braid group cryptography -- references

### Braid group cryptography
 {#BraidGroupCryptographyReferences}

Partly motivated by the possibility of [[quantum computation]] eventually breaking the security of [[cryptography]] based on [[abelian groups]], such as [[elliptic curves]], there are proposals to use [[non-abelian group|non-abelian]] [[braid groups]] for purposes of [[cryptography]] ("post-quantum cryptography").

#### Via Conjugacy Search

An early proposal was to use the _Conjugacy Search Problem_ in [[braid groups]] as a computationally hard problem for cryptography. This approach, though, was eventually found not to be viable.

Original articles:

* Iris Anshel, M. Anshel and D. Goldfeld, _An algebraic method for public-keycryptography_, Math. Research Letters 6 (1999), 287–291 ([pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0003/MRL-1999-0006-0003-a003.pdf))

* K.H. Ko, S.J. Lee, J.H. Cheon , J.W. Han, J. Kang, C. Park , _New Public-Key Cryptosystem Using Braid Groups_, In: M. Bellare (ed.) _Advances in Cryptology — CRYPTO 2000_  Lecture Notes in Computer Science, vol 1880. Springer 2000 ([doi:10.1007/3-540-44598-6_10](https://doi.org/10.1007/3-540-44598-6_10))

Review:

* Karl Mahlburg, _An Overview of Braid Group Encryption_, 2004 ([pdf](http://www.math.wisc.edu/~boston/mahlburg.pdf))

* Parvez Anandam, _Introduction to Braid Group Cryptography_, 2006 ([pdf](https://courses.cs.washington.edu/courses/csep590/06wi/finalprojects/anandam.pdf))

* David Garber, _Braid Group Cryptography_, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore ([arXiv:0711.3941](https://arxiv.org/abs/0711.3941), [doi:10.1142/9789814291415_0006](https://doi.org/10.1142/9789814291415_0006))

* Cryptowiki, _[Cryptosystems based on braid groups](http://cryptowiki.net/index.php?title=Cryptosystems_based_on_braid_groups)_

#### Via E-multiplication

A followup proposal was to use the problem of _reversing E-multiplication_ in braid groups, thought to remedy the previous problems.

Original article:

* Iris Anshel, Derek Atkins, Dorian Goldfeld and Paul E Gunnells, _WalnutDSA(TM): A Quantum-Resistant Digital Signature Algorithm_ ([eprint:2017/058](https://eprint.iacr.org/2017/058))

Review:

* Magnus Ringerud, _WalnutDSA: Another attempt at braidgroup cryptography_, 2019 ([pdf](https://ntnuopen.ntnu.no/ntnu-xmlui/bitstream/handle/11250/2622842/no.ntnu%3Ainspera%3A2452886.pdf?sequence=1&isAllowed=y))

But other problems were found with this approach, rendering it non-viable.

Original article:

* Matvei Kotov, Anton Menshov, Alexander Ushakov, _An attack on the Walnut digital signature algorithm_, Designs, Codes and Cryptography volume 87, pages 2231–2250 (2019) ([doi:10.1007/s10623-019-00615-y](https://doi.org/10.1007/s10623-019-00615-y))

Review:

*  José Ignacio Escribano Pablos, María Isabel González Vasco, Misael Enrique Marriaga and Ángel Luis Pérez del Pozo, _The Cracking of WalnutDSA: A Survey_, in: _Interactions between Group Theory, Symmetry and Cryptology_, Symmetry 2019, 11(9), 1072 ([doi:10.3390/sym11091072](https://doi.org/10.3390/sym11091072))

#### Further developments

The basic idea is still felt to be promising:

* Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, _A New Cryptosystem Based on Positive Braids_ ([arXiv:1910.04346](https://arxiv.org/abs/1910.04346))

* Garry P. Dacillo, Ronnel R. Atole,  _Braided Ribbon Group $C_n$-based Asymmetric Cryptography_, Solid State Technology Vol. 63 No. 2s (2020) ([JSST:5573](http://www.solidstatetechnology.us/index.php/JSST/article/view/5573))

But further attacks are being discussed:

* James Hughes, Allen Tannenbaum, *Length-Based Attacks for Certain Group Based Encryption Rewriting Systems* ([arXiv:cs/0306032](https://arxiv.org/abs/cs/0306032))

As are further ways around these:

* Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, *A New Cryptosystem Based on Positive Braids* ([arXiv:1910.04346](https://arxiv.org/abs/1910.04346))


braid group representations -- references

### Braid group representations (as topological quantum gates)

On [[linear representations]] of braid groups (see also at *[[braid group statistics]]* and interpretation as [[quantum gates]] in *[[topological quantum computation]]*):

* [[Ivan Marin]], *On the representation theory of braid groups*, Annales mathématiques Blaise Pascal, **20** 2 (2013) 193-260 ([arXiv:math/0502118](https://arxiv.org/abs/math/0502118), [dml:275607](https://eudml.org/doc/275607))

Review:

* [[Chen Ning Yang]], M. L. Ge (eds.). *Braid Group, Knot Theory and Statistical Mechanics*, Advanced Series in Mathematical Physics **9**, World Scientific (1991) $[$[doi:10.1142/0796](https://doi.org/10.1142/0796)$]$

  > (focus on [[quantum Yang-Baxter equation]])

* [[Camilo Arias Abad]], *Introduction to representations of braid groups*, Rev. colomb. mat. vol.49 no.1 (2015) ([arXiv:1404.0724](https://arxiv.org/abs/1404.0724), [doi:10.15446/recolma.v49n1.54160](https://doi.org/10.15446/recolma.v49n1.54160))

* [[Toshitake Kohno]], *Introduction to representation theory of braid groups*, Peking 2018 ([pdf](https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf), [[KohnoBraidRepresentations.pdf:file]])

in relation to [[modular tensor categories]]:

* [[Colleen Delaney]], *Lecture notes on modular tensor categories and braid group representations*, 2019 ([pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf), [[DelaneyModularTensorCategories.pdf:file]])

Braid representations from the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] on bundles of [[conformal blocks]] over [[configuration spaces of points]]:

* [[Ivan Todorov]], [[Ludmil Hadjiivanov]], *Monodromy Representations of the Braid Group*, Phys. Atom. Nucl. **64** (2001) 2059-2068; Yad.Fiz. **64** (2001) 2149-2158 $[$[arXiv:hep-th/0012099](https://arxiv.org/abs/hep-th/0012099), [doi:10.1134/1.1432899](https://doi.org/10.1134/1.1432899)$]$

* [[Ivan Marin]], *Sur les représentations de Krammer génériques*, Annales de l'Institut Fourier, **57** 6 (2007) 1883-1925 $[$[numdam:AIF_2007__57_6_1883_0](http://www.numdam.org/item/AIF_2007__57_6_1883_0)$]$

and understood in terms of [[anyon statistics]]:

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

Braid representations seen inside the [[topological K-theory]] of the [[braid group]]'s [[classifying space]]:

* [[Alejandro Adem]], [[Daniel C. Cohen]], [[Frederick R. Cohen]], *On representations and K-theory of the braid groups*, Math. Ann. **326** (2003) 515-542 ([arXiv:math/0110138](https://arxiv.org/abs/math/0110138), [doi:10.1007/s00208-003-0435-8](https://doi.org/10.1007/s00208-003-0435-8))

* [[Frederick R. Cohen]], Section 3 of: *On braid groups, homotopy groups, and modular forms*, in: J.M. Bryden (ed.), *Advances in Topological Quantum Field Theory*, Kluwer 2004,  275–288 ([pdf](https://link.springer.com/content/pdf/10.1007/978-1-4020-2772-7_11.pdf))

See also:

* R. B. Zhang, *Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials*, Journal of Mathematical Physics 33, 3918 (1992) ([doi:10.1063/1.529840](https://doi.org/10.1063/1.529840))

As [[quantum gates]] for [[topological quantum computation]] with [[anyons]]:

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Braiding Operators are Universal Quantum Gates*,  New Journal of Physics, Volume 6, January 2004 ([arXiv:quant-ph/0401090](https://arxiv.org/abs/quant-ph/0401090), [doi:10.1088/1367-2630/6/1/134](https://iopscience.iop.org/article/10.1088/1367-2630/6/1/134))

* [[Samuel J. Lomonaco]], [[Louis Kauffman]], _Topological Quantum Computing and the Jones Polynomial_, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) ([arXiv:quant-ph/0605004](https://arxiv.org/abs/quant-ph/0605004))

  > (braid group representation serving as a [[topological quantum computer|topological]] [[quantum gate]] to compute the [[Jones polynomial]])

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Topological quantum computing and $SU(2)$ braid group representations*, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) ([doi:10.1117/12.778068](https://doi.org/10.1117/12.778068), [rg:228451452](https://www.researchgate.net/publication/228451452))

* C.-L. Ho, A.I. Solomon, C.-H.Oh, *Quantum entanglement, unitary braid representation and Temperley-Lieb algebra*, EPL 92 (2010) 30002 ([arXiv:1011.6229](https://arxiv.org/abs/1011.6229))

* [[Louis H. Kauffman]], *Majorana Fermions and Representations of the Braid Group*, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) ([arXiv:1710.04650](https://arxiv.org/abs/1710.04650), [doi:10.1142/S0217751X18300235](https://doi.org/10.1142/S0217751X18300235))

* David Lovitz, *Universal Braiding Quantum Gates* [[arXiv:2304.00710](https://arxiv.org/abs/2304.00710)]

Introduction and review:

* [[Colleen Delaney]], [[Eric C. Rowell]], [[Zhenghan Wang]], *Local unitary representations of the braid group and their applications to quantum computing*, Revista Colombiana de Matemáticas(2017), 50 (2):211 ([arXiv:1604.06429](https://arxiv.org/abs/1604.06429), [doi:10.15446/recolma.v50n2.62211](http://dx.doi.org/10.15446/recolma.v50n2.62211))

* [[Eric C. Rowell]], *Braids, Motions and Topological Quantum Computing* $[$[arXiv:2208.11762](https://arxiv.org/abs/2208.11762)$]$

Realization of [[Fibonacci anyons]] on [[quasicrystal]]-states:

* Marcelo Amaral, [[David Chester]], Fang Fang, Klee Irwin, *Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing*, Symmetry **14** 9 (2022) 1780 $[$[arXiv:2207.08928](https://arxiv.org/abs/2207.08928), [doi:10.3390/sym14091780](https://doi.org/10.3390/sym14091780)$]$

Realization on [[supersymmetry|supersymmetric]] [[spin chains]]:

* Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, *Topological Quantum Computation on Supersymmetric Spin Chains* $[$[arXiv:2209.03822](https://arxiv.org/abs/2209.03822)$]$


braid group statistics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Anyon braiding
 {#Idea}

In [[quantum physics]], *braid group statistics* or *anyon statistics* (sometimes: *plektons*) refers to an exotic phenomenon where the [[braiding]] of the [[worldlines]] of certain effective [[particles]] ("anyons") in an effectively 2+1-dimensional [[spacetime]] has the effect of transforming the [[quantum state]] of the total [[quantum system]] by [[unitary operators]] which constitute a [[linear representation]] of the [[braid group]] -- a *[[braid representation]]*.

Often this is motivated as a generalization of the [[boson]]- or [[fermion]]-"statistics" which enters the *[[spin-statistics theorem]]*, see [below](#AsGeneralizedBosonFermionStatistics).

But the actual mathematical nature of anyons must be different from that of elements of [[boson]]/[[fermion]]-[[Fock spaces]] and often remains somewhat vague in existing discussions. One may recognize two different more concrete conceptualizations of *anyons* in the literature:

1. {#AnyonicQuanta} **anyonic quanta** much like [[boson]]/[[fermion]] [[quanta]] but subject to an additional global [[interaction]] by [[Aharonov-Bohm effect|Aharonov-Bohm phases]] due to a [[flat connection|flat]] *[[fictitious gauge field]]* which is sourced by and coupled to each of the quanta;

   > (this goes back to [Arovas, Schrieffer, Wilczek & Zee 1985](#ArovasSchriefferWilczekZee85), further developed in [Chen, Wilczek, Witten & Halperin 1989](#ChenWilczekWittenHalperin89), see [below](#AsAFictitiousAharonovBohmEffect))

1. {#AnyonicDefects} **anyonic defects** like [[vortices]] or other [[solitons]], whose position is a [[parameterized quantum system|classical parameter]] to the [[quantum system]], the *[[quantum adiabatic theorem|adiabatic movement]]* of which acts by [[Berry phases]] on the quantum [[ground state]].

    > (e.g. [Avros, Schrieffer & Wilczek 1984](#AvrosSchriefferWilczek84), see [further below](#AsBraidingOfDefects))

The concept of anyons is particularly well motivated in [[solid state physics]], where effectively 2-dimensional [[quantum materials]] are common place (eg. [[graphene]]) or where particles may otherwise be constrained to move in a [[plane]], such as in the [[quantum Hall effect]]. There is a multitude of *[[model (in theoretical physics)|models]]* in [[condensed matter theory]] (mostly [[lattice models]], such as [[string-net models]]) which *theoretically* realize anyon braid group statistics, and there are some first [[experiment|experimental]] indications of anyonic phenomena in actual materials (see the references under *[Experimental Realization](#ExperimentalRealization)*) below.

Specifically, in the context of [[topological phases of matter]], the (potential) presence of anyons has come to be known as the case of *[[topological order]]*, see there for more.

Besides general curiosity, much of the interest in anyonic braid group statistics lies in the fact that these [[braid representations]] are imagined to potentially serve as [[quantum gates]] in [[topological quantum computers]]. See there for more.

### As generalized boson/fermion statistics?
 {#AsGeneralizedBosonFermionStatistics}

In [[quantum field theory]], one speaks of the "statistics" of a [[particle]] species when referring to the [[linear representation]] that $n$-particle [[wavefunctions]] form under the the [[symmetric group]] $Sym(n)$ which [[permutation|permutes]] the $n$ particles.

While there is a rich [[representation theory of the symmetric group]],
the [[spin-statistics theorem]] says, when it applies, that for *[[field (physics)|field]] quanta* only the simplest two possibilities may occur:

* [[bosons]] (such as [[photons]]) transform under the [[trivial representation]] of $Sym(n)$,

* [[fermions]] (such as [[electrons]]) transform under the [[sign representation]] of $Sym(n)$.

  (see also at *[[Slater determinant]]*)

Now, the [[braid group]] on $n$ strands covers the symmetric group

$$
  Br(n)
  \twoheadrightarrow
  Sym(n)
$$

which allows one to regard any [[linear representation]] of the symmetric group also as a particular [[braid group representation]]. But by its definition, the [[braid group]] may be understood as the group of [[isotopy]]-classes of [[disjoint union|disjoint]] [[timelike]] [[worldlines]] in an effectively 2+1 dimensional [[spacetime]], with the group operation being [[concatenation]] of worldlines.

In this sense, one may imagine that *any* [[braid group representations]] may generalize the boson/fermion statistics in 2+1 dimensions. Texts typically suggest that this applies to *[[quasiparticle|quasiparticles]]*.

The term *anyon* (due to [Wilczek 1982b](#Wilczek82b)) is a pun on this state of affairs that *any* statistics "in between" boson- and fermion-statistics may be allowed.

On the other hand, anyonic braiding is conceptually different from boson/fermion statistics -- if it were on the same footing then the [[spin-statistics theorem]] would *rule out* anyonic braiding. This is acknowledged by [Chen, Wilczek, Witten & Halperin 1989, p. 352](#ChenWilczekWittenHalperin89) (cf. also [Wilczek 1990, §I.2, pp. 11](#Wilczek90)):

> Once the [[permutation group]] is replaced by the [[braid group]], the simple construction of passing from the solution to the one-particle problems to the solution of many-particle problems, familiar from the free bosons and free fermions, does not work anymore.

\linebreak

### Braiding of anyonic quanta -- via "fictitious" AB-phases
 {#AsAFictitiousAharonovBohmEffect}

A concrete model for *[anyonic quanta](#AnyonicQuanta)* via otherwise [[free field|free]] [[fermions]] in 2d [[interaction|interacting]] through a [[flat connection|flat]] "[[fictitious gauge field]]" was proposed in [Arovas, Schrieffer, Wilczek & Zee 1985](#ArovasSchriefferWilczekZee85) and developed in
[Chen, Wilczek, Witten & Halperin 1989](#ChenWilczekWittenHalperin89) and [Iengo & Lechner 1992](#IengoLechner92) (the model has been advertized in early reviews, e.g. [Wilczek 1990, §I.3](#Wilczek90) and [Wilczek 1991](#Wilczek91), but seems not to have been developed much since):

This model regards anyons as *a priori* [[free field|free]] [[fermions]], but equipped now with a non-local mutual [[interaction]] via a "fictitious gauge field" ([CWWH89, §2](#ChenWilczekWittenHalperin89)), in that
each of the particles is modeled as the singular source of a [[flat connection|flat]] [[circle bundle with connection|circle connection]] (a [[vector potential]] with vanishing [[field strength]]), which hence exerts no [[Lorentz force]] but has the effect that globally each other particle is subject to the same [[Aharonov-Bohm effect]] as would be caused by a tuple of infinite [[solenoids]] piercing through each of the other particle's positions.

For emphasis, from [CWWH89, p. 359](#ChenWilczekWittenHalperin89):

> Here the particles are to be regarded (in the absence of
interactions) as fermions; the interaction then makes them anyons with statistical
parameter $\theta = \pi(1 - 1/n)$.

It follows ([Wu 1984](#Wu84), [Imbo, Imbo & Sudarshan 1990](#ImboImboSudarshan90)) that (quoting from [Fröhlich, Gabbiani & Marchetti 1990, p. 20](#FroehlichGabbianiMarchetti90)):

> If $\theta \in\!\!\!\!\!/ \frac{1}{2}\mathbb{Z}$ the [[space of quantum states|Hilbert space]] of [[anyon]] [[wave functions]] must be chosen to be a space of [[multi-valued functions]] with half-[[monodromies]] given by the [[complex phase|phase]] factors $exp(2 \pi \mathrm{i} \theta)$. Such wave functions can be viewed as single-valued functions on the [[universal cover]] $\widetilde M_n$ of $M_n$ $[$the [[configuration space of points]]$]$.

Further discussion of anyon-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]], hence [[equivariant functions]] on its [[universal cover]]: [BCMS93, §1](#BCMS93), [Mund & Schrader 1995](#MundSchrader93), [DFT97, §1](#DFT97) [Myrheim 1999](#Myrheim99), [DMV03](#DMV03), [Murthy & Shankar 2009](#MurthyShankar09)

Incidentally, the [[quasi-particle]]-excitations *of* (or *in*) a gas of such Aharonov-Bohm phased anyons are argued to be [[vortices]] ([CWWH89, p. 457](#ChenWilczekWittenHalperin89)):

> we are led to conclude that *in anyon superconductivity, charged quasi-particles and vortices do not constitute two separate sorts of elementary excitations - they are one and the same.*

This seamlessly leads over to:

\linebreak

### Braiding of anyonic defects -- via adiabatic Berry phases
 {#AsBraidingOfDefects}

In practice, many (most?) incarnations of the concept of anyons are *[anyonic defects](#AnyonicDefects)* -- [[non-perturbative effect|non-perturbative]] [[soliton|solitonic]] [[defects]] (of [[codimension]]=2), akin to *[[vortices]]* in fluids:

> *Anyonic particles are best viewed as a kind of topological defects that reveal non-trivial properties of the ground state.* $[$[Kitaev 2006, p. 4](#Kitaev06)$]$

>  *Anyons can arise in two ways: as localised excitations of an interacting
quantum Hamiltonian or as defects in an ordered system.*  $[$[Das Sarma, Freedman & Nayak 2015, p. 1](topological+quantum+computation#DasSarmaFreedmanNayak15)$]$

(Compare also the original discussions in [Goldin, Menikoff & Sharp 1981, §III](#GoldinMenikoffSharp81), [Wilczek 1982a](#Wilczek82a) & [Wilczek 1990, p. 5](#Wilczek90), which offer a quantum particle "[[bound state|bound]]" to a classical & infinite [[solenoid]] -- hence a 2d [[magnetic monopole]] [[defect]] -- as a decent model for an anyon.)

But [[defects]] are a kind of [[boundary conditions]], hence *external parameters* or *[[background fields]]* for the actual [[quantum field]].

Concretely, a widely appreciated proposal ([Moore & Read 1991](#MooreRead91), [Read & Rezayi 1999](#ReadRezayi99)) identifies anyonic ground state [[wavefunctions]] with [[conformal blocks]] of a [[2d CFT]] -- [[AdS3-CFT2 and CS-WZW correspondence|hence]] with [[Chern-Simons theory]] [[quantum states|states]] -- with prescribed poles at the location of the anyons.

Now the [[quantum adiabatic theorem]] says that the sufficiently slow motion of such external parameters transforms the quantum [[ground state]] by [[unitary operators]] ("[[Berry phases]]", see also at *[[adiabatic quantum computation]]*). This suggests ([Avros, Schrieffer & Wilczek 1984, p. 1](#AvrosSchriefferWilczek84),  [Freedman, Kitaev, Larsen & Wang 2003, pp. 6](#FreedmanKitaevLarsenWang03), [Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6)](#NayakSimonSternFreedman08), [Cheng, Galitski & Das Sarma 2011, p. 1](adiabatic+quantum+computation#ChengGalitskiDasSarma11)) that:

\begin{definition}\label{DefectBraiding}
**(adiabatic defect braiding)**
\linebreak
*Anyon braiding statistics is the [[braid group representation]] on a quantum [[ground state]] induced by [[adiabatic theorem|adiabatic]] [[braiding]] of [[TQFT|topological]] [[codimension]]=2 [[defects]] in their [[configuration space of points|configuration space]].*

Here

1. for point-defects the "configuration space" is the [[configuration space of points]] in a [[surface]] (as briefly touched upon already in [Leinaas & Myrheim 1977, pp. 22](#LeinaasMyrheim77), [Wilczek 1982b, p. 959](#Wilczek82b)), such as in the [[plane]], in which case its fundamental group is the [[braid group]];

1. "topological" is meant as in [[topological quantum field theory]]: The induced [[adiabatic theorem|adiabatic]] [[unitary transformation]] is demanded/assumed to depend only on the [[isotopy]]-class of the defect-[[worldlines]], hence only on the underlying [[braid group|braid]]-pattern.

\end{definition}

This notion of anyon "statistics" is at least tacitly implicit in much of the literature on anyons in [[topological quantum computation]], such as in the popular graphics depicting anyon [[worldlines]] as the [[Wilson lines]] in [[Chern-Simons theory]]. (see the graphics [below](#BraidingOfNodalPointsInMomentumSpace)).

Indeed, the effects of [[adiabatic theorem|adiabatic]] braiding of [[defects]] in [[quantum materials]] has been understood and discussed before and in parallel to the term "anyon" becoming established: [Mermin 1979](#Mermin79), [Lo & Preskill 1993](#LoPreskill93).

A concrete realistic example of defect anyons are [[vortex]] anyons see [below](#VortexAnyons).
But the notion of codimension=2 defects subsumes situations that are quite different from the [[quasiparticle]]-excitations imagined in traditional texts on anyons, such as:

* [anyonic band nodes](#AnyonicBandNodes)

* [defect branes](#DefectBranes).

#### Vortex anyons
 {#VortexAnyons}

Specifically, *[[vortex]] anyons* are realized in [[Bose-Einstein condensates]] ([MPSS19](#MPSS19), following [PFCZ01](#PFCZ01)) and in (other) [[superfluids]] ([MMN21](#MMN21)).

{#VorticesInAnyonCondensate} In fact, defect-type *vortex anyons* generically appear in [[condensates]] *of non-defect anyons* ([CDLR19](#CDLR19)):

\begin{imagefromfile}
    "file_name": "VorticesInAnyonCondensate.jpg",
    "width": 430,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [CDLR19](#CDLR19)"
\end{imagefromfile}

A theoretical model of vortex anyons in a [[Higgs field]] [[interaction|coupled to]] [[Chern-Simons theory]] is discussed in [Fröhlich & Marchetti 1988](#FroehlichMarchetti88).
An instructive [[lattice model]] of vortex anyons is analyzed in detail in [Kitaev 2006](#Kitaev06).

Much attention in current efforts towards realizing [[topological quantum computation]] is being paid to anyons realized as [[Majorana zero modes]] [[bound state|bound]] to [[vortices]] ([Das Sarma, Freedman & Nayak 2015](topological+quantum+computation#DasSarmaFreedmanNayak15), cf. [MMBDRSC19](su2-anyon#MMBDRSC19)).

{#MayGeneralize} This situation may generalize to [[parafermion]]-[[su(2)-anyons]], where

> each (anti)[[soliton]] carries [[parafermion]] zero mode which supplies
it with the non-Abelian statistics $[$[Tsvelik 2014a](parafermion#Tsvelik2014a), [p. 2](https://arxiv.org/pdf/1404.2840.pdf#page=2), cf. [Borcherding 2018, pp. 3](parafermion#Borcherding18).
$]$

{#BraidingOfNodalPointsInMomentumSpace} **Braiding of nodal points in momentum space** (graphics from [MMBDRSC19](su2-anyon#MMBDRSC19), [Fig. 1](https://www.nature.com/articles/s41467-019-10397-5.pdf#page=2)):

\begin{imagefromfile}
    "file_name": "MZMVortex-220530.jpg",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

#### Anyonic band nodes?
 {#AnyonicBandNodes}

Around 2020 the view has been emerging that also defects "in momentum/reciprocal space" may behave as anyonic defects under braiding in momentum space. This applies concretely to nodal points (where [[electron bands]] touch or cross) in the momentum space [[Brillouin torus]] of [[topological semi-metals]]:

> here are band crossing points, henceforth called [[vortices]] $[$[Ahnm Park & Yang 2019](#AhnParkYang19)$]$

> a new type non-Abelian "braiding" of nodal-line rings inside the momentum space $[$[Tiwari & Bzdušek 2020](#TiwariBzdusek20)$]$

\begin{imagefromfile}
    "file_name": "BraidingOfBandNodes-220529.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

> (graphics from [[schreiber:Topological Quantum Computation in TED-K|SS22]])

Curiously, these reciprocal/momemtum space anyons lend themselves to tractable laboratory manipulation in a way that has remained notoriously elusive for "position space" anyons:

> Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding $[$[CBSM22](#CBSM22)$]$,

specifically

> it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field $[$[PBMS22](#PBMS22)$]$,

leading to:

> new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.

> Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding. $[$[PBSM22](#PBSM22)$]$.

#### Defect branes
 {#DefectBranes}

In [[string theory]], *[[defect branes]]* are [[D-branes]] or [[M-branes]] of [[codimension]]=2, such as [[D7-branes]] in [[type IIB string theory]] or [[M5-brane]]-[[brane intersection|intersections]] on "[[3-brane in 6d|M3-branes]]" in [[M-theory]].

It has been suggest in [deBoer & Shigemori 2012, p. 65](defect+brane#deBoerShigemori12) that these could behave like anyons. This is further substantiated in [SS22](defect+brane#SS22). See [there](defect+brane#RelationToAnyons) for more.

\linebreak

## Related concepts

* [[boson]], [[fermion]]

* [[parastatistics]]

* [[su(2)-anyon]]

* [[unitary fusion category]]

* [[topological order]]

* [[topological quantum computation]]

* [[quantum Hall effect]]

## References

### General

The concept of [[anyons]] satisfying [[braid group statistics]] originated independently in:

* {#LeinaasMyrheim77} [[Jon Magne Leinaas]], [[Jan Myrheim]], _On the theory of identical particles_, _К теории тождествениых частиц_, Nuovo Cim B 37, 1–23 (1977) ([doi:10.1007/BF02727953](https://doi.org/10.1007/BF02727953))

* {#GoldinMenikoffSharp81} G. A. Goldin, R. Menikoff, D. H. Sharp, *Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect*, J. Math. Phys. **22** 1664 (1981) $[$[doi:10.1063/1.525110](https://doi.org/10.1063/1.525110)$]$

* {#Wilczek82a} [[Frank Wilczek]], _Magnetic Flux, Angular Momentum, and Statistics_, Phys. Rev. Lett. **48** (1982) 1144 (reprinted in [Wilczek 1990, p. 163-165](#Wilczek90)) $[$[doi:10.1103/PhysRevLett.48.1144](https://doi.org/10.1103/PhysRevLett.48.1144)$]$

The term *anyon* was introduced in:

* {#Wilczek82b} [[Frank Wilczek]], *Quantum Mechanics of Fractional-Spin Particles*, Phys. Rev. Lett. **49** (1982) 957 (reprinted in [Wilczek 1990, p. 166-168](#Wilczek90)) $[$[doi:10.1103/PhysRevLett.49.957](https://doi.org/10.1103/PhysRevLett.49.957)$]$

Identification of anyon phases (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of an [[quantum adiabatic theorem|adiabatic transport]] of anyon positions:

* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

The "[[fictitious gauge field]]"-method for modelling anyons:

* {#ArovasSchriefferWilczekZee85} [[Daniel P. Arovas]], [[Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

* {#ChenWilczekWittenHalperin89} [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990, p. 342-408](#Wilczek90))  $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

* {#Wilczek91} [[Frank Wilczek]], *States of Anyon Matter*,  International Journal of Modern Physics B **05** 09 (1991) 1273-1312 [[doi:10.1142/S0217979291000626](https://doi.org/10.1142/S0217979291000626)]

and with specific emphasis on the resulting (abelian!) [[Chern-Simons theory]]:

* {#IengoLechner92} [[Roberto Iengo]], [[Kurt Lechner]], *Anyon quantum mechanics and Chern-Simons theory*, Physics Reports **213** 4 (1992) 179-269 [doi:10.1016/0370-1573(92)90039-3]

The suggestion that the anyonic [[ground state]]-[[wavefunctions]] are essentially [[conformal blocks]] of [[2d CFT]] (notably for [[su(2)-anyons]]):

* {#MooreRead91} [[Gregory Moore]], [[Nicholas Read]], *Nonabelions in the fractional quantum hall effect*, Nuclear Physics B **360** 2–3 (1991) 362-396 $[$doi:10.1016/0550-3213(91)90407-O, [pdf](https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)$]$

* {#ReadRezayi99} [[Nicholas Read]], [[Edward Rezayi]], *Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level*, Phys. Rev. B **59** (1999) 8084 $[$[doi:10.1103/PhysRevB.59.8084](https://doi.org/10.1103/PhysRevB.59.8084)$]$

More comprehensive accounts of anyons:

* {#Wilczek90} [[Frank Wilczek]], *Fractional Statistics and Anyon Superconductivity*, World Scientific (1990) [[doi:10.1142/0961](https://doi.org/10.1142/0961)]

* {#Kitaev06} [[Alexei Kitaev]], *Anyons in an exactly solved model and beyond*, Annals of Physics **321** 1 (2006) 2-111 [[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)]

* [[Eduardo Fradkin]], *Anyon superconductivity*, Chapter 11 in: *Field Theories of Condensed Matter Physics*, Cambridge University Press (2013) 414-431 [ISBN: 9781139015509, [doi:10.1017/CBO9781139015509.013](https://doi.org/10.1017/CBO9781139015509.013)]

See also:

* [[Martin Greiter]], [[Frank Wilczek]], *Fractional Statistics* [[arXiv:2210.02530](https://arxiv.org/abs/2210.02530)]

Rigorous discussion in terms of [[superselection sectors]] in [[algebraic quantum field theory]]:

* {#FroehlichMarchetti88} [[Jürg Fröhlich]], [[Pieralberto Marchetti]], *Quantum field theory of anyons*, Lett. Math. Phys. **16** (1988) 347–358 (reprinted in [Wilczek 1990, p. 202-213](#Wilczek90)) $[$[doi:10.1007/BF00402043](https://doi.org/10.1007/BF00402043)$]$

* [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], *Braid statistics in local quantum theory*, Reviews in Mathematical Physics, **2** 03 (1990) 251-353 $[$[doi:10.1142/S0129055X90000107](https://doi.org/10.1142/S0129055X90000107)$]$

* {#FroehlichGabbianiMarchetti90} [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2), [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- I: General theory_,   Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226.  ([euclid:cmp/1104179464](http://projecteuclid.org/euclid.cmp/1104179464))

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- II: Geometric aspects and conformal covariance_,  Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992)  ([doi:10.1142/S0129055X92000170](https://doi.org/10.1142/S0129055X92000170) [pdf](ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/92/superselection_sectors_II_RMP.pdf))

Discussion of anyon-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:

* {#Wu84} [[Yong-Shi Wu]], *Multiparticle Quantum Mechanics Obeying Fractional Statistics*, Phys. Rev. Lett. **53** (1984) 111 $[$[doi:10.1103/PhysRevLett.53.111](https://doi.org/10.1103/PhysRevLett.53.111), [pdf](https://core.ac.uk/download/pdf/276286925.pdf)$]$

* [Fröhlich, Gabbiani & Marchetti 1990, p. 20](#FroehlichGabbianiMarchetti90)

* {#ImboImboSudarshan90} Tom Imbo, Chandni Shah Imbo, E. C. G. Sudarshan, *Identical particles, exotic statistics and braid groups*, Physics Letters B **234** 1–2, (1990) 103-107 $[$doi:10.1016/0370-2693(90)92010-G, [pdf](https://lib-extopc.kek.jp/preprints/PDF/2000/0033/0033552.pdf)$]$

* {#BCMS93} Garth A. Baker, Geoff S. Canright, Shashikant B. Mulay, Carl Sundberg, *On the spectral problem for anyons*, Communications in Mathematical Physics **153** (1993) 277–295 $[$[doi:10.1007/BF02096644](https://doi.org/10.1007/BF02096644)$]$

* {#MundSchrader93} J. Mund, [[Robert Schrader]], *Hilbert Spaces for Nonrelativistic and Relativistic "Free" Plektons (Particles with Braid Group Statistics)*, in *Advances in dynamical systems and quantum physics* (Capri, 1993), World Sci. (1995) 235–259 $[$[arXiv:hep-th/9310054](https://arxiv.org/abs/hep-th/9310054)$]$

  followed up by:

  [[Klaus Fredenhagen]], [[Matthias Gaberdiel]] and Stefan M. Rüger, *Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory*, Communications in Mathematical Physics **175** (1996) 319–335 $[$[doi:10.1007/BF02102411](https://doi.org/10.1007/BF02102411)$]$

* {#DFT97} Gianfausto Dell'Antonio, Rodolfo Figari & Alessandro Teta, *Statistics in Space Dimension Two*, Letters in Mathematical Physics **40** (1997) 235–256 $[$[doi:10.1023/A:1007361832622](https://doi.org/10.1023/A:1007361832622)$]$

* {#Myrheim99} [[Jan Myrheim]], *Anyons*, p. 265-414 in: *Topological aspects of low dimensional systems*, Les Houches LXIX, Springer (1999) $[$[doi:10.1007/3-540-46637-1](https://doi.org/10.1007/3-540-46637-1), [[Myrheim-Anyons.pdf:file]]$]$

* {#MurthyShankar09} [[M.V.N. Murthy]], [[Ramamurti Shankar]], *Exclusion Statistics: From Pauli to Haldane* (1999, 2009) $[$[dspace:123456789/334](https://dspace.imsc.res.in/xmlui/handle/123456789/334), [pdf](https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1), [[MurthyShankar-ExclusionStatistics.pdf:file]]$]$

* {#DMV03} G. Date, [[M.V.N. Murthy]], [[Radhika Vathsan]], *Classical and Quantum Mechanics of Anyons* $[$[arXiv:cond-mat/0302019](https://arxiv.org/abs/cond-mat/0302019)$]$

The topic of [[quantum measurement]] of non-abelian anyons is crucial to their identification in [[experiment]] but has received little attenion, one exception being:

* {#BondersonShtengelSlingerland07} [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Decoherence of Anyonic Charge in Interferometry Measurements*, Phys. Rev. Lett. **98** (2007) 070401 [[doi:10.1103/PhysRevLett.98.070401](https://doi.org/10.1103/PhysRevLett.98.070401), [arXiv:quant-ph/0608119](https://arxiv.org/abs/quant-ph/0608119)]

* {#BondersonShtengelSlingerland08} [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Interferometry of non-Abelian Anyons*, Annals Phys. **323** (2008) 2709-2755 [[doi:10.1016/j.aop.2008.01.012](https://doi.org/10.1016/j.aop.2008.01.012), [arXiv:0707.4206](https://arxiv.org/abs/0707.4206)]

[[!include anyonic topological order via braided fusion categories -- references]]

[[!include anyons in the quantum Hall effect -- references]]

[[!include anyons in topological superconductors -- references]]

### Experimental detection of anyons
 {#ExperimentalRealization}

While the occurrence of [[anyon]]-excitations in the [[quantum Hall effect]] is a robust theoretical prediction (see the references [above](#AnyonsInTheQuantumHallEffectReferences)), and while the quantum Hall effect itself has long been established in [[experiment]], the actual observation of anyons in these systems is subtle:

An early claim of the observation of non-abelian anyons seems to remain unconfirmed:

* Sanghun An, P. Jiang, H. Choi, W. Kang, S. H. Simon, L. N. Pfeiffer, K. W. West, K. W. Baldwin, _Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect_ ([arXiv:1112.3400](https://arxiv.org/abs/1112.3400))

The claimed observation of abelian anyons is apparently more securely established:

* H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, G. Fève:

  *Fractional statistics in anyon collisions*, Science 368, 173-177 (2020) ([arXiv:2006.13157](https://arxiv.org/abs/2006.13157))

* James Nakamura, Shuang Liang, Geoffrey C. Gardner, Michael J. Manfra, _Direct observation of anyonic braiding statistics_, Nat. Phys. 16, 931–936 (2020). ([arXiv:2006.14115](https://arxiv.org/abs/2006.14115), [doi:10.1038/s41567-020-1019-1](https://doi.org/10.1038/s41567-020-1019-1))

* Bob Yirka, _Best evidence yet for existence of anyons_, PhysOrg News July 10, 2020  ([phys.org/news/2020-07](https://phys.org/news/2020-07-evidence-anyons.html))

[[!include defect anyons -- references]]

[[!include anyonic braiding in momentum space -- references]]

[[!include topological quantum computation with anyons -- references]]

[[!redirects braid statistics]]

[[!redirects anyon]]
[[!redirects anyons]]

[[!redirects anyon statistics]]

[[!redirects anyon braiding]]
[[!redirects anyonic braiding]]

[[!redirects abelian anyon]]
[[!redirects abelian anyons]]

[[!redirects non-abelian anyon]]
[[!redirects non-abelian anyons]]


braid lemma

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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #BraidLemma}
###### Proposition
**(braid lemma)**

Given a [[commuting diagram]] of [[abelian groups]] of the following form

$$
  \array{
    && A  && \longrightarrow && B && \longrightarrow && C  && \longrightarrow && D
    \\
    & \nearrow && \searrow && \nearrow && \searrow && \nearrow && \searrow && \nearrow
    \\
    E && && F && && G && && H
    \\
    & \searrow && \nearrow && \searrow && \nearrow && \searrow && \nearrow
    \\
    && I && \longrightarrow && J && \longrightarrow && K
  }
  \,.
$$

Consider the following four sequences inside the diagram

1. $E\to A \to B \to G \to K$;

1. $E \to I \to J \to G \to C \to D$;

1. $A \to F \to J \to K \to H \to D$;

1. $I \to F \to B \to C \to H$.

Then: if the first three of these are [[long exact sequences]] and the fourth is a [[chain complex]], then the fourth is also long exact.

=--

([Munkres, §26, Exc.](#Munkres))

## Applications

### Long exact sequence of a triple in homology

Given a [[generalized homology theory]] $H_\bullet$, then by definition for every inclusion $A \hookrightarrow X$ of topological spaces, there is an [[long exact sequence]]

$$
  \cdots \to H_{n+1}(X,A) \overset{\delta}{\longrightarrow} H_n(A) \longrightarrow H_n(X)\longrightarrow H_n(X,A) \to \cdots
  \,.
$$

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

For $H_\bullet$ an unreduced [[generalized homology theory]] and for two consecutive inclusions $Z \hookrightarrow Y \hookrightarrow X$ (a "triple" $(X, Y, Z)$) there is a long exact sequence of the form

$$
  \cdots H_{n+1}(X,Y) \to H_{n}(Y,Z) \to H_n(X,Z) \to H_n(X,Y) \to \cdots
  \,.
$$

Dually for [[generalized (Eilenberg-Steenrod) cohomology]].

=--

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

Consider the following braid diagram

<img src="http://www.ncatlab.org/nlab/files/BraidDiagramForHomologyOnTripled.jpg" width="500">

(graphics from [this Maths.SE comment](http://math.stackexchange.com/a/1180681/58526))

The blue, purple and the black sequence are the exact sequences of the pairs $(X,Y)$, $(Z,Y)$ and $(Z,X)$, respectively. The orange sequence is a chain complex by these exact sequences and using the commutativity of the diagram, and because the diagonals $H_\bullet(Y,Z) \to H_\bullet(X,Z) \to H_\bullet(X,Y)$ factor through  $H_\bullet(Y,Y) = 0$. Hence the braid lemma, prop. \ref{BraidLemma}, implies the claim.

=--

## References

* {#Munkres} Munkres, p. 148 of _Elements of algebraic topology_

[[!redirects braid lemmas]]


braid representation

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

## Idea

A *braid representation* is a [[linear representation]] of a [[braid group]].

In [[topological quantum computation]] on [[anyons]], braid representations serve as [[quantum gates]].

## Examples

* [[Burau representation]]

## Related concepts

* [[quantum Yang-Baxter equation]]

* [[braid group statistics]] ([[anyons]])

* [[Knizhnik-Zamolodchikov equation]]

* [[representation stability]]

## References

[[!include braid group representations -- references]]

[[!redirects braid representations]]

[[!redirects braid group representation]]
[[!redirects braid group representations]]


braided 2-group

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#Contents#
* table of contents
{:toc}

## Definition

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

A [[2-group]] $G$ is _braided_ if it is equipped with the following equivalent [[structure]]:

1. Regarded as a [[monoidal category]], $G$ is a [[braided monoidal category]].

1. The [[delooping]] [[2-groupoid]] $\mathbf{B}G$ is a [[3-group]].

1. The double [[delooping]] [[3-groupoid]] $\mathbf{B}^2 G$ exists.

1. The groupal [[A-∞ algebra]]/[[E1-algebra]] structure on $G$ refines to an [[E2-algebra]] structure.

1. $G$ is a [[k-tuply groupal n-groupoid|doubly groupal groupoid]].

1. $G$ is a groupal [[k-tuply monoidal (n,r)-category|doubly monoidal (1,0)-category]].

=--

## Related concepts

* [[braided monoidal groupoid]]

* [[k-tuply groupal n-groupoid]]

* [[symmetric 2-group]]

* [[braided ∞-group]]

* [[sylleptic 3-group]]

## References

Under the name "braided gr-categories" or "braided cat-groups" and thought of as a sub-class of [[braided monoidal categories]], the notion of braided 2-groups is considered in:

* [[André Joyal]], [[Ross Street]], Section 3 of: _Braided tensor categories_, Adv. Math. **102** (1993), 20--78 ([doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055))

* [[M. Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], *Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem*, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498))

* [[Antonio R. Garzón]], Jesus G Miranda, Sec. 1 of: *Homotopy theory for (braided) cat-groups*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 38 (1997) no. 2, pp. 99-139 ([numdam:CTGDC_1997__38_2_99_0](http://www.numdam.org/item/?id=CTGDC_1997__38_2_99_0))

* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Appendix E of: *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

As a special case of [[k-tuply groupal n-groupoids]]:

* [[Antonio R. Garzón]], Jesus G. Miranda, Section 6 of: _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

In the generality of [[braided ∞-group|braided ∞-group stacks]] the notion appears in

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Def. 4.28 of: _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 ([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

A discussion of [[∞-group extensions]] by braided 2-groups is in

* [[Evan Jenkins]], _Extensions of groups by braided 2-groups_ ([arXiv:1106.0772](http://arxiv.org/abs/1106.0772))

[[!redirects braided 2-groups]]


braided 3-group

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A braided 3-group is a [[braided ∞-group]] which is a [[3-group]]. For $G$ a 3-group, a braiding on it is the following equivalent structure

1. the structure of a [[2-group]] on the [[delooping]] $\mathbf{B}G$;

1. a doudle delooping $\mathbf{B}^2 G$;

1. a lift of tha [[A-infinity algebra|A-∞]]=[[En-algebra|E-1]]-algebra structure on $G$ to an [[En-algebra|E-2]] algebra structure.

## Examples

For $R$ a [[commutative ring]], and $Alg_R \simeq 2 Vect_R$ the [[braided monoidal 2-category]] of $R$-[[associative algebras|algebras]], [[bimodules]] and bimodule [[homomorphism]], the maximal [[3-group]]

$$
  \mathbf{Br}(R) \hookrightarrow Core(Alg_R)
$$

inside is a braided 3-group. Its [[homotopy groups]] are the [[Brauer group]], the [[Picard group]] and the [[group of units]] of $R$. See at _[Brauer group -- Relation to category of modules](Brauer+group#RelationToCatsOfModules)_ for more on this.

## Related concepts

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[braided ∞-group]],

  * [[braided 2-group]]

  * **braided 3-group**

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* [[abelian ∞-group]]

  * [[symmetric 2-group]]

## References

As a special case of [[k-tuply groupal n-groupoids]] the notion is at least implicit in:

* [[Antonio R. Garzón]], Jesus G. Miranda, Section 6 of: _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

In the generality of [[braided ∞-group|braided ∞-group stacks]] the notion appears in:

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Def. 4.28 of: _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 ([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

[[!redirects braided 3-groups]]


braided geometry

__Braided geometry__ is a flavour of [[noncommutative geometry]] in which the commutativity is relaxed to some sort of braiding.

Most obvious kind of braided geometry is to do geometry within a [[braided monoidal category]]. This has been developed by [[Vladimir Drinfel'd]] and then much further by [[Shahn Majid]].

This approach has some problems, for example, if the R-matrix is a deformation of the standard flip, then various objects in the category have their graded dimensions different from the classical case, i.e. "the deformation is not flat". If one replaces the FRT quantized function algebra by [[reflection equation algebra]] then, for Hecke R-matrix, the deformation
is flat.

Related entries: [[braid group]], [[quantum Yang-Baxter equation]], [[classical r-matrix]], [[bialgebra cocycle]], [[Drinfeld associator]], [[Drinfeld twist]], [[dynamical Yang-Baxter equation]], [[quasitriangular bialgebra]], [[quasi-Hopf algebra]], [[ribbon category]], [[braiding]], [[braided monoidal category]], [[reflection equation algebra]], [[double Hecke algebra]]


braided infinity-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
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+-- {: .hide}
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=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
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## Definition

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

An [[∞-group]] $G$ is _braided_ if it is equipped with the following equivalent [[structure]]

1. Regarded as a [[monoidal (∞,1)-category]], $G$ is a [[braided monoidal (∞,1)-category]].

1. The [[delooping]] [[∞-groupoid]] $\mathbf{B}G$ has the structure of an [[∞-group]].

1. The double [[delooping]] [[∞-groupoid]] $\mathbf{B}^2 G$ exists.

1. The groupal [[A-∞ algebra]]/[[E1-algebra]] structure on $G$ refines to an [[E2-algebra]] structure.

1. $G$ is a [[k-tuply groupal n-groupoid|doubly groupal ∞-groupoid]].

1. $G$ is a groupal [[k-tuply monoidal (n,r)-category|doubly monoidal (∞,0)-category]].

=--

## Examples

See the examples at _[[braided 2-group]]_, _[[braided 3-group]]_.

## Related concepts

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[May recognition theorem]]

* **braided ∞-group**,

  * [[braided 2-group]]

  * [[braided 3-group]]

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* [[abelian ∞-group]]

  * [[symmetric 2-group]]

## References

In the generality of [[braided ∞-group|braided ∞-group stacks]] the notion appears in:

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Def. 4.28 of: _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 ([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

[[!redirects braided ∞-group]]
[[!redirects braided ∞-groups]]
[[!redirects braided infinity-groups]]


braided monoidal (∞,1)-category

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#### Higher algebra
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## Definition

Where a [[monoidal (∞,1)-category]] is an [[E1-algebra]] in [[(∞,1)Cat]], a _braided monoidal $(\infty,1)$-category_ is an [[E2-algebra]] in [[(∞,1)Cat]].

If the [[(∞,1)-category]] $\mathcal{C}$ underlying a braided monoidal $(\infty,1)$-category $(\mathcal{C}, \otimes)$ is a [[1-category]], then $(\mathcal{C}, \otimes)$ is a [[braided monoidal category]].

## Related concepts

* [[monoidal category]], [[monoidal (∞,1)-category]]

* [[braided monoidal category]], **braided monoidal $(\infty,1)$-category

  * [[braided ∞-group]]

* [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]

* [[closed monoidal category]] ,  [[closed monoidal (∞,1)-category]]


braided monoidal 2-category

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* table of contents
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## Idea ##

A _braided monoidal (weak) 2-category_ is a [[monoidal 2-category]] with a categorified version of a [[braided monoidal category|braiding]].

That is, it is a [[2-category]] $C$ equipped with a [[tensor product]] $\otimes : C \times C \to C$ [[2-functor]] which satisfies the first in a hierarchy of conditions for being commutative up to equivalence.  In the language of [[k-tuply monoidal n-category|k-tuply monoidal n-categories]], a braided monoidal 2-category is a *doubly monoidal 2-category*.  As described there, this may be identified with a pointed [[4-category]] with a single [[object]] and a single [[1-morphism]].  We can also say that it is a monoidal 2-category whose [[En-algebra|E1-algebra]] structure is refined to an [[En-algebra|E2-algebra]] structure.

## Properties

### Coherence theorem

* [[coherence theorem for braided monoidal bicategories]]

### Picard 2-groupoid

The [[Picard 2-groupoid]] of a braided monoidal 2-category is a [[braided 3-group]].

## Related concepts

* [[monoidal 2-category]], [[3-group]]

* **braided monoidal 2-category**, [[braided 3-group]]

* [[sylleptic monoidal 2-category]], [[sylleptic 3-group]]

* [[symmetric monoidal 2-category]], [[symmetric 3-group]]

* [[k-tuply monoidal (n,r)-category]]

## References

* {#Breen88} [[Lawrence Breen]], _Une lettre à P. Deligne au sujet des $2$-catégories tressées_ (1988) ([pdf](http://www.math.univ-paris13.fr/~breen/deligne.pdf))

* [[André Joyal]], [[Ross Street]], Section 5 of: _Braided tensor categories_, Adv. Math. **102** (1993), 20--78 ([doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055))

* [[Mikhail Kapranov]], [[Vladimir Voevodsky]], *2-Categories and Zamolodchikov tetrahedra equations*, in: _Proc. Symp. Pure Math._ **56** Part 2 (1994), AMS, Providence, pp. 177--260 ([doi:10.1090/pspum/056.2](https://doi.org/10.1090/pspum/056.2))

* [[Mikhail Kapranov]], [[Vladimir Voevodsky]], *Braided monoidal 2-categories and Manin-Schechtman higher braid groups*, Journal of Pure and Applied Algebra Volume 92, Issue 3, 25 March 1994, Pages 241-267 (doi:10.1016/0022-4049(94)90097-3)

* [[John C. Baez]], [[Martin Neuchl]], *Higher-Dimensional Algebra I: Braided Monoidal 2-Categories*, Adv. Math. 121 (1996), 196-244 ([arXiv:q-alg/9511013](https://arxiv.org/abs/q-alg/9511013))

* [[Sjoerd E. Crans]], Section 2.2. of: *Generalized Centers of Braided and Sylleptic Monoidal 2-Categories*, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 ([doi:10.1006/aima.1998.1720](https://doi.org/10.1006/aima.1998.1720))

* Thibault D. Décoppet, Hao Xu, *Local Modules in Braided Monoidal 2-Categories* [[arXiv:2307.02843](https://arxiv.org/abs/2307.02843)]

[[!redirects braided monoidal 2-categories]]
[[!redirects braided monoidal bicategory]]
[[!redirects braided monoidal bicategories]]


braided monoidal category

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## Idea

Intuitively speaking, a braided monoidal category is a category with a tensor product and an isomorphism called the 'braiding' which lets us 'switch' two objects in a tensor product like $x \otimes y$.  Thus the tensor product is "commutative" in a sense, but not as coherently commutative as in a [[symmetric monoidal category]].

A braided monoidal category is a special case of the notion of [[braided pseudomonoid]] in a [[braided monoidal 2-category]].

## Definition
 {#Definition}

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

A **braided monoidal category**, or ("braided [[tensor category]]", but see there), is a [[monoidal category]] $\mathcal{C}$ equipped with a [[natural isomorphism]]

$$
  B_{x,y} : x \otimes y \to y \otimes x
$$

called the **[[braiding]]**, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved (called the **hexagon identities** encoding the compatibility of the braiding with the [[associator]] for the tensor product):

$$
  \array{
   (x \otimes y) \otimes z
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{B_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{B_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes B_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
$$

and

$$
  \array{
   x \otimes (y \otimes z)
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{B_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes B_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{B_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
$$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

=--

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

If the braiding in def. \ref{BraidedMonoidalCategory} "squares" to the identity in that $B_{y,x} \circ B_{x,y} = id_{x \otimes y}$, then the braided monoidal category is called a _[[symmetric monoidal category]]_.

=--

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

Intuitively speaking, the first hexagon identity in def. \ref{BraidedMonoidalCategory} says we may braid $x$ past $y \otimes z$ 'all at once' or in two steps. The second hexagon identity says that we may braid $x \otimes y$ past $z$ all at once or in two steps.

=--

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

From these axioms in def. \ref{BraidedMonoidalCategory}, it follows that the braiding is compatible with the left and right [[unitors]] $l_x : I \otimes x \to x$ and $r_x : x \otimes I \to x$.   That is to say, for all objects $x$ the diagram

$$
  \array{
    I \otimes x &&\stackrel{B_{I,x}}{\to}&& x \otimes I
    \\
    & {}_{l_x}\searrow && \swarrow_{r_x}
    \\
    && x
  }
$$

commutes.

=--

### In terms of higher monoidal structure

In terms of the language of [[k-tuply monoidal n-categories]] a braided monoidal category is a _doubly monoidal 1-category_ .

Accordingly, by [[delooping]] twice, it may be identified with a [[tricategory]] with a single [[object]] and a single 1-[[morphism]].

However, unlike the definition of a monoidal category as a [[bicategory]] with one object, this identification is not trivial; a doubly-degenerate tricategory is literally a category with two monoidal structures that [[interchange law|interchange]] up to isomorphism.  It requires the [[Eckmann-Hilton argument]] to deduce an equivalence with braided monoidal categories.

A commutative [[monoid]] is the same as a monoid [[internalization|in the category of]] monoids.
Similarly, a braided monoidal category is equivalent to a monoidal-category object (that is, a [[pseudomonoid]]) in the monoidal 2-category of monoidal categories.  This result goes back to the [1986 paper by Joyal and Street](http://maths.mq.edu.au/~street/JS1.pdf).  (There is also a notion of [[braided pseudomonoid]] that specializes directly in [[Cat]] to braided monoidal categories.)

A braided monoidal category is equivalently a category that is equipped with the structure of an [[algebra over an operad|algebra over]] the [[little cubes operad|little 2-cubes operad]].

Details are in example 1.2.4 of

* [[Jacob Lurie]], $\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]

### The 2-category of braided monoidal categories

There is a [[strict 2-category]] BrMonCat with:

* braided monoidal categories as objects,
* [[braided monoidal functor|braided monoidal functors]] as morphisms,
* [[braided monoidal natural transformation|braided monoidal natural transformations]] as 2-morphisms.

## Examples

* Any [[symmetric monoidal category]] is a braided monoidal category.

* The monoidal category $R$-$\mathcal{GMod}$ of [[graded modules]] over a [[commutative ring]] $R$ (with the usual [[tensor product of modules|tensor product of graded modules]]) can be made into a braided monoidal category in multiple ways, each one given by an invertible element $u$ of the base ring. Infact, all braidings on $R$-$\mathcal{GMod}$ arise in this way (as in [Joyal, Street](JoyalStreet86)). The braiding $B^u_{V,W} : V \otimes W \to W \otimes V$, is defined as
$$
  x \otimes y \mapsto u^{|x||y|} y \otimes x,
$$
where $|x|$ and $|y|$ denote the degrees. It's evident that the resulting braided monoidal category is symmetric if and only if $u^2 = 1$.

* The category of [[crossed G-set|crossed G-sets]].

* The [[braid category]] is the [[free construction|free]] [[braided monoidal category|braided]] [[strict monoidal category]] on an object.

* The category of [[vines]] is the [[free construction|free]] [[braided monoidal category|braided]] [[strict monoidal category]] containing a [[braided monoid]].

## Properties

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[k-tuply monoidal (n,r)-category]]

* [[monoidal category]], [[monoidal (∞,1)-category]]

* **braided monoidal category**, [[braided monoidal (∞,1)-category]]

  * [[modular tensor category]]

  * [[braided monoidal dagger category]]

  * [[braided monoidal 2-category]]

* [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]

* [[closed monoidal category]] ,  [[closed monoidal (∞,1)-category]]

* [[2-fold monoidal category]], [[duoidal category]]

* [[braided monoidal groupoid]]

  * [[braided 2-group]], [[braided ∞-group]]

## References

Textbook accounts:

* [[Saunders MacLane]], §XI of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux94} [[Francis Borceux]], Section 6.1 of: *Handbook of Categorical Algebra* Vol. 2: *Categories and Structures* $[$[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)$]$, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Chapter 2 of: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)$]$

  > (focus on [[tensor categories]])

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

* _[[geometry of physics -- categories and toposes]]_, Section 2: _[Basic notions of categorical algebra](geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra)_

The original papers on braided monoidal categories are by [[Andre Joyal|Joyal]] and [[Ross Street|Street]]. The published version does not completely supersede the _Macquarie Math Reports_ version, which has some nice extra results:

* {#JoyalStreet86} [[André Joyal]] and [[Ross Street]], *Braided monoidal categories*, _Macquarie Math Reports_ **860081** (1986) ([pdf](http://maths.mq.edu.au/~street/JS1.pdf))

* [[André Joyal]], [[Ross Street]], _Braided tensor categories_, Adv. Math. **102** (1993), 20--78 ([doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055))

Around the same time the same definition was also proposed independently by [[Lawrence Breen]] in a letter to [[Pierre Deligne]]:

* {#Breen88} [[Lawrence Breen]], _Une lettre à P. Deligne au sujet des $2$-catégories tressées_ (1988) ([pdf](http://www.math.univ-paris13.fr/~breen/deligne.pdf))

Textbook accounts include

* {#EGNO15} [[nLab:Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[nLab:Victor Ostrik]], chapter 2 of _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf))

For a review of definitions of braided monoidal categories, braided monoidal functors and braided monoidal natural transformations, see:

* [[John Baez]], [Some definitions everyone should know](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).

For an elementary introduction to braided monoidal categories using [[string diagram]]s, see:

* [[John Baez]] and [[Mike Stay]], [Physics, topology, logic and computation: a Rosetta Stone](http://math.ucr.edu/home/baez/rosetta.pdf), to appear in _New Structures in Physics_, ed. Bob Coecke.

A generalisation of braidings to *lax braidings*, where $B$ is not required to be invertible:

* [[Brian Day]], [[Elango Panchadcharam]], [[Ross Street]], _Lax braidings and the lax centre_, Contemporary Mathematics **441** 1 (2007) [[pdf](http://science.mq.edu.au/~street/laxcentre.pdf), [[DayPanchadcharamStreet-LaxBraidings.pdf:file]]]

[[!redirects braided monoidal category]]
[[!redirects braided monoidal categories]]
[[!redirects braided category]]
[[!redirects braided categories]]
[[!redirects braided tensor category]]
[[!redirects braided tensor categories]]

[[!redirects hexagon identity]]
[[!redirects hexagon identities]]


braided monoidal dagger category

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## Definition ##

A **braided monoidal dagger category** is a [[monoidal dagger category]] $(C, \otimes, \Iota)$ which is also a [[braided monoidal category]], in that:

* for objects $A \in Ob(C)$ and $B \in Ob(C)$, there is a [[natural unitary isomorphism]] called the **braiding** of $A$ and $B$

$$\beta_{A,B}: A\otimes B \cong^\dagger B\otimes A$$

* all objects $D \in Ob(C)$, $E \in Ob(C)$, and $F \in Ob(C)$ satisfy the **first hexagon identity**

$$\alpha_{E,F,D} \circ \beta_{D, E \otimes F} \circ \alpha_{D,E,F} = (id \otimes \beta_{D,F}) \circ \alpha_{E,D,F} \circ (\beta_{D, E} \otimes id)$$

* all objects $D \in Ob(C)$, $E \in Ob(C)$, and $F \in Ob(C)$ satisfy the **second hexagon identity**

$$\alpha_{F,D,E}^{-1} \circ \beta_{D \otimes E, F} \circ \alpha^{-1}_{D,E,F} = (\beta_{D, F} \otimes id) \circ \alpha^{-1}_{D,F,E} \circ (id \otimes \beta_{E, F})$$

## Examples ##

* [[symmetric monoidal dagger category]], [[symmetric monoidal groupoid]], [[2-group]]

* [[compact closed dagger category]]

## See also ##

* [[dagger category]]

* [[monoidal dagger category]]

* [[braided monoidal category]]


braided monoidal functor

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## Idea

A _braided monoidal functor_ is a [[functor]] $F : C \to D$ between [[braided monoidal categories]] that is a [[monoidal functor]] which respects the braiding on both sides.

## Properties

Between symmetric monoidal categories a braided monoidal functor is the same as a [[symmetric monoidal functor]].

## Related concepts

* [[monoidal functor]]

* [[strong monoidal functor]]

* [[lax monoidal functor]]

* [[oplax monoidal functor]]

* [[bilax monoidal functor]]

* [[Frobenius monoidal functor]]

* **braided monoidal functor**

* [[symmetric monoidal functor]]

## References

An exposition is in

* [[John Baez]], [Some definitions everyone should know](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).

[[!redirects braided monoidal functors]]

[[!redirects symmetric monoidal functor]]
[[!redirects symmetric monoidal functors]]


braided monoidal groupoid

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# Contents
* table of contents
{: toc}

## Idea

A *braided monoidal groupoid* is a [[braided monoidal category]] whose underlying [[category]] happens to be a [[groupoid]] (hence all whose [[morphisms]] are [[isomorphisms]].)

Equivalently: A [[monoidal groupoid]] with [[braiding]] that satisfies the [[hexagon identities]],

Equivalently: A [[1-type|1-truncated]] [[En-algebra|$E_2$-space]].

## Definitions

A **braided monoidal groupoid** is a [[monoidal groupoid]] $G$ with a [[natural isomorphism|natural]] [[unitary morphism]] $\beta_{A,B} : A \otimes B \cong^\dagger B \otimes A $ such that for all objects $A:G$, $B:G$, and $C:G$,

$$\alpha_{B,C,A} \circ \beta_{A, B \otimes C} \circ \alpha_{A,B,C} = (id_B \otimes \beta_{A,C}) \circ \alpha_{B,A,C} \circ (\beta_{A,B}\otimes id_C)$$

and

$$\alpha^{-1}_{C,A,B} \circ \beta_{A \otimes B, C} \circ \alpha^{-1}_{A,B,C} = (\beta_{A,C} \otimes id_B) \circ \alpha^{-1}_{A,C,B} \circ (id_A \otimes \beta_{B,C}$$

## See also

* [[commutative monoid]]
* [[En-algebra]]
* [[groupoid]]
* [[monoidal groupoid]]
* [[braided 2-group]]
* [[symmetric monoidal groupoid]]
* [[k-tuply monoidal n-groupoid]]
* [[braided monoidal category]]

[[!redirects braided monoidal groupoid]]
[[!redirects braided monoidal groupoids]]


braided monoidal natural transformation

#Contents#
* table of contents
{:toc}

## Idea

A _braided monoidal natural transformation_ is a [[natural transformation]] between [[braided monoidal functor]]s that respects the braided monoidal structure.

In fact, a braided monoidal natural transformation is the same as a [[monoidal natural transformation]] (as long as it goes between braided monoidal functors).

## Related concepts

* [[symmetric monoidal natural transformation]]

## References

See, for example,

* [[John Baez]], _Some definitions everyone should know_, [pdf](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).


braided object

A **braided object** in a [[monoidal category]] $(C,\otimes,I)$ is an object $V$ equipped with an invertible morphism $\sigma : V\otimes V \to V\otimes V$ satisfying the [[Yang-Baxter equation]]
$$(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)$$
More generally, if $\sigma$ is not necessarily invertible but still satisfies the Yang-Baxter equation, then $(V,\sigma)$ is called a **pre-braided object**.

## Related concepts

* [[group object]]
* [[monoid object]]
* [[virtual knot theory]]

## References

* {#LebedThesis} [[Victoria Lebed]], _Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles_, Thèse, Université Paris Diderot, 2012. ([pdf](http://www.maths.tcd.ie/~lebed/thesis_LEBED.pdf)) Note that the title is in French ("Braided objects: a unifying study of algebraic structures and a categorification of virtual braids") but the main text of the thesis is in English.

* {#Lebed13} [[Victoria Lebed]], Categorical Aspects of Virtuality and Self-Distributivity, _Journal of Knot Theory and its Ramifications_, 22 (2013), no. 9, 1350045, 32 pp. ([doi](http://dx.doi.org/10.1142/S0218216513500454))  According to the author, [arXiv:1206.3916](http://arxiv.org/abs/1206.3916) is "an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles".


braiding

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## Idea

The [[structure]] of a [[braided monoidal category]] on top of the [[underlying]] [[monoidal category]] with [[tensor product]] "$\otimes$" is a [[natural isomorphism]] of the form

$$
  B_{x,y}
    \;\colon\;
  x \otimes y
    \longrightarrow
  y \otimes x
  \,,
$$

called the **braiding**.

A braided monoidal category is called *[[symmetric monoidal category|symmetric]]* if and only if $B_{x,y}$ and $B_{y,x}$ are [[inverse morphisms]] to each other (while they are [[isomorphisms]] in any case).

## Examples

\begin{example}
\label{BraidingOnPlainModules}
In [[Vect]] (or generally [[Mod]]), the braiding maps elements $a\otimes b$ of a [[tensor product of vector spaces]] ([[tensor product of modules|of modules]]) $X \otimes Y$ to $b \otimes a$.
\end{example}

\begin{example}
The braiding for the [[tensor product of chain complexes]] and that of [[super vector spaces]] is as in Exp. \ref{BraidingOnPlainModules} up to multiplication by sign $a \otimes b \mapsto (-1)^{deg(a) deg(b)} (b \otimes a)$ (see at *[[signs in supergeometry]]*).
\end{example}

## Related concepts

* [[braid group]]

## References

See the references at *[[braided monoidal category]]*

[[!redirects braidings]]


Bram Geron

Bram Geron is a PhD student in Birmingham, working on the theory of programming languages.

* [website](https://bram.xyz/research)

category:people


Bram Mesland

* [homepage](http://www2.warwick.ac.uk/fac/sci/maths/people/staff/bram_mesland/)

* [genealogy page](http://genealogy.impa.br/id.php?id=136830)

category: people


branched cover

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## Idea

A "branched" or "ramified" covering space is much like a plain [[covering space]], only that over suitable subspaces, such as isolated points, in the base space -- called the _branch points_ or _ramification points_ -- sheets of the covering may merge, hence that the [[group]] of [[cover automorphisms]] has [[fixed points]] here.

The archetypical examples are in [[complex analysis]] and here the most archetypical example of all is the [[graph of a function|graph]] of (any choice of) the  [[square root]] function on the [[complex plane]], which is a [[double cover]] away from the point at the origin, but including the origin it is a branched double cover with branch point that origin

[[Isbell duality|Dually]] branching is reflected in [[rings of functions]] by [[ramification of ideals]].

Notably under the [[function field analogy]] one may also understand [[ramification of ideals]] in [[number fields]] as encoding branched coverings ("over Spec([[F1]])")

## Examples

### Branched covers of $n$-spheres
 {#BranchedCoversOfNSpheres}

Every $n$-[[dimension|dimensional]] [[PL manifold]] admits a branched covering of the [[n-sphere]] ([Alexander 20](#Alexander20)).

By the [[Riemann existence theorem]], every [[connected topological space|connected]] [[compact topological space|compact]] [[Riemann surface]] admits the [[structure]] of a branched cover by a [[holomorphic function]] to the [[Riemann sphere]]. See [there](branched+cover+of+the+Riemann+sphere#RiemannSurfaces) at _[[branched cover of the Riemann sphere]]_.

\begin{center}

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> graphics grabbed from [Chamseddine-Connes-Mukhanov 14, Figure 1](#ChamseddineConnesMukhanov14), [Connes 17, Figure 11](#Connes17)

For [[3-manifolds]] branched covering the [[3-sphere]] see ([Montesinos 74](#Montesinos74)).

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ branched covers of the  [[4-sphere]] ([Piergallini 95](#Piergallini95), [Iori-Piergallini 02](#IoriPiergallini02)).

But the [[n-torus]] for $n \geq 3$ is _not a [[cyclic group|cyclic]]_ branched over of the [[n-sphere]] ([Hirsch-Neumann 75](#HirschNeumann75))

## Related concepts

* [[ramified extension]]

* [[covering space]]

* [[Riemannian orbifold]]

## References

### General

* Wikipedia, _[Branched covering](http://en.wikipedia.org/wiki/Branched_covering)_

### Of arithmetic schemes

* {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_,  Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))

### Of manifolds
 {#ReferencesOfManifolds}

Every $n$-[[dimension|dimensional]] [[PL manifold]] is a [[branched cover]] of the [[n-sphere]]:

* {#Alexander20} [[James W. Alexander]], _Note on Riemann spaces_, Bull. Amer. Math. Soc. 26 (1920) 370–373

For [[3-manifolds]] branched covering the [[3-sphere]] see

*  {#Montesinos74} J. Montesinos, _A representation of closed orientable 3-manifolds as  3-fold branched coverings of $S^3$_, Bull. Amer. Math. Soc. 80 (1974), 845-846 ([Euclid:1183535815](https://projecteuclid.org/euclid.bams/1183535815))

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ branched covers of the  [[4-sphere]]:

* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))

* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))

For comments on this see also

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, section 4 of _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))

* {#Connes17} [[Alain Connes]], _Geometry and the Quantum_, Foundations of Mathematics and Physics One Century After Hilbert. Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470))

The [[n-torus]] for $n \geq 3$ is not a [[cyclic group|cyclic]] branched over of the [[n-sphere]]:

* {#HirschNeumann75} U. Hirsch, W. D. Neumann, _On cyclic branched coverings of spheres_, Math. Ann. (1975) 215: 289 ([doi:10.1007/BF01343895](https://doi.org/10.1007/BF01343895))

Relation to [[orbifolds]]:

* Benoît Claudon, _Gamma-reduction for smooth orbifolds_, Manuscripta Mathematica 127, 4 (2008) 521-532 ([arXiv:0801.4677](https://arxiv.org/abs/0801.4677))

[[!redirects branched covers]]

[[!redirects branched covering]]
[[!redirects branched coverings]]

[[!redirects branched covering space]]
[[!redirects branched covering spaces]]

[[!redirects ramified cover]]
[[!redirects ramified covers]]

[[!redirects ramified]]

[[!redirects ramification]]
[[!redirects ramifications]]

[[!redirects branch point]]
[[!redirects branch points]]

[[!redirects ramified extension]]
[[!redirects ramified extensions]]

[[!redirects unramified extension]]
[[!redirects unramified extensions]]


branched cover of the Riemann sphere

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## Definition

A _[[branched cover]] of the [[Riemann sphere]]_ $\mathbb{C}P^1$ is a [[compact topological space|compact]] [[connected topological space|connected]] [[Riemann surface]] $\Sigma$ equipped with a non-[[constant function|constant]] [[holomorphic function]]

$$
 \phi: \Sigma \longrightarrow \mathbb{C}P^1
  \,.
$$

If we think of $\Sigma$ as retrievable from its [[field]] of [[meromorphic functions]] (see for example [Mumford](#Mum)), then each such map $\phi$ is dual to a [[field extension]] $\mathbb{C}(t) \hookrightarrow Func(\Sigma)$; this is an _algebraic_ field extension. The dual map $\phi$ is [[surjective map|surjective]], and restricts to a [[covering space]] projection after removing all the (finitely many) [[ramification point|ramification points]] of $\phi$.

## Properties

### Riemann surfaces
 {#RiemannSurfaces}

By the [[Riemann existence theorem]], every [[connected topological space|connected]] [[compact topological space|compact]] [[Riemann surface]] admits the [[structure]] of a branched cover of the Riemann sphere. ([MO discussion](http://mathoverflow.net/a/53286/381))

## References

* Brian Ossermann, _Branched covers of the Riemann sphere_ ([pdf](https://www.math.ucdavis.edu/~osserman/rfg/290W/branched-covers.pdf))

* David Mumford, _Curves and their Jacobians_, U. Michigan Press (1977).
 {#Mum}

[[!redirects branched cover of Riemann sphere]]
[[!redirects branched covers of Riemann sphere]]
[[!redirects branched cover of a Riemann sphere]]
[[!redirects branched covers of a Riemann sphere]]
[[!redirects branched covers of Riemann spheres]]
[[!redirects branched cover of the Riemann sphere]]
[[!redirects branched covers of the Riemann sphere]]


branched manifold

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## Idea

Smooth _branched $n$-manifolds_ are a generalization of [[smooth manifolds]] which may have ill-defined [[tangent spaces]] at certain "branch loci", where however they are required to have a well defined tangent $n$-plane.

Branched $n$-manifolds arise for instance as [[quotients]] of [[foliations]].

## Definition

Let $X$ be a [[metrizable space]] with

: 1) a collection $\{U_i\}$ of [[closed subsets]] of $X$

: 2) for each $U_i$ a finite collection $\{D_{i j}\}$ of closed subsets of $U_i$

: 3). for each $i$ a map $\p_i:U_i\to D_i^n$ to a closed $n$-[[disk]] of class $C^k$ in $\mathbb{R}^n$

such that

: a) $\cup_i D_{i j}=U_i$ and $\cup_i U^\circ_i =X$

: b) $p_i |_{D_{i j}}$ is a [[homeomorphism]] onto its [[image]] which is a closed $C^k$ n-disk relative to $\partial D_i^n$.

: c) There is a "[[cocycle]]" of [[diffeomorphisms]] $\{\alpha_{i^' i}\}$ of class $C^k$ such that $p_{i^'}=\alpha_{i^',i}\circ p_i$ when defined. The domain of $\alpha_{i^' i}$ is $p_i(U_i\cap U_{i^'})$.

Then $X$ is called **branched n-manifold of class $C^k$**.

This appears as [Williams, def. 1.0 ns](#Williams).

If $X$ satisfies this set of axioms with b) replaced by

$b_{ns})$ $p_i |_{D_{i j}}$ is a homeomorphisms onto $D_i^n$

$X$ is called **nonsingular branched $n$-manifold of class** $C^k$.

## Related concepts

* [[orbifold]]

## References

* Robert F. Williams, _Expanding attractors_, Publications mathématique d l'IHÉS, tome 43 (1974) ([numdam](http://www.numdam.org/item?id=PMIHES_1974__43__169_0))
 {#Williams}

* [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 ([project euclid](http://projecteuclid.org/euclid.jsg/1180135649))

Discussion relating to [[orbifolds]] is in

* [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_ ([arXiv:math/0509664](http://arxiv.org/abs/math/0509664))

[[!redirects branched manifolds]]


branching

* [[branched cover]]

* [[branching rule]]

category: disambiguation

[[!redirects branchings]]


Brandenberger-Vafa mechanism

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## Idea

The _Brandenberger-Vafa mechanism_ is an argument, using ingredients of [[string theory]], to deduce that the [[observable universe]] is a [[spacetime]] of 3+1 _large_ [[dimensions]] (in addition to small higher dimensions in a [[Kaluza-Klein compactification]]). This is part of the issue of [[moduli stabilization]]. One also speaks of "brane gas cosmology".

The idea is that in the presence of [[strings]] and [[branes]], then the winding modes of these objects around [[cycles]] of the compactified dimensions (i.e. [[wrapped branes]]) induce an energy barrier for compact dimensions to grow large. A more detailed analysis of a "brane gas cosmology" is then argued to yield that exactly 3+1 spactime dimensions may grow large.

A similar claim, with very different methods, is made in ([Kim-Nishimura-Tsuchiya 12](#KimNishimuraTsuchiya12)), via computer simulation of the [[IKKT matrix model]] for [[type IIB string theory]].

## Related concepts

* [[cosmology]]

* [[moduli stabilization]]

* [[landscape of string theory vacua]]

## References

The original article is

* [[Robert Brandenberger]], [[Cumrun Vafa]], _Superstrings in the Early Universe_, Nucl.Phys. B316 (1989) 391 ([web](http://inspirehep.net/record/263348))

Review includes

* {#Easson00} Damien A. Easson, section 3 of _The Interface of Cosmology with String and M(ILLENNIUM) Theory_, Int.J.Mod.Phys.A16:4803-4843,2001 ([arXiv:hep-th/0003086](http://arxiv.org/abs/hep-th/0003086))

Further developments include

* {#WatsonBrandenberger03} Scott Watson, [[Robert Brandenberger]], _Stabilization of Extra Dimensions at Tree Level_, JCAP 0311 (2003) 008 ([arXiv:hep-th/0307044](http://arxiv.org/abs/hep-th/0307044))

* [[Brian Greene]], Daniel Kabat, Stefanos Marnerides, _On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism_,   10.1103/PhysRevD.88.043527  ([arXiv:1212.2115](http://arxiv.org/abs/1212.2115))

* [[Robert Brandenberger]], Damien A. Easson, Dagny Kimberly, _Loitering Phase in Brane Gas Cosmology_, Nucl.Phys. B623 (2002) 421-436 ([arXiv:hep-th/0109165](http://arxiv.org/abs/hep-th/0109165))

* Jin Young Kim, _Stabilization of the Extra Dimensions in Brane Gas Cosmology with Bulk Flux_, Phys.Lett.B652:43-52,2007 ([arXiv:hep-th/0608131](https://arxiv.org/abs/hep-th/0608131))

* [[Brian Greene]], Daniel Kabat, Stefanos Marnerides, _On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism_ ([arXiv:1212.2115](http://arxiv.org/abs/1212.2115))

For a very different argument coming to the same conclusion, within the [[IKKT matrix model]] for [[type IIB string theory]], see

* {#KimNishimuraTsuchiya12} S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions_, Phys. Rev. Lett. 108, 011601 (2012), ([arXiv:1108.1540](https://arxiv.org/abs/1108.1540)).

* S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Late time behaviors of the expanding universe in the IIB matrix model_, JHEP 10, 147 (2012), ([arXiv:1208.0711](https://arxiv.org/abs/1208.0711)).

[[!redirects brane gas cosmology]]


Brandon Doherty

* [Website](https://bcldoherty.github.io/personal-webpage/)

## Selected writings

On the [[model structure for quasi-categories|Joyal-type]] [[model structure for cubical quasi-categories]] on [[cubical sets]]:

[[connection on a cubical set|with connections]] (positive, negative, or both):

* [[Brandon Doherty]], [[Chris Kapulkin]], [[Zachery Lindsey]], [[Christian Sattler]], _Cubical models of (∞,1)-categories_, Memoirs of the AMS ([accepted 2022](https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo)) [[arXiv:2005.04853](https://arxiv.org/abs/2005.04853)]

and without connections:

* [[Brandon Doherty]], _Cubical models of higher categories without connections_ [[arXiv:2207.03636](https://arxiv.org/abs/2207.03636)]

category: people


Brandon Williams

* [webpage](http://www.math.sunysb.edu/~mbw/)

category: people


Brandt groupoid

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## History

[Brandt (1927)](#Brandt27) introduced (long before the notion of *[[category]]* was formulated) a class of partial binary algebraic structures and called them *[[groupoids]]* (German: *Gruppoide*).

Following [[Øystein Ore]], all binary algebraic structures were soon called *groupoids* (now we say either *binary algebraic structure* or, following [[Bourbaki]]: *[[magma]]*), hence *Brandt groupoids* are in general algebra often viewed as a class of partial groupoids.

Contemporary notion of a _[[connected groupoid|connected]]_ [[groupoid]] is informationally equivalent to a Brandt groupoid. Hence Brandt groupoids in the new categorical format, and usually without the connectedness assumption, took over the name in mainstream mathematics, regarding the importance of the notion. Wikipedia simply now redirects Brandt groupoid to [groupoid](https://en.wikipedia.org/wiki/Groupoid).

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###### Remark
**(usage of the terminology)**
In older literature, the specific class of groupoids, a [[codiscrete groupoid]] of a set $X$ is also sometimes called a Brandt groupoid (as mentioned in da Silva, Weinstein, Geometric models of noncommutative algebras).

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## Definition

A __Brandt groupoid__ $(M,\cdot)$ is a set with a partially defined binary operation $\cdot$ such that

1. (associativity) If $a\cdot b$ and $b\cdot c$ are defined then $(a\cdot b)\cdot c$ and $a\cdot (b\cdot c)$ are defined and they are equal
1. for each $a\in M$ there are unique elements $e,f\in M$ such that $e\cdot a = a\cdot f = a$, called respectively its left and right unit
1. if the left units of $a$ and $b$ agree then there is $x\in M$ such that $a \cdot x = b$; if the right units of $c$ and $d$ agree then there is $y\in M$ such that $y\cdot c = d$
1. (connectedness) if $e$ and $f$ are idempotents, then there is $m$ such
that $e\cdot m\cdot f$ is defined

## Properties

The first three properties imply that the idempotents $e$ in $M$ are precisely the left units of all elements $a$ such that $e\cdot a$ is defined; they are also precisely the right units of all elements $b$ such that $b\cdot e$ is defined.

If $(M,\cdot)$ is a Brandt groupoid then the set $M\coprod \{0\}$ can be made into an [[inverse semigroup]] by extending $\cdot$ so that the multiplication of any element with $0$ is $0$ and the product of any two elements in $M$ whose product was undefined is also $0$. Semigroups of that kind are called Brandt semigroups.

## Literature

* {#Brandt27} [[Heinrich Brandt]], _Über eine Verallgemeinerung des Gruppenbegriffes_, Mathematische Annalen **96** 1 (1927) 360-366 [[doi:10.1007/BF01209171](https://doi.org/10.1007%2FBF01209171)]

* G. B. Preston, _Congruences on Brandt semigroups_, Mathematische Annalen __139__:2 (1959) 91-94 [[doi:10.1007/BF01354867](http://dx.doi.org/10.1007/BF01354867)]

* A. H. Clifford, _Matrix representations of completely simple semigroups_
Amer. J. Math. 64, 327--342 (1942).

[[!redirects Brandt groupoids]]

[[!redirects Brandt semigroup]]