [[flavour anomalies]].

## Related experiments

* [[LHCb experiment]]

* [[Belle experiment]]

* [[XYZ particle]]

* [[precision experiment]]

* [[experiment]]

## References

* [BaBar Public webpage](https://www.slac.stanford.edu/BFROOT/)

* Wikpedia, _[BaBar experiment](https://en.wikipedia.org/wiki/BaBar_experiment)_

[[!redirects BaBar]]


Bachir Bekka

* [personal page](https://perso.univ-rennes1.fr/bachir.bekka/)

## Selected writings

On [[Kazhdan's property (T)]]:

* [[Bachir Bekka]], P. de la Harpe and A. Valette, _Kazhdan's Propert (T)_, 2007 ([pdf](https://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf), [[BekkaHarpeValetteOnKashdanPropertyT.pdf:file]])

Proof that the [[unitary group]] [[U(ℋ)]] on a [[separable Hilbert space]] in its strong [[operator topology]] satisfies [[Kazhdan's property (T)]]:

* [[Bachir Bekka]], *Kazhdan's Property (T) for the unitary group of aseparable Hilbert space*, Geom. funct. anal. 13, 509–520 (2003) ([doi:10.1007/s00039-003-0420-0](https://doi.org/10.1007/s00039-003-0420-0))

category: people


background field

[[!redirects background gauge field]]

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## Idea

### General

In [[physics]], the [[field (physics)|fields]] in ([[prequantum field theory|pre]]-)[[quantum field theory]] whose [[dynamics]] is described by the [[theory (physics)|theory]] may in general depend on certain fixed [[stuff, structure, property|structure]] which also has an interpretation of a [[field (physics)|field]], but one that is fixed and not regarded as having dynamics in the given [[model (in theoretical physics)]]. These are often called **background fields**.

### For sigma-models

In particuar a [[sigma-model]] [[quantum field theory]] describes the propagation of a [[brane]] on some [[target space]] $X$. Typically the dynamics is controled by a [[gauge field]] on $X$ under which the brane is [[charge]]d. This is the **background gauge field** of the $\sigma$-model. The term "background" alludes to the fact that this background gauge field is fixed and only serves to influence the dynamics of the brane coupled to it. More generally, one could also study the quantum dynamics of the possible background fields themselves. This would be given by the corresponding [[gauge theory]] on $X$.

## Definition

A formalization of the notion is discussed at _[[field (physics)|field]]_ in _[Definition -- Physical fields](field%20%28physics%29#DefinitionPhysicalField)_.

## Examples

* The electrically charged [[relativistic particle]] is described by a [[sigma-model]] with background gauge field the [[electromagnetic field]] on [[spacetime]].

* The [[string]] of [[string theory]] may be described by a [[sigma-model]] with background gauge field the [[Kalb-Ramond field]].

* [[Chern-Simons theory]] may be understood as the [[sigma-model]] of a [[membrane]] charged under a background field that is a [[Chern-Simons circle 3-bundle]].

## Related concepts

* [[field (physics)]]

* [[sigma model]]

  * [[worldvolume]]

  * [[target space]]

  * **background field**

  * [[genus]]

[[!redirects background field]]
[[!redirects background fields]]

[[!redirects background gauge fields]]


background field formalism

## References

* L.F. Abbott, _Introduction to the Background Field Method_ Acta Phys.Polon. B13 (1982) 33 ([inSPIRE:166273](http://inspirehep.net/record/166273))

* L.F. Abbott, _The Background Field Method Beyond One Loop_ Nucl.Phys. B185 (1981) 189 ([inSPIRE:155719](http://inspirehep.net/record/155719))

* L.F. Abbott, Marcus T. Grisaru, Robert K. Schaefer, _The Background Field Method and the S Matrix_ Nucl.Phys. B229 (1983) 372 ([inSPIRE:13462](http://inspirehep.net/record/13462))

[[!redirects background field formalism]]


backpropagation

## Idea

Reverse mode [[automatic differentiation]] is often referred to as backpropagation, when specialized to the context of [[neural networks]].

## See also

* [[neural network]]

* [[machine learning]]

* [[equilibrium propagation]]


backreaction

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## Idea

In [[physics]], the _backreaction_ of an object or [[field configuration]] is its effect on other objects/fields by their mutual [[interaction]].

Typically one speaks of backreaction only in situations where it has (previously) been _neglected_, in that all other fields have been regarded as a fixed [[background field]] for the given object/field. Given such a situation of a field in a [[background field]] one may then ask what the backreaction of the former on the latter is or would have been.

## Examples

### Particle mechanics

The orbit of an [[electron]] around a [[nucleus]] or of a planet around the sun is to good approximation computed by taking into account the [[force]] that the nucleus/sun exerts on the electron/planet, but neglecting the (much smaller) force exerted by the electron/planet on the nucleus/sun. The latter force is the _backreaction_; and one will speak about taking it into account or not.

### Cosmic inhomogeneities

Not all backreaction considered is between small and large point masses.

For example, in [[inhomogeneous cosmology]] one speaks about the backreaction (or not) of inhomogeneities in the matter density filling the [[observable universe]] on the [[gravitational field]] filling the universe.

### Branes

In [[string theory]] one speaks of _[[probe branes]]_ when the backreaction of a small number of [[branes]] on some large number of background [[black branes]] is neglected.

## References

See also

* Wikipedia, _[Backreaction](https://en.wikipedia.org/wiki/Back-reaction)_

[[!redirects backreactions]]

[[!redirects back-reaction]]
[[!redirects back-reactions]]


Badis Ydri

* [webpage](https://homepages.dias.ie/ydri/index1)

## Selected writings

Review of [[matrix models]] in [[string theory]]/[[M-theory]] ([[BFSS matrix model]], [[IKKT matrix model]], [[BMN matrix model]], ...) and related topics ([[light cone gauge quantization]], [[Penrose limit]] to [[pp-wave spacetimes]], [[D-brane geometry]], [[fuzzy spheres]], ...):

* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))

category: people


Baer sum

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## Idea

The _Baer sum_ is the natural addition operation on [[abelian group extensions]]
as well on the extensions of $R$-modules, for fixed ring $R$.

For $G$ a [[group]] and $A$ an [[abelian group]], the extensions of $G$ by $A$ are classified by the degree-2 [[group cohomology]]

$$
  H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A)
  \,.
$$

On [[cocycles]] $\mathbf{B}G \to \mathbf{B}^2 A$ there is a canonical addition operation coming from the additive structure of $A$, and the Baer sum is the corresponding operation on the extensions that these cocycles classify.

## Definition

Below are discussed several different equivalent ways to define the Baer sum

### On concrete cocycles

A [[cocycle]] in degree-2 [[group cohomology]] $H^2_{Grp}(G,A)$ is a [[function]]

$$
  c : G \times G \to A
$$

satisfying the cocycle property.

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

Given two cocycles $c_1, c_2 : G \times G \to A$ their **sum** is the composite

$$
  (c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A
$$

of

* the [[diagonal]] on $G\times G$;

* the [[direct product]] $(c_1,c_2)$;

* the group operation $+ \colon A \times A \to A$.

Hence for all $g_1, g_2 \in G$ this sum is the function that sends

$$
  (c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)
$$

=--

### On abstract cocycles

As discussed at [[group cohomology]], a cocycle $c \colon G \times G \to A$ is equivalently a morphism of [[2-groupoids]] from the [[delooping]] [[groupoid]] $\mathbf{B}G$ of $G$ to the double-delooping [[2-groupoid]] $\mathbf{B}^2 A$ of $A$:

$$
  c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A
  \,.
$$

Since $A$ is an abelian group, $\mathbf{B}^2 A$ is naturally an abelian [[infinity-group|3-group]], equipped with a group operation $+ \colon (\mathbf{B}^2 A) \times (\mathbf{B}^A)\to \mathbf{B}^2 A $.

With respect to this the sum operation is

$$
  c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times   \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A
$$

### On short exact sequences

In any category with products, for any object $C$ there is a [[diagonal]] morphism
$\Delta_C:C\to C\times C$; in a category with coproducts there is a codiagonal morphism $\nabla_C: C\coprod C\to C$ (addition in the case of modules). Every additive category is, in particular, a category with finite [[biproduct]]s, so both morphisms are there. Short exact sequences in
the category of $R$-modules, or in arbitrary abelian category $\mathcal{A}$,
form an additive category (morphisms are commutative ladders of arrows) in which the biproduct
$0 \to A_i \to \hat H_{i} \to G_i \to 0$ for $i = 1,2$ is
$0\to A_1\oplus A_2 \to H_1\oplus H_2\to G_1\oplus G_2\to 0$.

Now if $0\to M\to N\to P\to 0$ is any extension, call it $E$,
and $\gamma:P_1\to P$ a morphism,
then there is a morphism $\Gamma' = (id_M,\beta_1,\gamma)$ from an
extension $E_1$ of the form $0\to M\to N_1\to P_1\to 0$ to $E$, where the pair $(E_1,\Gamma_1)$ is unique up to isomorphism of extensions,
and it is called $E\gamma$. In fact, the diagram
$$\array{
N_1&\to &P_1\\
\downarrow\beta_1 && \downarrow\gamma\\
N&\to &P
}$$
is a pullback diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E'$ in a unique way decomposes as
$$
E\stackrel{(\alpha,\beta_a,id)}\longrightarrow E\gamma \stackrel{(id,\beta_ 1,\gamma)}\longrightarrow E'
$$
for some $\beta_a$ with $\beta_1$ as above. In short, the morphism of extensions factorizes through $E\gamma$.

Dually, for any morphism $\alpha:M\to M_2$, there is a morphism $\Gamma_2 = (\alpha,\beta_2,id_P)$ to an extension $E_2$ of the form
$0\to M_2\to N_2\to P$; the pair $(E_2,\Gamma_2)$ is unique up to isomorphism of
extensions and it is called $\alpha E$.

In fact, the diagram
$$\array{
M&\to &N\\
\downarrow\alpha && \downarrow\beta_2\\
M_2&\to &N_2
}$$
is a pushout diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E''$ in a unique way decomposes as
$$
E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E \stackrel{(id,\beta_ 2,\gamma)}\longrightarrow E''
$$
for some $\beta_a$, with $\beta_2$ as above. In short, the morphism of extensions factorizes through $\alpha E$.

There are the following isomorphisms of extensions: $(\alpha E)\gamma\cong \alpha (E\gamma)$, $id_M E \cong E$, $E id_P \cong P$,
$(\alpha'\alpha)E\cong\alpha' (\alpha E)$, $(E\gamma)\gamma' \cong E(\gamma\gamma')$.

The Baer's sum of two extensions $E_1,E_2$ of the form $0\to M\to N_i\to P\to 0$
(i.e. with the same $M$ and $P$) is given by $E_1+E_2 = \nabla_M (E_1\oplus E_2) \Delta_P$; this gives the structure of the abelian group on $Ext(P,M)$ and $Ext:\mathcal{A}^{op}\times\mathcal{A}\to Ab$ is a biadditive (bi)functor. This is also related to the isomorphisms of extensions
$\alpha (E_1+E_2)\cong \alpha E_1+\alpha E_2$, $(\alpha_1+\alpha_2) E \cong \alpha_1 E+ \alpha_2 E$, $(E_1+E_2)\gamma \cong E_1\gamma + E_2\gamma$, $E(\gamma_1+\gamma_2)\cong E\gamma_1 + E\gamma_2$.

In different notation, if $0 \to A \to \hat G_{i} \to G \to 0$ for $i = 1,2$
are two [[short exact sequences]] of [[abelian groups]], their **Baer sum** is

$$
  \hat G_1 + \hat G_2
  \coloneqq
  +_* \Delta^* \hat G_1 \times \hat G_2
$$

The first step forms the [[pullback]] of the short exact sequence along the diagonal on $G$:

$$
  \array{
    A \oplus A &\to& A \oplus A
    \\
    \downarrow && \downarrow
    \\
    \Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{\Delta_G}{\to}&  G\oplus G
  }
$$

The second forms the [[pushout]] along the addition map on $A$:

$$
  \array{
    A \oplus A &\stackrel{+}{\to}& A
    \\
    \downarrow && \downarrow
    \\
    \Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2)
    \\
    \downarrow && \downarrow
    \\
    G &\to&  G
  }
$$

## Related concepts

* [[cup product]]

## References

* S. MacLane, _Homology_, 1963

* Patrick Morandi, _Ext groups and Ext functors_ ([pdf](http://sierra.nmsu.edu/morandi/oldwebpages/math683fall2002/Ext.pdf))

[[!redirects Baer sums]]
[[!redirects Baer's sum]]


Baer's criterion

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#### Category theory
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#Contents#
* table of contents
{:toc}

## Idea

_Baer's criterion_ is a criterion for detecting [[injective objects]] in a [[category]] of [[modules]]: [[injective modules]].

## Statement

Let $R$ be a [[ring]] and  $C = R $[[Mod]] the category of $R$-[[modules]].

+-- {: .num_theorem}
###### Proposition
**([[Baer's criterion]])**
{#Baer}
An [[object]] $Q \in R Mod$ is [[injective object|injective]] precisely if for $I$ any left $R$-[[ideal]] regarded as an $R$-[[module]], any [[morphism]] $g : I \to Q$ in $C$ can be extended to all of $R$ along the inclusion $I \hookrightarrow R$.
=--

+-- {: .proof}
###### Sketch of proof
Let $i \colon M \hookrightarrow N$ be a mono in $R Mod$, and let $f \colon M \to Q$ be a map. We must extend $f$ to a map $h \colon N \to Q$. Consider the poset whose elements are pairs $(M', f')$ where $M'$ is an intermediate submodule between $M$ and $N$ and $f' \colon M' \to Q$ is an extension of $f$, ordered by $(M', f') \leq (M'', f'')$ if $M''$ contains $M'$ and $f''$ extends $f'$. By an application of [[Zorn's lemma]], this poset has a maximal element, say $(M', f')$. Suppose $M'$ is not all of $N$, and let $x \in N$ be an element not in $M'$; we show that $f'$ extends to a map $M'' = \langle x \rangle + M' \to Q$, contradiction.

The set $\{r \in R: r x \in M'\}$ is an ideal $I$ of $R$, and we have a module map $g \colon I \to Q$ defined by $g(r) = f'(r x)$. By hypothesis, we may extend $g$ to a module map $k \colon R \to Q$. Writing a general element of $M''$ as $r x + y$ where $y \in M'$, it may be shown that

$$f''(r x + y) = k(r) + f'(y)$$

is well-defined and extends $f'$, as desired.
=--

## Consequences

+-- {: .num_cor}
###### Corollary
{#DirectSumInjectives}
Let $R$ be a Noetherian ring, and let $\{Q_j\}_{j \in J}$ be a collection of injective modules over $R$. Then the direct sum $Q = \bigoplus_{j \in J} Q_j$ is also injective.
=--

+-- {: .proof}
###### Proof
By [Baer's criterion](#Baer), it suffices to show that for any ideal $I$ of $R$, a module map $f \colon I \to Q$ extends to a map $R \to Q$. Since $R$ is Noetherian, $I$ is finitely generated as an $R$-module, say by elements $x_1, \ldots, x_n$. Let $p_j \colon Q \to Q_j$ be the projection, and put $f_j = p_j \circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through

$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q$$

for some finite $J' \subset J$. But a product of injectives is injective, hence $f$ extends to a map $R \to \prod_{j \in J'} Q_j$, which completes the proof.
=--

Conversely, a result of Bass and Papp is that $R$ is Noetherian if direct sums of injective $R$-modules are injective. See [Lam](#Lam), Theorem 3.46.

## References

* {#Lam} T.-Y. Lam, _Lectures on modules and rings_, Graduate Texts in Mathematics 189, Springer Verlag (1999).


Baez-Crans 2-vector space

These are discussed in  [[2-vector space]].

[[!redirects Baez--Crans 2-vector space]]
[[!redirects Baez–Crans 2-vector space]]
[[!redirects Baez--Crans 2-vector spaces]]
[[!redirects Baez–Crans 2-vector spaces]]
[[!redirects Baez-Crans 2-vector spaces]]


bagdomain topos

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#### Topos Theory
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#Contents#
+ automatic table of contents goes here
{:toc}

## Idea
The **bagdomain topos construction** is the topos analogue of the powerdomain construction in domain-theoretic database theory. Roughly speaking, an object $X$ is replaced by an object $X'$ such that points of $X'$ correspond to sets (='bags') of points of $X$.

## Related entries

* [[symmetric topos]]

* [[Artin gluing]]

* [[lax-idempotent 2-monad]]

## References

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos_ , JPAA **113** (1996) pp.1-38.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos II_ , JPAA **130** (1998) pp.49-84.

* [[Marta Bunge]], [[Jonathon Funk]], _Singular Coverings of Toposes_ , Springer LNM **1890** Heidelberg 2006.

* M. Jibladze, _Lower Bagdomain as a Glueing_ , Proc. A. Razmadze Math. Inst. **118** (1998) pp.33-41. ([pdf](http://www.rmi.ge/~jib/baglue.pdf))

* [[Peter Johnstone]], _Partial Products, Bagdomains and Hyperlocal Toposes_ , pp.315-339 in Fourman, Johnstone, Pitts (eds.), _Applications of Categories in Computer Science - Proceedings of the LMS Symposium Durham 1991_ , Cambridge UP 1992.

* [[Peter Johnstone]], _Variations on the Bagdomain Theme_ , Theo. Comp. Sci. **136** (1994) pp.3-20.

* [[Peter Johnstone]], _Sketches of an Elephant I_ , Oxford UP 2002.  (sec. B4.4, pp.448-456) {#elephant}

* [[Steve Vickers]], _Geometric Theories and Databases_ , pp.288-314 in Fourman, Johnstone, Pitts (eds.), _Applications of Categories in Computer Science - Proceedings of the LMS Symposium Durham 1991_ , Cambridge UP 1992. ([preprint](http://www.cs.bham.ac.uk/~sjv/GeoTh+DBs.pdf))


Bai-Ling Wang

* [website](http://wwwmaths.anu.edu.au/~wangb/)

## Selected writings

On [[twisted K-theory]]:

* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393  ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))

On [[twisted differential K-theory]]:

* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))

On [[orbifold K-theory]]:

* Jianxun Hu, [[Bai-Ling Wang]], _Delocalized Chern character for stringy orbifold K-theory_, Trans. Amer. Math. Soc. 365 (2013), 6309-6341  ([arXiv:1110.0953](https://arxiv.org/abs/1110.0953), [doi:10.1090/S0002-9947-2013-05834-5](https://doi.org/10.1090/S0002-9947-2013-05834-5))

category: people


Baire category theorem

## Idea

The _Baire category theorem_ states sufficient conditions for a [[topological space]] to be a _[[Baire space]]_.

## References

* Wikipedia, _[Baire category theorem](https://en.wikipedia.org/wiki/Baire_category_theorem)_

[[!redirects Baire category theorems]]


Baire lattice

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#Contents#
* table of contents
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## Idea

Baire lattices are [[lattice|lattice-theoretic]] abstraction of the [[Baire space|Baire property]] among [[topological spaces]]. In parallel to the fact that every [[complete metric space]] is a Baire space every [[continuous poset|continuous lattice]] is a Baire lattice.

## Definitions

Recall that a [[complete lattice]] is a [[partial order|poset]] which has all small [[join|joins]] and [[meet|meets]]. For $a, b \in L$, we say that $a$ is __way below__ $b$ and write $a \ll b$ if whenever $S \subseteq L$ is a [[directed subset]] and $b \leq \bigvee S$ (where $\bigvee S$ denotes the [[join]] of $S$), then there exists $s \in S$ with $a \leq s$. Further we say that $L$ is __[[continuous lattice|continuous]]__ if for every $a\in L$, the subset
$$ \Downarrow (a) \coloneqq \{ b \in L | b \ll a \} $$
is directed and has join $a$.

For example the [[category of open subsets|lattice of open subsets]] of a topological space is a continuous lattice if and only if the [[sobrification]] of the topological space is [[locally compact]] (i.e. the topology has a basis of compact neighborhoods).

\begin{defn}
An element $p \in L$ is called __irreducible__ if $a \vee b = p$ implies $p = a$ or $p = b$.
\end{defn}

To illustrate this definition think of an [[irreducible topological space|irreducible subset]] of a topological space.

\begin{defn}
An element $d\in L$ is __dense__ if for all $a\in L$ the relation $ a \neq \bot $ implies that $ d \wedge a \neq \bot $.
\end{defn}

\begin{defn}
A [[complete lattice]] $L$ is called a __Baire lattice__ if for any countable family of dense elements $N \subset L$ and each nonzero element $u \in L$ there is an irreducible element $p \in L$ such that $ d \wedge u \nleq p $ for all $d \in N$.
\end{defn}

## Theorem

\begin{theorem}
Every continuous lattice is a Baire lattice.
\end{theorem}

## Example

Let $X$ be a [[sober space|sober]] [[topological space]]. Then the lattice of opens of $X$ is Baire if and only if the topological space $X$ has the [[Baire space|Baire property]].

## Related pages

 * [[complete lattice]]
 * [[Baire category theorem]]
 * [[Baire space]]

## References

The concept appeared in:

 * [[Karl Heinrich Hofmann]], _A note on Baire spaces and continuous lattices_ 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279. ([doi:10.1017/S0004972700006080]( https://doi.org/10.1017/S0004972700006080))

Textbook accounts:

 * G. Gierz, [[Karl Heinrich Hofmann]], K. Keimel, J. D. Lawson, [[Michael Mislove]], [[Dana Scott]], _Continuous Lattices and Domains_ 2003, Vol. 93 of _Encyclopedia of Mathematics and its Applications_ ([doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725))

[[!redirects Baire lattices]]


Baire space

> This entry is about the class of [[topological spaces]] satisfying the Baire category theorem. For *the* Baire space used in computable analysis, descriptive set theory, etc, see instead at _[[Baire space of sequences]]_.

# Baire spaces
* table of contents
{: toc}

## Idea

A Baire space is a [[topological space]] that satisfies the conclusion of the [[Baire category theorem]].

It should not be confused with the [[Baire space of sequences]] (which is an example of a Baire space in our sense but not a prominent one). Nor should it be confused with a [[Baire set]] (a [[subset]] somewhat analogous to a [[measurable set]] but defined by a topological property).

## Definition

A __Baire space__ is a [[topological space]] such that the [[intersection]] of any [[countable family]] of [[dense subspace|dense]] [[open subspace|open]] subspaces is also dense. Equivalently: a space such that a countable union of closed sets each with [[empty set|empty]] [[interior]] also has empty interior.

## Examples

The classical [[Baire category theorem]] states that:

*  Any [[complete metric space]] (or rather its [[forgetful functor|underlying]] topological space, hence any completely metrizable topological space) is a Baire space.

A second theorem, sometimes dubbed "BCT2" (as in Wikipedia):

*  Any [[locally compact Hausdorff space]], or indeed any locally compact [[sober space]] according to Wikipedia, is a Baire space. Moreover, any [[G-delta set]] of a locally compact Hausdorff space is a Baire space under the [[subspace]] topology.

Furthermore:

*  Any [[open subspace]] of a Baire space is also a Baire space.

*  Given a Baire space $X$, a _dense_ $G_\delta$ set in $X$ (i.e. a countable intersection of dense opens) is a Baire space under the subspace topology. See Dan Ma's blog, specifically Theorem 3 [here](https://dantopology.wordpress.com/2012/06/02/a-question-about-the-rational-numbers/).

*  As mentioned above, the space of [[infinite sequences]] of [[natural numbers]], or equivalently (up to topology) the space [[irrational numbers]], is also known as '[[Baire space of sequences|Baire space]]'.  It is a Baire space in the present sense (since it admits a complete metric), but the coincidence of names appears to be just a coincidence.  (It is much more important that Baire space is a [[Polish space]] than that Baire space is a Baire space.  Of course, every Polish space is a Baire space too.)

[[!redirects Baire space]]
[[!redirects Baire spaces]]


Baire space of sequences

> This entry is about *the* Baire space of sequences of natural numbers. For another concept of the same name in [[topology]] proper, see at _[[Baire space]]_.

+-- {: .rightHandSide}
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### Context
#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In the study of [[computability]], [[descriptive set theory]], etc, by _Baire space_ is meant the [[topological space]] $\mathbb{N}^{\mathbb{N}}$ of infinite sequences of [[natural numbers]] equipped with the [[product topology]].

The [[continuous functions]] from Baire space to itself serve the role of _computable functions_ in [[computable analysis]]. See at _[[computable function (analysis)]]_.

## Related concepts

* [[irrational number]]

[[!include computable mathematics -- table]]

## References

Lecture notes include

* {#Bauer05} [[Andrej Bauer]], page 5 and section 5.3.1 of _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

Textbook accounts include

* {#Weihrauch00} [[Klaus Weihrauch]], section 2.3 of _Computable Analysis_ Berlin: Springer, 2000

[[!redirects Baire space (computability)]]
[[!redirects Baire space (set theory)]]
[[!redirects Baire space (descriptive set theory)]]
[[!redirects Baire space of sequences]]
[[!redirects Baire space of irrational numbers]]
[[!redirects set-theoretic Baire space]]


Baire subset

# Contents
* table of contents
{: toc}

## Idea

Baire sets are certain [[subsets]] of a [[topological space]].  They form the Baire $\sigma$-[[sigma-algebra|algebra]] of the space, and they play an important role in [[measure theory]].

## Definition

Let $X$ be a [[topological space]].  Then there is a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{B}$ on $X$ generated by the [[open subsets]] of $X$ that are preimages of $(0,\infty)$ under some continuous map $X\to\mathbf{R}$. Elements of $\mathcal{B}$ are called the __Baire sets__ (or __Baire subsets__, or __Baire-measurable sets__, etc) of $X$, and $\mathcal{B}$ itself is called the __Baire $\sigma$-algebra__ on $X$.

## Properties

The Baire $\sigma$-algebra is a $\sigma$-subalgebra of the [[Borel sigma-algebra|Borel σ-algebra]] since every continuous map is [[measurable function|measurable]]. Often both σ-algebras even coincide.
This holds for [[perfectly normal spaces]],
such as [[metric spaces]] or [[regular]] [[hereditary property|hereditary]] [[Lindelöf topological space|Lindelöf]] spaces.

When working with [[locally compact topological space|locally compact spaces]] one can often instead use the following fact ([Dudley, Theorem 7.3.1](#Dudley02)):

\begin{theorem}
Let $ K $ be a compact Hausdorff space and $ \mu $ any finite Baire measure thereon. Then $ \mu $ has a unique extension to a regular Borel measure on $ K $.
\end{theorem}

## Related concepts

* [[Borel set]]

* [[Baire measure]]

## References

* {#Dudley02} Dudley, _Real analysis and probability_, 2002.

[[!redirects Baire subset]]
[[!redirects Baire subsets]]
[[!redirects Baire set]]
[[!redirects Baire sets]]
[[!redirects Baire-measurable subset]]
[[!redirects Baire-measurable subsets]]
[[!redirects Baire-measurable set]]
[[!redirects Baire-measurable sets]]

[[!redirects Baire sigma-algebra]]
[[!redirects Baire sigma-algebras]]
[[!redirects Baire ∞-algebra]]
[[!redirects Baire ∞-algebras]]


balanced category

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## Definition

\begin{definition}\label{BalancedCategory}
A category is **balanced** if every [[morphism]] which is both a [[monomorphism]] and an [[epimorphism]] is already an [[isomorphism]].
\end{definition}

\begin{remark}
The possibility of monic epics that are not isomorphisms does not survive any strengthening of "monic" or "epic."  Any monic [[extremal epimorphism]] is necessarily an isomorphism, and therefore so is any monic [[strong epimorphism]] or [[regular epimorphism]] (and dually).  It follows that if all epics, or all monos, are extremal, then the category is automatically balanced.

In an "unbalanced" category it is frequently the case that the monomorphisms, the epimorphisms, or both, are not the "right" notion to consider and should be replaced by their [[extremal epimorphism|extremal]], [[strong epimorphism|strong]], or [[regular epimorphism|regular]] counterparts.
\end{remark}

## Examples and non-examples
 {#Examples}

\begin{example}\label{SetIsBalanced}
  The category [[Set]] is balanced (Def. \ref{BalancedCategory}).
\end{example}
More generally:
\begin{example}
Any [[topos]] and in fact any [[pretopos]] is balanced.
\end{example}
(eg. [Johnstone 1977, Cor. 1.22](#Johnstone77))
\begin{remark}
 Beware the [[counterexample]]:
 A [[quasitopos]], need *not* be balanced.
\end{remark}

\begin{example}\label{AbelianCategoriesAreBalanced}
Any [[abelian category]] is balanced.

In particular categories [[Mod]] of [[modules]] and [[Vect]] of [[vector spaces]] are balanced.
\end{example}
\begin{remark}
  An [[additive category]] need not be balanced:  A [[counterexample]] is the category of [[torsion subgroup]]-free abelian groups, each nonzero homomorphism $\mathbb{Z} \to \mathbb{Z}$ is both monic and epic.
\end{remark}

\begin{example}
The [[Grp|the category of groups]] is balanced (see [here](regular+monomorphism#Examples) and [here](Grp#eq)).
\end{example}

\begin{remark}
Not all categories of [[algebra|algebraic]] [[structures]] are balanced.

As a [[counterexample]],
the [[category of rings]] is not balanced: $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is monic and epic but not an isomorphism.

On similar grounds, the [[category of commutative monoids]] is not balanced, as the inclusion $\mathbb{N} \hookrightarrow \mathbb{Z}$ is both monic and epic.
\end{remark}

\begin{remark}
Topological categories are rarely balanced. In [[Top]], for example, the monic epimorphisms are the continuous bijections.
\end{remark}
However:
\begin{example}
The category of [[compact Hausdorff spaces]] is balanced.
\end{example}

\begin{remark}
In a [[free category]] on a [[directed graph]], and also in any
[[partial order|poset]] and  generally in any _[[thin category]]_, _every_ morphism is both monic and epic while only the [[identity morphisms]] are invertible; thus such categories are "as far as possible from being balanced."
\end{remark}

## References

* {#Johnstone77} [[Peter Johnstone]], Cor. 1.22 in: _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press (1977), Dover (2014)

* [[Roy L. Crole]], p. 115 of: *Categories for types*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139172707](https://doi.org/10.1017/CBO9781139172707)]

[[!redirects balanced categories]]


balanced monoidal category

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# Balanced monoidal categories#
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{:toc}

## Definition

A __twist__, or __balance__, in a [[braided monoidal category]] $\mathscr{C}$ is a [[natural transformation | natural isomorphism]] $\theta$ from the [[identity functor]] on $\mathscr{C}$ to itself satisfying the following compatibility condition with the [[braiding]] $\beta$:

$$
  \theta_{A\otimes B}
  \;=\;
  \beta_{B,A}
    \circ
   \beta_{A,B}
     \circ
    (\theta_A \otimes \theta_B),
    \,\, \forall A, B \in \mathscr{C}
$$

A __balanced monoidal category__ is a braided monoidal category equipped with such a balance.

Beware that there is an un-related notion of *[[balanced categories]]*.

## Properties

In the language of [[string diagrams]], the balancing is represented by a 360-degree twist:

\begin{imagefromfile}
"file_name": "graphical-twist.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

Every [[symmetric monoidal category]] is balanced in a canonical way. In fact, the [[identity natural transformation]] on the identity functor of $\mathscr{C}$ is a balance on $\mathscr{C}$. In this way, the twist can be seen as a way of "controlling" the non-symmetric behavior of the braiding.

A braided [[rigid monoidal category]] is balanced if and only if it is a [[pivotal category]], but a balanced monoidal category need not be rigid (cf. [Selinger 2011,  Lem. 4.20](#Selinger11)).

## References

The original definition:

* [[André Joyal]], [[Ross Street]], *The geometry of tensor calculus I*,  Adv. Math.  __88__ 1 (1991) 55--112, [doi:10.1016/0001-8708(91)90003-P]

The above definition follows:

* [[Jeff Egger]], Appendix C in: *Of Operator Algebras and Operator Spaces* (2006) [[pdf](http://www.mscs.dal.ca/~jegger/4micah.pdf)]

See also:

* {#Selinger11} [[Peter Selinger]], *A survey of graphical languages for monoidal categories*,
Springer Lecture Notes in Physics **813** (2011) 289-355 [[arXiv:0908.3347](https://arxiv.org/abs/0908.3347), [doi:10.1007/978-3-642-12821-9_4](https://doi.org/10.1007/978-3-642-12821-9_4)]

[[!redirects balancing]]
[[!redirects balance]]

[[!redirects balanced monoidal category]]
[[!redirects balanced monoidal categories]]


balanced set

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## Definition

A subset $B$ of a [[vector space]] $V$ over $\mathbb{R}$ or $\mathbb{C}$ is **balanced** if $\lambda x \in B$ whenever $x \in B$ and ${|\lambda|} \le 1$.

## Properties

The intersection of two balanced sets is balanced. The [[closure]] of a balanced set is again balanced.

## Examples

* The [[unit ball]] of a [[seminorm|seminormed]] space is balanced
* Any subspace is balanced

# Related concepts

* [[convex set]]
* [[absolutely convex set]]
* [[norm]]
* [[absorbing set]]

[[!redirects balanced sets]]


ball

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

### Geometric

For $n \in \mathbb{N}$ a [[natural number]], the $n$-[[dimension|dimensional]] **ball** or **$n$-disk** in $\mathbb{R}^n$ is the [[topological space]]

$$
  D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\}
  \subset \mathbb{R}^n
$$

equipped with the [[induced topology]] as a subspace of the [[Cartesian space]] $\mathbb{R}^n$.

Its [[interior]] is the **open $n$-ball**

$$
  \mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \}
  \subset \mathbb{R}^n
  \,.
$$

Its [[boundary]] is the $(n-1)$-[[sphere]].

More generally, for $(X,d)$ a [[metric space]] then an open ball in $X$ is a subset of the form

$$
  B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \}
$$

for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$.
(The collection of all open balls in $X$ form the [[basis of a topology|basis]] of the [[metric topology]] on $X$.)

### Combinatorial

There are also combinatorial notions of _disks_. For instance that due to ([Joyal](#Joyal)), as entering the definition of the [[Theta-category]]. See for instance ([Makkai-Zawadowski](#MakkaiZawadowski)).

## Properties
 {#Properties}

### Closed balls

A simple result on the _homeomorphism_ type of _closed_ balls is the following:

+-- {: .num_theorem}
###### Theorem
A [[compact space|compact]] [[convex subset|convex]] [[subset]] $D$ in $\mathbb{R}^n$ with [[nonempty set|nonempty]] [[interior]] is [[homeomorphic]] to $D^n$.
=--

+-- {: .proof}
###### Proof
Without loss of generality we may suppose the origin is an interior point of $D$. We claim that the map $\phi: v \mapsto v/{\|v\|}$ maps the boundary $\partial D$ homeomorphically onto $S^{n-1}$. By convexity, $D$ is homeomorphic to the cone on $\partial D$, and therefore to the cone on $S^{n-1}$ which is $D^n$.

The claim reduces to the following three steps.

1. The restricted map $\phi: \partial D \to S^{n-1}$ is continuous.

1. It's surjective: $D$ contains a ball $B = B_{\varepsilon}(0)$ in its interior, and for each $x \in B$, the positive ray through $x$ intersects $D$ in a bounded half-open line segment. For the extreme point $v$ on this line segment, $\phi(v) = \phi(x)$. Thus every unit vector $u \in S^{n-1}$ is of the form $\phi(v)$ for some extreme point $v \in D$, and such extreme points lie in $\partial D$.

1. It's injective: for this we need to show that if $v, w \in \partial D$ are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have $w = t v$ for $t \gt 1$, say. Let $B$ be a ball inside $D$ containing $0$; then the convex hull of $\{w\} \cup B$ is contained in $D$ and contains $v$ as an interior point, contradiction.

So the unit vector map, being a continuous bijection $\partial D \to S^{n-1}$ between [[compact Hausdorff space]]s, is a homeomorphism.
=--

+-- {: .num_cor}
###### Corollary
Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.
=--

+-- {: .proof}
###### Proof
$D$ has nonempty interior relative to its affine span which is some $k$-plane, and then $D$ is homeomorphic to $D^k$ by the theorem.
=--

### Open Balls

Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the _smooth_ category:

+-- {: .num_lemma}
###### Observation

The open $n$-ball is [[homeomorphic]] and even [[diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^n$

$$
  \mathbb{B}^n \simeq \mathbb{R}^n
  \,.
$$

=--

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

For instance, the smooth map

$$
  x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n
$$

has smooth inverse

$$
  y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.
$$

=--

This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a ([[smooth manifold|smooth]]) [[manifold]]. Hence, each (smooth) $n$-dimensional manifold is locally isomorphic to both ${\mathbb{R}}^n$ and $\mathbb{B}^n$.

From general existence results about [[smooth structure]]s on [[Cartesian space]]s we have that

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

In [[dimension]] $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is [[homeomorphic]] to $\mathbb{B}^d$ is also [[diffeomorphic]] to it.

=--

See the first page of ([Ozols](#Ozols)) for a list of references.

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

In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$. See [De Michelis-Freedman](#DeMFreed).

=--

+-- {: .num_theorem #StarShapedOpenDiffeomorphicToOpenBall}
###### Theorem
**(star-shaped domains are diffeomorphic to open balls)**

Let $C \subset \mathbb{R}^n$ be a [[star-shaped]] [[open subset]] of a [[Cartesian space]]. Then $C$ is [[diffeomorphic]] to $\mathbb{R}^n$.

=--

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

Theorem \ref{StarShapedOpenDiffeomorphicToOpenBall} is a [[folk theorem]], but explicit **proofs** in the literature are hard to find. See the discussion in the References-section *[here](#ReferencesStarShapedReasonDiffeomorphicToOpenBall)*. An explicit proof has been written out by Stefan Born, and this appears as the proof of [theorem 237](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154) in ([Ferus 07](#Ferus07)). A simpler proof is given in [Gonnord-Tosel 98](#GonnordTosel98) reproduced [here](http://mathoverflow.net/a/212595/381).

=--

Here is another proof:

+-- {: .proof}
###### Proof
Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in [Lee 2009](#Lee09) proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying [[bump functions]] we can assume that $f\le 1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin; by rescaling the ambient space we can assume $\epsilon=2$.

The smooth vector field $V\colon x\mapsto f(x)\cdot x/{\|x\|}$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le 1$ such that $b=1$ for ${\|x\|} \gt 1/2$ and $b=0$ in a neighborhood of 0.
The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon \mathbb{R} \times T\to T$. (The parameter of the flow is all of $\mathbb{R}$ and not just some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt {\|x\|} \lt 2$ the vector field $V$ equals $x\mapsto x/{\|x\|}$. Also, all flow lines of $V$ are radial rays.

Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})$ for ${\|x\|} \gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms,

$${\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},$$

hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form $L: (A,B) \to T$, where $A$ and $B$ can be finite or infinite. If $B$ is finite and the limit of $L(t)$ as $t \to B$ exists, then the vector field $V$ vanishes at $B$. In our case $V$ can only vanish at the boundary of $T$, which is precisely what we want for surjectivity.)

Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for ${\|x\|} \lt 2$ and as $p$ for ${\|x\|}\gt 1/2$.
The map $g$ is smooth because for $1/2\lt {\|x\|} \lt 2$ both definitions give the same value.
=--

And here is another proof, due to Gonnord and Tosel, translated into English by Erwann Aubry and available on MathOverflow:

\begin{theorem}
Every open star-shaped set $\Omega$ in $\mathbb{R}^n$ is $C^\infty$-diffeomorphic to $\mathbb{R}^n$.
\end{theorem}

\begin{proof}
For convenience assume that $\Omega$ is star-shaped at $0$.

Let $F=\mathbf{R}^n\setminus\Omega$ and $\phi:\mathbf{R}^n\rightarrow\mathbb{R}_+$ (here $\mathbf{R}_+=[0,\infty)$) be a $C^\infty$-function such that $F=\phi^{-1}(\{0\})$.
(Such $\phi$ exists by the [[Whitney extension theorem]].)

Now we define $f:\Omega\rightarrow\mathbb{R}^n$ via the formula:
$$f(x)=\overbrace{\left[1+\left(\int_0^1\frac{dv}{\phi(vx)}\right)^2\|x\|^2\right]}^{\lambda(x)}\cdot x=\left[1+\left(\int_0^{\|x\|}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot x.$$
Clearly $f$ is smooth on $\Omega$.

We set $A(x)=\sup\{t\gt0\mid t\frac{x}{\|x\|}\in\Omega\}$.
$f$ sends injectively the segment (or ray) $[0,A(x))\frac{x}{\|x\|}$ to the ray $\mathbf{R}_+\frac{x}{\|x\|}$.
Moreover, $f(0\frac{x}{\|X\|})=0$ and
$$\lim_{r\rightarrow A(x)}\left\|f(r\frac{x}{\|x\|})\right\|=\lim_{r\to A(x)}\left[1+\left(\int_0^{r}\frac{dt}{\phi\left(t\cdot\frac{rx}{\|x\|}\cdot\left\|\frac{\|x\|}{rx}\right\|\right)}\right)^2\right]\cdot r= \left[1+\left(\int_0^{A(x)}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot A(x)=+\infty.$$
Indeed, if $A(x)=+\infty$, then it holds for obvious reason.
If $A(x)\lt+\infty$, then by definitions of $\phi$ and $A(x)$ we get that $\phi(A(x)\frac{x}{\|x\|})=0$.
Hence by the [[mean value theorem]] and the fact that $\phi$ is $C^1$ due to
$$\phi\left(r\frac{x}{\|x\|}\right)\le M(A(x)-r)$$
for some constant $M$ and every $r$.
As a result,
$$\int_0^{A(x)}\frac{dt}{\phi\left(t\frac{x}{\|x\|}\right)}$$
diverges.
Hence we infer that $f([0,A(x))\frac{x}{\|x\|})=\mathbf{R}_+\frac{x}{\|x\|}$ and so $f(\Omega)=\mathbf{R}^n$.

To end the proof we need to show that $f$ has a $C^\infty$-inverse.
But as a corollary from the [[inverse function theorem]] we get that it is sufficient to show that $df$ vanishes nowhere.

Suppose that $d_x f(h)=0$ for some $x\in\Omega$ and $h\neq 0$.
From definition of $f$ we get that
$$d_x f(h)=\lambda(x)h+d_x \lambda(h)x.$$
Hence $h=\mu x$ for some $\mu\neq 0$ and from that $x\neq 0$.
As a result $\lambda(x)+d_x \lambda(x)=0$.
But we have that $\lambda(x)\ge1$ and function $g(t):=\lambda(tx)$ is increasing, so $g'(1)=d_x \lambda(x)\gt0$, which gives a contradiction.
\end{proof}

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

Let $I(\Delta^n) \subset \mathbb{R}^n$ be the [[interior]] of the standard $n$-[[simplex]]. Then there is a diffeomorphism to $\mathbb{B}^n$ defined as follows:

Parameterize the $n$-simplex as

$$
  I(\Delta^n)
  =
  \left\{
    (x^1, \cdots, x^n) \in \mathbb{R}
    |
     (\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1)
  \right\}
  \,.
$$

Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by

$$
  (x^1, \ldots, x^n)
  \mapsto
  (\log(\frac{x^1}{1 - x^1 - \ldots -x^n}),
  \ldots,
  \log(\frac{x^n}{1 - x^1 - \ldots - x^n}))
  \,.
$$

=--

(Thanks to [[Todd Trimble]].) One way to think about it is that $I(\Delta^n)$ is the positive orthant of an open $n$-ball in $l^1$ norm, so that in the opposite direction we have a chain of invertible maps

$$\array{
\mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\
 & & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1)
}$$

which we simply invert to get the map $f$ above.

### Good covers by balls

One central application of balls is as building blocks for [[covering]]s. See [[good open cover]] for some statements.

## Related concepts

* [[unit ball]]

## References
 {#References}

### Geometric

* {#Ozols}  V. Ozols, _Largest normal neighbourhoods_ ,
Proceedings of the American Mathematical Society
Vol. 61, No. 1 (Nov., 1976), pp. 99-101  ([jstor](http://www.jstor.org/stable/2041672))

That an open subset $U \subseteq \mathbb{R}^4$ homeomorphic to $\mathbb{R}^4$ equipped with the smooth structure inherited as an open submanifold of $\mathbb{R}^4$ might nevertheless be non-diffeomorphic to $\mathbb{R}^4$, see

* De Michelis, Stefano; Freedman, Michael H. (1992) "Uncountably many exotic $\mathbb{R}^4$'s in standard 4-space", J. Diff. Geom. 35, pp. 219-254.
{#DeMFreed}

### Star-shaped regions diffeomorphic to open ball
 {#ReferencesStarShapedReasonDiffeomorphicToOpenBall}

The proof that open star-shaped regions are diffeomorphic to a ball appears as

* {#Ferus07} [[Dirk Ferus]], [theorem 237](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154) in: _Analysis III_ (2007) ([pdf](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf), [[Ferus_AnalysisIII.pdf:file]])

It is a lengthy proof, due to Stefan Born.

A simpler version of the proof appears in

* {#GonnordTosel98} Stéphane Gonnord, Nicolas Tosel, page 60 of: _Calcul Différentiel_, ellipses (1998) (English translation: [MO:a/212595](http://mathoverflow.net/a/212595), [[Aubry_reproducing_GonnordAndTosel.pdf:file]])

These proofs had remained obscure (see also [this Remark](good+open+cover#LiteratureOnExistenceOfDifferentiablyGoodOpenCovers) at [[good open cover]]):

For instance in a remark below lemma 10.5.5 of

* {#Conlon08} Lawrence Conlon, _Differentiable manifolds_, Birkhäuser 2001/2008 ([doi:10.1007/978-0-8176-4767-4](https://link.springer.com/book/10.1007/978-0-8176-4767-4))

it says:

> It seems that open star shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.

And in

* {#Lee09}  [[Jeffrey Lee]], _Manifolds and differential geometry_, Graduate Studies in Mathematics **107**  (2009) ([ISBN: 978-0-8218-4815-9](https://bookstore.ams.org/gsm-107), [doi:10.1090/gsm/107](https://doi.org/10.1090/gsm/107))

one finds the statement:

> Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to $\mathbb{R}^n$ is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).

Here "Grom" refers to

* [[Mikhail Gromov]], _Convex sets and Kähler manifolds_, Advances in differential geometry and topology. F. Tricerri ed., World Sci., Singapore,
(1990), 1-38. ([pdf](http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf))

where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of $C^1$ class, not $C^\infty$, so this is not a proof either.

A texbook account finally appears in

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], Thm. 5.3.2 and Appendix C of: _Differential Forms_, World Scientific (2019) ([doi:10.1142/11058](https://doi.org/10.1142/11058))

See also the Math Overflow discussion [here](http://mathoverflow.net/questions/41853/explicit-diffeomorphim-between-open-simplex-and-open-ball).

### Combinatorial

* {#Joyal} [[Andre Joyal]], _Disks, duality and Theta-categories_ ([[JoyalThetaCategories.pdf:file]])

* {#MakkaiZawadowski} [[Mihaly Makkai]], Marek Zawadowski, _Duality for Simple $\omega$-Categories and Disks_ ([TAC](http://www.emis.de/journals/TAC/volumes/8/n7/8-07abs.html))

[[!redirects ball]]
[[!redirects balls]]
[[!redirects  balls]]
[[!redirects n-ball]]
[[!redirects n-balls]]

[[!redirects open ball]]
[[!redirects open balls]]
[[!redirects open balll]]
[[!redirects open n-ball]]
[[!redirects open n-balls]]

[[!redirects closed ball]]
[[!redirects closed balls]]
[[!redirects closed n-ball]]
[[!redirects closed n-balls]]

[[!redirects disk]]
[[!redirects disks]]
[[!redirects disc]]
[[!redirects discs]]

[[!redirects open disk]]
[[!redirects open disks]]
[[!redirects open disc]]
[[!redirects open discs]]

[[!redirects closed disk]]
[[!redirects closed disks]]
[[!redirects closed disc]]
[[!redirects closed discs]]

[[!redirects n-disk]]
[[!redirects n-disks]]
[[!redirects n-disc]]
[[!redirects n-discs]]

[[!redirects open n-disk]]
[[!redirects open n-disks]]
[[!redirects open n-disc]]
[[!redirects open n-discs]]


Balázs Szendrői

* [webpage](https://people.maths.ox.ac.uk/szendroi/)

category: people

[[!redirects Balazs Szendroi]]


Ban > history

< [[Ban]]


Banach algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Banach algebras
* table of contents
{: toc}

## Definitions

An associative unital Banach algebra is [[monoid object]] in the [[closed monoidal category]] of [[Banach spaces]] (with [[short linear operators]] as [[morphisms]], and the usual [[internal hom]], or equivalently the [[projective tensor product]]).  However, Banach algebras are not usually assumed to be unital, making them [[semigroup]] objects (or even [[magma]] objects if not assumed to be associative).

Explicitly, this means a [[Banach space]] $A$ equipped with a [[bilinear map|bilinear]] _multiplication_ map
$$ m\colon A \times A \to A ,$$
which again is usually taken to be [[associativity|associative]] (and may even be unital), such that
$$ {\|a \cdot b\|} \leq {\|a\|} \cdot {\|b\|} ,$$
where $a \cdot b$ (or just $a b$) means $m(a, b)$.

In the unital case, we should also require ${\|1\|} \leq 1$, although some authors leave this out.  Other authors require ${\|1\|} = 1$, which is too strong, since it rules out the [[trivial ring|trivial algebra]].  (However, ${\|1\|} = 1$ follows from ${\|1\|} \leq 1$ and the existence of any element $a \ne 0$).  One can of course always formally adjoin a unit $e$ with ${\|e\|} = 1$, forming the Banach algebra $A \oplus \langle{e}\rangle$ (using the $l^1$-[[l-1-direct sum|direct sum]]).

The explicit description in terms of $m$ is of course earlier; but the abstract description as an internal monoid makes clear the most natural definition of [[Banach coalgebra]]: a [[comonoid]] in the same monoidal category.

+-- {: .query}
YC: An earlier version of this entry said that "the correct" definition of Banach coalgebra is as a comonoid in the usual monoidal category of Banach spaces and [[short linear operators]]. I would prefer that this be amended, with similar wording as what I've chosen, since experience has shown that the most fruitful candidates for "Banach spaces with coalgebraic structure" are NOT comonoids in this sense. One should instead only require a comultiplication which takes values in something like the injective tensor product, or if you are working with Cstar objects, in something like the spatial tensor product. Does anyone object to my rewording?
=--

## Examples

* A standard associative example is $L^1(\mathbb{R})$ with [[Lebesgue measure]], where the multiplication is taken to be [[convolution]]. (This lacks a unit for the multiplication, since there is no $L^1$ function $e$ that represents the [[Dirac functional]] $f \mapsto f(0)$, via
$$ f(0) = \int_{-\infty}^{\infty} e(x) f(x) \,\mathrm{d}x ,$$
on [[continuous functions]] $f\colon X \to \mathbb{C}$.) One can generalize this example in straightforward fashion, replacing $\mathbb{R}$ by any [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] $G$, and Lebesgue measure by a [[Haar measure]] on $G$; the algebra is unital if and only if $G$ is [[discrete space|discrete]].

* For *any* [[measure space]] $X$, $L^{\infty}(X)$ is a [[commutative ring|commutative]] unital associative Banach algebra (in fact a unital $C^*$-[[C-star-algebra|algebra]], in fact a [[von Neumann algebra]] if $X$ is [[localizable measure space|localizable]]) with respect to pointwise multiplication.

* If $A$ is a Banach space, the [[internal hom]] $hom(A, A)$ is a unital Banach algebra (by [[general abstract nonsense]]).

* Any $C^*$-[[C-star algebra|algebra]] (and thus every [[von Neumann algebra]]) is in particular a Banach algebra.

* The [[normed division algebras]] are (possibly nonassociative) Banach [[division algebras]] over $\mathbb{R}$.

* The only Banach division algebra over $\mathbb{C}$ is $\mathbb{C}$ itself, by the [[Gel'fand–Mazur theorem]].

* A $JB$-[[JB-algebra|algebra]] (or more generally a [[Jordan–Banach algebra]]) is a nonassociative (but commutative) kind of Banach algebra.  (The commutative associative Banach algebras also count as Jordan--Banach algebras.)

### An example of a 'nonunital' Banach algebra that has an identity element
Let $C_n$ be a cyclic group of order $n\geq 2$ and look at the Banach algebra (in the "strict" sense of a monoid object in $Ban$) that is obtained by equipping the Banach space $\ell^1(C_n)$ with the natural convolution product: $\delta_x * \delta_y = \delta_{x+y}$. There is a "short" homomorphism from $\ell^1(C_n)$ into the ground field which is just the unique linear extension of the group homomorphism $C_n \to \{1\}$ (by the free property of the $\ell^1$-functor) and we let $J$ be the kernel of this homomorphism. ($J$ is the so-called "augmentation ideal".) Now $J$ is a semigroup object in $Ban$ and as an algebra it has an identity element $p$, but a calculation/hindsight shows that $\delta_e-p$ must be the constant function $C_n \to \{1/n\}$, so that $p$ has norm $(1-1/n)+(n-1)/n = 2-2/n$.

## Arens products
If $A$ is a Banach algebra, its bidual $A^{**}$ has two naturally induced Banach algebra structures on it: these are the so-called Arens products on the second dual. These correspond to the left and right [[tensorial strength]]s for the bidual monad on the category of Banach spaces (whether with [[short linear operators]] as [[morphisms]], or all bounded linear operators). In different language, the two Arens multiplications arise from natural transformations

$$\alpha_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$$

$$\,$$

$$\beta_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$$

described at [[monoidal monad]]; putting $A = B$ and post-composing with $m^{\ast\ast}: (A \otimes A)^{\ast\ast} \to A^{\ast\ast}$ produces the two Arens products. They are named for Richard Arens, who has a 1955 paper which studies this construction in a more general setting . One can see Arens's "phyla" -- with hindsight and Whig history -- as a precursor of symmetric closed monoidal categories.

[Link to talk by F. E. J. Linton on Arens products](http://tlvp.net/~fej.math.wes/CMS-June2010/Arens01.htm)

### Arens regularity
Algebras where the two Arens products coincide are said to be Arens regular: since $B(H)$, the algebra of bounded linear operators on a Hilbert space $H$, has this property, so do all its closed subalgebras, in particular all $C^*$-[[C-star algebra|algebra]]s.
In contrast, if $G$ is an infinite locally compact group, a result of N. J. Young shows that $L^1(G)$ is not Arens regular. Consequently, it is not isomorphic as a topological algebra to any closed subalgebra of $B(H)$. Note that this consequence is significantly deeper than the observation that the norm on $L^1(G)$ does not satisfy the C-star identity with respect to the usual involution on $L^1(G)$. See also comments on [this MO question](http://mathoverflow.net/questions/134074/banach-algebra-counterexample).

In general, a Banach algebra $A$ is Arens regular if and only if, for each $\mu\in A^*$, the orbit maps $a\mapsto a\cdot\mu$ and $a\mapsto \mu\cdot a$ are [[weakly compact]] as linear maps $A\to A^*$. (Going from memory here, this result is due to J. S. Pym.) By using known results for factorization of [[weakly compact]] maps, it follows that if $A$ is Arens regular, one can construct a [[reflexive Banach space]] $E$ and an injective homomorphism of topological algebras $A\to B(E)$ which has closed range; this appears to have first been done explicitly by S. Kaijser.
+-- {: .query}
I suspect that here I mean something like a strict monomorphism in the category of monoid objects in TVS, but I am not sure of the details.
=--
However, this is not a characterization of Arens regularity; Young also observed that for any locally compact group $G$ one may build a reflexive Banach space $E$ and construct an injective homomorphism of topological algebras $L^1(G)\to B(E)$ which has closed range.

## Related concepts

* [[Banach module]]

[[!include analytic geometry ingredients -- table]]

## References

Named after [[Stefan Banach]].

* Zbigniew Semadeni, _Banach spaces of continuous functions_, vol. I, [gBooks](http://books.google.com/books/about/Banach_spaces_of_continuous_functions.html?id=vCDvAAAAMAAJ)

* N. Landsman, _Mathematical topics between classical and quantum mechanics_, Springer

* Walter Rudin, _Functional analysis_

* Richard V. Kadison, John R. Ringrose, _Fundamentals of the theory of operator algebras_

* F. F. Bonsall, J. Duncan, _Complete normed algebras_

* T. W. Palmer, _Banach Algebras and the General Theory of \* -Algebras_

Discussion over [[non-archimedean fields]] is in

* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], section 3.7 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

Discussion in the context of ([[global analytic geometry]]) [[analytic geometry]] is in the following articles.

Lecture notes on the concept of the [[analytic spectrum]] of a Banach ring include

* {#Brodsky12} Sarah  Brodsky, _Non-archimedean geometry_, 2012 ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf))

A more [[topos theory|topos-theoretic]] perspective:

* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 6.5 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))
category: analysis

[[!redirects Banach algebra]]
[[!redirects Banach algebras]]


Banach analytic space

#Contents#
* table of contents
{:toc}

## Idea

An infinite-dimensional [[analytic space]] equipped with [[Banach space]] structure.

## References

* Springer Encyclopedia of Mathematics, _[Banach analytic space](http://www.encyclopediaofmath.org/index.php/Banach_analytic_space)_

[[!redirects Banach analytic spaces]]