Zheng-Cheng Gu

* [personal page](http://www.phy.cuhk.edu.hk/people/teaching/zcgu.html)

* [GoogleScholar page
](https://scholar.google.com/citations?user=lbiF_2MAAAAJ&hl=en)

## Selected writings

On [[string-net models]] via [[tensor networks]]:

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

On [[braid group]]-effects in 2d [[symmetry protected topological phases]]:

* [[Michael Levin]], [[Zheng-Cheng Gu]], *Braiding statistics approach to symmetry-protected topological phases*, Phys. Rev. B **86** (2012) 115109 ([doi:10.1103/PhysRevB.86.115109](https://doi.org/10.1103/PhysRevB.86.115109))

Introducing the notion of [[symmetry protected topological phases]] and of [[topological order]]:

* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$

Suggestion that [[topological order]] goes along with [[long-range entanglement]]:

* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], Section V of: _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$

Claim that [[group cohomology]] classifies some [[symmetry protected topological phases]]:

* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114  $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$

* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*,  Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))

* {#GuWen14} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], *Symmetry-protected topological orders for interacting fermions -- fermionic topological non-linear sigma-models and a group super-cohomology theory*, Phys. Rev. B **90** 115141 (2014) $[$[arXiv:1201.2648](http://arxiv.org/abs/1201.2648), [doi:10.1103/PhysRevB.90.115141](https://doi.org/10.1103/PhysRevB.90.115141)$]$

* [[Juven Wang]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_, Phys. Rev. Lett. **114** (2015) 031601 [[arxiv:1405.7689](https://arxiv.org/abs/1405.7689), [doi:10.1103/PhysRevLett.114.031601](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)]

category: people


Zheng-Xin Liu

* [Institute page](https://wlibuaa.github.io/authors/zheng-xin-liu/)

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

## Selected writings

Claim that [[group cohomology]] classifies [[boson|bosonic]] [[symmetry protected topological phases]]:

* {#CGLW11} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B **87** (2013) 155114  $[$[arXiv:1106.4772](http://arxiv.org/abs/1106.4772)$]$

* [[Xie Chen]], [[Zheng-Cheng Gu]], [[Zheng-Xin Liu]], [[Xiao-Gang Wen]], *Symmetry protected topological orders and the group cohomology of their symmetry group*,  Science **338** (2012) 1604-1606 ([arXiv:10.1103/PhysRevB.87.155114](https://doi.org/10.1103/PhysRevB.87.155114))

On [[projective representations]] in the context of [[internal symmetry|internal]]-[[symmetry protected topological order]]:

* {#YangLiu18} Jian Yang, [[Zheng-Xin Liu]], *Irreducible Projective Representations and Their Physical Applications*, J. Phys. A: Math. Theor. **51** 025207 (2018) $[$[doi:10.1088/1751-8121/aa971a](https://doi.org/10.1088/1751-8121/aa971a), [arXiv:1605.05805](https://arxiv.org/abs/1605.05805)$]$

category: people


Zhenghan Wang

* [webpage](http://stationq.cnsi.ucsb.edu/~wang/)

## Selected writings

On [[modular tensor categories]]:

* [[Eric Rowell]], [[Richard Stong]], [[Zhenghan Wang]], *On classification of modular tensor categories*, Comm. Math. Phys. 292 (2009) no. 2, 343--389 ([arXiv:0712.1377](https://arxiv.org/abs/0712.1377))

On [[braiding]] of [[su(2)-anyons]] providing universal [[quantum gates]] for [[topological quantum computation]]:

* {#FreedmanLarsenWang00} [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics **227** (2002) 605–622 [[arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108), [doi:10.1007/s002200200645](https://doi.org/10.1007/s002200200645)]

On [[topological quantum computation]] with [[anyons]] in [[Chern-Simons theory]]:

* [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], _Topological quantum computation_,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

* [[Zhenghan Wang]], *Topological Quantum Computation*, CBMS Regional Conference Series in Mathematics **112**, AMS 2010 ([ISBN-13: 978-0-8218-4930-9](http://www.ams.org/publications/authors/books/postpub/cbms-112), [pdf](http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf))

* [[Eric Rowell]], [[Zhenghan Wang]], _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

and with focus on [[abelian anyons]] in terms of [[modular tensor categories]]:

* Wade Bloomquist, [[Zhenghan Wang]], *On Topological Quantum Computing With Mapping Class Group Representations*, J. Phys. A: Math. Theor. **52** (2019) 015301 ([arXiv:1805.04622](https://arxiv.org/abs/1805.04622), [doi:10.1088/1751-8121/aaeea1](https://iopscience.iop.org/article/10.1088/1751-8121/aaeea1))

* Liang Wang, [[Zhenghan Wang]], *In and around Abelian anyon models*, J. Phys. A: Math. Theor. **53** 505203 (2020) ([doi:10.1088/1751-8121/abc6c0](https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0))

On identifying [[topological order]] via [[topological entanglement entropy]] of the ground state:

* Hong-Chen Jiang, [[Zhenghan Wang]], [[Leon Balents]], *Identifying Topological Order by Entanglement Entropy*, Nature Physics 8, 902-905 (2012) ([arXiv:1205.4289](https://arxiv.org/abs/1205.4289))

Introducing the [[4d TQFT]] [[Walker-Wang model]]:

* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))

On [[number theory|number theoretic]] aspects of [[modular tensor categories]]:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

On [[braid group representations]] (specifically the [[Burau representation]] and its relation to the [[Alexander polynomial]]) and their application to [[topological quantum computation]]:

* [[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))

A proposal for classifying [[symmetry protected topological phases|symmetry-protected]] [[anyon|anyonic]] [[topological order]] in terms of [[2-group|2-groupoidal]]-actions on [[modular tensor categories]]:

* [[Zhenghan Wang]], §2.2 of: *Beyond Anyons*,  Modern Physics Letters A **33** 28 (2018) 1830011  [[arXiv:1710.00464](https://arxiv.org/abs/1710.00464), [doi:10.1142/S0217732318300112](https://doi.org/10.1142/S0217732318300112)]

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) $[$[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147), [talk pdf](http://helper.ipam.ucla.edu/publications/stq2015/stq2015_12401.pdf)$]$

On [[quantum adiabatic theorem|adiabatic transformations]] of certain 2d [[topological insulators]]:

* David Aasen, [[Zhenghan Wang]], [[Matthew B. Hastings]], *Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes* [[arXiv:2203.11137](https://arxiv.org/abs/2203.11137)]

## Related entries

* [[Walker-Wang model]]

category: people


Zhi Lu

* [webpage](http://homepage.fudan.edu.cn/zhilu/)

## Selected writings

On [[Calabi-Yau manifolds]] in [[SU-bordism theory]] (see [[Calabi-Yau manifolds in SU-bordism theory]]):

* [[Ivan Limonchenko]], [[Zhi Lu]], [[Taras Panov]], _Calabi-Yau hypersurfaces and SU-bordism_, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 ([arXiv:1712.07350](https://arxiv.org/abs/1712.07350))

category: people


Zhih-Ahn Jia

* [Blog](https://zhihahnjia.wordpress.com/author/zhihahnjia/)

* [Publication list](https://arxiv.org/search/?query=Jia%2C+Zhih-ahn&searchtype=author&abstracts=show&order=-announced_date_first&size=50)

category: people


Zhiqin Lu

* [webpage](http://www.math.uci.edu/~zlu/)

## related $n$Lab entries

* [[moduli space of Calabi-Yau manifolds]]

category: people


Zhiwei Yun

* [Institute page](https://math.mit.edu/~zyun/)

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

## Selected writings

Relating the [[Pontrjagin algebra]] on [[loop groups]] of [[compact Lie groups]] to their [[Langlands dual groups]]:

* [[Zhiwei Yun]], [[Xinwen Zhu]], *Integral homology of loop groups via Langlands dual groups*, Represent. Theory **15** (2011) 347-369 [[arXiv:0909.5487](https://arxiv.org/abs/0909.5487), [doi:10.1090/S1088-4165-2011-00399-X](https://doi.org/10.1090/S1088-4165-2011-00399-X)]

category: people


Zhouli Xu

[Website](https://sites.google.com/view/xuzhouli/)

## Selected writings

On [[homotopy groups of spheres]] and the framed [[bordism ring]] ([[MFr]]):

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], Section 2.6 of: _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([pdf](http://math.mit.edu/~guozhen/homotopy%20groups.pdf), [[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

On computation of [[stable homotopy groups of spheres]] via the [[Adams spectral sequence]]:

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _Stable homotopy groups of spheres_, PNAS October 6, 2020 117 (40) 24757-2476 ([arXiv:2001.04247](https://arxiv.org/abs/2001.04247), [doi:10.1073/pnas.2012335117](https://doi.org/10.1073/pnas.2012335117))

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _More stable stems_ ([arXiv:2001.04511](https://arxiv.org/abs/2001.04511))

category: people


Zhuan Li

* [personal page](https://www.physicsandastronomy.pitt.edu/people/zhuan-li)

## Selected writings

On detection of [[topological order]] by observing [[modular transformations]] on the [[ground state]]:

* [[Zhuan Li]], [[Roger S. K. Mong]], *Detecting topological order from modular transformations of ground states on the torus*, Phys. Rev. B **106** (2022) 235115 $[$[doi:10.1103/PhysRevB.106.235115](https://doi.org/10.1103/PhysRevB.106.235115), [arXiv:2203.04329](https://arxiv.org/abs/2203.04329)$]$

category: people


Zhuo Chen

## Zhuo Chen ##

Professor of Mathematics

![Alt text](http://p0.qhimg.com/dr/200_200_/t015898f97b746081fd.jpg)

###Institute:###

Department of Mathematics, Tsinghua University

###Address:###

Tsinghua University Campus,
100084
Beijing, China. P.R.China

###Research Interests:###

[[Poisson geometry]], non-linear [[Lie theory|Lie theories]].

### Selected writings

On [[Atiyah Lie algebroids]]:

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _From Atiyah classes to homotopy Leibniz algebras_, Commun. Math. Phys. __341__ (2016) 309--349 [doi](https://doi.org/10.1007/s00220-015-2494-6)

On [[Poisson Lie 2-groups]]:

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _Poisson 2-groups_ ([arXiv](https://arxiv.org/abs/1202.0079))

category: people


Zi-zhi Wang

* [spire page](https://inspirehep.net/authors/1700357)

## Selected writings

On [[quantum information theory]] in view of [[AdS-CFT duality]] ([[holographic entanglement entropy]]):

* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]

category: people


Ziegler spectrum

Let $k$ be a fixed field. Consider associative $k$-algebra $A$ and its category of right modules $Mod_A$.

Recall that a monomorphism $f:M\to N$ in $Mod_A$-modules is __pure__ if after tensoring with any left $A$-module $L$ gives a mono $M\otimes_A L\stackrel{f\otimes L}\to N\otimes_A L$. A module $M$ in $Mod_A$ is __pure-injective__ if every pure mono $M\to N$ splits. This is clearly a weaker property than being an [[injective object]].

Given an associative algebra $A$, its __Ziegler spectrum__ $Zsp(A)$ is the topological space whose points are the isomorphism classes $[M]$ of indecomposable pure-injective $A$-modules $M$ and the topology is defined in terms of pp-formulas (or finite matrices) over $A$. Here pp stands for "positive primitive in the usual language for $A$-modules"

The importance of Ziegler spectrum is in the

__Ziegler's theorem.__ There is a correspondence between the definable classes in $Mod_A$ and closed subsets of $Zsp(A)$.

There are applications to the spectra of theories of modules.

It is introduced in

* Martin Ziegler,  _Model theory of modules_, Ann. Pure Appl. Logic __26__ (1984), no. 2, 149–213, [MR86c:03034](http://www.ams.org/mathscinet-getitem?mr=739577) doi

and generalized to locally coherent [[Grothendieck categories]] in

* Ivo Herzog, _The Ziegler spectrum of a locally coherent Grothendieck category_, Proc. London Math. Soc. __74__ (3): 503–558 (1997) [doi](https://doi.org/10.1112/S002461159700018X)

* Krause [pdf](http://www2.math.uni-paderborn.de/fileadmin/Mathematik/AG-Krause/presentations_krause/ziegler.pdf)
* Grigory Garkusha, Mike Prest (2005) _Triangulated categories and the Ziegler spectrum_, Algebras and Representation Theory, 8 (4). pp. 499-523, [doi](http://dx.doi.org/10.1007/s10468-005-8147-2), [pdf](http://eprints.ma.man.ac.uk/290/01/covered/MIMS_ep2006_122.pdf)
* Mike Prest, [slides](http://www.slideshare.net/guest9ab010/zieglers-spectrum)
* Mike Prest, Topological and geometric aspects of the Ziegler spectrum (1998)
* Lorna Gregory, Thesis, [pdf](http://www.maths.manchester.ac.uk/~mprest/GregoryThesis.pdf)
* Mike Prest, _Purity, spectra and localisation_, Enc. of Math. and its Appl. __121__, Camb. Univ. Press 2009

On the relation between the Ziegler's spectrum of the category of finitely generated left $A$-modules and the Gabriel's spectrum of the category of Abelian presheaves on it see

* Ivo Herzog, _Contravariant functors on the category of finitely presented modules_, Israel J. Math. __167__, 347–410 (2008) [doi](https://doi.org/10.1007/s11856-008-1052-8)


zigzag

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Zigzags
* table of contents
{: toc}

## Zigzags of morphisms

In a [[category]] $C$ a **zigzag of morphisms** is a finite collection of [[morphism]]s $(f_i)$ in $C$ of the form

$$
  \array{
    && x_1 &&&& x_3 & \cdots
    \\
    & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}}
    && {}^{\mathllap{f_3}}\swarrow
    && \searrow^{\mathrlap{f_4}} & \cdots
    \\
    x_0 &&&& x_2 &&&& x_4 & \cdots
  }
  \,.
$$

A zigzag consisting just out of two morphisms is a _roof_ or [[span]].

General such zig-zags of morphisms represent ordinary morphisms in the _groupoidification_ of $C$ -- the [[Kan fibrant replacement]] of its [[nerve]], its [[simplicial localization]] or its 1-categorical [[localization]] at all its morphisms.

More generally, if in these zig-zags the left-pointing morphisms are restricted to be in a class $S \subset Mor(C)$, then these zig-zags represent morphisms in the simplicial localizaton or localization of $C$ at $S$.

## Related concepts

* [[resolution]], [[derived functor]]

* [[span]], [[correspondence]]

* [[Burnside category]]

* [[zigzag persistence module]]

[[!redirects zigzag]]
[[!redirects zigzags]]
[[!redirects zig-zag]]
[[!redirects zig-zags]]
[[!redirects zig zag]]
[[!redirects zig zags]]


zigzag category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _zigzag category_ is the [[category]] whose objects are [[diagram]]s that look like

$$
  \{
    t_0 \leftarrow t_1 \leftarrow t_2 \rightarrow t_3 \leftarrow t_4
    \cdots
  \}
  \,.
$$

## Definition

Let $\mathbf{T}$ be the [[category]] defined as follows:

* An [[object]] is given by a triple of data $t \coloneqq (n,t_+,t_-)$, which is a [[partition]] of the [[poset]] $\{i \;|\; 1\leq i\leq n\}$ for some [[natural number]] $n \geq 0$ into two [[subset]] parts $t_+$ and $t_-$. (The corresponding diagram as above has nodes $t_0, \ldots, t_n$, and the arrow between $t_{i-1}$ and $t_i$ points forward to $t_i$ if $i \in t_+$, and backward if $i \in t_-$.)

* A [[morphism]] is a [[monotone map]] preserving the partitions.

We will call $\mathbf{T}$ the __zigzag category__, or the __category of zigzag types__.

Also, notice that we have cleverly hidden the [[empty set]] among the objects.  We pat ourselves on the backs for doing this. (Here the zigzag type consists of a single node.)

There is a functor $Z : \mathbf{T}^{op} \to \mathbf{RelCat}$ that realizes each zigzag as a [[relative category]]. It is given by:

   * The realization of a type $(n, t_+, t_-)$, is the relative category on objects $\{0, 1, \ldots, n \}$ and generated by arrows $(k-1) \to k$ whenever $k \in t_+$ and $(k-1) \xleftarrow{\sim} k$ whenever $k \in t_-$
   * The realization of a morphism $\tau$ sends the arrow between $(k-1)$ and $k$ to the product of the arrows between $(i-1)$ and $i$ for all $i$ with $\tau(i) = k$.

In particular, in this realization, the morphisms of zigzag types are precisely the relative functors that are also nondecreasing on object labels and preserve bottom and top objects.

Then, the category of zigzags of type $t$ from $X$ to $Y$ in a [[relative category]] $\mathbf{C}$ is nothing more than the category of relative functors $Z(t) \to \mathbf{C}$.

By forgetting the marked arrows, we get a similar realization $\mathbf{T}^{op} \to \mathbf{Cat}$ for talking about zigzags in ordinary categories.

## Applications

Such zig-zag [[diagram]]s serve to model morphisms in an [[(∞,1)-category]] in terms of a presentation by a [[category with weak equivalences]]. See [[simplicial localization of a homotopical category]].

[[!redirects zigzag category]]
[[!redirects zigzag categories]]
[[!redirects zig-zag category]]
[[!redirects zig-zag categories]]
[[!redirects category of zigzag types]]
[[!redirects categories of zigzag types]]
[[!redirects category of zig-zag types]]
[[!redirects categories of zig-zag types]]
[[!redirects zigzag type]]
[[!redirects zigzag types]]


zigzag identity 1 - SVG

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    <semantics>
     <mrow>
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   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>η</mi>
     </mrow>
     <annotation encoding="application/x-tex">\eta</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject id="svg_42715_10" height="20" width="19" font-size="16" y="91" x="152.000001">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mo>=</mo>
     </mrow>
     <annotation encoding="application/x-tex">=</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
</svg>

category: svg


zigzag identity 1 implies 2 - SVG

<svg width="900" height="300" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="24191">
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  <path id="svg_24191_2" d="m127,114l0,-68c0.333328,-31.333332 70.666672,-31.666668 70,-1c2,42.333328 -23,48.666672 -24,25c-1.333328,-34 -22.666672,-13 -16,18c-1.333328,37 62.333328,44 62,0l-1,-66" fill-opacity="0" stroke-width="2" stroke="#000000" fill="#ff0000"/>
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category: svg


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category: svg


zigzag persistence module

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

#Contents#
* table of contents
{:toc}

## Idea

Where a linear [[persistence module]], as traditionally considered in [[persistent homology theory]], is a [[diagram]] of [[vector space]]/[[modules]] of the directed form

\begin{tikzcd}[column sep=small]
  \cdots
  \ar[r]
  &
  V_1
  \ar[r]
  &
  V_2
  \ar[r]
  &
  V_3
  \ar[r]
  &
  V_4
  \ar[r]
  &
  \cdots
  \,,
\end{tikzcd}

in a *zig-zag persistence module* one allows the [[linear maps]] to go in either direction, such as in regular [[zigzags]]

\begin{tikzcd}[column sep=small]
  \cdots
  \ar[r]
  &
  V_1
  \ar[from=r]
  &
  V_2
  \ar[r]
  &
  V_3
  \ar[from=r]
  &
  V_4
  \ar[r]
  &
  \cdots
  \,,
\end{tikzcd}

or in more general zigzags such as

\begin{tikzcd}[column sep=small]
  \cdots
  \ar[from=r]
  &
  V_1
  \ar[from=r]
  &
  V_2
  \ar[r]
  &
  V_3
  \ar[from=r]
  &
  V_4
  \ar[r]
  &
  \cdots
  \,,
\end{tikzcd}

In the language of [[quiver theory]], zigzag persistence modules are precisely the [[quiver representations]] of [[ADE-classification|A-type]] [[quivers]]. But in [[quiver representation theory]] the classical [[Gabriel's theorem|Gabriel theorem]] says (see [there](Gabriel's+theorem#ATypeQuivers)) that for all possible zig-zag patterns as above the [[indecomposable object|indecomposable]] [[quiver representations]] are still "interval modules", ie. those for which the $V_i$ are [[zero object|zero]] except in some [[interval]] where they are [[dimension of a vector space|1-dimensional]] and connected by ([[zigzags]] of) [[identity maps]].

This means that the key property of [[persistence modules]] -- namely that they are entirely characterized by [[persistence barcodes]], i.e. by [[multisets]] of such intervals -- immediately generalizes to zigzag persistence modules.

A typical type of a zigzag persistence module appearing in the practice of [[topological data analysis]] consists of [[homology groups]] $H(X_i)$ of stages $\cdots \hookrightarrow X_{i - 1} \hookrightarrow X_i \hookrightarrow X_{i + 1} \hookrightarrow \cdots$ of a [[filtered topological space]], as usual, but evaluating now on the [[zigzag]] of inclusions into the [[unions]]

\begin{tikzcd}
  &
  X_i \cup X_j
  \ar[from=dr, hook', "{\iota_j^r}"{swap}]
  \\
  X_i
  \ar[ur, hook, "{\iota_i^l}"]
  &
  &
  X_j
\end{tikzcd}

of consecutive filter stages:

\begin{tikzcd}[column sep=small]
  \cdots
  \ar[from=dr, hook']
  &
  &
  H(X_i \cup X_{i+1})
  \ar[from=dr, hook', "{H(\iota_{i+1}^r)}"{swap}]
  &
  &
  H(X_{i+1} \cup X_{i+2})
  \ar[from=dr, hook', "{H(\iota_{i+2}^r)}"{swap}]
  &
  &
  \cdots
  \\
  &
  H(X_i)
  \ar[ur, hook, "{H(\iota_i^l)}"]
  &
  &
  H(X_{i+1})
  \ar[ur, hook, "{H(\iota_{i+1}^l)}"]
  &
  &
  H(X_{i+2})
  \ar[ur, hook]
  &
\end{tikzcd}

It is claimed (...) that such zigzag persistence modules still retain the same persistent information of interest, but are more robust.

## Examples

### Well modules

As the [[level set|level]] varies, the [[well groups]] of a [[continuous function]] naturally form a zigzag persistence module, called a *[[well module]]* ([Edelsbrunner, Morozov & Patel 2011](well +group#EMP11)).

## Properties

* [[stability of persistence diagrams]]

* [[diamond principle]]

## Related concepts

* [[well group]]

## References

The concept of zigzag persistence modules as a tool in [[topological data analysis]] and their relation to [[quiver representation theory]] (including a re-proof of [[Gabriel's theorem]] for the case of A-type quivers) is due to:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

Application to [[level sets]]:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Dmitriy Morozov]], *Zigzag persistent homology and real-valued functions*, in: *SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$[doi:10.1145/1542362.1542408](https://doi.org/10.1145/1542362.1542408)$]$

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

Review:

* [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf), [ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/)$]$

The [[stability of persistence diagrams|algebraic stability theorem]] for zigzag persistence modules:

* {#BotnanLesnick18} [[Magnus Bakke Botnan]], [[Michael Lesnick]], *Algebraic Stability of Zigzag Persistence Modules*, Algebr. Geom. Topol. **18** (2018) 3133-3204 $[$[arXiv:1604.00655](https://arxiv.org/abs/1604.00655), [doi:10.2140/agt.2018.18.3133](https://doi.org/10.2140/agt.2018.18.3133)$]$

[[!redirects zigzag persistence modules]]

[[!redirects zig-zag persistence module]]
[[!redirects zig-zag persistence modules]]

[[!redirects zigzag persistence]]


Zinbiel algebra

Zinbiel is a mathematical person (in the sense like [[Nicolas Bourbaki]] whose mirror person is Leibniz.

A __Zinbiel algebra__ or __dual Leibniz algebra__ (in $Vec$) is a [[nonassociative algebra]] with a product satisfying

$$
(x y) z = x (z y + y z),
$$
Equivalently, it is an algebra over the operad which is Koszul dual to the [[quadratic operad]] of [[Leibniz algebra]]s.

It is also natural to consider dg-algebras over a dg-version of the operad of dual Leibniz algebras.

Dually, a __Zinbiel coalgebra__ is a vector space $C$ equipped with a linear map $\delta:C\to C\otimes C$
satisfying: $\delta\otimes id)\circ\delta = (id\otimes \delta)
\circ \delta + (id\otimes \tau\delta)\circ\delta$ where $\tau$ is the flip of tensor factors.

* wikipedia [Zinbiel algebra](http://en.wikipedia.org/wiki/Zinbiel_algebra)
* Guillaume W. Zinbiel, _Encyclopedia of types
of algebras 2010_, in Guo, Li; Bai, Chengming; Loday, Jean-Louis, Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics 9, pp. 217–298, [arxiv/1101.0267](http://arxiv.org/abs/1101.0267)

* Emily Burgunder, _A symmetric version of Kontsevich [[graph complex]] and Leibniz homology_,
Journal of Lie Theory __20__ (2010), No. 1, 127--165 [arxiv/0804.2052](http://arxiv.org/abs/0804.2052)
* J-L. Loday, _Algebraic K-theory and the conjectural Leibniz K-theory_, K-Theory 09/2003; 30(2):105-127, [pdf](http://www-irma.u-strasbg.fr/~loday/PAPERS/2003Loday%28LeibnizConj%29.pdf) [doi](http://dx.doi.org/10.1023/B:KTHE.0000018382.90150.ce)

Manon Didry, _Construction of groups associated to Lie- and to Leibniz-algebras_, Journal of Lie Theory __17__ (2007), No. 2, 399--426

> We describe a method for associating to a Lie algebra $\mathfrak{g}$ over a ring $K$ a sequence of groups $(G_{n}(\mathfrak{g}))_{n\in\N}$, which are {\it polynomial groups} in the sense that will be explained in Definition 5.1. Using a description of these groups by generators and relations, we prove the existence of an action of the symmetric group $\Sigma_{n}$ by automorphisms. The subgroup of fixed points under this action, denoted by $J_{n}(\mathfrak{g})$, is still a polynomial group and we can form the projective limit $J_{\infty}(\mathfrak{g})$ of the sequence $(J_{n}(\mathfrak{g}))_{n\in\N}$. The formal group $J_{\infty}(\g)$ associated in this way to the Lie algebra $\mathfrak{g}$ may be seen as a generalisation of the formal group associated to a Lie algebra over a field of characteristic zero by the Campbell-Haussdorf formula.

[[!redirects Zinbiel algebra]]
[[!redirects dual Leibniz algebra]]
[[!redirects Zinbiel coalgebra]]


Zinovy Diskin

\rightarrow ( = \to), \longrightarrow, \Rightarrow ( = \implies), \hookrightarrow ( = \embedsin), \mapsto ( = \map), \leftarrow, \longleftarrow, \Leftarrow ( = \impliedby), \hookleftarrow, \leftrightarrow, \Leftrightarrow, \Longleftrightarrow ( = \iff), \nearrow ( = \nearr), \nwarrow ( = \nwarr), \searrow ( = \searr), \swarrow ( = \swarr), \neArrow ( = \neArr), \nwArrow ( = \nwArr), \seArrow ( = \seArr), \swArrow ( = \swArr), \darr, \Downarrow, \uparr, \Uparrow, \downuparrow ( = \duparr = \updarr), \Updownarrow, \leftsquigarrow, \rightsquigarrow, \leftrightsquigarrow, \upuparrows, \rightleftarrows, \rightrightarrows, \dashleftarrow, \dashrightarrow, \curvearrowleft, \curvearrowbotright, \downdownarrows, \leftleftarrows, \leftrightarrows, \righttoleftarrow, \lefttorightarrow, \circlearrowleft, \circlearrowright, \curvearrowright, \leftarrowtail, \rightarrowtail, \leftrightsquigarrow, \Lleftarrow, \Rrightarrow, \looparrowleft, \looparrowright, \Lsh, \Rsh, \twoheadleftarrow, \twoheadrightarrow, \nLeftarrow, \nleftarrow, \nLeftrightarrow, \nleftrightarrow, \nRightarrow, \nrightarrow, \leftharpoonup, \leftharpoondown, \rightharpoonup, \rightharpoondown, \downharpoonleft, \downharpoonright, \leftrightharpoons, \rightleftharpoons, \upharpoonleft, \upharpoonright


Ziv Ran

* [webpage](http://mathdept.ucr.edu/faculty/ran.html)

## Related pages

* [[Ran space]]

category: people


Zoe Wyatt

* [webpage](https://zoewyatt.wordpress.com/)

## Selected writings

On a rigorous [[proof]] of [[moduli stabilization]] of vacuum [[spacetimes]], refuting [Penrose 03](moduli+stabilization#Penrose03) (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics):

* {#AndersonBlueWyattYau20} [[Lars Andersson]], [[Pieter Blue]], [[Zoe Wyatt]], [[Shing-Tung Yau]], _Global stability of spacetimes with supersymmetric compactifications_ ([arXiv:2006.00824](https://arxiv.org/abs/2006.00824))

category: people


Zohar Komargodski

* [webpage at Weizmann institute](http://www.weizmann.ac.il/particle/Zohar/)

* [webpage at IAS](https://www.ias.edu/scholars/zohar-komargodski)

## Selected writings

On the [[theta angle]] in [[QCD]]:

* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))

## Related $n$Lab entries

* [[theta-angle]]

category: people


Zoltán Zimborás

* [institute page](https://sites.google.com/site/gedentqopt/Members/ZoltanZimboras)

## Selected writings

Discussion of [[asymptotic boundaries]] of [[hyperbolic space|hyperbolic]] [[tensor networks]] as conformal [[quasicrystals]] (see also at [[AdS/CFT in solid state physics]]):

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Tensor network models of AdS/qCFT_ ([arXiv:2004.04173](https://arxiv.org/abs/2004.04173))

category: people

[[!redirects Zoltan Zimboras]]


Zoran Skoda > history

< [[Zoran Skoda]]
[[!redirects Zoran Skoda -- history]]


Zoran Škoda

I am a mathematical physicist/mathematician from Zagreb.
My longer scientific biography is [[zoranskoda:careerpage]]. My Ph.D. is from University of Wisconsin-Madison, 2002 (title: _Coset spaces for quantum groups_ (abstract)). I currently teach mathematics as an associate professor at the University of Zadar, Croatia; 2015-2021 I worked part time on a research project as an associate professor at the University of Hradec Králové, Czech Republic. I supervise some graduate students at the University of Zagreb, Croatia (where I have an adjunct title of docent) and maintain some scientific ties and collaboration with Theoretical Physics Division of the Institute Rudjer Bošković, Zagreb, Croatia, where I was previously employed. In Croatia there is a national accreditation of the scientific titles: I hold the titles of a senior scientist in mathematics (2020) and a research associate in physics (2016).

My mathematical interests include
geometric aspects of mathematical physics (TQFT, [[geometric
quantization]], [[coherent states]] etc.),
[[noncommutative algebraic geometry]] and [[noncommutative localization]], [[Hopf algebra]]s, [[bialgebroid]]s, categories, [[cohomology]],
in particular nonabelian cohomology/[[descent]] theory.

My other scientific interests include historical linguistics (general principles and Indo-European), interface between language and computation (computational [[linguistics]], construction of compilers and computer language design), [[blockchain]] technology and medical herbs (herbal tea) but most of these past hobbies are latent now for a long time. I used to play piano [accordion](http://en.wikipedia.org/wiki/Accordion). My native tongue is the [kajkavian dialect](http://en.wikipedia.org/wiki/Kajkavian_dialect) of Croatian. In Croatian we use diacritics for the sch-sound: Škoda.

View a list of some of my mathematical/physical [[zoranskoda:my writings|articles]] and [[zoranskoda:talks|talks]]. My other web page is [here](https://www.irb2.hr/korisnici/zskoda) I have a (low-activity) blog [here](http://mathlight.wordpress.com).

I have also a personalized part of $n$lab [[zoranskoda:HomePage|here]] with some coursepages for students, and other pages not in close accord with purposes and collaborative nature of $n$lab proper.

category: people

[[!redirects Zoran Skoda]]
[[!redirects Z. Skoda]]
[[!redirects Z. Škoda]]


Zorn's lemma

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

# Zorn\'s Lemma
* table of contents
{: toc}

## Idea

_Zorn\'s lemma_, also known as Kuratowski-Zorn lemma, is a [[proposition]] in [[set theory]] and [[order theory]]. It says that: _A [[preorder]] in which every sub-[[total order]] has an [[upper bound]] has a [[maximal element]]._

Depending on the chosen perspective on [[foundations]]
this is either an [[axiom]] or a [[deduction|consequence]] of the [[axiom of choice]]; or neither if neither the axiom of choice nor Zorn's lemma is assumed as an axiom.
Conversely, Zorn\'s lemma and the [[axiom of excluded middle]] together imply the [[axiom of choice]].  Although it doesn\'t imply excluded middle itself, Zorn\'s lemma is not generally accepted in [[constructive mathematics]] as an [[axiom]].  Sometimes the maximal element is only a convenience, and it's enough to use the entire poset or perhaps the collection of all chains instead; sometimes, the use of Zorn's Lemma is essential.

In as far as it holds or is assumed to hold, Zorn's lemma is a standard method for constructing (or at least, [[proof|proving]] the [[existence]] of) objects that are 'maximal' in an appropriate sense.  It is commonly used in [[algebra]] and named after the algebraist [[Max Zorn]] (although he himself protested the naming after him).

## Statement and proofs

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

* Given a [[preorder|preordered]] set $(S, \leq)$, an element $x$ of $S$ is _[[maximal element|maximal]]_ if $y \leq x$ whenever $x \leq y$.

* A _[[chain]]_ in $S$ is a [[subset]] $A \hookrightarrow S$ such that $\leq$ is a [[total order]] on $A$.

* An _[[upper bound]]_ of a subset $A$ of $S$ is an element $x$ of $S$ such that $y \leq x$ whenever $y \in A$.
=--

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

The [[proposition]] __Zorn\'s Lemma__ states that a preordered set $S$ has a maximal element if every chain in $S$ has an upper bound.
=--

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

Suppose the [[axiom of choice]] holds. Then Zorn\'s Lemma holds.
=--

+-- {: .proof}
###### Proof
By contradiction. (We freely use the [[axiom of excluded middle]], which in any case follows from AC, the axiom of choice. The one use made of AC is noted below. WLOG we prove the result for [[posets]].)

Suppose $S$ has no maximal element. Then every chain $C \subseteq S$ has a *strict* upper bound: an element $x$ such that $c \lt x$ for all $c \in C$. (Take an upper bound $y$ of $C$, then $y$ is not maximal by supposition, so any $x$ with $y \lt x$ will serve.) For each $W \subseteq S$ that is well-ordered by $\leq$, use AC to choose a strict upper bound $f(W)$ of $W$. Given a such a function $f: Well(S) \to S$ from well-ordered subsets of $S$ to $S$, let us say $W \in Well(S)$ is *$f$-inductive* if for every $x \in W$, we have $x = f(\{y \in W: y \lt x\})$.

Let $Y, Z \in Well(S)$ be $f$-inductive sets; we will show one is an [[lower set|initial segment]] of the other. Let $I$ be the union of all subsets of $S$ that are initial segments of both $Y$ and $Z$; then $I$ is maximal among such initial segments of both. Suppose $I$ is *properly* contained in both; let $y$ be the least element of $Y \setminus I$, and $z$ the least element of $Z \setminus I$. Then $\{x \in Y: x \lt y\} = I = \{x' \in Z: x' \lt z\}$, so by $f$-inductivity of $Y, Z$, we have $y = f(I) = z$. Thus $I$ extends to an initial segment $I \cup \{y\} = I \cup \{z\}$, contradicting maximality of $I$. Therefore we must have $I = Y$ or $I = Z$, so one of $Y, Z$ is an initial segment of the other.

So the collection of $f$-inductive sets is totally ordered by inclusion of initial segments, making their union $U$ the maximal $f$-inductive set (Lemma \ref{wo} below). Appending its strict upper bound $f(U)$, the chain $U \cup \{f(U)\}$ is a larger $f$-inductive set, contradiction. The proof is complete.
=--

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

The [[empty set]] (with its unique structure as preordered set) is not an exception to Zorn\'s lemma, as the chain $\emptyset$ does not have an upper bound.
=--

+-- {: .num_remark}
###### Remark
The proof above (originally by [Kneser](#Kneser)) can be seen as an application of general recursion theory à la Paul Taylor's [[Practical Foundations of Mathematics|book]], which in turn is inspired in large part by [Osius](#Osius). In particular, initial segments are the same as $P$-coalgebra maps between [[well-founded coalgebras]], and the maximal $U$ constructed is a maximal attempt with respect to a partial algebra structure $f: P(S) \rightharpoonup S$, following Taylor's formulation of the recursion scheme. For a slightly different arrangement of the facts, one very commonly found in the literature, see the account of the Bourbaki-Witt theorem below.
=--

+-- {: .num_lemma #wo}
###### Lemma
Let $P_\alpha$ be a collection of subsets of a set $S$, each equipped with a well-ordering $\leq_\alpha$, such that for any $\alpha, \beta$, one of $P_\alpha, P_\beta$ (each with their orderings) is an [[lower set|initial segment]] of the other. Let $P$ be the union $\cup_\alpha P_\alpha$, equipped with the ordering $x \leq y$ if $x \leq_\alpha y$ in some $P_\alpha$ containing them both. Then $P$ is well-ordered by $\leq$, with each $P_\alpha$ an initial segment of $P$.
=--

+-- {: .proof}
###### Proof
Observe that $\leq$ is well-defined, and is a [[total order]]: if $x, y \in P$, then $x \in P_\alpha$ and $y \in P_\beta$ for some $\alpha, \beta$, where one of $P_\alpha, P_\beta$ contains the other, say $P_\alpha \subseteq P_\beta$, whence $x, y$ are comparable in $P_\beta$. If $T$ is an inhabited subset of $P$, then $T \cap P_\alpha$ is inhabited for some $\alpha$, so there is a minimal element $t \in T \cap P_\alpha$ with respect to $\leq_\alpha$. This $t$ is minimal in $T$ with respect to $\leq$, for if $s \in T$, $s \leq t$, then $s \in P_\alpha$ by the initial segment condition, and then $t \leq s$ by definition of $t$.
=--

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

If Zorn\'s Lemma and the [[principle of excluded middle]] hold, then so do the [[well-ordering principle]] and the [[axiom of choice]].
=--

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

We first show that Zorn's lemma implies the classical [[well-ordering principle]]. The [[axiom of choice]] easily follows from the well-ordering principle.

1. Let $A$ be a set, and consider the [[poset]] whose elements are pairs $(X, R)$, where $X \in P A$ and $R$ is a (classical) [[well-order]]ing on $X$, ordered by $(X, R) \leq (Y, S)$ if $X$ as a well-ordered set is an initial segment of $Y$. If $C$ is a chain in the poset consisting of subsets $X_\alpha$, then their union $X$ is a well-order by Lemma \ref{wo}, and so the hypothesis of Zorn's lemma is satisfied.

2. By Zorn's lemma, conclude that the poset above contains a maximal element $(X, R)$, where $X \subseteq A$. We claim $X = A$; suppose not (here is where we need [[excluded middle]]), and let $x$ be a member of the complement $\neg X$. Well-order $X \cup \{x\}$, extending $R$ by deeming $x$ to be the final element. This shows $(X, R)$ was not maximal, contradiction. Hence any set $A$ may be well-ordered.

3. The [[axiom of choice]] follows: suppose given a [[surjection]] $p: E \to B$, so that every [[fiber]] $p^{-1}(b)$ is [[inhabited set|inhabited]]. Consider any well-ordering of $E$, and define $s: B \to E$ by letting $s(b)$ be the least element in $p^{-1}(b)$ with respect to the well-ordering. This gives a section $s$ of $p$.
=--

## Bourbaki-Witt theorem

Many accounts of the proof of Zorn's lemma start by establishing first the so-called Bourbaki-Witt theorem, which does not require AC and is of interest in its own right. However, it too does not admit a constructive proof; see [Bauer](#Bauer) for a demonstration that it is not valid in the [[effective topos]]. That said, the issue is subtle enough that the Bourbaki-Witt theorem nonetheless holds in topos with a [[geometric morphism]] to $Set$, for instance any [[Grothendieck topos]], assuming B-W holds in $Set$ ([Bauer-Lumsdaine 2013](#Bauer-Lumsdaine_13)).

+-- {: .num_theorem}
###### Theorem
**(Bourbaki-Witt)**
Let $P$ be an inhabited [[poset]] such that every chain has a [[supremum|least upper bound]], and $s: P \to P$ a function that is *inflationary*: satisfies $x \leq s(x)$ for all $x$. Then $s$ has a [[fixed point]]: an $x$ such that $s(x) = x$.
=--

+-- {: .proof}
###### Proof
Without loss of generality, assume $P$ has a bottom element $0$; otherwise just pick an element $a \in P$ and replace $P$ with the upward set $\uparrow a$.

Say that a subset $I \subseteq P$ is *$s$-inductive* if: $0 \in I$, $I$ is closed under $s$ ($s(I) \subseteq I$), and $I$ contains the sup of any chain in $I$. The intersection of any family of $s$-inductive sets is also $s$-inductive, so the intersection $M$ of all $s$-inductive sets is $s$-inductive.

The idea is that $M$ is totally ordered (is a chain) by the following intuition:
the elements of $M$ are $0, s(0), s s(0), \ldots s^n(0), \ldots$, where we continue by transfinite induction: at limit ordinals $\alpha$ define $s^\alpha(0)$ to be the sup of $\{s^\beta(0): \beta \lt \alpha\}$, and at successors define $s^{\alpha + 1}(0) = s(s^\alpha(0))$. We cannot have a strictly increasing chain that goes on forever, by cardinality considerations, so at some point we hit an $\alpha$ such that $s^{\alpha + 1}(0) = s^\alpha(0)$, which provides a fixed point. (What could be nonconstructive about that? See this [comment](https://golem.ph.utexas.edu/category/2012/10/the_zorn_identity.html#c042428) by Lumsdaine.) In any case, once we prove $M$ is totally ordered, the sup of $M$ is obviously a fixed point.

Without getting into ordinals, one may prove $M$ is totally ordered as follows.
Call an element $c \in M$ a *cap* if $x \lt c$ for $x\in M$ implies $s(x) \leq c$. For each $c \in M$, put $M_c = \{x \in M: x \leq c \vee s(c) \leq x\}$. A routine verification shows $M_c$ is $s$-inductive if $c$ is a cap, so in fact $M_c = M$ by minimality of $M$. Then, show that the set of caps is $s$-inductive. This is also routine if we avail ourselves of the just-proven fact that $M_c = M$ for any cap (but consult Appendix 2 in Lang's Algebra if you get stuck). So again by minimality of $M$, every element of $M$ is a cap. Finally, if $c, d \in M$, use the fact that $c$ is a cap to conclude either that $d \leq c$ or $s(c) \leq d$, whence $c \leq s(c) \leq d$; thus we see any two elements are comparable.
=--

For a discussion of how to get from Bourbaki-Witt to Zorn, see either Lang or this $n$-Category Café [post](https://golem.ph.utexas.edu/category/2012/10/the_zorn_identity.html) by [[Tom Leinster]], especially the main post which gives three very closely related results bearing on Zorn's lemma, and a follow-up comment [here](https://golem.ph.utexas.edu/category/2012/10/the_zorn_identity.html#c042422) which brings in Bourbaki-Witt.

+-- {: .num_remark}
###### Remark
An alternative proof of Bourbaki-Witt would be along lines similar to those used to show AC implies Zorn's lemma. Given the inflationary function $s$, consider the function $f: Well(S) \to S$ which takes a well-ordered subset $W$ to $s(\sup W)$. If $s$ has no fixed points, then $f(W)$ will always be a strict upper bound of $W$, and from there one derives a contradiction exactly as in the proof of Zorn's lemma.
=--

+-- {: .num_remark}
###### Remark
The Bourbaki-Witt theorem is an example of a fixed-point theorem. We should point out its kinship with the quite remarkable fixed-point theorem due to Pataraia, who observed that the conclusion of the Bourbaki-Witt theorem may be strengthened quite considerably, and proved constructively (!), if we change the hypothesis to say that $s$ is a monotone operator (preserves the order $\leq$) on an inhabited [[dcpo]]. See [[fixed point]] for a brief account, and this [blog post](https://projects.lsv.ens-cachan.fr/topology/?page_id=176) for some appreciative commentary.
=--

## Well-ordered formulation

[[Terry Tao]] [points out](https://terrytao.wordpress.com/2009/01/28/245b-notes-7-well-ordered-sets-ordinals-and-zorns-lemma-optional/) (Remark 14) that the proof of Zorn's Lemma uses only the well-ordered chains, allowing a weakened hypothesis.  This is most naturally stated in the context of [[quosets]] rather than [[posets]]:

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

* Given a [[quasiordered set]] $(S, \lt)$, an element $x$ of $S$ is _[[maximal element|maximal]]_ if $x \lt y$ never holds.

* A _[[well-ordered set|well-ordered]] chain_ (or a _well-ordered subset_ or simply a _well-chain_) in $S$ is a [[subset]] $A \hookrightarrow S$ such that $\lt$ is a [[well order]] on $A$.

* A _[[strict upper bound]]_ of a subset $A$ of $S$ is an element $x$ of $S$ such that $y \lt x$ whenever $y \in A$.
=--

The change to *strict* upper bounds is natural in a quasiordered context; it makes no difference in the presence of excluded middle, since we wish to conclude that a maximal element exists; if there is no maximal element, then any set with an upper bound immediately gets a strict upper bound along with it.

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

The well-ordered __Zorn\'s Lemma__ states that any quasiordered set $S$ has a maximal element if every well-chain in $S$ has a strict upper bound, or more simply that it is not possible in a quasiordered set that every well-chain has a strict upper bound.  Or from another perspective, if every well-ordered subset of a quasiordered [[class]] $S$ has a strict upper bound, then $S$ is a [[proper class]].
=--

The ‘more simply’ formulation is equivalent (even constructively) because, on the one hand, if $m$ is a maximal element, then ${m}$ is a well-chain with no strict upper bound, contradicting the hypothesis; while on the other hand, anything whatsoever is true of a nonexistent quoset $S$.  The ‘another perspective’ is then immediate if by a ‘proper class’ one simply means a class that is not a set.

Assuming excluded middle, this is (a priori) stronger than the usual Zorn's Lemma, since only well-ordered chains are required to have upper bounds.  (Of course, once all the proofs are in, they are both simply equivalent to the Axiom of Choice.)  In [[constructive mathematics]], however, it is not stronger, for two reasons: just because $S$ has no maximal element, that doesn't mean that every upper bound has a strict upper bound; and even if this could be fixed, we would only conclude that $S$ does [[not not]] have a maximal element.  However, I intend to argue that this version of Zorn's Lemma should be regarded as more constructively acceptable; some relativized versions are flat-out true.

## Usage/applications

It is very common, when starting with a preordered set $S$, to apply Zorn\'s lemma not to $S$ itself but to an [[up-set]] (an [[under category]]) in $S$.  That is, one starts with an element $x$ of $S$ and proves the existence of a maximal element comparable to $x$.

Zorn\'s lemma may be used to prove all of the following:

* The [[Hausdorff maximal principle]]: Every chain in $S$ is contained in a maximal chain.
* The [[ultrafilter theorem]]: Every proper [[filter]] on a set may be extended to an [[ultrafilter]].
* The [[maximal ideal theorem]]: Every [[ideal]] in a [[ring]] may be extended to a [[maximal ideal]].
* The [[basis theorem]]: Every [[vector space]] (over a [[field]]) has a basis.
* etc ...

Some of these are equivalent to Zorn\'s lemma, while some are weaker; conversely, some additionally require excluded middle.

## References

Due Kuratowski and, independently later, Zorn:

* Casimir Kuratowski, _Une méthode d'élimination des nombres transfinis des raisonnements mathématiques_, Fundamenta Mathematicae __3__ (1922) 76--108 [doi](https://doi.org/10.4064/fm-3-1-76-108)
* Max Zorn, _A remark on method in transfinite algebra_, Bull. Amer. Math. Soc. __41__ (10): 667--670 (1935)[doi](https://doi.org/10.1090/S0002-9904-1935-06166-X)

* Serge Lang, _Algebra_ (third edition), Addison-Wesley 1993.

* {#Osius} Gerhard Osius, _Categorical set theory: a characterization of the category of sets_, Jour. Pure Appl. Alg. 4 (1974), 79--119. doi:[10.1016/0022-4049(74)90032-2](https://doi.org/10.1016/0022-4049%2874%2990032-2)

* {#Bauer} [[Andrej Bauer]], _On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos_, Electronic Notes in Theoretical Computer Science, **249** (2009) 157--167, doi:[10.1016/j.entcs.2009.07.089](https://doi.org/10.1016/j.entcs.2009.07.089) _and_
Theoretical Computer Science, **430** (2012) 43--50, doi:[10.1016/j.tcs.2011.12.005](https://doi.org/10.1016/j.tcs.2011.12.005).
arXiv:[0911.0068](https://arxiv.org/abs/0911.0068).

* {#Bauer-Lumsdaine_13} [[Andrej Bauer]], [[Peter LeFanu Lumsdaine]], _On the Bourbaki-Witt Principle in Toposes_, Math. Proc. Cam. Phil. Soc. 155 (2013), no. 1, 87--99 doi:[10.1017/S0305004113000108](https://doi.org/10.1017/S0305004113000108), arXiv:[1201.0340](https://arxiv.org/abs/1201.0340).

* {#Kneser} Hellmuth Kneser, *Das Auswahlaxiom und das Lemma von Zorn*, Mathematische Zeitschrift, 96:62--63, 1967; available [here](http://resolver.sub.uni-goettingen.de/purl?PPN266833020_0096)

On Zorn's lemma and [[Boolean algebra]] in [[intuitionistic type theories]]:

* {#Bell} [[John Lane Bell]], *Zorn’s lemma and complete Boolean algebras in intuitionistic type theories*, The Journal of Symbolic Logic, 62(4):1265--1279, 1997 ([doi:10.2307/2275642](https://doi.org/10.2307/2275642))

[[!redirects Zorn lemma]]
[[!redirects Zorn's lemma]]
[[!redirects Zorn\'s lemma]]
[[!redirects Zorn's lemma]]
[[!redirects Zorn's Lemma]]


Zuckerman induction

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

#Contents#
* table of contents
{:toc}

## Idea

_Zuckerman induction_ is a special case of [[induced representations|co-induced representations]]:

For $\mathfrak{g}$ a [[semisimple Lie algebra]] and $K$ a suitable [[algebraic]] group, write $(\mathfrak{g}, K)Mod$ for the [[category]] of [[Harish-Chandra modules]] over $(\mathfrak{g}, K)$. Then for $i \colon T \hookrightarrow K$ an algebraic [[subgroup]] there is the corresponding [[forgetful functor]] $ i^* \colon (\mathfrak{g}, K)Mod \to (\mathfrak{g}, T)Mod$ which restricts the representation along the inclusion.

This functor has a [[right adjoint]] [[coinduced representation]] functor

$$
  i_* \colon (\mathfrak{g}, T)Mod \to (\mathfrak{g}, K)Mod
$$

and this is called the _Zuckerman functor_. ([MP, 1](#MP)).

The discussion of the [[derived functor]] of this is sometimes called _[[cohomological induction]]_.

## References

* Dragan Miličić, Pavle Pandžić, _Equivariant derived categories, Zuckerman functors and localization_,  from  Geometry and Representation Theory of real and p-adic Lie Groups , J. Tirao, D. Vogan, J.A. Wolf, editors, Progress in Mathematics __158__, Birkhäuser, Boston, 1997, 209-242, [pdf](http://www.math.utah.edu/~milicic/Eprints/cordoba.pdf)
 {#MP}

[[!redirects Zuckerman functor]]
[[!redirects Zuckerman functors]]
[[!redirects Zuckerman induction functor]]

[[!redirects Zuckermann induction]]


Zuoqin Wang

* [personal page](http://staff.ustc.edu.cn/~wangzuoq/)

## Selected writings

On [[Cartan's closed subgroup theorem]]:

* [[Zuoqin Wang]], *Cartan's closed subgroup theorem*, Lecture 11 in: [李群引论](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Lie.html), 2013 ([pdf](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Notes/Lec%2011.pdf))

category: people


Zurab Janelidze

[Website](https://zurab.online/)

## Selected writings

* {#Janelidze2005} [[Zurab Janelidze]], _Subtractive categories_, Appl. Categ. Struct 13, No. 4, 343-350 (2005) ([doi:10.1007/s10485-005-0934-8](https://doi.org/10.1007/s10485-005-0934-8))

* [[Zurab Janelidze]], _Closedness properties of internal relations_.  A series of papers.

  * I. A unified approach to Malʹtsev, unital and subtractive categories.

  * II. Bourn localization.

  * III. Pointed protomodular categories.

  * IV. Expressing additivity of a category via subtractivity.

  * V. Linear Malʹtsev conditions.

  * VI. Approximate operations.


Zvi Bern

* [webpage](http://personnel.physics.ucla.edu/directory/faculty/bern)

## Selected writings

On the [[worldline formalism]] for [[perturbative quantum field theory]]:

* {#BernKosower91} [[Zvi Bern]], D. Kosower, _Efficient calculation of one-loop QCD amplitudes_ Phys. Rev. D 66 (1991),([journal](http://prl.aps.org/abstract/PRL/v66/i13/p1669_1))

* {#BernKosower92} [[Zvi Bern]], D. Kosower, _The computation of loop amplitudes in gauge theories_, Nucl. Phys. B379 (1992)  ([journal](http://ccdb4fs.kek.jp/cgi-bin/img_index?9108076))

On [[on-shell recursion]]-methods in [[QCD]]:

* [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], _On-Shell Methods in Perturbative QCD_ ([arXiv:0704.2798](http://arxiv.org/abs/0704.2798))

On [[perturbative quantum gravity]]

* _Do I have to draw you a diagram? A tale of quantum gravity_, KITP Public Lecture 2016 ([pdf](http://online.kitp.ucsb.edu/plecture/zbern16/pdf/Zbern16_PublicLecture_KITP.pdf))

* [[Zvi Bern]], _Perturbative quantum gravity and its relation to gauge theory_, Living reviews in relativity, [www.livingreviews.org/Articles/Volume5/2002-5bern](www.livingreviews.org/Articles/Volume5/2002-5bern)

On the [[KLT relations]]:

* [[Zvi Bern]], _Perturbative Quantum Gravity and its Relation to Gauge Theory_, LivingRev.Rel.5:5,2002 ([arXiv:gr-qc/0206071](http://arxiv.org/abs/gr-qc/0206071))

* N. E. J. Bjerrum-Bohr, K. Risager, _String theory and the KLT-relations between gravity and gauge theory including external matter_ ([arXiv:hep-th/0407085](http://arxiv.org/abs/hep-th/0407085))

* [[Zvi Bern]], _The S-matrix reloaded: Twistors, Unitarity, Gauge theories and Gravity_, 2005 ([pdf](http://online.kitp.ucsb.edu/online/strings05/bern/))

in [[D=4 N=8 supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

On the "[[double copy]]" [[KLT relations]]-approach to [[scattering amplitudes]] in [[perturbative quantum gravity]] and [[supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Henrik Johansson]], _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_, Phys. Rev. Lett. **105** 061602 (2010) [[arXiv:1004.0476](https://arxiv.org/abs/1004.0476)]

* [[Zvi Bern]], D.C. Dunbar, T. Shimada, _String-Based Methods in Perturbative Gravity_, Phys. Lett. B **312** (1993) [[arXiv:hep-th/9307001](https://arxiv.org/abs/hep-th/9307001)]

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Marco Chiodaroli]], [[Henrik Johansson]], [[Radu Roiban]], *Supergravity amplitudes, the double copy and ultraviolet behavior*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.07392](https://arxiv.org/abs/2304.07392)]

On using [[KLT relation]]/[[double copy]] for computing [[gravitational wave]]-signatures of [[relativistic binary]] mergers for use with [[LIGO]]:

* {#BCRSSZ19} [[Zvi Bern]], Clifford Cheung, [[Radu Roiban]], Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_, Phys. Rev. Lett. 122, 201603 (2019) ([arXiv:1901.04424](https://arxiv.org/abs/1901.04424))

On evidence for high [[loop order]] finiteness of [[D=4 N=8 supergravity]] as in [[perturbative quantum gravity]]:

* [[Zvi Bern]], [[Lance Dixon]], [[Radu Roiban]], _Is $N = 8$ Supergravity Ultraviolet Finite?_, Phys.Lett.B644:265-271,2007 ([arXiv:hep-th/0611086](http://arxiv.org/abs/hep-th/0611086))

* [[Zvi Bern]], J. J. Carrasco, [[Lance Dixon]], H. Johansson, [[David Kosower]], [[Radu Roiban]], _Three-Loop Superfiniteness of N=8 Supergravity_, Phys.Rev.Lett.98:161303,2007 ([arXiv:hep-th/0702112](http://arxiv.org/abs/hep-th/0702112))

* [[Zvi Bern]], J. J. Carrasco, [[Lance Dixon]], H. Johansson, R. Roiban, _The Ultraviolet Behavior of $N=8$ Supergravity at Four Loops_, Phys. Rev. Lett.103:081301, 2009 ([arXiv:0905.2326](https://arxiv.org/abs/0905.2326))

and via [[KLT relations]] in

* {#BCDJR11} [[Zvi Bern]], John Joseph Carrasco, [[Lance Dixon]], Henrik Johansson, [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_ ([arXiv:1103.1848](http://arxiv.org/abs/1103.1848))

On [[scattering amplitudes]] in [[D=5 supergravity]]:

* [[Zvi Bern]], John Joseph Carrasco, [[Lance Dixon]], [[Michael Douglas]], Matt von Hippel, Henrik Johansson, _$D = 5$ maximally supersymmetric Yang-Mills theory diverges at six loops_ ([arXiv:1210.7709](http://arxiv.org/abs/1210.7709))

## Related $n$Lab entries

* [[worldline formalism]]

* [[4d supergravity]]

* [[KLT relations]], [[double copy]]

* [[gravitational wave]]

* [[string theory results applied elsewhere]]

category: people


Zvonko Iljazovic

* [Mathematics Genealogy Page](http://genealogy.math.ndsu.nodak.edu/id.php?id=157587)

## Related $n$Lab entries

* [[computable set]]

category: people


ZX-calculus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The "ZX-calculus" is an elaboration for the purpose of [[qbit]]-based [[quantum computation]] of the [[string diagram]]-calculus used in [[quantum information theory via dagger-compact categories]].

The basic idea is to formalize the co-existence of the ubiquitous pair of [[quantum measurement|measurement]] [[linear basis|bases]] of [[qbits]] $QBit \simeq \mathbb{C}^2$, namely

1. the "computational basis"

   $$
    \big\{
      \left\vert 0 \right\rangle
      ,\,
      \left\vert 1 \right\rangle
    \big\}
   $$

   which consists (by common convention) of the [[eigenstates]] of the [[Pauli-Z gate]];

1. the [[Hadamard gate|Hadamard]] basis

   $$
     \Big\{
       \tfrac{1}{\sqrt{2}}
       \big(
         \left\vert 0 \right\rangle
         \pm
         \left\vert 1 \right\rangle
       \big)
     \Big\}
   $$

   which consists of the [[eigenstates]] of the [[Pauli-X gate]];

by observing [[Coecke & Duncan, p. 1](#CoeckeDuncan)] that:

1. either equips $QBit = \mathbb{C}^2$ with a [[Frobenius algebra]]-[[structure]] (associated to the [[Frobenius monad]]-[[structure]] of the corresponding [[quantum reader monad]] as originally observed by [[Coecke & Pavlović (2008)](quantum+measurement#CoeckePavlović08)]);

1. jointly these make a certain [[bialgebra]]-structure, algebraically reflecting the fact that they are [[mutually unbiased bases]].

It is this bi-algebraic formalizaton of the interaction between the "Pauli-Z basis" and the "Pauli-X basis" for [[qbits]] which gives the ZX-calculus its name.

Graphically, the ZX-calculus proceeds to declare that:

1. the [[comultiplication|(co-)]][[multiplication]] and [[counit|(co)-]][[unit]] [[string diagrams]] of these two [[Frobenius algebra]]-structures are to be denoted

   1. in green for the Z-basis

   1. in red for the X-basis

1. [[connected topological spaces|connected]] sub-[[string diagrams]] formed from (co-)multiplications and (co-)units  all of the same color are shown as a single box/circle with the given number of in/outputs -- then called *spiders*

   (since the [[Frobenius algebra]]-property ensures --- recalled as [Coecke & Duncan (2011), Theorem 1](#CoeckeDuncan11) --- that any connected diagrams are equal iff they have the same number of in/outputs).

* ...

(...)

## References

Landing page:

* [zxcalculus.com](https://zxcalculus.com/)

The ZX-calculus has its origin as a tool for organizing [[measurement-based quantum computation]]-protocols, following [Danos, Kahsefi & Panangaden (2007)](measurement-based+quantum+computation#DanosKahsefiPanangaden07):

* [[Ross Duncan]], [[Simon Perdrix]], *Rewriting Measurement-Based Quantum Computations with Generalised Flow*, in: *Automata, Languages and Programming. ICALP 2010*, Lecture Notes in Computer Science **6199**, Springer (2010) [[doi:10.1007/978-3-642-14162-1_24](https://doi.org/10.1007/978-3-642-14162-1_24)]

* [[Ross Duncan]], *A graphical approach to measurement-based quantum computing* [[arXiv:1203.6242](https://arxiv.org/abs/1203.6242) video exposition:[YT](https://youtu.be/E_zH9C0ds8c?t=648)]

The official introduction of the ZX-calculus:

* {#CoeckeDuncan} [[Bob Coecke]], [[Ross Duncan]], *A graphical calculus for quantum observables* [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/GreenRed.pdf)]

* {#CoeckeDuncan08} [[Bob Coecke]], [[Ross Duncan]], *Interacting Quantum Observables*, in *Automata, Languages and Programming. ICALP 2008*, Lecture Notes in Computer Science **5126**, Springer (2008) [[doi:10.1007/978-3-540-70583-3_25](https://doi.org/10.1007/978-3-540-70583-3_25)]

* {#CoeckeDuncan11} [[Bob Coecke]], [[Ross Duncan]], *Interacting Quantum Observables: Categorical Algebra and Diagrammatics*, New J. Phys. **13** (2011) 043016 [[arXiv:0906.4725](http://arxiv.org/abs/0906.4725), [doi:10.1088/1367-2630/13/4/043016](https://doi.org/10.1088/1367-2630/13/4/043016)]

Textbook account:

* [[Bob Coecke]], [[Stefano Gogioso]], *Quantum in Pictures*, [Quantinuum Publications](https://www.quantinuum.com/publications) (2023) [[ISBN 978-1739214715](https://www.amazon.co.uk/dp/1739214714), [Quantinuum blog](https://www.quantinuum.com/news/quantum-in-pictures)]

Introduction and review:

* John van de Wetering, *ZX-calculus for the working quantum computer scientist* [[arXiv:2012.13966](https://arxiv.org/abs/2012.13966)]

* [[Bob Coecke]], *Basic ZX-calculus for students and professionals* [[arXiv:2303.03163](https://arxiv.org/abs/2303.03163)]

Further developments:

* Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, Sam Roberts, *Unifying flavors of fault tolerance with the ZX calculus* [[arXiv:2303.08829](https://arxiv.org/abs/2303.08829)]

Relation to [[braided fusion categories]] for [[anyon]] [[braiding]]:

* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$

See also:

* Razin A. Shaikh, Quanlong Wang, Richie Yeung, *How to sum and exponentiate Hamiltonians in ZXW calculus* [[arXiv:2212.04462](https://arxiv.org/abs/2212.04462)]

(...)

[[!redirects ZX calculus]]

[[!redirects zx-calculus]]

[[!redirects zxCalculus]]


|cocomplete category > history

< [[|cocomplete category]]
[[!redirects |cocomplete category -- history]]


|identity assigning morphism > history

< [[|identity assigning morphism]]
[[!redirects |identity assigning morphism -- history]]


Álvaro Pelayo

[Homepage](http://www.math.ucsd.edu/~alpelayo/)

## Publications

On [[homotopy type theory]] and [[univalent foundations]]:

* _Homotopy type theory and Voevodsky's univalent foundations._ [[Álvaro Pelayo]] and [[Michael A. Warren]], 2012. (Bulletin of the AMS, forthcoming) [arXiv](http://arxiv.org/abs/1210.5658)

* _Voevodsky's Univalence Axiom in homotopy type theory._ [[Steve Awodey]], [[Álvaro Pelayo]], and [[Michael A. Warren]], October 2013, Notices of the American Mathematical Society 60(08), pp.1164-1167. [arXiv](http://arxiv.org/abs/1302.4731)

category: people


Ángel Murcia

* [inspire page](https://inspirehep.net/authors/1739706)

## Selected writings

On [[contact geometry]] in [[D=6 supergravity]]:

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity*, Journal of Geometry and Physics **158** (2020) [[arXiv:1912.08723](https://arxiv.org/abs/1912.08723), [doi:10.1016/j.geomphys.2020.103868](https://doi.org/10.1016/j.geomphys.2020.103868)]

Discussion of [[Killing spinors]] on [[globally hyperbolic Lorentzian manifolds]]:

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Parallel spinors on globally hyperbolic Lorentzian four-manifolds*, Ann Glob Anal Geom **61** (2022) 253–292 [[arXiv:2011.02423](https://arxiv.org/abs/2011.02423), [doi:10.1007/s10455-021-09808-y](https://doi.org/10.1007/s10455-021-09808-y)]

category: people

[[!redirects Angel Murcia]]


Ángel Uranga

[[!redirects Angel Uranga]]

* [webpage](http://members.ift.uam-csic.es/auranga/intro.html)

## Selected writings

On [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

* {#ParkUranga98} [[Jaemo Park]], [[Angel Uranga]], _A Note on Superconformal $\mathcal{N}=2$ theories and Orientifolds_, Nucl. Phys. B542:139-156, 1999 ([arXiv:hep-th/9808161](https://arxiv.org/abs/hep-th/9808161))

On [[intersecting D-brane models]] (on [[orbifolds]]/[[orientifolds]] with [[RR-field tadpole cancellation]]):

* {#AFIRU01a} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _D=4 Chiral String Compactifications from Intersecting Branes_, J. Math. Phys. 42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _Intersecting Brane Worlds_, JHEP 0102:047, 2001 ([arXiv:hep-ph/0011132](https://arxiv.org/abs/hep-ph/0011132))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

On IIA [[NS5-branes]] and their [[T-duality|T-dual]] [[orientifolds]] with [[RR-field tadpole cancellation]]:

* [[Angel Uranga]], _A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points_, Nucl. Phys. B577:73-87, 2000 ([arXiv:hep-th/9910155](https://arxiv.org/abs/hep-th/9910155))

* {#FengHeKarchUranga01} [[Bo Feng]], [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))

On [[bottom-up model building]] for [[intersecting D-brane models]]:

* {#AldazabalIbanezQuevedoUranga00} G. Aldazabal, [[Luis Ibáñez]], [[Fernando Quevedo]], [[Angel Uranga]], _D-Branes at Singularities: A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067v2](https://arxiv.org/abs/hep-th/0005067))

On [[toric duality]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))

## Related $n$Lab entries

* [[string phenomenology]]

* [[intersecting D-brane models]]

* [[Higgs mechanism]]

category: people


Æ code

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]], *Absorption-Emission codes* or *Æ codes* are a family of [[quantum error correcting codes]] designed to encode [[quantum information]] within a diatomic [[molecule]]'s orientation, allowing changes in [[angular momentum]] and error correction from interference from torques, Raman scattering and spontaneous emission.

Æ codes are more practical in theory than molecular codes in the sense that they require lower (median) [[momentum]] and can protect against photonic processes.

They target molecular systems but can also be applied or extended to [[atoms]] and atomic scale ions as long as a Zeeman [[manifold]] with a large enough [[angular momentum]] is present, or any system in which multiple $2J+1$-dimensional [[linear representation|representations]] of [[SU(2)]].

It is formed by defining logical-qubit encoding codewords and theorem proving through Knill-Laflamme conditions, yielding highly tunable symmetric codes which can then be tuned to form a basic error-detecting version.

## Related entries

* [[bit-flip code]]

## References

* Shubham Jain, Eric Hudson, Wesley Campbell, [[Victor V. Albert]], *Æ codes* [[arxiv:2311.12324](https://arxiv.org/abs/2311.12324)]

[[!redirects Æ codes]]

[[!redirects absorption-emission code]]
[[!redirects absorption-emission codes]]

[[!redirects Absorption-Emission code]]
[[!redirects Absorption-Emission codes]]


Édouard Goursat

* [[Goursat theorem]]
* [[Stokes theorem]]

* [MacTutor bio](http://www-history.mcs.st-andrews.ac.uk/Biographies/Goursat.html)

category: people

[[!redirects Édouard Goursat]]
[[!redirects Edouard Goursat]]
[[!redirects Edward Goursat]]
[[!redirects É. Goursat]]
[[!redirects É Goursat]]
[[!redirects E. Goursat]]
[[!redirects E Goursat]]
[[!redirects Goursat]]


Élie Cartan

**Élie Joseph Cartan** was a French [[differential geometry|differential geometer]]. His results include the classification of complex semisimple [[Lie algebras]] ("Cartan classification"), extension of these results to a class of symmetric spaces, the proof of the Lie--Cartan theorem (after Serre sometimes called "[[Lie's three theorems|Lie's third theorem]]") on integration of Lie algebras to Lie groups (Lie proved just the integration to [[local Lie group]]s), the method of moving frames, the introduction of Cartan's connection, numerous results in [[Riemannian geometry]], results related to the formal integrability of PDEs (Cartan involutive equations, Pfaffian system), etc.

Father of [[Henri Cartan]].

* wikipedia [en](http://en.wikipedia.org/wiki/%C3%89lie_Cartan), [fr](http://fr.wikipedia.org/wiki/%C3%89lie_Cartan), [ru](http://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D1%82%D0%B0%D0%BD,_%D0%AD%D0%BB%D0%B8_%D0%96%D0%BE%D0%B7%D0%B5%D1%84)
* wikipedia  [integrability conditions for  differential systems](http://en.wikipedia.org/wiki/Integrability_conditions_for_differential_systems)

## Related $n$Lab entries

* [[Cartan calculus]]

* [[Cartan's magic formula]]

[[!redirects Elie Cartan]]
[[!redirects É. Cartan]]

category: people


Émile Borel

* [Wikipedia entry](http://en.wikipedia.org/wiki/Émile_Borel)

## related $n$Lab entries

* [[Borel's theorem]]

category: people

[[!redirects Emile Borel]]
[[!redirects Borel]]


Éric Goubault

Éric Goubault is a research director at CEA (since 2003), and professor at École Polytechnique since 2005.  He is in charge of the Laboratory for the Modelling and Analysis of Interactive Systems (LIST) at CEA, as well as a joint research team (with the same name) between CEA, CNRS and École Polytechnique.

 His research interests include fundamental computer science, semantics, abstract interpretation, static analysis and verification (both theory and practise), numerical domains, concurrency theory and [[directed algebraic topology]].

[webpage](http://www.lix.polytechnique.fr/~goubault/)

## Selected writings

On [[S4 modal logic]]:

* [[Jean Goubault-Larrecq]], [[Éric Goubault]], _On the geometry of intuitionistic S4 proofs_, Homology, Homotopy and Applications **5** 2 (2003) 137-209 [[eudml:50340](https://eudml.org/doc/50340), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-5/issue-2/On-the-geometry-of-intuitionistic-S4-proofs/hha/1088453324.full)]

category: people

[[!redirects Éric Goubault]]
[[!redirects E. Goubault]]
[[!redirects Eric Goubault]]


étale (infinity,1)-site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _étale (∞,1)-site_ is an [[(∞,1)-site]] whose [[(∞,1)-topos]] encodes the [[derived geometry]] version of the geometry encoded by the [[topos]] over the [[étale site]].

Its underlying [[(∞,1)-category]] is the [[opposite (∞,1)-category]] $sCAlg_k^{op}$ of commutative [[simplicial algebras]] over a commutative ring $k$, whose [[covering]] families are essentially those which under [[decategorification]] become coverings in the [[étale site]] of ordinary $k$-algebras.

## Definition

Let $k$ be a commutative ring. Let $T$ be the [[Lawvere theory]] of commutative [[associative algebra]]s over $k$.

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

Let

$$
  \infty CAlg_k := T Alg_\infty
$$

be the [[(∞,1)-category]] of [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] over $T$ regarded as an [[(∞,1)-algebraic theory]].

=--

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

Let $sCAlg_k = (T Alg)^{\Delta^{op}}$ be the [[sSet]]-[[enriched category]] of [[simplicial algebra|simplicial commutative associative k-algebras]] equipped with the standard [[model structure on simplicial T-algebras]]. Write $sCAlg_k^\circ$ for the $(\infty,1)$-category [[presentable (∞,1)-category]]. Then we have an [[equivalence of (∞,1)-categories]]

$$
  \infty CAlg_k \simeq (sCAlg_k)^\circ
  \,.
$$

=--

This is a special case of the general statement discussed at [[(∞,1)-algebraic theory]]. See also ([Lurie, remark 4.1.2](#Lurie)).

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

For $X \in \infty CAlg_k^{op}$ we write $\mathcal{O}(X)$ for the corresponding object in $\infty CAlg_k$ and conversely for $A \in \infty CAlg_k$ we write $Spec A$ for the corresponding object in $\infty CAlg_k^{op}$.

So $Spec \mathcal{O} X = X$ and $\mathcal{O} Spec A = A$, by definition of notation.

=--

Notice from  the discussion at [[model structure on simplicial algebras]] the [[homotopy group]] functor

$$
  \pi_* : sCAlg_k \to CAlg_k
  \,.
$$

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

A morphism $Spec A  \to Spec B$ in $\infty CAlg_k^{op}$ is an **étale morphism** if

1. The underlying morphism $Spec \pi_0(A) \to Spec \pi_0(B)$ is an [[étale morphism]] of [[schemes]];

1. for each $i \in \mathbb{N}$ the canonical morphism

   $$
     \pi_i(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_i(B)
   $$

   is an [[isomorphism]].

=--

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

The **étale $(\infty,1)$-site** is the [[(∞,1)-site]] whose underlying $(\infty,1)$-category is the [[opposite (∞,1)-category]] $\infty CAlg_k^{op}$ and whose [[covering]] famlies $\{Spec A_i \to Spec B\}_{i \in I}$ are those collections of morphisms such that

1. every $Spec A_i \to Spec B$ is an étale morphism

2. there is a [[finite set|finite subset]] $J \subset I$  such that the underlying decategorified family $\{Spec \pi_0(A_j) \to Spec \pi_0(B)\}_{j \in J}$ is a covering family in the 1-[[étale site]].

=--

This appears as ([ToënVezzosi, def. 2.2.2.12](#ToenVezzosi)) and as ([Lurie, def. 4.3.3; def. 4.3.13](#Lurie)).

## Properties

### Derived étale geometry {#AsDerivedGeometry}

The following definition and theorem show how the étale $(\infty,1)$-site arises naturally from the [[étale site|étale 1-site]], and naturally encodes the [[derived geometry]] induced by the étale site.

+-- {: .num_defn}
###### Definition
**(étale pregeometry)**

Let $\mathcal{T}_{et}$ be the 1-[[étale site]] regarded as a [[pregeometry (for structured (∞,1)-toposes)]] as follows.

* the underlying [[(∞,1)-category]] is the 1-[[category]]

  $$
    (CAlg_k^{sm})^{op} \hookrightarrow CAlg_k ^{op}
    \,,
  $$

  which is the full [[subcategory]] of $CAlg_k$ on those objects $A \in CAlg_k$ for which there exists an [[étale morphism]] $k[x^1, \cdots, x^n] \to A$ from the [[polynomial]] algebra in $n$ generators for some $n \in \mathbb{N}$;

* the _admissible morphisms_ in the pregeometry are the [[étale morphism]]s;

* a collection of admissible morphisms is a [[covering]] family if it is so as a family of morphisms in the [[étale site]].

=--

This is ([Lurie, def. 4.3.1](#Lurie)).

+-- {: .num_defn}
###### Definition
**(étale geometry)**

Let $\mathcal{G}_{et}$ be the [[geometry (for structured (∞,1)-toposes)]] given by

* the underlying [[(∞,1)-site]] is the étale $(\infty,1)$-site;

* the admissible morphisms are the étale morphisms.

=--

This is ([Lurie, def. 4.3.13](#Lurie)).

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

The [[geometry (for structured (∞,1)-toposes)|geometry]] generated by the étale pregeometry $\mathcal{T}_{et}$ is the étale geometry $\mathcal{G}_{et}$.

=--

This is ([Lurie, prop. 4.3.15](#Lurie)).

## Related concepts

* [[étale morphism]], [[étale site]], [[étale cohomology]]

* [[étale morphism of E-∞ rings]]

* **étale $(\infty,1)$-site**

## References

In its [[presentable (∞,1)-category|presentation]] as a [[model site]] the étale $(\infty,1)$-site is given in definition 2.2.2.12 of

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications _ ([arXiv](http://arxiv.org/abs/math/0404373))
{#ToenVezzosi}.

A discussion in the context of [[structured (∞,1)-toposes]] is

* [[Jacob Lurie]], section 4.3 of _[[Structured Spaces]]_
{#Lurie}

* [[Jacob Lurie]], section 3 and 4 of, _Descent theorems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))

See also

* [[Benjamin Antieau]], [[David Gepner]], _Brauer groups and étale cohomology in derived algebraic geometry_ ([arXiv:1210.0290](http://arxiv.org/abs/1210.0290))
 {#AntieauGepner12}

[[!redirects étale (∞,1)-site]]
[[!redirects etale (∞,1)-site]]

[[!redirects etale (infinity,1)-site]]


étale algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An $R$-[[associative algebra|algebra]] $A$ is _étale_ if the definition [[homomorphism]] $R \to A$ is, dually, an [[étale morphism of schemes]].

A more elementary definition for algebras over a discrete field exists in constructive algebra.

## Related concepts

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

## References

Henri Lombardi & Claude Quitté, _Constructive Algebra, Constructive Methods_, 2014,
[PDF](http://hlombardi.free.fr/CACM-english.pdf)

[[!redirects etale algebra]]

[[!redirects etale algebras]]
[[!redirects étale algebras]]