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Viktor Vassiliev | https://ncatlab.org/nlab/source/Viktor+Vassiliev | __Viktor A. Vassiliev__, a Russian geometer, a pupil of [[Vladimir Arnold]]. He is also active in the questions of education and the well-being of the academic community in contemporary Russia.
* [webpage](http://www.hse.ru/en/org/persons/1297545) at Higher School of Economics, Moscow
## Selected writings
Introducing [[Vassiliev knot invariants]]:
* [[Viktor Vassiliev]], _Complements of discriminants of smooth maps: topology and applications_, Amer. Math. Soc. 1992 ([ams:mmono-98](https://bookstore.ams.org/mmono-98))
Further:
* _Ramified integrals, singularities and lacunas_, Kluwer 1994, 289+xvii pp. Russian version: Ветвящиеся интегралы, Moscow 2002, 432 pp.
* _Applied Picard-Lefschetz theory_, Mathematical Surveys and Monographs __97__. Amer. Math. Soc. 2002. xii+324 pp. [MR2003k:32043](http://www.ams.org/mathscinet-getitem?mr=1930577)
* _Stratified Picard-Lefschetz theory_,
Selecta Math. (N.S.) 1 (1995), no. 3, 597–621, [MR96i:32037](http://www.ams.org/mathscinet-getitem?mr=1366625), [doi](http://dx.doi.org/10.1007/BF01589499)
* _Complements of discriminants of smooth maps: topology and applications_, Amer. Math. Soc. 1992.
* _Complexes of connected graphs_, [arxiv/1409.5999](http://arxiv.org/abs/1409.5999)
On [[cocycle spaces]] in ([[equivariant cohomotopy|equivariant]]) [[Cohomotopy]]:
* [[Victor Vassiliev]], _Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials_ ([arXiv:1809.05632](https://arxiv.org/abs/1809.05632))
* [[Victor Vassiliev]], *Cohomology of spaces of Hopf equivariant maps of spheres* ([arXiv:2102.07157](https://arxiv.org/abs/2102.07157))
## Related entries
* [[Vassiliev knot invariant]]
* [[Picard-Lefschetz theory]]
* [[singularity theory]]
* [[Kontsevich integral]]
category: people
[[!redirects V. A. Vassiliev]]
[[!redirects Viktor A. Vassiliev]]
[[!redirects Victor Vassiliev]] |
Viktoriya Ozornova | https://ncatlab.org/nlab/source/Viktoriya+Ozornova |
* [website](https://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Lehrstuhl-XIII/ozornova.html)
## Selected writings
A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]
On classes of [[categories]] which are [[fibrant object|fibrant]] in the [[Thomason model structure]]:
* [[Lennart Meier]], [[Viktoriya Ozornova]], _Fibrancy of partial model categories_, Homology, Homotopy and Applications, Volume 17 (2015) Number 2
([arXiv:1408.2743](https://arxiv.org/abs/1408.2743), [doi:10.4310/HHA.2015.v17.n2.a5](http://dx.doi.org/10.4310/HHA.2015.v17.n2.a5))
On the [[adjunction unit]] of [[total décalage]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *The unit of the total décalage adjunction*, Journal of Homotopy and Related Structures **15** (2020) 333–349 [[doi.org/10.1007/s40062-020-00257-1](https://doi.org/10.1007/s40062-020-00257-1)]
A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]
A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics **421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980)]
category:people |
Ville Lahtinen | https://ncatlab.org/nlab/source/Ville+Lahtinen |
* [GoogleScholar page](https://scholar.google.com/citations?user=fOosCKgAAAAJ&hl=en)
## Selected writings
On [[topological quantum computation]]:
* {#LahtinenPachos17} [[Ville Lahtinen]], [[Jiannis K. Pachos]], _A Short Introduction to Topological Quantum Computation_, SciPost Phys. **3** 021 (2017) [[arXiv:1705.04103](https://arxiv.org/abs/1705.04103), [doi:10.21468/SciPostPhys.3.3.021](https://scipost.org/SciPostPhys.3.3.021)]
category: people |
Vin de Silva | https://ncatlab.org/nlab/source/Vin+de+Silva |
* [institute page](https://www.pomona.edu/directory/people/vin-de-silva)
## Selected writings
Introducing [[zigzag persistence]], connecting [[persistent homology]] to [[quiver representation]]-theory, re-proving [[Gabriel's theorem]] for the case of A-type [[quivers]] and introducing the [[diamond principle]]:
* {#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)$]$
and applied 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)$]$
On [[persistence modules]] and [[stability of persistence diagrams]]:
* [[Frédéric Chazal]], [[Vin de Silva]], [[Marc Glisse]], [[Steve Oudot]], *The structure and stability of persistence modules*, SpringerBriefs in Mathematics, Springer (2016) $[$[arXiv:1207.3674](https://arxiv.org/abs/1207.3674), [doi:10.1007/978-3-319-42545-0](https://doi.org/10.1007/978-3-319-42545-0)$]$
category: people |
Vincent Bouchard | https://ncatlab.org/nlab/source/Vincent+Bouchard |
* [webpage](http://www.perimeterinstitute.ca/people/vincent-bouchard)
## related $n$Lab entries
* [[topological recursion]]
* [[topological string]]
category: people |
Vincent Braunack-Mayer | https://ncatlab.org/nlab/source/Vincent+Braunack-Mayer | [[!redirects Vincent Schlegel]]
> (formerly V. Schlegel)
* [webpage](http://www.math.uzh.ch/index.php?assistenten&key1=9218)
## Selected writings
On dg-algebraic models for [[rational parameterized stable homotopy theory]]:
* [[Vincent Braunack-Mayer]]
_**[[schreiber:thesis Braunack-Mayer|Rational parametrised stable homotopy theory]]**_
Zurich 2018
[pdf](https://ncatlab.org/schreiber/files/VBM_RPSHT.pdf)
* [[Vincent Braunack-Mayer]],
**Strict algebraic models for rational parametrised spectra I**
Algebraic & Geometric Topology 21 (2021) 917–1019
[arXiv:1910.14608](https://arxiv.org/abs/1910.14608)
[doi:10.2140/agt.2021.21.917](https://doi.org/10.2140/agt.2021.21.917)
On [[combinatorial model category|combinatorial]] [[model structures for parameterized spectra]]:
* [[Vincent Braunack-Mayer]],
**Combinatorial parametrised spectra**
Algebr. Geom. Topol. **21** (2021) 801-891
[[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]
On [[rational parameterized stable homotopy theory]] applied to the mathematical analysis of the [[duality between M-theory and type IIA string theory]]:
* [[nLab:Vincent Braunack-Mayer]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]]
_**[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes rational parameterized stable homotopy theory]]**_
Communications in Mathematical Physics (2019)
[doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4)
[arXiv:1806.01115](https://arxiv.org/abs/1806.01115)
* [[nlab:Vincent Braunack-Mayer]]
_**[[schreiber:Parametrised homotopy theory and gauge enhancement]]**_
talk at
_[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_
Durham Symposium 2018
Fortschritte der Physik (2019)
[doi:10.1002/prop.201910003](https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201910003)
[arXiv:1903.02862](https://arxiv.org/abs/1903.02862)
## Related $n$Lab entries
* [[tangent (∞,1)-topos]], [[parametrized homotopy theory]]
* [[rational parameterized stable homotopy theory]]
* [[twisted differential cohomology]]
category: people |
Vincent Danos | https://ncatlab.org/nlab/source/Vincent+Danos |
* [GoogleScholar page](https://scholar.google.com/citations?user=Q0aaJVUAAAAJ&hl=en)
## Selected writings
On [[measurement-based quantum computation]]:
* [[Vincent Danos]], [[Elham Kashefi]], [[Prakash Panangaden]], *The Measurement Calculus*, Journal of the ACM, **54** 2 (2007) [[arXiv:0704.1263](https://arxiv.org/abs/0704.1263), [doi:10.1145/1219092.1219096](https://doi.org/10.1145/1219092.1219096)]
category: people
|
Vincent Lafforgue | https://ncatlab.org/nlab/source/Vincent+Lafforgue |
* [webpage](http://www.math.jussieu.fr/~vlafforg/)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Vincent_Lafforgue)
category: people
|
Vincent Laurence Rouleau | https://ncatlab.org/nlab/source/Vincent+Laurence+Rouleau |
* [webpage](http://137.122.43.51/cgi-bin/mat/people/create_list?lastname=Laurence%20Rouleau&firstnames=Vincent&language=Francais)
## related entries
* [[transcendental syntax]]
## writings
* _Towards an understanding of Girard's transcendental syntax: Syntax by testing_, PhdD thesis 2013 ([pdf](https://www.ruor.uottawa.ca/fr/bitstream/handle/10393/23680/Laurence_Rouleau_Vincent_2013_thesis.pdf?sequence=3))
category: people
|
Vincent Mourik | https://ncatlab.org/nlab/source/Vincent+Mourik |
* [institute page](https://www.fqt.unsw.edu.au/staff/vincent-mourik-0)
* [GoogleScholar page](https://scholar.google.com/citations?user=VbOBK0YAAAAJ&hl=en)
## Selected writings
On (non-)detection of [[Majorana zero modes]] in [[experiment]]:
* {#FrolovMourik22} [[Sergey M. Frolov]], [[Vincent Mourik]], *We cannot believe we overlooked these Majorana discoveries* [[arXiv:2203.17060](https://arxiv.org/abs/2203.17060), [doi:10.5281/zenodo.6364928](https://doi.org/10.5281/zenodo.6364928), conclusion on: [p. 7](https://arxiv.org/ftp/arxiv/papers/2203/2203.17060.pdf#page=7)]
* {#Frolov22} [[Sergey M. Frolov]], [[Vincent Mourik]], *Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review* [[video](https://www.youtube.com/watch?v=RnYghkDaHH0), conclusion at: [1:01:31](https://www.youtube.com/watch?v=RnYghkDaHH0&t=3691s)]
category: people |
Vincent Pasquier | https://ncatlab.org/nlab/source/Vincent+Pasquier |
* [GoogleScholar page](https://scholar.google.com/citations?user=QQywOzkAAAAJ&hl=fr)
## Selected writings
On [[Drinfeld doubles]] of [[quasi-Hopf algebras]], motivated from [[rational conformal field theory|rational]] [[orbifold]] [[conformal field theories]]:
* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], *QuasiHopf algebras, group cohomology and orbifold models*, Nucl. Phys. B Proc. Suppl **18** (1990) 60-72 [<a href="https://doi.org/10.1016/0920-5632(91)90123-V">doi:10.1016/0920-5632(91)90123-V</a>]
* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], _Quasi-quantum groups related to orbifold models_, *International Colloquium on Modern Quantum Field Theory* (Bombay, 1990), 375-383, World Sci. (1991) [[cds:206306](https://cds.cern.ch/record/206306?ln=ka), [[DPR-QuasiQuantumGroups.pdf:file]]]
category: people |
Vincent Vargas | https://ncatlab.org/nlab/source/Vincent+Vargas |
* [personal page](https://www.math.ens.fr/~vargas/)
## Selected writings
On the rigorous construction of [[Liouville theory]]:
* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))
* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]
* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics, Pages 81-166 from Volume 191 (2020), Issue 1, ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2))
* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *The DOZZ formula from the path integral*, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) <a href="https://doi.org/10.1007/JHEP05(2018)094">doi:10.1007/JHEP05(2018)094</a>)
review:
* [[Vincent Vargas]], *Lecture notes on Liouville theory and the DOZZ formula* ([arXiv:1712.00829](https://arxiv.org/abs/1712.00829))
and via the [[conformal bootstrap]]:
* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))
and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88):
* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]
category: people
|
vine | https://ncatlab.org/nlab/source/vine |
+-- {: .rightHandSide}
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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea
[Lavers (1997)](#Lavers97) introduced the [[monoid]] $V_n$ of *$n$-vines*, whose elements are thought of as paths between two sets of $n$ distinct points in $\mathbb{R}^3$ which are allowed to merge into a single path, but not separate again. Some examples, from Lavers' original paper, are depicted below.
\begin{imagefromfile}
"file_name": "vinesJPEG.jpg",
"width": 300
\end{imagefromfile}
Notice that $V_n$ is not a group because a general vine cannot be untangled to give the trivial vine which is $n$ paths straight down. However, via pre- and post-composition, the vine monoid is acted on by the [[braid group|braid group on $n$ strands]]. When two $n$-vines are composed any resulting strands which are only attached to one endpoint get retracted down to that endpoint, as in the following image (again from [Lavers 97](#Lavers97)):
\begin{imagefromfile}
"file_name": "vinescompose.jpg",
"width": 300
\end{imagefromfile}
More generally, one can consider vines from $n$ points to $m$ points. As a result vines can be assembled into a category, as described below.
## Definition
What is called the *category of vines* is the [[free construction|free]] [[braided monoidal category|braided]] [[strict monoidal category]] containing a [[braided monoid]] (i.e. an [[E2-algebra|$E_2$-algebra]]). It is also the [[PRO|PROB]] for braided monoids.
Concretely:
\begin{defn}
Let
* $P_I \coloneqq \big\{ (i,0,1) \,\big\vert\, i=1, 2, \ldots, m \big\}$
* $P_T \coloneqq \big\{ (i,0,0) \,\big\vert\, i=1, 2, \ldots,n \big\}$
be [[linear order|linearly ordered]] collections of points in [[Cartesian space|$\mathbb{R}^3$]].
Then an *$(m,n)$-vine* is a set of arcs $\{v_1,\ldots,v_m\}$ (i.e. piecewise linear maps from $[0,1]$ to $\mathbb{R}^3$) with the following properties:
1. $v_i(0)=(i,0,1)$,
1. $v_i(1)\in P_T$,
1. for all $h\in [0,1]$, $v_i(h)$ has $z$-coordinate $1-h$,
1. if $v_i(t)=v_j(t)$ for any $t\in (0,1]$ then $v_i(s)=v_j(s)$ for all $t\leq s\leq 1$.
\end{defn}
\begin{defn}\label{CategoryV}
Let $\mathbb{V}$ be the category with set of [[objects]] the [[natural numbers]] with set of [[morphisms]] $\mathbb{V}(m,n)$ being $(m,n)$-vines modulo a suitable notion of ambient isotopy that allows arcs to be deformed up to homotopy but not pass through one another.
\end{defn}
##Properties
The [[monoidal category|monoidal structure]] of $\mathbb{V}$ is given by addition of [[natural numbers]] and juxtaposition of vines. Because $\mathbb{V}$ contains the [[braid category]] as a [[subcategory]], it cannot be [[symmetric monoidal]], but it is [[braided monoidal category|braided monoidal]]. The braiding is the same as that of the braid category, the "$m$-over-$n$" braid depicted below.
\begin{imagefromfile}
"file_name": "braiding.jpg",
"width": 300
\end{imagefromfile}
\begin{exercise}
Show that the above is a braided monoidal structure on $\mathbb{V}$.
\end{exercise}
## References
* {#Lavers97} T. G. Lavers, *The theory of vines*, Comm. Algebra **25** 4 (1997) 1257–1284 [[doi:10.1080/00927879708825919](https://doi.org/10.1080/00927879708825919)]
* {#Weber2015}[[Mark Weber]], §6.3 of: _Internal algebra classifiers as codescent objects of crossed internal categories_. Theory and Applications of Categories, **30** 50 (2015) 1713–1792 [[tac:30/50](http://www.tac.mta.ca/tac/volumes/30/50/30-50abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/30/50/30-50.pdf)]
[[!redirects vines]]
[[!redirects vine monoid]]
[[!redirects vine monoids]]
[[!redirects vine category]]
[[!redirects vine categories]]
|
Vipul Naik | https://ncatlab.org/nlab/source/Vipul+Naik |
* [webpage](https://vipulnaik.com/)
## Selected writings
On [[separation axioms]]:
* {#Naik} [[Vipul Naik]], _Topology: The journey into separation axioms_ [[[Naik-SeparationAxioms.pdf:file]]]
category: people |
Virasoro algebra | https://ncatlab.org/nlab/source/Virasoro+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The __Virasoro algebra__ or _Virasoro [[Lie algebra]]_ is the nontrivial [[central extension]] of the [[Witt Lie algebra]] (the Lie algebra of the [[diffeomorphism group|group of diffeomorphisms]] of the [[circle]]). It is of central importance in some questions of [[complex analysis]], in [[conformal field theory]] and the study of [[affine Lie algebras]].
## Definition
The generators $L_n$ of the Virasoro algebra are indexed by an integer $n \in \mathbb{Z}$, and they satisfy the commutation relation
$$
[L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0}.
$$
Here, $c$ denotes the element of the algebra known as the **[[central charge]]**; it commutes with each generator,
$$
[L_n, c] = 0 \forall n.
$$
The factor of 1/12 is conventional and chosen for normalisation purposes.
## Related concepts
* [[W-algebra]]
[[!redirects Virasoro Lie algebra]] |
Virasoro-Shapiro amplitude | https://ncatlab.org/nlab/source/Virasoro-Shapiro+amplitude |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Virasoro-Shapiro amplitude_ is the [[string scattering amplitude]] [[n-point function|4-point function]] for four [[tachyon]] field insertions.
## Related entries
* [[Veneziano amplitude]]
* [[vacuum amplitude]]
## References
E.g. (16.47) in
* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013
[[!redirects Virasoro-Shapiro amplitudes]]
[[!redirects Virasoro amplitude]]
[[!redirects Virasoro amplitudes]]
|
virtual cohomological dimension | https://ncatlab.org/nlab/source/virtual+cohomological+dimension | #Contents#
* table of contents
{:toc}
## Idea
In [[group cohomology]] theory, if $G$ is a [[group]] which has [[torsion-free module|torsion-free]] subgroups of [[finite index subgroup|finite index]], then all such subgroups have the same [[cohomological dimension]]; this common dimension is called the **virtual cohomological dimension** of $G$ and denoted $vcd(G)$.
For example, $SL_n(\mathbb{Z})$ has infinite cohomological dimension, and yet
$$
vcd(SL_n(\mathbb{Z})) = \binom{n}{2}.
$$
## Related concepts
* [[Farrell-Tate cohomology]]
[[!include notions of dimension -- contents]]
|
virtual double category | https://ncatlab.org/nlab/source/virtual+double+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* tic
{: toc}
## Idea
A **virtual double category** or **$fc$-multicategory** is a common generalization of a [[monoidal category]], a [[bicategory]], a [[double category]], and a [[multicategory]]. It contains:
* objects
* vertical arrows, which form a category
* horizontal arrows, which do not have identities or composites, and
* 2-cells which have
* a horizontal source and target, which are vertical arrows,
* a vertical target, which is a horizontal arrow, and
* a vertical source, which is a composable string of horizontal arrows.
2-cells are usually drawn like this:
[[virtual-double-category-cell.png:pic]]
Note that this includes the case when $n=0$, i.e. a cell of "nullary" source. In this case, we must have $X_0 = X_n$.
Note that the "empty" case of a string of horizontal arrows this has a single object $X$ in which case the 2-cell looks like:
\begin{tikzcd}% https://q.uiver.app/?q=WzAsNCxbMSwwLCJYIl0sWzAsMiwiWV8wIl0sWzIsMiwiWV8xIl0sWzEsMSwiXFxEb3duYXJyb3cgXFxhbHBoYSJdLFswLDEsImYiLDJdLFswLDIsImciXSxbMSwyLCJxIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn19fV1d
& X \\
& {\Downarrow \alpha} \\
{Y_0} && {Y_1}
\arrow["f"', from=1-2, to=3-1]
\arrow["g", from=1-2, to=3-3]
\arrow["q"', "\shortmid"{marking}, from=3-1, to=3-3]
\end{tikzcd}
Finally, the 2-cells can be composed in a more or less evident way, akin to composition in a multicategory:
[[virtual-double-category-composite.png:pic]]
Virtual double categories are related to double categories precisely as ordinary multicategories are related to monoidal categories (see [[generalized multicategory]] and [[tensor product]]).
## Definition
A virtual double category can be defined in two equivalent ways:
* It is a $T$-[[generalized multicategory|multicategory]], in the sense of Leinster, relative to the monad $T$ on [[quiver|directed graphs]] whose algebras are categories. For this reason, Leinster originally called them **fc-multicategories**, where "fc" is a name for this monad $T$ which stands for "free-category."
* It is a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad $T$ on graphs-internal-to-Cat whose algebras are double categories. This is the origin of the name "virtual double category," in line with the general terminology "virtual $T$-algebra" of Cruttwell-Shulman for such generalized multicategories.
We can also give an explicit definition, which was more or less already given in the "Idea" section: all that is missing are identities and associativity for 2-cell composition.
## Examples
* Any double category is an example, and thus also any bicategory
viewing the arrows as horizontal.
* For any [[monoidal category]] $V$, there is a virtual double category
of $V$-matrices whose objects are sets, vertical arrows are functions
and a horizontal arrow $p : X \to Y$ is a family of objects $p_{y,x}
\in V$ for each $x\in X, y \in Y$, and a 2-cell from $X_0
\overset{p_1}{\to} X_1 \to\dots \to X_n$ to $Y_0 \overset{q}{\to} Y_1$
along $f : X_0 \to Y_0, g : X_n \to Y_n$ is a family of arrows
$\alpha_{x_0,...} : p_1(x_1,x_0)\otimes p_2(x_2,x_1)\otimes\dots \to
q(g(x_n),f(x_0))$ in $V$ (using the unit of the monoidal category if the source string is empty).
If $V$ has certain colimits that are preserved by $\otimes$ then
composites exist and this virtual double category is pseudo.
* Given a monad on a virtual double category, the [[generalized multicategory| horizontal kleisli
double category]] produces a virtual double
category that is only pseudo under strong conditions on the monad. In
particular, "free monoid" monad on the double category of sets and
spans does not produce a pseudo double category.
## Higher categories of virtual double categories
There are notions of **functor**, **transformation**, and **profunctor** between virtual double categories. The neatest way to define all of these notions at once is to use the general framework of [[generalized multicategories]]: from the monad $fc$ on the [[virtual equipment]] $Span = Span(Set)$ we can construct a new virtual equipment $vDblProf = KMod(Span,fc)$ whose objects are virtual double categories, whose arrows are functors between them, whose proarrows are profunctors between them, and whose cells are transformations. But we can also give explicit definitions of all of these notions.
### Functors and transformations
A **functor** of virtual double categories is fairly obvious; it takes each kind of morphism/cell to the same kind, preserving sources, targets, composition, and identities.
The relevant **transformations** are a "virtual" version of [[vertical transformations]] between ordinary double categories. Specifically, a transformation $\alpha\colon F\to G$ has a vertical arrow component $\alpha_X\colon F X\to G X$ for each object $X$ of the domain, and a cell component
$$\array{F X & \overset{F p}{\to} & F Y\\
^{\alpha_X}\downarrow & \Downarrow ^{\alpha_p}& \downarrow^{\alpha_Y}\\
G X& \underset{G p}{\to} & G Y}$$
for each horizontal arrow $p\colon X\to Y$ in the domain. These must be natural with respect to vertical composition of arrows and of 2-cells, where we must of course allow composites with arbitrary arities in the latter case.
Virtual double categories, functors, and transformations form a [[strict 2-category]], and thus we can apply all notions of 2-category theory to it. In particular, we have a notion of a [[monad]] *on* a virtual double category, which is the starting point for one theory of [[generalized multicategories]].
### Profunctors
The **profunctors** between virtual double categories are a similar "virtualization" of the notion of [[double profunctor]] between double categories. Explicitly, a profunctor $H\colon C ⇸ D$ consists of:
* An ordinary [[profunctor]] $H_0\colon C_0 ⇸ D_0$ between the categories of objects and vertical arrows.
* For each string of horizontal arrows $X_0 \overset{p_1}{\to} X_1 \to\dots \to X_n$ in $D$, each horizontal arrow $Y_0 \overset{q}{\to} Y_1$ in $C$, and each pair of elements $f \in H_0 (X_0,Y_0)$ and $g\in H_0(X_n,Y_1)$, a set of "hetero-cells" of shape
[[virtual-double-category-cell.png:pic]]
* The hetero-cells are acted on by the 2-cells of $D$ on the top, and by the 2-cells of $C$ on the bottom, in an evident way, respecting the given action of vertical arrows of $D$ and $C$ on the elements of $H_0$.
Every [[double profunctor]] induces such a profunctor in an evident way, but even if $C$ and $D$ are (non-virtual) double categories, not every "virtual double profunctor" from $C$ to $D$ need be a double functor; only those for which the "hetero-cells" also factor uniquely through the opcartesian cells in $D$ which make it "representable."
As mentioned above in the context of the abstract definition, virtual double categories, functors, transformations, and profunctors form another virtual double category, which is in fact a [[virtual equipment]].
### Monads on virtual double categories {#Monads}
+-- {: .un_defn}
###### Definition
A _monad on a virtual double category_ is a [[monad]] in the [[2-category]] [[vDbl]].
=--
So a monad on a $X \in vDbl$ consists of a functor
$$
T : \mathbb{X} \to \mathbb{X}
$$
and transformations $\eta : Id \to T$ and $\mu : T T \to T$ satisfying [[associativity]] and [[unitality]].
#### Monoids and modules
+-- {: .un_defn}
###### Definition
For $T$ a monad on $\mathbb{X} \in $ [[vDbl]], a **$T$-monoid** is
* an object $X_0 \in \mathbb{X}$;
* a horizontal morphism $X_0 \stackrel{X}{⇸} T X_0$
* an [[action]] [[2-morphism]]
$$
\array{
X_0 &\stackrel{X}{⇸} & T X_0
& \stackrel{T X}{⇸} T^2 & X_0
\\
{}^{\mathllap{=}}\downarrow && \Downarrow^{\bar x}
&& \downarrow^{\mathrlap{\mu}}
\\
X_0 && \underset{X}{⇸} && T X_0
}
$$
and a [[unit]] 2-morphism
$$
\array{
&& X_0
\\
& {}^{\mathllap{=}}\nearrow &\Downarrow^{\bar x}& \searrow^{\mathrlap{\eta}}
\\
X_0 &&\underset{X}{⇸}&& T X_0
}
$$
satisfying the evident compatibility conditions.
=--
This is ([CruttwellShulman, def. 4.2](#CruttwellShulman)).
+-- {: .un_defn}
###### Definition
A $T$-monoid $X_0 \stackrel{X}{⇸} T X_0$ is called **normalized** if its unit 2-morphism
$$
\array{
X_0 &\stackrel{U_{X_0}}{\to}& X_0
\\
{}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{\eta}}
\\
X_0 &\underset{X}{⇸}& T X_0
}
$$
is a [cartesian 2-morphism](#...).
=--
#### Generalized multicategories
+-- {: .un_defn}
###### Definition
A **[[generalized multicategory]]** is a normalized $T$-monoid for some monad $T$ on a [[virtual equipment]] $\mathbb{X} \in $ [[vDbl]].
=--
This is ([CruttwellShulman, page 7](#CruttwellShulman)).
## Enriching categories
Virtual double categories can be viewed as "the natural place in which to enrich categories." Specifically, for any set $A$, there is a virtual double category $A_{ch}$ which has $A$ as its objects, only identity vertical arrows, exactly one horizontal arrow from every object to every other object, and exactly one 2-cell in every possible niche. For any other virtual double category $W$, a functor $A_{ch}\to W$ of virtual double categories is the same as a $W$-enriched category with object set $A$.
## Related pages
* [[double category]]
* [[virtual equipment]]
* [[hypervirtual double category]]
## References
Virtual double categories were first considered by [[Albert Burroni]] under the name **multicatégorie** on page 61 of:
* [[Albert Burroni]], _$T$-catégories (catégories dans un triple)_, Cahiers de topologie et géométrie différentielle catégoriques **12**.3 (1971) 215-321 [[dml:91097](https://eudml.org/doc/91097), [pdf](http://www.numdam.org/article/CTGDC_1971__12_3_215_0.pdf)]
Later, they were considered in:
* [[Tom Leinster]], _Higher Operads, Higher Categories_, [link](http://www.maths.gla.ac.uk/~tl/book.html), [arXiv:0305049](http://arxiv.org/abs/math/0305049)
* [[Tom Leinster]], **fc**-multicategories, [arxiv](https://arxiv.org/abs/math/9903004) (1999)
* {#CruttwellShulman} [[Geoff Cruttwell]], [[Mike Shulman]], _A unified framework for generalized multicategories_ [[arXiv:0907.2460](http://arxiv.org/abs/0907.2460)]
On a [[string diagram]]-calculus for ([[virtual double category|virtual]]) [[double categories]] with ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:
* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]
* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]
[[!redirects virtual double categories]]
[[!redirects fc-multicategory]]
[[!redirects fc-multicategories]]
|
virtual equipment | https://ncatlab.org/nlab/source/virtual+equipment |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _proarrow equipment on a virtual double category_ or _virtual equipment_ for short is a [[virtual double category]] equipped with the same kind of structure that makes an ordinary [[double category]] into a [[proarrow equipment]].
Virtual equipments are the structures that support structures of [[generalized multicategories]].
## Definition
+-- {: .un_defn}
###### Definition
A **virtual equipment** is a [[virtual double category]] in which all units and all restrictions exist.
=--
This is ([CruttwellShulman, def. 7.6](#CruttwellShulman)).
## References
* {#CruttwellShulman} [[Geoff Cruttwell]], [[Mike Shulman]], Section 7 of: _A unified framework for generalized multicategories_, [[arXiv:0907.2460](http://arxiv.org/abs/0907.2460)]
On a [[string diagram]]-calculus for ([[virtual double category|virtual]]) [[double categories]] with ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:
* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]
* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]
[[!redirects virtual equipments]]
[[!redirects virtual proarrow equipment]] |
virtual fundamental class | https://ncatlab.org/nlab/source/virtual+fundamental+class |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _virtual fundamental class_ is a generalization of that of _[[fundamental class]]_ from [[manifolds]] to more general spaces in [[higher geometry]], notably to [[orbifolds]] and their equivalent incarnations as [[stacks]], as well as to [[derived manifolds]].
This plays a central role when these stacks serve as [[moduli stacks]] for certain structures on some space (certain maps into that space regarded as a [[target space]], notably) and one is interested in the relevant "[[path integral]]" over all these structures (to produce [[invariants]] of target space). This is manifestly so for instance in the application to [[Gromov-Witten invariants]]. In these cases the pairing of [[cocycles]] against the virtual fundamental class plays the role of [[integration]] over the given moduli stack.
## Definition
### For derived Schemes
Given a [[locally noetherian scheme|locally Noetherian]] [[derived scheme]], $(X, \mathcal{O}_X)$ with underlying [[scheme]] $t_0X$, $\pi_i\mathcal{O}_X$ [[coherent module|coherent]], and $\pi_i\mathcal{O}_X = 0$ for $i\gg0$, the _virtual fundamental class_ is defined by first constructing a class in $G_0(t_0X)$ (the [[K-theory]] of [[coherent sheaves]]) and then using this to produce an element of the [[Chow homology]] of $t_0X$. In our hearts we all know that the fundamental class should be something tautological, just like how the fundamental class of a triangulated manifold in simplicial homology is _the manifold itself_. We can use this idea once we have the following theorem:
+-- {: .num_theorem #Devissage}
###### Theorem
**(Devissage)**.
The natural map of [[derived schemes]] $j: t_0X \rightarrow X$ induces an [[isomorphism]] $j_*: G_0(t_0X) \rightarrow G_0(X)$.
=--
A proof of this theorem and a definition of [[G-theory]] for derived schemes in ([Barwick](#Barwick)).
+-- {: .num_defn #KTheoryVirtualFundamentalClass}
###### Definition
The $K$-theory _virtual fundamental class_ of a derived scheme $(X, \mathcal{O}_X)$ is the unique element $[X]$ in $G_0(t_0X)$ such that $j_*[X] = [\mathcal{O}_X]$ where $j: t_0X \rightarrow X$ is the natural map.
=--
If you trace through the definition of $G_0$ and the identification of the [[heart]] of $Coh(X)$ with $Coh(t_0X)$, you can show that $[X] = \sum (-1)^i [\pi_i\mathcal{O}_X]$.
To get from here to a virtual fundamental class in cohomology we need some more assumptions on $X$. For example, let's assume $X$ is finitely-presented over a field $k$ and that $\mathbb{L}_{X/k}$ is concentrated in degrees $0$ and $-1$, locally up to quasi-isomorphism, and a perfect complex. Then $j^*\mathbb{L}_{X/k}$ is a perfect complex over $t_0X$ and the dual is called the virtual tangent sheaf $\mathbb{T}^{vir}$.
+-- {: .num_defn}
###### Definition
With the above assumptions and notation, the _virtual fundamental class_ of $X$ is the evaluation of the inverse of the [[Todd class]] $Td(\mathbb{T}^{vir})$ on $\tau([X])$, where $[X]$ is as in def. \ref{KTheoryVirtualFundamentalClass} and $\tau$ is the [[Grothendieck-Riemann-Roch theorem|Grothendieck-Riemann-Roch transformation]].
=--
([Toën 14, p. 23](#Toen14))
## Examples
### Moduli space of stable maps
{#ModuliSpaceOfStableMaps}
Although the [[moduli space]] of [[stable maps]] is sometimes referred to as a compactifiaction of the space of maps, in analogy with the [[Deligne-Mumford compactification]] of the [[moduli space]] of curves, in
fact it typically has boundary components of higher dimension than the space it was supposed to compactify!
Take for example $\bar M_{1,0}(P^2,3)$. It ought to be a compactification of the space of degree-3 maps from genus-1 curves to $P^2$, and indeed one of its components has a Zariski open subset birational to the $P^9$ of all plane cubics. But there is also a 'boundary component' of
higher dimension, namely the boundary component consisting of maps whose domain is a genus-1 curve glued to a nodal rational curve: the nodal curve maps to a rational cubic in $P^2$, while the $g=1$ component contracts to a point on that nodal cubic.
This boundary component has
dimension 10: namely, there are 8 parameters to specify the image nodal cubic, 1 paramenter to determine the point to which the $g=1$ component contracts, and finally there is 1 paramenter for the
[[j-invariant]] for the $g=1$ component. The topological fundamental class lives in dimension 10 so it is rather useless to integrate against if all your cohomology classes are codimension 9 --- which is the
expected dimension.
The virtual fundamental class always lives in the
expected dimension.
(The expected dimension is often the one you would expect(!) from
naive counts like the above. More formally it can be computed as dim
$H^0(C,N_f)$, where $f:C\to P^2$ is a moduli point (with [[normal bundle]]
$N_f$) such that $H^1(C,N_f)=0$ (this is to say that the first order [[infinitesimal space|infinitesimal]] deformations are unobstructed).)
The situation is analogous (possibly in fact a special case of) the standard situation in [[intersection theory]] when a section of a [[vector bundle]] is not regular: its zero locus is then of too high dimension
and is of little use to intersect against. The correct class to work with is then the top [[Chern class]] of the vector bundle (cf.
[Fulton] ch.14), which could be called the virtual class of the zero locus.
In the example above, I don't know right now if the virtual class in fact appears as a top Chern class of a vector bundle --- I think it should, because the excess is just a variation of the standard example $\bart M_{1,1}(X,0)$, and in that example it is true that the virtual class appears as a top [[Chern class]]: there is a so-called [[obstruction bundle]] which in this case is the dual of the [[Hodge bundle]]
from the factor $\bar M_{1,1}$ tensored with the [[tangent bundle]] from $X$.
(The Hodge bundle is the direct image bundle of the canonical bundle of the universal curve, hence of rank $g$, hence just a line bundle in this case.)
The virtual fundamental class is the top Chern class of
the obstruction bundle (cap the topological fundamental class).
In this case, $dim \bar M_{1,1}(X,0) = 1 + dim X$, and the obstruction bundle has rank $dim X$, hence the virtual class has dimension 1.
Perhaps it should be mentioned also that the moduli space of maps can have components of too high dimension even before it is 'compactified', and even without involving contracting curves. A
famous example is $M_{0,0}(Q,d)$ (no bar needed for this argument) where
$Q$ is a quintic three-fold. Let's say $d=2$, so we are talking about conics on the quintic three-fold. Since $Q$ has trivial [[canonical class]] it follows that the expected dimension is always 0 (i.e. in every
degree there ought to be a finite number of rational curves on Q).
But now, $M_{0,0}(Q,2)$ is a space of maps, not a space of curves, and for every one of the famous 2875 lines on $Q$ there is a 2-dimensional family of double covers of the line, which clearly count as stable
degree-2 maps, so $M_{0,0}(Q,2)$ contains 2875 components of dimension 2, in contrast to the virtual dimension 0.
## Related entries
* [[fundamental class]]
* [[Poincaré duality algebra]]
* [[intersection theory]]
* Virtual fundamental classes play a central role in the theory of [[Gromov-Witten invariants]].
## References
The idea of virtual fundamental classes and corresponding picture of derived moduli spaces comes from
* [[Maxim Kontsevich]], _Enumeration of rational curves via torus actions_, in: The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. __129__, Birkhäuser 1995. MR1363062 (97d:14077), [hep-th/9405035](http://arxiv.org/abs/hep-th/9405035)
A quick overview of virtual fundamental classes for [[algebraic stacks]] is in section 4.4.3 of
* [[Bertrand Toën]], _Higher and derived stacks: a global overview_ ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504))
A more detailed overview with many pointers to the literature is in section 8 "Introduction to the virtual fundamental class" of
* Nicola Pagani, _Introduction to Gromov-Witten theory and quantum cohomology_ ([pdf](http://pcwww.liv.ac.uk/~pagani/DOC/corsogw.pdf))
A review of virtual fundamental classes of [[orbifolds]] in [[differential geometry]] is around page 7 of
* Andrés Angel, _When is a differentiable manifold the boundary of an orbifold?_ ([pdf](http://ems.math.uni-bonn.de/people/aangel79/files/boundary.pdf))
Detailed disucssion for the [[branched manifold]]-variant of [[orbifolds]] is in section 3.4 of
* [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_ ([arXiv:math/0509664](http://arxiv.org/abs/math/0509664))
Virtual fundamental classs of [[derived manifolds]] are discussed in section 13.2 of
* [[Dominic Joyce]], _D-manifolds and d-orbifolds: a theory of derived differential geometry_ ([pdf](http://people.maths.ox.ac.uk/joyce/dmbook.pdf))
with applications to [[moduli spaces]] of coherent sheaves on [[Calabi-Yau space|Calabi-Yau 4-folds]] in
* [[Dominic Joyce]], [[Dennis Borisov]], _Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds_ (2014) ([wev](https://docs.google.com/file/d/0B4KuPQcvke9sOS1mTDBhNWxocnc/edit))
Specifically for applications to [[Gromov-Witten theory]] see for instance
* Ch. Okonek, A. Teleman, _Computing virtual fundamental classes: gauge theoretical Gromov-Witten invariants for toric varieties_ ([arXiv:math/0205137](http://arxiv.org/abs/math/0205137))
The material in _[Moduli space of stable maps](#ModuliSpaceOfStableMaps)_ above originates in a blog discussion [here](http://golem.ph.utexas.edu/category/2009/09/a_seminar_on_gromovwitten_theo.html#c027000)
The definition and description above for derived schemes derives from p.23 of
* [[Bertrand Toën]] Derived Algebraic Geometry, 2013. ([http://arxiv.org/pdf/1401.1044v1.pdf](http://arxiv.org/pdf/1401.1044v1.pdf))
{#Toen14}
The devissage result needed is proven as Proposition 8.2 here:
* [[Clark Barwick]] _On Exact $\infty$-categories and the Theorem of the Heart ([pdf](http://dl.dropboxusercontent.com/u/1741495/papers/k2-1.pdf))
{#Barwick}
[[!redirects virtual fundamental classes]] |
virtual knot theory | https://ncatlab.org/nlab/source/virtual+knot+theory | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
One fundamental tool in a knot theorist's toolbox is the [[knot diagram]].
Classically, a [[knot]] is an embedding of the circle $S^1$ into Euclidean space $\mathbb{R}^3$, and knot diagrams arise by projecting back down to the [[plane]] $\mathbb{R}^2$ to obtain a 4-valent [[plane graph]] (referred to as the _shadow_ of the knot), while keeping track of whether each vertex corresponds to an under-crossing or an over-crossing.
Although the knot is traditionally seen as embedded in $\mathbb{R}^3$, it could equally well be realized inside of a "thickened" [[sphere]] $S^2 \times [0,1]$, with its shadow embedded in $S^2$.
From that perspective, _virtual knot theory_ generalizes classical knot theory by considering knots as embeddings of circles into thickened [[orientable]] surfaces $X \times [0,1]$ of arbitrary [[genus of a surface|genus]].
Abstractly, the shadow of such a knot is a 4-valent graph embedded in $X$ (i.e., a [[topological map]]).
However, if one tries to project the knot onto the plane, the corresponding diagram might contain crossings that do _not_ represent places where the knot passes over/under itself inside the thickened surface, but rather are artifacts of the knot's non-planar shadow.
So, in a _virtual knot diagram_ such crossings are explicitly indicated as "virtual", using a distinct notation from that for under/overcrossings.
## Definitions
There are various equivalent definitions of virtual knots/links:
* As equivalence classes of virtual knot/link diagrams, modulo the "virtual" [[Reidemeister moves]] [(Kauffman 1999)](#Kauffman1999).
* As equivalence classes of "abstract" knot/link diagrams embedded in orientable surfaces, modulo the Reidemeister moves [(Kamada and Kamada 2000)](#KamadaKamada).
* As embeddings of circles into thickened orientable surfaces, considered up to [[isotopy]] [(Kuperberg 2003)](#Kuperberg2003).
* As equivalence classes of [[Gauss codes]], modulo relations corresponding to the Reidemeister moves [(Kauffman 1999)](#Kauffman1999).
## Categorifications
The original work on virtual knot theory was not expressed in categorical language, but a first attempt at categorifying the _virtual braid group_ $VB_n$ was made by [Kauffman and Lambropoulou](#KL2011). [[Victoria Lebed]] studied the question extensively in her thesis [(Lebed 2012)](#LebedThesis), and developed a categorical approach based on the notion of a _[[braided object]]_ in a [[symmetric monoidal category]], that is an object $V$ equipped with an invertible morphism $\sigma : V\otimes V \to V\otimes V$ satisfying the [[Yang-Baxter equation]]
$$(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)$$
In particular, Lebed shows that the virtual braid group $VB_n$ is isomorphic to the group of endomorphisms $End_{\mathcal{C}_{2br}}(V^{\otimes n})$, where $\mathcal{C}_{2br}$ is the free symmetric monoidal category generated by a single braided object $V$.
The intuition here is that the (symmetric) braiding of the ambient symmetric monoidal category represents virtual crossings "for free", while the braiding $\sigma$ on the object $V$ represents "real" over- and under-crossings. (Compare some [remarks by John Baez](#Baez08), which are similar in spirit.)
## Terminology
Warning: a virtual knot/link has a genus in the sense of the genus of the underlying thickened surface into which it embeds (or equivalently, the genus of its shadow as a topological map), but this is unrelated to the classical notion of _knot genus_, in the sense of the minimal genus of a [[Seifert surface]] whose [[boundary]] is the knot.
## Related concepts
* [[Gauss diagram]]
* [[embedded graph]]
## References
The original paper on virtual knot theory and some early followup work:
* {#Kauffman1999} [[Louis Kauffman]], Virtual Knot Theory, _European Journal of Combinatorics_ (1999) 20, 663-691. [pdf](http://homepages.math.uic.edu/~kauffman/VKT.pdf)
* {#KamadaKamada} Naoko Kamada and Seiichi Kamada, Abstract Link Diagrams and Virtual Knots, _Journal of Knot Theory and its Ramifications_, Vol. 9, No. 1 (2000), 93-106. [doi](http://dx.doi.org/10.1142/S0218216500000049)
* {#Kuperberg2003} [[Greg Kuperberg]], What is a virtual link?, _Algebraic & Geometric Topology_ Volume 3 (2003), 587-591. [pdf](http://arxiv.org/pdf/math/0208039v2.pdf)
Miscellaneous papers:
* {#Kauffman2012} [[Louis Kauffman]], Introduction to Virtual Knot Theory. July 2012. [arXiv](http://arxiv.org/abs/1101.0665)
* {#Viro} Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference, pp. 184–209, 2005. [pdf](http://www.pdmi.ras.ru/~olegviro/ggt05-viro.pdf)
* {#ManturovIlyutko} Vassily Olegovich Manturov and Denis Petrovich Ilyutko, _Virtual Knots: The State of the Art_. Series on Knots and Everything (vol. 51), World Scientific, 2013.
* {#Kauffman2015} [[Louis Kauffman]], Rotational Virtual Knots and Quantum Link Invariants, 14 Oct 2015. [arxiv:1509.00578](http://arxiv.org/abs/1509.00578)
On the question of categorifying virtual knot theory, see:
* {#Baez08} [[John Baez]], [comment at the n-Café](https://golem.ph.utexas.edu/category/2008/10/categorification_in_new_scient.html#c019490) following a post titled "Categorification in New Scientist", October 2008.
* {#KL2011} [[Louis Kauffman]] and Sofia Lambropoulou, A Categorical Structure for the Virtual Braid Group, _Comm. Algebra_, 39(12):4679–4704, 2011. ([pdf](http://www.math.ntua.gr/~sofia/publications/A22%20VirtualPureMia-my%20pdf.pdf))
* {#LebedThesis} [[Victoria Lebed]], _Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles_, Thèse, Université Paris Diderot, 2012. ([pdf](http://www.maths.tcd.ie/~lebed/thesis_LEBED.pdf)) Note that the title is in French ("Braided objects: a unifying study of algebraic structures and a categorification of virtual braids") but the main text of the thesis is in English.
* {#Lebed13} [[Victoria Lebed]], Categorical Aspects of Virtuality and Self-Distributivity, _Journal of Knot Theory and its Ramifications_, 22 (2013), no. 9, 1350045, 32 pp. ([doi](http://dx.doi.org/10.1142/S0218216513500454)) According to the author, [arXiv:1206.3916](http://arxiv.org/abs/1206.3916) is "an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles". |
virtual machine | https://ncatlab.org/nlab/source/virtual+machine | In [[computer science]], a virtual machine is a simulation of a fixed architecture executing some sort of signaling or software. It can abstract an entire or part of hardware, firmware, a file system with some ports, devices and/or CPU and installed operating system, but it can be simply an isolated environment executing some specified class of software, or simply any installation in hardware or in another virtual machine of a specified execution model. For example, Java Virtual Machine executes formally specified "bytecode" to ensure identical execution on all hardware.
See also [[certified programming]], [[WebAssembly]]
## Virtual machines for cloud
### Virtual machine simulating machine with operating system
...
### Containers
...
### Blockchain runtimes for multiple blockchains
Microsoft Azure develops a runtime framework intended to work on various blockchains. This is the CoCo project (see whitepaper 2017 at github [pdf](https://github.com/Azure/coco-framework/blob/master/docs/Coco%20Framework%20whitepaper.pdf), [announcement](https://azure.microsoft.com/en-us/blog/announcing-microsoft-s-coco-framework-for-enterprise-blockchain-networks) and a presentation on [Medium](https://medium.com/@jrodthoughts/with-coco-framework-microsoft-wants-to-become-the-redhat-of-the-blockchain-5b35f8ac8707)).
## Languages for simulation of devices
...
## Bytecode virtual machines
### Java Virtual Machine
Java VM has a specification which executes Java bytecode. Several languages compile to JVM including Java, Kotlin and Scala.
* [java (language)](http://en.wikipedia.org/wiki/Java_%28programming_language%29), [java (platform)](http://en.wikipedia.org/wiki/Java_%28software_platform%29)
* [kotlin](https://kotlinlang.org)
* [scala](http://en.wikipedia.org/wiki/Scala_%28programming_language%29)
* [scalaz](https://github.com/scalaz/scalaz) - library for functional programming in scala
### WebAssembly VM
WebAssembly is optimized for small compiling time and near native execution time on major architectures (like 86 series). It appeared first as new VM standard
on web browsers, backed by major internet companies; it is also used or planned on a number of [[blockchain]] projects.
* WebAssembly [wikipedia](https://en.wikipedia.org/wiki/WebAssembly), [spec](https://webassembly.github.io/spec), [webassembly.org](https://webassembly.org), [msdn](https://developer.mozilla.org/en-US/docs/WebAssembly) docs, [rust-to-wasm](https://developer.mozilla.org/en-US/docs/WebAssembly/Rust_to_wasm), [github](tps://github.com/WebAssembly)
* [WebAssembly-Links](https://wiki.parity.io/WebAssembly-Links) in Parity Tech Documentation
* AssemblyScript (maps a subset of javascript code to wasm) [github](https://github.com/AssemblyScript/assemblyscript), [news](https://www.infoworld.com/article/3224006/assemblyscript-compiles-typescript-to-webassembly.html)
[[Rust]] has small runtime, which is desirable in common applications of WebAssembly. Thus Rust commonly compiles either to native code or to wasm.
* Rust & WebAssembly with Nick Fitzgerald [yt](https://www.youtube.com/watch?v=ZiiTRxWk8gA)
Ethereum flavoured version of wasm VM specification is at github/[ewasm](https://github.com/ewasm), see also github/[ewasm/design](https://github.com/ewasm/design). eWasm has a [testnet](http://ewasm.ethereum.org/explorer). According to article _[ewasm explained](https://www.mycryptopedia.com/ewasm)_,
> The ewasm specification consists of a subset of WebAssembly components suitable for Ethereum’s needs, namely determinism and relevant features. It also includes a number of system smart contracts that provide access to Ethereum platform features.
### IELE
Part of [[zoranskoda:Cardano]] project, IELE executes and verifies [[smart contract]]s as well as providing a human-readable language for blockchain developers.
* IELE, _Semantics based compilation_, at [iohkdev.io](https://testnet.iohkdev.io/iele/about/semantics-based-compilation)
### Ethereum VM
Primarily used for smart contract execution on Ethereum blockchain (and some others blockchains, like [[zoranskoda:Hyperledger]] Fabric). Ethereum community has plans to supercede it with Ethereum version of wasm VM (ewasm).
### TON VM
Used for executing smart contracts on [[Telegram Open Network]]. See
* [[Nikolai Durov]], _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf)
### CKB VM (Nervos)
This is a [RISC-V](https://en.wikipedia.org/wiki/RISC-V) VM for a Nervos blockchain design. "Uses rv64imc architecture: it is based on core RV64I ISA with M standard extension for integer multiplication and division, and C standard extension for RCV(RISC-V Compressed Instructions)." No floating point.
* github [mb](https://github.com/nervosnetwork/rfcs/blob/master/rfcs/0002-ckb/0002-ckb.md), [code](https://github.com/nervosnetwork/ckb)
[[!redirects Virtual Machine]]
|
virtual particle | https://ncatlab.org/nlab/source/virtual+particle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[perturbative quantum field theory]], it is suggestive to think of the [[edges]] in the [[Feynman diagrams]] as [[worldlines]] of "virtual particles" and of the [[vertices]] as the points where they collide and transmute. (Care must be exercised not to confuse this with concepts of real [[particles]].)
This intuition is made precise by the _[[worldline formalism]]_ of [[perturbative quantum field theory]] ([Strassler 92](worldline+formalism#Strassler92)). This is the perspective on [[perturbative QFT]] which directly relates [[perturbative QFT]] to [[perturbative string theory]] ([Schmidt-Schubert 94](worldline+formalism#SchmidtSchubert94)). In fact the [[worldline formalism]] for [[perturbative QFT]] was originally found by taking thre point-particle limit of [[string scattering amplitudes]] ([Bern-Kosower 91](worldline+formalism#BernKosower91), [Bern-Kosower 92](worldline+formalism#BernKosower92)).
## Related concepts
* [[relativistic particle]], [[spinning particle]], [[superparticle]]
* [[fundamental particle]], [[antiparticle]]
* [[radiative correction]]
## References
* [[Arnold Neumaier]], _[The Physics of Virtual Particles](https://www.physicsforums.com/insights/physics-virtual-particles/)_
* [[Arnold Neumaier]], _[Misconceptions about Virtual Particles](https://www.physicsforums.com/insights/misconceptions-virtual-particles/)_
See also
* Wikipedia, _[Virtual particle](https://en.wikipedia.org/wiki/Virtual_particle)_
[[!redirects virtual particles]]
|
virtual representation | https://ncatlab.org/nlab/source/virtual+representation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _virtual representation_ of a [[group]] is a formal difference with respect to [[direct sum]] of two ordinary [[representations]], hence the [[isomorphism class]] of a virtual representation is an element of the [[Grothendieck group]] of $(G Rep, \oplus)$. Equivalence classes of virtual representations form the elements of the [[representation ring]] of the group, see there for more.
If we regard an ordinary [[representation]] as an [[equivariant vector bundle]] over the point, then a virtual representation is a corresponding equivariant [[virtual vector bundle]]. Accordingly the [[representation ring]] of a [[finite group]] is its [[equivariant K-theory]] of the point.
## Related concepts
* [[representation ring]]
* [[Burnside ring]]
[[!redirects virtual representations]]
|
virtual vector bundle | https://ncatlab.org/nlab/source/virtual+vector+bundle |
#Contents#
* table of contents
{:toc}
## Idea
A formal difference (with respect to [[direct sum]]/[[Whitney sum]]) of [[vector bundles]]. Appears as representative for classes in [[topological K-theory]]. A virtual vector bundle $[E] - [F]$ has a _virtual rank_ $rk([E]-[F])$, which is the integer $rk(E) - rk(F)$.
## Properties
+-- {: .num_prop}
###### Proposition
Let X be a space that is locally compact and regular. Then any virtual vector bundle of virtual rank 0 has some tensor power that is 0 in K-theory.
=--
The following argument comes from ([Karoubi 1978](#Karoubi78), Theorem II.5.9), but generalised to take into account non-compact spaces which are introduced in the following pages.
+--{: .proof}
###### Proof
For closed subsets $Y_1$ and $Y_2$, then the ring structure on K-theory means
$$
K_{Y_1}(X)\cdot K_{Y_2}(X) \subset K_{Y_1\cup Y_2}(X)
$$
where $K_{Y_i}(X) := ker(K(X) \to K(Y_i))$. We can likewise consider the same for a sequence $Y_1,\ldots ,Y_n$.
If $e = [E] - [\underline{rk(E)}]$ is in $ker(rk)$ (i.e. reduced K-theory $\tilde K(X)$) with $supp(e)$ in some compact set $C$ (recall that vector bundle K-theory elements are differences of vector bundles that are isomorphic outside a compact set. Without loss of generality such an element can be represented as $[E] - [\underline{V}]$ where $\underline{V}$ is a trivial bundle with fibre $V$), cover $C$ by finitely many closed neighbourhoods $Y_1,\ldots ,Y_n$ over which $E$ trivialises. Thus $e$ is in $K_{Y_i}(X)$ for each $i=1,\ldots n$.
Thus $e$ is in the intersection of all the $K_{Y_i}(X)$ as well as $K_Z(X)$ for $Z$ the complement of $supp(e)$. Thus $e^{n+1}$ is in $K_{\cup Y_i \cup Z}(X) = K_X(X) = 0$.
Hence for every element in $ker(rk)$, some power of it is zero, hence $ker(rk)$ is a [[ideal|nil ideal]].
=--
## Related concepts
* [[virtual representation]]
## References
* {#Karoubi78} [[Max Karoubi]], _K-Theory: An Introduction_, Grundlehren der mathematischen Wissenschaften 226, (1978) Springer-Verlag
[[!redirects virtual vector bundles]]
[[!redirects virtual vector space]]
[[!redirects virtual vector spaces]] |
virtually fibered conjecture | https://ncatlab.org/nlab/source/virtually+fibered+conjecture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _virtually fibered conjecture_ says that every [[closed manifold|closed]], [[irreducible manifold|irreducible]], [[atoroidal 3-manifold|atoroidal]] [[3-manifold]] with infinite [[fundamental group]] has a [[finite cover]] which is a [[surface]] [[fiber bundle]] over the [[circle]].
A proof was announced by [[Ian Agol]] in 2012, based on work by [[Daniel Wise]].
## Related concepts
* [[arithmetic topology]]
* [[Alexander polynomial]]
## References
* Wikipedia, _[Virtually fibered conjecture](https://en.m.wikipedia.org/wiki/Virtually_fibered_conjecture)_
The final step of the proof after the work of Wise is contained in
* [[Ian Agol]], Daniel Groves, Jason Manning, _The virtual Haken conjecture_ [arXiv:1204.2810](http://arxiv.org/abs/1204.2810)
See also
* [[Ian Agol]], _The virtual Haken conjecture_, Talk at ICM 2014 ([video](https://www.youtube.com/watch?v=Y6ACJYvfs5U))
|
viscosity | https://ncatlab.org/nlab/source/viscosity | [[!redirects shear viscosity]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
## Examples
* The discussion of shear viscosity of [[quark-gluon plasmas]] turns out to be subtle. See there.
## Related concepts
* [[elasticity]]
* [[solidity]]
* [[cohesion]]
## References
See also
* Wikipedia, _[Viscosity](https://en.wikipedia.org/wiki/Viscosity)_ |
Vishal Lama | https://ncatlab.org/nlab/source/Vishal+Lama | Hi, I co-author the mathematical blog [Topological Musings](http://topologicalmusings.wordpress.com) with [Todd Trimble](http://ncatlab.org/nlab/show/Todd+Trimble).
I hope to fill up more of this page as time progresses.
**Books**
There are at least three books that I would highly recommend to an undergraduate student who may have the requisite "mathematical maturity" but perhaps has not tasted the elixir of [Category Theory](http://ncatlab.org/nlab/show/category+theory) yet.
* [Conceptual Mathematics: A First Introduction to Categories](http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&s=books&qid=1255426142&sr=8-1) (Paperback) - F. William Lawvere and Stephen H. Schanuel.
* [Sets for Mathematics](http://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521804442/ref=sr_1_3?ie=UTF8&s=books&qid=1255426295&sr=1-3) - F. William Lawvere and Robert Rosebrugh
* [Topoi: The Categorial Analysis of Logic](http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&s=books&qid=1255426408&sr=1-1) - Robert Goldblatt.
I intend to write reviews (containing details on each chapter, hopefully) for all the aforementioned books some time in the future. It will be a work in progress.
category: people |
Vitalik Buterin | https://ncatlab.org/nlab/source/Vitalik+Buterin | __Vitalik Buterin__ is the principal founder of [[zoranskoda:Ethereum]] [[blockchain]]. He was considering the limitations of [[zoranskoda:bitcoin|Bitcoin]] blockchain, where new functionalities were added one by one and came up with the idea of developing a new blockchain with Turing complete runtime for [[smart contract]]s on blockchain, enabling also the development of distributed applications, turning the blockchain into a computer-like platform where new applications could easily be added. Buterin is considered as a prodigy who founded Ethereum at age of 19 and now considered as one of main gurus of blockchain technology. He is one of the authors of the Plasma algorithm for layer 2 network over Ethereum-like blockchains and promotes the Casper proof of stake protocol for Ethereum 2.0.
Other leading figures behind early Ethereum network include Charles Hoskinson and Gavin Wood. Hoskinson is now behind the [[zoranskoda:Cardano]] project, leading company IOHK, and also contributing to [polymath](https://polymath.network) blockchain for financial securities. Gavin Wood introduced Solidity, the major higher level language for smart contracts which compiles to Ethereum [[virtual machine]] bytecode, in August 2014. He is now leading the web 3.0 project and Parity Technologies which is behind Parity client for Ethereum, Polkadot and Substrate projects.
* site [vitalik.ca](https://vitalik.ca)
* V. Buterin, _Ethereum in 25 minutes_, [youtube](https://www.youtube.com/watch?v=66SaEDzlmP4); _Blockchain and Ethereum security on the higher level_, [youtube](https://www.youtube.com/watch?v=UFDAtStVXbc) 1:21h
* V. Buterin, Ethereum: A next-generation smart contract and decentralized application platform, [pdf](https://github.com/ethereum/wiki/wiki/White-Paper.pdf)
* V. Buterin, _Blockchain and Ethereum security on the higher level_, [youtube](https://www.youtube.com/watch?v=UFDAtStVXbc) 1:21h
* Joseph Poon, Vitalik Buterin, _Plasma: scalable autonomous smart contracts_
[pdf](https://www.plasma.io/plasma.pdf) draft whitepaper
* V. Buterin, _Chain interoperability_, R3 reports, Sep 2016 [pdf](https://www.r3.com/wp-content/uploads/2018/04/Chain_Interoperability_R3.pdf)
* _Sidechains vs Plasma vs Sharding_, [html](https://vitalik.ca/general/2019/06/12/plasma_vs_sharding.html)
Vitalik recently proposed a privacy mixer for Ethereum to enhance anonymity via groupings
* _Minimal mixer design_, proposal at [hackmd](https://hackmd.io/s/rJj9hEJTN); [news at coindesk](https://www.coindesk.com/vitalik-roposes-mixer-to-anonymize-one-off-transactions-on-ethereum)
category: people, computer science, applications |
Vitaly Ginzburg | https://ncatlab.org/nlab/source/Vitaly+Ginzburg |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Vitaly_Ginzburg)
## Selected writings
Introducing [[Landau-Ginzburg models]] in [[superconductivity]]:
* [[Vitaly Ginzburg]], [[Lev Landau]], _On the Theory of Superconductivity_, reprinted In: _On Superconductivity and Superfluidity_ Springer (2009) ([doi:1007/978-3-540-68008-6_4](https://doi.org/10.1007/978-3-540-68008-6_4))
category: people
[[!redirects В. Л. Гинзбург]] |
Vitaly Lorman | https://ncatlab.org/nlab/source/Vitaly+Lorman |
* [webpage](http://www.math.jhu.edu/~vlorman/)
## Related entries
* [[image of J]]
category: people
|
VitalyR | https://ncatlab.org/nlab/source/VitalyR | ## VitalyR ##
I'm a student in China. This is [my home page](https://vitalyr.com). |
Vittoria Bussi | https://ncatlab.org/nlab/source/Vittoria+Bussi |
* [webpage](https://www.maths.ox.ac.uk/people/profiles/vittoria.bussi)
category: people
|
Vittorio Giovannetti | https://ncatlab.org/nlab/source/Vittorio+Giovannetti |
* [personal page](https://sites.google.com/site/giovannettivittorio/home)
* [Institute page](https://www.sns.it/en/persona/vittorio-giovannetti)
## Selected writings
Introducing the notion of [[quantum random access memory]]:
* {#GiovanettiyLloydMaccone08} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Quantum random access memory*, Phys. Rev. Lett. **100** 160501 (2008) [[doi:10.1103/PhysRevLett.100.160501](https://doi.org/10.1103/PhysRevLett.100.160501), [arXiv:0708.1879](https://arxiv.org/abs/0708.1879)]
* {#GiovanettiyLloydMaccone08b} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Architectures for a quantum random access memory*, Phys. Rev. A **78** 052310 (2008) [[doi:10.1103/PhysRevA.78.052310](https://doi.org/10.1103/PhysRevA.78.052310)]
category: people
|
Vivek Kumar Singh | https://ncatlab.org/nlab/source/Vivek+Kumar+Singh |
* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/researchers/vivek-kumar-singh.html) at NYU Abu Dhabi
* [GoogleScholar page](https://scholar.google.com/citations?user=lL8i4AsAAAAJ&hl=en&oi=sra)
* [inspire page](https://inspirehep.net/authors/1418741)
Member of [[CQTS]] at NYU Abu Dhabi.
## Selected writings
On [[topological quantum computation ]]:
* [[Vivek Kumar Singh]], Akash Sinha, Pramod Padmanabhan, Indrajit Jana, *Dyck Paths and Topological Quantum Computation*, [[arXiv:2306.16062v1](https://arxiv.org/abs/2306.16062)]
On [[topological entanglement entropy]] in [[Chern-Simons theory]]:
* Siddharth Dwivedi, [[Vivek Kumar Singh]], Saswati Dhara, P. Ramadevi, Yang Zhou, Lata Kh Joshi, *Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups*, J. High Energy Phys.02 (2018) 163 [[arXiv:1711.06474](https://arxiv.org/abs/1711.06474), <a href="
https://doi.org/10.1007/JHEP02%282018%29163">doi:10.1007/JHEP02(2018)163</a>]
* Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, [[Vivek Kumar Singh]], *Topological entanglement and hyperbolic volume*, J. High Energy Phys. **21021** 172 (2021) 172 [[arXiv:2106.03396](https://arxiv.org/abs/2106.03396), <a href="https://doi.org/10.1007/JHEP10(2021)172">doi:10.1007/JHEP10(2021)172</a>]
On the [[knots-quivers correspondence]]:
* [[Vivek Kumar Singh]], S. Chauhan, A. Dwivedi, P. Ramadevi, B.P. Mandal, S. Dwivedi, *Knot-Quiver correspondence for double twist knots*, Phys. Rev. D 108 (2023) 10, **106023** [[arXiv:2303.07036](https://arxiv.org/abs/2303.07036),<a href="https://doi.org/10.1103/PhysRevD.108.106023"> doi:10.1103/PhysRevD.108.106023</a>]
On [[knot theory]]:
* Satoshi Nawata, P. Ramadevi,[[Vivek Kumar Singh]], *Colored HOMFLY polynomials that distinguish mutant knots
*, J.Knot Theor.Ramifications 26 (2017) 14, **1750096** [[arXiv:1504.00364](https://arxiv.org/abs/1504.00364),<a href="https://doi.org/10.1142/S0218216517500961"> doi:10.1142/S0218216517500961</a>]
* Saswati Dhara, A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, [[Vivek Kumar Singh]], A. Sleptsov, *Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations*, Annales Henri Poincare 20 (2019) 12, **4033-4054** [[arXiv:1805.03916](https://arxiv.org/abs/1805.03916),<a href="https://doi.org/10.1007/s00023-019-00841-z"> doi:10.1007/s00023-019-00841-z</a>]
On [[topological string]]:
A. Mironov, A. Morozov, An. Morozov, P. Ramadevi,[[Vivek Kumar Singh]], A. Sleptsov, *Checks of integrality properties in topological strings
*, JHEP 08 (2017) 139 [[arXiv:1702.06316](https://arxiv.org/abs/1702.06316),<a href="
https://doi.org/10.1007/JHEP08%282017%29139"> doi:10.1007/JHEP08%282017%29139</a>]
category: people
[[!redirects Vivek Singh]]
[[!redirects Vivek Kr. Singh]]
|
Vivien M. Kendon | https://ncatlab.org/nlab/source/Vivien+M.+Kendon |
* [institute page](https://www.durham.ac.uk/staff/viv-kendon/)
* [GoogleScholar page](https://scholar.google.com/citations?user=hF8meRwAAAAJ&hl=en)
## Selected writings
On "continuous variable" or *analog* [[quantum computation]]:
* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]
category: people
[[!redirects Vivien Kendon]]
[[!redirects Viv Kendon]]
|
Vlad Patryshev | https://ncatlab.org/nlab/source/Vlad+Patryshev |
Vlad Patryshev vpatryshev@gmail.com
programmer, amateur categorist, organizer of [Bay Area Categories and Types meetup](http://www.meetup.com/Bay-Area-Categories-And-Types/).
category: people
[[!redirects vpatryshev]]
[[!redirects Vlad Patryshev]] |
Vladimir A. Fateev | https://ncatlab.org/nlab/source/Vladimir+A.+Fateev |
* [institute page](https://www.itp.ac.ru/en/persons/fateev-vladimir-aleksandrovich/)
## Selected writings
A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:
* [[Vladimir S. Dotsenko]], [[Vladimir A. Fateev]], *Conformal algebra and multipoint correlation functions in 2D statistical models*, Nuclear Physics B **240** 3 (1984) 312-348 $[$<a href="https://doi.org/10.1016/0550-3213(84)90269-4">doi:10.1016/0550-3213(84)90269-4</a>$]$
Introducing [[parafermion]] [[2d CFT]]:
* [[Alexander B. Zamolodchikov]], [[Vladimir A. Fateev]] *Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems*, Sov. Phys. JETP **62** 2 (1985) 215-225 $[$[pdf](http://www.jetp.ras.ru/cgi-bin/dn/e_062_02_0215.pdf), [osti:5929972](https://www.osti.gov/biblio/5929972)$]$
category: people
|
Vladimir Abramovich Rokhlin | https://ncatlab.org/nlab/source/Vladimir+Abramovich+Rokhlin |
* [Wikipedia page](https://en.m.wikipedia.org/wiki/Vladimir_Abramovich_Rokhlin)
* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=42580&fChrono=1)
* V I Arnol'd et al., _Vladimir Abramovich Rokhlin (obituary)_, Russ. Math. Surv. 41 189 ([doi:10.1070/RM1986v041n03ABEH003331](https://iopscience.iop.org/article/10.1070/RM1986v041n03ABEH003331))
## Selected writings
Computing the [[third stable homotopy group of spheres]]:
* [[Vladimir Abramovich Rokhlin]], _On a mapping of the $(n+3)$-dimensional sphere into the $n$-dimensional sphere_, (Russian) Doklady Akad. Nauk SSSR (N.S.) 80, (1951). 541–544
with a mistake (in the unstable range) corrected in
* [[Vladimir Abramovich Rokhlin]], _New results in the theory of four-dimensional manifolds_, (Russian) Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.
French translation in:
* Lucien Guillou, Alexis Marin (eds.), _A la Recherche de la Topologie Perdue: I. Du côté de chez Rohlin. II. Le côté de Casson_, Progress in Mathematics 62, Birkhäuser Boston 1985 (ISBN:0817633294, 9780817633295)
category: people
[[!redirects V. A. Rokhlin]]
[[!redirects V. A. Rohlin]] |
Vladimir Aleksandrovich Fock | https://ncatlab.org/nlab/source/Vladimir+Aleksandrovich+Fock |
> disambiguation: [[Vladimir Fock]], mathematician at Strasbourg
***
Soviet Rusiian physicist. His name is attached to _[[Fock space]]_, _[[Hartree-Fock method]]_ etc.
* [Wikipedia entry](http://en.wikipedia.org/wiki/Vladimir_Fock)
## Selected writings
Introducing the [[adiabatic theorem]]:
* [[Max Born]], [[Vladimir A. Fock]], *Beweis des Adiabatensatzes*, Zeitschrift für Physik **51** (1928) 165–180 ([doi:10.1007/BF01343193](https://doi.org/10.1007/BF01343193))
[[!redirects Vladimir A. Fock]]
category: people
|
Vladimir Arnold | https://ncatlab.org/nlab/source/Vladimir+Arnold | <div style="float:right;margin:-70px 0px 10px 20px;"><img width = "200" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/42/Vladimir_Arnold-1.jpg/180px-Vladimir_Arnold-1.jpg" alt="Vladimir Arnold in 2008" /></div>
Владимир Игоревич Арнольд (Vladimir Igorevich Arnol'd) was a Russian mathematician and mathematical physicist, working in Moscow and Paris.
* [Wikipedia entry](http://en.wikipedia.org/wiki/Vladimir_Arnold),
* B. Khesin & S. Tabachnikov: *Vladimir Igorevich Arnold. 12 June 1937 - 3 June 2010*, Biogr. Mems Fell. R. Soc. **64** (2018) 7–26 [[doi:10.1098/rsbm.2017.0016](https://doi.org/10.1098/rsbm.2017.0016), [pdf](https://royalsocietypublishing.org/doi/pdf/10.1098/rsbm.2017.0016)]
* biography at MacTutor [history of math archive](http://www-history.mcs.st-andrews.ac.uk/Biographies/Arnold.html)
## Selected writings
On the [[real cohomology]] of [[configuration spaces of points]]:
* {#Arnold69} [[Vladimir Arnold]], _The cohomology ring of the colored braid group_, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 ([mathnet:mz6827](http://mi.mathnet.ru/eng/mz6827))
On [[classical mechanics|classical]] [[Hamiltonian mechanics]] formulated via [[symplectic geometry]] (ie. with [[phase space]] understood as a [[symplectic manifold]]):
* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate Texts in Mathematics **60**, Springer (1978) [[doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)]
The [[Arnold-Kuiper-Massey theorem]]:
* [[Vladimir Arnold]], _Ramified covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology_, Siberian Math. Journal 1988, V. 29, N 5, P.36-47
* [[Vladimir Arnold]], _On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms_, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.
On methods of [[topology]] and [[homotopy theory]] in [[hydrodynamics]]:
* [[Vladimir Arnold]], [[Boris Khesin]], _Topological methods in hydrodynamics_, Applied Mathematical Sciences __125__, Springer 1998 ([doi:10.1007/b97593](https://link.springer.com/book/10.1007/b97593))
On [[mathematics]] and its relation to [[physics]]:
* {#Arnold95} [[Vladimir I. Arnold]], *Will mathematics survive? Report on the Zurich Congress*, The Mathematical Intelligencer **17** (1995) 6–10 [[doi:10.1007/BF03024363](https://doi.org/10.1007/BF03024363)]
> "At the beginning of this century a self-destructive democratic principle was advanced in mathematics [which] led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste of a dying religion, like Druids, parasitic on science and technology, recruiting acolytes in the mathematical schools by Zombie-like mental subjection. Bizarre questions like [[Fermat's last theorem|Fermat's problem]] or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."
* {#Arnold98} [[Vladimir I. Arnold]], *On teaching mathematics*, Russ. Math. Surv. **53** 1 (1998) 229 [[doi:10.1070/RM1998v053n01ABEH000005](https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005), [web](https://www.math.fsu.edu/~wxm/Arnold.htm)]
> "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
> The Jacobi identity (which forces the altitudes of a triangle to meet in a point) is an experimental fact in the same way as the fact that the earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
> In the middle of the twentieth century an attempt was made to separate physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the sun).
> Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was a universal hatred of mathematicians, both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.
> [...]
> A teacher of mathematics who has not got to grips with at least some of the volumes of the [[Course of Theoretical Physics|course]] by Landau and Lifshitz will then become a relic like the person nowadays who does not know the difference between an open and a closed set."
* {#Arnold00} [[Vladimir I. Arnold]], *Polymathematics: Is mathematics a single science or a set of arts?*, in: [[Vladimir Arnold|V. Arnold]], [[Peter Lax|P. Lax]], [[Barry Mazur|B. Mazur]] (eds.), *Mathematics: Frontiers and Perspectives*, AMS (2000) 403-416 [[ISBN: 978-0-8218-2697-3](https://bookstore.ams.org/mfp), [pdf](https://math.ucr.edu/home/baez/Polymath.pdf)]
> "All mathematics is divided into three parts: [[cryptography]] (paid for by CIA, KGB and the like), [[hydrodynamics]] (supported by manufacturers of atomic submarines) and [[classical mechanics|celestial mechanics]] (financed by military and other institutions dealing with missiles, such as NASA).
> Cryptography has generated [[number theory]], [[algebraic geometry]] over [[finite fields]], [[algebra]]", [[combinatorics]] and [[computers]].
> Hydrodynamics procreated [[complex analysis]], [[partial differential equations]], [[Lie groups]] and algebra theory, [[cohomology theory]] and scientific [[computing]].
> Celestial mechanics is the origin of [[dynamical systems]], [[linear algebra]], [[topology]], [[variational calculus]] and [[symplectic geometry]].
> The existence and mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation)."
> [...]
> "[[David Hilbert|Hilbert]] tried to predict the future development of mathematics and to influence it by [[Hilbert's problems|his Problems]]. The development of mathematics in the 20th century has followed a different path. The most important achievements -- the flourishing of [[homotopy theory]] and of [[differential topology]], the [[geometry|geometrisation]] of all branches of mathematics, its fusion with [[theoretical physics]], the discovery of the algorithmically undecidable problems and the appearance of [[computers]] -- all this went in a different (if not opposite) direction.
> The influence of [[Henri Poincaré|H. Poincaré]] and of [[Hermann Weyl|H. Weyl]] on the science of the 20th century was much deeper. To Poincaré, who created modern mathematics, [[topology]] and [[dynamical systems]] theory, the future of mathematics lay in the development of [[mathematical physics]], oriented to the description of the [[relativity|relativistic]] and [[quantum physics|quantum]] phenomena."
> [...]
## Related entries
* [[symplectic geometry]]
* [[Hamiltonian mechanics]]
category: people
[[!redirects Владимир Игоревич Арнольд]]
[[!redirects Владимир Арнольд]]
[[!redirects Vladimir Arnold]]
[[!redirects Vladimir Arnol'd]]
[[!redirects Vladimir Arnol'd]]
[[!redirects Vladimir I. Arnold]]
[[!redirects Vladimir I. Arnol'd]]
[[!redirects Vladimir I. Arnol'd]]
[[!redirects V. I. Arnold]]
[[!redirects V. I. Arnol'd]]
[[!redirects V. I. Arnol'd]]
|
Vladimir Baranovsky | https://ncatlab.org/nlab/source/Vladimir+Baranovsky | __Vladimir Baranovsky__ is a mathematician at the University of California at Irvine. Main area: algebraic geometry.
* [web](http://www.math.uci.edu/~vbaranov)
* _Orbifold cohomology as periodic cyclic homology_, Internat. Journal of Math, 14 (2003), no 8, 791-812, [pdf](http://math.uci.edu/%7Evbaranov/math/08orbicyclic.pdf)
* _A universal enveloping for L-infinity algebras_, Math Res Lett __15__ (2008) [pdf](http://math.uci.edu/%7Evbaranov/math/12uofl.pdf)
category: people
|
Vladimir Berkovich | https://ncatlab.org/nlab/source/Vladimir+Berkovich |
* [personal webpage](http://www.wisdom.weizmann.ac.il/~vova/), [institute webpage](http://www.wisdom.weizmann.ac.il/math/profile/scientists/berkovich-profile.html)
## related entries
* [[non-archimedean analytic geometry]], [[Berkovich analytic space]]
category: people |
Vladimir Bogachev | https://ncatlab.org/nlab/source/Vladimir+Bogachev | Vladimir Bogachev (Владимир Игоревич Богачев) is a Russian mathematician.
## Selected writings
## References
[Math-Net.Ru page](http://www.mathnet.ru/eng/person8592).
|
Vladimir Danilov | https://ncatlab.org/nlab/source/Vladimir+Danilov |
* [webpage](http://mathecon.cemi.rssi.ru/en/danilov/)
category: people
|
Vladimir Dobrev | https://ncatlab.org/nlab/source/Vladimir+Dobrev |
* [webpage](http://theo.inrne.bas.bg/~dobrev/)
category: people
|
Vladimir Drinfel'd | https://ncatlab.org/nlab/source/Vladimir+Drinfel%27d |
**Vladimir Gershonovich Drinfel'd** (Russian: Владимир Гершонович Дринфельд, Ukrainian: Володимир Гершонович Дрінфельд) is an Ukrainian-born (Feb 4, 1954) mathematician, recipient of Fields Medal (1990), now a professor at the University of Chicago. He was student of [[Yuri Manin]] at Moscow; his main specialty is algebraic geometry, including applications to arithmetic, [[automorphic form]]s, [[representation theory]] and mathematical physics. His famous discoveries and new concepts Drinfel'd modular varieties, Drinfel'd shtukas (Drinfel'd elliptic modules), ADHM-construction of instantons, [[Drinfel'd-Sokolov reduction]] in the theory of [[integrable systems]], algebraic approach to quantum inverse scattering method in [[integrable systems]], and related subject of [[quantum group]]s including Yangians and quasi-Hopf algebras, geometric Langlands program, chiral algebras (with A. Beilinson), certain category of infinite-dimensional bundles in algebraic geometry (now sometimes called Drinfel'd bundles), and the general construction of quotient [[dg-categories]] in [[homological algebra]]. His ideas also influenced the beginnings of the new subject of [[derived algebraic geometry]].
* [wikipedia](http://en.wikipedia.org/wiki/Vladimir_Drinfel%27d)
## Selected writings
Introducing the [[ADHM construction]] for [[Yang-Mills instantons]]:
* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65** 3 (1978) 185-187 [<a href="https://doi.org/10.1016/0375-9601(78)90141-X">doi:10.1016/0375-9601(78)90141-X</a>]
Introducing [[quantum groups]] and the [[Drinfeld double]]-construction:
* {#Drinfeld87} [[Vladimir Drinfeld]], _Quantum groups_, in: A. Gleason (ed.) *[Proceedings of the](https://archive.org/details/proceedingsofint0002inte_v5c3/mode/2up) [1986 International Congress of Mathematics](https://inspirehep.net/conferences/966215?ui-citation-summary=true)* **1** (1987) 798-820 [[pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf)]
expanded version:
Journal of Soviet Mathematics **41** (1988) 898–915 [[doi:10.1007/BF01247086](https://doi.org/10.1007/BF01247086)]
On [[chiral algebras]]:
* A. A. Beilinson, V. Drinfeld, _[[Chiral Algebras]]_, AMS 2004 (the final form of a preprint in various forms since around 1995, cf. [here](http://www.math.uchicago.edu/~mitya/langlands.html)).
On [[braided monoidal category|braided]] [[fusion categories]]:
* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))
On [[prismatic cohomology]]
* [[Vladimir Drinfeld]], _Prismatization_ ([arXiv:2005.04746](https://arxiv.org/abs/2005.04746))
category: people
[[!redirects Vladimir Drinfeld]]
[[!redirects Vladimir Drinfel'd]]
[[!redirects V. Drinfeld]]
[[!redirects V. Drinfel'd]]
[[!redirects V. G. Drinfel'd]]
[[!redirects Drinfeld]]
[[!redirects Drinfel'd]]
[[!redirects Владимир Дринфельд]]
[[!redirects Володимир Дрінфельд]]
|
Vladimir Fock | https://ncatlab.org/nlab/source/Vladimir+Fock |
Professor of mathematics at Institut de Recherche Mathématique Avancée in Strasbourg. Collaborating with [[Alexander Goncharov]].
* [webpage](http://www-irma.u-strasbg.fr/~fock/)
> disambiguation: [[Vladimir Aleksandrovich Fock]], soviet russian physicist
category: people |
Vladimir Glaser | https://ncatlab.org/nlab/source/Vladimir+Glaser |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Vladimir_Jurko_Glaser)
## writings
* [[Henri Epstein]], Vladimir Glaser, _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.
## related $n$Lab entries
* [[renormalization]]
* [[causal perturbation theory]],
* [[lcoally covariant perturbative quantum field theory]]
category: people |
Vladimir Gribov | https://ncatlab.org/nlab/source/Vladimir+Gribov |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Vladimir_Gribov)
* Ya.I. Azimov, _Vladimir Naumovich Gribov: Pieces of biography_ ([arXiv:1608.05727](http://arxiv.org/abs/1608.05727))
## related $n$Lab entries
* [[Gribov ambiguity]]
* [[S-matrix]]
category: people
|
Vladimir Guletskii | https://ncatlab.org/nlab/source/Vladimir+Guletskii |
* [web](http://pcwww.liv.ac.uk/~guletski/)
* grant [Lambda-structures in stable categories](http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/I034017/1)
## Selected writings
On [[symmetric powers]] in [[higher algebra|higher]] [[homotopical algebra]] (and on [[monoidal localization|monoidal]] [[Bousfield localization of model categories]]):
* {#GorchinskiyGuletskii16} [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 [[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)]
[[zeta function|Zeta function]]s in triangulated context
* Vladimir Guletskii, _Zeta functions in triangulated categories_, Mathematical Notes __87__, 3 (2010) 369--381, [math/0605040](https://arxiv.org/abs/math/0605040)
category:people |
Vladimir Hinich | https://ncatlab.org/nlab/source/Vladimir+Hinich |
* [website](http://math.haifa.ac.il/hinich/WEB/hinich.htm)
* [recent publications](http://math.haifa.ac.il/hinich/WEB/publications.htm)
## Selected writings
On [[model category]] [[structures]] on [[algebra over an operad|algebras over operads]] in [[chain complexes]] (and introducing the [[model structure on unbounded chain complexes]]):
* {#Hinich97} [[Vladimir Hinich]], *Homological algebra of homotopy algebras*, Communications in Algebra **25** 10 (1997) 3291-3323 [[arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), [doi:10.1080/00927879708826055](https://doi.org/10.1080/00927879708826055), Erratum: [arXiv:math/0309453](http://arxiv.org/abs/math/0309453)]
The [[model structure on dg-coalgebras]] (in [[characteristic zero]]) as a [[model structure for L-infinity algebras|model structure for $L_\infty$-algebras]] and the [[Quillen equivalence]] between [[dg-Lie algebras]] as well as the interpretation in terms of formal $\infty$-stacks ([[L-infinity algebras|$L_\infty$-algebras]]):
* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, Journal of Pure and Applied Algebra **162** 2 (2001) 209-250 [[arXiv:9812034](http://arxiv.org/abs/math/9812034), <a href="https://doi.org/10.1016/S0022-4049(00)00121-3">doi:10.1016/S0022-4049(00)00121-3</a>]
On the [[enriched Yoneda lemma]]:
* {#Hinich16} [[Vladimir Hinich]], *Enriched Yoneda lemma*, Theory and Applications of Categories **31** 29 (2016) 833-838 [[tac:31-29](http://www.tac.mta.ca/tac/volumes/31/29/31-29abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/31/29/31-29.pdf)]
On ([[Lie algebra weight system|Lie algebra-]])[[weight systems]] on [[chord diagrams]]:
* [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197))
On [[(infinity,1)-colimits|$\infty$-colimits]] and [[Day convolution]] in the context of [[enriched (infinity,1)-categories|enriched $\infty$-categories]]:
* [[Vladimir Hinich]], *Colimits in enriched ∞-categories and Day convolution*, Theory and Applications of Categories **39** 12 (2023) 365-422 [[tac:39-12](http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html), [arXiv:2101.09538](https://arxiv.org/abs/2101.09538)]
category: people
[[!redirects V. Hinich]]
[[!redirects Hinich]]
|
Vladimir Lifschitz | https://ncatlab.org/nlab/source/Vladimir+Lifschitz |
* [website](https://www.cs.utexas.edu/~vl/)
## Selected publications
On [[Lifschitz realizability]]:
* [[Vladimir Lifschitz]], *$CT_0$ is stronger than $CT_0!$*. Proceedings of the American Mathematical Society, Volume 73, Number 1, January 1979, pp. 101–106. [[pdf](https://www.ams.org/journals/proc/1979-073-01/S0002-9939-1979-0512067-X/S0002-9939-1979-0512067-X.pdf)]
category: people |
Vladimir Retakh | https://ncatlab.org/nlab/source/Vladimir+Retakh | __Vladimir Retakh__ is a professor of mathematics at Rutgers University, formerly in Moscow, specialized in combinatorics, algebra and their applications.
* [web](http://www.math.rutgers.edu/~vretakh)
## Selected writings
On [[quasideterminants]] and [[Cohn localization]]:
* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_ (2007) [[[RetakhWilson-Quasideterminants.pdf:file]]]
[[!redirects V. Retakh]] |
Vladimir Rubtsov | https://ncatlab.org/nlab/source/Vladimir+Rubtsov | __Vladimir Rubtsov__ (also spelled Roubtsov in France) is a mathematician and mathematical physicist in Angers, France ([[integrable system]]s, geometry in physics, Painlevé transcendents, character varieties, [[tau function]]s, [[Lie algebroid]]s, noncommutative algebra, Yang-Baxter equations).
* [webpage](https://www.math.univ-angers.fr/~volodya/index_uk.html)
* [[A. Odesskii]], V. Rubtsov, V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935) [arxiv/1208.2935](https://arxiv.org/abs/1208.2935); _Bi-hamiltonian ordinary differential equations with matrix variables_, Theor. and Math. Phys. 171(1): 442--447 (2012)
* Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, _Painlevé monodromy manifolds, decorated character varieties, and cluster algebras_, IMRN __24__ (2017) 7639–7691 [arxiv/1511.03851](https://arxiv.org/abs/1511.03851) [doi](https://doi.org/10.1093/imrn/rnw219); _Algebras of quantum monodromy data and decorated character varieties_, in: Geometry and physics: A Festschrift in honour of Nigel Hitchin, Oxford Univ. Press 2018 [arxiv/1705.01447](https://arxiv.org/abs/1705.01447); _Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras_, Adv. Math. __376__ (2021) 107442 [arxiv/1905.02772](https://arxiv.org/abs/1905.02772) [doi](https://doi.org/10.1016/j.aim.2020.107442)
* Marta Mazzocco, Vladimir Rubtsov, _Confluence on the Painlevé monodromy manifolds, their Poisson structure and quantisation_, [arXiv/1212.6723](https://arxiv.org/abs/1212.6723)
* U. Bruzzo, I. Mencattini, V. Rubtsov, P. Tortella, _Nonabelian Lie algebroid extensions_, [arXiv:1305.2377](https://arxiv.org/abs/1305.2377)
* D. Gurevich, V. Rubtsov, _Yang-Baxter equation and deformation of associative and Lie algebras_, in: Quantum Groups, Springer Lecture Notes in Math. __1510__ (1992) 47-55 [doi](https://doi.org/10.1007/BFb0101177)
* B. Enriquez, V. Rubtsov, _Hitchin systems, higher Gaudin Hamiltonians and r-matrices_, Math. Res. Letters __3__ (1996) n° 3, 343-357, [alg-geom/9503010](https://arxiv.org/abs/alg-geom/9503010)
* [[Vladimir Retakh]], Vladimir Rubtsov, _Noncommutative Toda chains, Hankel quasideterminants and [[Painleve transcendent|Painlevé]] II equation_, [arxiv/1007.4168](http://arxiv.org/abs/1007.4168)
category: people
[[!redirects V. Rubtsov]]
|
Vladimir S. Dotsenko | https://ncatlab.org/nlab/source/Vladimir+S.+Dotsenko |
* [institute page](https://www.itp.ac.ru/en/persons/dotsenko-vladimir-stepanovich/)
## Selected writings
A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:
* [[Vladimir S. Dotsenko]], [[Vladimir A. Fateev]], *Conformal algebra and multipoint correlation functions in 2D statistical models*, Nuclear Physics B **240** 3 (1984) 312-348 $[$<a href="https://doi.org/10.1016/0550-3213(84)90269-4">doi:10.1016/0550-3213(84)90269-4</a>$]$
category: people
|
Vladimir S. Retakh | https://ncatlab.org/nlab/source/Vladimir+S.+Retakh |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Vladimir_Retakh)
category: people
[[!redirects Vladimir Retakh]]
|
Vladimir Turaev | https://ncatlab.org/nlab/source/Vladimir+Turaev | [[!redirects Turaev]]
Vladimir Turaev is a mathematician of Russian origin, who has made fundamental contributions to [[quantum field theory|Quantum Field Theory]], low dimensional [[topology]], [[knot theory]], and their interactions with [[mathematical physics]] and [[algebra]].
He is currently based at Indiana University ([Home page](https://math.indiana.edu/about/faculty/touraev-vladimir.html) which is not very informative!)
## Selected writings
* _On certain enumeration problems in two-dimensional topology_ ,
Math. Res. Lett.16(3) (2009), 515-529.
Introducing the [[Reshetikhin-Turaev construction]]:
* {#ReshetikhinTuraev91} [[Nikolai Reshetikhin]], [[Vladimir Turaev]], _Invariants of 3-manifolds via link polynomials and quantum groups_. Invent. Math. 103 (1991), no. 3, 547–597. ([doi:10.1007/BF01239527](https://doi.org/10.1007/BF01239527), [pdf](http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf))
On [[knot theory]] and [[3-manifolds]]:
* [[Vladimir Turaev]], *Quantum invariants of knots and 3-manifolds*, de Gruyter Studies in Mathematics **18** Walter de Gruyter & Co. (1994) [[doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221)]
On [[HQFT|homotopy quantum field theory]]:
* _Homotopy Quantum Field Theory_ , EMS Tracts in Math.10, European Math. Soc. Publ. House, Zurich 2010.
On [[Dijkgraaf-Witten theory]]:
* Dijkgraaf-Witten invariants of surfaces and projective representations of groups,
J. Geom. Phys. 57(11) (2007), 2419-2430.
On [[braid groups]]:
* [[Christian Kassel]], [[Vladimir Turaev]], _Braid Groups_, GTM **247** Springer Heidelberg 2008 ([doi:10.1007/978-0-387-68548-9](https://link.springer.com/book/10.1007/978-0-387-68548-9), [webpage](http://irma.math.unistra.fr/~kassel/Braids-bk.html))
## Related entries
* [[Reshetikhin-Turaev construction]]
* [[Reshetikhin-Turaev invariant]]
* [[Turaev-Viro model]]
* [[unitary functorial field theory]]
[[!redirects Turaev]]
category: people |
Vladimir Vapnik | https://ncatlab.org/nlab/source/Vladimir+Vapnik | * [website](https://datascience.columbia.edu/people/vladimir-vapnik/)
### Selected writings
On [[statistical learning theory]]:
* [[Vladimir Vapnik]], _Statistical Learning Theory_, Wiley (1998) [[ISBN:978-0-471-03003-4](https://www.wiley.com/en-us/Statistical+Learning+Theory-p-9780471030034)]
* [[David Corfield]], [[Bernhard Schölkopf]], [[Vladimir Vapnik]], _Falsificationism and statistical learning theory: Comparing the Popper and Vapnik-Chervonenkis dimensions_, Journal for General Philosophy of Science **40** 1 (2009) 51-58 [[doi:10.1007/s10838-009-9091-3](https://doi.org/10.1007/s10838-009-9091-3)]
category: people |
Vladimir Vershinin | https://ncatlab.org/nlab/source/Vladimir+Vershinin |
* [MathNet page](http://www.mathnet.ru/eng/person8972)
## Selected writings
On the [[group homology]] and [[group cohomology]] of [[braid groups]]:
* [[Vladimir Vershinin]], *Homology of Braid Groups and their Generalizations*, Banach Center Publications (1998) **42** 1 421-446 ([pdf](https://hopf.math.purdue.edu/Vershinin/hobr.pdf), [dml:208821](https://eudml.org/doc/208821))
category: people |
Vladimir Voevodsky | https://ncatlab.org/nlab/source/Vladimir+Voevodsky | Владимир Воеводский (who published in English as Vladimir Voevodsky) was a Russian mathematician working in the Institute for Advanced Study.
* IAS [obituary](https://www.ias.edu/news/2017/vladimir-voevodsky-obituary)
* former [web site](http://www.math.ias.edu/~vladimir/Site3/home.html)
* [Wikipedia article](http://secure.wikimedia.org/wikipedia/en/wiki/Vladimir_Voevodsky) .
Voevodsky received a [[Fields medal]] in 2002 for a [[proof]] of the [[Milnor conjecture]]. The proof crucially uses [[A1-homotopy theory]] and [[motivic cohomology]] developed by Voevodsky for this purpose. In further development of this in 2009 Voevodsky announced a proof of the [[Bloch-Kato conjecture]].
After this work in [[algebraic geometry]], [[cohomology]] and [[homotopy theory]] Voevodsky turned to the [[foundations of mathematics]] and worked on [[homotopy type theory]] which he described as a new "[[univalence|univalent]] foundations" for modern mathematics with its emphasis on [[homotopy theory]] and [[higher category theory]].
## Selected writings
Introducing the modern notion of [[equivalence in type theory]] (namely via [[contractible type|contractible]] [[fiber type|fibers]]) and thereby fixing the [[univalence axiom]] of [Hofmann & Streicher (1998), §5.4](univalence+axiom#HofmannStreicher98) (due to the [subtlety with quasi-inverses](equivalence+in+type+theory#TheIssueWithQuasiInverses)):
* {#UnivalentFoundationsProject} [[Vladimir Voevodsky]], *Univalent Foundations Project* (2010) [[pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/univalent_foundations_project.pdf), [[Voevodsky-UFP2010.pdf:file]]]
Proposal for a "[[Homotopy Type System]]" (cf. *[[semisimplicial type]]*):
* [[Vladimir Voevodsky]], _A type system with two kinds of identity types_ (Feb. 2013) [[[Voevodsky-HTS.pdf:file]]]
* [[Vladimir Voevodsky]], _A simple type system_ (Jan 2013) [[[Voevodsky-TTS.pdf:file]], [Implementation](https://github.com/DanGrayson/checker)]
## Selected Talks
* {#Talk14} _The origins and motivations for univalent foundations_, IAS 2014 ([adapted transcript](https://www.ias.edu/ideas/2014/voevodsky-origins), [video]( https://video.ias.edu/voevodsky14/))
* list of video-recorded talks on [[homotopy type theory]]: [here](http://video.ias.edu/taxonomy/term/42).
## Selected Interviews
* An interview is [here](http://www.youtube.com/watch?v=vcDaQTPH-Rc).
* {#InterviewWithVoevodskyByMikhailov} [[Roman Mikhailov]], *Интервью Владимира Воеводского*, Princeton 2012 ([Part 1](https://baaltii1.livejournal.com/198675.html), [Part 2](https://baaltii1.livejournal.com/200269.html))
* _Le bifurcation de Vladimir Voevodsky_, interview conducted by Fondation Sciences Mathématiques de Paris, 2014, [video](https://vimeo.com/99586217), [transcript in French](http://smf4.emath.fr/Publications/Gazette/2014/142/smf_gazette_142_87-94.pdf).
## Related $n$Lab entries
* [[homotopy theory]]
* [[algebraic geometry]]
* [[algebraic topology]]
* [[K-theory]]
* [[A1-homotopy theory]]
* [[Voevodsky motive]]
* [[Kapranov–Voevodsky 2-vector space]]
* [[homotopy type theory]]
* etc ...
category: people
[[!redirects Владимир Воеводский]]
[[!redirects Vladimir Voevodsky]]
[[!redirects Vladimir Voevodskij]]
[[!redirects Vladimir Voevodskiĭ]]
[[!redirects V. Voevodsky]]
[[!redirects Voevodsky]] |
Vladimiro Sassone | https://ncatlab.org/nlab/source/Vladimiro+Sassone |
* [Home Page](http://www.ecs.soton.ac.uk/people/vs)
category: people
|
Vladimír Souček | https://ncatlab.org/nlab/source/Vladim%C3%ADr+Sou%C4%8Dek |
* [webpage](http://www.karlin.mff.cuni.cz/~soucek/)
## related $n$Lab entries
* [[BGG sequence]]
* [[parabolic geometry]], [[Cartan geometry]]
category: people
[[!redirects Vladimir Soucek]]
[[!redirects Vladimir Souček]]
[[!redirects Vladimír Soucek]] |
Vladislav Kupriyanov | https://ncatlab.org/nlab/source/Vladislav+Kupriyanov |
* [MPI page](https://www.mpp.mpg.de/en/news/news/detail/max-planck-institut-fuer-physik-begruesst-vladislav-kupriyanov-als-capes-humboldt-stipendiaten/)
## Selected writings
* [[Vladislav Kupriyanov]]
**$L_\infty$-Bootstrap Approach to Non‐Commutative Gauge Theories**
in: Proceedings of _[[Higher Structures in M-Theory 2018]]_
Fortschritte der Physik, Special Issue Volume 67, Issue 8-9
[doi:1903.02867](https://arxiv.org/abs/1903.02867) [doi:10.1002/prop.201910010](https://doi.org/10.1002/prop.201910010)
on [[L-infinity algebras]] in [[noncommutative geometry|noncommutative]] [[gauge theory]]
category: people |
Voevodsky motive | https://ncatlab.org/nlab/source/Voevodsky+motive |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[A. Suslin]] and [[Vladimir Voevodsky|V. Voevodsky]] have realized a [[triangulated category]] $D^b(\mathcal{MM}_k)$ which is supposed to be the bounded [[derived category]] of the hypothetical [[abelian category]] of [[mixed motives]] over $k$, predicted by Grothendieck--Beilinson--Deligne.
There are variants developed by Hanamura and M. Levine. There is a different "derived" approach to mixed motives, namely the $A^1$-homotopy theory of [[F. Morel]] and [[V. Voevodsky]].
See at _[[motive]]_ the section _[Contructions of the derived category of mixed motives](http://ncatlab.org/nlab/show/motive#DerivedMotives)_.
## Applications
Voevodsky used the derived category of mixed motives to solve [[Milnor's conjecture]] in [[algebraic K-theory]].
## Related concepts
See also
* [[motivic homotopy theory]]
* [[motive]]
* [[pure motive]]
* [[Chow motive]], [[numerical motive]]
* [[mixed motive]]
* [[noncommutative motive]]
## References
* <a href="http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)">Wikipedia article on motives</a> and references therein.
* A. Beilinson, V. Vologodsky, _A DG guide to Voevodsky's motives_, [math.AG/0604004](http://de.arxiv.org/abs/math/0604004)
* M.V. Bondarko, _Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura_, [math.AG/0601713](http://de.arxiv.org/abs/math/0601713)
* M.V. Bondarko, _Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey_, [arxiv:0903.0091](http://de.arxiv.org/abs/0903.0091)
* [[V. Voevodsky]], _Motives over [[simplicial scheme|simplicial schemes]]_, Journal of K-Theory, Volume 5, Issue 01 , pp 1 - 38,
(preliminary version in K-theory preprint archive: [here](http://www.math.uiuc.edu/K-theory/0638/).
Models for Voevodsky motives:
* Peter Bonart, *Triangulated Categories of Big Motives via Enriched Functors* (2023) [[arXiv:2310.17349](https://arxiv.org/abs/2310.17349)]
[[!redirects Voevodsky motives]]
|
Vojta's conjecture | https://ncatlab.org/nlab/source/Vojta%27s+conjecture |
#Contents#
* table of contents
{:toc}
## Idea
Vojta introduced a dictionary between [[value distribution theory]]
of Nevanlinna and [[Diophantine approximation theory]] of Roth and suggested that this dictionary should continue to hold in higher dimensions.
This leads to qualitative conjectures in
[[arithmetic geometry]] which cover almost every important conjecture in the field; notably the [[abc conjecture]],
[[Mordell's conjecture]], some of [[Lang's conjecture]]s. The perspective given by [[Arakelov geometry|Arakelov theory]] is central in Vojta's conjectures.
## Related concepts
* [[abc conjecture]]
## References
The original article is
* [[Paul Vojta]], _Diophantine approximations and value distribution theory_, Lecture Notes in Mathematics, 1239, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0072989, ISBN 978-3-540-17551-3, MR883451
See also
* David McKinnon, _Vojta's main conjecture for blowup surfaces_ ([pdf](http://www.math.uwaterloo.ca/~dmckinno/Papers/Vojta.Main.pdf))
[[!redirects Vojta conjecture]]
|
Vojtěch Pravda | https://ncatlab.org/nlab/source/Vojt%C4%9Bch+Pravda |
* [webpage](http://users.math.cas.cz/~pravda/)
## related $n$Lab entries
* [[universal spacetime]]
category: people
[[!redirects Vojtech Pravda]] |
Volker Braun | https://ncatlab.org/nlab/source/Volker+Braun |
* [webpage](http://www.sas.upenn.edu/~vbraun/)
## Selected writings
On [[torsion subgroup|torsion]] in the [[D-brane charge quantization in topological K-theory]]:
* [[Volker Braun]], *K-Theory Torsion* [<a href="https://arxiv.org/abs/hep-th/0005103">arXiv:hep-th/0005103</a>]
Computation of the [[twisted ad-equivariant K-theory]]-groups for all [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[simple Lie groups]]:
* [[Volker Braun]], _Twisted K-Theory of Lie Groups_, JHEP 0403 (2004) 029 ([arXiv:hep-th/0305178](https://arxiv.org/abs/hep-th/0305178))
On [[single trace operators]]/[[BMN operators]] in [[super Yang-Mills theory]] regarded as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation operator]] and application to [[AdS-CFT duality]]:
* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))
On [[D-branes]] in [[Gepner models]] via [[boundary conformal field theory]]:
* {#BraunSchaeferNameki05} [[Volker Braun]], [[Sakura Schafer-Nameki]], _D-Brane Charges in Gepner Models_, J. Math. Phys. 47 (2006) 092304 ([arXiv:hep-th/0511100](https://arxiv.org/abs/hep-th/0511100))
On [[duality in string theory|duality]] of [[M-theory on G2-manifolds]] with [[heterotic string theory on CY3-manifolds]]:
* {#BraunSchaeferNameki17} [[Andreas Braun]], [[Sakura Schafer-Nameki]], _Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality_, JHEP04(2018)126 ([arXiv:1708.07215](https://arxiv.org/abs/1708.07215))
## Related pages
* [[string phenomenology]]
category: people
|
Volker Puppe | https://ncatlab.org/nlab/source/Volker+Puppe |
* [webpage](http://www.math.uni-konstanz.de/~puppe/)
category: people
|
Volker Schomerus | https://ncatlab.org/nlab/source/Volker+Schomerus |
* [webpage](http://wwwiexp.desy.de/sfb676/researchers/schomerus/)
category: people
|
Volodin model | https://ncatlab.org/nlab/source/Volodin+model | #Volodin model for K-theory
This is the simplicial set / complex constructed by Volodin, using a construction similar to that of the [[Vietoris complex]]. It is the [[Volodin space]] of the family of subgroups of the stable general linear group described as follows:
We let $T_n^\sigma(R)$ be the subgroup of $G\ell_n(R)$ formed by the $\sigma$-triangular matrices, (discussed at [[higher generation by subgroups]]), and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell(R)$ as the colimit of the nested sequence of all the $G\ell_n(R)$, take $G = G\ell(R)$. Considering the family, $\mathcal{H}$, of all the $T_n^\sigma(R)$, form the corresponding [[Volodin space]].
##References
* A. A. [[Suslin]] and M. Wodzicki, _Excision in algebraic K-theory_, The Annals of Mathematics,
136, (1992), 51 – 122.
* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative
rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887)
|
Volodin space | https://ncatlab.org/nlab/source/Volodin+space | [[!redirects Volodin spaces]]
#Volodin space
* table of contents
{: toc}
##Idea
_Volodin spaces_ are the [[Vietoris complex]] analogues of the nerve of a family of subgroups discussed in the entry, [[higher generation by subgroups]]. They provide a way of building a geometric object that provides a means of comparing the information on the 'big group' that is 'stored' by subgroups within the family.
They were essentially introduced by [[Volodin]] as part of his approach to higher [[algebraic K-theory]]. We will discuss them via another approach that is explicit in work by [[Suslin]], on the equivalence of the Volodin K-theory with that of [[Quillen]].
##Preliminaries
Let $X$ be a non-empty set, and denote by $E(X)$, the [[simplicial set]] having $E(X)_p = X^{p+1}$, so a $p$-simplex is a $p+1$ tuple, $\underline{x}= (x_0,\ldots, x_p)$, each $x_i \in X$, and in which
$$d_i(\underline{x}) = (x_0,\ldots, \hat{x_i}, \ldots x_p),$$and
$$s_j(\underline{x}) = (x_0,\ldots, x_j, x_j, \ldots x_p),$$
so $d_i$ omits $x_i$, whilst $s_j$ repeats $x_j$.
+--{: .un_lemma}
######Lemma
The simplicial set, $E(X)$, is contractible.
=--
The proof is fairly easy to construct and is 'well known'.
The case we are really interested in is when we replace the general set, $X$, by the underlying set of a group, $G$. (As is often done, we will not introduce a special notation for the underlying set of $G$, just writing $G$ for it.) In this case, we have the simplicial set $E(G)$ and the group, $G$, acts freely on $E(G)$ by
$$g\cdot(g_0,\ldots , g_p) = (gg_0,\ldots, gg_p).$$
(Here we have used a left action of $G$, and leave you to check that the evident right action could equally well be used.)
The quotient simplicial set of orbits, will be denoted $G\backslash E(G)$. It is often useful to write $[g_1,\ldots,g_p]$ for the orbit of the $p$-simplex $(1,g_1,g_1g_2,\ldots, g_1g_2\ldots g_p)\in E(G)_p$.
It is 'instructive' to calculate the faces and degeneracy maps in this notation. We will only look at $[g_1,g_2]$ in detail. This element has representative $(1,g_1,g_1g_2)$. We thus have:
* $d_0(1,g_1,g_1g_2) = (g_1,g_1g_2) \equiv (1,g_2)$,
so $d_0[g_1,g_2] = [g_2]$;
* $d_1(1,g_1,g_1g_2) = (1,g_1g_2)$,
so $d_1[g_1,g_2] = [g_1g_2]$;
* $d_2(1,g_1,g_1g_2) = (1,g_1)$,
so $d_2[g_1,g_2] = [g_1]$.
(That looks familiar!)
For the degeneracies,
* $s_0(1,g_1,g_1g_2) = (1,1,g_1,g_1g_2)$, so $s_0 [g_1,g_2] =[1,g_1,g_2] $;
* $s_1(1,g_1,g_1g_2) = (1,g_1,g_1,g_1g_2)$, so $s_1 [g_1,g_2] = [g_1,1,g_2] ;$
and similarly $s_2 [g_1,g_2] = [g_1,g_2,1]$.
The general formulae are now easy to guess and to prove - so they will be left to you, and then the following should be obvious.
+--{: .un_lemma}
######Lemma
There is a natural simplicial isomorphism,
$$G\backslash E(G)\xrightarrow{\cong}Ner(G[1])= BG.$$
=--
We thus have that $G\backslash E(G)$ is a [[classifying space]] for $G$.
This construction of $E(G)$ is exactly that of the nerve of the [[action groupoid]] of the action of $G$ on itself by left multiplication.
##Volodin spaces
We put ourselves in the context of a group, $G$, and a family, $\mathcal{H}$, of subgroups of $G$ as in the context of [[higher generation by subgroups]]. We suppose that $\mathcal{H}= \{H_i\mid i\in I\}$ for some indexing set, $I$.
+--{.un_defn}
(cf. Suslin-Wodzicki, (ref. below) p. 65.)
We denote by $V(G,\mathcal{H})$, or $V(\mathfrak{H})$, the simplicial subset of $E(G)$ formed by simplices, $(g_0,\ldots,g_p)$, that satisfy the condition that there is some $i\in I$ such that, for all $0\leq j,k\leq p$, $g_j g_k^{-1}\in H_i$.
The simplicial set, $V(G,\mathcal{H})$, will be called the _Volodin space_ of $(G,\mathcal{H})$.
=--
##References
* A. A. [[Suslin]] and M. Wodzicki, _Excision in algebraic K-theory_, The Annals of Mathematics,
136, (1992), 51 – 122.
* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative
rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887)
|
Volodymyr Lyubashenko | https://ncatlab.org/nlab/source/Volodymyr+Lyubashenko | **Volodymyr Lyubashenko** (Володимир Любашенко) is a [[mathematical physics|mathematical physicist]] and [[mathematics|mathematician]] from Kiev. He has been working in [[potential theory]], conformal field theory, [[Hopf algebras]], quantum groups, low-dimensional topology, category theory, operads and higher category theory, and [[homological algebra|homological]] (and most recently homotopical) [[algebra]].
* a list of papers [html](http://www.math.ksu.edu/~lub/papers.html)
* webpages at [Kiev](http://www.imath.kiev.ua/~lub), [Kansas State U.](http://www.math.ksu.edu/~lub)
## selected writings
* Yu. Bespalov, V. Lyubashenko, O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))
(on [[pretriangulated A-∞ categories]])
* notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are [here](http://www.math.ksu.edu/~lub/cmcMoCor.pdf).
## related $n$Lab entries
* [[hypersimplex]]
* [[Serre functor]]
* [[Tannaka duality]]
* [[A-∞ category]]
* [[pretriangulated A-∞ category]]
* [[(∞,1)-bimodule]]
* [[Atiyah 2-framing]]
category: people
[[!redirects Володимир Любашенко]] |
volume | https://ncatlab.org/nlab/source/volume |
#Contents#
* table of contents
{:toc}
## Idea
For $(X, \mu)$ a [[measure space]], the [[integral]] $\int_X \mu$ is, if it exists, the _volume_ of $X$ as seen by the [[measure]] $\mu$.
## Related concepts
* [[length]]
* [[area]]
* [[density]], [[volume form]]
* [[complex volume]]
[[!redirects volumes]] |
volume conjecture | https://ncatlab.org/nlab/source/volume+conjecture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _volume conjectures_ are a class of [[conjectures]] (slightly differing in generality and assumptions), saying that on a suitable (in particular: [[hyperbolic 3-manifold|hyperbolic]]) [[3-manifold]] $X$ a [[large N limit]] of [[SU(n)]]-[[Chern-Simons theory]] [[quantum observables]] ($N$-[[colored Jones polynomials]] or more generally [[RT-invariants]], [[TV-invariants]]) is [[equality|equal]] to the [[volume]] or [[complex volume]] of $X$.
For a few special cases of 3-manifolds there are explicit [[proofs]] of the volume conjecture(s). Besides this there is an abundance of [[computer experiment|numerical evidence]] for the volume conjectures, using computer algebra such as [[SnapPy]] (see also [Zickert 07](#Zickert07)). In fact experimentation with these numerics is what has been driving the formulation of further variants of the volume conjecture.
Hence [[experimental mathematics]] strongly suggests that the volume conjectures are true. But a conceptual explanation (let alone [[proof]]) in terms of [[quantum field theory]] has remained open ([Witten 14, bottom of p. 4](#Witten14)).
But an explanation via [[duality in string theory]], combining [[AdS/CFT duality]] with the [[3d/3d correspondence]] for [[wrapped brane|wrapped]] [[M5-branes]], is argued for in [Gang-Kim-Lee 14, 3.2](#GangKimLee14), [Gang-Kim 18 (21)](#GangKim18), see [below](#AsAdSCFTPlus3d3dDuality).
### For $SU(2)$ on knot complements
The original _volume conjecture_ (also "Kashaev's conjecture", due to [Kashaev 95](#Kashaev95), and understood in terms of the $N$-[[colored Jones polynomial]] by [Murakami-Murakami 01](#MurakamiMurakami01)) states that the [[large N limit]] of the $N$-[[colored Jones polynomial]] (for [[gauge group]] [[SU(2)]]) of a [[knot]] $K$ gives the simplicial [[volume]] of its [[complement]] in the [[3-sphere]] (for [[hyperbolic knots]] this is the volume of the complementary [[hyperbolic 3-manifold]])
\[
\label{KashaevConjecture}
lim_{N \to \infty}
\left(
\frac{
2 \pi log
}
{N}
\left\vert
V_N(K; q = e^{\frac{2 \pi i}{N}})
\right\vert
\right)
\;=\;
vol(K).
\]
Here $V_N(K; q)$ is the ratio of the values of the $N$-[[colored Jones polynomial]] of $K$ and of the [[unknot]]
$$
V_N(K; q) = \frac{J_N(K; q)}{J_N(\bigcirc; q)}.
$$
The simplicial volume of a knot complement can be found via its unique **torus** decomposition into hyperbolic pieces and Seifert fibered pieces by a system of tori. The simplicial volume is then the sum of the hyperbolic volumes of the hyperbolic pieces of the decomposition.
If one omits the [[absolute value]] in (eq:KashaevConjecture) then the volume conjecture instead involves the [[complex volume]] ([MMOTY 02, Conjecture 1.2](#MMOTY02)).
### For $SU(2)$ on general 3-manifolds
{#IdeaGenerally}
More generally, volume conjectures state [[convergence of a sequence|convergence]] of the [[Turaev-Viro invariants]] or [[Reshetikhin-Turaev invariants]] on general [[hyperbolic 3-manifolds]] to the [[volume]] or [[complex volume]], respectively.
See ([Chen-Yang 15](#ChenYang15))
### For $SU(n)$
Generalization from [[gauge group]] [[SU(2)]] to [[SU(n)]]: [Chen-Liu-Zhu 15](#ChenLiuZhu15)
## Proof strategies
### As combined AdS/CFT + 3d/3d duality for wrapped M5-branes
{#AsAdSCFTPlus3d3dDuality}
In [Gang-Kim-Lee 14b, 3.2](#GangKimLee14b), [Gang-Kim 18 (21)](#GangKim18) it is argued that the [[volume conjecture]] for [[Chern-Simons theory]] on [[hyperbolic 3-manifolds]] $\Sigma^3$ is the combined statement of two [[dualities in string theory]]:
1. [[AdS/CFT duality]]
1. [[3d-3d correspondence]]
for the situation of [[M5-branes]] [[wrapped brane|wrapped on]] $\Sigma^3$ ([DGKV 10](3d-3d+correspondence#DGKV10)):
<center>
<img src="https://ncatlab.org/nlab/files/VolumeConjectureAsAdSCFTPlus3d3dDuality.jpg" width="660">
</center>
For review of the literature see also [Dimofte 16, Section 4.1](#Dimofte16).
## Related concepts
* [[hyperbolic manifold]]
* [[analytically continued Chern-Simons theory]]
* [[Borel regulator]]
* [[membrane instanton]]
## References
### General
Original articles include
* {#Kashaev95} [[Rinat Kashaev]], _A Link Invariant from Quantum Dilogarithm_, Modern Physics Letters AVol. 10, No. 19, pp. 1409-1418 (1995) ([arXiv:q-alg/9504020](https://arxiv.org/abs/q-alg/9504020))
* {#Kashaev95} [[Rinat Kashaev]], _The Hyperbolic Volume Of Knots From The Quantum Dilogarithm_ Lett. Math. Phys. 39 (1997) 269-275 ([arXiv:q-alg/9601025](https://arxiv.org/abs/q-alg/9601025))
* {#KashaevTirkkonen99} [[Rinat Kashaev]], O. Tirkkonen, _Proof of the volume conjecture for torus knots_, Journal of Mathematical Sciences (2003) 115: 2033 ([arXiv:math/9912210](http://arxiv.org/abs/math/9912210))
* {#MurakamiMurakami01} [[Hitoshi Murakami]], [[Jun Murakami]], _The Colored Jones Polynomial And The Simplicial Volume Of A Knot_, Acta Math. 186 (2001) 85-104 ([euclid.acta/1485891370](https://projecteuclid.org/euclid.acta/1485891370))
* {#MMOTY02} [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, _Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links_, Experiment. Math. 11 (2002) 427-435 ([arXiv:math/0203119](https://arxiv.org/abs/math/0203119))
* {#Murakami04} [[Hitoshi Murakami]], _Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot_, Internat. J. Math. 15 (2004) 547-555.
Generalization to [[Reshetikhin-Turaev construction]] on closed manifold, to the [[Turaev-Viro construction]] on [[manifolds with boundary]], and to more general [[roots of unity]] than considered before is in
* {#ChenYang15} [[Qingtao Chen]], [[Tian Yang]], _A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary_ ([arXiv:1503.02547](http://arxiv.org/abs/1503.02547))
* Dongmin Gang, Mauricio Romo, Masahito Yamazaki, _All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory_, Commun. Math. Phys. (2018) 359: 915. ([arXiv:1704.00918](https://arxiv.org/abs/1704.00918), [doi:10.1007/s00220-018-3115-y](https://doi.org/10.1007/s00220-018-3115-y))
Generalization to [[SU(n)]]:
* {#ChenLiuZhu15} [[Qingtao Chen]], [[Kefeng Liu]], Shengmao Zhu, _Volume conjecture for $SU(n)$-invariants_ ([arXiv:1511.00658](https://arxiv.org/abs/1511.00658))
Review includes
* {#Murakami11} [[Hitoshi Murakami]], _An Introduction to the Volume Conjecture_ ([arXiv:1002.0126](https://arxiv.org/abs/1002.0126)) In: _Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory_, Contemporary Mathematics Volume 541, AMS 2011 ([doi:10.1090/conm/541](http://dx.doi.org/10.1090/conm/541))
* {#Witten14} [[Edward Witten]], pp. 4 of _Two Lectures On The Jones Polynomial And Khovanov Homology_ ([arXiv:1401.6996](http://arxiv.org/abs/1401.6996))
* Wikipedia, _[Volume conjecture](http://en.wikipedia.org/wiki/Volume_conjecture)
See also
* {#Neumann03} [[Walter Neumann]], _Extended Bloch group and the Cheeger-Chern-Simons class_, Geom. Topol. 8 (2004) 413-474 ([arXiv:math/0307092](http://arxiv.org/abs/math/0307092))
* {#Zickert07} [[Christian Zickert]], _The volume and Chern-Simons invariant of a representation_, Duke Math. J., 150 (3):489-532, 2009 ([arXiv:0710.2049](http://arxiv.org/abs/0710.2049))
* {#Neumann11} [[Walter Neumann]], _Realizing arithmetic invariants of hyperbolic 3-manifolds_, Contemporary Math 541 (Amer. Math. Soc. 2011), 233--246 ([arXiv:1108.0062](http://arxiv.org/abs/1108.0062))
* {#GaroufalidisThurstonZickert11} [[Stavros Garoufalidis]], [[Dylan Thurston]], [[Christian Zickert]], _The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds_ ([arXiv:1111.2828](http://arxiv.org/abs/1111.2828), [Euclid](http://projecteuclid.org/euclid.dmj/1259332507))
### Via string theory
#### General
Speculative discussion in terms of [[quantum field theory]] or [[string theory]] includes
* {#Gukov03} [[Sergei Gukov]], _Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial_, Commun. Math. Phys. 255 (2005) 577-627 ([arXiv:hep-th/0306165](https://arxiv.org/abs/hep-th/0306165))
* {#DijkgraafFuji09} [[Robbert Dijkgraaf]], Hiroyuki Fuji, _The Volume Conjecture and Topological Strings_ ([arXiv:0903.2084](http://arxiv.org/abs/0903.2084))
* {#DimofteGukov10} [[Tudor Dimofte]], [[Sergei Gukov]], _Quantum Field Theory and the Volume Conjecture_, Contemporary Mathematics 541 (2011), p.41-67 ([arxiv:1003.4808](http://arxiv.org/abs/1003.4808))
* {#Dimofte16} [[Tudor Dimofte]], Section 4.1 of: _Perturbative and nonperturbative aspects of complex Chern-Simons Theory_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02961](https://arxiv.org/abs/1608.02961))
A conceptual explanation of the volume conjecture via [[analytically continued Chern-Simons theory]] was proposed in
* {#Witten10} [[Edward Witten]], _Analytic Continuation Of Chern-Simons Theory_, AMS/IP Stud. Adv. Math 50 (2011): 347 ([arXiv:1001.2933](https://arxiv.org/abs/1001.2933))
(but it seems that as a sketch or strategy for a rigorous proof, it didn't catch on).
#### As AdS/CFT + 3d/3d duality for wrapped M5-branes
Suggestion that the statement of the [[volume conjecture]] is really [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:
* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, Section 3.2 of: _Holography of 3d-3d correspondence at Large $N$_, JHEP 04 (2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))
* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], around (21) of: _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))
[[!redirects volume conjectures]]
|
volume form | https://ncatlab.org/nlab/source/volume+form |
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#### Riemannian geometry
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Contents
* table of contents
{: toc}
## Idea
The _volume form_ on a finite-[[dimension|dimensional]] [[orientation|oriented]] (pseudo)-[[Riemannian manifold]] $(X,g)$ is the [[differential form]] whose [[integral]] over pieces of $X$ computes the [[volume]] of $X$ as measured by the [[metric]] $g$.
If the manifold is unoriented, then we get a _volume [[pseudoform]]_ instead, or equivalently a volume [[density]] (of weight $1$). We can also consider volume (pseudo)-forms in the absence of a metric, in which case we have a choice of volume forms.
## Definition
### General
For $X$ a general [[smooth manifold]] of finite [[dimension]] $n$, a __volume form__ on $X$ is a nondegenerate (nowhere vanishing) differential $n$-[[differential form|form]] on $X$, equivalently a nondegenerate [[section]] of the [[canonical line bundle]] on $X$. A __volume pseudoform__ or __volume element__ on $X$ is a positive definite [[density]] (of rank $1$) on $X$, or equivalently a positive definite differential $n$-[[pseudoform]] on $X$.
A volume form defines an [[orientation]] on $X$, the one relative to which it is positive definite. If $X$ is already oriented, then we require the orientations to agree (to have a volume form on $X$ qua oriented manifold); that is, a volume form on an oriented manifold must be positive definite (just as a volume pseudoform on any manifold must be). In this situation, there is essentially no difference between a form and a pseudoform, hence no difference between a volume form and a volume pseudoform or volume element.
### For Riemannian manifolds
More specifically, for $(X,g)$ a (pseudo)-[[Riemannian manifold]] of [[dimension]] $n$, *the* __volume pseudoform__ or __volume element__ $vol_g$ is a specific differential $n$-[[pseudoform]] that measures the volume as seen by the metric $g$. If $X$ is [[orientation|oriented]], then we may interpret $vol_g \in \Omega^n(X)$ as a differential $n$-form, also denoted $vol_g$.
This $vol_g$ is characterized by any of the following equivalent statements:
* The symmetric square $vol_g \cdot vol_g$ is equal to the $n$-fold [[wedge product]] $g \wedge \cdots \wedge g$, as elements of $\Omega^n(X) \otimes \Omega^n(X)$, and $vol_g$ is [[positive n-form|positive]] (meaning that its [[integral]] on any [[open submanifold]] is nonnegative).
* The volume form is the image under the [[Hodge star]] operator $\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X)$ of the [[smooth function]] $1 \in \Omega^0(X)$
$$
vol_g = \star_g 1
\,.
$$
* In local oriented coordinates, $vol_g = \sqrt{|det(g)|}$, where $det(g)$ is the [[determinant]] of the [[matrix]] of the coordinates of $g$. In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to $vol_g = \sqrt{\det(g)}$. (Note that local coordinates for a pseudoform include a local orientation, so this makes sense regardless of whether $X$ is oriented.)
* For $(E, \Omega)\colon T X \to iso(n)$ the [[Lie algebra valued differential form]] on $X$ with values in the [[Poincare Lie algebra]] $iso(n)$ that encodes the metric and orientation (the _[[spin connection]]_ $\Omega$ with the _[[vielbein]]_ $E$), the volume form is the image of $(E,\Omega)$ under the canonical volume [[Lie algebra cohomology|Lie algebra cocycle]] $vol \in CE(iso(n))$:
$$
vol_g = vol(E)
\,.
$$
See [[Poincare Lie algebra]] for more on this.
### Degenerate cases
If we allow a volume (pseudo)form to be degenerate, then most of this goes through unchanged. In particular, a degenerate (pseudo)-[[Riemannian metric]] defines a degenerate volume pseudoform (and hence a degenerate volume form on an oriented manifold).
However, a degenerate $n$-form $\omega$ does not specify an orientation in general, so there is not necessarily a good notion of volume form on an unoriented manifold. On the other hand, if the open submanifold on which $\omega \ne 0$ is [[dense subspace|dense]], then there is at most one compatible orientation, although there still may be none. Of course, on an oriented manifold, forms are equivalent to pseudoforms, so we still know what a degenerate volume form is there.
To remove the requirement of positivity is much more drastic; an arbitrary $n$-pseudoform is simply a $1$-[[density]] (and an arbitrary $n$-form is a $1$-pseudodensity). This is at best a notion of *signed* volume, rather than volume.
## Properties
Since an $n$-(pseudo)form is [[positive n-form|positive]] iff its [[integration of differential forms|integral]] on any [[open submanifold]] is nonnegative and nondegenerate iff its integral on sufficiently small [[inhabited subspace|inhabited]] open submanifolds is nonzero, a volume (pseudo)form may be defined as one whose integral on any inhabited open submanifold is (strictly) positive.
A volume (pseudo)form is also equivalent to an [[absolutely continuous measure|absolutely continuous]] positive [[Radon measure]] on $X$. Here, nondegeneracy corresponds precisely to absolute continuity.
If I remember correctly, every volume (pseudo)form comes from a metric, which is unique iff $n \leq 1$.
## Related concepts
* [[density]]
* [[divergence]]
[[!redirects volume form]]
[[!redirects volume forms]]
[[!redirects volume pseudoform]]
[[!redirects volume pseudoforms]]
[[!redirects volume element]]
[[!redirects volume elements]]
|
volume of a Lie groupoid | https://ncatlab.org/nlab/source/volume+of+a+Lie+groupoid | #Idea#
One expects a notion of volume measure on [[Lie groupoid]]s (also known as [[differentiable stack]]s), which generalizes the notion of [[groupoid cardinality]] of finite groupoids in that it reduces the volume of the object space by the degree to which [[automorphism]]s encode [[homological resolution|weak quotient]]s.
A solution to this was proposed by Alan Weinstein, and re-interpreted in terms of 2-vector bundles by Richard Hepworth.
One would expect some relation of this to the [[BV theory|Lagrangian BV formalism]], which is also a formalism for integration over $L_\infty$-[[Lie infinity-algebroid|algebroid]]s.
#References#
* [[Alan Weinstein]], _The volume of a differentiable stack_ ([arXiv](http://arxiv.org/abs/0809.2130))
* Richard Hepworth, _2-Vector Bundles and the Volume of a Differentiable Stack ([pdf](http://www.hepworth.staff.shef.ac.uk/files/2VB.pdf))
[[!redirects volume of Lie groupoids]] |
von Neumann algebra | https://ncatlab.org/nlab/source/von+Neumann+algebra |
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#### Algebra
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#### Functional analysis
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#### Measure and probability theory
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#### AQFT
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\tableofcontents
## Idea
A _von Neumann algebra_ or _$W^*$-algebra_ is an important and special kind of [[operator algebra]], relevant in particular to [[measure theory]] and [[quantum mechanics]]/[[quantum field theory]] in its algebraic formulation as [[AQFT]]. Specifically, (non-commutative) von Neumann algebras can be understood as the [[Isbell duality|formal duals]] of ([[non-commutative geometry|non-commutative]]) [[localizable measurable spaces]] (or [[measurable locales]]); see the section _[Relation to measurable spaces](#RelationToMeasureSpaces)_ below.
## History and terminology
Since terminology varies in the literature, we will say something about this first. There are no precise definitions here; see below for those.
(Of course, _$W^*$-algebras_ should not be confused with _[[W-algebras]]_ in (logarithmic) [[conformal field theory]].)
[[John von Neumann]] originally studied certain [[operator algebras]] (back then they were called _[[rings]] of [[linear operator|operators]]_), defined as unital $*$-subalgebras of the algebra $B(H)$ of [[bounded operators]] on some [[Hilbert space]] $H$ that are [[closed subspace|closed]] in any of the several [[operator topology|operator topologies]] on $B(H)$ (except for the norm topology, which gives $C^*$-[[C-star-algebra|algebras]]); the ultraweak topology is most convenient for our purposes.
One disadvantage of such a definition is that it makes it difficult to separate properties of von Neumann algebras from properties of their [[representations]] on Hilbert spaces. For example, all faithful representations induce the same ultraweak topology on a von Neumann algebra, but different representations induce different weak topologies. Furthermore, not all von Neumann algebras come automatically
equipped with a representation on a Hilbert space, such as the [[coproduct]] of two von Neumann algebras (although such a representation can always be constructed). Finally, this definition unnecessarily confuses two very distinct notions: algebras and [[modules]] (or representations).
Therefore, we may use the modern abstract terminology in which a von Neumann algebra is defined as an [[associative algebra|algebra]] with certain [[extra structure|structures]] and [[extra property|properties]]. It then becomes a theorem that every von Neumann algebra has a free representation on a Hilbert space (such as Haagerup\'s standard form), so we may study von Neumann algebras in the historical concrete sense if we wish; but now we think of these as particular *representations* of algebras.
In the old terminology, morphisms of representations
of von Neumann algebras (von Neumann algebras in the historical concrete sense) are sometimes called _spatial morphisms of von Neumann algebras_ (as opposed to the _abstract morphisms_ that we will define below). Similarly, the concrete von Neumann algebras themselves are sometimes called _von Neumann algebras_, whereas the abstract von Neumann algebras are called _$W^*$-algebras_. Compare the historic definitions of $C^*$-algebras, as well as other examples of [[concrete and abstract structures]] such as [[manifolds]].
The [[nPOV]] dictates that a clear distinction between
the categories of algebras and modules must be maintained,
in particular, modules should not be confused with algebras.
Hence we stick to the modern terminology, which also seems
to be preferred in new papers on von Neumann algebras,
see for example [arXiv:1110.5671v1](http://arxiv.org/abs/1110.5671).
## Definitions
For completeness, we give both the modern abstract and historical concrete definitions.
### Abstract von Neumann algebras {#abstractdefn}
We build on the concepts of [[Banach space]] and (abstract) $C^*$-[[C-star-algebra|algebra]]. In this definition, a Banach space is a [[complex number|complex]] Banach space and a [[morphism]] of Banach spaces is a [[short linear map]] (a complex-linear map of norm at most $1$); a $C^*$-algebra is a complex unital $C^*$-algebra, and a morphism of $C^*$-algebras is a unital $*$-homomorphism (which is necessarily also a short linear map). Note in particular that an [[isomorphism]] of either must be an [[isometry]].
Given a Banach space $A$, a __[[predual]]__ of $A$ is a Banach space $V$ whose [[dual Banach space]] $V^*$ is isomorphic to $A$:
$$ V^* \overset{i}\to A $$
(or more properly, equipped with such an isomorphism $i$). Similarly, given a morphism $f\colon A \to B$ (properly, with $A$ and $B$ so equipped), a __predual__ of $f$ is a morphism $t\colon W \to V$ whose dual is isomorphic to $f$:
$$ \array {
V^* & \overset{i}\to & A \\
\mathllap{t^*}\downarrow & & \downarrow\mathrlap{f} \\
W^* & \underset{j}\to & B
} .$$
With these preliminaries, a __$W^*$-algebra__ or ("abstract") *von Neumann algebra* is a $C^*$-algebra that admits a predual (or more properly, equipped with one), and a $W^*$-[[homomorphism]] is a $C^*$-homomorphism that admits a predual. In this way, the [[category]] of $W^*$-algebras becomes a [[subcategory]] of the category of $C^*$-algebras.
It is a theorem (see [below](#preduals)) that the predual of a $W^*$-algebra or $W^*$-homomorphism is essentially unique; we speak of [[generalized the|the]] predual of $A$, write it $A_*$, and identify $A$ with $(A_*)^*$ (and similarly for morphisms). (So in fact we don\'t need the word 'equipped'; being a $W^*$-algebra is an [[extra property]], not an [[extra structure]], on a $C^*$-algebra.)
### Concrete von Neumann algebras
Fix a [[complex number|complex]] [[Hilbert space]] $H$ and consider the [[associative algebra|algebra]] $B(H)$ of [[bounded operators]] on $H$. A ("concrete") __von Neumann algebra__ on $H$ is a unital $*$-subalgebra of $B(H)$ that is [[closed subspace|closed]] in the [[weak operator topology]], or equivalently in the [[ultraweak topology]] or in the [[strong topology]]. As such is automatically [[closed subspace|closed]] in the [[norm topology]], the von Neumann algebras form a (particularly nice) class of concrete $C^*$-[[C-star-algebra|algebras]] on $H$, where the latter are defined as unital $*$-subalgebras of $B(H)$ closed under the norm topology.
We equip a von Neumann algebra with the topology induced by its inclusion into $B(H)$ equipped with the ultraweak topology. An __abstract morphism__ of von Neumann algebras can then be defined as a unital $*$-homomorphism that is continous in the ultraweak topology. Here we are disregarding the data of the inclusion of a von Neumann algebra into $B(H)$ and treating it as an algebra on its own.
Alternatively, we can define a von Neumann algebra $A$
as a unital $*$-algebra that admits an injective morphism into $B(H)$ for some Hilbert space $H$ such that the image of the inclusion is closed in the ultraweak topology on $B(H)$. One can then prove that the topology induced on $A$ by the ultraweak topology on $B(H)$ does not depend on the choice of $H$ or the particular inclusion of $A$ into $B(H)$. Hence one can define an abstract morphism of von Neumann algebras as a unital morphism of $*$-algebras that is continuous in the ultraweak topology.
It is a theorem that the category of (concrete) von Neumann algebras and abstract morphisms is [[equivalence of categories|equivalent]] to the category of (abstract) $W^*$-algebras and $W^*$-homomorphisms. Similarly, we get the category of [[representations]] of $W^*$-algebras on Hilbert spaces using instead the __spatial morphisms__ of concrete von Neumann algebras.
## Sakai's theorem and properties of preduals
{#preduals}
Sakai's theorem states that preduals considered in the abstract definition
are necessarily unique.
More precisely, given a von Neumann algebra $A$
we consider the category whose objects are pairs $(V,f)$,
where $V$ is a Banach space and $f\colon V^*\to A$ is an isomorphism
of Banach spaces. A morphism from $(V,f)$ to $(W,g)$
is a morphism $h\colon V\to W$ of Banach spaces
such that $f h^* = g$.
Sakai's theorem then states that in the above category
there is exactly one morphism
between any pair of objects, which is necessarily an isomorphism.
In particular, the category of preduals is canonically isomorphic
to the [[terminal category]].
Sakai's theorem can be extended to morphisms of von Neumann algebras.
Thus preduals of von Neumann algebras and their morphisms
are unique up to a unique isomorphism, in particular we can talk
about [[generalized the|the]] predual of a von Neumann algebra and [[generalized the|the]] predual
of a morphism of von Neumann algebras.
The weak topology induced on a von Neumann algebra by its predual
is called the **ultraweak topology**.
The role of the ultraweak topology for von Neumann algebras
is analogous to the role of the norm topology for C\*-algebras.
In particular, morphisms of von Neumann algebras are precisely
those morphisms of C\*-algebras that are continuous
in the ultraweak topology.
Consider the [[dual vector space|dual space]] $V$ of a von Neumann algebra $A$ equipped
with the ultraweak topology.
The [[topological vector space]] $V$ canonically embeds
into the dual of $A$ as a Banach space, the embedding map being
induced by the canonical continuous map from $A$
equipped with the norm topology to $A$
equipped with the ultraweak topology.
Thus $V$ is also a Banach space.
There is a canonical morphism of Banach spaces from $A$ to $V^*$
given by evaluating an element of $V$ on the given element of $A$.
This morphism is in fact an isomorphism, hence $V$ is the predual of $A$.
In other words, the predual of a von Neumann algebra is canonically
isomorphic to its dual in the ultraweak topology.
Similarly, the predual of a morphism of von Neumann algebras
is canonically isomorphic to its dual in the ultraweak topology.
## Elementary examples
The easiest example of a von Neumann algebra is given by the $C^*$-algebra $B(H)$
of bounded operators on a complex [[Hilbert space]] $H$.
The predual can be canonically identified with the Banach space
of [[trace class operator|trace class operators]].
Any $C^*$-subalgebra of $B(H)$ closed in the ultraweak topology
is again a von Neumann algebra.
Another example is $L^\infty(X)$ under pointwise almost everywhere multiplication, where $X$ is a σ-finite [[measure space]] or a [[localizable measurable space]]. These are (up to isomorphism) all of the _commutative_ von Neumann algebras, according to a specialized version of the [[Gelfand duality theorem]].
A faithful representation (in fact, the standard from) of $L^\infty(X) \hookrightarrow B(L^2(X))$ is given by considering $L^2(X)$ as an $L^\infty(X)$-module given by pointwise almost everywhere multiplication.
## Properties of morphisms of von Neumann algebras
## The category of von Neumann algebras
The category of von Neumann algebras is not a [[locally presentable category]]
since its [[small objects]] are precisely von Neumann algebras of dimension 0 and 1.
See Theorem 4.2 in [Chirvasitu–Ko](#CK).
The [[forgetful functor]] from von Neumann algebras
to [[sets]] that sends a von Neumann algebra
to its [[unit ball]] is a [[right adjoint functor]].
In fact, it is a [[monadic functor]] and preserves all [[sifted colimits]].
Thus, [[limits]] and [[sifted colimits]] of von Neumann algebras can be computed
on the level of underlying unit balls.
Small [[coproducts]] of von Neumann algebras exist.
There is also a “reduced” version of small [[coproducts]],
known as [[free products]],
which can be defined in a manner analogous to the [[spatial tensor product]].
## Monoidal structures
There are two different [[tensor products]] one can define on von Neumann algebras.
First, one can use the usual universal property of tensor products
and postulate that morphisms $A\otimes B\to M$
are in a natural bijection with pairs of morphisms
$A\to M$ and $B\to M$ whose images commute in $M$.
This yields a [[symmetric monoidal structure]] on von Neumann algebras.
This monoidal structure is not [[closed]].
Secondly, one can also define a “reduced” version, known as the _spatial tensor product_.
Given two von Neumann algebras $A$ and $B$, their spatial tensor product
is the von Neumann algebra generated by $A\otimes 1$ and $1\otimes B$
in the von Neumann algebra $B(L^2 A\otimes L^2 B)$,
where $L^2 A$ and $L^2 B$ are the Haagerup standard form of $A$ and $B$ respectively.
This also results in a [[symmetric monoidal structure]].
Furthermore, passing to the [[opposite category]]
yields a [[closed monoidal structure]].
## $W^*$-categories
## Modules over von Neumann algebras
## Bimodules over von Neumann algebras and Connes fusion
## Modular algebra and Tomita--Takesaki theory
* [[modular theory]]
## Gelfand duality for commutative von Neumann algebras
See the article [[commutative von Neumann algebra]].
## Relevance
The [[Gelfand duality theorem]] states that there is a contravariant [[equivalence of categories|equivalence]] between the [[category]] of commutative von Neumann algebras and the category of compact strictly localizable enhanced [[measurable space|measurable spaces]]; that is, the [[opposite category]] of one is equivalent to the other. See [Relation to Measurable Spaces](#RelationToMeasureSpaces) below. General von Neumann algebras are seen then as a 'noncommutative' measurable spaces in a sense analogous to [[noncommutative geometry]].
The importance of von Neumann algebras for ([[higher category theory|higher]]) [[category theory]] and topology lays in the evidence that von Neumann algebras are deeply connected with the low dimensional [[quantum field theory]] (2d [[CFT]], [[TQFT]] in low dimensions, inclusions of [[von Neumann algebra factor|factor]]s, [[quantum group]]s and [[knot theory]]; [[elliptic cohomology]]: works of Wenzl, Vaughan Jones, Anthony Wasserman, Kerler, Kawahigashi, Ocneanu, Szlachanyi etc.).
The highlights of their structure theory include the results on classification of [[von Neumann algebra factor|factors]] ([[Alain Connes]], 1970s) and theory of inclusions of subfactors (V. Jones). (Hilbert) [[bimodule]]s over von Neumann algebras have a remarkable tensor product due Connes ([[Connes fusion]]). Following Segal's manifesto
* Graeme Segal, _Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)_. Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162 (1988), Exp. No. 695, 4, 187--201 (1989).
and its update
* [[Graeme Segal]], _What is an elliptic object?_ Elliptic cohomology, 306--317, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007.
on hypothetical connections between [[CFT]] and [[elliptic cohomology]], Stolz and Teichner have made a case for a role von Neumann algebras seem to play in a partial realization of that program:
* [[Stefan Stolz]] and [[Peter Teichner]], _What is an elliptic object?_ ([pdf](http://math.berkeley.edu/~teichner/Papers/Oxford.pdf))
See also the [Wikipedia entry](http://en.wikipedia.org/wiki/Von_Neumann_algebra) entry for more on von Neumann algebras and a list of references and links.
## General
The _[[bicommutant theorem]]_ (as known as the _double commutant theorem_ , or _von Neumann's double commutant theorem_ ) is the following result.
Let $A \subseteq B(H)$ be a sub-[[star-algebra]] of the [[C-star algebra]] of [[bounded linear operator|bounded linear operators]] on a [[Hilbert space]] $H$. Then $A$ is a [[von Neumann algebra]] on $H$ if and only if $A = A''$, where $A'$ denotes the [[commutant]] of $A$.
Notice that the condition of $A$ being a von Neumann algebra (being closed in the [[weak operator topology]]; "weak" here can be replaced by "strong", "ultrastrong", or "ultraweak" as described in [[operator topology]]), which is a [[topology|topological]] condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).
## Relation to measurable spaces
{#RelationToMeasureSpaces}
The [[Gelfand-Naimark theorem|Gel'fand–Naimark theorem]] states that the category of [[localizable measurable spaces]] is contravariantly [[equivalence of categories|equivalent]] to (that is equivalent to the [[opposite category|opposite]] of) the category of commutative von Neumann algebras. As such, arbitrary von Neumann algebras may be interpreted as 'noncommutative' measurable spaces in a sense analogous to [[noncommutative geometry]]. See at _[[noncommutative probability space]]_.
## Topics of interest for the understanding of AQFT
This paragraph will collect some facts of interest for the aspects of [[AQFT]].
In this paragraph $\mathcal{M}$ will always be a von Neumann algebra acting on a [[Hilbert space]] $\mathcal{H}$ with [[commutant]] $\mathcal{M}'$.
### Vectors
+-- {: .un_defn}
###### Definition
A [[vector]] $x \in \mathcal{H}$ is a **[[cyclic vector]]** if $\mathcal{M}x$ is dense in $\mathcal{H}$.
=--
+-- {: .un_defn}
###### Definition
A [[vector]] $x \in \mathcal{H}$ is a **[[separating vector]]** if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.
=--
+-- {: .un_theorem}
###### Theorem
The notions of cyclic and separating are dual with respect to the commutant, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.
=--
### Projections in von Neumann algebras
One crucial feature of von Neumann algebras is that they contain
"every projection one could wish for": there are three points that make this statement precise:
* the linear combinations of projections are norm dense in a von Neumann algebra
* [[Gleason's theorem]]
* Murray--von Neumann classification of [[von Neumann algebra factor|factors]]
#### Projections are norm dense
First let us note that every element $A$ of a von Neumann algebra can trivially be written as a linear combination of two selfadjoint elements:
$$
A = \frac{1}{2} (A + A^*) + i\frac{1}{2i} (A - A^*)
$$
Then, by the [[spectral theorem]] every selfadjoint element A is represented by it's spectral measure E via
$$
A = \integral_{-\|A\|}^{\|A\|} \lambda E(d\lambda)
$$
The integral converges in norm to A and all spectral projections are elements of the von Neumann algebra. It immediatly follows that the set of finite sums of multiples of projections is norm dense in every von Neumann algebra.
#### Gleason's theorem {#GleasonsTheorem}
See [[Gleason's theorem]].
#### Murray--von Neumann classification of factors
To be done...
### Miscellaneous
* [[split inclusion of von Neumann algebras]]
## Related concepts
* [[enveloping von Neumann algebra]]
* [[von Neumann algebra factor]]
* [[W-star category]]
* [[quantum relation]]
## References
{#References}
The original definition is due to [[John von Neumann]]:
* [[J. v. Neumann]], _Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren_, Mathematische Annalen 102:1 (1930), 370-427, [doi](http://dx.doi.org/10.1007/bf01782352).
Monographs:
* [[Masamichi Takesaki]]_, _Theory of Operator Algebras_, Encyclopaedia of Mathematical Sciences **124**, **125**, **127** (1979, 2002, 2003)
* {#Blackadar06} [[Bruce Blackadar]], *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]
On [[von Neumann algebras]] in [[algebraic quantum field theory]]:
* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *From quantum fields to local von Neumann algebras*, Rev. Math. Phys. **4** spec01 (1992) 15-47 [[doi:10.1142/S0129055X92000145](https://doi.org/10.1142/S0129055X92000145)]
See also:
* [[Jacob Lurie]], _von Neumann algebras_, lecture series (2011) ([web](http://www.math.harvard.edu/~lurie/261y.html))
* Abraham Westerbaan, _The Category of von Neumann Algebras_, [1804.02203](https://arxiv.org/abs/1804.02203) 2018
* {#CK} [[Alexandru Chirvasitu]], Joanna Ko, _Monadic forgetful functors and (non-)presentability for C∗- and W*-algebras_, [arXiv:2203.12087](https://arxiv.org/abs/2203.12087).
In [[quantum physics]]/[[quantum computing]]:
as [[categorical semantics]]
for the [[quantum lambda-calculus]]:
* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]
and for [[QPL]]:
* [[Kenta Cho]], *Semantics for a Quantum Programming Language by Operator Algebras*, EPTCS **172** (2014) 165-190 [[arXiv:1412.8545](https://arxiv.org/abs/1412.8545), [doi:10.4204/EPTCS.172.12](https://doi.org/10.4204/EPTCS.172.12)]
[[!redirects W*-algebra]]
[[!redirects W*-algebras]]
[[!redirects W* algebra]]
[[!redirects W* algebras]]
[[!redirects W-star-algebra]]
[[!redirects W-star-algebras]]
[[!redirects W-star algebra]]
[[!redirects W-star algebras]]
[[!redirects von Neumann algebra]]
[[!redirects von Neumann algebras]]
[[!redirects von Neumann-algebra]]
[[!redirects von Neumann-algebras]]
[[!redirects category of von Neumann algebras]]
[[!redirects categories of von Neumann algebras]]
|
von Neumann algebra factor | https://ncatlab.org/nlab/source/von+Neumann+algebra+factor |
+-- {: .rightHandSide}
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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $A$ a [[von Neumann algebra]] write $A'$ for its [[commutant]] in the ambient algebra $B(\mathcal{H})$ of [[bounded operator]]s.
+-- {: .num_defn}
###### Definition
A von Neumann algebra $A$ is called a **factor** if its [[center]] is trivial
$$
Z(A) := A \cap A' = \mathbb{C}1
\,.
$$
Equivalently: if $A$ and its [[commutant]] $A'$ generate the full algebra of [[bounded operators]] $B(\mathcal{H})$.
=--
## Properties
Every von Neumann algebra may be written as a [[direct integral]] over factors. ([von Neumann 49](#vNeumann49))
## Classification
Factors are classified in terms of the [[K-theory]] of their categories of finite [[W\*-modules]].
A [[W\*-module]] over a factor $A$ is _finite_ if it is not isomorphic to its proper submodule.
### Type I
Type I factors are characterized by the condition that the K-theory of finite modules
is isomorphic to $\mathbf{Z}$, the group of integers.
The only factors of this type are of the from $B(H)$, bounded operators on a Hilbert space $H$.
### Type II
Type II factors are characterized by the condition that the K-theory of finite modules
is isomorphic to $\mathbf{R}$, the group of real numbers.
Type II factors are subdivided into two classes: type II$_1$ factors are characterized
by the condition that $A$ is a finite $A$-module,
whereas for a type II$_\infty$ factor $A$ is not a finite $A$-module.
### Type III
Type III factors are characterized by the condition that the K-theory of finite modules
is trivial, i.e., only the zero module is finite.
Type III factors are further subdivided into three classes,
according to the structure of the center of their [[modular algebra]],
which is a commutative von Neumann algebra graded by purely imaginary numbers,
whose graded components are [[noncommutative L^p-spaces]].
By the [[von Neumann duality]] for commutative von Neumann algebras,
the spectrum of this center is a [[measurable space]] equipped with a σ-ideal
of negligible sets and the grading yields an action of $\mathbf{R}$, the group of real numbers.
This object is known as the [[noncommutative flow of weights]].
If the center is trivial (so the spectrum is a point), the factor has type III$_1$.
If the action of $\mathbf{R}$ is not periodic, then the factor has type III$_0$.
If the action is periodic with period $\lambda$, a positive real number,
then the factor has type III$_{\exp(-\lambda)}$.
## Related concepts
* [[C-star algebra]]
* [[uniformly hyperfinite algebra]]
## References
### General
{#ReferencesGeneral}
The original articles:
The classification of factors into types I, II, III and the construction of examples not of type I:
* [[Francis J. Murray]], [[John von Neumann]], *On Rings of Operators*, Annals of Mathematics, Second Series, **37** 1 (1936) 116-229 [[doi:10.2307/1968693](https://doi.org/10.2307/1968693), [jstor:1968693](https://www.jstor.org/stable/1968693)]
Discussion of [[traces]] on these factors:
* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. II*, Trans. Amer. Math. Soc. **41** (1937) 208-248 [[doi:10.2307/1989620](https://doi.org/10.2307/1989620), [jstor:1989620](https://www.jstor.org/stable/1989620)]
On [[isomorphism]] of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:
* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. IV*, Annals of Mathematics, Second Series, **44** 4 (1943) 716-808 [[doi:10.2307/1969107](https://doi.org/10.2307/1969107), [jstor:1969107](https://www.jstor.org/stable/1969107)]
On decomposing [[von Neumann algebras]] as a [[direct integral]] of factors:
* {#vNeumann49} [[John von Neumann]], _On rings of operators, reduction theory_, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [[jstor:1969463]( http://www.jstor.org/stable/1969463)]
Recollection of the history which made von Neumann turn to discussion of these "factors", motivated from considerations in the foundations of [[quantum mechanics]] and [[quantum logic]]:
* {#Rédei96} [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [<a href="https://doi.org/10.1016/S1355-2198(96)00017-2">doi:10.1016/S1355-2198(96)00017-2</a>]
Lecture notes:
* V.S. Sunder, _von Neumann algebras, $II_1$-factors, and their subfactors_ [[pdf](https://www.imsc.res.in/~sunder/iitb3.pdf)]
* Hideki Kosaki, _Type III factors and index theory_ (1993) [[pdf](http://pages.uoregon.edu/njp/lec-f.pdf)]
### Subfactors
The mathematics of inclusions of _subfactors_ is giving deep structural insights. See also at _[[planar algebra]]_.
* [[Vaughan Jones]],
_Index for subfactors_, Invent. Math. __72__, I (I983);
_A polynomial invariant for links via von Neumann algebras_, Bull. AMS __12__, 103 (1985);
_Hecke algebra representations of braid groups and link polynomials_, Ann. Math. __126__, 335 (1987)
* [[Vaughan Jones]], [[Scott Morrison]], [[Noah Snyder]], _The classification of subfactors of index at most 5_ ([arXiv:1304.6141](http://arxiv.org/abs/1304.6141))
* Vaughan F. R. Jones, David Penneys, _Infinite index subfactors and the GICAR categories_, [arxiv/1410.0856](http://arxiv.org/abs/1410.0856)
Symmetries of depth two inclusions of subfactors may be described via associative [[bialgebroid]]s,
* Lars Kadison, Kornél Szlachányi, _Bialgebroid actions on depth two extensions and duality_, Adv. Math. __179__:1 (2003) 75-121 <a href="https://doi.org/10.1016/S0001-8708(02)00028-2">doi</a>
### Relation to quantum field theory
{#ReferencesInQuantumFieldTheory}
On von Neumann algebra factors as algebras of [[quantum observables]] in [[quantum physics]] and [[quantum field theory]] (which was their original motivation, cf. [Rédei 1996](#Rédei96)):
* [[Jakob Yngvason]], _The role of type III factors in quantum field theory_ [[arXiv:math-ph/0411058](https://arxiv.org/abs/math-ph/0411058)]
* [[Jonathan Sorce]], *Notes on the type classification of von Neumann algebras* [[arXiv:2302.01958](https://arxiv.org/abs/2302.01958)]
* [[Roberto Longo]], [[Edward Witten]], *A note on continuous entropy* [[arXiv:2202.03357](https://arxiv.org/abs/2202.03357), [spire:2029393](https://inspirehep.net/literature/2029393)]
and particularly in [[quantum field theory on curved spacetime]]:
* [[Edward Witten]], *Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?*, in *Dialogues Between Physics and Mathematics*, Springer (2022) [[arXiv:2112.11614](https://arxiv.org/abs/2112.11614), [doi:10.1007/978-3-031-17523-7_11](https://doi.org/10.1007/978-3-031-17523-7_11)]
* Edward Witten, *Algebras, Regions, and Observers* [[arXiv:2303.02837](https://arxiv.org/abs/2303.02837)]
such as on [[de Sitter spacetime]]:
* [[Venkatesa Chandrasekaran]], [[Roberto Longo]], [[Geoff Penington]], [[Edward Witten]], *An Algebra of Observables for de Sitter Space*, Journal of High Energy Physics **2023** 82 (2023) [[arXiv:2206.10780](https://arxiv.org/abs/2206.10780), <a href="https://doi.org/10.1007/JHEP02(2023)082">doi:10.1007/JHEP02(2023)082</a>]
and potential application of von Neumann algebra factors to [[quantum gravity]]:
* [[Edward Witten]], *Gravity and the Crossed Product*, Journal of High Energy Physics **2022** 8 (2022) [[arXiv:2112.12828](https://arxiv.org/abs/2112.12828), <a href="https://doi.org/10.1007/JHEP10(2022)008">doi:10.1007/JHEP10(2022)008</a>]
* [[Edward Witten]], *A Note On The Canonical Formalism for Gravity* [[arXiv:2212.08270](https://arxiv.org/abs/2212.08270), [inspire:2615434](https://inspirehep.net/literature/2615434)]
[[!redirects von Neumann algebra factors]]
[[!redirects von Neumann algebra subfactor]]
[[!redirects von Neumann algebra subfactors]]
[[!redirects subfactor]]
[[!redirects subfactors]]
|
von Neumann hierarchy | https://ncatlab.org/nlab/source/von+Neumann+hierarchy | # Idea
The **von Neumann hierarchy** is a way of "building up" all [[pure set]]s recursively, starting with the [[empty set]], and indexed by the [[ordinal number]]s.
# Definition
Using [[transfinite recursion]], define a hierarchy of well-founded sets $V_\alpha$, where $\alpha\in\mathbf{Ord}$ is an [[ordinal number]], as follows:
* $V_0 = \emptyset$
* $V_{\alpha+1} = P(V_\alpha)$ (the [[power set]] of $V_\alpha$)
* $V_\alpha = \cup_{\beta\lt\alpha} V_\beta$ if $\alpha$ is a limit ordinal.
The formula for $0$ is actually a special case of the formula for a limit ordinal. Alternatively, you can do them all at once:
* $V_\alpha = \cup_{\beta \lt \alpha} P(V_\beta)$
The [[axiom of foundation]] in [[ZFC]] is equivalent to the statement that every set is an element of $V_\alpha$ for some ordinal $\alpha$. The **rank** of a set $x$ is defined to be the least $\alpha$ for which $x\in V_\alpha$ (this is well-defined since the ordinals are [[well-order|well-ordered]]).
|
von Neumann regular category | https://ncatlab.org/nlab/source/von+Neumann+regular+category | +-- {: .rightHandSide}
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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
[[!redirects Von Neumann regular categories]]
[[!redirects Von Neumann regular category]]
[[!redirects von Neumann regular categories]]
## Idea
**Von Neumann regular categories**, or _regular categories_ for short, are a class of [[small categories]] introduced by [[William Lawvere]] ([2002](#Law02)) in the context of his theory of dimension and [[unity of opposites]] in order to provide [[presheaf toposes]] with well-behaved [[level of a topos|levels]] which in particular admit the [[Aufhebung]] of each level.
As shown in ([Kelly-Lawvere 1989](#KL89)) essential subtoposes of presheaf toposes $\mathcal{S}^{\mathcal{C}^{op}}$ correspond to idempotent two-sided ideals $J$ of the underlying small category $\mathcal{C}$. What causes the problems for the general existence of Aufhebung at each level is the fact that an infinite intersection of such $J$ is not necessarily idempotent itself. The condition occurring in the definition of a von Neumann regular category is a simple way to enforce this.
## Definition
A small category $\mathcal{C}$ is called _von Neumann regular_ if all two-sided ideals are [[idempotent]].
## Remark
The terminology is presumably chosen in view of the concept of a _von Neumann regular ring_ $R$ i.e. one such that every $a \in R$ has a 'weak inverse' $\bar{a}$ with $a = a \bar{a} a$ as the following property illustrates.
## Properties
* $\mathcal{C}$ is von Neumann regular iff for any morphism $a$ in $\mathcal{C}$ there exists a reverse morphism $\bar{a}$ and two endomorphisms $x,y$ with $a=y a \bar{a} a x$. ([Lawvere 2002](#Law02))
## Related entries
* [[Aufhebung]]
* [[graphic category]]
* [[level of a topos]]
## References
* [[Francis Borceux|F. Borceux]], [[Jiri Rosicky|J. Rosicky]], _On Von Neumann Varieties_ , TAC **13** no. 1 (2004) pp.5-26. ([pdf](http://www.tac.mta.ca/tac/volumes/13/1/13-01.pdf))
* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the Complete Lattice of Essential Localizations_ , Bull.Soc.Math. de Belgique **XLI** (1989) 261-299 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]
* {#Law89b} [[F. W. Lawvere]], _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_ , Proceedings of the first international conference on algebraic methodology and software technology University of Ioowa, May 22-24 1989, Iowa City, pp.51-74.
* {#Law02} [[F. W. Lawvere]], _Linearization of graphic toposes via Coxeter groups_ , JPAA **168** (2002) pp.425-436. |
von Neumann–Bernays–Gödel set theory | https://ncatlab.org/nlab/source/von+Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del+set+theory | # Contents
* automatic table of contents goes here
{: toc}
## Idea
NBG or von Neumann–Bernays–Gödel set theory is a [[material set theory]]. It is a conservative extension of [[ZFC]] and its ontology includes [[proper classes]], like [[MK]]. But unlike [[ZFC]] and [[MK]], NBG can be finitely axiomatized.
## History
[[Georg Cantor]] was well acquainted with the phenomenon that some collections are 'too big' to be sets and in his late years made a distinction between _consistent_ and _inconsistent multitudes_, the former comprising sets, but his ideas were confined to private letters to (Jourdain and) Dedekind that were not published until 1932.
[[John von Neumann]] began $NBG$ in the 1920s ([von Neumann 1925](#vN1925), [von Neumann 1928](#vN1928)), but his version was unwieldy. [[Paul Bernays]] and [[Kurt Gödel]] simplified it later. It also illustrates Gödel's theorem that any [[first-order theory]] has a [[conservative extension]] with a finite axiomatization.
## Axioms
NBG is a [[material set theory]], based on a global binary membership [[predicate]] $\in$. The objects of NBG are called [[classes]]. If a class $x$ is a member of another class $A$, i.e., $x \in A$, then $x$ is called [[set]]. A class which is not a set is called [[proper class]]. NBG can be presented as a two-sorted theory, with lowercase letters denoting sets and uppercase letters denoting classes.
1. [[axiom of extensionality|Extensionality]]: Two classes are equal if and only if the have the same members.
2. [[axiom of pairing|Pairing]], [[axiom of union|Union]], [[axiom of power sets|Power Sets]] and [[axiom of infinity|Infinity]]: regard sets and are identical to the [[ZFC]] counterparts.
3. [[axiom of foundation|Foundation]]: For each nonempty class $A$ there exists a set $x \in A$ such that $x \cap A = \varnothing$.
4. [[axiom of comprehension|Class Comprehension schema]]: For any formula $\phi$ containing no quantifiers over classes (it may contain quantifiers over sets and it may contain both class and set parameters), there is a class $A$ whose elements are precisely those sets $x$ satisfying $\phi(x)$. In particular, taking $\phi(x)$ as $x = x$ it follows that there exists the class $V$ of all sets.
5. Limitation of Size: A class $A$ is a set if and only if there is no bijection between $A$ and the class $V$ of all sets.
From the axiom of Limitation of Size, it turn out that $V$ is not a set but a proper class, avoiding [[Russell's paradox]]. Moreover, since one can prove that the [[ordinal number|ordinal numbers]] form a proper class, there is a bijection beetween $V$ and $\mathrm{Ord}$, i.e., the class of all sets can be well-ordered which implies the [[axiom of global choice]].
## Finite axiomatization
Every instance of class comprehension can be built out of a few, based on the logical connnectives. This is similar to the finite axiomatization of [[bounded separation|bounded]] $ZFC$ (or, in other language, [[ETCS]]); indeed, $NBG$ is essentially bounded [[MK]].
## References
* [[John von Neumann]], "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik 154 (1925) 219–240, doi:[10.1515/crll.1925.154.219](http://dx.doi.org/10.1515/crll.1925.154.219), [GDZ](http://gdz.sub.uni-goettingen.de/index.php?id=resolveppn&PPN=GDZPPN002169606)
{#vN1925}
* [[John von Neumann]], "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift 27 (1928) 669–752, doi:[10.1007/bf01171122](http://dx.doi.org/10.1007/bf01171122), [GDZ](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0027&DMDID=DMDLOG_0042)
{#vN1928}
* [[Gerhard Osius]], _Kategorielle Mengenlehre: Eine Charakterisierung der Kategorie der Klassen und Abbildungen_ , Math. Ann. **210** (1974) pp.171-196. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002309939))
[[!redirects NBG]]
[[!redirects NBG set theory]]
[[!redirects Neumann-Bernays-Gödel set theory]]
[[!redirects Neumann–Bernays–Gödel set theory]]
[[!redirects Neumann--Bernays--Gödel set theory]]
[[!redirects von Neumann-Bernays-Gödel set theory]]
[[!redirects von Neumann–Bernays–Gödel set theory]]
[[!redirects von Neumann--Bernays--Gödel set theory]]
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[[!redirects Neumann--Bernays--Gödel class theory]]
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[[!redirects von Neumann–Bernays–Gödel class theory]]
[[!redirects von Neumann--Bernays--Gödel class theory]]
|
von Staudt | https://ncatlab.org/nlab/source/von+Staudt | __Karl Georg Christian von Staudt__ was one of the major [[geometry of 19th century|geometers of the 19th century]].
* Jeremy Gray, _World out of nothing. A course in history of geometry in the 19th century_, Springer Undergraduate Mathematics Series (2007)
* wikipedia [Karl Georg Christian von Staudt](http://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt)
[[!redirects Karl Georg Christian von Staudt]] |
Vopěnka's principle | https://ncatlab.org/nlab/source/Vop%C4%9Bnka%27s+principle |
+-- {: .rightHandSide}
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Vopěnka\'s principle
* table of contents
{: toc}
## Idea
**Vopěnka's principle** is a [[large cardinal]] axiom which implies a good deal of simplification in the theory of [[locally presentable categories]].
It is fairly strong as large cardinal axioms go: Its consistency follows from the existence of [[huge cardinal]]s, and it implies the existence of arbitrarily large [[measurable cardinal]]s.
## Statements
### The Vopěnka principle
Vopěnka's principle has many equivalent statements. Here are a few:
+-- {: .num_theorem}
###### Theorem
The VP is equivalent to the statement:
Every [[discrete category|discrete]] [[full subcategory]] of a [[locally presentable category]] is [[small category|small]].
=--
+-- {: .num_theorem}
###### Theorem
The VP is equivalent to the statement:
For every [[proper class]] sequence $\langle M_\alpha | \alpha \in Ord\rangle$ of [[logic|first-order structures]], there is a pair of [[ordinals]] $\alpha\lt\beta$ for which $M_\alpha$ [[elementary embedding|embeds elementarily]] into $M_\beta$.
=--
+-- {: .num_theorem #ColimitsCoreflective}
###### Theorem
The VP is equivalent to the statement:
For $C$ a [[locally presentable category]], every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[colimit]]s is a [[coreflective subcategory]].
=--
This is ([AdamekRosicky, theorem 6.28](#AdamekRosicky)).
+-- {: .num_theorem}
###### Theorem
The VP is equivalent to the statement:
Every [[cofibrantly generated model category]] (in a slightly more general sense than usual) is a [[combinatorial model category]].
=--
This is in ([Rosicky](#Rosicky))
+-- {: .num_remark}
###### Remark
If one insists on the traditional stricter definition of cofibrant generated model category, then the VP still implies that these are all combinatorial. But the VP is slightly stronger than this statement.
=--
+-- {: .num_theorem}
###### Theorem
The VP is equivalent to both of the statements:
1. For every $n$, there exists a [[C(n)-extendible cardinal]].
1. For every $n$, there exist arbitrarily large [[C(n)-extendible cardinals]].
=--
This is in ([BCMR](#BagariaCasacubertaMathiasRosicky)).
### The weak Vopěnka principle
The Vopěnka principle implies the weak Vopěnka principle.
+-- {: .num_theorem}
###### Theorem
The weak VP is equivalent to the statement:
For $C$ a [[locally presentable category]], every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[limit]]s is a [[reflective subcategory]].
=--
This is [AdamekRosicky, theorem 6.22 and example 6.23](#AdamekRosicky)
### Relativized versions of Vopěnka's principle
Vopěnka's principle can be relativized to levels of the [[Lévy hierarchy]] by restricting the complexity of the (definable) classes to which it is applied. The following theorems are from ([BCMR](#BagariaCasacubertaMathiasRosicky)).
+-- {: .num_theorem}
###### Theorem
For any $n\ge 1$, the following statements are equivalent.
1. There exists a [[C(n)-extendible cardinal]].
1. Every proper class of first-order structures that is defined by a conjunction of a $\Sigma_{n+1}$ formula and a $\Pi_{n+1}$ formula contains distinct structures $M$ and $N$ and an [[elementary embedding]] $M\hookrightarrow N$.
=--
The "$n=0$ case" of this is:
+-- {: .num_theorem}
###### Theorem
For any $n\ge 1$, the following statements are equivalent.
1. There exists a [[supercompact cardinal]].
1. Every proper class of first-order structures that is defined by a $\Sigma_2$ formula contains distinct structures $M$ and $N$ and an [[elementary embedding]] $M\hookrightarrow N$.
=--
Many more refined results can be found in ([BCMR](#BagariaCasacubertaMathiasRosicky)).
## Motivation
From a category-theoretic perspective, Vopěnka's principle can be motivated by applications and consequences, but it can also be argued for somewhat *a priori*, on the basis that *large discrete categories* are rather pathological objects. We can't avoid them entirely (at least, not without restricting the rest of mathematics fairly severely), but maybe at least we can prevent them from occurring in some nice situations, such as full subcategories of locally presentable categories. See [this MO answer](http://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent/29473#29473).
## Consequences {#Consequences}
+-- {: .num_theorem #ConsequenceForBousfieldLoc}
###### Theorem
The VP implies the statement:
Let $C$ be a [[left proper model category|left proper]] [[combinatorial model category]] and $Z \in Mor(C)$ a [[class]] of [[morphism]]s. Then the [[Bousfield localization of model categories|left Bousfield localization]] $L_Z W$ exists.
=--
This is theorem 2.3 in ([RosickyTholen](#RosickyTholen))
+-- {: .num_corollary #ConsequenceForReflectiveInfCatLoc}
###### Corollary
The VP implies the statement:
Let $C$ be a [[locally presentable (∞,1)-category]] and $Z$ a class of morphisms in $C$. Then the reflective [[localization of an (∞,1)-category|localization]] of $C$ at $W$ extsts.
=--
+-- {: .proof}
###### Proof
By the facts discussed at [[locally presentable (∞,1)-category]] and [[combinatorial model category]] and [[Bousfield localization of model categories]] we have that every locally presentable $(\infty,1)$-category is presented by a combinatorial model category and that under this correspondence reflective localizations correspond to left Bousfield localizations. The claim then follows with the ([above theorem](#ConsequenceForBousfieldLoc)).
=--
## Set-theoretic notes
### First- versus second-order
As usually stated, Vopěnka's principle is not formalizable in first-order [[ZF]] set theory, because it involves a "second-order" [[quantifier|quantification]] over [[proper classes]] ("...there does not exist a large discrete subcategory..."). It can, however, be formalized in this way in a class-set theory such as [[NBG]].
On the other hand, it can be formalized in ZF as a first-order axiom schema consisting of one axiom for each class-defining formula $\phi$, stating that "$\phi$ does not define a class which is a large discrete subcategory..." We might call this axiom schema the **Vopěnka axiom scheme**. As in most situations of this sort, the first-order Vopěnka scheme is appreciably weaker than the second-order Vopěnka principle. See, for instance, [this MO question](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably) and answer.
### Vopěnka cardinals
Unlike some large cardinal axioms, Vopěnka's principle does not appear to be merely an assertion that "there exist very large cardinals" but rather an assertion about the precise size of the "universe" (the "boundary" between sets and proper classes). In other words, the universe could be "too big" for Vopěnka's principle to hold, in addition to being "too small."
(The equivalence of Vopěnka's principle with the existence of [[C(n)-extendible cardinals]] may appear to contradict this. However, the property of being $C(n)$-extendible itself "depends on the size of the whole universe" in a sense.)
More precisely, if $\kappa$ is a cardinal such that $V_\kappa$ satisfies ZFC + Vopěnka's principle, then knowing that $\lambda\gt\kappa$ does not necessarily imply that $V_\lambda$ also satifies Vopěnka's principle. By contrast, if $V_\kappa$ satisfies ZFC + "there exists a [[measurable cardinal]]" (say), then there must be a measurable cardinal less than $\kappa$, and that measurable cardinal will still exist in $V_\lambda$ for any $\lambda\gt\kappa$. On the other hand, large cardinal axioms such as "there exist arbitrarily large measurable cardinals" have the same property that Vopěnka's principle does: even if measurable cardinals are unbounded below $\kappa$, they will not be unbounded below $\lambda$ if $\lambda$ is the next greatest [[inaccessible cardinal]] after $\kappa$.
Relativizing Vopěnka's principle to cardinals also raises the same first- versus second-order issues as above. We say that a **Vopěnka cardinal** is one where Vopěnka's principle holds "in $V_\kappa$" where the quantification over classes is interpreted as quantification over all subsets of $V_\kappa$. By contrast, we could define an **almost-Vopěnka cardinal** to be one where $V_\kappa$ satisfies the first-order Vopěnka scheme. Then one can show, using the Mahlo reflection principle (see [here](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538) again), that every Vopěnka cardinal $\kappa$ is a limit of $\kappa$-many almost-Vopěnka cardinals, and in particular the smallest almost-Vopěnka cardinal cannot be Vopěnka. Thus, being Vopěnka is much stronger than being almost-Vopěnka.
### Definable counterexamples
If Vopěnka's principle fails, then there exist counterexamples to all of its equivalent statements, such as a large discrete full subcategory of a locally presentable category. If Vopěnka's principle fails but the first-order Vopěnka scheme holds, then no such counterexamples can be explicitly definable.
On the other hand, if the Vopěnka scheme also fails, then there will be explicit finite formulas one can write down which define counterexamples. However, there is no "universal" counterexample, in the following sense: if Vopěnka's principle is consistent, then for any class-defining formula $\phi$, there is a model of set theory in which Vopěnka's principle fails (and even in which the first-order Vopěnka scheme fails), but in which $\phi$ does not define a counterexample to it. See [here](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538) yet again.
## References
The relation to the theory of [[locally presentable categories]] is the contents of chapter 6 of
* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, London Mathematical Society Lecture Note Series 189
{#AdamekRosicky}
The relation to [[combinatorial model categories]] is discussed in
* [[Jiří Rosický]], _Are all cofibrantly generated model categories combinatorial?_ ([ps](http://www.math.muni.cz/~rosicky/papers/cof1.ps))
{#Rosicky}
The implication of VP on [[homotopy theory]], [[model categories]] and [[cohomology localization]] are discussed in the following articles
* [[Jiří Rosický]], [[Walter Tholen]], _Left-determined model categories and universal homotopy theories_ Transactions of the American Mathematical Society
Vol. 355, No. 9 (Sep., 2003), pp. 3611-3623 ([JSTOR](http://www.jstor.org/stable/1194855)).
{#RosickyTholen}
* [[Carles Casacuberta]], Dirk Scevenels, [[Jeff Smith]], _Implications of large-cardinal principles in homotopical localization_ Advances in Mathematics
Volume 197, Issue 1, 20 October 2005, Pages 120-139
* Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, [[Jiří Rosicky]] _Definable orthogonality classes in accessible categories are small_, [arXiv](http://arxiv.org/abs/1101.2792)
{#BagariaCasacubertaMathiasRosicky}
* Giulio Lo Monaco, _Vopěnka's principle in ∞-categories_, [arxiv:2105.04251](https://arxiv.org/abs/2105.04251)
category: foundational axiom
[[!redirects Vopěnka's principle]]
[[!redirects Vopenka's principle]]
[[!redirects Vopěnka's principle]]
[[!redirects Vopenka's principle]]
[[!redirects Vopěnka s principle]]
[[!redirects Vopenka s principle]]
[[!redirects Vopěnka principle]]
[[!redirects Vopěnka's axiom]]
[[!redirects Vopenka's axiom]]
[[!redirects Vopěnka's axiom]]
[[!redirects Vopenka's axiom]]
[[!redirects Vopěnka s axiom]]
[[!redirects Vopenka s axiom]]
[[!redirects Vopěnka's axiom scheme]]
[[!redirects Vopenka's axiom scheme]]
[[!redirects Vopěnka's axiom scheme]]
[[!redirects Vopenka's axiom scheme]]
[[!redirects Vopěnka s axiom scheme]]
[[!redirects Vopenka s axiom scheme]]
[[!redirects Vopěnka's axiom schema]]
[[!redirects Vopenka's axiom schema]]
[[!redirects Vopěnka's axiom schema]]
[[!redirects Vopenka's axiom schema]]
[[!redirects Vopěnka s axiom schema]]
[[!redirects Vopenka s axiom schema]]
[[!redirects Vopěnka cardinal]]
[[!redirects Vopěnka cardinals]]
[[!redirects Vopenka cardinal]]
[[!redirects Vopenka cardinals]]
|
vortex | https://ncatlab.org/nlab/source/vortex |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A kind of [[soliton]].
\begin{imagefromfile}
"file_name": "VorticesSchematics.jpg",
"width": 300,
"unit": "px",
"margin": {
"top": -40,
"bottom": -10,
"right": 0,
"left": 10
},
"caption": "From [Thuneberg](#Thuneberg)"
\end{imagefromfile}
## Examples
[[!include flux quantization in superconductors -- section]]
### Vortex anyons
See *[Votex anyons](braid+group+statistics#VortexAnyons)* at *[[braid group statistics]]*.
## Related concepts
* [[instanton]]
* [[monopole]]
* [[vortex string]]
* [[Skyrmion]]
* [[domain wall]]
* [[cosmic string]]
## References
### General
Reviews and surveys include
* [[David Tong]], _TASI Lectures on Solitons_ ([web](http://www.damtp.cam.ac.uk/user/tong/tasi.html)), part 3: _Vortices_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/vortex.pdf))
On water vortices forming [[knots]]:
* Dustin Kleckner and William T. M. Irvine, *Creation and dynamics of knotted vortices*, Nature Physics **9** (2013) 253–258 [[doi:10.1038/nphys2560](https://doi.org/10.1038/nphys2560)]
On quantum vortices as in [[superconductors]] or [[Bose-Einstein condensates]]/[[superfluids]]:
* Rufus Phillips, *[Quantum Vortices](https://www.physics.umd.edu/courses/Phys726/The_Quantum_Vortex.htm)*.
* {#Thuneberg} Erkki Thuneberg, *[Superfluidity and Quantized Vortices](http://ltl.tkk.fi/research/theory/vortex.html)*
* K. R. Sreenivasan, *Vortices, particles and superfluidity* ([pdf](https://higherlogicdownload.s3.amazonaws.com/APS/22d184d0-d6d2-4493-97da-847d341d3d82/UploadedImages/Documents/Sreenivasan_DFD07.pdf), [[SreenivasanVortices.pdf:file]])
For more see at _[[vortex string]]_.
A historical precursor with an abandoned but influential speculation is
* [[Lord Kelvin]], _[[On Vortex Atoms]]_
[[!include defect anyons -- references]]
[[!redirects vortices]] |
vortex string | https://ncatlab.org/nlab/source/vortex+string |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By a _vortex string_ one typically means a [[string]]-shaped [[soliton]] threading through a [[vortex]]-solution in a 4-dimensional [[gauge theory]].
The original example are the magnetic flux tubes through the [[Abrikosov vortices]] seen in [[type II superconductors]] at critical values of the [[magnetic field]]. The idea to think of these as [[string]]-like dynamical objects in themselves, much as in [[string theory]], is due to [Nielsen-Olesen 73](#NielsenOlesen73), hence one also speaks of _Nielsen-Oleson vortices_ or _Nielsen-Olesen strings_.
More generally, such string-shaped vortices are considered in [[Yang-Mills theories]] with [[non-abelian group|non-abelian]] [[gauge group]] (see e.g. [Tong 09](#Tong09)).
## Details
[[!include flux quantization in superconductors -- section]]
## Related cocepts
* [[Abrikosov vortex]]
* [[cosmic string]]
* [[defect brane]]
## References
### In electromagnetism
Vortex strings were originally considered in the abelian gauge theory of [[electromagnetism]] as [[Abrikosov vortices]] in [[type II superconductors]].
* {#Abrikosov57a} [[Alexei Abrikosov]], _On the Magnetic properties of superconductors of the second group_, Sov. Phys. JETP 5 (1957) 1174-1182, Zh. Eksp. Teor. Fiz. 32 (1957) 1442-1452 ([spire:9138](https://inspirehep.net/literature/9138))
The suggestion that these vortex flux tube could be thought of as dynamical [[strings]] much as in [[string theory]]:
* {#NielsenOlesen73} [[Holger Bech Nielsen]], [[Poul Olesen]], *Vortex-line models for dual strings*, Nuclear Physics B **61** (1973) 45-61 \[<a href="https://doi.org/10.1016/0550-3213(73)90350-7">doi:10.1016/0550-3213(73)90350-7</a>\]
* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]
* {#Polyakov08} [[Alexander M. Polyakov]], *From Quarks to Strings* [[arXiv:0812.0183](https://arxiv.org/abs/0812.0183)]
published as: *Quarks, strings and beyond*, section 44 in: [[Paolo Di Vecchia]] et al. (ed.), *The Birth of String Theory*, Cambridge University Press (2012) 544-551 [[doi:10.1017/CBO9780511977725.048](https://doi.org/10.1017/CBO9780511977725.048)]
> "I remember that in the late sixties to early seventies Tolya Larkin and I discussed (many times) whether Abrikosov’s vortices could be viewed as elementary particles. Nothing concrete came out of this at that time, but it helped me with my later work. With some imagination we could have related the vortex lines with strings but we missed it."
See also:
* Wikipedia, _[Nielsen-Olesen vortex](https://en.wikipedia.org/wiki/Nielsen%E2%80%93Olesen_vortex)_
* Wikipedia, *[Quantum vortex](https://en.wikipedia.org/wiki/Quantum_vortex)*
### In Yang-Mills theory
A more general picture of vortex strings in [[Yang-Mills theory]] with [[non-abelian group|nonabelian]] [[gauge group]] (in fact in [[D=4 N=2 super Yang-Mills theory]]):
* [[Amihay Hanany]], [[David Tong]], _Vortices, Instantons and Branes_, JHEP 0307 (2003) 037 ([arXiv:hep-th/0306150](https://arxiv.org/abs/hep-th/0306150))
* [[Amihay Hanany]], [[David Tong]], _Vortex Strings and Four-Dimensional Gauge Dynamics_, JHEP 0404 (2004) 066 ([arXiv:hep-th/0403158](https://arxiv.org/abs/hep-th/0403158))
reviewed in:
* {#Tong09} [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060))
* [[David Tong]], _Vortices, Strings, and Vortex Strings_ ([[TongVortexStrings.pdf:file]])
As [[probe brane|probe]] [[D1-branes]] of [[D1-D5 brane bound states]] in a context of [[AdS-QCD duality]]:
* [[Joseph Polchinski]], [[Matthew Strassler]], p. 34 of: _The String Dual of a Confining Four-Dimensional Gauge Theory_ ([arXiv:hep-th/0003136](https://arxiv.org/abs/hep-th/0003136))
* Roberto Auzzi, S. Prem Kumar, Section 5.2 in: _Non-Abelian k-Vortex Dynamics in $\mathcal{N} = 1^\ast$ theory and its Gravity Dual_, JHEP 0812:077, 2008 ([arXiv:0810.3201](https://arxiv.org/abs/0810.3201))
On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:
* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)
* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]
* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]
* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]
[[!redirects vortex strings]]
[[!redirects Abrikosov vortex]]
[[!redirects Abrikosov vortices]]
[[!redirects Abrikosov vortex string]]
[[!redirects Abrikosov vortex strings]]
[[!redirects Nielsen-Olesen vortex]]
[[!redirects Nielsen-Olesen vortices]]
[[!redirects Nielsen-Olesen string]]
[[!redirects Nielsen-Olesen strings]]
[[!redirects Nielsen-Olesen vortex string]]
[[!redirects Nielsen-Olesen vortex strings]]
|
vpatryshev > history | https://ncatlab.org/nlab/source/vpatryshev+%3E+history | < [[vpatryshev]] |
vukovinski | https://ncatlab.org/nlab/source/vukovinski | Spirited Systems Redesigner |
Vyacheslav Soroka | https://ncatlab.org/nlab/source/Vyacheslav+Soroka |
* [memorial page](http://ivv5hpp.uni-muenster.de/u/douplii/soroka)
* [[Steven Duplij]], _Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn't it?_, East Eur. J. Phys., v3, p. 81-82 (2019) ([arXiv:1910.03259](https://arxiv.org/abs/1910.03259))
## Selected writings
Introducing [[supergravity]] ([[D=4 supergravity]]):
* [[Dmitry Volkov]], [[Vyacheslav Soroka]], _Higgs effect for Goldstone particles with spin 1/2_, ZhETF Pis. Red. (JETP Letters, AIP translation), 18, n.8 (1973) 529 ([pdf](https://www.jetpletters.ac.ru/ps/1568/article_24038.shtml))
On [[D=4 supergravity]] formulated in [[superspace]]:
* V. Akulov, [[Dmitry Volkov]] and [[Vyacheslav Soroka]], _Generally covariant theories of gauge fields on superspace_, Theor. Math. Phys. 31 (1977) 285 ([doi:10.1007/BF01041233](https://doi.org/10.1007/BF01041233))
On the history of [[supergravity]]:
* [[Vyacheslav Soroka]], _The Sources of Supergravity_ in [The Supersymmetric World. The Beginnings of the Theory, G. Kane and M. Shifman (Eds.), World Scientific, 2000](http://www.worldscientific.com/doi/10.1142/9789812385505_0011) or [arXiv:hep-th/0203171](https://arxiv.org/abs/hep-th/0203171)
category: people |
Véronique Bernard | https://ncatlab.org/nlab/source/V%C3%A9ronique+Bernard |
## Selected writings
On [[chiral perturbation theory]]:
* [[Véronique Bernard]], [[Ulf-G. Meissner]], _Chiral perturbation theory_, Ann. Rev. Nucl. Part. Sci.57:33-60, 2007 ([arXiv:hep-ph/0611231](https://arxiv.org/abs/hep-ph/0611231))
On [[baryon chiral perturbation theory]]:
* [[Véronique Bernard]], _Chiral Perturbation Theory and Baryon Properties_, Prog. Part. Nucl. Phys. 60:82-160, 2008 ([arXiv:0706.0312](https://arxiv.org/abs/0706.0312))
category: people |
Věra Trnková | https://ncatlab.org/nlab/source/V%C4%9Bra+Trnkov%C3%A1 |
Věra Trnková (1934 - 2018) was a pure [[category theory|category theorist]]. She was a student of [[Eduard Čech]]. The list of her students includes [[Jiří Adámek]], Jiří Velebil, Libor Barto, Václav Koubek.
* [Věra Trnková's homepage (archived)](https://web.archive.org/web/20180325022120/http://www.karlin.mff.cuni.cz/~trnkova/)
* [short obituary (in Czech)](https://www.mff.cuni.cz/verejnost/konalo-se/2018-05-trnkova/)
* J. Adámek, M. Katětov: Věra Trnková's unbelievable 60, in Mathematica Bohemica [dml.cz](https://dml.cz/handle/10338.dmlcz/126082), 1994.
* [Věra Trnková's Wikipedia entry](https://en.wikipedia.org/wiki/V%C4%9Bra_Trnkov%C3%A1)
## related entries
* [[category theory]]
category: people
[[!redirects Vera Trnkova]] |
W algebra | https://ncatlab.org/nlab/source/W+algebra |
## Idea
A __W algebra__ is a higher spin extension of the [[Virasoro algebra]]. It is an extended symmetry algebra
in [[conformal field theory]].
## References
An early survey is
* [[Peter Bouwknegt]], Kareljan Schoutens, _W symmetry in conformal field theory_, Phys. Rep. __223__:4 (1993) 183--276 <a href="https://doi.org/10.1016/0370-1573(93)90111-P">doi</a>
* [[Peter Bouwknegt]], Jim McCarthy, [[Krzysztof Pilch]], _The $W_3$ algebra: modules, semi-infinite cohomology, and BV algebras_, Lec. Notes in Phys. __42__, Springer 1996 ([doi:10.1007/978-3-540-68719-1](https://link.springer.com/book/10.1007/978-3-540-68719-1))
* Wikipedia, _[W-algebra](https://en.wikipedia.org/wiki/W-algebra#CITEREFBouwknegt1993)_
Relation to [[Jordan algebra]] is discussed in
* L. J. Romans, _Realisations of classical and quantum $W_3$ symmetry_, Nuclear Physics B __352__:3 (1991) 829--848 <a href="https://doi.org/10.1016/0550-3213(91)90108-A">doi</a>
Relation to [[L-infinity algebras]] is discussed in
* {#BlumenhagenFuchsTraube17} [[Ralph Blumenhagen]], Michael Fuchs, Matthias Traube, _$\mathcal{W}$-Algebras are $L_\infty$-algebras_ ([arXiv:1705.00736](https://arxiv.org/abs/1705.00736))
category: physics, algebra
[[!redirects W algebra]]
[[!redirects W algebras]]
[[!redirects W-algebra]]
[[!redirects W-algebras]]
|
W*-category | https://ncatlab.org/nlab/source/W%2A-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of $W^\ast$-categories is the special case of that of [[C-star category|$C^\ast$-categories]] like [[von Neumann algebras]] (aka $W^\ast$-algebras) are a special case of [[C-star algebra|$C^\ast$-algebras]]. Hence a more systematic name for $W^\ast$-categories would be *$W^\ast$-algebroids* or *von Neumann algebroids*.
## Definition
A __W\*-category__ is a [[C*-category]] $C$
such that for any objects $A,B\in C$, the [[hom-object]] $Hom(A,B)$
admits a [[predual]] as a [[Banach space]].
That is, there is a [[Banach space]]
$Hom(A,B)_*$ such that $(Hom(A,B)_*)^*$ is isomorphic to $Hom(A,B)$
in the category of [[Banach spaces]] and [[contractive maps]] (alias _short maps_).
## W\*-functors
A __W\*-functor__ is a [[functor]] $F\colon C\to D$
such that $F(f^*)=F(f)^*$ for any [[morphism]] $f$ in $C$
and the map of [[Banach spaces]] $Hom(A,B)\to Hom(F(A),F(B))$
admits a [[predual]] for any objects $A$ and $B$ in $C$.
## Bounded natural transformation
The good notion of natural transformations between W\*-categories
is given by __bounded natural transformations__:
a [[natural transformation]] $t\colon F\to G$ between W\*-functors
is bounded if the norm of $t_X\colon F(X)\to G(X)$
is bounded with respect to the object $X$ of $C$.
## The bicategory of W\*-categories
W\*-categories, W\*-functors, and bounded natural transformations
form a [[bicategory]].
This [[bicategory]] is a good setting to work with objects
like [[Hilbert spaces]], [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]], etc.
In particular, in this [[bicategory]],
the [[category of Hilbert spaces]] has infinite [[direct sums]] (generalizing the definition of a [[biproduct]] to infinite families of objects),
unlike in the usual [[bicategory]]
of [[categories]], [[functors]], and [[natural transformations]],
where it only has [[finite limits]] and [[finite colimits]].
The same is true for [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]].
## Related concepts
* [[von Neumann algebra]]
* [[C-star category|$C^\ast$-category]]
## References
* [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], _W\*-categories_, Pacific Journal of Mathematics 120:1 (1985), 79–109 [[doi:10.2140/pjm.1985.120.79](https://doi.org/10.2140/pjm.1985.120.79)]
[[!redirects W*-categories]]
[[!redirects W-star category]]
[[!redirects W-star+categories]]
|
W*-representation | https://ncatlab.org/nlab/source/W%2A-representation | ## Definition
A __W\*-representation__, or simply a __representation__
of a [[von Neumann algebra]] $A$ on a [[Hilbert space]] $H$
is a morphism of [[von Neumann algebras]] $A\to B(H)$,
where $B(H)$ is the [[von Neumann algebra]]
of [[bounded operators]] on the [[Hilbert space]] $H$.
## Warning
If the map $A\to B(H)$ is injective,
the resulting notion coincides with that of a “concrete von Neumann algebra”,
as opposed to an (abstract) [[von Neumann algebra]].
An isomorphism of representations is sometimes referred to as
a “spatial isomorphism of concrete von Neumann algebras”.
Such terminology is confusing, but is present in some sources, especially older ones.
## Properties
The category of [[W*-representations]] of $A$
is equivalent to the category of [[Hilbert W*-modules]] over $A$.
See the article [[Hilbert W*-module]] for more information.
## Related concepts
* [[Hilbert W*-module]]
* [[von Neumann algebra]]
* [[duality between geometry and algebra]]
[[!redirects W*-representations]]
[[!redirects representation of a von Neumann algebra]]
[[!redirects representations of von Neumann algebras]] |
W-boson | https://ncatlab.org/nlab/source/W-boson |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The W- and Z-[[bosons]] are the quanta of the [[electroweak field]],[[vector boson]] in the [[standard model of particle physics]].
A hypothetical high-[[mass]] cousin of the Z-boson is the [[Z'-boson]].
## Related concepts
* [[charged current]]
* [[electroweak symmetry breaking]], [[Higgs effect]]
* [[leptonic decay]]
## References
### General
(...)
### Mass discrepancy
{#ReferencesMassDiscrepancy}
* {#CDF22} CDF Collaboration, *High-precision measurement of the W boson mass with the CDF II detector*, Science **376** (2022) 170-176 [[doi:10.1126/science.abk1781](https://www.science.org/doi/10.1126/science.abk1781)]
The following article claims that this mass discrepancy is compatible with (in fact predicted by, see footnote 3) the observed [[flavour anomalies]]:
* {#ACMM22} Marcel Algueró, [[Andreas Crivellin]], Claudio Andrea Manzari, Joaquim Matias, *Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$ Mass*, Physical Review D (2022) [[arXiv:2201.08170](https://arxiv.org/abs/2201.08170)]
>
[[!redirects W-bosons]]
[[!redirects Z-boson]]
[[!redirects Z-bosons]]
[[!redirects B-boson]]
[[!redirects B-bosons]]
|
w-contractible ring | https://ncatlab.org/nlab/source/w-contractible+ring |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #wContractible}
###### Definition
A [[commutative ring]] $R$ is **w-contractible** if every [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$ has a [[section]].
=--
([Bhatt-Scholze 13, def. 2.4.1](#BhattScholze13))
## Properties
+-- {: .num_prop}
###### Proposition
For every [[commutative ring]] $R$, there is a w-contractible $A$, def. \ref{wContractible},
equipped with a [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$.
=--
([Bhatt-Scholze 13, lemma 2.4.9](#BhattScholze13))
+-- {: .num_prop}
###### Proposition
For $R$ w-contractible, the [[profinite set]] $\pi_0(Spec R)$
is an [[extremally disconnected profinite set]].
=--
part of ([Bhatt-Scholze 13, theorem 1.8](#BhattScholze13))
## Related concepts
* [[pro-étale site]]
## References
* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
{#BhattScholze13}
[[!redirects w-contractible rings]]
|
W-star category > history | https://ncatlab.org/nlab/source/W-star+category+%3E+history | see at *[[W-star category]]* |
W-suspension | https://ncatlab.org/nlab/source/W-suspension |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
\tableofcontents
## Idea
A W-suspension is similar to a [[W-type]], except instead of being inductively generated by [[term]] constructors and [[function]] constructors, W-suspensions are inductively generated by term constructors and [[identification]] constructors.
## Definition
There are multiple possible definitions of a (directed, pseudo-) [[graph]] in dependent type theory:
* a graph consists of a type $A$ and a binary type family $R(x, y)$ indexed by $x:A$ and $y:A$. $A$ is the type of vertices of a graph and each $R(x, y)$ is the type of edges of a graph.
* a graph consists of types $A$ and $B$, a type family $R(z)$ indexed by $z:B$, and functions $s:B \to A$ and $t:B \to A$. $A$ is the type of vertices of a graph, $B$ is the type of endpoint configurations of edges in the graph, and $R(x)$ is the type of edges in the graph given an endpoint configuration $x:B$. Given elements $x:A$ and $y:A$, the type of edges between $x:A$ and $y:A$ in the graph is the type
$$\sum_{z:B} R(z) \times (s(z) =_A x) \times (t(z) =_A y)$$
* a graph consists of a type $A$ and a type $R$ and two functions $s:R \to A$ and $t:R \to A$. $A$ is the type of vertices in $A$, and the type of edges the type of edges between $x:A$ and $y:A$ in the graph is the type
$$\sum_{r:R} (s(r) =_A x) \times (t(r) =_A y)$$
These definitions of graph result in different possible definitions of a W-suspension, whose points are generated by the vertices of the graph and whose paths are generated by the edges of the graph.
### W-suspensions of binary type families
The first notion of graph leads to the following higher inductive type:
Given a graph $(A, R)$, one could construct the **W-suspension** $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ which is generated by the graph in the following sense: there are point constructors
$$\mathrm{points}:A \to \underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$$
and for each $x:A$ and $y:A$, path constructors
$$\mathrm{paths}(x, y):R(x, y) \to \left(\mathrm{points}(x) =_{\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)} \mathrm{points}(y)\right)$$
The recursion principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given a type $C$ together with
* a function $c_\mathrm{points}:A \to C$
* a dependent function $c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} R(x, y) \to (c_\mathrm{points}(x) =_C c_\mathrm{points}(y))$
there exists a function $c:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \to C$ such that
* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{ap}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)$
Similarly, the induction principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given a type family $C(z)$ indexed by $z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)$ together with
* a dependent function $c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))$
* a dependent function $c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} \prod_{r:R(x, y)} c_\mathrm{points}(x) =_C^{\mathrm{paths}(x,y,r)} c_\mathrm{points}(y)$
there exists a dependent function $c:\prod_{z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)} C(z)$ such that
* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{apd}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)$
The large recursion principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given
* a type family $x:A \vdash C(x)$
* a dependent function $c_\mathrm{equiv}:\prod_{x:A} \prod_{y:A} R(x, y) \to (C(x) \simeq C(y))$
there exists a type family $z:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \vdash D(z)$ such that
* for all $x:A$, $D(c(x)) \equiv C(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{tr}_C(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{equiv}(x, y, r):C(x) \simeq C(y)$
### Sojakova W-suspensions
The second notion of graph leads to the following higher inducive type defined in [Sojakova15](#Sojakova15):
Given types $A$ and $B$, a type family $R(x)$ indexed by $x:B$, and functions $s:B \to A$ and $t:B \to A$, one could construct the **W-suspension** $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ which is generated by the graph in the following sense:
there are point constructors
$$\mathrm{points}:A \to \underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$$
and for each endpoint configuration $x:B$, path constructors
$$\mathrm{paths}(x):R(x) \to \left(\mathrm{points}(s(x)) =_{\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} \mathrm{points}(t(x))\right)$$
The recursion principle of $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ says that given a type $C$ together with
* a function $c_\mathrm{points}:A \to C$
* a dependent function $c_\mathrm{paths}:\prod_{x:B} R(x) \to (c_\mathrm{points}(s(x)) =_C c_\mathrm{points}(t(x)))$
there exists a function $c:\left(\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)\right) \to C$ such that
* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:B$, and $r:R(x)$, $\mathrm{ap}_c(\mathrm{paths}(s,r)) \equiv c_\mathrm{paths}(s, r)$
Similarly, the induction principle of $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ says that given a type family $C(z)$ indexed by $z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ together with
* a dependent function $c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))$
* a dependent function $c_\mathrm{paths}:\prod_{x:B} \prod_{r:R(x)} c_\mathrm{points}(s(x)) =_C^{\mathrm{paths}(x,r)} c_\mathrm{points}(t(x))$
there exists a dependent function $c:\prod_{z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} C(z)$ such that
* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:B$, and $r:R(x)$, $\mathrm{apd}_c(\mathrm{paths}(x,r)) \equiv c_\mathrm{paths}(x, r)$
### W-suspensions of parallel functions
For the final notion of graph, the resulting definition of W-suspension is just the [[coequalizer type]] of $s:R \to A$ and $t:R \to A$.
$$\mathrm{Wsus}_{R,A}(s, t) \coloneqq \mathrm{coeq}_{R,A}(s, t)$$
### Relation between the definitions
W-suspensions in the first sense could be constructed as the coequalizer of the left and middle dependent pair projections $\pi_{\sum}^A$ and $\pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)}$ of the [[dependent pair type]] $\sum_{x:A} \sum_{y:A} R(x, y)$, both of which have [[domain]] $\sum_{x:A} \sum_{y:A} R(x, y)$ and [[codomain]] $A$:
$$\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) \coloneqq \mathrm{coeq}_{\sum_{x:A} \sum_{y:A} R(x, y),A}(\pi_{\sum}^A, \pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)})$$
Alternatively, given two [[parallel morphisms|parallel]] functions $s:B \to A$ and $t:B \to A$, the coequalizer of $s$ and $t$ could be defined as the W-suspension (in the first sense) generated by the graph defined by the family of dependent sum types $\sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$
$$\underset{x:A,y:A}{\mathrm{Wsus}} \sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$$
## Related concepts
* [[coequalizer type]]
* [[W-type]]
## References
* {#Sojakova15} [[Kristina Sojakova]], *Higher Inductive Types as Homotopy-Initial Algebras*, ACM SIGPLAN Notices **50** 1 (2015) 31–42 [[arXiv:1402.0761](http://arxiv.org/abs/1402.0761), [doi:10.1145/2775051.2676983](https://doi.org/10.1145/2775051.2676983)]
[[!redirects W-suspension]]
[[!redirects W-suspensions]] |
W-topical dagger 2-poset | https://ncatlab.org/nlab/source/W-topical+dagger+2-poset | [[!redirects W-topical dagger 2-posets]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
## Contents ##
* table of contents
{:toc}
## Idea ##
A W-topical dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[W-topos]].
## Definition ##
A **W-topical dagger 2-poset** $C$ is an [[elementarily topical dagger 2-poset]] with an object $\mathbb{N} \in Ob(C)$ and [[map in a dagger 2-poset|maps]] $0 \in Map_C(\mathcal{P}(0),\mathcal{N})$ and $s \in Map_C(\mathbb{N},\mathbb{N})$, such that for every object $A$ with maps $0_A \in Map_C(\mathcal{P}(0),A)$ and $s_A \in Map_C(A,A)$, there is a map $f \in Map_C(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.
## Examples ##
The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a W-topical dagger 2-poset.
## See also ##
* [[dagger 2-poset]]
* [[elementarily topical dagger 2-poset]] |
W-type | https://ncatlab.org/nlab/source/W-type |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--
# W-types
* table of contents
{: toc}
## Idea
{#Idea}
In [[type theory]], by a *$\mathcal{W}$-type* [[Martin-Löf (1982), pp. 171](#Martin-Löf82), [(1984)](#Martin-Löf84), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49)] one means a [[type]] which is defined [[induction|inductively]] in a *[[well-founded relation|$\mathcal{W}$ell-founded]]* way based on a type $C$ of "*c*onstructors" and a type of $A$ of "*a*rities". As such, $\mathcal{W}$-types are special kinds of [[inductive types]] (see [below](#WTypesInTypeTheory)).
The same terminology "$\mathcal{W}$-type" is used [[Moerdijk & Palmgren (2000)](MoerdijkPalmgren00)] for [[objects]] in (suitable [[pretopos|pre]]-)[[toposes]] which provide [[categorical semantics]] for the type-theoretic notion (see [further below](#WTypesInCategories)).
Concretely, the [[terms]]/[[elements]] of a $\mathcal{W}$-type may be thought of as "rooted well-founded [[trees]]" with a certain branching type; different $\mathcal{W}$-types are distinguished by their branching signatures: In [[categorical semantics]] in [[Sets]], a branching signature is represented by a [[family of sets]] $\{A_c\}_{c \in C}$ such that
* each *node* of the tree is labeled with an element $c \colon C$ -- referring to a *constructor*;
* if a node is labeled by $c$ then it has exactly ${|A_c|}$ outgoing edges, each labeled by some $a \colon A_c$ -- the *arity* of the constructor $c$.
If one discards the requirement that the trees be well-founded, then the notion of $\mathcal{W}$-type becomes that of a [[coinductive type]] called an *[[M-type]]* (presumably since "M" is like a "W" upside down).
In practice, $\mathcal{W}$-types are used to:
1. provide a [[constructive mathematics|constructive]] counterpart of the [[classical logic|classical]] notion of a [[well-ordering]]
1. uniformly define a variety of well-behaved [[inductive types]].
More complex [[inductive types]], with multiple constructors that are assumed only to be strictly [[positive type|positive]], can be reduced to $\mathcal{W}$-types, at least in the presence of other structure such as [[sum types]] and [[function extensionality]]; see for instance [Abbott, Altenkirch & Ghani (2004)](#AbbottAltenkirchGhani04). This can even be extended to [[inductive families]].
{#InMost} In most [[set theory|set theoretic]] [[mathematical foundations]], $\mathcal{W}$-types can be proven to exist, but in [[predicative mathematics]] and in [[type theory]], where this is not the case, they are often introduced [[axiom|axiomatically]] (as usual for [[inductive types]] more generally).
## Definition
{#Definition}
There are two slightly different formulations of W-types:
1. [$\mathcal{W}$-types in type theory](#WTypesInTypeTheory)
1. [$\mathcal{W}$-types in categories](#WTypesInCategories)
### $\mathcal{W}$-types in type theory
{#WTypesInTypeTheory}
\begin{definition}\label{WTypeInferenceRules}
**([[inference rules]] for $\mathcal{W}$-types)**
\linebreak
(1) **[[type formation rule]]:**
$$
\frac
{
C \,\colon\, Type
\;;
\;\;\;
c \,\colon\, C
\;\;\vdash\;\;
A(c) \,\colon\,Type
\mathclap{\phantom{\vert_{\vert}}}
}
{
\mathclap{\phantom{\vert^{\vert}}}
\underset{c \colon C}{\mathcal{W}}\, A(c)
\,\colon\,
Type
}
$$
(2) **[[term introduction rule]]:**
$$
\frac{
\vdash\; root \,\colon\, C
\;;
\;\;
subtr
\,\colon\,
A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
}{
\mathclap{\phantom{\vert^{\vert}}}
tree\big(root ,\, subtr\big)
\,\colon\,
\underset{c \colon C}{\mathcal{W}}\, A(c)
}
$$
(3) **[[term elimination rule]]:**
$$
\frac{
\begin{array}{l}
t \,\colon\, \underset{c \colon C}{\mathcal{W}}\, A(c)
\;\vdash\;
D(t) \,\colon\, Type
\;;
\\
root \,\colon\, C
\,,
\;
subt
\,\colon\,
A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
\,,
\;
subt_D
\,\colon\,
\underset{a \colon A(root)}{\prod}
D\big(subt(a)\big)
\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\vdash\;\;
tree_D\big(root,\,subt ,\, subt_D\big)
\,\colon\,
D\big(tree(c,\, subt)\big)
\mathclap{\phantom{\vert_{\vert}}}
\end{array}
}{
\mathclap{\phantom{\vert^{\vert}}}
t
\,\colon\,
\underset{c \colon C}{\mathcal{W}}\, A(c)
\;\vdash\;
wrec_{(D,tree_D)}(t) \,\colon\, D(t)
}
$$
(4) **[[computation rule]]:**
$$
\frac{
\begin{array}{l}
t
\,\colon\,
\underset{c \colon C}{\mathcal{W}}\, A(c)
\;\vdash\;
D(t) \,\colon\, Type
\;;
\\
root \,\colon\, C
\,,\;
subt
\,\colon\,
A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
\,,
\;
subt_D
\,\colon\,
\underset{a\colon A(c)}{\Pi} D\big(subt(a)\big)
\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\vdash\;\;
tree_D\big(root,\,subt,\,subt_D\big)
\,\colon\,
D\big(tree(root,\, subt)\big)
\mathclap{\phantom{\vert_{\vert}}}
\end{array}
}{
\begin{array}{l}
\mathclap{\phantom{\vert^{\vert}}}
root \,\colon\, C
\,,
\;\;
subtr
\,\colon\,
A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
\\
\;\;\;\;\;\;\;\;\;\;\;
\;\;\vdash\;\;
wrec_{(D,tree_D)}\big( tree(root, \, subtr) \big)
\;=\;
tree_D
\Big(
root
,\,
subt
,\,
\lambda a . wrec_{(D,tree_D)}\big(subtr(a)\big)
\Big)
\end{array}
}
$$
\end{definition}
(due to [Martin-Löf (1984)](#Martin-Löf84), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49); streamlined review e.g. in [Awodey, Gambino & Sojakova (2012, §2)](#AwodeyGambinoSojakova12))
\begin{remark}
If the type theory is *[[extensional type theory|extensional]]*, i.e. [[identity type|identities]] have unique proofs and (dependent) [[function types]] are [[function extensionality|extensional]] ($f=g$ if and only if $f(x)=g(x)$ for all $x$), then this is more or less equivalent to the [[1-category]]-[[category theory|theoretic]] definition [below](#WTypesInCategories). The type theories that are the [[internal logic]] of familiar kinds of categories are all extensional in this sense.
The main distinction from the naive categorical theory below is that the map $f$ (eq:TheDisplayMap) must be assumed to be a [[display map]], i.e. to exhibit $A$ as a [[dependent type]] over $B$, in order that the dependent product $\Pi_f$ be defined.
In the case of dependent polynomial functors, it seems that $q$ must also be a display map, in order to define $\Sigma_q$. However, using [[adjunction|adjointness]], one can still define the W-type even if $q$ is not a display map. This more general version is what in type theory is called an *indexed W-type*; if $q$ is a display map then one sometimes refers to *non-uniform parameters* instead of *indices*. (By contrast, *uniform parameters* are the kind discussed above where $g f = h$, so that the entire construction takes place in a single slice category This is an instance of the [[red herring principle]], since non-uniform parameters are not really parameters at all, but a special kind of indices.) For example, [[identity type]]s are indexed W-types but not parametrized ones (even non-uniformly); see [this blog post](http://homotopytypetheory.org/2011/04/18/whats-special-about-identity-types/).)
\end{remark}
\begin{remark} In [[intensional type theory]], a $\mathcal{W}$-type is only an initial algebra with respect to [[propositional equality]], not [[definitional equality]]. In particular, the constructors are injective only propositionally, not definitionally. This applies already for the [[natural numbers type]] (Exp. \ref{NaturalNumbersAsWType}).
\end{remark}
### $\mathcal{W}$-types in categories
{#WTypesInCategories}
#### Plain version
We discuss the [[categorical semantics]] of $\mathcal{W}$-types, due to [Moerdijk & Palmgren (2000)](#MoerdijkPalmgren00).
Here one describes a $\mathcal{W}$-type as an [[initial algebra|initial]] [[algebra for an endofunctor]] on an ambient category $\mathcal{C}$. The family $\{A_c\}_{c\in C}$ can be thought of as a [[morphism]]
\[
\label{TheDisplayMap}
\array{
A
\\
\big\downarrow\mathrlap{^{f}}
\\
C
}
\]
in some [[category]] $\mathcal{C}$ (such that the [[fiber]] over $c\in C$ is $A_c$), and the [[endofunctor]] in question is the [[composition|composite]]
$$
\mathcal{C}
\overset{A^*}{\longrightarrow}
\mathcal{C}_{/A}
\overset{\Pi_f}{\longrightarrow}
\mathcal{C}_{/C}
\overset{\Sigma_C}{\longrightarrow}
\mathcal{C}
\,,
$$
where
* $A^*$ denotes [[context extension]], hence the [[pullback]] functor (a.k.a. $A\times -$);
* $\Pi_f$ denotes the interpretation of the [[dependent product]], i.e. the right [[base change]] along $f$;
* $\Sigma_C$ the denotes the interpretation of the [[dependent sum]], i.e. the left [[base change]] given by the [[forgetful functor]] from $\mathcal{C}_{/C}$ to $\mathcal{C}$.
Equivalently, it is the composite
$$
\mathcal{C}
\overset{C^*}{\longrightarrow}
\mathcal{C}_{/C}
\overset{\bigcirc_f}{\longrightarrow}
\mathcal{C}_{/C}
\overset{\Sigma_C}{\longrightarrow}
\mathcal{C}
\,,
$$
where $\bigcirc_f$ is the [[reader monad]] $\Pi_f \circ f^*$. In other words, the dependent product is not actually dependent.
Such a composite is called a _[[polynomial endofunctor]]_. Explicitly, it is the functor $X \mapsto \sum_{c : C} X^{A_c}$.
This definition makes sense in any [[locally cartesian closed category]], although the W-type (the initial algebra) may or may not exist in any given such category. (A non-elementary construction of them is given by the [[transfinite construction of free algebras]].)
The above definition is most useful when the category $\mathcal{C}$ is not just
[[locally cartesian closed category|locally cartesian closed]] but is a [[Π-pretopos]], since often we want to use at least [[coproducts]] in constructing $A$ and $C$. For example, a [[natural numbers object]] (Exp. \ref{NaturalNumbersAsWType}) is a $\mathcal{W}$-type specified by one of the coproduct inclusions $1\to 1+1$, and the [[list object]] $L X$ is a $\mathcal{W}$-type specified by $X\to X+1$. More generally, endofunctors that look like [[polynomials]] in the traditional sense:
$$
F(Y) = A_n \times Y^{\times n} + \dots + A_1 \times Y + A_0
$$
can be constructed as [[polynomial endofunctors]] in the above sense for any [[lextensive category]], and hence in any $\Pi$-pretopos. A $\Pi$-pretopos in which all W-types exist is called a **[[ΠW-pretopos]]**.
In a [[topos]] with a [[natural numbers object]] (NNO), W-types for any polynomial endofunctor can be constructed as certain sets of well-founded trees; thus every topos with a NNO is a [[ΠW-pretopos]]. This applies in particular in the topos [[Set]] (unless one is a [[predicative mathematics|predicativist]], in which case $Set$ is not a topos and W-types in it must be postulated explicitly).
#### Dependent $\mathcal{W}$-types
{#DependentWTypesCategoricalSemantics}
The above has a natural generalization to [[dependent type|dependent]] or *indexed* W-types ([Gambino & Hyland (2004)](#GambinoHyland04)) with a type $C$ of *indices*:
given a [[diagram]] of the form
$$
\array{
A &\stackrel{f}{\longrightarrow}& B
\\
\downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{g}}
\\
C && C
}
$$
there is the _[[dependent polynomial functor]]_
$$
\mathcal{C}_{/C} \overset{h^\ast}{\longrightarrow}
\mathcal{C}_{/A}
\overset{\Pi_f}{\longrightarrow}
\mathcal{C}_{/B}
\overset{\Sigma_g}{\longrightarrow}
\mathcal{C}_{/C}
\,,
$$
This reduces to the above for $C = \ast$ the [[terminal object]].
Notice that we do not necessarily have $g f = h$, so this is not just a [[polynomial endofunctor]] of $\mathcal{C}/_{C}$ considered as a lccc in its own right. If we *do* have $g f = h$, then $C$ is called a type of *parameters* instead of indices.
## Examples
\begin{example}\label{NaturalNumbersAsWType}
**([[natural numbers type]] as a [[W-type|$\mathcal{W}$-type]])**
\linebreak
The [[natural numbers type]] $(\mathbb{N},\, 0,\, succ)$ is equivalently the [[W-type|$\mathcal{W}$-type]] (Def. \ref{WTypeInferenceRules}) with
* $C \,\coloneqq\, \{0, succ\} \,\simeq\, \ast \sqcup \ast$;
* $A_0 \,\coloneqq\, \varnothing$ ([[empty type]]);
$A_{succ} \,\coloneqq\, \ast$ ([[unit type]])
\end{example}
[[Martin-Löf (1984)](#Martin-Löf84), [pp. 45](/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=51), [Dybjer (1997, p. 330, 333)](#Dybjer97)]
## Properties
{#Properties}
\begin{proposition}
**([Danielsson (2012)](#Danielsson12))**
\linebreak
In [[homotopy type theory]], if $C$ has [[h-level]] $n\geq -1$, then any $\mathcal{W}$-type of the form $\underset{C}{\mathcal{W}} B$ has h-level $n$ (as it should be for [[(infinity,1)-colimits|$\infty$-colimits]]).
\end{proposition}
## Related concepts
* [[M-type]]
* [[predicative topos]]
[[!include notions of type]]
## References
### General
The original definition in [[type theory]] is due to
* {#Martin-Löf82} [[Per Martin-Löf]], pp. 171 of: *Constructive Mathematics and Computer Programming*, in: *Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979)*, Studies in Logic and the Foundations of Mathematics **104** (1982) 153-175 $[$<a href="https://doi.org/10.1016/S0049-237X(09)70189-2">doi:10.1016/S0049-237X(09)70189-2</a>, [ISBN:978-0-444-85423-0](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/104/suppl/C)$]$
* {#Martin-Löf84} [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49) of: _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]
In [[homotopy type theory]]:
* [[Jasper Hugunin]], *Why Not W?*, Leibniz International Proceedings in Informatics (LIPIcs) **188** (2021) [[doi:10.4230/LIPIcs.TYPES.2020.8](https://doi.org/10.4230/LIPIcs.TYPES.2020.8), [pdf](https://jashug.github.io/papers/whynotw.pdf)]
On the [[h-level]] of W-types:
* {#Danielsson12} Nils Anders Danielsson, _Positive h-levels are closed under W_ (2012) [[web](https://homotopytypetheory.org/2012/09/21/positive-h-levels-are-closed-under-w/)]
* [[Jasper Hugunin]], _IWTypes_, <https://github.com/jashug/IWTypes>
which also computes the [[identity types]] of W-types (and more generally [[indexed W-types]]).
W-types in [[Coq]]: [wiki](https://coq.inria.fr/cocorico/WTypeInsteadOfInductiveTypes)
[[!include semantics of W-types -- references]]
[[!redirects W-types]] |