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Zhenbin Yang | https://ncatlab.org/nlab/source/Zhenbin+Yang |
* [webpage](https://profiles.stanford.edu/zhenbin-yang?releaseVersion=7.16.0)
## Selected writings
General aspects of [[nearly AdS2-CFT1]] ([[JT-gravity]]/matrix models):
* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ([arXiv:1606.01857](https://arxiv.org/abs/1606.01857))
Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:
* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]
On the [[SYK model]]:
* Biao Lian, S. L. Sondhi, [[Zhenbin Yang]], _The chiral SYK model_ ([arXiv:1906.03308](https://arxiv.org/abs/1906.03308))
Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:
* [[Geoff Penington]], [[Stephen Shenker]], [[Douglas Stanford]], [[Zhenbin Yang]], _Replica wormholes and the black hole interior_ ([arXiv:1911.11977](https://arxiv.org/abs/1911.11977))
category: people |
Zheng Yin | https://ncatlab.org/nlab/source/Zheng+Yin |
* [SemanticScholar page](https://www.semanticscholar.org/author/Zheng-Yin/2048400828)
## Selected writings
On a [[matrix model]] for the [[3-brane in 6d]] arising as the [[brane intersection|intersection]] of two [[M5-branes]]:
* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))
category: people |
Zheng-Cheng Gu | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Zhiqin+Lu |
* [webpage](http://www.math.uci.edu/~zlu/)
## related $n$Lab entries
* [[moduli space of Calabi-Yau manifolds]]
category: people |
Zhiwei Yun | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Zhuo+Chen | ## Zhuo Chen ##
Professor of Mathematics
###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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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) <a href="http://dx.doi.org/10.1016/0168-0072(84)90014-9">doi</a>
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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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]]
|
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|
zigzag identity 1 implies 2 - SVG | https://ncatlab.org/nlab/source/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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|
zigzag of real numbers - SVG | https://ncatlab.org/nlab/source/zigzag+of+real+numbers+-+SVG |
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|
zigzag persistence module | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Ziv+Ran | * [webpage](http://mathdept.ucr.edu/faculty/ran.html)
## Related pages
* [[Ran space]]
category: people |
Zoe Wyatt | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/Zolt%C3%A1n+Zimbor%C3%A1s |
* [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 | https://ncatlab.org/nlab/source/Zoran+Skoda+%3E+history | < [[Zoran Skoda]]
[[!redirects Zoran Skoda -- history]] |
Zoran Škoda | https://ncatlab.org/nlab/source/Zoran+%C5%A0koda | 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_ (<a href="https://www.irb2.hr/korisnici/zskoda/abs1">abstract</a>)). 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 | https://ncatlab.org/nlab/source/Zorn%27s+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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/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 | https://ncatlab.org/nlab/source/%7Ccocomplete+category+%3E+history | < [[|cocomplete category]]
[[!redirects |cocomplete category -- history]] |
|identity assigning morphism > history | https://ncatlab.org/nlab/source/%7Cidentity+assigning+morphism+%3E+history | < [[|identity assigning morphism]]
[[!redirects |identity assigning morphism -- history]] |
Álvaro Pelayo | https://ncatlab.org/nlab/source/%C3%81lvaro+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 | https://ncatlab.org/nlab/source/%C3%81ngel+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 | https://ncatlab.org/nlab/source/%C3%81ngel+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 | https://ncatlab.org/nlab/source/%C3%86+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 | https://ncatlab.org/nlab/source/%C3%89douard+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 | https://ncatlab.org/nlab/source/%C3%89lie+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 | https://ncatlab.org/nlab/source/%C3%89mile+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 | https://ncatlab.org/nlab/source/%C3%89ric+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 | https://ncatlab.org/nlab/source/%C3%A9tale+%28infinity%2C1%29-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 | https://ncatlab.org/nlab/source/%C3%A9tale+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]]
|
étale cohomology | https://ncatlab.org/nlab/source/%C3%A9tale+cohomology | [[!redirects etale cohomology]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
Traditional _étale cohomology_ (e.g. [Deligne 77](#Deligne77)) is the [[abelian sheaf cohomology]] for [[sheaf|sheaves]] on the [[étale site]] of a [[scheme]] -- which is an analog of the [[category of open subsets]] of a [[topological space]] $X$ , or rather the analog of the category of [[étale spaces]] over $X$, with [[finite set|finite]] [[fibers]].
A certain [[inverse limit]] over étale cohomology groups for different [[coefficients]] yields [[ℓ-adic cohomology]], which is a [[Weil cohomology theory]].
More generally, there is étale [[generalized cohomology theory]] with [[coefficients]] in [[sheaves of spectra]] on the [[étale site]] ([Jardine 97](#Jardine97)). Still more generally, there is étale generalized cohomology on the [[étale (∞,1)-site]] ([Antieau-Gepner 12](#AntieauGepner12), [Lurie](#Lurie)).
## Definition ##
+-- {: .num_prop }
###### Proposition
Given a [[scheme]] $X$ of finite type, the small [[étale site]] $X_{et}$ is the [[category]] whose [[objects]] are [[étale morphisms]] $Spec R \to X$ and whose morphisms $(f:Spec R\to X)\to (f':Spec R'\to X)$ are morphisms $\alpha: Spec(R)\to Spec(R')$ of schemes completing triangles: $f'\circ\alpha=f$ (notice that the morphisms between étale morphisms are automatically étale). This category naturally carries a [[Grothendieck topology]] that makes it a [[site]], the [[étale site]].
For $A \in Sh(X_{et}, Ab)$ an [[abelian sheaf]] on $X$, the **étale cohomology** $H_{et}^\bullet(X,A)$ of $X$ with [[coefficients]] in $A$ is the [[abelian sheaf cohomology]] with respect to this site.
=--
## Basic properties
{#BasicProperties}
The following are some basic properties of étale [[cohomology groups]] for various standard choices of [[coefficients]].
### Relation to Zariski cohomology
{#RelationZariskiEtaleCohomology}
+-- {: .num_remark }
###### Remark
A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zariski sheaf, but not necessarily conversely.
=--
+-- {: .num_remark #LerayForInclusionOfZariskiIntoEtale}
###### Remark
For $X$ a [[scheme]], the inclusion
$$
\epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}
$$
of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]] which computes étale cohomology in terms of Zariski cohomology
$$
E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F})
\Rightarrow
E^{p+q} = H^{p+q}(X_{et}, \mathcal{F})
\,.
$$
=--
This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include ([Tamme, II 1.3](#Tamme)).
### With coefficients in coherent modules
{#WithCoefficientsInCoherentModules}
+-- {: .num_prop #CohomologyWithCoeffsInCoherentModules}
###### Proposition
For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced étale sheaf (by the discussion at [étale topos -- Quasicohetent sheaves](etale+topos#QuasiCoherentModules)),
then the [[edge morphism]]
$$
H^p_{Zar}(X, N)
\longrightarrow
H^p_{et}(X,N_{et})
$$
of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an [[isomorphism]] for all $p$, itentifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the étale cohomology with coefficients in $N_{et}$.
Moreover, for $X$ affine we have
$$
H^p_{et}(X, N_{et}) \simeq 0
\,.
$$
=--
This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance ([Tamme, II (4.1.2)](#Tamme)).
+-- {: .proof}
###### Proof
By the discussion at _[[edge morphism]]_ it suffices to show that
$$
R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0
\,.
$$
By the discussion at _[[direct image]]_ (also at _[[abelian sheaf cohomology]]_), $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by
$$
U \mapsto H^q(X_{et}|U, N)
\,,
$$
hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]].
By the existence of [cofinal affine étale covers](etale+site#CofinalAffineCovers) the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ with the induced [[coverage]] is a [[dense subsite]] of affines. Therefore it suffices to show the statement there.
Moreover, by the finiteness condition on [[étale morphisms]] every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map
$$
Spec(B) \longrightarrow Spec(A)
\,.
$$
It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex
$$
N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots
$$
is of the form
$$
0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots
\,.
$$
known as the [[Amitsur complex]].
Since $A \to B$ is a [[faithfully flat morphism]] it follows by the [[descent theorem]] that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes.
=--
### With coefficients in a cyclic group
{#WithCoefficientsInACyclicGroup}
+-- {: .num_prop}
###### Proposition
If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is
$$
H^q(X, (\mathbb{Z}/p\mathbb{Z})_X)
\simeq
\left\{
\array{
A/(F - id)A & if\; q = 1
\\
0 & if \; q \gt 0
}
\right.
\,.
$$
=--
+-- {: .proof}
###### Proof
Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form
$$
\begin{aligned}
0 & \to
H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X)
\\
& \to
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X)
\\
& \to
H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots
\end{aligned}
\,,
$$
where $F(-) = (-)^p$ is the [[Frobenius endomorphism]].
By prop. \ref{CohomologyWithCoeffsInCoherentModules}
the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish,
and so from [[exact sequence|exactness]] we find an [[isomorphism]]
$$
H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X))
\stackrel{\simeq}{\to}
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
\,,
$$
hence the claimed isomorphism
$$
A/(F-id)(A)
\stackrel{\simeq}{\to}
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
\,.
$$
By the same argument all the higher cohomology groups vanish, as claimed.
=--
### With coefficients in the multiplicative group
the étale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names:
* $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]];
* $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], [Tamme, II 4.3.1](#Tamme));
* $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]];
### With coefficients in groups of roots of unity
* [[Kummer sequence]]
([Tamme, II, 4.4](#Tamme))
(...)
## Main theorems
{#MainTheorems}
The following are the main theorems characterizing properties of étale cohomology. Together these theorems imply that étale cohomology, in its variant as [[l-adic cohomology]], is a [[Weil cohomology theory]].
### Proper base change theorem
* [[proper base change theorem]]
([Milne, section 17](#Milne))
### Comparison theorem: Relation to singular cohomology
* [[comparison theorem (étale cohomology)]]
([Milne, section 21](#Milne))
### Künneth formula
{#KünnethFormula}
([Milne, section 22](#Milne))
### Cycle map
{#CycleMap}
([Milne, section 23](#Milne))
### Poincaré duality
{#PoincareDuality}
([Milne, section 24](#Milne))
### Lefschetz fixed-point formula
[Künneth formula](#KünnethFormula) + [cycle map](#CycleMap) + [Poincaré duality](#PoincareDuality) $\Rightarrow$ [[Lefschetz fixed-point formula]]
([Milne, section 25](#Milne))
\section{Relation to motivic cohomology}
We have the following, which is Theorem 10.2 in [MazzaVoevodskyWeibel2006](#MazzaVoevodskyWeibel2006).
\begin{thm} Let $X$ be a smooth, separated [[scheme]] over a [[field]] $k$. Let $n$ be an integer prime to the characteristic of $k$. Then for every integer $p$ and every integer $q \geq 0$, we have that $H^{p,q}_{L}(X, \mathbb{Z} / n)$, the étale (or Lichtenbaum) motivic cohomology of $X$, is isomorphic to $H^{p}_{et}\left(X, \mathbb{Z}/n(q) \right)$, the [[étale cohomology]] of $X$ with coefficients in the $q$-th [[Tate twist]] of $\mathbb{Z}/n$. \end{thm}
The inverting of smashing with the [[Tate sphere]] as opposed to just with $S^{1}$ in the construction of the [[stable motivic homotopy category]] allows the $q$ part of [[motivic cohomology]] to be represented in it (by the motivic Eilenberg-MacLane spectrum), as opposed to just the $(p,0)$ part. Combining this with the above theorem, we see that inverting the Tate sphere as opposed to just $S^{1}$ allows the Tate twists of étale cohomology to be represented.
## Related concepts
* [[basics of étale cohomology]]
* [[étale morphism]], [[étale site]], étale cohomology
* [[étale (∞,1)-site]], [[étale topos]]
* [[étale homotopy]]
* [[Weil conjecture]]
* [[continuous étale cohomology]]
## References
### History, motivation and original accounts
Étale cohomology was conceived by [[Artin]], [[Deligne]], [[Alexander Grothendieck|Grothendieck]] and [[Verdier]] in 1963. It was used by Deligne to prove the [[Weil conjectures]]. Some useful (and also funny) remarks on this are in the beginning of
* [[Spencer Bloch]], Review of [[Milne]]'s _[[Étale Cohomology]]_ ([pdf](http://www.ams.org/bull/1981-04-02/S0273-0979-1981-14894-1/S0273-0979-1981-14894-1.pdf); publisher's [book page](http://www.worldscibooks.com/mathematics/7773.html))
See also
* MathOverflow _[Étale cohomology — Why study it?](http://mathoverflow.net/questions/6070/etale-cohomology-why-study-it)_
The classical references include [[SGA]], esp.
* [[Pierre Deligne]] et al., _Cohomologie étale_ , Lecture Notes in Mathematics __569__, Springer-Verlag, 1977.
{#Deligne77}
* [[James Milne]], _[[Étale Cohomology]]_, Princeton Mathematical Series __33__, 1980. xiii+323 pp.
See also
* [[Barry Mazur]], _Notes on étale cohomology of number fields_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 ([Numdam](http://www.numdam.org/item?id=ASENS_1973_4_6_4_521_0), [pdf](http://modular.math.washington.edu/edu/2010/582e/refs/mazur-notes_on_etale_cohomology_of_number_fields_original.pdf))
### Reviews and modern accounts
* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_, 1994
{#Tamme}
A modern textbook, though largely based on the material in SGA is
* Lei Fu, _Étale cohomology theory_, Nankai Tracts in Math. __13__, World Sci. 2011; ([toc pdf](http://www.worldscibooks.com/etextbook/7773/7773_toc.pdf); Preface [pdf](http://www.worldscibooks.com/etextbook/7773/7773_preface.pdf); chap. 1 Descent theory [pdf](http://www.worldscibooks.com/etextbook/7773/7773_chap01.pdf))
Lecture notes include
* [[James Milne]], _[[Lectures on Étale Cohomology]]_ ([html](http://www.jmilne.org/math/CourseNotes/lec.html), [pdf](http://www.jmilne.org/math/CourseNotes/LEC.pdf))
{#Milne}
* [[Aise Johan de Jong]], _Étale cohomology_ 2009, in _[[The Stacks Project]]_ ([pdf](http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf))
{#deJong}
* [[Evan Jenkins]], _Étale cohomology seminar_ ([web](http://math.uchicago.edu/~ejenkins/etale/))
* Donu Arapura, _An introduction to Étale cohomology_ ([pdf](http://www.math.purdue.edu/~dvb/preprints/etale.pdf))
* Antoine Ducros, _Étale cohomology of schemes and analytic spaces_, ([pdf](http://www.math.jussieu.fr/~ducros/Cohetale.pdf))
See also
* Edgar Costa, _Étale cohomology_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/686086/1/tese.pdf))
* Thomas H. Geisser, _Weil-etale motivic cohomology_, [K-th archive](http://www.math.illinois.edu/K-theory/0565)
Discussion of [[generalized cohomology theory]] on the [[étale site]] but with [[coefficients]] in [[sheaves of spectra]] is in
* [[Rick Jardine]], _Generalized Étale cohomology theories_, 1997 Progress in mathematics volume 146
{#Jardine97}
Discussion of generalized étale cohomology over the [[étale (∞,1)-site]] (hence in [[higher topos theory]]/[[higher algebra]]) is in
* [[Benjamin Antieau]], [[David Gepner]], _Brauer groups and étale cohomology in derived algebraic geometry_ ([arXiv:1210.0290](http://arxiv.org/abs/1210.0290))
{#AntieauGepner12}
* [[Jacob Lurie]], _Descent theorems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))
{#Lurie}
\subsection{Other}
* {#MazzaVoevodskyWeibel2006} Mazza, [[Vladimir Voevodsky|Voevodsky]], [[Chuck Weibel|Weibel]], _Lecture notes on motivic cohomology_, Clay Mathematics Monographs 2. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. xiv, 216 p. (2006). [zentralblatt](https://zbmath.org/?q=an%3A1115.14010) [pdf](http://www.claymath.org/library/monographs/cmim-2-voevodsky-cov1.pdf)
[[!redirects étale cohomology]]
[[!redirects generalized étale cohomology]]
[[!redirects generalized étale cohomologies]]
[[!redirects generalized etale cohomology]]
[[!redirects generalized etale cohomologies]] |
étale cover | https://ncatlab.org/nlab/source/%C3%A9tale+cover | [[!redirects etale cover]]
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn}
###### Definition
An **étale cover** of an algebraic [[scheme]] $X$ is a set
$\{p_i : U_i \to X\}$ of [[étale morphism]]s locally of finite type
which are jointly surjective in the sense that $X$ equals the union of
set-theoretic images:
$$
X = \union_i p_i(U_i).
$$
=--
+-- {: .num_remark}
###### Remark
The condition of being locally of finite type is just strengthening the variant of the notion of étale: in the case of non-Noetherian schemes Grothendieck requires instead that étale morphisms be locally of finite presentation; for the purpose of [[étale topology]] locally of finite type is required.
=--
+-- {: .num_remark}
###### Remark
The [[étale site]] has [[covering]]s given by the étale covers.
=--
## Properties
+-- {: .num_prop}
###### Proposition
Every étale cover is a [[cover]] in the [[fpqc topology]].
=--
This appears for instance as ([tag 03PH](https://stacks.math.columbia.edu/tag/03PH) of the stacks project).
## References
* [The Stacks project](https://stacks.math.columbia.edu)
[[!redirects etale covers]]
[[!redirects étale cover]]
[[!redirects étale covers]]
[[!redirects etale covering]]
[[!redirects etale coverings]]
[[!redirects étale covering]]
[[!redirects étale coverings]]
|
étale geometric morphism | https://ncatlab.org/nlab/source/%C3%A9tale+geometric+morphism | [[!redirects etale geometric morphism]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[sheaf]] $F$ on (the [[site]] [[category of open subsets|of open subsets]] of) a [[topological space]] $X$ corresponds to an [[étalé space]] $\pi_F : Y_F \to X$. This space $Y_F$ has itself a [[sheaf topos]] associated to it, and the map $Y_F \to X$ induces a [[geometric morphism]] of [[sheaf topos]]es
$$
\pi_F : Sh(Y_F) \to Sh(X)
\,.
$$
Due to the special nature of $Y_F$, the topos on the left is equivalent to the [[slice topos]] $Sh(X)/F$, and the projection morphism above factors through a canonical standard [[geometric morphism]] $Sh(X)/F \to Sh(X)$
$$
\pi_F : Sh(Y_F) \stackrel{\simeq}{\to} Sh(X)/F \to Sh(X)
\,.
$$
And conversely, every [[local homeomorphism]] $Y \to X$ of [[topological space]]s corresponds to a [[geometric morphism]] of sheaf toposes of this form.
This motivates calling a [[geometric morphism]]
$$
\mathcal{X} \to \mathcal{Y}
$$
a **local homeomorphism of toposes** or **étale geometric morphism** if it factors as an equivalence followed by a projection out of an overcategory topos.
If the topos is a locally ringed topos, or more generally a [[structured (∞,1)-topos]], it makes sense to require additionally that the local homeomorphism is compatible with the extra structure.
## Definition
For $\mathbf{H}$ a [[topos]] (or [[(∞,1)-topos]], etc.) and for $X \in \mathbf{H}$ an [[object]], the [[overcategory]] $\mathbf{H}_{/X}$ is also a topos ($(\infty,1)$-topos, etc), the _[[slice topos]]_ ([[slice (∞,1)-topos]], ...).
The canonical projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ is part of an [[essential geometric morphism|essential]] (in fact, [[locally connected geometric morphism|locally connected]]/ [[locally ∞-connected (∞,1)-topos|locally ∞-connected]]) geometric morphism:
$$
\pi = (\pi_! \dashv \pi^* \dashv \pi_*) :
\mathbf{H}_{/X}
\stackrel{\overset{\pi_!}{\longrightarrow}}{\stackrel{\overset{\pi^*}{\longleftarrow}}{\underset{\pi_*}{\longrightarrow}}}
\mathbf{H}
\,.
$$
This is the [[base change geometric morphism]] for the terminal morphism $X \to *$.
### For toposes
+-- {: .num_defn}
###### Definition
A [[geometric morphism]] $\mathbf{K} \to \mathbf{H}$ is called a
**local homeomorphism of toposes**, or an **étale geometric morphism**, if it is equivalent to such a projection--- in other words, if it factors by geometric morphisms as $\mathbf{K} \stackrel{\simeq}{\to} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ for some $X \in \mathbf{H}$ .
=--
### For structured toposes
If the [[(∞,1)-topos]]es in question are [[structured (∞,1)-toposes]], then this is refined to the following
+-- {: .num_defn}
###### Definition
A morphism $f : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ of [[structured (∞,1)-topos]]es is an **étale morphism** if
1. the underlying morphism of $(\infty,1)$-toposes is an étale geometric morphism;
1. the induced map $f^* \mathcal{O}_\mathcal{Y} \to \mathcal{O}_\mathcal{X}$
is an equivalence.
=--
This is [[Structured Spaces|StSp, Def. 2.3.1]].
## Examples
If $\mathbf{H}$ is a [[localic topos]] $Sh(S)$ over a [[topological space]] $S$ we have that $X \in Sh(S)$ corresponds to an [[étalé space]] over $X$ and $\mathbf{H}/X \to \mathbf{H}$ to an [[étale map]].
If $\mathcal{G}$ is a [[geometry (for structured (∞,1)-toposes)]]
then for $f : U \to X$ an _admissible_ morphism in $\mathcal{G}$,
the induced morphism of [[structured (∞,1)-topos]]es
$$
Spec^\mathcal{G} U \to Spec^{\mathcal{G}} X
$$
is an étale geometric morphism of structured $(\infty,1)$-toposes.
This is [[Structured Spaces|StrSp, example 2.3.8]].
## Properties
+-- {: .num_prop}
###### Proposition
The [[inverse image]] of an étale geometric morphism is a [[cartesian closed functor]].
=--
See at _[[cartesian closed functor]]_ for proof.
Therefore
+-- {: .num_prop}
###### Proposition
An étale geometric morphism is a cartesian [[Wirthmüller context]].
=--
+-- {: .num_prop}
###### Proposition
**(recognition of étale geometric morphisms)**
A [[geometric morphism]] $(f^* \dashv f_*) : \mathbf{K} \to \mathbf{H}$ is étale precisely if
1. it is [[essential geometric morphism|essential]];
1. $f_!$ is a [[conservative functor]];
1. For every [[diagram]] $X \to Y \leftarrow f_! Z$ in $\mathbf{H}$ the
induced diagram
$$
\array{
f_!(f^* X \times_{f^* Y} Z) &\to& f_! Z
\\
\downarrow && \downarrow
\\
X &\to& Y
}
$$
is a [[pullback]] diagram.
=--
For [[(∞,1)-topos]]es this is [[Higher Topos Theory|HTT, prop. 6.3.5.11]].
+-- {: .num_prop #RecoveringAToposFromItsSlices}
###### Proposition
**(Recovering a topos from its etale overcategory)
For $\mathbf{H}$ an $(\infty,1)$-topos we have
$$
\mathbf{H} \simeq ((\infty,1)Topos/\mathbf{H})_{et}
\,,
$$
where $((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H}$ is the full [[sub-(∞,1)-category]] of the [[over-(∞,1)-category]] on the etale geometric morphisms $\mathbf{K} \to \mathbf{H}$.
=--
This is [[Higher Topos Theory|HTT, remark 6.3.5.10]].
## Related concepts
* [[slice topos]], [[slice (∞,1)-topos]]
## References
The notion of local homeomorphisms of toposes is page 651 (chapter C3.3) of
* [[Peter Johnstone]], _[[Elephant|Sketches of an Elephant]]_ .
The notion of étale geometric morphisms between [[(∞,1)-topos]]es is introduced in section 6.3.5 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
Discussion of the refinement to [[structured (∞,1)-topos]]es is in section 2.3 of
* [[Jacob Lurie]], _[[Structured Spaces]]_ .
[[!redirects etale geometric morphism]]
[[!redirects etale geometric morphisms]]
[[!redirects étale geometric morphisms]]
[[!redirects locally homeomorphic geometric morphism]]
[[!redirects local homeomorphism of toposes]]
[[!redirects local homeomorphism of topoi]]
[[!redirects local homeomorphisms of toposes]]
[[!redirects local homeomorphisms of topoi]] |
étale groupoid | https://ncatlab.org/nlab/source/%C3%A9tale+groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #PlainDefinition}
###### Definition
A [[topological groupoid]] or [[Lie groupoid]] $C$ is called an **étale groupoid** if the [[domain|source]]-map $s : Mor C \to Obj C$ is a [[local homeomorphism]] or [[local diffeomorphism]], respectively, and hence exhibits the space of morphisms as an [[étale space]] over the space of objects.
=--
+-- {: .num_remark}
###### Remark
In the situation of def. \ref{PlainDefinition} it follows that all the other structure maps ([[codomain|target]], [[identity]], [[composition]]) are also local homeomorphisms, resp. local diffeomorphisms.
This means that an étale groupoid is equivalently an [[internal groupoid]] in the [[category]] whose objects are [[topological spaces]]/[[smooth manifold]]s and whose [[morphisms]] are local homeomorphisms/diffeomorphisms.
=--
+-- {: .num_remark}
###### Remark
Definition \ref{PlainDefinition} is not invariant under the general notion of [[equivalence]] of Lie groupoids, the equivalence between them regarded as [[smooth groupoids]], specifically as [[differentiable stacks]] ("[[Morita equivalence]]").
But it does make sense to take an **étale smooth groupoid** to be a [[smooth groupoid]]/[[differentiable stack]] which is equivalent, as such, to, hence is presented by an étale Lie groupoid as in def. \ref{PlainDefinition}.
This notion has been called **folitation groupoid** in ([Crainic-Moerdijk 00](#CrainicMoerdijk)).
=--
The following characterizes foliation groupoids
+-- {: .num_theorem}
###### Theorem
For a [[Lie groupoid]] $\mathcal{G}_\bullet$ the following are equivalent
1. $\mathcal{G}$ is a foliation groupoid, hence is equivalent, as a [[differentiable stack]] to an étale groupoid as in def. \ref{PlainDefinition};
1. The [[Lie algebroid]] $(\mathfrak{g},\mathcal{G}_0)$ which corresponds to $\mathcal{G}$ under [[Lie differentiation]] has an [[injection|injective]] [[anchor map]];
hence the [[orbits]] of $\mathcal{G}$ form the [[leaves]] of a [[foliation]], the foliation whose leaves are tangent to the vectors in the image of this anchor map;
1. All [[isotropy groups]] of $\mathcal{G}_\bullet$ are [[discrete groups]].
=--
This is ([Crainic-Moerdijk 00, theorem 1](#CrainicMoerdijk)).
## Properties
### Cohomology and homology
In the literature one finds, roughly speaking, two different approaches to the study of étale groupoids. One approach is based on the construction of the convolution algebras associated to an étale groupoid, in the spirit of Connes' [[noncommutative geometry]], and involves the study of [[cyclic homology|cyclic]] and [[Hochschild homology]] and [[Hochschild cohomology|cohomology]] of these algebras. The other approach uses methods of algebraic topology such as the construction of the [[classifying space]] of an
étale groupoid and its [[abelian sheaf cohomology|(sheaf) cohomology groups]].
### Relation to Haefliger groupoids
{#RelationToHaefligerGroupoids}
For $X_\bullet$ an étale groupoid, there is a canonical morphism
$$
X_\bullet \longrightarrow \mathcal{H}(X_0)
$$
to the [[Haefliger groupoid]], example \ref{HaefligerGroupoid}, of its manifold of objects. The [[kernel]] of this map is the _ineffective part_ of $X_\bullet$. If the kernel vanishes, then $X$ is called an [[effective Lie groupoid]].
(e.g. [Carchedi 12, section 2.2](Carchedi12))
### Characterization by convolution Hopf algebroids (Gelfand duality)
{#GelfandDuality}
The [[groupoid convolution algebra]] $C^\ast(\mathcal{G}_\bullet)$ of a [[Lie groupoid]] with its canonical [[atlas]] remembered has the structure of a [[Hopf algebroid]].
In ([Mrčun 99](#Mrcun99), [Kališnik-Mrčun 07](#KalisnikMrcun07)) étale Lie groupoids are characterized dually by their Hopf algebroids (a refinement of [[Gelfand duality]] to [[noncommutative topology]]).
### Characterization by site of manifolds and étale maps
{#CharacterizationBySiteOfManifolds}
+-- {: .num_prop}
###### Proposition
The [[2-category]] of [[étale stacks]] with [[étale maps]] between them is equivalent to the [[2-topos]] over the [[site]] of [[smooth manifolds]] with [[local diffeomorphisms]] between them
=--
([Carchedi 12, theorem 3.4, corollary 3.3](Carchedi12))
+-- {: .num_prop #EtaleStackAsProlongations}
###### Proposition
A [[smooth stack]] is an étale stack precisely if it is in the [[essential image]] of the [[left Kan extension]] along the non-full inclusion of [[sites]]
$$
SmthMfd^{et} \to SmthMfd
$$
of [[smooth manifolds]], with [[local diffeomorphisms]] on the left and all [[smooth functions]] on the right.
=--
([Carchedi 12, theorem 3.5, corollary 3.4](Carchedi12))
In particular:
+-- {: .num_prop}
###### Proposition
A [[smooth stack]] is an [[effective Lie groupoid|effective]] étale stack precisely if under the prolongation of prop. \ref{EtaleStackAsProlongations} it is equivalent to the image of a [[sheaf]] (i.e. of a [[0-truncated]] stack).
=--
([Carchedi 12, corollary 4.1, corollary 4.2](Carchedi12))
### Formalization in differential cohesion
{#FormalizationInDifferentialCohesion}
See at _[[differential cohesion]]_ the section _[Etale objects](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)_.
## Examples
Étale groupoids arise naturally as models for [[leaf spaces]] of [[foliations]], for [[orbifold]]s, and for [[orbit spaces]] of [[discrete group]] [[actions]].
+-- {: .num_example}
###### Example
Every [[topological space]] may be regarded as an étale groupoid with only identity morphisms.
=--
+-- {: .num_example}
###### Example
For $X$ a [[topological space]] and $\Gamma$ a [[discrete group]] with a continuous [[action]] $X \times \Gamma \to X$ on $X$, the [[action groupoid]] $X//\Gamma$ is étale.
=--
+-- {: .num_example #HaefligerGroupoid}
###### Example
The [[Haefliger groupoid]] $\Gamma^q$ has the [[Cartesian space]] $\mathbb{R}^q$ as its space of objects. A [[morphism]] $x \to y$ is a [[germ]] of a [[diffeomorphism]] $(\mathbb{R}^q,x) \to (\mathbb{R}^q, y)$.
This groupoid, and its [[geometric realization]] play a central role in [[foliation theory]].
=--
+-- {: .num_example}
###### Example
Every [[orbifold]] is an étale [[Lie groupoid]].
=--
## Related concepts
* [[étale space]]
* [[orbifold]], [[Deligne-Mumford stack]]
* [[étale ∞-groupoid]]
* [[proper Lie groupoid]], [[effective Lie groupoid]]
## References
A standard textbook account is section 5.5. of
* [[Ieke Moerdijk]], [[Janez Mrčun]], _[[Introduction to foliations and Lie groupoids]]_, Cambridge (2003)
The relation between étale groupoid and [[foliations]] is analyzed in detail in
* [[Marius Crainic]], [[Ieke Moerdijk]], _Foliation groupoids and their cyclic homology_ ([arXiv:math/0003119](http://arxiv.org/abs/math/0003119))
{#CrainicMoerdijk}
See also at _[[orbifold]]_ for basic and introductory literature.
Further discussion of étale groupoids and their properties includes
* [[Marius Crainic]], [[Ieke Moerdijk]], _A Homology Theory for Étale groupoids_ ([journal](http://www.math.uiuc.edu/K-theory/0284/))
* [[David Carchedi]], _Sheaf Theory for Étale Geometric Stacks_ ([arXiv:1011.6070](http://arxiv.org/abs/1011.6070))
* {#Carchedi12} [[David Carchedi]], section 2.2, section 3 of _Étale Stacks as Prolongations_ ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))
The [[groupoid convolution algebra|convolution]]-[[Hopf algebroids]] of étale Lie groupoids have been characterized in
* [[Janez Mrčun]], _On spectral representation of coalgebras and Hopf algebroids_ ([arXiv:math/0208199](http://arxiv.org/abs/math/0208199))
{#Mrcun99}
* [[Jure Kališnik]], [[Janez Mrčun]], _Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids_ ([arXiv:math/0703374](http://arxiv.org/abs/math/0703374))
{#KalisnikMrcun07}
[[!redirects étale groupoids]]
[[!redirects etale groupoid]]
[[!redirects etale groupoids]]
[[!redirects étale Lie groupoid]]
[[!redirects étale Lie groupoids]]
[[!redirects etale Lie groupoid]]
[[!redirects etale Lie groupoids]]
[[!redirects foliation groupoid]]
[[!redirects foliation groupoids]]
[[!redirects étale stack]]
[[!redirects étale stacks]]
[[!redirects etale stack]]
[[!redirects etale stacks]] |
étale homotopy | https://ncatlab.org/nlab/source/%C3%A9tale+homotopy |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _étale homotopy_ can be understood as a vast generalization of the following classical fact.
The [[nerve theorem]] says that for $X$ a [[paracompact topological space]] and $\{U_i \to X\}$ a [[good cover]] of $X$ by open subsets, then the [[simplicial set]] obtained from the [[Cech nerve]] of the covering by degreewise contracting all connected components to a point, presents the [[homotopy type]] of $X$.
If $X$ here is more generally a [[locally contractible space]] there is in general no notion of "good" enough open cover anymore. Instead, one can consider the above kind of construction for _all_ [[hypercovers]] and take the [[limit]] over the resulting simplicial sets. The classical theorem by [Artin-Mazur](#ArtinMazur) states that this still gives the [[homotopy type]] of $X$.
The construction itself, however, makes sense for arbitrary topological spaces and in fact for arbitrary [[sites]].
In the literature, particularly the _[[étale site]]_ is often considered and "étale homotopy" is often implicitly understood to take place over this site.
But the concept is much more general. In particular, one can understand the construction of the limit over contractions of [[hypercovers]] as a presentation of naturally defined [[(∞,1)-functors]] in [[(∞,1)-topos theory]].
Notably, if the given site is a a [[locally ∞-connected site]], then the étale homotopy construction computes precisely the [[derived functor]] that presents the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]. Many constructions in the literature can be understood as being explicit realizations of this simple general concept. Detailed discussion of this is at _[[geometric homotopy groups in an (∞,1)-topos]]_.
Even more generally, étale homotopy give the notion of [[shape of an (∞,1)-topos]]. (...)
## Examples
{#Examples}
### Chevalley groups and Galois groups
For the special case of [[fundamental groups]], the concept of étale homotopy groups also goes by the name of _[[Chevalley fundamental groups]]_.
The étale fundamental group of a [[scheme]] is its [[absolute Galois group]]. See at _[Galois theory -- Statement of the main result](Galois+theory#StatementOfMainTheorem)_.
### Étale contractibility
+-- {: .num_example}
###### Example
For $k$ a [[field]] of [[characteristic]] 0, then the [[affine line]] $\mathbb{A}^1_k$ is étale [[contractible homotopy type|contractible]]. This is no longer the case in [[positive number|positive]] [[characteristic]].
=--
([HSS 13, section 1](#HSS13))
+-- {: .num_prop}
###### Proposition
Let $k$ be an [[algebraically closed field]] of [[positive number|positive]] [[characteristic]]. Then the only [[smooth variety]] over $k$ which is étale [[contractible homotopy type|contractible]] is the point $Spec(k)$. In fact this is the only smooth variety which is [[n-connected object of an (infinity,1)-topos|2-connected]].
=--
([HSS 13, theorem 1](#HSS13))
## Related concepts
* [[Čech homology]]
* [[algebraic fundamental group]]
* [[anabelian geometry]]
* [[étale topos]], [[pro-étale topos]]
* [[étale cohomology]]
## References
### General
Original articles include
* [[Michael Artin]], [[Barry Mazur]], _Homotopy of varieties in the etale topology_, Proceedings of a Conference on Local Fields, Driebergen 1966, Springer.
* [[Michael Artin]], [[Barry Mazur]], _Étale Homotopy_, Springer Lecture Notes in Mathematics 100, Berlin (1969)
{#ArtinMazur}
* [[Eric Friedlander]], _Fibrations in étale homotopy theory [(numdam)](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1973__42_/PMIHES_1973__42__5_0/PMIHES_1973__42__5_0.pdf)_, Publ. Math.
Inst. des Haut. Études Scient., 42, (1973), 5 – 46.
The modern perspective from the point of view of [[model structure on simplicial presheaves|model structures on simplicial presheaves]] is in
* {#Isaksen01} [[Daniel Isaksen]], _Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes_, 2001 ([K-theory archive](http://www.math.uiuc.edu/K-theory/0495/))
* [[G. Quick]], _Stable étale realization and étale cobordism_, Adv. in Math.,
214 (2007), 730–760.
and fully abstractly from the point of view of [[(∞,1)-topos]]-theory ([[shape of an (∞,1)-topos]]) in
* {#Hoyois13a} [[Marc Hoyois]], _Higher Galois theory_, ([pdf](http://math.mit.edu/~hoyois/papers/highergalois.pdf))
and ([Hoyois 13b, section 1](#Hoyois13b)).
and in, where a comparison theorem over the complex numbers is also proven:
* {#Carchedi15} [[Dave Carchedi]], _On the étale homotopy type of higher stacks_ ([arXiv:1511.07830](http://arxiv.org/abs/1511.07830))
An introduction is in
* Tomer Schlank, Alexei Skorobogatov, _A very brief introduction to étale homotopy_. In: "Torsors, étale homotopy and applications to rational points". LMS Lecture Note Series 405, Cambridge University Press, 2013. ([pdf](https://www.ma.imperial.ac.uk/~anskor/SCHLANK-SKOROBOGATOV.pdf))
Lecture notes on the étale fundamental group are in
* [[James Milne]], section 4 of _[[Lectures on Étale Cohomology]]
Generalization to [[simplicial schemes]] is discussed in
* [[Eric Friedlander]], 1982, _Étale homotopy of simplicial schemes_ , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J.
More on this is in
* Michael Misamore, _Étale homotopy types and bisimplicial hypercovers_, Homology, Homotopy and Applications, Vol. 15 (2013), No. 1, pp.27-49. ([web](http://www.intlpress.com/HHA/v15/n1/a2/))
### Examples and applications
* {#Hoyois13b} [[Marc Hoyois]], _The étale-symmetric Künneth theorem_, 2013 ([pdf](http://math.mit.edu/~hoyois/papers/etalekunneth.pdf))
Discussion in [[positive characteristic]] is in
* {#HSS13} Armin Holschbach, Johannes Schmidt, [[Jakob Stix]], _Étale contractible varieties in positive characteristic_ ([arXiv:1310.2784](http://arxiv.org/abs/1310.2784))
Étale homotopy type of [[moduli stacks of curves]] is discussed in
* Paola Frediani, Frank Neumann, _Étale homotopy types of moduli stacks of algebraic curves with symmetries_, K-Theory 30: 315-340, 2003 ([arXiv:math/0404387](http://arxiv.org/abs/math/0404387))
[[!redirects étale homotopy theory]]
[[!redirects etale homotopy]]
[[!redirects etale homotopy theory]]
[[!redirects etale homotopy group]]
[[!redirects étale homotopy group]]
[[!redirects etale homotopy groups]]
[[!redirects étale homotopy groups]]
[[!redirects étale fundamental group]]
[[!redirects étale fundamental groups]]
[[!redirects etale fundamental group]]
[[!redirects etale fundamental groups]]
[[!redirects étale homotopy type]]
[[!redirects étale homotopy types]]
[[!redirects etale homotopy type]]
[[!redirects etale homotopy types]] |
étale infinity-groupoid | https://ncatlab.org/nlab/source/%C3%A9tale+infinity-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _étale ∞-groupoid_ is meant to be an [[∞-groupoid]]-analog to an [[étale groupoid]].
For details see at _[[V-manifold]]_.
## Examples
* classifying étale stack for [[symplectic forms]]: [Carchedi 12, Example 2](#Carchedi12)
* classifying étale stack for [[complex structures]]: [Carchedi 12, p. 38](#Carchedi12)
## References
A formalization of the [[petit topos|petit]] [[(∞,1)-toposes]] corresponding to étale ∞-groupoids is in
* [[Jacob Lurie]], _[[Structured Spaces]]_.
A characterization of étale ∞-groupoids as objects in a [[big topos|big]] [[(∞,1)-topos]] is given in
* {#Carchedi12} [[David Carchedi]], _Étale Stacks as Prolongations_, Advances in Mathematics Volume 352, 20 August 2019, Pages 56-132 ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))
* [[David Carchedi]], _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi_, Memoirs of the American Mathematical Society 2020; 120 ([arXiv:1312.2204](https://arxiv.org/abs/1312.2204), [ISBN:978-1-4704-5810-2](https://bookstore.ams.org/memo-264-1282))
and aspects of their [[geometric realization]]/[[shape modality]] are discussed in
* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_, Advances of Mathematics, Volume 294, 2016, Pages 756-818 ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))
* [[David Carchedi]], _On the étale homotopy type of higher stacks_ ([arXiv:1511.07830](https://arxiv.org/abs/1511.07830))
A formalization as [[V-manifolds]] in terms of [[differential cohesion]] is discussed at _[differential cohesion -- Structures -- cohesive étale ∞-groupoids](differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)_.
The formalization of this in [[homotopy type theory]] is discussed in
* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017
[[!redirects étale infinity-groupoids]]
[[!redirects étale ∞-groupoid]]
[[!redirects étale ∞-groupoids]]
[[!redirects étale ∞-stack]]
[[!redirects étale ∞-stacks]]
[[!redirects etale infinity-groupoid]]
[[!redirects etale infinity-groupoids]]
[[!redirects étale infinity-stack]]
[[!redirects étale infinity-stacks]]
[[!redirects etale infinity-stack]]
[[!redirects etale infinity-stacks]]
[[!redirects etale ∞-groupoid]]
[[!redirects etale ∞-groupoids]]
|
étale map | https://ncatlab.org/nlab/source/%C3%A9tale+map |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Étale morphisms
+-- {: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The notion of _étale map_ is an abstraction of that of [[local homeomorphism]] in [[topology]]. The concept is usually found in places with a geometric or topological flavour.
## Examples
### Between topological spaces
An étale map between [[topological spaces]] is a [[local homeomorphism]]; see [[étalé space]] (which is the total space of such a map viewed as a [[bundle]]).
### Between smooth manifolds
An étale map between [[smooth spaces]] is a [[local diffeomorphisms|local diffeomorphism]], which is in particular a [[local homeomorphism]] on the underlying [[topological spaces]].
### Between schemes (affine schemes / rings)
For an étale map between [[schemes]] see _[[étale morphism of schemes]]_.
Restricted to [[affine schemes]], this yields, [[Isbell duality|dually]], a notion of étale morphisms between [[rings]]. Étale maps between noncommutative [[rings]] have also been considered.
### Between $E_\infty$-rings
* [[étale morphism of E-∞ rings]]
### Between analytic spaces
* Étale maps between [[analytic spaces]] are closely related to étale maps between schemes, while also (when the spaces are smooth) a special case of an étale map between smooth spaces.
+-- {: .query}
Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth spaces.
Toby: Is it correct now?
=--
### Between toposes
* [[étale geometric morphism]]
### Between anabelioids
* [[finite étale morphism of anabelioids]]
## Axiomatization
The idea of _étale morphisms_ can be axiomatized in any [[topos]]. This idea goes back to lectures by [[André Joyal]] in the 1970s. See ([Joyal-Moerdijk 1994](#JoyalMoerdijk)) and ([Dubuc 2000](#Dubuc)).
## Related concepts
* [[open morphism]], [[closed morphism]]
* [[separated morphism]]
* [[formally étale morphism]]
* [[log-étale morphism]]
## References
Axiomatizations of the notion of étale maps in general [[toposes]] are discussed in
* [[André Joyal]], [[Ieke Moerdijk]], _A completeness theorem for open maps_, Annals of Pure and Applied Logic __70__, Issue 1, 18 November 1994, p. 51-86, [MR95j:03104](http://www.ams.org/mathscinet-getitem?mr=1303663), <a href="http://dx.doi.org/10.1016/0168-0072(94)90069-8">doi</a>
{#JoyalMoerdijk}
* [[Eduardo Dubuc]], _Axiomatic etal maps and a theory of spectrum_, Journal of pure and applied algebra __149__ (2000) 15--45
{#Dubuc}
[[!redirects étale map]]
[[!redirects étale maps]]
[[!redirects etale map]]
[[!redirects etale maps]]
[[!redirects etalé map]]
[[!redirects etalé maps]]
[[!redirects étalé map]]
[[!redirects étalé maps]]
[[!redirects étale morphism]]
[[!redirects étale morphisms]]
[[!redirects etale morphism]]
[[!redirects etale morphisms]]
|
étale morphism of E-∞ rings | https://ncatlab.org/nlab/source/%C3%A9tale+morphism+of+E-%E2%88%9E+rings |
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn}
###### Definition
A morphism $f \colon \mathbf{A} \to \mathbf{B}$ of [[E-∞ rings]] is an _étale morphism_ if the underlying homomorphism of [[commutative rings]] $\pi_0 f\colon \pi_0 \mathbf{A}\to \pi_0 \mathbf{B}$ is an [[étale morphism]], and the map $\pi_n(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_n(B)$ is an isomorphism of abelian groups for every integer $n$.
=--
([Lurie, def.7.5.1.4](#Lurie))
## Properties
étale morphisms of underlying rings lift essentially uniquely to étale morphisms of [[E-∞ rings]]:
+-- {: .num_prop #EssentiallyUniqueLiftsFromOrdinaryRings}
###### Proposition
For $\mathbf{A}$ an [[E-∞ ring]] and $\pi_0 \mathbf{A} \to B$ a homomorphism to an ordinary ring $B$, then there is an essentially unique $E_\infty$-ring $\mathbf{B}$ with $\pi_0 \mathbf{B} \simeq B$ and étale morphism $\mathbf{A}\to \mathbf{B}$.
=--
([Lurie, theorem 7.5.0.6](#Lurie))
+-- {: .num_remark }
###### Remark
Proposition \ref{EssentiallyUniqueLiftsFromOrdinaryRings} is a central ingredient in the characterization of the moduli stack of [[derived elliptic curves]] as having underlying it the ordinaty [[moduli stack of elliptic curves]].
=--
+-- {: .num_remark #LocalizationOfRings}
###### Remark
**(localization of $E_\infty$-rings)**
Proposition \ref{EssentiallyUniqueLiftsFromOrdinaryRings} serves to lift [[localization of rings]] from rings to $E_\infty$-rings: for $\mathbf{A}$ an [[E-∞ ring]] and $a\in \pi_0 A$ an element, then the map $\pi_0 \mathbf{A} \to (\pi_0 \mathbf{A})[a^{-1}]$ of [[localization of a ring]] away from $a$ lifts to yield an [[E-∞ ring]] $\mathbf{A}[a^{-1}]$ with étale morphism $\mathbf{A} \to \mathbf{A}[a^{-1}]$. See also at _[[localization of a module]]_ for more on this.
=--
## References
* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_
[[!redirects étale morphisms of E-∞ rings]]
[[!redirects etale morphism of E-∞ rings]] |
étale site | https://ncatlab.org/nlab/source/%C3%A9tale+site | [[!redirects etale site]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _étale site_ of a [[scheme]] is an analog of the [[category of open subsets]] of a [[topological space]]. The corresponding [[cohomology]] is [[étale cohomology]].
The étale topology has similar cohomological properties as the [[complex analytic geometry|complex analytic topology]], and in particular it is much finer for cohomological purposes than the [[Zariski topology]].
## Definition
+-- {: .num_defn}
###### Definition
Let $X$ be a [[scheme]].
The **[[big site|big]] étale [[site]]** $Sch_{/X,et}$ of $X$ is the [[over category]] $Sch_{/X}$ of [[scheme]]s over $X$ equipped with the [[coverage]] given by [[étale covers]] (after forgetting the maps to $X$).
The **[[small site|small]] étale [[site]]** $i : X_{et} \hookrightarrow Sch/{X,et}$ is the full [[subcategory]] of $Sch_{/X}$on the [[étale morphism]]s $U \to X$.
=--
The [[abelian sheaf cohomology]] of the étale site is called [[étale cohomology]].
## Properties
### Cofinal affine covers
{#CofinalAffineCovers}
+-- {: .num_prop}
###### Proposition
For $X = Spec(A)$ an [[affine scheme]] and $\{Y_i \to X\}$
an étale cover, then there exists a refinement to an étale cover
$\{U_i \to X\}$ such that each $U_i$ is an [[affine scheme]].
=--
### Cohomology
+-- {: .num_prop}
###### Proposition
The [[inverse image]] restriction functor $i^* Sh(Sch_{/X,et}, Ab) \to Sh(X_{et}, Ab)$ on the [[categories of sheaves]] with values in [[Ab]]
* is an [[exact functor]]
* maps [[injective object]]s to injective objects.
=--
+-- {: .num_cor}
###### Corollary
For $X$ a [[scheme]] and $F \in Sh(Sch_{/{X,et}}, Ab)$ an [[abelian sheaf]] on its big site, then the [[etale cohomology]] of $X$ with [[coefficients]] in $F$ may equivalently be computed on the small site:
$$
H^p(X_{et}, F|_{et}) \simeq H^p(X,F)
\,.
$$
=--
This appears for instance in ([deJong, prop. 3.4](#deJong)).
### Derived geometry
The [[derived geometry]] of the étale site is the [[étale (∞,1)-site]]. The precise statement is at <a href="http://nlab.mathforge.org/nlab/show/%C3%A9tale+(infinity%2C1)-site#AsDerivedGeometry">derived étale geometry</a>.
## Related concepts
* [[étale morphism]], **étale site**, [[étale topos]], [[étale cohomology]], [[étale homotopy]]
* The [[lisse-étale site]] of some $X$ consists of [[smooth morphisms]] $U \to X$.
[[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ **étale site** $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]
* [[pro-étale site]]
* [[Weil-étale topology for arithmetic schemes]]
* [[étale (∞,1)-site]]
* [[constructible sheaf]]
* [[étale homotopy]]
## References
The classical references are
* [[Pierre Deligne]] et al., _Cohomologie étale_ , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.
{#Deligne}
Textbooks include
* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_
* [[James Milne]], _Etale cohomology_, Princeton Mathematical Series __33__, 1980. xiii+323 pp.
A detailed survey is in chapter 34 of
* [[Aise Johan de Jong]], _[[The Stacks Project]]_
Lecture notes include
* [[James Milne]], _Lectures on etale cohomology_ ([pdf](http://www.jmilne.org/math/CourseNotes/LEC.pdf))
* {#deJong} [[Aise Johan de Jong]], _Étale cohomology_ ([pdf](http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf))
A variant, the _[[pro-étale site]]_ (locally contractible in some sense) is discussed in
* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
[[!redirects étale site]]
[[!redirects etale sites]]
[[!redirects étale sites]]
[[!redirects small étale site]]
[[!redirects small etale sites]]
[[!redirects small étale sites]]
[[!redirects small etale sites]]
[[!redirects big étale site]]
[[!redirects big etale sites]]
[[!redirects big étale sites]]
[[!redirects big etale sites]]
[[!redirects étale topology]]
[[!redirects etale topology]]
[[!redirects étale topologies]]
[[!redirects etale topologies]] |
étale space | https://ncatlab.org/nlab/source/%C3%A9tale+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### étale morphisms
+-- {: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
Let [[Top]] be a category of [[topological space]]s and $B$ an object in $Top$ (the 'base' space). The [[over category|slice category]] $Top/B$ is called the category of (topological) spaces over $B$ (or sometimes simply [[bundles]]).
An __étale space__ (or _[[étale map]]_, sometimes *étalé space*) over $B$ is an object $p:E\to B$ in $Top/B$ such that $p$ is a [[local homeomorphism]]: that is, for every $e\in E$, there is an open set $U \ni e$ such that the [[image]] $p(U)$ is open in $B$ and the restriction of $p$ to $U$ is a [[homeomorphism]] $p|_U: U \to p(U)$.
The set $E_x = p^{-1}(x)$ where $x\in B$ is called the __[[stalk]]__ of $p$ over $x$.
The underlying set of the _total space_ $E$ is the union of its stalks (notice that we do not say fiber!). $p$ is sometimes refered to as the _projection_.
## Definition for arbitrary toposes
The following is adapted from a [MathOverflow answer](https://mathoverflow.net/questions/78681/etal%c3%a9-space-construction-for-presheaves-on-a-grothendieck-site/111293#111293) by [[David Carchedi]]:
> As long as $C$ has a [[small set]] of topological generators (i.e., as long as $Sh(C,J)$ isn't too large to be a [[topos]]), there always exists a certain version of an étale space: If $F$ is a [[sheaf]] on $(C,J)$, the [[slice topos]] $Sh(C,J)/F$ has a canonical étale projection $\pi_F:Sh(C,J)/F \to Sh(C,J)$. This map is a [[local homeomorphism of topoi]]. This topos with this local homeomorphism is the étale space of $F$. Indeed, we may make this construction for each object $c \in C$, call it $U(c) \coloneqq Sh(C,J)/y(c)$, where $y(c)$ is the (possibly sheafified) [[Yoneda embedding|Yoneda embedded]] object. Then, "[[sections]] of $\pi_F$ over $U(c) \to Sh(C,J)$" are in [[bijection]] with [[elements]] of $F(c)$. If the Grothendieck [[site]] $(C,J)$ happened to be the [[canonical site]] of a [[topological space]], then each [[slice]] $Sh(C,J)/F$ is equivalent to [[category of sheaves|sheaves]] on the étale space of that [[sheaf]], and the projection corresponds to the usual one. In particular, $U(c) \to Sh(C,J)$ corresponds to the inclusion of an [[open subset]]. So, this is reduces to the usual construction for spaces. Another example is, if $(C,J)$ were the [[small étale site]] of some [[scheme]] $S$, then each $Sh(C,J)/F$ is the small étale site of some algebraic space (with no separation conditions) étale over $S$, with $\pi_F$ corresponding to the étale map from this algebraic space to $S$.
## Properties
### Relation to sheaves
{#RelationToSheaves}
Let $p:E\to B$ be in $Top/B$. The (local) __[[sections]]__ of $p$ over an open set $U\subseteq B$ are the [[continuous maps]] $s:U\to E$ such that $p\circ s = \mathrm{id}_U$. It is an elementary but central fact that for an étale map $p$, _the images of local sections form a base for the topology_ of the total space $E$. The topology of $E$ is then typically non-[[Hausdorff space|Hausdorff]].
The set of sections of $p$ over $U$ is denoted by $\Gamma_U p = (\Gamma p)(U) = \Gamma_U E = (\Gamma E)(U)$ and may be shown to extend to a [[functor]] $\Gamma : Top/B\to PShv_B$ where $PShv_B$ is the [[category of presheaves]] over $B$. The functor $\Gamma$ has a [[left adjoint]] $L : PShv_B\to Top/B$, whose [[essential image]] is the [[full subcategory]] $Et/B$ of étale spaces over $B$. The [[essential image]] of the functor $\Gamma$ is the [[category of sheaves]] $Shv_B$ over $B$, and this [[adjunction]] restricts to an [[equivalence of categories]] between $Et/B$ and $Shv_B$ (that is, it is an [[idempotent adjunction]]).
If $P:Open(B)^{op}\to Set$ is a [[sheaf]], then one sometimes calls the total space $E(P)$ of the étale space $L(P) = (E(P)\to B)$ the __space of the sheaf__ $P$, having in mind the adjoint equivalence above. (This is also called the __sheaf space__ or the __display space__ (alias __étale space__, or in French: __espace étalé__); compare also a [[display morphism]] of [[contexts]].) The associated sheaf functor $a:PShv_B\to Shv_B\hookrightarrow PShv_B$ decomposes as $a = \Gamma\circ L$, and $a$ may be considered as an endofunctor part of an [[idempotent monad]] in $PShv_B$ whose corresponding [[reflective subcategory]] is $Shv_B$.
(e.g. [MacLane-Moerdijk, section II.5, II.6](#MacLaneMoerdijk))
### Relation to covering spaces
Every [[covering space]] (even in the more general sense not requiring any connectedness axiom) is étale but not vice versa:
* for a covering space the [[inverse image]] of some [[open subset]] in the base $B$ needs to be, by the definition, a [[disjoint union]] of homeomorphic open sets in $E$; however the 'size' of the [[open neighborhoods]] over various $e$ in the same [[stalk]] required in the definition of étale space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.
* even if the the stalks of the étale space are finite, it need not be locally trivial. For instance the [[disjoint union]] $\coprod_i U_i$ of a collection of [[open subsets]] of a topological space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is étale, but does not have a typical fiber: the fiber over a given point has [[cardinality]] the number of open sets $U_i$ that contain this particular point.
## Grammar/pronunciation note
{#grammar}
### Grammar and spelling
In French, the verb 'étaler' means, roughly, to spread out; '-er' becomes '-é' to make a past participle. So an 'espace étalé' is a space that has been spread out over $B$. On the other hand, 'étale' is a (relatively obscure, distantly related) nautical adjective that can be translated as 'calm' or 'slack'.
To quote from the [Wiktionnaire français](https://fr.wiktionary.org/wiki/%C3%A9tale):
> 'étale' _qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant_
('flot' = 'flow' and 'jusant' = 'ebb').
* [Further reference](https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive11#french_spelling)
### Pronunciation
The French pronunciation of "étalé" is three syllables; that of "étale" is two syllables, with the final "e" silent. Many English-speaking mathematicians follow the respective French pronunciations; see for example [this nForum thread](https://nforum.ncatlab.org/discussion/11283/etale-space/?Focus=84522#Comment_84522).
Others use the two-syllable pronunciation (corresponding to French "étale") for both written forms "étale" and "étalé". This group included [[Grothendieck]] himself, when speaking English. (See [[Colin McLarty#McLarty18|McLarty18]], from 4:45 to 6:05 and particularly [at 5:26](https://www.youtube.com/watch?v=5AR55ZsHmKI&t=326s).)
### Poem
There is an interesting stanza from a song of Léo Ferré:
> Et que les globules figurent
> Une mathématique bleue,
> Sur cette mer jamais étale
> D'où me remonte peu à peu
> Cette mémoire des étoiles.
> — (Léo Ferré, La mémoire et la mer)
He also mentions geometry and 'théorème' elsewhere in the song.
## Related concepts
* [[étale groupoid]], [[étale infinity-groupoid]]
## References
* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], sections II.5 and II.6 of _[[Sheaves in Geometry and Logic]]_
For generalizations to étale spaces of stacks in groupoids see
* [[David Carchedi]], _An étalé space construction for stacks_, Algebr. Geom. Topol. 13 (2013), no. 2, 831–903.
[[!redirects etale space]]
[[!redirects etale spaces]]
[[!redirects étalé space]]
[[!redirects étalé spaces]]
[[!redirects étale space]]
[[!redirects étale spaces]]
[[!redirects etalé space]]
[[!redirects etalé spaces]]
[[!redirects espace étalé]]
[[!redirects espaces étalés]]
[[!redirects espace etale]]
[[!redirects espaces etales]]
[[!redirects sheaf space]]
[[!redirects sheaf spaces]]
[[!redirects display space]]
[[!redirects display spaces]]
|
étale theta function | https://ncatlab.org/nlab/source/%C3%A9tale+theta+function | A concept in [[arithmetic geometry]] introduced by [[Shinichi Mochizuki]] in the context of his [[inter-universal Teichmüller theory]].
###Related concepts
* [[Frobenioid]]
## References
* [[Shinichi Mochizuki]], (2009), 'The étale theta function and its Frobenioid-theoretic manifestations', Kyoto University. Research Institute for Mathematical Sciences. Publications 45 (1): 227–349. ([pdf](http://www.kurims.kyoto-u.ac.jp/~prims/pdf/45-1/45-1-7.pdf))
* [[Minhyong Kim]], _[What is an étale theta function?](http://mathoverflow.net/q/195841/447)_
[[!redirects étale theta functions]] |
étale topos | https://ncatlab.org/nlab/source/%C3%A9tale+topos | [[!redirects etale topos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the context of the [[geometry]] of [[schemes]] there is a traditional notion of [[étale morphism of schemes]] and an _étale topos_ is a [[category of sheaves]] on the [[étale site]] of a [[scheme]], consisting of [[covers]] by such [[étale morphism of schemes|étale morphisms]]. This traditional notion we discuss in
* [Étale topos of a scheme](#EtaleToposOfAScheme).
More abstractly, given that [[étale morphisms of schemes]] may be characterized as modal morphisms with respect to an [[infinitesimal shape modality]], one can consider étale toposes in every context of [[differential cohesion]]. This we discuss in
* [Étale topos of a differentially cohesive object](#GeneralAbstract)
### Étale topos of a scheme
{#EtaleToposOfAScheme}
An **étale topos** is the [[sheaf topos]] over an [[étale site]], hence over a site whose "open subsets" are [[étale morphisms]] into the base [[space]]. The intrinsic [[cohomology]] of an étale [[(∞,1)-topos]] is _[[étale cohomology]]_.
More generally there is the pro-étale topos over a [[pro-étale site]], which is a bit better behaved. In particular the intrinsic [[cohomology]] of a pro-étale [[(∞,1)-topos]] includes the [[Weil cohomology theory]] [[ℓ-adic cohomology]].
Generally, given that an [[étale morphism of schemes]] is a [[formally étale morphism]] subject to a size constraint on its [[fibers]] -- for an actual [[étale morphism of schemes]] the fibers are [[finite sets]] in the suitable sense (formal duals to [[étale algebras]]) while for a [[pro-étale morphism of schemes]] they are [[pro-objects]] of such fibers --
in a suitable ambient context ("[[differential cohesion]]") one can drop all finiteness conditions and consider just opens given by [[formally étale morphisms]] as encoded by an [[infinitesimal shape modality]]. This we discuss [below](#GeneralAbstract).
### Étale topos of a differentially cohesive object
{#GeneralAbstract}
We discuss how in [[differential cohesion]] $\mathbf{H}_{th}$ every object $X$ canonically induces its étale topos $Sh_{\mathbf{H}_{th}}(X)$.
For $X \in \mathbf{H}_{th}$ any object in a [[differential cohesion|differential cohesive]] $\infty$-topos, we formulate
1. the [[(∞,1)-topos]] denoted $\mathcal{X}$ or $Sh_\infty(X)$ of [[(∞,1)-sheaves]] over $X$, or rather of formally étale maps into $X$;
1. the [[structure (∞,1)-sheaf]] $\mathcal{O}_{X}$ of $X$.
The resulting structure is essentially that discussed ([Lurie, Structured Spaces](#Lurie)) if we regard $\mathbf{H}_{th}$ equipped with its formally étale morphisms, ([def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)), as a ([[large category|large]]) [[geometry for structured (∞,1)-toposes]].
One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive [[∞-group]] with [de Rham coefficient object](cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology) $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection
$$
X \times \flat_{dR} \mathbf{B}G \to X
$$
regarded as an object of the [[slice (∞,1)-topos]] $(\mathbf{H}_{th})_{/X}$ _almost_ qualifies as a "bundle of flat $\mathfrak{g}$-valued differential forms" over $X$: for $U \to X$ an cover (a [[1-epimorphism]]) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$.
This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like -- but only _if_ $U \to X$ is sufficiently _spread out_ over $X$, hence sufficiently [[étale map|étale]]. Because, on the extreme, if $X$ is the point (the [[terminal object]]), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$.
In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be _restricted_ to plots out of suficiently étale maps into $X$. In order to correctly test differential form data, "suitable" here should be "formally", namely infinitesimally. Hence the restriction should be along the full inclusion
$$
(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}
$$
of the formally étale maps (see [def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)) into $X$. Since on formally étale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding "cotangent bundle" must be the [[coreflective subcategory|coreflection]] along this inclusion. The following proposition establishes that this coreflection indeed exists.
+-- {: .num_defn #EtaleSlice}
###### Definition
For $X \in \mathbf{H}_{th}$ any object, write
$$
(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}
$$
for the full [[sub-(∞,1)-category]] of the [[slice (∞,1)-topos]] over $X$ on those maps into $X$ which are formally étale, (see [def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)).
We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$.
=--
+-- {: .num_prop #EtalificationIsCoreflection}
###### Proposition
The inclusion $\iota$ of def. \ref{EtaleSlice} is both [[reflective sub-(∞,1)-category|reflective]] as well as [[coreflective subcategory|coreflective]], hence it fits into an [[adjoint triple]] of the form
$$
(\mathbf{H}_{th})_{/X}^{fet}
\stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}}
(\mathbf{H}_{th})_{/X}
\,.
$$
=--
+-- {: .proof}
###### Proof
By the general discussion at _[[reflective factorization system]]_, the reflection is given by sending a morphism $f \colon Y \to X$ to
$X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in
$$
\array{
Y &\to& X \times_{\mathbf{\Pi}_{inf}(Y)} \mathbf{\Pi}_{inf}(Y) &\to& \mathbf{\Pi}_{inf}(Y)
\\
& \searrow & \downarrow^{} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}}
\\
&& X &\to& \mathbf{\Pi}_{inf}(X)
}
\,.
$$
Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, being a reflective subcategory of a [[locally presentable (∞,1)-category]], is (as discussed there) itself locally presentable. Hence by the [[adjoint (∞,1)-functor theorem]] it is now sufficient to show that the inclusion preserves all small [[(∞,1)-colimits]] in order to conclude that it also has a right [[adjoint (∞,1)-functor]].
So consider any [[diagram]] [[(∞,1)-functor]] $I \to (\mathbf{H}_{th})_{/X}^{fet}$ out of a [[small (∞,1)-category]]. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{fet}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at _[slice category - Colimits](overcategory#LimitsAndColimits)_).
Therefore we are reduced to showing that the square
$$
\array{
\underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i
\\
\downarrow && \downarrow
\\
X &\to& \mathbf{\Pi}_{inf}(X)
}
$$
is an [[(∞,1)-pullback]] square. But since $\mathbf{\Pi}_{inf}$ is a [[left adjoint]] it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to
$$
\array{
\underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i
\\
\downarrow && \downarrow
\\
X &\to& \mathbf{\Pi}_{inf}(X)
}
\,.
$$
This diagram is now indeed an [[(∞,1)-pullback]] by the fact that we have [[universal colimits]] in the [[(∞,1)-topos]] $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the [[(∞,1)-pullback]] of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.
=--
+-- {: .num_prop }
###### Proposition
The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an [[(∞,1)-topos]] and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a [[geometric embedding]].
=--
+-- {: .proof}
###### Proof
By prop. \ref{EtalificationIsCoreflection} the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is [[reflective sub-(infinity,1)-category|reflective]] with reflector given by the $(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed)$ factorization system. Since $\mathbf{\Pi}_{inf}$ is a [[right adjoint]] and hence in particular preserves [[(∞,1)-pullbacks]], the $\mathbf{\Pi}_{inf}$-equivalences are stable under pullbacks. By the discussion at _[[stable factorization system]]_ this is the case precisely if the corresponding reflector preserves [[finite (∞,1)-limits]]. Hence the embedding is a [[geometric embedding]] which exhibits a [[sub-(∞,1)-topos]] inclusion.
=--
+-- {: .num_defn #TheStructureSheafOfX}
###### Definition
For $X \in \mathbf{H}_{th}$ we speak of
$$
\mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X)
\coloneqq
(\mathbf{H}_{th})_{/X}^{fet}
$$
also as the ([[petit (∞,1)-topos|petit]]) [[(∞,1)-topos]] of $X$, or the **étale topos** of $X$.
Write
$$
\mathcal{O}_X
\colon
\mathbf{H}_{th}
\stackrel{(-) \times X}{\to}
(\mathbf{H}_{th})_{/X}
\stackrel{Et}{\to}
(\mathbf{H}_{th})_{/X}^{fet}
$$
for the composite [[(∞,1)-functor]] that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. \ref{EtalificationIsCoreflection}, of the projection $A \times X \to X$.
We call $\mathcal{O}_X$ the **[[structure sheaf]]** of $X$.
=--
+-- {: .num_remark }
###### Remark
For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a [[formally étale morphism]] in $\mathbf{H}_{th}$ (hence like an [[open subset]] of $X$), we have that
$$
\begin{aligned}
\mathcal{O}_{X}(A)(U)
& \coloneqq
Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) )
\\
& \coloneqq
Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) )
\\
& \simeq
(\mathbf{H}_{th})_{/X}(U, X \times A)
\\
& \simeq
\mathbf{H}_{th}(U,A)
\\
& \simeq
A(U)
\end{aligned}
\,,
$$
where we used the [[adjoint (∞,1)-functor|∞-adjunction]] $(\iota \dashv Et)$ of prop. \ref{EtalificationIsCoreflection} and the [[(∞,1)-Yoneda lemma]].
This means that $\mathcal{O}_{X}(A)$ behaves as the _sheaf of $A$-valued functions over $X$_.
Since $\mathcal{O}_{X}$ is [[right adjoint]] to the forgetful functor
$$
Sh_{\mathbf{H}}(X)
\simeq
(\mathbf{H}_{th})_{/X}^{fet}
\hookrightarrow
(\mathbf{H}_{th})_{/X}
\stackrel{\underset{X}{\sum}}{\to}
\mathbf{H}_{th}
$$
it preserves [[(∞,1)-limits]]. Therefore this is a [[structure sheaf]] which exhibits $Sh_{\mathbf{H}_{th}}(X)$ as a [[structured (∞,1)-topos]] over $\mathbf{H}_{th}$ regarded as a (large) [[geometry (for structured (∞,1)-toposes)]], with the formally étale morphisms being the "admissible morphisms".
=--
+-- {: .num_example #CotangentBundle}
###### Example
Let $G \in Grp(\mathbf{H}_{th})$ be an [[∞-group]] and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object.
Then
$$
\mathcal{O}_X(\flat_{dR}\mathbf{B}G)
\in
Sh_{\mathbf{H}}(X)
$$
we may call the **$G$-valued flat cotangent sheaf** of $X$.
=--
+-- {: .num_remark }
###### Remark
For $U \in \mathbf{H}_{th}$ a test object (say an object in a [[(∞,1)-site]] of definition, under the [[Yoneda embedding]]) a formally étale morphism $U \to X$ is like an [[open map]]/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams
$$
\array{
U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G)
\\
& \searrow && \swarrow
\\
&& X
}
\,.
$$
By the fact that $Et(-)$ is [[right adjoint]], such diagrams are in bijection to diagrams
$$
\array{
U &&\to&& X \times \flat_{dR} \mathbf{B}G
\\
& \searrow && \swarrow
\\
&& X
}
$$
where we are now simply including on the left the formally étale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$.
In other words, the sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the _domain_ of the section is constrained to be a formally é patch of $X$.
But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$.
=--
+-- {: .num_prop #StructuredPetitToposesAreLocallyContractible}
###### Proposition
For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive
$\infty$-topos, then its petit structured $\infty$-topos
$Sh_{\mathbf{H}_{th}}(X)$, according to def. \ref{TheStructureSheafOfX},
is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]].
=--
+-- {: .proof}
###### Proof
We need to check that the composite
$$
\infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th}
\stackrel{(-) \times X}{\longrightarrow}
(\mathbf{H}_{th})_{/X}
\stackrel{L}{\longrightarrow}
Sh_{\mathbf{H}}(X)
$$
preserves [[(∞,1)-limits]], so that it has a further
[[left adjoint]]. Here $L$ is the
reflector from prop. \ref{EtalificationIsCoreflection}.
Inspection shows that this composite sends an object $A \in \infty Grpd$ to
$\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X$:
$$
\array{
\mathbf{\Pi}_{inf}(Disc(A)) \times X
&\longrightarrow&
\mathbf{\Pi}_{inf}(Disc(A) \times X) & \simeq \mathbf{\Pi}_{inf}(Disc(A)) \times \mathbf{\Pi}_{inf}(X)
\\
\downarrow &{}^{(pb)}& \downarrow
\\
X &\longrightarrow& \mathbf{\Pi}_{inf}(X)
}
\,.
$$
By the discussion at [slice (∞,1)-category -- Limits and colimits](slice+infinity-category#LimitsAndColimits) an [[(∞,1)-limit]] in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an [[(∞,1)-limit]] in $\mathbf{H}$ of the [[diagram]] with the slice [[cocone]] adjoined. By [[right adjoint|right adjointness]] of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$.
Now for $A \colon J \to \infty Grpd$ a [[diagram]], it is taken to the diagram $j \mapsto \mathbf{\Pi}_{inf}(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form
$$
\array{
X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
&&\longrightarrow&&
X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
\\
& \searrow && \swarrow
\\
&& X
}
\simeq
\array{
X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
&&\longrightarrow&&
X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
\\
& \searrow && \swarrow
\\
&& X \times \ast
}
\,.
$$
Since $\infty$-limits commute with each other this limit is the product of
1. $\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))$
1. $\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$).
For the first of these, since the [[infinitesimal shape modality]] $\mathbf{\Pi}_{inf}$
is in particular a [[right adjoint]] (with [[left adjoint]] the
[[reduction modality]]), and since $Disc$ is also [[right adjoint]] by [[cohesion]], we have a [[natural equivalence]]
$$
\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))
\simeq
\mathbf{\Pi}_{inf}(Disc(\underset{\leftarrow}{\lim}_j(A_j)))
\,.
$$
For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is
$$
\begin{aligned}
\underset{\leftarrow}{\lim}_{J \star \Delta^0} X
& \simeq
\underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X]
\\
& \simeq
[\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X]
\\
& \simeq
[{\vert {J \star \Delta^0}\vert}, X]
\\
& \simeq
[\ast, X] \simeq X
\end{aligned}
\,,
$$
where the last line follows since ${J \star \Delta^0}$
has a terminal object and hence contractible geometric realization.
In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$.
=--
## Properties
### Sheaf condition and examples of étale sheaves
{#SheafConditionAndExamples}
+-- {: .num_prop #EtaleDescentDetectedOnOpenImmersionCovers}
###### Proposition
For $X$ a [[scheme]], and $A \in PSh(X_{et})$ a [[presheaf]],
for checking the [[sheaf]] condition it is sufficient to
check [[descent]] on the following two kinds of [[covers]]
in the [[étale site]]
1. jointly surjective collections of [[open immersions of schemes]];
1. single surjective/[[étale morphism of schemes|étale]] morphisms between [[affine schemes]]
(all over $X$).
=--
([Tamme, II Lemma (3.1.1)](#Tamme), [Milne, prop. 6.6](#Milne))
+-- {: .proof}
###### Proof (sketch)
Since [[covers]] by standard [[open immersions of schemes]] in the [[Zariski topology]] are also [[étale morphisms of schemes]] and étale covers, we may take any étale cover $\{Y_i \to Y\}$ over $X$, find an Zariski cover $\{U_i \to X\}$ of $X$, pull back the original cover to that and in turn cover the pullbacks themselves by Zariski covers. The result is still a cover and is so by a collection of [[open immersions of schemes]].
Now using compactness assumptions we find finite subcovers of all these covers. This makes their [[disjoint union]] be a single morphisms of affines.
=--
+-- {: .num_prop #XSchemesRepresentSheaves}
###### Proposition
For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]]
$$
(U_Y \to X) \mapsto Hom_X(U_Y, Z)
$$
is a [[sheaf]].
=--
This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in ([Tamme, II theorem (3.1.2)](#Tamme), [Milne, 6.2](#Milne)).
+-- {: .proof}
###### Proof
By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} we are reduced to
showing that the represented presheaf satisfies [[descent]] along collections of open immersions and along surjective maps of affines. For the first this is clear (it is [[Zariski topology]]-descent). For the second case of a [[faithfully flat]] cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the correspomnding [[Amitsur complex]]. See there for details.
=--
+-- {: .num_remark}
###### Remark
This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$.
Unique representatives are given by [[étale morphism of schemes|étale schemess]] over $X$.
=--
(e.g. [Tamme, II theorem 3.1](#Tamme))
We consider some examples of [[sheaves of abelian groups]]
induced by prop. \ref{XSchemesRepresentSheaves} from [[group schemes]]
over $X$.
+-- {: .num_example}
###### Example
The [[additive group]] over $X$ is the [[group scheme]]
$$
\mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X
\,.
$$
By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment
$$
\begin{aligned}
(\mathbb{G}_a)_X(U_X \to X) & =
Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X)
\\
& =
Hom(U_X, Spec(\mathbb{Z}[t]))
\\
& =
Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X}))
\\
& = \Gamma(U_X, \mathcal{O}_{U_X})
\end{aligned}
\,.
$$
In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$.
=--
Similarly one finds
+-- {: .num_example}
###### Example
The [[multiplicative group]] over $X$
$$
\mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X
$$
represents the sheaf $(\mathbb{G}_m)_X$ given by
$$
(\mathbb{G}_m)_X(U_X)
\mapsto
\Gamma(U_X, \mathcal{O}_{U_X})^\times
\,.
$$
=--
(e.g. [Tamme, II, 3](#Tamme))
### Base change and sheaf cohomology
{#BaseChange}
+-- {: .num_defn #BaseChangeOnSites}
###### Definition
For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]]
$$
f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}
$$
given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$
$$
f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X)
\,.
$$
=--
+-- {: .num_prop #DirectAndInverseImageAlongMapOfBases}
###### Proposition
The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]]
and hence induces a [[geometric morphism]] between the étale toposes
$$
(f^\ast \dashv f_\ast)
\;\colon\;
Sh(X_{et})
\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}
Sh(Y_{et})
\,.
$$
Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by
$$
f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)
$$
while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by
$$
f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y)
\,.
$$
=--
By the discussion at _[morphisms of sites -- Relation to geometric morphisms](morphism+of+sites#RelationToGeometricMorphisms)_.
See also for instance ([Tamme I 1.4](#Tamme)).
+-- {: .num_defn}
###### Definition
For $X_{et}$ an [[étale site]], write $\mathcal{D}(X_{et})$ for the [[derived category]] of the [[abelian category]] $Ab(Sh(X_{et}))$ of [[abelian sheaves]] on $X$.
=--
+-- {: .num_prop}
###### Proposition
The $q$th [[derived functor]] $R^q f_\ast$ of the [[direct image]] functor of def. \ref{DirectAndInverseImageAlongMapOfBases} sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]
$$
(U_Y \to Y)
\mapsto
H^q(X \times_Y U_Y, \mathcal{F})
\,,
$$
where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]).
=--
By the discussion at _[[direct image]]_ and at _[[abelian sheaf cohomology]]_. See e.g. ([Tamme, II (1.3.4)](#Tamme), [Milne prop. 12.1](#Milne)).
+-- {: .num_remark}
###### Remark
For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$
two composable [[morphisms of sites]],
the [[Leray spectral sequence]] for the corresponding [[direct images]] exists and is of the form
$$
E^{p,q}_2 = R^p f_\ast(R^q g_\ast(\mathcal{F}))
\Rightarrow
E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F})
\,.
$$
For the special case that $S_Z = \ast$ and $g^{-1}$ includes an [[étale morphism of schemes|étale morphism]] $U_Y \to Y$ this yields
$$
E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F})
\Rightarrow
E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F})
\,.
$$
=--
### Quasi-coherent modules
{#QuasiCoherentModules}
+-- {: .num_prop }
###### Proposition
For $X$ a [[scheme]] and $N$ a [[quasicoherent module]] over its [[structure sheaf]] $\mathcal{O}_X$, then this induces an [[abelian sheaf]] on the [[étale site]] by
$$
N_{et} \;\colon\; (U_X \to X)
\mapsto
\Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_X})
\,.
$$
=--
(e.g. [Tamme, II 3.2.1](#Tamme))
### Relation to Zariski topos
+-- {: .num_remark }
###### Remark
A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zariski sheaf, but not necessarily conversely. Expressed in a different way, the étale topos is a [[subtopos]] of the Zariski topos.
=--
For more see at _[étale cohomology -- Properties -- Relation to Zariski cohomology](etale+cohomology#RelationZariskiEtaleCohomology)_.
### As a classifying topos
The étale topos over the big étale site of [[commutative rings]] is the [[classifying topos]] for [[strict local rings]].
## Related concepts
* [[basics of étale cohomology]]
* [[étale homotopy]]
* [[étale cohomology]]
## References
### Etale topos of a schemes
* [[Günter Tamme]], section II 1 of _[[Introduction to Étale Cohomology]]_
{#Tamme}
* [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_
{#Milne}
### Etale topos of a differentially cohesive object
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))
[[!redirects etale toposes]]
[[!redirects etale topoi]]
[[!redirects étale topos]]
[[!redirects étale toposes]]
[[!redirects étale topoi]]
[[!redirects étale (∞,1)-topos]]
[[!redirects etale (∞,1)-topos]]
[[!redirects étale (∞,1)-toposes]]
[[!redirects etale (∞,1)-toposes]]
[[!redirects étale (infinity,1)-topos]]
[[!redirects etale (infinity,1)-topos]]
[[!redirects étale (infinity,1)-toposes]]
[[!redirects etale (infinity,1)-toposes]] |
étendue | https://ncatlab.org/nlab/source/%C3%A9tendue |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* automatic table of contents goes here
{:toc}
## Idea
An **étendue** (also 'etendue', 'étendu', or 'etendu'; from French 'étendue' (fem.)- _extent_) is a [[topos]] $\mathcal{Y}$ that locally looks like the category of sheaves on a space:
>Briefly, the slogan is that $\mathcal{Y}$ is locally a topological space. ([Lawvere 1976](#Lawvere76), p.129)
In some papers, [[topological spaces]] are replaced by [[locales]]
in the definition of [[étendues]].
For example, see Definition 1.1 in [Rosenthal](#Rosenthal) or Definition 1.12 in [Kock and Moerdijk](#KockMoerdijk).
Originally defined by [[A. Grothendieck]] in one of the famous 'exercises' of [SGA4](#SGA4) (ex. 9.8.2) as a [[Grothendieck topos]] $\mathcal{Y}$ that has a well-supported object $X$ such that the [[slice topos]] $\mathcal{Y}/X$ is equivalent to a sheaf topos on a topological space, the definition was generalized by [[Lawvere]] (1975,1976) by dropping the spatiality of the slice and require only that $\mathcal{Y}/X$ is a [[localic topos]].
Several characterizations of étendues are known and the _Ur_-example of an étendue, the presheaf topos $\mathcal{S}^G$ of group actions, exhibits one in terms of sites rather directly: it has a [[site]] where every morphism is monic. Other characterizations involve (local) [[equivalence relations]] and yield connections to [[orbifolds]], [[foliations]], and [[stacks]], which are instrumental for the generalization to _$\infinity$-étendues_ (cf. [Carchedi 2013](#Carchedi13)).
Étendues play an important role in Lawvere's approach to [[cohesion]] and the distinction between [[petit and gros toposes]] where they provide one of the classes of **petit** toposes (generalized spaces). In this context, Lawvere (1989,1991) interprets the cancellative property of the site as enabling an interpretation of étendues as _categories of processes_.
## Definition
A topos $\mathcal{Y}$ is called an _étendue_ if there is an object $X\in|\mathcal{Y}|$ such that the unique $X\rightarrow 1$ is epic and the [[slice topos]] $\mathcal{Y}/X$ is a [[localic topos]].[^fine]
[^fine]: An epic $k:X\rightarrow Y$ induces a [[geometric morphism]] $k_\ast:\mathcal{Y}/X\rightarrow \mathcal{Y}/Y$ whose inverse image part, the change of base functor, $k^\ast:\mathcal {Y}/Y\rightarrow\mathcal{Y}/X$ is faithful, which says by definition that $k_\ast$ is a surjection, and in case $Y=1$, one says that $\mathcal{Y}/X$ _covers_ $\mathcal{Y}$. $k_\ast$ is an [[étale geometric morphism]].
## Examples
> The first example of an étendue seems to have been the [[space of moduli]] of algebraic curves, which is prevented from being globally a space due to the action of the [[Galois groups]] within each point. Yes, something vaguely reminiscent of particle spin is going on in such spaces, and the most naked form is that for any group G, the category $\mathcal{S}^G$ is an étendue with only one point! This is easily seen from the observations that $\mathcal{S}^G/G\cong\mathcal{S}^G$ and that $G\twoheadrightarrow 1$ where the last two $G$'s denote the regular representation. ([Lawvere 1976](#Lawvere76), pp.129-130)
* The [[Sierpinski topos]] $\mathcal{S}^{\cdot\rightarrow\cdot}$, as the sheaf topos on the [[Sierpinski space]], is an étendue.
* The topos $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ of _directed [[graphs]]_ (aka [[quivers]]; Lawvere calls them *irreflexive graphs*) is an étendue, as it is locally equivalent to the sheaf topos on a three point space (Lawvere 1986). The contrast between $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ and the topos $\mathcal{S}^{\Delta_1^{op}}$ of _reflexive graphs_ is a paradigmatic example of the distinction between a _petit_ and a _gros_ topos.
* The [[Jónsson-Tarski topos]] $\mathcal{J}_2$ is an étendue, as it is locally equivalent to the sheaf topos on the [[Cantor space]]. It is discussed as a petit topos for _labeled graphs_ in (Lawvere 1989).
## Properties
* **Proposition**. A Grothendieck topos $\mathcal{Y}$ is an étendue iff there exists a [[site]] $(\mathcal{C}, J)$ for $\mathcal{Y}$ such that every morphism of $\mathcal{C}$ is monic.
* In particular for small $\mathcal{C}$, the presheaf topos $\mathcal{S}^{\mathcal{C}^{op}}$ is an étendue iff all morphisms in $\mathcal{C}$ are monic. In particular for [[monoid]]s $\mathcal{C}$ this is sometimes called [[left cancellative category|left cancellative]] (e.g. each free monoid is left cancellative).
* [[Subtoposes]] of étendues are étendues. K. Rosenthal uses this together with the preceding remark on $\mathcal{S}^{\mathcal{C}^{op}}$ for all-monic $\mathcal{C}$ in order to construct further étendues $Sh_j(\mathcal{S}^{\mathcal{C}^{op}})$ via a topology $j$ from a suitable functor $H:\mathcal{C}\to\mathcal{S}$ (for further details see [[Jónsson-Tarski topos]] or Rosenthal(1981)).
* A Grothendieck topos is a _[[boolean topos|Boolean]] étendue_ precisely if it satisfies the _internal [[axiom of choice]]_ (Freyd&Scedrov 1990). An example of such a Boolean étendue is $\mathcal{S}^G$, for $G$ a group.
* Étendues are 'locally co-decidable' in the sense that for a small $\mathcal{C}$ the functor category $[\mathcal{C},Set]$ is a [[locally decidable topos]] precisely if $[\mathcal{C}^{op},Set]$ is an [[étendue]]. Also the all-monic-site property is dual to the all-epic-site property of locally decidable toposes. Both concepts are subsumed under the notion of having a (sub canonical) site representation with no (non-trivial) [[idempotents]] (McLarty 2006, Lawvere 2007).
## Slice toposes of étendues are étendues
One can use the site characterization to show that being an étendue topos is a _local property_.
+-- {: .num_prop #etendue_slices}
###### Proposition
Let $Sh(\mathcal{C},J)$ be an étendue topos with $(\mathcal{C}, J)$ being an all-monic-site presentation and $P:C^{op}\to Set$ be a presheaf on $\mathcal{C}$ that is a $J$-sheaf. Then $Sh(\mathcal{C},J)/P$ is an étendue topos.
=--
+-- {: .proof}
###### Proof
It suffices to show that $Sh(\mathcal{C},J)/P$ has an all-monic site presentation. By exercise III.8 in [Mac Lane-Moerdijk (1994, p.157)](#MacLaneMoerdijk) there exists a topology $J'$ on the [[category of elements]] $\int_\mathcal{C} P$ such that
$$Sh(\mathcal{C},J)/P\simeq Sh(\int_\mathcal{C} P,J')\; .$$
Whence it suffices to show that $\int_\mathcal{C} P$ is all-monic: Let $f:(C,x)\to (D,y)$ be a morphism in $\int_\mathcal{C} P$ and $g,h:(B,z)\rightrightarrows (C,x)$ such that $f\cdot g=f\cdot h$ then $g=h$ since composition in $\int_\mathcal{C} P$ is inherited from $\mathcal{C}$ and the latter was all-monic by assumption.
=--
+-- {: .num_prop #IAC_local}
###### Corollary
Let $\mathcal{E}$ be a Grothendieck topos satisfying the [[axiom of choice|internal axiom of choice]] (IAC). Then any [[slice topos]] $\mathcal{E}/X$ satisfies the internal axiom of choice as well.
=--
+-- {: .proof}
###### Proof
By Freyd-Scedrov ([1990](#FreydScedrov90), p.181) a Grothendieck topos $\mathcal{E}$ satisfies IAC iff $\mathcal{E}$ is a Boolean étendue. But being [[Boolean category|Boolean]] is a local property whence by the preceding proposition all slices $\mathcal{E}/X$ are Boolean étendues.
=--
## Related entries
* [[localic topos]]
* [[petit topos]]
* [[locally decidable topos]]
* [[Deligne-Mumford stack]]
* [[bicategory of fractions]]
## References
* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, Springer LNM vol.269 (1972), pp.479-484.
* {#Carchedi13}[[David Carchedi]], _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi_ , arXiv1312.2204 (2013). ([pdf](http://arxiv.org/pdf/1312.2204v1.pdf))
* Darian De Wolf, [[Dorette Pronk|D. Pronk]], _A Double-Categorical View on Representations of Etendues_, Cah. Top. Géom. Diff. Cat. **LXI** 1 (2020) 3-56 [, [pdf](http://cahierstgdc.com/wp-content/uploads/2020/01/DEWOLF-PRONK-61-1.pdf), [arXiv:1910.02540](https://arxiv.org/abs/1910.02540)]
* {#FreydScedrov90}[[Peter Freyd|P. J. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.
* [[Peter Johnstone]], _Sketches of an [[Elephant]] II_, Oxford UP 2002, pp.769-775.
* {#KockMoerdijk} [[A. Kock]], [[I. Moerdijk]], _Presentations of étendues_ , Cah. Top. Géom. Diff. Cat. **XXXII** 2 (1991) pp.145-164. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_2/CTGDC_1991__32_2_145_0/CTGDC_1991__32_2_145_0.pdf))
* [[A. Kock]], [[I. Moerdijk]], _Every étendue comes from a local equivalence relation_ , JPAA **82** (1992) pp.155-174.
* [[M. V. Lawson]], [[B. Steinberg]], _Ordered groupoids and étendues_ , Cah. Top. Géom. Diff. Cat. **XXXXV** 2 (2004) pp.82-108. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_2004__45_2/CTGDC_2004__45_2_82_0/CTGDC_2004__45_2_82_0.pdf))
* [[F. William Lawvere]], _Variable sets etendu and variable structure in topoi_ , Lecture notes University of Chicago 1975.
* {#Lawvere76}[[F. William Lawvere]], _Variable quantities and variable structures in topoi_ , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.
* [[F. William Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary in TAC **9** (2005) pp.1-7. ([pdf](http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf))
* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.
* [[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM vol. 1488 (1991).
* [[F. William Lawvere]], _Axiomatic Cohesion_ , TAC **19** no. 3 (2007) pp.41-49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))
* [[F. William Lawvere]], _Cohesive Toposes: Combinatorial and Infinitesimal Cases_, Como Ms. 2008. ([pdf](http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf))
* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994.
* [[Colin McLarty]], _Every Grothendieck Topos has a One-Way Site_ , TAC **16** no. 5 (2006) pp.123-126. ([pdf](http://www.tac.mta.ca/tac/volumes/16/5/16-05.pdf))
* [[Ieke Moerdijk]], _Foliations, groupoids and Grothendieck étendues_ , Rev. Acad. Cienc. Zaragoza (2) **48** (1993) pp.5-33.
* [[Dorette A. Pronk]], _Etendues and stacks as bicategory of fractions_ , Comp. Math. **102** 3 (1996) pp.243-303. ([pdf](http://archive.numdam.org/ARCHIVE/CM/CM_1996__102_3/CM_1996__102_3_243_0/CM_1996__102_3_243_0.pdf))
* [[Pedro Resende]], _Groupoid Sheaves as Quantale Modules_ , arXiv.0807.4848v3 (2011). ([pdf](http://arxiv.org/pdf/0807.4848v3.pdf))
* {#Rosenthal} [[Kimmo I. Rosenthal]], _Étendues and Categories with Monic Maps_ , JPAA **22** (1981) pp.193-212.
* [[Kimmo I. Rosenthal]], _Sheaves and Local Equivalence Relations_ , Cah. Top. Géom. Diff. Cat. **XXV** 2 (1984) pp.179-206. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1984__25_2/CTGDC_1984__25_2_179_0/CTGDC_1984__25_2_179_0.pdf))
* [[Kimmo I. Rosenthal]], _Covering Étendues and Freyd's Theorem_ , Proc. AMS **99** 2 (1987) pp.221-222. ([pdf](http://www.ams.org/journals/proc/1987-099-02/S0002-9939-1987-0870775-X/S0002-9939-1987-0870775-X.pdf))
[[!redirects etendue]]
[[!redirects etendues]]
[[!redirects étendues]] |
Évariste Galois | https://ncatlab.org/nlab/source/%C3%89variste+Galois |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Évariste_Galois)
* [MacTutorBiographies page](https://mathshistory.st-andrews.ac.uk/Biographies/Galois/)
## Selected writings
* [[Évariste Galois]], *[[Galois' last letter|letter to Auguste Chevallier]]* (29 May 1832) [scan of original: [wikipedia](https://commons.wikimedia.org/wiki/File:E._Galois_Letter.jpg), French original text: [web](http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament/), English translation: [pdf](https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100)]
## Related entries
* [[group]], [[group theory]]
* [[Galois theory]]
* [[Galois connection]]
category: people
[[!redirects Galois]]
[[!redirects Evariste Galois]] |
Øystein Ore | https://ncatlab.org/nlab/source/%C3%98ystein+Ore | __Øystein Ore__ was a Norwegian algebraist and graph theorist. His Norwegian name in his mathematical articles was however spelled as Oystein Ore. He introduced [[Ore extension]]s and Ore conditions for [[Ore set]]s (in the context of study of linear equations over noncommutative [[Ore domain]]s) of rings. Ore introduced name groupoid for what we now call [[magma]] (cf. [[historical notes on quasigroups]]).
* wikipedia: [Øystein Ore](http://en.wikipedia.org/wiki/%C3%98ystein_Ore), [Ore extension](http://en.wikipedia.org/wiki/Ore_extension), [Ore condition](http://en.wikipedia.org/wiki/Ore_condition), [Ore's theorem](http://en.wikipedia.org/wiki/Ore%27s_theorem)
* [biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Ore.html) at MacTutor History of Mathematics archive, University of St Andrews
category: people
[[!redirects Oystein Ore]] |
Čech cohomology | https://ncatlab.org/nlab/source/%C4%8Cech+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Čech cohomology#
* table of contents
{:toc}
## Idea ##
### General idea
{#GeneralIdea}
Čech cohomology is a tool, or an algorithm, which, when it applies, computes [[abelian sheaf cohomology]] (of some $X$ with [[coefficients]] in some $A$) by use of [[coverings]] and systems of [[coefficients]] on the covering and all its non-empty finite intersections. More generally, it applies also to [[nonabelian cohomology]] and may hence for instance (in degree 1) be used to compute classes of [[principal bundles]] and generally (in higher degree) those of [[principal ∞-bundles]]. Quite generally Čech cohomology is the way to express the intrinsic [[cohomology]] of [[(∞,1)-sheaf]] [[(∞,1)-toposes]] by mapping out of [[Čech nerves]] of [[covers]].
Hence Čech cohomology is more an algorithm for computing [[cohomology]] (see also at _[[Čech methods]]_) than a cohomology theory in itself. In particular it applies (when it applies) to all sorts of [[coefficient]] [[sheaves]]/[[stacks]]/[[∞-stacks]] $A$.
When no further qualification of the coefficients $A$ is given then one usually has in mind by default either that $A = \mathbb{Z}$ is the group of [[integers]] (or a [[delooping]] thereof, in which case one computes [[ordinary cohomology]] by Čech methods) or, if working on a [[ringed space]]/[[ringed topos]]/[[structured (∞,1)-topos]] that $A = \mathcal{O}^\times$ is the [[group of units]] in the [[structure sheaf]].
Notice that there are technical conditions on a [[site]] and a [[cover]] to ensure that the result of the Čech cohomology algorithm really coincides with the more abstractly defined [[abelian sheaf cohomology]] -- and generally with the [[hom-spaces]] in the given [[(∞,1)-topos]]. (Technically the condition is that the [[Čech nerve]] is a [[cofibrant resolution]] in a [[model structure on simplicial presheaves]] for which the coefficient sheaf is a [[fibrant object]]). If these conditions do not apply, then the definition of Čech cohomology groups still works as such, but typically no longer qualifies as a "cohomology theory", axiomatically. They are just some groups. Strictly speaking one should maybe not use the term "Čech cohomology" in such cases, but beware that some authors do.
### More technical idea
From the discussion at [[abelian sheaf cohomology]] we know that the [[right derived functor]] definition computes the hom-set in the [[homotopy category of an (infinity,1)-category|homotopy category]] of an [[(infinity,1)-topos]] $\mathbf{H}$ that may alternatively be computed as a [[colimit]] over [[resolutions]] of the [[domain]] object
$$
H(X,A)
\coloneqq
\pi_0 \mathbf{H}(X,A)
= colim_{Y \stackrel{\in W \cap F}{\longrightarrow} X}
C_H(Y,A)/_{homotopy}
$$
where the colimit is over all acyclic fibrations $Y \stackrel{\in W \cap F}{\longrightarrow} X$ in an appropriate [[model category]] $C_H$ that [[presentable (infinity,1)-category|presents]] $\mathbf{H}$. For $\mathbf{H}$ an [[infinity-stack]] [[(infinity,1)-topos]] this $C_H$ is a [[model structure on simplicial presheaves]] and the acyclic fibrations $Y \stackrel{\in W \cap F}{\to} X$ for $X$ an ordinary space are the [[hypercovers]].
Now, for some coefficient objects $A$ it is sufficient to take the colimit here not over all [[hypercovers]], but just over [[Čech covers]]. The resulting formula
$$
H(X,A)
= colim_{Y \stackrel{\mathop{Čech} cover}{\to} X}
C_H(Y,A)/_{homotopy}
$$
is then called the formula for **Čech cohomology** on $X$ with values in $A$.
Here a [[Čech cover]] is a simplicial presheaf that arises from an ordinary covering map $U \to X$ of $X$ by another space $U$ as the corresponding [[Čech nerve]] simplicial presheaf
$$
\begin{aligned}
Y := \left(
\cdots
U \times_X U \times_X U
\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U
\right)
\simeq
hocolim_{[k] \in \Delta} U^{\times_X k}
\end{aligned}
\,.
$$
See [[descent for simplicial presheaves]] for more on the manipulations involved here.
To amplify, a general [[hypercover]] would start in degree 0 with a $U$ as above, but then in the next degree would have a cover $V \to U \times_X U$ of the fiber product, and so on, each fiber product in turn being covered by another space.
If $Y$ is not simply a [[Čech cover]] but also not the most general [[hypercover]] in that this iterative choice of further covering stops in degree $n$, then one also speaks of **Čech cover of level $n$** and of the corresponding cohomology formula as **higher Čech cohomology**. See for instance the reference by Tibor Beke below.
## General ("nonabelian") Čech cohomology
{#NonabelianČechCohomology}
We start with describing the general " _nonabelian_ " Čech cohomology (compare the terminology and remarks at [[cohomology]] and [[nonabelian cohomology]]), i.e. the plain unwrapping of the above definition, before assuming that our coefficient object is [[abelian sheaf cohomology|abelian]] and before applying the [[Moore complex]] functor that sends the following simplicial computation to the maybe more familiar one in [[chain complexes]].
The reasoning parallels that described at [[descent]] to some extent, but is maybe still worthwhile repeating here.
So let $C$ be some [[site]] and let $SSh(C)$ the category of [[simplicial presheaf|simplicial (pre)sheaves]] on $C$. Let $C(U)$ be the Čech nerve for a cover $\pi : U \to X$ of some simplicial presheaf $X$ and let $A$ be any other simplicial presheaf that serves as the coefficient object.
Using [[end]] and [[coend]] notation we compute the required hom
$$
\begin{aligned}
SSh(C(U), A)
&
\simeq
SSh( \int^{[k] \in \Delta} \Delta^k \cdot C(U)_k, A )
\\
&
\simeq
\int_{[k] \in \Delta} SSh(\Delta^k \cdot C(U)_k, A )
\\
&
\simeq
\int_{[k] \in \Delta} SSet(\Delta^k , SSh(C(U)_k,A) )
\\
&
\simeq
\int_{[k] \in \Delta} SSet(\Delta^k , A(U \times_X \cdots \times_X U) )
\\
&
\simeq
\int_{[k] \in \Delta} SSet(\Delta^k , \prod_{i_0, \cdots, i_k} A(U_{i_0 i_1, \cdots ,i_k}) )
\end{aligned}
\,.
$$
Here in the last line we have assumed that $U = \sqcup_i U_i$ for $\{U_i \to X\}$ an open cover of some space $X$ and we abbreviate as usual with
$
U_{i_0, i_1, \cdots, i_k}
$ the $(k+1)$-fold intersections
$\cap_{0 \leq r \leq k} U_{i_r}$.
An object in this last expression -- an $A$-valued cocycle on $X$ relative to $U$ -- is a collection of morphisms in [[SSet]] of the form
$$
\left(
\array{
\vdots && \vdots
\\
\Delta^2
&\stackrel{f}{\longrightarrow}&
\prod_{i j k} A(U_{i j k})
\\
\Delta^1
&\stackrel{g}{\longrightarrow}&
\prod_{i j}A(U_{ i j})
\\
\Delta^0
&\stackrel{a}{\longrightarrow}&
\prod_i A(U_i)
}
\right)
$$
that make all diagrams of the form
$$
\array{
\vdots && \vdots
\\
\uparrow
&&
\uparrow
\\
\Delta^2
&\stackrel{f}{\longrightarrow}&
\prod_{i j k} A(U_{i j k})
\\
\uparrow
&&
\uparrow
\\
\Delta^1
&\stackrel{g}{\to}&
\prod_{i j}A(U_{ i j})
\\
\uparrow^{\delta_{\cdot}}
&&
\uparrow^{A(d_{C(U)})_\cdot}
\\
\Delta^0
&\stackrel{a}{\to}&
\prod_i A(U_i)
}
$$
commute. Here the vertical morphism on the right, when one traces their origin through the derivation above, are the various inclusion and restriction maps of elements of $A$ evaluated on a $k$-fold intersection to some other intersection.
For instance the three possible vertical maps at the
bottom have components on the cartesian factors given by
$$
A(d_1)|_{U_i} := (-)|_{U_{i j}}
:
A(U_i)
\to
A(U_{i j})
$$
and
$$
A(d_0)|_{U_i} := (-)|_{U_{i j}}
:
A(U_j)
\to
A(U_{i j})
$$
and
$$
A(s_0)|_{U_{i i}}
:
A(U_{i i})
\to
A(U_{i })
\,.
$$
This picks
* an object $a \in A(U)$
* a morphism $g : \pi_1 ^* a \to \pi_2^* a in A(U \times_X U)$
* a 2-morphism
$$
\array{
&& \pi_{2}^* a
\\
& {}^{\pi_{12}^* g}\nearrow &\Downarrow^{f}& \searrow^{\pi_{23}^* g}
\\
\pi_{1}^* a
&&\stackrel{\pi_{13}^* g}{\to}&&
\pi_3^* a
}
$$
in $A(U \times_X U \times_X U)$
* etc.
If we follow tradition and write
* $U = \sqcup_i U_i$;
* and $U \times_X U = \sqcup_{i j} U_{i j} := \sqcup_{i j} U_i \cap U_j$
* etc
this is
* an collection of objects $(a_i \in A(U_i))$
* a collection of morphisms
$(g_{i j} : a_i|_{U_{ij}} \to a_{j}|_{U_{i j}}
in A(U_{i j}))$
* a collection of 2-morphisms
$$
\left(
\array{
&& a_{j}|_{U_{i j k}}
\\
& {}^{g_{ij}|_{U_{i j k}}}\nearrow
&\Downarrow^{f_{i j k}}&
\searrow^{g_{j k}|_{U_{i j k}}}
\\
a_i|_{U_{i j k}}
&&\stackrel{g_{i k}|_{U_{i j k}}}{\to}&&
a_k|_{U_{i j k}}
}
\right)
$$
in $A(U_{i j k})$
* etc.
This is a **Čech-cocycle** on $X$ with values in $A$ relative to $U$.
A **transformation/homotopy/coboundary** between two such cocycles is a cylinder over these diagrams, i.e.
* a collection of morphism
$ h_i : a_i \to a'_i in A(U_i)$;
* a collection of 2-morphism
$$
\array{
a_i|_{U_{i j}}
&\stackrel{g_{i j}}{\to}&
a_j|_{U_{i j}}
\\
\;\;\;\downarrow^{h_i|_{U_{i j}}}
&\Downarrow^{a_{i j}}&
\;\;\;\downarrow^{h_j|_{U_{i j}}}
\\
a'_i|_{U_{i j}}
&\stackrel{g_{i j}}{\to}&
a'_j|_{U_{i j}}
}
$$
on $U_{i j}$
* a collection of 3-morphisms
$$
\array{
&& a_j|_{U_{i j k}}
\\
& {}^{g_{ij}|_{U_{i j k}}}\nearrow
&\Downarrow^{f_{i j k}}&
\searrow^{g_{j k}|_{U_{i j k}}}
\\
a_i|_{U_{i j k}}
&&\stackrel{g_{i k}}{\to}&&
a_k|_{U_{i j k}}
\\
\;\;\;\downarrow^{h_i|_{U_{i j}}}
&&\Downarrow^{a_{i k}}&&
\;\;\;\downarrow^{h_j|_{U_{i j}}}
\\
a'_i|_{U_{i j k}}
&&\stackrel{g_{i k}}{\to}&&
a'_k|_{U_{i j k}}
}
\;\;
\;\;
\;\;
\;\;
\;\;
\stackrel{\rho_{i j k}}{\Rightarrow}
\;\;
\;\;
\;\;
\;\;
\;\;
\array{
&& a_j|_{U_{i j k}}
\\
&
{}^{g_{ij}|_{U_{i j k}}}\nearrow
&\Downarrow^{a_{i j}\cdot a_{j k}|_{U_{i j k}}}&
\searrow^{g_{j k}|_{U_{i j k}}}
\\
a_i|_{U_{i j k}}
&& a'_j|_{U_{i j k}} &&
a_k|_{U_{i j k}}
\\
\downarrow^{h_{i}|_{U_{i j k}}}
&
{}^{g'_{i j}|_{U_{i j k}}}\nearrow
&
\Downarrow^{f'_{ijk}}
&
\searrow^{g'_{j k}|_{U_{i j k}}}
&
\downarrow^{h_{k}|_{U_{i j k}}}
\\
a'_i|_{U_{i j k}}
&&\stackrel{g'_{i k}|_{U_{i j k}}}{\to}&&
a'_k|_{U_{i j k}}
}
$$
We now plug in some concrete coefficient object
$n$-types for low $n$ and reproduce some concrete formulas
from this.
### Nonabelian 1-cocycles: principal bundles
For $G$ a [[group]] let
$\mathbf{B} G$ by abuse of notation denote
* first of all the corresponding
[[delooping|delooped]] [[groupoid]],
* then the corresponding [[nerve]]
* then finally the corresponding simplicial sheaf that for each object $U$ assigns the [[nerve]] for the group $Hom(U,G)$:
$$
\mathbf{B}G : U \mapsto
\left\{
\array{
\mathbf{B}G(U)_0 = Hom(U,{*}) = {*}
\\
\mathbf{B}G(U)_1 = Hom(U,G)
\\
\mathbf{B}G(U)_2 = Hom(U,G)\times Hom(U,G)
\\
\vdots
}
\right.
$$
The above then says that
* a $\mathbf{B}G$-Čech cocycle is
* a collection of $G$-valued functions
$$
(g_{ij} \in Hom(U_{i j}, G))
$$
* a collection of identities between $G$-valued functions
$$
g_{ij}|_{U_{i j k }}
\cdot
g_{j k}|_{U_{i j k }}
=
g_{i k}|_{U_{i j k }}
$$
* a $\mathbf{B}G$-Čech coboundary is
* a collection of $G$-valued functions
$$
(h_i \in Hom(U_i , G))
$$
* a collection of identites between $G$-valued functions
$$
g_{ij} \cdot h_j|_{U_{i j}}
=
h_i|_{U_{i j}} \cdot g'_{j k}
\,.
$$
Remembering that the Čech cohomology is the colimit over refinement of covers over cohomology classes defined this way, one sees the standard
+-- {: .num_theorem }
###### Theorem
Čech cohomology with coefficients in $\mathbf{B}G$
(as above) classies $G$-[[principal bundles]]
$$
colim_{U \stackrel{\mathop{Čech} cover}{\to} X}
SSh(C(U), \mathbf{B}G)/_{homtopy}
\simeq
G Bund(X)/_\sim
$$
=--
### Nonabelian 2-cocycles: gerbes and principal 2-bundles
In one degree higher the general [[homotopy n-type|homotopy 2-type]] coefficient object is modeled using a strict [[2-group]] $H$ coming from a [[crossed module]]
$G_2 \stackrel{\delta}{\to} G_1$.
The [[nerve]] of the corresponding [[delooping|delooped]]
[[2-groupoid]], regarded immediately as a simplicial sheaf starts out like
$$
\mathbf{B}(G_2 \to G_1)
:
U
\mapsto
\left\{
\array{
\mathbf{B}(G_2 \to G_1)(U)_{0}
=
Hom(U, G_1)
\\
\mathbf{B}(G_2 \to G_1)(U)_{1}
=
Hom(U, G_1) \times Hom(U, G_1)
\times Hom(U, G_2)
\\
\vdots
}
\right.
\,.
$$
One finds that a Čech-cocycle now is
(see also the diagrams at [[group cohomology]] for this)
* a collection of functions
$$
(g_{i j} \in Hom(U_{i j}, G_1))
$$
* a collection of functions
$$
(f_{i j k }\in Hom(U_{i j k}, G_2) )
$$
* a collection of identities
$$
\delta(f_{i j k}|_{U_{i j k}})
g_{i j}|_{U_{i j k}}
g_{j k}|_{U_{i j k}}
=
g_{i k}|_{U_{i j k}}
$$
* and a collection of identities
$$
f_{i k l}|_{U_{i j k l}}
\cdot
f_{i j k}|_{U_{i j k l}}
=
f_{i j l}|_{U_{i j k l}}
\cdot
\alpha(g_{i j}|_{U_{i j k l}})(f_{j k l}|_{U_{i j k l}}),
$$
where $\alpha:G_1\to Aut(G_2)$ is the homomorphism associated with the crossed-module description of the 2-group.
This is a **nonabelian Čech 2-cocycle**.
Reading off the formulas for the coboundaries is
left as an excercise for the reader.
For the specical case the
$$
(G_2 \to G_1) = (G \to Aut(G))
$$
this is the nonabelian cohomology classifying
[[gerbes]].
## Abelian Čech cohomology
{#AbelianČechCohomology}
In much of the literature _Čech cohomology_ denotes exclusively the abelian case, which we now describe.
In the special case that the coefficient [[simplicial presheaf]] $A$ is in the image of the [[nerve]] operation
$$
N : Ch_+ \to sAb \to SSet
$$
on [[chain complexes]] with values in [[simplicial sets]] that happen to be abelian [[simplicial groups]], the computation of Čech cocycles may be entirely pulled back to the world of [[homological algebra]] by making use of the [[Dold-Kan correspondence]] that provides an [[adjoint equivalence]] between simplicial sets with values in abelian groups and non-negatively graded chain complexes.
### Definition
{#AbelianČechCohomologyDefinition}
We discuss the [[double complex]] which gives the traditional definition of Čech cohomology with coefficients in [[sheaves of abelian groups]].
Let
* $X$ be a [[topological space]];
* $A_\bullet = (\cdots \stackrel{\partial}{\to} A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0)$ be a [[chain complex]] of [[abelian sheaves]] on $X$ (i.e. on the [[category of open subsets]] $S \colon Op(X)$ of $X$, regarded as a [[site]] with the usal [[covering]] [[Grothendieck topology]]);
* $\{U_i \to X\}$ be an [[open cover]] of $X$.
We write
$$
U_{i_0,\cdots i_k}
\coloneqq
U_{i_0}
\underset{X}{\times}
U_{i_1}
\underset{X}{\times}
\cdots
\underset{X}{\times}
U_{i_k}
$$
for the $k$-fold intersections of these open subsets. Given an inclusion of open subsets $U_1 \to U_2$ and given any group element $a\in A_\bullet(U_2)$ we write $a|_{U_1}$ for its restriction along this map.
More generally, we may allow $S$ to be any [[site]]. For simplicity of the following formulas assume that $S$ has [[finite products]] (which in the case that $S$ is a [[category of open subsets]] are the [[fiber products]] above.) Then $A_\bullet$ is a chain complex of [[abelian sheaves]] on that site.
+-- {: .num_defn #CechComplex}
###### Definition
**(Čech complex)**
The **Čech cochain complex** $C^\bullet((X,\{U_i\}),A_\bullet)$
of $X$ with respect to the cover $\{U_i \to X\}$ and with [[coefficients]] in $A_\bullet$ is in degree $k \in \mathbb{N}$ given by the [[abelian group]]
$$
C^k((X,\{U_i\}),A_\bullet)
\coloneqq
\oplus_{{l,n} \atop {k = n-l}}
\oplus_{i_0, i_1, \cdots, i_n}
A_l(U_{i_0, \cdots, i_n})
$$
which is the [[direct sum]] of the values of $A_\bullet$ on the given intersections as indicated; and whose [[differential]]
$$
d
\colon
C^{k}((X,\{U_i\}),A_\bullet)
\longrightarrow
C^{k+1}((X,\{U_i\}),A_\bullet)
$$
is defined componentwise (see at [[matrix calculus]] for conventions on maps between [[direct sums]]) by
$$
\begin{aligned}
(d a)_{i_0, \cdots, i_{k+1}}
& \coloneqq
(\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}}
\\
& \coloneqq
\partial_A a_{i_0, \cdots, i_{k+1}}
+
(-1)^k
\sum_{0 \leq j \leq k+1} (-1)^{j}
a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}}
|_{U_{i_0, \cdots, i_{k+1}}}
\end{aligned}
$$
where on the right the sum is over all components of $a$ obtained via the canonical restrictions obtained by discarding one of the original $(k+1)$ subscripts.
The **Čech cohomology** groups of $X$ with coefficients in $A_\bullet$ __relative to the given cover__ are the [[chain homology]] groups of the Čech complex
$$
H_{Cech}^k((X,\{U_i\}), A_\bullet)
\coloneqq
H^k(C^\bullet((X,\{U_i\}),A_\bullet))
\,.
$$
The **Čech cohomology** groups as such are the [[colimit]] ("[[directed limit]]") of these groups over refinements of covers
$$
H^k_{Cech}(X, A_\bullet)
\coloneqq
\underset{\longrightarrow}{\lim}_{\{U_i \to X\}}
H_{Cech}^k((X,\{U_i\}), A_\bullet)
\,.
$$
=--
+-- {: .num_remark}
###### Remark
Often Čech cohomology is considered for the case that $A_\bullet$ is concentrated in a single degree, in which case the first term in the sum defining the differential in def. \ref{CechComplex} disappears.
When $A_\bullet$ is not concentrated in a single degree, then for emphasis and following terminology of _[[hypercohomology]]_
one may speak of the **Čech hypercomplex** computing **Čech hypercohomology**.
=--
+-- {: .num_remark}
###### Remark
The Čech chain complex in def. \ref{CechComplex} is the [[total complex]] of the [[double complex]] whose vertical differential is that of $A_\bullet$ and whose horizontal differential is the _Čech differential_ $\delta$ given by alternating sums over restrictions along patch inclusions
$$
\array{
\vdots && \vdots
\\
\downarrow^{\mathrlap{\partial_A}}
&&
\downarrow^{\mathrlap{\partial_A}}
\\
\oplus_i A_1(U_i)
&\stackrel{\delta}{\longrightarrow}&
\oplus_{i_1, i_2} A_1(U_{i_1, i_2})
&\stackrel{\delta}{\longrightarrow}&
\cdots
\\
\downarrow^{\mathrlap{\partial_A}}
&&
\downarrow^{\mathrlap{\partial_A}}
\\
\oplus_i A_0(U_i)
&\stackrel{\delta}{\longrightarrow}&
\oplus_{i_1, i_2} A_0(U_{i_1, i_2})
&\stackrel{\delta}{\longrightarrow}&
\cdots
}
$$
=--
### Relation to nonabelian Čech cohomology
{#RelationAbelianNonabelian}
We discuss how the abelian Čech complex of def. \ref{CechComplex} arises as a special case of the simplicial nonabelian cocycle complex of general nonabelian Čech cohomology [above](#NonabelianČechCohomology), under the [[Dold-Kan correspondence]].
+-- {: .num_remark}
###### Remark
It may be helpful to keep the following pictures in mind to match the signs to the orientations of simplices.
In dimension one we have:
$$
(0 \to 1)
\;\;\;
\mapsto
\;\;\;
(a_0 \stackrel{a_{0 1}}{\to} a_1)
\;\;\;
\leftrightarrow
\;\;\;
(a_1 = a_0 + d_A a_{01})
$$
In dimension 2 we have:
$$
\array{
&& 1
\\
& \nearrow &\Downarrow& \searrow
\\
0 &&\to&& 2
}
\;\;\;
\mapsto
\array{
&& a_1
\\
& {}^{a_{0 1}}\nearrow &\Downarrow^{a_{0 1 2}}
& \searrow^{a_{1 2}}
\\
a_0 &&\stackrel{a_{0 2}}{\to}&& a_2
}
\;\;\;
\leftrightarrow
(a_{0 2} = a_{0 1} + a_{1 2} + d a_{0 1 2})
\,.
$$
Here the relation on the right is the
[[Dold-Kan correspondence]] relating coboundaries in the
complex $A_\bullet$ to simplices .
=--
+-- {: .num_remark}
###### Remark
The following derivation of abelian Čech cohomology from nonabelian Čech cohomology restricted to nerves of chain complexes needs of the [[Dold-Kan correspondence]] only that it is an [[adjunction]], not that it is an [[adjoint equivalence]].
=--
+-- {: .num_prop }
###### Proposition (recovering abelian Čech cohomology)
Let $A_\bullet$ be a sheaf with values in $Ch_+$
and write $N(A_\bullet)$ for the corresponding
simplicial sheaf under the [[nerve]] operation
of the [[Dold-Kan correspondence]].
Then the general (nonabelian) Čech cohomology
of $N(A_\bullet)$ as defined above coincides
with the cohomology of the Čech complex of
$A_\bullet$:
$$
H(X, N(A_\bullet))
\simeq
colim_U H_0(C(U,A_\bullet))
\,.
$$
=--
+-- {: .proof}
###### Proof
The underlying idea is to use the
adjunction of the [[Dold-Kan correspondence]]
to move the nerve operation $N(A_\bullet)$
on the right to the
free abelian chain complex operation
$C_bullet(F(C(U)))$
on the
[[Čech cover]] simplicial sheaf $C(U)$ and then
use the [[Yoneda lemma]] to evaluate $A_\bullet$
on $N_\bullet(F(C(U)))$. The result is the
Čech complex of $A_\bullet$.
Cocycles and homotopies/coboundaries
are in bijection on both sides.
Spelled out in full detail this looks a bit more lengthy,
but is nothing but this simple idea.
Before starting the computation notice the following
observation on the image of the [[Čech cover]]
$C(U)$ under the [[Dold-Kan correspondence]]:
For $Y = C(U)$ a [[Čech cover]] on a sieve
$\{U_i \to X\}_{i \in I}$, and for $W$ any test object,
the non-degenerate $n$-simplices in $C(U)(W)$ are the
$$
U_{i_0, i_1, \cdots, i_n}(W)
$$
with all indices pairwise different, $i_k \neq i_l$.
Accordingly the free abelian simplicial group-valued
sheaf $F(C(U))$ is for each $W$ and $n$ the
free abelian group generated from these
$U_{i_0, i_1, \cdots, i_n}(W)$ with pairwise distinct elements, i.e. that given by
formal interger combinations of these element.
In turn, the normalized [[Moore complex]] sheaf
$$
N_\bullet(F(C(U)))
$$
assigns to a test domain $W$ the complex that in
each degree is the free abelian group on these
elements.
With this in hand,
we compute the set of $N(A_\bullet)$-valued cocycles on $U$ as follows:
$$
\begin{aligned}
Hom_{SPSh}(C(U), N(A_\bullet))
&\simeq
\int_W Hom_{SSet}( C(U)(W) , N(A_\bullet)(W) )
\\
& \simeq
\int_W Hom_{SSet}( C(U)(W) , N(A(W)_\bullet) )
\\
& \simeq
\int_W Hom_{SAb}( F(C(U)(W)) , N(A(W)_\bullet) )
\\
& \simeq
\int_W Hom_{Ch_+}( N_\bullet(F(C(U)(W))) ,
A(W)_\bullet )
\\
& \simeq
\int_W \int_{n}
Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n )
\\
& \simeq
\int_{n}
\int_W
Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n )
\\
& \simeq
\int_{n}
\int_W
Hom_{Set}( \coprod_{i_0, i_1, \cdots, i_k}
U_{i_0,i_1, \dots, i_n}(W) , A(W)_n ) )
\\
& \simeq
\int_{n}
\prod_{i_0, i_1, \cdots, i_n}
A(U_{i_0,i_1, \dots, i_n})_n
\\
& =
ker d_{C(U, A_\bullet)} \subset C(U,A_\bullet)_0
\end{aligned}
\,.
$$
Here
* The first step is the definition of morphism of [[presheaf|presheaves]] using the [[end]] notation;
* the second step is the definition of the [[nerve]] of of [[chain complexes]] applied to chain complex valued sheaves;
* the third step uses that the free simplicial abelian group functor is [[left adjoint]] to the forgetful one that remembers the underlying simplicial set;
* the fourth step then uses the [[Dold-Kan correspondence]], or actually just that the normalized [[Moore complex]] functor is [[left adjoint]] to the [[nerve]] of [[chain complexes]];
* the fifth step expresses the set of morphisms of [[chain complexes]] as an [[end]] (being itself a [[natural transformation]]);
* the sixth step uses the [[Fubini theorem]] of [[enriched category theory]] to commute the two [[ends]];
* the seventh step uses that the chain complex in the left argument is generated freely on elements of a set to rewrite the hom of abelian groups into one of sets;
* this finally allows to apply the [[Yoneda lemma]] in step eight
* and in step nine, by inspection, one notices that the result thus obtained is the set of 0-cycles in the Čech complex $C(U,A_\bullet)$ as previously defined.
An entirely analogous argument shows that dividing out homotopies is respected.
One starts by observing that the cohomology coboundaries are given by homotopies in the hom-simplicial set, i.e. by
$$
Hom_{SPSh}(C(U) \times \Delta^1, N(A_\bullet))
\,.
$$
With this one goes in the above computation.
After applying the
[[Dold-Kan correspondence|Dold-Kan adjunction]]
we now have on the left in the integrand the term
$$
N_\bullet(F(C(U)(W) \times \Delta^1))
\,.
$$
Using first that the free simplicial group functor is monoidal and then that the normalized [[Moore complex]]
functor is lax monoidal (as described at
[[Dold-Kan correspondence]])
> what to do about laxness versus pseudoness??
we get a morphism to that from
$$
N_\bullet(F(C(U)(W))) \otimes N_\bullet(F(\Delta^1))
\,.
$$
Using that the normalized Moore complex is
isomorphic to the Moore complex divided by the part
generated by degenerate cells,
on the right we identify the [[interval object]] complex
$$
( \cdots \to 0 \to I \stackrel{1 \oplus -1}{\to} I \oplus I)
\,.
$$
To see that one just needs to observe that the normalized [[Moore complex]] of the 1-[[simplex]] serves as an [[interval object]] in chain complexes.
=--
### Relation to abelian sheaf cohomology
For $A$ a complex of sheaves, there is a canonical morphism
$$
\check{H}(X,A) \to H(X,A)
$$
from the Čech cohomology to the full (hypercompleted) cohomology, which is [[abelian sheaf cohomology]] in the case that $A$ is in the image of the [[Dold-Kan correspondence|Dold-Kan map]] from [[chain complexes]]. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of [[Čech covers]] into all [[hypercovers]].
+-- {: .num_theorem }
###### Theorem
If $X$ is a [[paracompact space]] the canonical morphism
$$
\check{H}(X,A) \to H(X,A)
$$
from Čech cohomology to [[abelian sheaf cohomology]] is an isomorphism from every $A \in Sh(X, Ch_+)$.
=--
+-- {: .proof}
###### Proof
Recalled as theorem 1.3.13 in
* Brylinski, _Loop spaces, characteristic classes and geometric quantization_
=--
When $X$ is not paracompact, we still have the following condition under which Čech cohomology computes [[abelian sheaf cohomology]]-.
For $A$ a complex of sheaves, there is a canonical morphism
$$
H_0(C(U,A)_\bullet) \to H(X,A)
$$
from the cohomology of the Čech complex with respect to a cover $U$ with coefficients in $A$ to the [[abelian sheaf cohomology]] of $X$ with values in $A$. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of [[Čech covers]] into all [[hypercovers]].
+-- {: .num_theorem }
###### Theorem
Let $A$ be a complex of sheaves on $X$ concentrated
in a single degree $p \geq 0$,
and let $\{U_i \to X\}$ be a [[cover]]
such that the $A$-cohomology of all intersections vanishes,
$$
\forall U_{i_0 , \cdots, i_k} H(U_{i_0, \cdots, i_k},A) = 0
$$
Then the canonical morphism
$$
H_0(C(U,A)_\bullet) \to H(X,A)
$$
is an [[isomorphism]].
=--
+-- {: .proof}
###### Proof
One considers the spectral sequence associated with the Čech double complex. Details are on page 28 of
* Brylinski, _Loop spaces, characteristic classes and geometric quantization_
=--
### Examples ###
We now list examples for abelian Čech cohomology expressed in terms of the Čech complex described above.
#### Line bundles ####
Let $G = U(1)$ be the circle group, an abelian
group. The [[nerve]] $N(\mathbf{B}G)$
of its [[delooping]] $\mathbf{B}G$ is the
[[bar construction]] of $G$. This is equivalently
the nerve of the [[chain complex]]
$$
U(1)[1]
:= ( \cdots \to 0 \to U(1) \to 0 )
\,.
$$
Equivalently we get the corresponding simplicial sheaves
and complexes of sheaves, eg
$$
U(1)[1] : W \mapsto
( \cdots \to 0 \to Hom(W,U(1)) \to 0 )
\,.
$$
So given a cover $\{U_i \to X\}$ a Čech cocycle is a collection
$$
c = ( \{g_{i j} \in U(1)[1](U_{i j})\}|_{i,j})
$$
such that the Čech differential evaluated on it
$$
\begin{aligned}
(d_{C(U,U(1)[1])} c)_{i j k}
&=
g_{j k}|_{U_{i j k}} - g_{i k}|_{U_{i j k}}
+ g_{i j}|_{U_{i j k}} )
\end{aligned}
$$
vanishes:
$$
g_{i j}|_{U_{i j k}} + g_{j k}|_{U_{i j k}} = g_{i k}|_{U_{i j k}}
$$
for all $i, j, k$.
Here now we write plus signs for the operation in the abelian group $U(1)$ which above we have written by juxtaposition or using a dot. So up to notation for the group operation this is
$$
g_{i j} g_{j k} = g_{i k}
$$
on $U_{i j k}$ as before in the nonabelian case of
Čech cocycles for
$U(1)$-[[principal bundles]].
Here and from now on we shall notationally suppress the restriction maps $(-)|_{U_{i_0, i_1, \cdots, i_n}}$
as they are unambiguously obviuous in every case.
#### Line bundle gerbes
Similarly by shifting $U(1)$ ever higher in chain degree,
one finds Čech cocycles for [[bundle gerbes]], bundle 2-gerbes, etc.
A Čech cocycle for
$$
U(1)[2]
:= ( \cdots \to 0 \to U(1) \to 0 \to 0 )
\,.
$$
is
$$
c = ( \{g_{i j k} \in U(1)[2](U_{i j k})\}|_{i,j, k})
$$
such that on $U_{i,j,k,l}$ we have
$$
g_{i j k} - g_{i j l} + g_{i k l} - g_{j k l} = 0
$$
which in the nonabelian context would be written as
$$
g_{i j k} g_{i k l} = g_{i j l} g_{j k l}
\,.
$$
#### Čech-Deligne cohomology
When refining the complexes of sheaves $U(1)[n]$
to the [[Deligne cohomology|Deligne complex]]
$$
U(1)[n] \hookrightarrow \mathbb{Z}(n+1)_D^\infty
$$
and then evaluating Čech cohomology with coefficients
in the Deligne complex, we obtain the formulas for
Čech-Deligne cohomology.
* For $n = 1$ a cocycle is a collection
$$
( A_i \in \Omega^1(U_i),
g_{i j} \in C^\infty(U_{i j}, U(1)) )
$$
such that for all $i, j$
$$
d log g_{i j} + A_i - A_j - = 0
$$
and for all $i,j ,k$
$$
g_{i j} g_{j k} = g_{i k}
\,.
$$
Such cocycles classify $U(1)$-[[principal bundles]]
with [[connection on a bundle|connection]].
These $n=1$ Čech-Deligne cocycles appear naturally in the study of the [[electromagnetic field]].
* For $n = 2$ a cocycle is a collection
$$
( B_i \in \Omega^2(U_i),
A_{i j} \in \Omega^1(U_{i j}),
g_{i j k} \in C^\infty(U_{i j k}, U(1)) )
$$
such that for all $i, j$
$$
d A_{i j} + B_i - B_j = 0
$$
and for all $i,j ,k$
$$
A_{i j} - A_{i k} + A_{j k} + d log g_{i j k} = 0
$$
and for all $i, j, k ,l$
$$
g_{i j k} g_{i k l} = g_{i j l} g_{j k l}
\,.
$$
Such cocycles classify $U(1)$-[[bundle gerbes]]
with connection.
## Related entries
* For discussion of the isomorphism between isomorphism classes of [[topological vector bundles]] and Čech cohomology on [[topological spaces]] with coefficients in continuous functions with values in the [[general linear group]] see [there](topological+vector+bundle#TransitionFunctionsAndCechCohomology).
## References
Classical accounts (in the generality of [[nonabelian cohomology]]):
* [[Alexander Grothendieck]], *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]
* [[Jean Frenkel]], *Cohomologie à valeurs dans un faisceau non abélien*, C. R. Acad. Se., t. 240 (1955) 2368-2370
* [[Jean Frenkel]], *Cohomologie non abélienne et espaces fibrés*, Bulletin de la Société Mathématique de France, **85** (1957) 135-220 [[numdam:BSMF_1957__85__135_0](http://www.numdam.org/item/BSMF_1957__85__135_0)]
* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]
Textbook account:
* [[Torsten Wedhorn]], *Torsors and Non-abelian Čech Cohomology*, chapter 7 in: *Manifolds, Sheaves, and Cohomology*, Springer (2016) [[doi:10.1007/978-3-658-10633-1](https://doi.org/10.1007/978-3-658-10633-1)]
A historical survey (of some aspects) is in
* [[David Edwards]], Harold M. Hastings, _Cech Theory: Its past, present, and future_, Rocky Mountain J. Math. Volume 10, Number 3 (1980), 429-468. ([Euclid](http://projecteuclid.org/euclid.rmjm/1250128825))
A motivational introduction from within [[complex analytic geometry]] is in
* Michael Weiss, _The Search for a Global Primitive -- Cech Cohomology with Coefficients in a Sheaf_ ([pdf](http://files.meetup.com/4699592/cech-ipad.pdf))
A discussion of Čech cohomology in the wider context of [[cohomology]] particularly realized in terms of the [[model structure on simplicial presheaves]] and with an emphasis on the shades of notions between [[Čech cover]] and [[hypercover]] is
* [[Tibor Beke]], _Higher Čech theory_ ([journal](http://www.math.uiuc.edu/K-theory/0646/), [pdf](http://www.math.uiuc.edu/K-theory/0646/cech.pdf))
Abelian Čech cohomology is discussed in some detail in section I.3 of
* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_
A discussion of the [[model structure on simplicial presheaves]] with an eye towards the distinction between Čech and hyperlocalization is in
* [[Daniel Dugger]], _Sheaves and Homotopy theory_ ([web](http://math.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))
(But beware that, while providing useful insights, these are unfinished abandoned notes that seem to have some gaps right at this point.)
A long list of reasons why higher Čech cohomology might
is after all better behaved than its [[hypercompletion]] where a cycle is with respect to an arbitrary [[hypercover]] is in section 6.5.4, **Descent versus Hyperdescent** of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
The relation between smooth and continuous Čech cohomology is discussed in
* Christoph Müller, [[Christoph Wockel]], _Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups_, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)
[[!redirects {?}ech cohomology]]
[[!redirects ?ech cohomology]]
[[!redirects ?ech cohomology]]
[[!redirects Cech cohomology]] |
Čech cohomology > history | https://ncatlab.org/nlab/source/%C4%8Cech+cohomology+%3E+history | < [[Čech cohomology]] |
Čech cover | https://ncatlab.org/nlab/source/%C4%8Cech+cover | # Čech covers
* table of contents
{: toc}
## Idea
A _Čech cover_ is a [[Čech nerve]] $C(U)$ that comes from a [[cover]] $U \to X$.
## Definition
Let $C$ be a [[site]] and $\{U_i \to X\}$ a covering [[sieve]]. Write $U = \sqcup_i Y(U_i)$ for the [[coproduct]] of the patches in the [[presheaf]] category $PSh(C)$ ($Y$ is the [[Yoneda embedding]]).
Write $X$ for $Y(X)$, for short.
Then the [[Čech nerve]] $C(U)$ of $U \to X$ in $PSh(C)$, i.e. the [[simplicial presheaf]]
$$
\cdots
U \times_X U \times_X U
\stackrel{\stackrel{\to}{\to}}{\to}
U \times_X U \stackrel{\to}{\to} U
$$
is called a **Čech cover**.
## Properties
Consider the [[local model structure on simplicial presheaves]] on $C$. If the [[sheaf]] [[topos]] $Sh(C)$ has [[point of a topos|enough points]], then the weak equivalences (called local weak equivalences for emphasis) are the [[stalk]]-wise weak equivalences of [[simplicial set ]] (with respect to the standard [[model structure on simplicial sets]]).
+-- {: .un_defn}
###### Definition
Write
$$
\pi_0(C(U)) = colim_{[n] \in \Delta^{op}} C(U)_n
$$
for the presheaf of connected components (see [[simplicial homotopy group]]) of $C(U)$. Regard this as a simplicially constant [[simplicial presheaf]].
=--
**Remark**. By the discussion in the section "Interpretation in terms of descent and codescent" at [[sieve]] this $\pi_0(C(U))$, regarded as an ordinary presheaf, is precisely the subfunctor of $Y(X)$ that corresponds to the [[sieve]] $\{U_i \to X\}$.
+-- {: .un_lemma }
###### Lemma
For every Čech cover $C(U)$ the morphism of simplicial presheaves
$$
C(U) \to \pi_0(C(U))
$$
is a local weak equivalence.
=--
+-- {: .proof}
###### Proof
Being a simplicially discrete simplicial sheaf, for every test object $V$ $\pi_0(C(U))(V)$ has all [[simplicial homotopy group]]s trivial except possibly the set of connected components.
But by the very definition of $\pi_0(C(U))$ the morphism
$C(U) \to \pi_0(C(U))$ is a bijection on $\pi_0$.
Over each test domain $V$ the [[simplicial set]]
$C(U)(V)$ is just the [[nerve]] of the [[Cech groupoid|Čech groupoid]]
$$
\left(
C(V,U)\times_{C(V,U \times_X U)}
\stackrel{\to}{\to}
C(V,U)
\right)
\,.
$$
The nerve of that groupoid is readily seen to have vanishing first [[simplicial homotopy group]]. Being the [[nerve]] of a 1-[[groupoid]], also all higher [[simplicial homotopy group]]s vanish.
So $C(U) \to \pi_0(C(U))$ induces for each object $V \in C$ an isomorphism of [[simplicial homotopy group]]s. It therefore is an objectwise weak equivalence of simplicial sets.
See also for instance lemma 3.3.5 in
* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))
=--
+-- {: .un_prop}
###### Proposition
Every Čech cover
$$
C(U) \to X
$$
is a stalkwise weak equivalence.
=--
+-- {: .proof}
###### Proof
From the above we know that $C(U) \to X$ factors as
$$
C(U) \to \pi_0(C(U)) \to X
$$
and that the first morphism is an objectwise, hence also a [[stalk]]wise weak equivalence. It therefore suffices to show that $\pi_0(C(U)) \to X$ is a stalkwise weak equivalence.
But by the remark above, $\pi_0(C(U)) \to X$ is actually the [[local isomorphism]] corresponding to the [[cover]] $U$. It is therefore even a [[stalk]]wise [[isomorphism]].
See also for instance lemma 3.4.9 in
* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))
=--
+-- {: .un_remark}
###### Remark
So this says that every Čech cover is a [[hypercover]]. But not conversely. [[localization|Localization]] of [[simplicial presheaf|simplicial presheaves]] at Čech covers yields [[Čech cohomology]].
=--
[[!redirects Čech cover]]
[[!redirects Čech covers]]
[[!redirects Cech cover]]
[[!redirects Cech covers]]
|
Čech descent | https://ncatlab.org/nlab/source/%C4%8Cech+descent | ## Idea
The **Čech descent** condition is used to refer to a specific [[∞-descent]] condition for [[∞-presheaves]].
Concretely, Čech descent can be implemented by performing the [[left Bousfield localization]] of the [[model category]] of [[simplicial presheaves]] equipped with the [[projective model structure]] (say) at [[Čech nerves]] of [[covering families]].
Then [[fibrant objects]] in the resulting [[model structure]] will be precisely [[presheaves]] of [[Kan complexes]] that satisfy the Čech descent condition.
Čech descent should be contrasted with [[hyperdescent]].
The [[hyperdescent]] condition is stronger than the Čech descent condition, since it uses arbitrary [[hypercovers]] instead of [[Čech nerves]] of [[covering families]].
However, for [[hypercomplete sites]] such as [[Zariski site]], [[Nisnevich site]], [[CartSp]], [[Stein site]], and most of the other sites used in differential and complex geometry, the [[hyperdescent]] condition coincides with the Čech descent condition.
## Related concepts
* [[hyperdescent]]
* [[∞-presheaf]], [[∞-sheaf]], [[simplicial presheaf]]
* [[Čech nerve]] |
Čech groupoid | https://ncatlab.org/nlab/source/%C4%8Cech+groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For $\{U_i \to X\}$ a [[cover]] of a [[space]] $X$, the corresponding **Čech groupoid** is the [[internal groupoid]]
$$
C(\{U_i\}) = (\coprod_{i j} U_i \cap U_j \rightrightarrows \coprod_i U_i)
$$
whose set of objects is the [[disjoint union]] $\coprod_i U_i$ of the covering patches, and the set of morphisms is the disjoint union of the [[intersections]] $U_i \cap U_j$ of these patches.
This is the $2$-[[coskeleton]] of the full [[Čech nerve]]. See there for more details.
If we speak about [[generalized element|generalized points]] of the $U_i$ (which are often just ordinary points, in applications), then
* an [[object]] of $C(\{U_i\})$ is a pair $(x,i)$ where $x$ is a point in $U_i$;
* there is a unique [[morphism]] $(x,i) \to (x,j)$ for all pairs of objects labeled by the same $x$ such that $x \in U_i \cap U_j$;
* hence the composition of morphism is of the form
$$
\array{
&& (x,j)
\\
& \nearrow &=& \searrow
\\
(x,i) &&\to&& (x,k)
}
\,.
$$
## Definition
{#Definition}
+-- {: .num_defn #CechGroupoid}
###### Definition
**(Cech groupoid)**
Let $\mathcal{C}$ be a [[site]], and $X \in \mathcal{C}$ an [[object]] of that site. For each [[covering]] family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given [[coverage]], the _[[Cech groupoid]]_ is the [[presheaf of groupoids]]
$$
C(\{U_i\})
\;\in\;
[\mathcal{C}^{op}, Grpd]
\;\simeq\;
Grpd\left( [\mathcal{C}^{op}, Set] \right)
$$
which, regarded as an [[internal category]] in the [[category of presheaves]] over $\mathcal{C}$, has as [[presheaf]] of [[objects]] the [[coproduct]]
$$
Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod} y(U_i)
$$
of the [[representable presheaf|presheaves represented]] (under the [[Yoneda embedding]]) by the [[covering]] objects $U_i$, and as [[presheaf]] of [[morphisms]] the [[coproduct]] over all [[fiber products]] of these:
$$
Mor_{C(\{U_i\})}
\;\coloneqq\;
\underset{i,j}{\coprod} y(U_i) \times_{y(X)} y(U_j)
\,.
$$
This means that for any $V \in \mathcal{C}$ the [[groupoid]] assigned by $C(\{U_i\})$ has as set of objects [[pairs]] consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects
$$
\kappa_i \longrightarrow \kappa_j
$$
precisely if
\[
\label{CechMatchingCondition}
\iota_i \circ \kappa_i
\;=\;
\iota_j \circ \kappa_j
\phantom{AAAAAAAA}
\array{
&& V
\\
& {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
\\
U_i && && U_j
\\
& {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
\\
&& X
}
\]
=--
## Properties
### Codescent
{#Codescent}
We discuss (Prop. \ref{CechGroupoidCoRepresents} below) how the Cech groupoid co-represents _[[matching families]]_ as they appear in the definition of [[sheaves]].
For reference, we first recall that definition:
+-- {: .num_defn #CompatibleElements}
###### Definition
**([[matching family]] -- [[descent object]])**
Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.
Given an [[object]] $X \in \mathcal{C}$ and a [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it we say that a _[[matching family]]_ (of probes of $\mathbf{Y}$) is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and [[pairs]] of [[morphisms]] $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying
\[
\label{MatchingCondition}
\iota_i \circ \kappa_i
\;=\;
\iota_j \circ \kappa_j
\phantom{AAAAAAAA}
\array{
&& V
\\
& {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
\\
U_i && && U_j
\\
& {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
\\
&& X
}
\]
we have
\[
\label{GluingCondition}
\mathbf{Y}(\kappa_i)(\phi_i)
\;=\;
\mathbf{Y}(\kappa_j)(\phi_j)
\,.
\]
We write
\[
\label{SetOfMatching}
Match\big(
\{U_i\}_{i \in I}
\,,\,
\mathbf{Y}
\big)
\subset
\underset{i}{\prod} \mathbf{Y}(U_i)
\;\in\;
Set
\]
for the set of [[matching families]] for the given presheaf and covering.
This is also called the _[[descent object]]_ of $\mathbf{Y}$ for _[[descent]]_ along the [[covering]] $\{U_i \overset{\iota_i}{\to}X\}$.
=--
+-- {: .num_example #CechGroupoidCoRepresents}
###### Proposition
**([[Cech groupoid]] co-represents [[matching families]] -- [[codescent]])**
For [[Grpd]] regarded as a [[cosmos]] for [[enriched category theory]], via its [[cartesian closed category]]-structure, and $\mathcal{C}$ a [[site]], let
$$
\mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]
$$
be a [[presheaf]] on $\mathcal{C}$, regarded as a [[Grpd]]-[[enriched presheaf]], let $X \in \mathcal{C}$ be any [[object]] and $\{U_i \overset{\iota_i}{\to} X\}_i$ a [[covering]] family with induced [[Cech groupoid]] $C(\{U_i\}_i)$ (Def.\ref{CechGroupoid}).
Then there is an [[isomorphism]]
$$
[\mathcal{C}^{op},Grpd]
\left(
C\left(\{U_i\}_i\right), \, \mathbf{Y}
\right)
\;\simeq\;
Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
$$
between the [[hom-object|hom-groupoid]] of [[Grpd]]-[[enriched presheaves]] and the set of [[matching families]] (Def. \ref{CompatibleElements}).
Since therefore the Cech-groupoid co-represents the [[descent object]], it is sometimes called the _[[codescent object]]_ along the given covering.
Moreover, under this identification the canonical morphism $C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} X$ induces the comparison morphism
$$
\array{
[\mathcal{C}^{op}, Grpd]\left( X, \, \mathbf{Y} \right)
& \simeq &
\mathbf{Y}(X)
\\
{}^{
\mathllap{
[\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y})
}
}\downarrow && \downarrow
\\
[\mathcal{C}^{op},Grpd]
\left(
C\left(\{U_i\}_i\right), \, \mathbf{Y}
\right)
&\simeq&
Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
}
\,.
$$
In conclusion, this means that the [[presheaf]] $\mathbf{Y}$ is a [[sheaf]] (Def. \ref{Sheaf}) precisely if homming Cech groupoid projections into it produces an isomorphism.
=--
+-- {: .proof}
###### Proof
The hom-groupoid is computed as the [[end]]
$$
[\mathcal{C}^{op},Grpd]
\left(
C\left(\{U_i\}_i\right), \, \mathbf{Y}
\right)
\;=\;
\int_{V \in \mathcal{C}}
\left[
C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
\right]
\,,
$$
where the "integrand" is the [[functor category]] (here: a [[groupoid]]) from the [[Cech groupoid]] at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.
Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the [[functors]] $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those [[functions]] on sets of objects
$$
\underset{i}{\coprod} y(U_i)(V)
=
Obj_{C\left(\{U_i\}_i\right)(V)}
\longrightarrow
Obj_{\mathbf{Y}(V)}
=
\mathbf{Y}(V)
$$
which respect the [[equivalence relation]] induced by the morphisms in the Cech groupoid at $V$.
Hence the hom-groupoid is a subset of the [[end]] of these [[function sets]]:
$$
\begin{aligned}
\int_{V \in \mathcal{C}}
\left[
C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
\right]
& \hookrightarrow
\int_{V \in \mathcal{C}}
\left[
\underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V)
\right]
\\
& \simeq
\int_{V \in \mathcal{C}}
\underset{i}{\prod}
\left[
y(U_i)(V), \, \mathbf{Y}(V)
\right]
\\
& \simeq
\underset{i}{\prod}
\int_{V \in \mathcal{C}}
\left[
y(U_i)(V), \, \mathbf{Y}(V)
\right]
\\
& \simeq
\underset{i}{\prod}
\mathbf{Y}(U_i)
\end{aligned}
$$
Here we used: first that the [[internal hom]]-functor turns colimits in its first argument into limits (see at _[[internal hom-functor preserves limits]]_), then that [[limits commute with limits]], hence that in particular [[ends]] commute with [[products]] , and finally the [[enriched Yoneda lemma]], which here comes down to just the plain [[Yoneda lemma]] . The end result is hence the same [[Cartesian product]] set that also the set of matching families is defined to be a subset of, in (eq:SetOfMatching).
This shows that an element in
$ \int_{V \in \mathcal{C}}
\left[
C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
\right]
$ is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment
$$
\array{
C\left(\{U_i\}_i\right)(V)
& \longrightarrow &
\mathbf{Y}(V)
\\
(V \overset{\kappa_i}{\to} U_i)
&\mapsto&
\kappa_i^\ast \phi_i
=
\mathbf{Y}(\kappa_i)(\phi_i)
}
$$
constitutes a [[functor]] of [[groupoids]].
By definition of the [[Cech groupoid]], and since the [[codomain]] is a just [[set]] regarded as a [[groupoid]], this is the case precisely if
$$
\mathbf{Y}(\kappa_i)(\phi_i)
\;=\;
\mathbf{Y}(\kappa_j)(\phi_j)
\phantom{AAAA}
\text{for all}\, i,j
\,,
$$
which is exactly the condition (eq:GluingCondition) that makes $(\phi_i)_i$ a matching family.
=--
## Examples
For $X$ a [[smooth manifold]] and $\{U_i \to X\}$ an [[atlas]] by [[coordinate chart]]s, the Čech groupoid is a [[Lie groupoid]] which is equivalent to $X$ as a Lie groupoid:
$C(\{U_i\}) \stackrel{\simeq}{\to} X$
For $\mathbf{B}G$ the Lie groupoid with one object coming from a [[Lie group]] $G$ morphisms of Lie groupoids of the form
$$
\array{
C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G
\\
\downarrow^{\mathrlap{\simeq}}
\\
X
}
$$
are also called [[anafunctor]]s from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-[[principal bundle]]s on $X$.
## Related concepts
* [[Cech nerve]]
## References
For instance
* {#Jardine06} [[J. F. Jardine]], Example 5 in: _Homotopy classification of gerbes_ ([arXiv:math/0605200](https://arxiv.org/abs/math/0605200))
[[!redirects Cech groupoid]]
[[!redirects Cech groupoids]]
[[!redirects Čech groupoids]]
|
Čech homology | https://ncatlab.org/nlab/source/%C4%8Cech+homology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
#Čech homology#
* table of contents
{:toc}
## Idea
Given a [[topological space]] $X$, its *Čech homology* is the [[limit]] of [[simplicial homology]]-[[homology group|groups]] of the [[Čech nerves]] of its [[open covers]], under [[refinement of open covers]].
Compare to the [[singular homology]] of $X$, which is the [[simplicial homology]] of its [[singular simplicial complex]]. For well behaved topological spaces the two notions agree and are jointly known as computing the *[[ordinary homology]]* of $X$.
## Definition
Given a [[topological space]] $X$ with an [[open cover]],
$$
\mathcal{U}
\;=\;
\Big\{
U_i \xhookrightarrow{open} X
\Big\}_{i \in I}
$$
we write
\[
\label{CechNerve}
C(X,\mathcal{U})
\;\coloneqq\;
\pi_0
\big(
U^{\times^\bullet_X}
\big)
\;\in\;
sSet
\]
for the [[simplicial set]] which is its [Čech+nerve](Čech+nerve#FromACover):
whose $n$-[[simplex|simplices]] are the [[inhabited set|inhabited]] $(n + 1)$-fold [[intersections]] of the [[open subsets]] $U_i$ in $\mathcal{U}$.
\begin{remark}
For sufficiently well-behaved topological spaces $X$ ([[paracompact spaces]]) and *[[good open cover|good]]* open covers, the Čech nerve (eq:CechNerve) is [[simplicial homotopy equivalence|simplicially homotopy equivalent]] to the [[singular simplicial complex]] of $X$ -- this is the statement of the *[[nerve theorem]]*.
However, in general $C(X,\mathcal{U})$ may differ from the [[weak homotopy type]] of $X$ (even in the [[limit]] below) in which case Čech homology may *differ* from the [[singular homology]] of $X$. (See for instance the discussion at *[[well group]]*.)
\end{remark}
If $\mathcal{U}$ is a [[refinement of an open cover|refinement]] open cover, i.e. such that for each $U' \in \mathcal{U}$, there is a $U \in \mathcal{U}$ with $U' \subseteq U$, then these inclusions induce a [[morphism]] of [[simplicial sets]]
$$
C(X,\mathcal{U}) \longrightarrow C(X,\mathcal{U}')
$$
This yields an [[inverse system]] of [[simplicial sets]].
\begin{definition}
The $n$th **Čech homology group** of the space $X$ is the [[limit]]
$$
\check{H}_n(X)
\;\coloneqq\;
\underset{
\underset{\mathcal{U}}{\longleftarrow}
}{lim}
H_n(X,\,\mathcal{U})
$$
over the [[inverse system]] of [[open covers]] $\alpha$, of the [[simplicial homology]] [[homology group|groups]] of the [Čech nerve](Čech+nerve#FromACover) $C(X,\alpha)$:
$$
H_n(X,\,\mathcal{U})
\;\coloneqq\;
H_n\big(C(X,\,\mathcal{U})\big)
\,.
$$
\end{definition}
\begin{remark}
It is to be noted that these groups do not constitute a [[homology theory]] in the sense of the [[Eilenberg-Steenrod axioms]] as the _exactness axiom_ fails in general. There is a "corrected" theory known under the name [[strong homology]].
\end{remark}
## Related concepts
* [[Čech methods]], [[nerve theorem]]
* [[singular homology]], [[cellular homology]]
* [[ordinary homology]]
* [[étale homotopy type]]
* [[well group]]
## References
Exposition:
* [[David H. Fremlin]], Section 2 of: _Singular homology for amateurs_ (2016) $[$[pdf](https://www1.essex.ac.uk/maths/people/fremlin/n16703.pdf), [[FremlinSIngularHomology.pdf:file]]$]$
On [[closed covers]] in Čech homology:
* [[Edwin E. Floyd]], *Closed coverings in Čech homology theory*, Trans. Amer. Math. Soc. **84** (1957) 319-337 [[doi:10.1090/S0002-9947-1957-0087100-2](https://doi.org/10.1090/S0002-9947-1957-0087100-2), [pdf](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0087100-2/S0002-9947-1957-0087100-2.pdf)]
[[!redirects Cech homology]]
[[!redirects Čech homology]]
[[!redirects Čech homology theory]]
[[!redirects Cech homology]] |
Čech homotopy | https://ncatlab.org/nlab/source/%C4%8Cech+homotopy | # Čech homotopy
* table of contents
{: toc}
##Idea and discussion##
In Čech cohomology, in its *traditional* form, you resolve the space by an inverse system or [[pro-object]], $\check{C}(X,-)$, of nerves of open covers (or if you prefer the nerves of the corresponding sheaf of groupoids). For constant and Abelian coefficients (and we will keep to those for the moment), $A$, we throw $H^n(-,A)$ at this pro-object. As cohomology switches 'variance' we get a _direct system_ or _ind-object_ of Abelian groups, and usually we then take its (direct) limit (i.e. its colimit) to get $\check{H}^n(X,A)$.
Now dualise ... . Replace cohomology by homotopy $[A,-]$, but although there will be some very useful long exact (Puppe) sequences available in homotopy, we cannot go this way to get them! In dualising, things can go slightly 'wrong'. We have $[A,\check{C}(X,-)]$ is a pro-object and so we might want to take limits not colimits to get the thing we want, but $lim$ is not an exact functor on the usual categories available here, such as groups, sets, modules, etc.
There is another problem, coherence! $\check{C}(X,-)$ is a pro-object in the homotopy category of simplicial sets, not in the category of simplicial sets itself. It is a homotopy coherent system, but the choices of covering refinement maps cause the problem.
We are very near the topics discussed in [[shape theory]], [[Čech methods]] and [[strong shape theory]] (when that has been written!).
One interpretation of Čech homotopy is [[shape theory]], but that does not usually handle the 'homotopy is dual to cohomology' idea so this will need more work here.
##Handling the problem of coherence##
###Giving homotopy coherence directly:###
As explained in [[Čech methods]], given an open cover $\beta$ of a space $X$ and a refinement $\alpha$ of $U$
for a given set in $V$ there may be many different sets in $U$ containing it, so the seemingly natural way to get a simplicial map
$$\check{C}(X,\alpha)\to \check{C}(X,\beta)$$
involves a choice. We choose a 'refinement mapping' $\varphi : \alpha \to \beta$ so that
for each $V\in \alpha$, $V\subseteq \varphi(V)$. There is an obvious extension to $n$-simplices. We send $\langle V_0, \ldots, V_n\rangle$ to $\langle \varphi(V)_0, \ldots, \varphi(V)_n\rangle$. This still does not get us a functor from $Cov(X)$ to $SSets$. The fact that we had to choose make it unlikely in the extreme that given three (or more) covers, each a refinement of the next, that the resulting diagram of simplicial sets will turn out (by some miracle) to be commutative. However once we choose a refinement maps for each pair $(\alpha,\beta)$, we _do_ get a [[homotopy coherent diagram]]
$$\check{C}(X,-) : Cov(X)\to SSets.$$
with the coherence being able to be spelled out completely, (see _Coverings, Hypercoverings and Homotopy Coherence_, thesis (1999) from Bangor, by Alinor Abdul Kadir, available [here](http://www.maths.bangor.ac.uk/research/ftp/theses/kadir.pdf)).
###Using the Vietoris complex: ###
The [[Vietoris complex]] is an alternative way of obtaining a simplicial complex (or simplicial set) from a space $X$ and an open cover $\alpha$. The disadvantage of it is that whilst the nerve of the open cover would be a finite simplicial set if the open cover is finite (for instance this is useful when handling compact spaces as we can replace the indexing set of _all_ open covers by that of _finite open covers_ without changing the result up to isomorphism), the Vietoris complex has as many vertices as there are points in $X$. It has the advantage that the induced maps between the simplicial complexes are independent of the choice of refinement maps so we do get a pro-simplicial set, $V(X,-) : Cov(X)\to SSet$. We therefore get a well defined [[pro-homotopy type]] $V(X)$ in the homotopy category $Ho(pro-SSet)$. (The category $pro-SSet$ has various [[model category]] structures and [[cofibration category]] structures, with related basic notions of homotopy equivalence. These are discussed in the entry on [[pro-homotopy theory]].)
How are the two ways around the coherence problem related?
There is a result [[Dowker's Theorem]] which shows that the geometric realisations of $V(X,\alpha)$ and $\check{C}(X,\alpha)$ are homotopy equivalent. (This explains why Čech and Vietoris homology are isomorphic.) Given a commutative diagram of spaces (or simplicial sets), if you change some of the
spaces to homotopy equivalent spaces then you can build from those new spaces, a homotopy coherent diagram equivalent to the original one. Using Dowker's theorem we get that $\check{C}(X,-)$ is a homotopy coherent diagram, (by a second route).
## Homotopy groups or homotopy progroups?##
One obvious case to investigate of the above is to take a pointed space $X$, apply the $n^th$ homotopy group functor $\pi_n = [S^n, -]$ to the Čech or Vietoris complex to get a progroup, $\pi_n(\check{C}(X))$. This will lead to long exact sequences for pairs, fibrations etc., but these will be of progroups so are less amenable to study. If, on the other hand, you take the limit of these progroups, as mentioned above, you destroy the exactness of the sequences and thus, to a large extent, the simplicity of their use.
### Definition
The **$n^{th}$ Čech homotopy group** of a pointed space $X$ is defined to be $\check{\pi_n}(\check{C}(X)) = lim \pi_n(\check{C}(X))$.
### Examples
If $X$ is compact metric then it has countable cofinal subcategories of covers, so the progroups
$\pi_n(\check{C}(X))$ can be replaced up to isomorphism by inverse sequences, indexed by the natural numbers. (These are sometimes called *towers*.)
Here are two specific examples:
1. The [[Warsaw circle]], $S_W$: The progroup $\pi_1(\check{C}(S_W))$ is isomorphic to the constant tower with value $C_\infty$, the bonding maps being the identity maps.
1. The [[dyadic solenoid]]: This time the progroup is isomorphic in $Pro(Groups)$ to the inverse sequence
$$C_\infty\stackrel{a}{\leftarrow} C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}\ldots $$
where if $c$ is the generator of $C_\infty$, $a(c) = c^2$.
This second example is certainly not isomorphic to a constant one. Note that it has trivial limit group, so is it possible to get some 'useful' limiting information from the progroup by taking some other type of 'limiting' approach? Yes. There are two related ways.
* **The [[derived functors of lim]]**. Applied to the above progroup you get a definite measure of the non-triviality of the progroup. (This example is in that entry.)
* (A second method will be put here but will need additional work over on [[proper homotopy theory]].)
+--{: .query}
That is a start on it anyhow!
=--
[[!redirects Čech homotopy]]
[[!redirects Cech homotopy]]
|
Čech methods | https://ncatlab.org/nlab/source/%C4%8Cech+methods |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
The construction of Čech homology used coverings of the space by families of open sets. The way the open sets overlap gives 'combinatorial' information on the space.
This has been abstracted and extended to [[homotopy]] and also in the direction of algebraic geometry. This article aims to introduce some of the main ways this idea has interacted with some of the many themes in the nLab.
##History (Čech 1930s)##
The term '[[Čech nerve|nerve of a covering]]' has been used from the late
1920s for a construction that starts with a space and an open cover of it and builds a [[simplicial complex]].
Historically these data were organized as a [[simplicial complex]], rather than a [[simplicial set]]. The idea had been independently worked out by [[Eduard Cech|Čech]] and [[P. S. Aleksandrov]]
so one often talks about Aleksandrov--Čech (co)homology. There was an alternative approach described by Vietoris (1927), which led to what is known as [[Vietoris homology]] There are separate entries in nlab for [[Cech homology|Čech homology]]
and [[Čech cohomology]]. The first is not exact, and so is not really
a homology theory in the sense of Steenrod--Eilenberg axioms. It is also not the (Alexander--Spanier) dual of Čech cohomology. The two problems can be avoided at the same time using coherent homotopy theory. The resulting homology is one which is exact. It also is one that satisfies the wedge axiom, and then it is the unique such theory, nowadays sometimes called [[strong homology]] and in special cases it reduces to what was called [[Steenrod-Sitnikov homology]] (but which was originally constructed in a different way).
##Nerve of an open cover##
+--{: .un_defn}
######Definition######
([[Pavel Sergeevič Aleksandrov]], 1926)
Given a (compact) space $X$ and a (finite) open cover, $\alpha$, of $X$, we can
form a [[simplicial set]], $C(X,\alpha)$, called the *nerve of the cover* whose $n$-[[simplex|simplices]] are $(n + 1)$-strings
of open sets from $\alpha$, i.e. $\langle U_0, \ldots, U_n\rangle$, each $U_i \in \alpha$,
satisfying $\cap U_i \neq \emptyset$.
If $\beta$ is another cover such that for each $V\in \beta$, there is a $U\in
\alpha$ with $V\subseteq U$, then the assignment $V\to U$ in this case defines
a map
$$C(X,\alpha)\to C(X,\beta)$$
dependent on the choice of $U$ for each
$V$, but independent 'up to homotopy' (see below).
=--
##Application of Čech methods##
Applying one's favourite homotopy
functor, $F : \mathbf{S}\to \mathbf{A}$, to each $C(X,\alpha)$ and to these homotopy
classes of induced transition maps, yields an inverse system of objects in
$\mathbf{A}$, i.e., a
[[pro-object]] in $\mathbf{A}$, but the $C(X,\alpha)$ by themselves do _not_ define a pro-object in the category of simplicial sets. They do form a pro-object in $Ho(sSets)$, or alternative, a 'coherent pro-object' in $sSets$, (see [[Čech homotopy]] for more discussion on this point.)
### Čech homology and cohomology ###
Aleksandrov and Čech in the 1930s, applied
homology and cohomology in this situation to extend simplicially-based
homology to a much wider class of spaces. Lefschetz, without the language of
category theory, studied, again in the 1930s, the formal properties of inverse
systems of polyhedra and maps between them and his student Christie looked at the homotopy groups in this setting.
* see [[Čech cohomology]] and [[Čech homotopy]] for more on this.
### Shape theory as a Čech homotopy theory ###
Borsuk (1968) developed [[shape theory]], which although initially very geometric in flavour turned out also to be described in terms of Christie's
theory of the "Čech extensions" of homotopy theory.
The first papers on this approach were by [[Tim Porter]]. They used the [[Vietoris complex]] of a space relative to an open cover as well as the [[Čech nerve]] itself. (By [[Dowker's Theorem]] the two complexes give the same information up to homotopy, but the [[Vietoris complex]] is a functor on the category of covers, having values in the category of simplicial sets, whilst the Čech complex does not give so nice a structure as most naturally it takes values in the homotopy category of simplicial sets. In fact, that functor can be rigidified but this requires a certain amount of work.)
Porter also gave a partial solution to the problem of the stability of a space, that is whether it has the same Čech homotopy as a polyhedron. The methods developed for that problem then suggested the existence of a homotopy theory for pro-simplicial sets related to homotopy coherence. That observation was also made by Edwards and Hastings, who proposed a strong shape theory based on what they termed [[Steenrod homotopy]]. This relates to [[Alexander-Čech duality|Steenrod-Sitnikov homology]].
(See [[shape theory]] and [[Čech homotopy]] for references and more on this area).
### Čech-like methods in algebraic geometry ###
The use of modified Čech methods in algebraic geometry was well
established when Grothendieck and his collaborators in Paris started adapting
it to work with a [[Grothendieck topology]]. Verdier in SGA4,
introduced hypercoverings and in Artin and Mazur (SLN 100), you can find their use
in homotopy theory. (The work of Lubkin (1967) should also be
mentioned here as it contains much that is parallel to the development by
Verdier, Artin and Mazur and is sometimes much easier to decipher for the
non-specialist algebraic geometer. A summary of his construction is given under [[Lubkin's construction]].)
Given that a [[Grothendieck topology]] is essentially about abstracting a notion
of '[[cover|covering]]', it is not surprising that
modified Čech methods can be applied. Artin and Mazur
used Verdier's idea of a [[hypercover|hypercovering]] to get, for each Grothendieck topos,
$\mathbb{E}$, a pro-object in $Ho(\mathbf{S})$ (i.e. an inverse system of
simplicial sets), which they call the étale [[homotopy type]] of the topos
$\mathbb{E}$ (which for them is 'sheaves for the étale topology on a
variety').
Applying homotopy group functors gives pro-groups $\pi_i(\mathbb{E})$
such that $\pi_1(\mathbb{E})$ is essentially the same as Grothendieck's [[algebraic fundamental group]],
$\pi_1(\mathbb{E})$. (Here you need to know that the category of [[profinite
group]]s in the topological sense, is equivalent to the category of [[pro-object]]s in the category of
finite groups (as explained in [[profinite group]]). (If you remove
'finite' the result does not hold, but you can recover it in part by
working with "pro-discrete [[localic groups]]" instead of topological groups,
i.e. take limits of finite groups within the category of '[[locale|localic]]' rather than
'[[topological group|topological]]' groups, remembering that 'locales' are almost 'spaces without
points'.
Grothendieck's nice $\pi_1$ has thus an interpretation as a formal limit of a Čech
type, or shape theoretic, system of $\pi_1$s of 'hypercoverings'.
Can [[shape
theory]] (or its more powerfully structured 'strong' or 'homotopy coherent' version,
cf. Lisica and Mardešić, Edwards and Hastings, Porter
(papers 1976--78) be useful for studying étale homotopy
type? Not without extra work, since the Artin--Mazur--Verdier approach leads
one to look at inverse systems in $proHo(\mathbf{S})$, i.e. inverse systems
(diagrams) in a homotopy category not a homotopy category of inverse systems
as in [[strong shape theory]]. Attempts to 'rigidify' the hypercovering approach,
so as to get into $Hopro(\mathbf{S})$ have been made (e.g. by Lubkin, (1967)
or using simplicial schemes in Friedlander, (1982),
but it is not completely clear if one of them is
the definitive method.
+-- {: .query}
What is the consensus on this here? I have sometimes seen talks which claim to have THE method but none has been clearly THE ONE. Perhaps I did not go to the right conferences! (Some is needed here, e.g. for more ways of getting around the difficulty.)
=--
One of the difficulties with this hypercovering approach is that
'hypercovering' is a difficult concept and to the 'non-expert' seem
non-geometric and lacking in intuition. Thankfully for us, there is an
alternative approach put forward by Ken Brown (1973), (see [[BrownAHT]]).
As the Grothendieck topos $\mathbb{E}$ 'pretends to be' the category of
$Sets$, but with a strange logic, we can 'do' simplicial set theory in
$Simp(\mathbb{E})$ as long as we take care of the arguments we use. To see a
bit of this in action we can note that the object $[0]$ in $Simp(\mathbb{E})$
will be the constant simplicial sheaf with value the ordinary $[0]$,
"constant" here taking on two meanings at the same time,
(a) constant sheaf, i.e. not varying 'over $X$' if $\mathbb{E}$ is thought of as $Sh(X)$, and
(b)
constant simplicial object, i.e. each $K_n$ is the same and all face and
degeneracy maps are identities. Thus $[0]$ interpreted as an étale space
is the identity map $X\to X$ as a space over $X$. Of course not all
simplicial objects are constant and so $Simp(\mathbb{E})$ can store a lot of
information about the space (or site) $X$.
One can look at the homotopy
structure of $Simp(\mathbb{E})$. Ken Brown in [[BrownAHT]] showed it had a
fibration category structure and if we look at
those fibrant objects $K$ in which the natural map
$$p : K \to [0]$$
is a weak
equivalence, we find that these $K$ are _exactly_ the [[hypercover|hypercoverings]]. Global
sections of $p$ give a simplicial set, $\Gamma(K)$ and varying $K$ amongst the
hypercoverings gives a pro-simplicial set (still in $proHo(\mathbf{S})$ not in
$Hopro(\mathbf{S})$ unfortunately) which determines the Artin--Mazur
[[etale homotopy type|pro-homotopy type]] of $\mathbb{E}$
###Links with derived category theory###
This makes it clear there is a link between Čech methods and [[derived category]]
theory. In the first, the 'space' is resolved using '[[cover]]ings'
and these, in a [[sheaf]] theoretic setting, lead to [[simplicial object]]s in $Sh(X)$
that are weakly equivalent to $[0]$; in the second, to evaluate the
[[derived functor]] of some functor $F : \mathbf{C} \to \mathbf{A}$, say, on an object $C$,
one takes the 'average' of the values of $F$ on objects weakly equivalent to
$G$, i.e. one works with the functor
$$F' : \mathbf{W}(C) \to \mathbf{A}$$
(where $ \mathbf{W}(C)$ has objects, $\alpha : C \to C'$, $\alpha$ a
weak equivalence, and maps, the commuting 'triangles', and this has a 'domain'
functor $\delta :\mathbf{W}(C) \to \mathbf{C}$, $\delta(\alpha) = C'$
and $F'$ is the composite $F\delta$).
This is in many cases a pro-object
in $\mathbf{A}$ -- unfortunately standard derived functor theory interprets
'commuting triangles' in too weak a sense and thus corresponds to shape rather
than strong shape theory -- one thus, in some sense, arrives in $pro
Ho(\mathbf{A})$ instead of in $Ho(pro \mathbf{A})$. This problem has been in part resolved with Grothendieck's theory of _Derivateurs_ (see the volume by Maltsiniotis.)
* see also [[abelian sheaf cohomology]]
## Related entries
* [[Čech nerve]]
* [[Čech homology]], [[Čech cohomology]]
* [[nerve theorem]]
*(...)
[[!redirects Čech methods]]
[[!redirects Čech method]]
[[!redirects Cech method]]
[[!redirects Cech methods]]
|
Čech model structure on simplicial presheaves | https://ncatlab.org/nlab/source/%C4%8Cech+model+structure+on+simplicial+presheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Čech model structure on simplicial presheaves** on a [[site]] $C$ is a model for the [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]].
It is obtained from the the [[global model structure on simplicial presheaves]] on $C$ by [[Bousfield localization of model categories|left Bousfield localization]]s at [[Cech cover]]s: its fibrant objects are [[∞-stack]]s that satisfy [[descent]] over [[Cech cover]]s but not necessarily over [[hypercover]]s.
Further [[Bousfield localization of model categories|left Bousfield localization]] at [[hypercover]]s leads from the Čech model structure to the Joyal-Jardine [[local model structure on simplicial presheaves]] that presents the [[hypercomplete (∞,1)-topos]] which is the [[hypercompletion]] of that presented by the Čech model structure.
## Definition
Let $C$ be a small [[site]] and write $[C^{op}, sSet]_{proj}$ and $[C^{op}, sSet]_{inj}$ for the projective and injective [[global model structure on simplicial presheaves]], respectively.
For $\{U_i \to V\}_i$ a covering family in the [[site]] $C$, let
$$
C(\{U_i\})
:=
\left(
\cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i
\right)
$$
be the corresponding [[Cech nerve]], regarded as a [[simplicial presheaf]] on $C$. This comes canonically with a morphism
$$
C(\{U_i\}) \to V
$$
of simplicial presheaves, the corresponding _Čech cover morphism_ .
Notice that by the discussion at [model structure on simplicial presheaves - fibrant and cofibrant objects](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#FibAndCofibObjects) this is a morphism between cofibrant objects.
+-- {: .un_defn}
###### Definition
The injective (projective) **Čech model structure on simplicial presheaves** $[C^{op},sSet]_{Cech}$ on $C$ is the [[Bousfield localization of model categories|left Bousfield localization]] of $[C^{op}, sSet]_{inj}$ ($[C^{op}, sSet]_{proj}$) at the set of Čech cover morphisms.
=--
By the general poperties of [[Bousfield localization of model categories|Bousfield localization]] this means that the fibrant-cofibrant objects $A$ of $[C^{op},sSet]_{Cech}$ are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the [[descent]] condition that for all covers $\{U_i \to V\}$ the morphism of simplicial sets
$$
A(U) \simeq [C^op,sSet](V,U) \to [C^{op},sSet](C(\{U_i\}), A)
$$
is a weak equivalence in the standard [[model structure on simplicial sets]].
This is the model for the $\infty$-analog of the [[sheaf]] condition, modelling the [[topological localization]] of an $(\infty,1)$-presheaf $(\infty,1)$-topos.
+--{: .query}
[[Mike Shulman]]: Two questions, one (hopefully) easy and one (perhaps) hard:
1. Is there a Quillen equivalent Čech model structure on simplicial *sheaves*? Can we just lift the model structure for simplicial presheaves along the sheafification adjunction?
1. Is there a characterization of the weak equivalences in either Čech model structure?
I am particularly interested in this for the following reason. According to Beke in *Sheafifiable homotopy model categories*, the weak equivalences in the [[local model structure on simplicial sheaves]] are precisely those maps $f\colon X\to Y$ of simplicial objects in the corresponding 1-topos of sheaves of sets such that the statement "$f$ is a weak equivalence of simplicial sets" is true in the [[internal logic]] of the topos (at least, interperiting "$f$ is a weak equivalence of simplicial sets" by one particular set of geometric sentences whose interpretation in $Set$ is equivalent to saying that a simplicial map is a weak equivalence). But if, as [[HTT]] teaches us, Čech descent is often to be preferred to hyperdescent, then we should be interested in Čech weak equivalences instead. So I would really like to know what it means for a map of simplicial sheaves to be a Čech weak equivalence, in the *internal logic* of the 1-topos of sheaves of sets. If nothing else, I think such a characterization would help me understand the real meaning of [[hypercompletion]]. But any sort of characterization of them would be better than none.
[[Urs Schreiber]]: below is a reply to the first question.
[[Mike Shulman]]: Thanks for attacking this. I thought I should also mention, for anyone listening in, that this question is evidently also relevant to what the correct notion of [[internal ∞-groupoid]] may be.
=--
> check
We may form the [[transferred model structure]] on simplicial _[[sheaf|sheaves]]_ by transferring along the degreewise [[sheafification]] [[adjunction]]
$$
Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}}
PSh(C)
\,.
$$
This defines fibrations and weak equivalences in $sSh(C)$ to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in $sPSh(C)_{Cech} = [C^{op},sSet]_{Cech}$.
As discussed there, sufficient conditions for this to be a model structure is that
* the inclusion $Sh(C) \hookrightarrow PSh(C)$ preserves [[filtered colimit]]s;
* $sSh(C)$ has functorial fibrant replacement and functorial [[path object]]s for fibrant objects.
Since [[sheafification]] does preserve [[filtered colimit]]s the first condition is satisfied degreewise and hence is satisfied.
+--{: .query}
[[Mike Shulman]]: I believe that sheafification preserves $\kappa$-filtered colimits for some sufficiently large $\kappa$, but if the site has covers of infinite cardinality, I don't see why sheafification would preserve $\omega$-filtered colimits. But I think this is enough for the proof to work.
=--
Since the [[small object argument]] holds in $sSh(C)$ for generating acyclic cofibrations we have functorial fibrant replacement. And a path object is obtained just by forming objectwise the standard path object in [[sSet]], as in $[C^{op}, sSet]$.
+--{: .query}
[[Mike Shulman]]: The small object argument doesn't automatically produce functorial fibrant replacements in this context... isn't the whole question whether the map to the "fibrant replacement" is still a weak equivalence (in the underlying category)? I.e. whether $F(J)$-cell complexes are still weak equivalences.
=--
## References
A detailed though unfinished account of the Čech model structure is given in
* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))
But beware of this document is unfinished. Some aspects of this appeared in
* [[Daniel Dugger]], _[[DuggerUniv.pdf:file]]_
[[!redirects ?ech model structure on simplicial presheaves]]
[[!redirects Cech model structure on simplicial presheaves]]
[[!redirects ?ech model structure on simplicial presheaves]]
|
Čech model structure on simplicial sheaves | https://ncatlab.org/nlab/source/%C4%8Cech+model+structure+on+simplicial+sheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Čech model structure on simplicial sheaves** on a [[site]] $C$ is a model by [[simplicial sheaves]] for the [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]].
It is obtained from the the [[Čech model structure on simplicial presheaves]] on $C$ by [[transferred model structure|transfer]] along the [[sheafification]] [[adjunction]].
Further [[Bousfield localization of model categories|left Bousfield localization]] at "internal" weak homotopy equivalences leads from the Čech model structure to the [[model structure on simplicial sheaves]] that presents the [[hypercomplete (∞,1)-topos]] which is the [[hypercompletion]] of that presented by the Čech model structure.
## Definition
Let $C$ be a small [[site]], let $sSh (C)$ be the [[category]] of [[simplicial sheaves]] on $C$, and write $[C^{op}, sSet]_{\check{C},proj}$ and $[C^{op}, sSet]_{\check{C},inj}$ for the projective and injective [[Čech model structure on simplicial presheaves]], respectively.
+-- {: .num_defn}
###### Definitions
The injective **Čech model structure on simplicial sheaves** on $C$ is the unique model structure on $sSh (C)$ with the following properties:
* The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},inj}$.
* The cofibrations are the monomorphisms.
* The fibrations are the morphisms in $sSh (C)$ that are fibrations in $[C^{op}, sSet]_{\check{C},inj}$.
=--
+-- {: .num_defn}
###### Definition
The projective **Čech model structure on simplicial sheaves** on $C$ is the unique [[model category]] structure on $sSh (C)$ with the following properties:
* The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},proj}$.
* The trivial fibrations are the morphisms in $sSh (C)$ that are trivial fibrations in the projective Čech model structure on $[C^{op}, sSet]_{\check{C},proj}$, i.e. the componentwise trivial Kan fibrations.
=--
## Constructions
To construct the injective Čech model structure on $sSh (C)$, we use the following facts:
* The inclusion $sSh (C) \hookrightarrow [C^{op}, sSet]$ is fully faithful.
* The left adjoint (i.e. sheafification) $[C^{op}, sSet] \to sSh (C)$ preserves monomorphisms, and the class of monomorphisms in $sSh (C)$ is closed under pushouts, transfinite composition, and retracts.
* The adjunction unit is a natural weak equivalence with respect to the (injective) Čech model structure on $[C^{op}, sSet]$: see Theorem A.2 in [DHI](#DuggerHollanderIsaksen02).
We may then apply Kan's recognition principle for [[cofibrantly generated model category|cofibrantly generated model structures]] to transfer the injective Čech model structure from $[C^{op}, sSet]$ to $sSh (C)$. By construction, the sheafification adjunction becomes a Quillen equivalence.
On the other hand, to construct the projective Čech model structure on $sSh (C)$, we use Smith's recognition principle for [[combinatorial model category|combinatorial model structures]] and build it like a [[mixed model structure]].
## References
* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], [_Hypercovers and simplicial presheaves_](http://dx.doi.org/10.1017/S0305004103007175) ([arXiv:math/020502](http://arxiv.org/abs/math/0205027)).
[[!redirects Cech model structure on simplicial sheaves]] |
Čech nerve | https://ncatlab.org/nlab/source/%C4%8Cech+nerve |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
# Čech nerve
* table of contents
{:toc}
## Definition
### In category theory
+-- {: .num_defn}
###### Definition
In a [[category]] $C$ with [[pullbacks]] (possibly [[homotopy pullbacks]]), given a [[morphism]] $U \to X$ in $C$ its corresponding **Čech nerve** $C(U)$ is the [[simplicial object]] in $C$ that in degree $k$ is given by the $(k+1)$-fold [[fiber product]] of $U$ over $X$ with itself :
$$
C(U) \coloneqq
\left(
\cdots
U \times_X U \times_X U
\overset{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U
\right)
\,.
$$
=--
+-- {: .num_remark}
###### Remark
This is the [[internal nerve]] of the [[internal groupoid]] corresponding to the [[kernel pair]] of the morphism $U \to X$.
=--
### In $(\infty,1)$-category theory
The notion of Čech nerve makes sense in any [[(∞,1)-category]] with [[limit in a quasi-category|(∞,1)-pullbacks]].
Consider a morphism $f: A \rightarrow B$ in an $(\infty,1)$-category $M$. The Cech nerve of that morphism is, by definition, the [[augmented simplicial object]] in $M$ obtained by the [[right Kan extension]] of that morphism (regarded as a functor from the obvious category into $M$) along the inclusion of the first two objects into the augmented simplicial category. See Lurie HTT 6.1.2.11. One can show that this gives a [[groupoid object]] in $M$. The Cech nerve is essentially unique by uniqueness of adjoint $\infty-$ functors.
See [[groupoid object in an (∞,1)-category]].
## Applications and occurences
* The [[cohomology]] theory obtained by mapping out of [[Čech covers]] instead of general [[hypercovers]] is [[Čech cohomology]].
* A [[groupoid object in an (infinity,1)-category]] that is a Čech nerve $U \to X$ exhibits $X$ as a [[delooping]].
* In an [[infinity-stack]] [[(infinity,1)-topos]] every [[groupoid object in an (infinity,1)-category]] is a Čech nerve.
## Examples
+-- {: .num_example #FromACover}
###### Example
For $U = \coprod_i U_i$ the [[disjoint union]] of a [[covering]] [[sieve]] $\{U_i \to X\}$ with respect to a [[coverage]], the objectwise [[simplicial homotopy group|connected components]] of the Čech nerve is the [[subfunctor]] corresponding to the [[sieve]]
$$
\Pi_0 C(U) = \bigcup_i hom(-,U_i)
\,.
$$
This is described in more detail in the section "Interpretation in terms of higher descent and codescent" at [[sieve]]. This example is important in understanding the construction of the [[etale homotopy type]] of a scheme or more generally of objects in certain types of topos.
=--
+-- {: .num_remark}
###### Remark (Historical and pedagogic note)
This example is more or less the way that [[Eduard Čech]] gave the original form of the construction that now carries his name. More on this can be found in the entry on [[Čech methods]], and the discussion there of the **nerve of an open cover**. For the case of triangulable spaces (polyhedra), for the cover by open stars of vertices of a triangulation, one retrieves the simplicial complex used to triangulate the space. This is one of the strong roots of the modern theory of cohomology as nerves of open covers can be seen as analogues of triangulations, and then [[Čech cohomology]] is seen to extend simplicial cohomology to spaces that are locally nice. That is one of the first steps on the long route to [[Grothendieck]]'s definition of topos as a generalisation of space, expressly so as to define a cohomology further extending Čech cohomology to the geometric objects studied in algebraic geometry.
=--
##Related entries
* [[Čech groupoid]]
* [[effective epimorphism]]
* [[nerve theorem]]
## References
An early appearance (in the form of the 0-coskeleton) is in Enlightenment 8.5(b) of
* [[Michael Artin]], [[Barry Mazur]], _Etale Homotopy_, Lecture Notes in Mathematics 100 (1969), [doi](http://dx.doi.org/10.1007/bfb0080957).
An early appearance of the term “Čech nerve” is in
* [[Eric M. Friedlander]], _Etale Homotopy of Simplicial Schemes_, Annals of Mathematics Studies 104 (1983), [doi](http://dx.doi.org/10.1515/9781400881499).
[[!redirects Cech nerve]]
[[!redirects Cech-nerve]]
[[!redirects Čech nerve]]
[[!redirects Čech-nerve]]
[[!redirects Čech nerve]]
[[!redirects Čech-nerve]]
[[!redirects Cech complex]]
[[!redirects Čech nerves]]
[[!redirects Cech nerves]] |
Čedomir Crnković | https://ncatlab.org/nlab/source/%C4%8Cedomir+Crnkovi%C4%87 | [[!redirects Cedomir Crnković]]
* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=206648)
* [inspire page](https://inspirehep.net/authors/2295325)
## Selected writings
On the [[covariant phase space]] for [[field theories]]:
* [[Čedomir Crnković]], [[Edward Witten]], *Covariant Description of Canonical Formalism in Geometrical Theories*, chapter 16 in: S. W. Hawking and W. Israel (eds.) *Three Hundred Years of Gravitation*, Cambridge University Press (1987) 676-684 [[pdf](https://www.ias.edu/sites/default/files/sns/files/CovariantPaper-1987.pdf)]
* [[Čedomir Crnković]], *Symplectic Geometry of the Covariant Phase Space*, Class. Quant. Grav. **5** (1988) 1557-1575 [[doi:10.1088/0264-9381/5/12/008](https://doi.org/10.1088/0264-9381/5/12/008), [inspire:247290](https://inspirehep.net/literature/247290)]
category: people
[[!redirects Cedomir Crnkovic]]
|
Łoś ultraproduct theorem | https://ncatlab.org/nlab/source/%C5%81o%C5%9B+ultraproduct+theorem | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The Łoś [[ultraproduct]] theorem characterizes the [[first-order theory]] of an ultraproduct of $\mathcal{L}$-structures (for $\mathcal{L}$ some [[signature (in logic)|signature]]). It generalizes the _transfer principle_ from [[nonstandard analysis]], by replacing the hyperreals $^*\mathbb{R}$ with an ultraproduct and formulas from the standard reals $\mathbb{R}$ with formulas which are true on an "ultrafilter-large" subset of factors of the ultraproduct.
## Definition
The theorem is usually given in this form:
+-- {: .num_theorem}
###### Theorem
let $(A_i)_{i \in I}$ be a family of nonempty $\mathcal{L}$-structures, and let $\mathcal{U}$ be an [[ultrafilter]] on $I$. Let $^* A$ be the ultraproduct of $A_i$ with respect to $\mathcal{U}$. Since each $A_i$ was nonempty, $^* A$ is the quotient of the product $\prod_{i \in I} A_i$ by the equivalence relation which identifies sequences which coincide on a subset of indices belonging to $\mathcal{U}$. Let $(a_i)_{i \in I}$ be such a sequence, and let $[(a_i)]$ denote its equivalence class. Then for each $\mathcal{L}$-formula $\varphi(x)$,
$$^* A \models \varphi([a_i]) \iff \{j \in I | A_j \models \varphi(a_j)\} \in \mathcal{U}.$$
=--
The above follows from a more fundamental fact:
+-- {: .num_prop}
###### Proposition
Any functor
$$[\mathcal{U}] : \mathbf{Set}^I \to \mathbf{Set}$$
which specifies a choice of ultraproduct (and hence comparison maps, since ultraproducts are colimits) is elementary (logical in the terminology of Makkai-Reyes, i.e. a [[pretopos]] morphism.)
=--
+-- {: .proof}
###### Proof
Recall that the ultraproduct sends the family of sets $A_i$ to the colimit of $s \mapsto \prod_{i \in s} A_i$, indexed by $s \in U^op$. $U^op$ (the opposite category of the sub-poset $U \subseteq \mathcal{P}(I)$) is filtered because $U$ is a filter.
A functor is a pretopos morphism precisely when it preserves finite limits, initial objects, disjoint sums, and quotients by internal equivalence relations.
Finite limits: a $s$-indexed product of $J$-indexed limits is a $J$-indexed limit of $s$-indexed products, and finite limits commute with filtered colimits.
Initial objects: a $s$-indexed product of empty sets is empty (as each $s$ is nonempty by properness of $U$), and a colimit of empty sets is empty.
Disjoint sums: a $s$-indexed product of disjoint sums $\prod_{i:s} (A_{0i} + A_{1i})$ is a disjoint sum of products $\sum_{f : s \to 2} \prod_{i:s} A_{f(i)i}$. For any $(f,a)$ in this set, there is a $U$-large subset $t \subset s$ such that the restriction of $f$ to $t$ is constantly $b$ ($b \in \{0,1\}$). Therefore the restriction map identifies $(f,a)$ with $(const_b,a|_t)$, and the $U^op$-indexed colimit of the $\sum_{f : s \to 2} \prod_{i:s} A_{f(i)i}$ is equivalent to the $U^op$-indexed colimit of the $\sum_{b : 2} \prod_{i:s} A_{bi} = \prod_{i:s} A_{0i} + \prod_{i:s} A_{1i}$, and colimits commute with colimits.
Quotients: Given $X_i$ and equivalence relations $E_i \to X_i \times X_i$, a $s$-indexed product of quotients $\prod_{i:s} (X_i / E_i)$ is a quotient of products $(\prod_{i:s} X_i) / (\prod_{i:s} E_i)$ (because the natural map from the latter to the former is an iso, using the axiom of choice), and colimits commute with colimits.
=--
Now we can show the theorem:
+-- {: .proof}
###### Proof of theorem
Since the ultraproduct functor is elementary, then the process of taking points $M(X)$ of a definable set $X$ inside models $M$ commutes with taking ultraproducts. In symbols,
$$\left(\prod_{i \in I} M_i / \mathcal{U}\right) (X) \simeq \left(\prod_{i \in I} M_i(X) / \mathcal{U}\right).$$
Since this is a filtered colimit, a sequence $\overline{x}$ satisfies that its germ $[\overline{x}]$ is in $\prod_{i \in I} M_i / \mathcal{U}(X)$ if and only if there is some $J \in \mathcal{U}$ such that the restriction of $\overline{x}$ to $J$ is in $\prod_{j \in J} M_j(X)$. i.e. if $x_j \in M_j(X)$ for each $j \in J$.
=--
## Examples
An immediate consequence of the Łoś theorem is the [[nonstandard analysis|transfer principle]] for the hyperreals.
The [[compactness theorem]] also follows quickly from the Łoś theorem, so anything that you [build](http://math.stackexchange.com/questions/413770/most-astonishing-applications-of-compactness-theorem-outside-logic) using compactness can be realized a bit more concretely as an ultraproduct.
## Remarks
The proposition should be interpreted as saying that [[Set]] simultaneously carries a [[pretopos]] structure and an [[ultracategory]] structure, and that these two structures commute. Indeed, one can view [[Set]] as a [[dualizing object]] for a generalized [[Stone duality]] between pretoposes and ultracategories; this is the starting point of [[Makkai duality]].
There is an analogous statement for ultraproducts of structures in [[continuous logic]].
## Related concepts
* [[ultrafilter]]
* [[ultrapower]]
* [[stalk]]
## References
* Hisashi Aratake, _Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Łoś's Theorem_ , arXiv:2012.04317 (2020). ([abstract](https://arxiv.org/abs/2012.04317))
* [[Michael Makkai]], _Stone duality for first order logic_, Advances in Mathematics, 65(2):97–170, 1987.
[[!redirects Los theorem]]
[[!redirects Los ultraproduct theorem]]
[[!redirects transfer principle]]
|
Łukasiewicz logic | https://ncatlab.org/nlab/source/%C5%81ukasiewicz+logic |
\tableofcontents
## Idea
(...)
## Related concepts
* [[fuzzy logic]]
* [[linear logic]]
* [[substructural logic]]
* [[MV-algebra]]
## References
* Wikipedia, [Łukasiewicz logic](https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic)
Łukasiewicz logic is discussed in section 5 of:
* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html)) |
δ-ring | https://ncatlab.org/nlab/source/%CE%B4-ring | \tableofcontents
## Definition
(Definition 2.1 in Bhatt–Scholze.)
Fix a prime $p$.
A __δ-ring__ is a pair $(R,\delta)$,
where $R$ is a [[commutative ring]]
and $\delta\colon R\to R$ is a map of underlying sets
such that $\delta(0)=0$, $\delta(1)=0$,
$$\delta(x y)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$$
and
$$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$$
## Properties
If $(R,\delta)$ is a δ-ring,
then the map $\phi\colon R\to R$
given by $\phi(x)=x^p + p\delta(x)$
is a ring homomorphism
that lifts the [[Frobenius endomorphism]] on $R/p$.
For $p$-torsionfree rings,
the above correspondence between δ-structures
and lifts of the Frobenius endomorphism on $R/p$ to $R$
is bijective.
This motivates the identities in the definition of a δ-structure.
## Related entries
* [[perfectoid ring]]
* [[perfectoid space]]
* [[prism]]
* [[prismatic cohomology]]
## References
The original notion is due to [[André Joyal]]:
* [[André Joyal]], δ-anneaux et vecteurs de Witt, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), no. 3, 177–182.
Recent developments can be found in
* [[Bhargav Bhatt]], [[Peter Scholze]], _Prisms and prismatic cohomology_, [arXiv](http://arxiv.org/abs/1905.08229v3). |
κ-ary exact category | https://ncatlab.org/nlab/source/%CE%BA-ary+exact+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Regular and Exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# $\kappa$-ary regular and exact categories
* table of contents
{: toc}
## Idea
The notions of [[regular category]], [[exact category]], [[coherent category]], [[extensive category]], [[pretopos]], and [[Grothendieck topos]] can be nicely unified in a theory of "familial regularity and exactness." This was apparently first noticed in [Street 1984](#Street), and expanded on by [Shulman 2012](#Shulman) with a generalized theory of [[exact completion]].
## Sinks and relations
### Definition
Let $C$ be a [[finitely complete category]]. By a *[[sink]]* in $C$ we mean a [[family]] $\{f_i\colon A_i \to B\}_{i\in I}$ of [[morphisms]] with common [[target]]. A sink $\{f_i\colon A_i \to B\}$ is **[[extremal epimorphism|extremal epic]]** if it doesn't factor through any proper [[subobject]] of $B$. The _[[pullback]]_ of a sink along a morphism $B' \to B$ is defined in the evident way.
By a (many-object) **[[relation]]** in $C$ we will mean a family of objects $\{A_i\}_{i\in I}$ together with, for every $i,j\in I$, a monic span $A_i \leftarrow R_{i j} \to A_j$ (that is, a [[subobject]] $R_{i j}$ of $A_i \times A_j$. We say such a relation is:
* *[[reflexive relation|reflexive]]* if $R_{i i}$ contains the [[diagonal morphism|diagonal]] $A_i \to A_i \times A_i$, for all $i$,
* *[[transitive relation|transitive]]* if the pullback $R_{i j} \times_{A_j} R_{j k}$ factors through $R_{i k}$, for all $i,j,k$,
* *[[symmetric relation|symmetric]]* if $R_{i j}$ contains, hence is equal to, the transpose of $R_{j i}$ for all $i,j$, and
* a *[[congruence]]* if it is reflexive, transitive, and symmetric; this is an internal notion of (many-object) [[equivalence relation]].
Abstractly, reflexive and transitive relations can be identified with categories [[enriched category|enriched]] in a suitable [[bicategory]]; see [Street 1984](#Street). Congruences can be identified with enriched $\dagger$-[[dagger-categories|categories]].
A __quotient__ for a relation is a [[colimit]] for the diagram consisting of all the $A_i$ and all the spans $A_i \leftarrow R_{i j} \to A_j$. And the __kernel__ of a sink $\{f_i\colon A_i\to B\}$ is the relation on $\{A_i\}$ with $R_{i j} = A_i \times_B A_j$. It is evidently a congruence.
Finally, a sink is called **[[effective epimorphism|effective-epic]]** if it is the quotient of its kernel. It is called **universally effective-epic** if any pullback of it is effective-epic.
### Examples
* If ${|I|} = 1$, a congruence is the same as the ordinary internal notion of [[congruence]]. In this case [[quotient object|quotients]] and [[kernel pair|kernels]] reduce to the usual notions.
* If ${|I|} = 0$, a congruence contains no data and a sink is just an object in $C$. The empty congruence is, trivially, the kernel of the empty sink with any target $B$, and a quotient for the empty congruence is an [[initial object]].
* Given a family of objects $\{A_i\}$, define a congruence by $R_{i i}=A_i$ and $R_{i j}=0$ (an initial object) if $i \neq j$. Call a congruence of this sort _trivial_ (empty congruences are always trivial). Then a quotient for a trivial congruence is a [[coproduct]] of the objects $A_i$, and the kernel of a sink $\{f_i\colon A_i\to B\}$ is trivial iff the $f_i$ are disjoint monomorphisms.
## $\kappa$-ary regularity and exactness
### Definition
Let $\kappa$ be an [[arity class]]. We call a sink or relation **$\kappa$-ary** if the cardinality ${|I|}$ is $\kappa$-small. As usual for arity classes, the cases of most interest have special names:
* When $\kappa = \{1\}$ we say **unary**.
* When $\kappa = \omega$ is the set of finite cardinals, we say **finitary**.
* When $\kappa$ is the class of all cardinal numbers, we say **infinitary**.
+-- {: .num_theorem #KRegular}
###### Theorem
For a category $C$, the following are equivalent:
1. $C$ has [[finite limits]], every $\kappa$-ary sink in $C$ factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic $\kappa$-ary sink is extremal epic.
1. $C$ has finite limits, and the kernel of any $\kappa$-ary sink in $C$ is also the kernel of some universally effective-epic sink.
1. $C$ is a [[regular category]] and has [[pullback-stable colimit|pullback-stable]] [[joins]] of $\kappa$-small families of [[subobjects]].
=--
When these conditions hold, we say $C$ is **$\kappa$-ary regular**, or alternatively **$\kappa$-ary coherent**. There are also some other more technical characterizations; see [Shulman 2012](#Shulman).
+-- {: .num_theorem #KExact}
###### Theorem
For a category $C$, the following are equivalent:
1. $C$ has [[finite limits]], and every $\kappa$-ary congruence is the [[kernel]] of some universally effective-epic sink.
1. $C$ is $\kappa$-ary regular, and every $\kappa$-ary congruence is the kernel of some sink.
1. $C$ is both [[exact category|exact]] and $\kappa$-ary [[extensive category|extensive]].
=--
\begin{definition}\label{KappaAryExactCategory}
When these conditions hold, we say that $C$ is called a **$\kappa$-ary exact category**, or alternatively a **$\kappa$-ary pretopos**.
\end{definition}
\begin{definition}\label{KappaAryExactFunctor}
A [[functor]] $F \,\colon\, C\to D$ between $\kappa$-ary exact categories (Def. \ref{KappaAryExactCategory}) is called a **$\kappa$-ary exact functor** if it [[preserved limit|preserves]] [[finite limits]] and $\kappa$-small effective-epic (or equivalently extremal-epic) families.
\end{definition}
\begin{definition}\label{2CategoryOfKappaAryExactCategories}
**(The 2-category of $\kappa$-ary exact categories)**
\linebreak
The [[2-category]]
$EX_\kappa$
of $\kappa$-ary exact categories (Def. \ref{KappaAryExactCategory}),
$\kappa$-ary exact functors (Def. \ref{KappaAryExactFunctor})
and [[natural transformations]]
is a [[reflective subcategory|reflective]] [[full sub-2-category]] of the 2-category $SITE_\kappa$ of [[∞-ary sites]]. The reflector is called *[[exact completion]]*.
\end{definition}
### Examples
1. $C$ is [[regular category|regular]] iff it is unary regular.
2. $C$ is [[coherent category|coherent]] iff it is finitary regular.
3. $C$ is [[coherent category|infinitary-coherent]] iff it is infinitary regular.
4. $C$ is [[exact category|exact]] iff it is unary exact.
5. $C$ is a [[pretopos]] iff it is finitary exact.
6. $C$ is an [[pretopos|infinitary pretopos]] iff it is infinitary exact.
Some other sorts of [[exactness properties]] (especially [[lex-colimits]]) can also be characterized in terms of congruences, kernels, and quotients. For instance:
7. $C$ is $\kappa$-ary [[extensive category|lextensive]] iff every $\kappa$-ary _trivial_ congruence has a pullback-stable quotient of which it is the kernel.
In [Street](#Street), there is also a version of regularity and exactness that applies even to some _large_ sinks and congruences, and implies some small-generation properties of the category as well.
### Properties
In a $\kappa$-ary regular category,
* Every extremal-epic $\kappa$-ary sink is the quotient of its kernel.
* Any $\kappa$-ary congruence that is a kernel has a quotient.
Thus, in a $\kappa$-ary exact category,
* Every $\kappa$-ary congruence has a quotient.
In a $\kappa$-ary regular category, the class of all $\kappa$-small and effective-epic families generates a [[topology]], called its $\kappa$-canonical topology. This topology makes it a [[∞-ary site]].
## References
* {#Street} [[Ross Street]], *The family approach to total cocompleteness and toposes*, Transactions of the AMS **284** no. 1, 1984. ([doi:10.1090/S0002-9947-1984-0742429-3](https://doi.org/10.1090/S0002-9947-1984-0742429-3))
* {#Shulman} [[Michael Shulman]], *Exact completions and small sheaves*, *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173. ([tac:27-07](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html))
[[!redirects familial regularity and exactness]]
[[!redirects familial regularity]]
[[!redirects familial exactness]]
[[!redirects familially exact category]]
[[!redirects familially regular category]]
[[!redirects ∞-ary exact category]]
[[!redirects ∞-ary exact categories]]
[[!redirects k-ary exact category]]
[[!redirects k-ary exact categories]]
[[!redirects ∞-ary exactness]]
[[!redirects k-ary exactness]]
[[!redirects ∞-ary regular category]]
[[!redirects ∞-ary regular categories]]
[[!redirects k-ary regular category]]
[[!redirects k-ary regular categories]]
[[!redirects ∞-ary regularity]]
[[!redirects k-ary regularity]]
[[!redirects ∞-ary pretopos]]
[[!redirects ∞-ary pretoposes]]
[[!redirects ∞-ary pretopoi]]
[[!redirects k-ary pretopos]]
[[!redirects k-ary pretoposes]]
[[!redirects k-ary pretopoi]]
[[!redirects ∞-ary coherent category]]
[[!redirects ∞-ary coherent categories]]
[[!redirects k-ary coherent category]]
[[!redirects k-ary coherent categories]]
[[!redirects ∞-ary regular and exact categories]]
|
κ-ary site | https://ncatlab.org/nlab/source/%CE%BA-ary+site |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# $\kappa$-ary sites
* table of contents
{: toc}
## Idea
A *$\kappa$-ary site* is a [[site]] whose covering sieves are determined by $\kappa$-small covering families, and which has a very weak sort of finite limits. These conditions get weaker as $\kappa$ gets larger, until when $\kappa$ is the size of the universe, every [[small category|small]] site is $\kappa$-ary.
$\kappa$-ary sites are a very general (perhaps the most general) appropriate input for $\kappa$-ary [[exact completion]].
## Definition
Let $\kappa$ be an [[arity class]].
+-- {: .un_defn}
###### Definition
A [[site]] $C$ is **weakly $\kappa$-ary** if for any covering sieve $R$ of an object $V$ in $C$, there exists a $\kappa$-small family $\{p_i: U_i \to V\}_i$ in $C$ such that (1) each $p_i \in R$, and (2) the sieve generated by $\{p_i\}$ is a covering sieve.
=--
This definition can also be rephrased purely in terms of the covering families; see ([Shulman](#Shulman)).
+-- {: .un_defn}
###### Definition
Let $C$ be a site and $G:D\to C$ a [[functor]]. A **local $\kappa$-prelimit** of $G$ is a $\kappa$-small family of [[cones]] $\{q_i: \Delta L \to G \}_i$ in $C$ such that for any cone $r:\Delta u \to G$, the sieve $\{ p: v\to u | r p$ factors through some $q_i \}$ is a covering sieve of $u$.
=--
+-- {: .un_defn}
###### Definition
A **$\kappa$-ary site** is a weakly $\kappa$-ary site which has all *finite* local $\kappa$-prelimits (i.e. whenever $D$ is a [[finite category]]).
=--
## Examples
* If $C$ has a [[trivial topology]], then a local *unary* prelimit (i.e. $\kappa=\{1\}$) is precisely a [[weak limit]]. The trivial topology is always weakly $\kappa$-ary, so a trivial site is unary just when it has weak limits.
* Any limit is, in particular, a local $\kappa$-prelimit. Thus, any weakly $\kappa$-ary site with finite limits is $\kappa$-ary.
* If the class of *all* cones over $G$ is $\kappa$-ary, then it is a local $\kappa$-prelimit. Thus, any $\kappa$-small and weakly $\kappa$-ary site is $\kappa$-ary. In particular, any small site is an *infinitary* site.
* The [[regular topology]] on a [[regular category]] (including an [[exact category]]) is unary.
* The [[coherent topology]] on a [[coherent category]] (including a [[pretopos]]) is finitary.
* Generalizing the previous two examples, the class of all $\kappa$-small and effective-epic families on a [[∞-ary regular category]] (including a [[∞-ary exact category]]) is a $\kappa$-ary topology. This is called its *$\kappa$-canonical topology*.
* The [[extensive topology]] on a (finitary) [[extensive category]] is finitary.
* The [[canonical topology]] on any [[Grothendieck topos]] is infinitary.
* The [[Zariski topology]] on $CRing^{op}$ is finitary.
## The 2-category of $\kappa$-ary sites
The 2-category $SITE_\kappa$ has $\kappa$-ary sites as its objects, and [[morphisms of sites]] as its morphisms, where we use the more general [[covering-flat functor|covering-flat]] definition of a morphism of sites.
### Properties
* $SITE_\kappa$ is equivalent to a 2-category of [[framed allegory|framed allegories]]; see ([Shulman](#Shulman)).
* $SITE_\kappa$ contains, as a full [[reflective subcategory|reflective]] sub-2-category, the 2-category of [[∞-ary exact categories]] with their $\kappa$-canonical topologies. The reflector is called [[exact completion]]. When $\kappa$ is the size of the universe, this reflector applied to a small (hence infinitary) site constructs its [[topos of sheaves]].
## References
* [[Michael Shulman]], "Exact completions and small sheaves". *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173. [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)
{#Shulman}
[[!redirects ∞-ary site]]
[[!redirects ∞-ary sites]]
[[!redirects k-ary site]]
[[!redirects k-ary sites]]
[[!redirects unary site]]
[[!redirects unary sites]]
[[!redirects finitary site]]
[[!redirects finitary sites]]
[[!redirects infinitary site]]
[[!redirects infinitary sites]]
|
π-Kan condition | https://ncatlab.org/nlab/source/%CF%80-Kan+condition |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The *$\pi_\ast$-Kan condition* ([Bousfield & Friedlander 1978, B.3](#BousfieldFriedlander78)) on a [[bisimplicial set]] guarantees that its [[geometric realization]] behaves well with respect to [[homotopy pullback]] ([Bousfield & Friedlander 1978, Thm. B.4](#BousfieldFriedlander78)).
See [this Prop.](geometric+realization+of+simplicial+topological+spaces#SufficientConditionsForRealizationToPreserveHomotopyPullback) at *[[geometric realization of simplicial topological spaces]]*.
## Related entries
* [[commutativity of homotopy limits and colimits]]
## References
* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])
* {#Anderson78} [[Donald Werner Anderson]], _Fibrations and geometric realization_, Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. ([euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139))
* {#Lurie11} [[Jacob Lurie]], *Simplicial spaces*, Lecture 7 of: *[Algebraic L-theory and Surgery](https://www.math.ias.edu/~lurie/287x.html)*, 2011 ([pdf](https://www.math.ias.edu/~lurie/287xnotes/Lecture7.pdf))
* {#Rezk14} [[Charles Rezk]], _When are homotopy colimits compatible with homotopy base change?_, 2014 ([pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]])
* Edoardo Lanari, _Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves_ ([pdf](http://algant.eu/documents/theses/lanari.pdf), [[LanariHomotopyColimitsBaseChange.pdf:file]])
> (expanded version of [Rezk 14](#Rezk14))
* {#MazelGee14} [[Aaron Mazel-Gee]], _Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces_ ([arXiv:1412.8411](https://arxiv.org/abs/1412.8411))
* {#LurieHigherAlgebra} [[Jacob Lurie]], around Lem. 5.5.6.17 in: *[[Higher Algebra]]* (~2017)
* [[Jacob Lurie]], Thms. A.5.4.1, *[[Spectral Algebraic Geometry]]* (2018) [[pdf](https://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf)]
* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida]], [[Dmitri Pavlov]], §3 of: _Classifying spaces of infinity-sheaves_, Algebraic & Geometric Topology [[arXiv:1912.10544](https://arxiv.org/abs/1912.10544)]
[[!redirects π-Kan+conditions]]
[[!redirects pi-Kan condition]]
[[!redirects pi-Kan conditions]]
|
Π-pretopos | https://ncatlab.org/nlab/source/%CE%A0-pretopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
A **$\Pi$-pretopos** is a [[pretopos]] that is also a [[locally cartesian closed category]].
## Related concepts
* A [[ΠW-pretopos]] is a $\Pi$-pretopos that also has all [[W-types]].
* A [[predicative topos]] is $\Pi W$-pretopos with [[axiom of multiple choice]].
## References
See at _[[predicative topos]]_.
[[!redirects Pi-pretopos]]
[[!redirects Pi-pretoposes]]
[[!redirects Pi-pretopoi]]
[[!redirects Π-pretopos]]
[[!redirects Π-pretoposes]]
[[!redirects Π-pretopoi]]
|
ΠW-pretopos | https://ncatlab.org/nlab/source/%CE%A0W-pretopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
A **$\Pi W$-pretopos** is a [[Π-pretopos]] with all [[W-types]].
## Examples
{#Examples}
* In [[homotopy type theory]] the [[h-sets]] form a ΠW-pretopos ([Rijke-Spitter 13](#RijkeSpitter13)). See also at _[[structural set theory]]_.
* The [[inequality spaces]] and [[strongly extensional functions]] form a ΠW-pretopos.
## Related concepts
* A [[predicative topos]] is a $\Pi W$-pretopos satisfying the [[axiom of multiple choice]].
## References
See also at _[[predicative topos]]_.
That the [[h-sets]] in [[homotopy type theory]] form a ΠW-pretopos is discussed in
* {#RijkeSpitters13} [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics) ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))
which became one chapter in
* [[Univalent Foundations Project]], _[[Homotopy Type Theory – Univalent Foundations of Mathematics]]_
[[!redirects arithmetic Π-pretopos]]
[[!redirects arithmetic Π-pretopoi]]
[[!redirects arithmetic Π-pretoposes]]
[[!redirects arithmetic Pi-pretopos]]
[[!redirects arithmetic Pi-pretoposes]]
[[!redirects arithmetic Pi-pretopoi]]
[[!redirects ΠW-pretopos]]
[[!redirects ΠW-pretopoi]]
[[!redirects ΠW-pretoposes]]
[[!redirects Pi-W-pretopos]]
[[!redirects Pi-W-pretoposes]]
[[!redirects Pi-W-pretopoi]]
[[!redirects ∞-W-pretopos]]
[[!redirects ∞-W-pretoposes]]
[[!redirects ∞-W-pretopoi]]
|
σ-meson | https://ncatlab.org/nlab/source/%CF%83-meson |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _sigma-meson_, now also called the _$f_0(500)$-resonance_, is the [[light meson|lightest]] [[Lorentz group|Lorentz]]-[[scalar]] and [[isospin]]-[[scalar]] [[meson]].
\begin{imagefromfile}
"file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": 0,
"right": 10,
"bottom": 0,
"left": 20
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\end{imagefromfile}
Together with the [[omega-meson]] the sigma is responsible for most of the long-range [[interaction]] between [[baryons]], exhibiting the residual [[strong nuclear force]] between them, at least as modeled by the [[Walecka model]] and [[quantum hadrodynamics]].
Other models of [[quantum hadrodynamics]] do not contain the sigma at all. See [Pelaez 16](#Pelaez16) for review of the controversial and inconclusive understanding of the sigma-meson.
## Related concepts
* [[Walecka model]]
## References
### General
Original articles:
* [[Murray Gell-Mann]], M. Lévy, _The axial vector current in beta decay_, Il Nuovo Cimento (1955-1965) volume 16, pages 705–726 (1960) ([doi:10.1007/BF02859738](https://doi.org/10.1007/BF02859738))
* [[Julian Schwinger]], _A Theory of the Fundamental Interactions_, Annals Phys. 2 (1957) 407-434 ([spire:12286](http://inspirehep.net/record/12286/))
Review:
* {#Pelaez16} J. R. Pelaez, _From controversy to precision on the sigma meson: a review on the status of the non-ordinary $f_0(500)$ resonance_, Physics Reports 658 (2016) 1 ([arXiv:1510.00653](https://arxiv.org/abs/1510.00653))
See also:
* Eef van Beveren, Frieder Kleefeld, George Rupp, Michael D. Scadron, _Remarks on the $f_0(400-1200)$ scalar meson as the dynamically generated chiral partner of the pion_, Mod. Phys. Lett. A17:1673, 2002 ([arXiv:hep-ph/0204139](https://arxiv.org/abs/hep-ph/0204139))
* M. Albaladejo, J.A. Oller, _On the size of the sigma meson and its nature_, Phys. Rev. D 86, 034003 – Published 3 August 2012 ([arXiv:1205.6606](https://arxiv.org/abs/1205.6606), [doi:10.1103/PhysRevD.86.034003](https://doi.org/10.1103/PhysRevD.86.034003))
* S. S. Afonin, T. D. Solomko, _The sigma meson from QCD sum rules for large-$N_c$ Regge spectra_, Eur. Phys. J. C 76, 678 (2016) ([arXiv:1608.08131](https://arxiv.org/abs/1608.08131))
* S. S. Afonin, T. D. Solomko, _The linear radial spectrum of scalar mesons within the QCD sum rules in the planar limit_, Eur. Phys. J. C 76, 678 (2016) ([arXiv:1608.08131](https://arxiv.org/abs/1608.08131))
* Physics.SE discussion, _[What the heck is the sigma (f0) 600?](https://physics.stackexchange.com/q/13458/5603)_
* Christian S. Fischer, Paul C. Wallbott, Richard Williams, Nico Santowsky, Gernot Eichmann, _The $\sigma$-meson: four-quark vs. two-quark components and decay width in a Bethe-Salpeter approach_ ([arXiv:2007.06495](https://arxiv.org/abs/2007.06495))
Inclusion of the [[sigma-meson]] into the [[Skyrme model]] for [[baryons]]:
* Thomas D. Cohen, _Explicit $\sigma$ meson, topology, and the large-$N$ limit of the Skyrmion_, Phys. Rev. D 37 (1988) ([doi:10.1103/PhysRevD.37.3344](https://doi.org/10.1103/PhysRevD.37.3344))
### In holographic QCD
Discussion in [[holographic QCD]]:
On [[sigma-mesons]] and [[omega-mesons]] mediating [[baryon]] [[interaction]], discussed in [[holographic QCD]] via [[D3-D7 brane intersections]]:
* Anatoly Dymarsky, Dmitry Melnikov, [[Jacob Sonnenschein]], _Attractive Holographic Baryons_, JHEP 06 (2011) 145 ([arXiv:1012.1616](https://arxiv.org/abs/1012.1616))
and discussed in the [[Witten-Sakai-Sugimoto model]] on [[D4-D8 brane intersections]]:
* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], _Searching for an Attractive Force in Holographic Nuclear Physics_, JHEP 05 (2011) 058 ([arXiv:1003.2621](https://arxiv.org/abs/1003.2621))
[[!redirects f0-meson]]
[[!redirects σ-mesons]]
[[!redirects sigma-meson]]
[[!redirects sigma-mesons]]
|
ω-category > history | https://ncatlab.org/nlab/source/%CF%89-category+%3E+history | < [[∞-category]]
[[!redirects ∞-category/history]]
[[!redirects ∞-category -- history]] |
ω-meson | https://ncatlab.org/nlab/source/%CF%89-meson |
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### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[nuclear physics]], specifically in the [[chiral perturbation theory]] of [[quantum chromodynamics]], the _omega-meson_ is the [[isospin]]-[[trivial representation|singlet]] [[vector meson]] [[field (physics)|field]] in the first-[[generation of fermions]], i.e. a [[bound state]] of an [[up quark]] and a [[down quark]] (a [[light meson]]), the [[chiral partner]] of the [[f1-meson]]:
\begin{imagefromfile}
"file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": 0,
"right": 10,
"bottom": 0,
"left": 20
}
\end{imagefromfile}
\linebreak
[[!include flavors of fundamental particles -- table]]
## Properties
### Nuclear binding
Together with the [[sigma-meson]] the omega is responsible for most of the long-range [[interaction]] between [[baryons]], exhibiting the residual [[strong nuclear force]] between them (as modeled by [[Walecka model]] and [[quantum hadrodynamics]]).
### Couplings
#### The $\omega$-$3 \pi$-coupling
{#TheOmega3PiCoupling}
The [[interaction]] term of the omega-meson three [[pions]] is, in the [[Lagrangian density]], given by contraction
$$
\omega_\mu B^\mu
$$
with the [[chiral anomaly|chirally anomalous]] [[baryon current]] $B \coloneqq \star tr( (U^{-1} d U) \wedge (U^{-1} d U) \wedge (U^{-1} d U) )$, with $U$ is the exponential of the [[pion]]-[[field (physics)|field]] ([Adkins-Nappi 84, (1) and (2)](#AdkinsNappi84), [Park-Vento 09, (5.5.43) and above (5.5.50)](#ParkVento09)).
This gives a [[decay mode]]
$$
\omega \to \pi^+ + \pi^- + \pi^0
$$
(the "charged decay", e.g. [Rudaz 84, (2)](#Rudaz84)).
Or rather, this is the direct (contact term) decay. The net process $\omega \to 3 \pi$ is dominated by the successive decay
$$
\omega \to \rho + \pi
\to
(2 \pi) + \pi
$$
#### The $\omega$-$\rho$-$\pi$-coupling
{#TheOmegaRhoPiCoupling}
Then there is an [[omega-meson|ω]]-[[rho-meson|ρ]]-[[pion|π]]-coupling given by the anomalous part of the [[chiral perturbation theory|chiral]] [[WZW-term]]:
$$
g_{\omega \rho \pi}
\epsilon^{\mu \nu \kappa \lambda}
\partial_\mu \omega_\nu \partial_\kappa \rho_\lambda \cdot \pi
$$
(e.g. [Renard 69](#Renard69), [Meissner-Kaiser-Weise 87, (2.18)](#MeissnerKaiserWeise87) [Volkov-Ebert-Nagy 97, p. 12](#VolkovEbertNagy97), [Guetta-Singer 00, (1)](#GuettaSinger00),
[Kaiser 00, (12)](#Kaiser00), [GKSY 03, (1)](#GKSY03) [Gudino-Sanchez 12, (1)](#GudinoSanchez12))
#### The radiative decays
Then there is the "neutral decay"
$$
\omega \to \pi^0 + \gamma
$$
seen in [[experiment]] as
$$
\begin{aligned}
\omega & \to \pi^0 + \gamma
\\
& \to \gamma + \gamma + \gamma
\end{aligned}
$$
([Nambu 57, (a)](#Nambu57), [FFHNR 67](#FFHNR67), [Dolinsky et al. 89,
(5)](#Dolinsky89))
## Related concepts
* [[pion]], [[rho-meson]]
## References
### General
The $\omega$-meson was first postulated by
* {#Nambu57} [[Yoichiro Nambu]], _Possible Existence of a Heavy Neutral Meson_, Phys. Rev. 106, 1366 (1957) ([doi:10.1103/PhysRev.106.1366](https://doi.org/10.1103/PhysRev.106.1366))
as reviewed in
* [[Jun John Sakurai]], p. 48-49 of: _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))
(in the context of [[vector meson dominance]])
See also:
* Wikipedia, _[Omega meson](https://en.wikipedia.org/?title=Omega_meson)_
See also
* Reinhard Alkofer, H. Reinhardt, H. Weigel, U. Zuckert, _The Isoscalar vector meson omega in the Nambu-Jona-Lasinio soliton_, Phys. Lett. B298 (1992) 132-140 ([spire:343052](http://inspirehep.net/record/343052), <a href="https://doi.org/10.1016/0370-2693(93)91719-4">doi:10.1016/0370-2693(93)91719-4</a>)
Phenomenology:
* Cheng-Qun Pang, Ya-Rong Wang, Jing-Fu Hu, Tian-Jie Zhang, Xiang Liu, _Study of the $\omega$ meson family and newly observed $\omega$-like state $X(2240)$_ ([arXiv:1910.12408](https://arxiv.org/abs/1910.12408))
* M. K. Volkov, A. A. Pivovarov, K. Nurlan, _On the mixing angle of the vector mesons $\omega(782)$ and $\phi(1020)$_ ([arXiv:2005.00763](https://arxiv.org/abs/2005.00763))
### Decays
The direct decay $\omega \to \pi^0 + \pi^+ + \pi^-$:
* {#Rudaz84} S. Rudaz, _Anomalies, vector mesons and the $\omega \to 3 \pi$ contact term_, Phys. Lett. B 145 (1984) 281-284 ([spire:208193](https://inspirehep.net/literature/208193), <a href="https://doi.org/10.1016/0370-2693(84)90355-1">doi:10.1016/0370-2693(84)90355-1</a>)
* E. A. Kuraev, Z. K. Silagadze, _Once more about the $\omega \to 3 \pi$ contact term_, Phys. Atom. Nucl. 58:1589-1596, 1995 ([arXiv:hep-ph/9502406](https://arxiv.org/abs/hep-ph/9502406))
* M. Albaladejo, I. Danilkin, S. Gonzalez-Solis, D. Winney, C. Fernandez-Ramirez, A. N. Hiller Blin, V. Mathieu, M. Mikhasenko, A. Pilloni, A. Szczepaniak, _$\omega \to 3\pi$ and $\omega \pi^0$ transition form factor revisited_ ([arXiv:2006.01058](https://arxiv.org/abs/2006.01058))
The $\omega \pi \rho$-coupling
* {#GudinoSanchez12} D. Garcia Gudino, G. Toledo Sanchez, _The $\omega \rho \pi$ coupling in the VMD model revisited_, Int. J. Mod. Phys. A 27, 1250101 (2012) ([arXiv:1106.1467](https://arxiv.org/abs/1106.1467))
On [[Dalitz decays]] of omega-mesons:
* {#Wachs00} Mirko Wachs, _Die Selbstenergie des Omega-Mesons_, 2000 ([epda:000050](http://elib.tu-darmstadt.de/diss/000050))
* {#Berghauser10} Henning Berghäuser, _Investigation of the Dalitz decays and the electromagnetic form factors of the $\eta$ and $\pi^0$-meson_, 2010 ([spire:1358057](https://inspirehep.net/literature/1358057))
[[!include Skyrme hadrodynamics with vector mesons -- references]]
### Couplings
On [[omega-meson]] [[interactions]] and [[decay modes]]:
* Stanley M. Flatté, Darrell O. Huwe, Joseph J. Murray, Janice Button-Shafer, Frank T. Solmitz, M. Lynn Stevenson, and Charles Wohl, _Decay Properties of the $\omega$ Meson_, Phys. Rev. 145, 1050 – Published 27 May 1966 ([doi:10.1103/PhysRev.145.1050](https://doi.org/10.1103/PhysRev.145.1050))
* {#FFHNR67} M. Feldman, W. Frati, R. Gleeson, J. Halpern, M. Nussbaum, S. Richert, _Neutral Decay of the $\omega$ Meson_, Phys. Rev. 159, 1219 (1967) ([doi10.1103/PhysRev.159.1219](https://doi.org/10.1103/PhysRev.159.1219), [spire:52556](http://inspirehep.net/record/52556))
* W. Deinet A. Menzione H.Müller, H. M.Staudenmaier, S.Buniatov, D.Schmitt, _Neutral decay modes of the $\omega^0$-meson_, Physics Letters B
Volume 30, Issue 6, 10 November 1969, Pages 426-429 (<a href="https://doi.org/10.1016/0370-2693(69)90479-1">doi:10.1016/0370-2693(69)90479-1</a>)
* {#Renard69} F. M. Renard, _The reaction $e^+ e^- \to \pi^0 + \omega(\pi^+ \pi^- \pi^0) $ and the $\omega$-$\rho$-$\pi$ coupling_, Nuovo Cimento A (1965-1970) 64, 979–984 (1969) ([doi:10.1007/BF02758844](https://doi.org/10.1007/BF02758844))
* {#VolkovEbertNagy97} M. K. Volkov, D. Ebert, M. Nagy, _Excited pions, $\rho$- and $\omega$-mesons and their decays in a chiral $SU(2) \times SU(2)$ Lagrangian_, Int. J. Mod. Phys. A13 (1998) 5443-5458 ([arXiv:hep-ph/9705334](https://arxiv.org/abs/hep-ph/9705334))
* {#Dolinsky89} S. I. Dolinsky, et al., _Radiative Decays of $\rho$ and $\omega$ Mesons_, Z. Phys. C42 (1989) 511 ([spire:264694](http://inspirehep.net/record/264694), [doi:10.1007/BF01557655](https://doi.org/10.1007/BF01557655))
* J. T. Dakin, M. G. Hauser, M. N. Kreisler, R. E. Mischke, _Measurement of the Branching Ratios for ω Neutral Decays_, Phys. Rev. D 6, 2321 (1972) ([doi:10.1103/PhysRevD.6.2321](https://doi.org/10.1103/PhysRevD.6.2321))
* {#GuettaSinger00} Dafne Guetta, Paul Singer, _$\omega$-$\rho$ Mixing and the $\omega \to \pi \pi \gamma$ Decay_, Phys. Rev. D63 (2001) 017502 ([arXiv:hep-ph/0005059](https://arxiv.org/abs/hep-ph/0005059))
* {#Kaiser00} Roland Kaiser, equation (12) of: _Anomalies and WZW-term of two-flavour QCD_, Phys. Rev. D63:076010, 2001 ([arXiv:hep-ph/0011377](https://arxiv.org/abs/hep-ph/0011377))
* {#GKSY03} A. Gokalp, A. Kucukarslan, S. Solmaz, O. Yilmaz, _$\sigma$-Meson and $\omega$-$\rho$ mixing effects in $\omega \to \pi^+ \pi^- \gamma$ decay_, Acta Phys.Polon. B34 (2003) 4095-4104 ([arXiv:hep-ph/0306044](https://arxiv.org/abs/hep-ph/0306044))
* [[Jeffrey Harvey]], Christopher T. Hill, Richard J. Hill, Section II.B of: _Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions_, Phys. Rev. D77:085017, 2008 ([arXiv:0712.1230](https://arxiv.org/abs/0712.1230))
* S. Leupold, M. F. M. Lutz, _Hadronic three-body decays of light vector mesons_, Eur. Phys. J. A39:205-212, 2009 ([arXiv:0807.4686](https://arxiv.org/abs/0807.4686))
* Florian Jonas, _Measurement of $\omega$ and $\eta$ mesons via their three pion decay with ALICE in pp collisions at $\sqrt{s} = t TeV$_, 2018 ([cds:2653176](https://cds.cern.ch/record/2653176))
### In holographic QCD
The [[omega-meson]] in [[holographic QCD]] ([[Witten-Sakai-Sugimoto model]]):
* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Section 4.3 of: _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))
### Mediating baryon interaction
On [[sigma-mesons]] and [[omega-mesons]] mediating [[baryon]] [[interaction]], discussed in [[holographic QCD]] via [[D3-D7 brane intersections]]:
* Anatoly Dymarsky, Dmitry Melnikov, [[Jacob Sonnenschein]], _Attractive Holographic Baryons_, JHEP 06 (2011) 145 ([arXiv:1012.1616](https://arxiv.org/abs/1012.1616))
[[!redirects ω-mesons]]
[[!redirects omega-meson]]
[[!redirects omega-mesons]]
[[!redirects omega meson]]
[[!redirects omega mesons]]
[[!redirects Omega-meson]]
[[!redirects Omega-mesons]]
[[!redirects Omega meson]]
[[!redirects Omega mesons]]
|
Е. Б. Богомольный | https://ncatlab.org/nlab/source/%D0%95.+%D0%91.+%D0%91%D0%BE%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9 | [[!redirects Е. Б. Богомольний]]
Evgeny Bogomolny
* [webpage](http://www.itp.ac.ru/en/persons/bogomolny-evgeny-borisovich)
## Papers and related $n$Lab entries
* [[BPS state]]
* E. B. Bogomolny, _Stability of slassical solutions_, Sov. J. Nucl. Phys. 24 (1976), 449; Yad. Fiz. 24 (1976), 861
category: people
[[!redirects Evgeny Bogomolny]]
[[!redirects Evgeny Bogomolnyj]]
[[!redirects E. B. Bogomolny]]
|
Павел Мнёв | https://ncatlab.org/nlab/source/%D0%9F%D0%B0%D0%B2%D0%B5%D0%BB+%D0%9C%D0%BD%D1%91%D0%B2 | [[!redirects Pavel Mnev]]
* [webpage](http://www.pdmi.ras.ru/~pmnev/)
## Selected writings
On [[BV quantization]]:
* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]
category: people |
שנה טובה > history | https://ncatlab.org/nlab/source/%D7%A9%D7%A0%D7%94+%D7%98%D7%95%D7%91%D7%94+%3E+history | < [[שנה טובה]] |
’t Hooft operator | https://ncatlab.org/nlab/source/%E2%80%99t+Hooft+operator |
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#### Physics
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[[!include physicscontents]]
=--
#### Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[gauge theory]], where a [[Wilson line]] is a curve in ambient [[spacetime]] with a [[gauge field]] [[holonomy]] around the curve, dually a _'t Hooft operator_ is a curve with [[Dirac monopole]]-like [[singularity]] of the ambinent [[gauge field]] along it (hence they may be thought of as 1-dimensional distributions of [[magnetic charge]]).
review includes ([Kapustin-Witten 06, section 6.2](#KapustinWitten06))
At least in the additional presence of [[Higgs bundle]] fields the singularity makes the [[field strength]] [[curvature]] that of a [[differential form with logarithmic singularities]] along the specified curve
([Kapustin-Witten 06, (6.8), (6.9)](#KapustinWitten06))
## Properties
### Relation to S-duality and geometric Langlands correspondence
Under the identification of the [[geometric Langlands correspondence]] with aspects of [[S-duality]] in [[super Yang-Mills theory]], the t Hooft operators correspond to [[Hecke operator]] ([Kapustin-Witten 06, section 9](#KapustinWitten06)).
[[!include geometric Langlands QFT -- table]]
## References
The original definition is due to
* [[Gerard 't Hooft]], _On the phase transition towards permanent quark confinement_, Nuclear Physics : B, volume: 138, issue: 1 (1978), pp. 1 - 25 ([igitur](http://dspace.library.uu.nl/handle/1874/4701))
Review includes
* Wikipedia, _['t Hooft operator](http://en.wikipedia.org/wiki/'t_Hooft_operator)_
Discussion in the context of [[S-duality]] is in
* {#Kapustin06} [[Anton Kapustin]], _Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality_, Phys.Rev. D74 (2006) 025005 ([arXiv:hep-th/0501015](http://arxiv.org/abs/hep-th/0501015))
and further discussion of this relating to the [[geometric Langlands correspondence]] is in
* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_, Communications in Number Theory and Physics Volume 1 (2007) Number 1 ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))
[[!redirects 't Hooft operators]]
[[!redirects 't Hooft operator]]
[[!redirects 't Hooft operators]]
[[!redirects 't Hoof line]]
[[!redirects 't Hoof lines]]
|
ℓ-adic cohomology | https://ncatlab.org/nlab/source/%E2%84%93-adic+cohomology |
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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
$\ell$-Adic cohomology is a [[cohomology theory]] on suitable [[varieties]] which is constructed as an [[inverse limit]] of [[étale cohomology]] over different [[coefficients]].
$\ell$-Adic cohomology is a [[Weil cohomology theory]]. It is endowed with a Galois action and therefore gives us a way of obtaining [[Galois modules]] from [[algebraic varieties]].
## Definition
### Over the étale site
{#OverEtaleSite}
Let $X$ be a (smooth?) [[proper map|proper]] [[variety]] over a field. Fix $\ell$ a [[prime number]] different from the [[characteristic]] of $k$. The $\ell$-adic cohomology is defined to be the [[cohomology theory]] on the [[étale site]] given by the [[inverse limit]] over $n \in \mathbb{N}$
$$
H^j_{et}(X_{\overline{k}}, \mathbb{Q}_\ell)
\coloneqq
\lim_n
H^j_{et}(X_{\overline{k}}, \mathbb{Z}/\ell^n\mathbb{Z})\otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_\ell
$$
of [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/\ell^n\mathbb{Z}$.
The key insights into getting finite dimensionality with coefficients in a [[field]] of [[characteristic]] $0$ when $k$ has positive characteristic is to first base change to $\overline{k}$ to make the theory "geometric." Then only work with torsion sheaves so that appropriate finiteness theorems for [[étale cohomology]] of proper varieties can be used, and then pass to the limit.
Notice that on the left $\mathbb{Q}_{\ell}$ is not an actual sheaf whose actual [[sheaf cohomology]] is being computed instead the expression on the left is defined by the genuine sheaf cohomology groups on the right.
### Over the pro-étale site
This is rectified by passing to [[pro-étale cohomology]]. Here $\mathbb{Q}_{\ell}$ exists as an actual sheaf and its genuine [[abelian sheaf cohomology]] is $\ell$-adic cohomology.
([Bhatt-Scholze 13, section 6.8](#BhattScholze13))
## Related concepts
* [[Weil cohomology]]
* [[étale cohomology]], [[basics of étale cohomology]]
* [[pro-étale cohomology]]
* [[p-adic integers]]
* [[p-adic geometry]]
* [[p-adic homotopy]]
* [[p-adic cohomology]]
* [[p-adic physics]]
## References
A textbook account is in
* [[James Milne]], section 19 of _[[Lectures on Étale Cohomology]]_ ([web] (http://www.jmilne.org/math/CourseNotes/lec.html))
Surveys include
* [[Akhil Mathew]], _$l$-adic cohomology and exponential sums_ ([web](http://amathew.wordpress.com/tag/l-adic-cohomology/))
A variant of the [[étale site]], well adapted to the needs of $\ell$-adic cohomology, the _[[pro-étale site]]_ (locally contractible in some sense) is discussed in
* Bhargav Bhatt, [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
{#BhattScholze13}
[[!redirects l-adic cohomology]] |
∞-category > history | https://ncatlab.org/nlab/source/%E2%88%9E-category+%3E+history | < [[∞-category]]
[[!redirects ∞-category/history]]
[[!redirects ∞-category -- history]] |
∞-group cohomology | https://ncatlab.org/nlab/source/%E2%88%9E-group+cohomology | #Contents#
* table of contents
{:toc}
## Idea
For $\mathbf{H}$ an [[∞-topos]] and for $G \in Grp(\mathbf{H})$ an [[∞-group]] in $\mathbf{H}$, the higher analog of [[group cohomology]] of $G$ is the [[cohomology]] of the [[delooping]] object $\mathbf{B}G \in \mathbf{H}$.
Given a [[cocycle]] $\mathbf{B}G \longrightarrow \mathbf{B}^{n+1} A$ in the $\infty$-group cohomology of $G$, then its [[homotopy fiber]] (the [[principal ∞-bundle]] over $\mathbf{B}G$ that it modulates) is the corresponding [[∞-group extension]].
## Examples
* [[∞-Lie groupoid cohomology]]
## Related concepts
[[!include homotopy type representation theory -- table]]
[[!redirects infinity-group cohomology]]
[[!redirects ∞-group cocycle]]
[[!redirects ∞-group cocycles]]
[[!redirects infinity-group cocycle]]
[[!redirects infinity-group cocycles]] |
∞-group ∞-ring | https://ncatlab.org/nlab/source/%E2%88%9E-group+%E2%88%9E-ring |
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[group object]] in the context of [[homotopy theory]] then the [[free construction]] of a [[ring object]] from it in [[stable homotopy category]] is the corresponding "[[group ring]]" construction generalized to homotopy theory.
## Definition
One may consider the construction at various levels of algebraic structure:
* for [[∞-groups]] and [[E-∞ rings]]: _[∞-Group E-∞ rings](InfinityGroupInfintyRings)_
* for [[H-groups]] and [[ring spectra]]: _[H-group ring spectra](#HGroupRingSpectra)_
### $\infty$-Group $\infty$-rings
{#InfinityGroupInfintyRings}
+-- {: .num_defn #GroupOfUnitsFunctor}
###### Definition
Write
$$
gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty
$$
for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].
=--
+-- {: .num_theorem}
###### Theorem
The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a right-[[adjoint (∞,1)-functor]]
$$
CRing_\infty
\stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
AbGrp_\infty
\,.
$$
=--
This is ([ABGHR 08, theorem 2.1/3.2, remark 3.4](#ABGHR08)).
+-- {: .num_remark}
###### Remark
The [[left adjoint]]
$$
\mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty
$$
is a higher analog of forming the [[group ring]] of an ordinary [[abelian group]] over the [[integers]]
$$
\mathbb{Z}[-] \colon Ab \to CRing
\,,
$$
which is indeed [[left adjoint]] to forming the ordinary [[group of units]] of a ring.
We might call $\mathbb{S}[A]$ the **[[∞-group ∞-ring]]** of $A$ over the [[sphere spectrum]].
=--
### H-Group ring spectra
{#HGroupRingSpectra}
We consider here the simpler concept after passage to [[equivalence classes]].
Recall
the [[classical homotopy category]] $(Ho(Top), \times, \ast)$ which is a [[symmetric monoidal category]] with respect to forming [[Cartesian product|Cartesian]] [[product spaces]] ([[tensor unit]] is the [[point space]])
its [[pointed objects]] version $(Ho(Top^{\ast/}), \wedge, S^0)$, which is a [[symmetric monoidal category]] with respect to [[smash product]] of [[pointed topological spaces]] ([[tensor unit]] is the [[0-sphere]])
the [[stable homotopy category]] $(Ho(Spectra), \wedge, \mathbb{S})$ which is a [[symmetric monoidal category]] with respect to the [[smash product of spectra]] ([[tensor unit]] is the [[sphere spectrum]])
There is a [[free-forgetful adjunction]]
$$
Ho(Top^{\ast/})
\underoverset
{\underset{U}{\longrightarrow}}
{\overset{(-)_+}{\longleftarrow}}
{\bot}
Ho(Top)
$$
The [[left adjoint]] functor $(-)_+$ adjoins basepoint. This is a [[strong monoidal functor]] (by [this example](pointed+object#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces)) in that there is a [[natural isomorphism]]
$$
(X \times Y)_+ \simeq X_+ \wedge Y_+
\,.
$$
Then there is the [[stabilization]] [[adjunction]]
$$
Ho(Spectra)
\underoverset
{\underset{\Omega^\infty}{\longrightarrow}}
{\overset{\Sigma^\infty}{\longleftarrow}}
{}
Ho(Top^{\ast/})
$$
(by [this prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsIndeedStabilizationOfClassicalHomotopyCategory))
Again the left adjoint is a [[strong monoidal functor]] in that there is a [[natural isomorphism]]
$$
\Sigma^\infty( (X,x) \wedge (Y,y) )
\simeq
\left(\Sigma^\infty (X,x)\right)
\wedge
\left(
\Sigma^\infty (Y,y)
\right)
$$
(by [this prop](Introduction+to+Stable+homotopy+theory+--+1-2#SuspensionSpectrumStructuredStrictQuillenAdjunction))
Accordingly also the composite functor
$$
\mathbb{S}[-] \coloneqq \Sigma^\infty(-)_+
$$
given by
$$
Ho(Spectra)
\underoverset
{\underset{\Omega^\infty}{\longrightarrow}}
{\overset{\Sigma^\infty}{\longleftarrow}}
{}
Ho(Top^{\ast/})
\underoverset
{\underset{U}{\longrightarrow}}
{\overset{(-)_+}{\longleftarrow}}
{\bot}
Ho(Top)
$$
are strong monoidal.
That $\mathbb{S}[-] = \Sigma^\infty((-)_+)$ is [[strong monoidal functor]] for a [[monoid]] $(A,\mu)$ in $(Ho(Top), \times , \ast)$ (an [[H-space]]), then also
$$
\Sigma^\infty (A_+)
\;\in\;
Ho(Spectra)
$$
is a monoid. This is the "monoid ring spectrum" of $A$.
If $A = G$ is in fact a [[group object]] in $(Ho(Top),\times, \ast)$ (hence an [[H-group]]), then we may call the ring spectrum
$$
\Sigma^\infty(G_+)
\;\in \;
Ho(Spectra)
$$
the _H-group ring spectrum_ of $G$.
+-- {: .num_remark #HGroupRingSpectrumSplitsAsDirectSumWithSphereSpectrum}
###### Remark
**(H-group ring spectrum is a direct sum with the [[sphere spectrum]])**
Notice that an [[H-group]] $G$ already is canonically a [[pointed object]] itself, pointed by its [[neutral element]] $e \colon \ast \to G$. Regarded as an object $(G,e) \in Ho(Top^{\ast/})$ this way then the pointed object $G_+$ above is equivalently the [[wedge sum]] of $G$ with the [[0-sphere]]:
$$
G_+ \simeq (G,e) \vee S^0
\,.
$$
Since $\Sigma^\infty$ preserves [[wedge sum]], this means that there is an [[isomorphism]] in $Ho(Spectra)$
$$
\begin{aligned}
\Sigma^\infty(G_+)
& \simeq
\Sigma^\infty
\left(
(G,e)
\vee
S^0
\right)
\\
& \simeq
\left( \Sigma^\infty(G,e) \right) \vee \mathbb{S}
\\
& \simeq
\left( \Sigma^\infty(G,e) \right) \oplus \mathbb{S}
\end{aligned}
$$
(where the last isomorphism exhibits that [[wedge sum]] is the [[direct sum]] in the [[additive category]] $Ho(Spectra)$ (by [this lemma](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryHasCoproducts))).
=--
## Examples
* The localization of the $E_\infty$-group ring $\Sigma^\infty(BU(1)_+)$ of the [[circle 2-group]] at the [[Bott element]] $\beta$ is equivalently the [[Brown representability theorem|representing]] [[spectrum]] [[KU]] of complex [[topological K-theory]]:
$\mathbb{S}[B U(1)][\beta^{-1}] \simeq KU$
This is _[[Snaith's theorem]]_.
## References
* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))
[[!redirects H-group ring spectrum]]
[[!redirects H-group ring spectra]]
[[!redirects H-space ring spectrum]]
[[!redirects H-space ring spectra]]
[[!redirects ∞-group ∞-rings]]
[[!redirects ∞-group E-∞-ring]]
[[!redirects ∞-group E-∞-rings]]
[[!redirects ∞-group E-∞ ring]]
[[!redirects ∞-group E-∞ rings]]
[[!redirects ∞-group E-infinity-ring]]
[[!redirects ∞-group E-infinity-rings]]
[[!redirects ∞-group E-infinity ring]]
[[!redirects ∞-group E-infinity rings]]
[[!redirects ∞-group ring]]
[[!redirects ∞-group rings]]
[[!redirects group ∞-ring]]
[[!redirects group ∞-rings]]
[[!redirects infinity-group infinity-ring]]
[[!redirects infinity-group infinity-rings]]
[[!redirects infinity-group ring]]
[[!redirects infinity-group rings]]
[[!redirects group infinity-ring]]
[[!redirects group infinity-rings]]
|
∞-groupoid > history | https://ncatlab.org/nlab/source/%E2%88%9E-groupoid+%3E+history | < [[∞-groupoid]]
[[!redirects ∞-groupoid/history]]
[[!redirects ∞-groupoid -- history]] |
∞-stack > history | https://ncatlab.org/nlab/source/%E2%88%9E-stack+%3E+history | < [[∞-stack]]
[[!redirects ∞-stack/history]]
[[!redirects ∞-stack -- history]] |
∞Grpd > history | https://ncatlab.org/nlab/source/%E2%88%9EGrpd+%3E+history | < [[∞Grpd]]
[[!redirects ∞Grpd/history]]
[[!redirects ∞Grpd -- history]] |
道德經 | https://ncatlab.org/nlab/source/%E9%81%93%E5%BE%B7%E7%B6%93 | [[!redirects 道德经]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
# Tao Te Ching
* table of contents
{: toc}
(...)
## On nothing, being and becoming
On [[being]], [[nothing]] and [[becoming]] (English translation following [[Xiao-Gang Wen]] [here](http://physics.stackexchange.com/a/94612/5603)):
> The nameless nonbeing is the origin of universe;
> The named being is the mother of all observed things.
> Within nonbeing, we enjoy the mystery of the universe.
> Among being, we observe the richness of the world.
> Nonbeing and being are two aspects of the same mystery.
> From nonbeing to being and from being to nonbeing is the gateway to all understanding.
Notice the similarity to ([Hegel 12](#Hegel12)), for more on this see ([Hegel 27](#Hegel27)).
## Related entries
* [[ontology]], [[metaphysics]]
* [[logos (in philosophy)]], [[Nous]]
* [[hermeticism]]
## References
* Wikipedia, _[Tao Te Ching](http://en.wikipedia.org/wiki/Tao_Te_Ching)_
* Bureau of Public Secrets _[Lao Tzu: Tao Te Ching](http://www.bopsecrets.org/gateway/passages/tao-te-ching.htm)_
* {#Hegel27} [[Georg Hegel]], _[[Lectures on the Philosophy of Religion]]_, Volume II: Determinate Religion. Edited by Peter C. Hodgson; translated by R.F. Brown, P.C. Hodgson, and J.M. Stewart, with the assistance of J.P. Fitzer and H.S. Harris. Berkeley: University of California Press, 1995 (orig. 1987). (Translation of: Vorlesungen über die Philosophie der Religion.) This extract (pp. 556-561) is from the Lectures of 1827; A. Immediate Religion, or Nature Religion; 1. The Religion of Magic; c. The State Religion of the Chinese Empire and the Dao. ([web](http://www.autodidactproject.org/quote/hegel-tao1.html))
* {#Hegel12} [[Georg Hegel]], _[[Science of Logic]]_ (1812)
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[[!redirects Dào Dé J?ng]]
[[!redirects Dàodéj?ng]]
[[!redirects Dao De Jing]]
[[!redirects Daodejing]]
[[!redirects Tao Te Ching]]
[[!redirects Taoteching]]
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