Konrad Voelkel

* [webpage](https://www.konradvoelkel.com/)

## Selected writings

On the [[h-principle]] and [[formal immersion of smooth manifolds]]:

* [[Konrad Voelkel]], Helene Sigloch, _Homotopy sheaves and h-principles_, ([pdf](https://www.konradvoelkel.com/wp-content/uploads/homotopy-sheaves.pdf))

On [[projective spaces]] over [[split quaternion|split quaternions]]:

* [[Konrad Voelkel]], _Motivic cell structures for projective spaces over split quaternions_, 2016 ([freidok:11448](https://freidok.uni-freiburg.de/data/11448), [pdf](https://freidok.uni-freiburg.de/fedora/objects/freidok:11448/datastreams/FILE1/content))

category: people

[[!redirects Konrad Völkel]]


Konrad Waldorf

* [website](http://math.konradwaldorf.de/)

## Selected writings

On [[bundle gerbes]] over [[Lie groups]]:

* [[Christoph Schweigert]], [[Konrad Waldorf]], _Gerbes and Lie Groups_, In: KH. Neeb, A. Pianzola  (eds) _Developments and Trends in Infinite-Dimensional Lie Theory_, Progress in Mathematics, vol 288. Birkhäuser Boston 2011 ([arXiv:0710.5467](https://arxiv.org/abs/0710.5467), [doi:10.1007/978-0-8176-4741-4_10](https://doi.org/10.1007/978-0-8176-4741-4_10))

Characterizing the [[image]] of the [[transgression]] operation from [[bundle gerbes]] ([[bundle gerbe with connection|with connection]]) to [[complex line bundles]] ([[connection on a vector bundle|with connection]]) on the [[free loop space]] of their base space as consisting of *[[fusion bundles]]*:

* {#Waldorf09} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps*, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 [[arXiv:0911.3212](https://arxiv.org/abs/0911.3212), [cahierstgdc:LIII](http://cahierstgdc.com/index.php/volumes/volume-liii-2012)]

* {#Waldorf10a} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection*, Asian Journal of Mathematics **20** 1 (2016) 59-116 [[arXiv:1004.0031](https://arxiv.org/abs/1004.0031), [doi:10.4310/AJM.2016.v20.n1.a4](https://dx.doi.org/10.4310/AJM.2016.v20.n1.a4)]

* {#Waldorf10b} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles*, Advances in Mathematics **231** (2012) 3445-3472 [[arXiv:1109.0480](https://arxiv.org/abs/1109.0480), [doi:10.1016/j.aim.2012.08.016](https://doi.org/10.1016/j.aim.2012.08.016)]

On [[T-folds]] via [[principal 2-bundles]] for the [[T-duality 2-group]]:

* {#NikolausWaldorf18} [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_, Commun. Math. Phys. **374**  (2020) 317-366 [[arXiv:1804.00677](https://arxiv.org/abs/1804.00677), [doi:10.1007/s00220-019-03496-3](https://doi.org/10.1007/s00220-019-03496-3)]

On a smooth open/closed [[functorial field theory]] exhibiting the [[string]]'s [[WZW term]] in a background with [[D-branes]]:

* {#BunkWaldorf21a} [[Severin Bunk]], [[Konrad Waldorf]], *Transgression of D-branes*, Adv. Theor. Math. Phys. **25** 5 (2021) 1095-1198 [[arXiv:1808.04894] (https://arxiv.org/abs/arXiv:1808.04894), [doi:10.4310/ATMP.2021.v25.n5.a1](https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1)]

* {#BunkWaldorf21b} [[Severin Bunk]], [[Konrad Waldorf]], *Smooth functorial field theories from B-fields and D-branes*, J. Homot. Rel. Struc. **16** 1 (2021) 75-153 [[doi:10.1007/s40062-020-00272-2](https://doi.org/10.1007/s40062-020-00272-2), [arXiv:1911.09990] (https://arxiv.org/abs/arXiv:1911.09990)]

On geometric [[T-duality]]:

* [[Konrad Waldorf]], *Geometric T-duality: Buscher rules in general topology* [[arXiv:2207.11799](https://arxiv.org/abs/2207.11799)]

On [[2-vector bundles]] for [[2-vector spaces]] regarded ([here](2-vector+space#ReferencesAsAlgebrasWithBimodules)) as algebras with bimodules between them:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *The insidious bicategory of algebra bundles* [[arXiv:2204.03900](https://arxiv.org/abs/2204.03900)]

On [[2-representations]] of the [[string 2-group]] on [[2-vector spaces]] and the construction of the [[stringor bundle]]:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]

Reviewed in:

* [[Konrad Waldorf]], *The stringor bundle*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#WaldorfMar2023), video:[YT](https://www.youtube.com/watch?v=wX99ZbMuyMA)]

category: people


Konrad Zuse

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

category: people


Konstadinos Sfetsos

* [webpage](http://users.uoa.gr/~ksfetsos/)

## Selected writings

On [[Dp-D(p+4) brane bound states]] as [[black holes in string theory]] and their [[black hole entropy]]:

* [[Konstadinos Sfetsos]], [[Kostas Skenderis]], _Microscopic derivation of the Bekenstein-Hawking entropy formula for non-extremal black holes_, Nucl. Phys. B517 (1998) 179-204 ([arXiv:hep-th/9711138](https://arxiv.org/abs/hep-th/9711138))

## Related $n$Lab entries

* [[nonabelian T-duality]]

category: people

[[!redirects Konstantinos Sfetsos]]


Konstadinos Siampos

* [personal page](http://users.uoa.gr/~ksiampo/)

* [InSpire page](https://inspirehep.net/authors/1049785)

## Selected writings

On [[black brane]] [[M2-M5 brane bound states]]:

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *M2-M5 blackfold funnels*, Journal of High Energy Physics **2012** 175 (2012) [[arXiv:1205.1535](https://arxiv.org/abs/1205.1535), doi:10.1007/JHEP06(2012)175]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *Entropy of the self-dual string soliton*, Journal of High Energy Physics **2012** 134 (2012) [[arXiv:1206.2935](https://arxiv.org/abs/1206.2935), doi:10.1007/JHEP07(2012)134]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *The black M2-M5 ring intersection spins* [[arXiv:1302.0854](https://arxiv.org/abs/1302.0854)]

category: people


Konstantin Eder

* [institute page](https://www.physik.nat.fau.de/department/staff/univisid/21779927_show=zeige_sprungmarken%2Ctelefon%2Cmail%2Csprungmarken%2Ccall&hide=/)

* [InSpire page](https://inspirehep.net/authors/1648703)

## Selected writings

On [[supergravity]] as [[super Cartan geometry]]:

* [[Konstantin Eder]], _Super Cartan geometry and the super Ashtekar connection_ ([arXiv:2010.09630](https://arxiv.org/abs/2010.09630))

* [[Konstantin Eder]], _Super fiber bundles, connection forms and parallel transport_ ([arXiv:2101.00924](https://arxiv.org/abs/2101.00924))

On [[D=3 supergravity]] via [[super Cartan geometry]] for [[super L-infinity algebras]]:

* [[Konstantin Eder]], [[John Huerta]], [[Simone Noja]], §16 in: *Poincaré Duality for Supermanifolds, Higher Cartan Geometry and Geometric Supergravity* [[arXiv:2312.05224](https://arxiv.org/abs/2312.05224)]

category: people


Konstantin Novoselov

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

## Selected writings

Announcing the isolation of [[graphene]]:

* [[Konstantin Novoselov]], [[Andre Geim]], S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, _Electric field effect in atomically thin carbon films_, Science __306__, no. 5696, pp. 666-669 (2004) ([doi:10.1126/science.1102896](http://dx.doi.org/10.1126/science.1102896), [arXiv:cond-mat/0410550](https://arxiv.org/abs/cond-mat/0410550))

and discussion of its [[electron band structure]]:

* A. H. Castro Neto, F. Guinea, N. M. R. Peres, [[Konstantin S. Novoselov]], and [[Andre K. Geim]], *The electronic properties of graphene*, Rev. Mod. Phys. **81** (2009) 109 $[$[doi:10.1103/RevModPhys.81.109](https://doi.org/10.1103/RevModPhys.81.109)$]$

category: people

[[!redirects Konstantin S. Novoselov]]


Konstantin Zarembo

## Selected writings

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation opetator]], and the identification of their spectrum with [classical field theory|classical]] [[string theory]] under the [[AdS/CFT correspondence]]:

* {#MinahanZarembo02}  J. A. Minahan, [[Konstantin Zarembo]], _The Bethe-Ansatz for $N=4$ Super Yang-Mills_, JHEP 0303 (2003) 013

Interpretation of the [[DBI-action]] on [[flavor branes]] as witnessing the [[holographic Schwinger effect]]:

* {#SemenoffZarembo11} [[Gordon Semenoff]], [[Konstantin Zarembo]], _Holographic Schwinger Effect_, Phys. Rev. Lett. 107, 171601 (2011) ([arXiv:1109.2920](https://arxiv.org/abs/1109.2920),  [doi:10.1103/PhysRevLett.107.171601](https://doi.org/10.1103/PhysRevLett.107.171601))

category: people


Konstantinos Rigatos

* [Spire page](https://inspirehep.net/authors/1762168?ui-citation-summary=true)

## Selected writings

On the [[Higgs field]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), doi:10.1007/JHEP02(2021)058]

category: people


Kontsevich formality

+-- {: .rightHandSide}
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#### Higher algebra
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[[!include higher algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

### Original version for associative algebras

The **formality theorem** ([Kontsevich 97](#Kontsevich97)) states that there is an [[L-∞-algebra]] [[quasi-isomorphism]] from the [[dg-Lie algebra]] of [[polyvector fields]] (with zero differential and [[Schouten-Nijenhuis bracket]]) to the [[dg-Lie algebra]] of the shifted [[Hochschild cohomology|Hochschild cochain complex]] (with Hochschild differential and Gerstenhaber bracket), whose first Taylor coefficient is the [[Hochschild-Kostant-Rosenberg theorem|HKR]] quasi-isomorphism.

Tamarkin alternatively proves the formality of the [[little disks operad]] (see also Kontsevich 1999) and proves that it implies the Kontsevich formality.

The Kontsevich formality theorem implies that every [[Poisson manifold]] has a [[deformation quantization]], unique up to an element in the freely-acting piece of the [[automorphism infinity-group]] of the [[E1-operad]]. This is the [[Grothendieck-Teichmüller group]] (see there for more).

### General version for higher $E_n$-algebras
  {#GeneralVersionForEnAlgebras}

More generally, a result of Kontsevich and Tamarkin (...) says that over a [[field]] $\mathbb{F}$ of [[characteristic]] 0 the canonical functor

$$
  E_n Algebras_{\mathbb{F}}
    \stackrel{\simeq}{\longrightarrow}
  P_n Algebras_{\mathbb{F}}
$$

from [[E-n algebras]] to [[Poisson n-algebras]] is an [[equivalence]] (since $P_n$ is the [[homology]] of $E_n$, this says that _The $E_n$" operad is formal_ over a field of [[characteristic zero]], see at _[[the little n-disk operad is formal]]_)

But of course the [[automorphism infinity-group]] of both $E_n$ and $P_n$ acts on both sides and makes the space of all possible such equivalences a [[torsor]] over this group. A $P_n$ algebra may be thought of as encoding a [[prequantum field theory]] of higher dimension, of sorts, and so formality says that the [[deformation quantization]] of a prequantum field theory always exists and that the choices are being acted on by the corresponding higher analog of the [[Grothendieck-Teichmüller group]].

## Related entries

* [[graph complex]]

* [[deformation quantization]], [[Grothendieck-Teichmüller group]]

* [[the little n-disk operad is formal]]

* [Hochschild cohomology -- Differential calculus](Hochschild+cohomology#DifferentialCalculus).

## References

* {#Kontsevich97} [[Maxim Kontsevich]], _Deformation quantization of Poisson manifolds_, [q-alg/9709040](http://arxiv.org/abs/q-alg/9709040), Lett. Math. Phys. __66__ (2003), no. 3, 157–216 [doi](http://dx.doi.org/10.1023/B:MATH.0000027508.00421.bf); _Operads and motives in deformation quantization_, Lett. Math. Phys. __48__ (1999), no. 1, 35–72.

* [[Dmitry Tamarkin]], _Another proof of M. Kontsevich formality theorem_, [math.QA/9803025](http://arxiv.org/abs/math/9803025); _Formality of chain operad of little discs_, Lett. Math. Phys. __66__ (1-2):65–72, 2003.

* [[Bernhard Keller]], _Notes for an Introduction to Kontsevich's quantization theorem_, [pdf](https://webusers.imj-prg.fr/~bernhard.keller/publ/emalca.pdf)

* Dan Petersen, _Minimal models, GT-action and formality of the little disk operad_, [arxiv/1303.1448](http://arxiv.org/abs/1303.1448)
* Damien Calaque, Carlo A. Rossi, _Lectures on Duflo isomorphisms in Lie algebra and complex geometry_, European Math. Soc. 2011 [MPI pdf](http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf)

* [[Vladimir Hinich]], _Tamarkin's proof of Kontsevich's formality theorem_, [math.QA/0003052](http://arxiv.org/abs/math/0003052)

* [[Vasily Dolgushev]], _All coefficients entering Kontsevich's formality quasi-isomorphism can be replaced by rational numbers_, [arxiv/1306.6733](http://arxiv.org/abs/1306.6733)
* [[Damien Calaque]], [[Thomas Willwacher]], _Triviality of the higher Formality Theorem_, [arxiv/1310.4605](http://arxiv.org/abs/1310.4605)

In agreement with Tsygan's philosophy of noncommutative differential calculus and its relations to braces algebra, Willwacher extends the Kontsevich formality to a homotopy braces morphism and to a $G_\infty$-morphism in

* [[Thomas Willwacher]], _A note on Br-infinity and KS-infinity formality_, [arxiv/1109.3520](http://arxiv.org/abs/1109.3520)

Campos has extended Kontsevich formality to a $BV_\infty$ morphism using a cyclic analog of the minimal operad

* [[Ricardo Campos]], _BV Formality_, [arxiv/1506.07715] (https://arxiv.org/abs/1506.07715)

category: algebra, geometry

[[!redirects Kontsevich formality theorem]]
[[!redirects Tamarkin formality]]


Kontsevich integral

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#### Integration theory
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[[!include topology - contents]]
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# The Kontsevich Integral
* table of contents
{: toc}

## Idea

The _Kontsevich integral_ is the [[Dyson formula]] for the [[parallel transport]] or [[holonomy]] of the [[Knizhnik-Zamolodchikov connection]] on [[ordered configuration spaces of points]] in the [[plane]]. As such it is a ([[universal Vassiliev invariant|universal]]) [[Vassiliev invariant]] of [[braids]] and, with due care, of [[knots]] and [[links]], essentially coinciding with the [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbatively quantized]] [[Chern-Simons theory]].

The Kontsevich integral generalises the [[Gauss integral formula]] which computes the [[linking number]] of two [[embedding of topological spaces|embedded]] [[circles]] via [[integration]].

## Definition

### On braids
 {#OnBraids}

[[!include Knizhnik-Zamolodchikov-Kontsevich construction -- definition]]

### On knots
 {#OnKnots}

+-- {: .num_definition #kint}
###### Definition

Let $K$ be a [[strict Morse knot]].  Let $\widehat{\mathcal{A}}$ be the [[graded completion]] of the [[algebra of chord diagrams]] with $1$-term relations.  The **Kontsevich integral** of $K$ is given by:

$$
  Z(K)
  \;=\;
  \sum_{m = 0}^\infty \frac{1}{(2 \pi i)^m} \int_{t_{\min} \lt t_m \lt \cdots \lt t_1 \lt t_{\max} \over t_j\; \text{non-critical}} \sum_{P = \{(z_j,z_j')\}} (-1)^{\downarrow P} D_p \bigwedge_{j=1}^m \frac{d z_j - d z_j'}{z_j - z_j'}
$$

=--

In this definition:

* $t_{\min}$ and $t_{\max}$ are the minimum and maximum of the $t$-coordinate in the [[Morse knot]] $K$.
* The integration is over the points in the simplex of $m$ points in the interval $[t_{\min},t_{\max}]$ where no coordinate is critical on $K$.
* Upon removing the critical values (note: _values_ not _points_, so we remove a point if it is on the same level as a critical point), the knot decomposes into a set of arcs which can be parametrised by height.  Each arc therefore defines a function $z \colon I \to \mathbb{C}$ where $I$ is the corresponding interval of height values.  In fact, $I$ must be the open interval between two successive critical values of the height function.  For a particular such interval, there must be an even number of arcs with that domain.  Given a point in the simplex (with no critical values), each coordinate in that point lies in an interval between critical values, and then for that interval we choose an unordered pair of arcs.  A choice of pair for each coordinate is called a **pairing**, and is written $P \coloneqq \{(z_j,z_j')\}$.
* For a pairing, $P$, the symbol $\downarrow P$ denotes the number of arcs that are oriented downwards when equipped with the inherited orientation from $K$.
* Putting the knot back together as a circle, we join the ends of the pairing to make a [[chord diagram]] with $m$ chords.  This defines an element in the [[algebra of chord diagrams]] which we denote by $D_P$.

## Invariance

The Kontsevich integral is an invariant of [[Morse knots]] but is not quite a [[knot invariant]].  When a "hump" is introduced to the knot then it is multiplied by $Z(H)$ where $H$ is the "humped" unknot.  Therefore, it can be made in to a genuine knot invariant via the formula
$$
I(K) = \frac{Z(K)}{Z(H)^{c/2}}
$$
where $c$ is the number of critical points of $K$.  To distinguish this from the Kontsevich integral, it is sometimes called the **final** Kontsevich integral (and the other the **preliminary** one).

## Related concepts

* [[Vassiliev invariant]]

* [[universal Vassiliev invariant]]

* [[graph complex]]

* [[Chern-Simons theory]]

## References

### General

Review:

* {#Lescop00} [[Christine Lescop]], _Introduction to the Kontsevich Integral of Framed Tangles_, 2000 ([[LescopKontsevichIntegral.pdf:file]])

* {#Kohno14} [[Toshitake Kohno]], Section 5 of: _Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks_,  Acta Math Vietnam 39, 575–598 (2014)  ([doI;10.1007/s40306-014-0088-6](doi:10.1007/s40306-014-0088-6), [pdf](https://www.ms.u-tokyo.ac.jp/~kohno/papers/kohno_config.pdf))

Textbook account:

* {#Lescop20} [[Christine Lescop]], _Invariants of links and 3-manifolds from graph configurations_ ([arXiv:2001.09929](https://arxiv.org/abs/2001.09929))

[[!include perturbative quantization of Chern-Simons theory -- references]]

[[!redirects Kontsevich integrals]]


Kontsevich-Rosenberg principle

__Kontsevich-Rosenberg principle__ is a heuristic that a good analogue of a commutative geometric structure on a noncommutative space (associative algebras or related objects) should induce its classical counterpart on the associated  commutative representation schemes (or [[derived representation scheme]]s).
An example is a [[double Poisson structure]] of [[Michel Van den Bergh]] which induces ordinary Poisson bracket on representation spaces.

* [[M. Kontsevich]], [[Alexander Rosenberg|A. Rosenberg]], _Noncommutative smooth spaces_,  The Gelfand Mathematical Seminars, 1996--1999,  85--108, Gelfand Math. Sem., Birkhäuser Boston 2000; [arXiv:math/9812158](https://arxiv.org/abs/math/9812158) (published version is slightly updated with respect to arXiv version)

This came as continuation/generalization of some ideas from

* [[Maxim Kontsevich]], _Formal (non)-commutative symplectic geometry_, The Gelfand Mathematical. Seminars, 1990-1992, Ed. L.Corwin, I.Gelfand, J.Lepowsky, Birkhauser 1993, 173-187, [pdf](http://193.51.104.7/~maxim/TEXTS/Formal%20non-commutative%20symplectic%20geometry.pdf)

Some formalizations of the principle include so called Van den Bergh's functor.

* [[Yu. Berest]], X. Chen, F. Eshmatov, A. Ramadoss, _Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras_, Contemp. Math. __583__ (2012) 219--246 [arXiv:1202.2717](https://arxiv.org/pdf/1202.2717)
* [[Yu. Berest]], [[G. Felder]], A. Ramadoss, _Derived representation schemes and noncommutative geometry_, [arXiv:1304.5314](https://arxiv.org/abs/1304.5314)
* George Khachatryan, _Derived representation schemes and non-commutative geometry_, Cornell PhD thesis under guidance of Yuri Berest [online](https://hdl.handle.net/1813/29138]
* Yuri Berest, George Khachatryan, Ajay Ramadoss, _Derived representation schemes and cyclic homology_, Adv. Math.
__245,__ (2013) 625--689 [arXiv:1112.1449](https://arxiv.org/abs/1112.1449)
* David Fernández, _The Kontsevich-Rosenberg principle for bi-symplectic forms_, [arXiv:1708.02650](https://arxiv.org/abs/1708.02650)
* Stefano D'Alesio, _Noncommutative derived Poisson reduction_, [arXiv:2012.04451](https://arxiv.org/abs/2012.04451)
* R. Bocklandt, L. Le Bruyn, _Necklace Lie algebras and noncommutative symplectic geometry_, Math Z 240, 141–167 (2002) [doi](https://doi.org/10.1007/s002090100366)
* H. Zhao, _Commutativity of quantization and reduction for quiver representations_, Math. Z. 301, 3525–3554 (2022). [doi](https://doi.org/10.1007/s00209-022-03028-1)
* Maxime Fairon, [[David Fernández]], _On the noncommutative Poisson geometry of certain wild character varieties_, [arXiv:2103.10117](https://arxiv.org/abs/2103.10117); _Euler continuants in noncommutative quasi-Poisson geometry_, [arXiv:2105.04858](https://arxiv.org/abs/2105.04858)


Kontsevich-Vishik trace

Kontsevich-Vishik canonical trace and determinant

* [[Maxim Kontsevich]], S. Vishik, _Determinants of elliptic pseudo-differential operators_, hep-th/9404046, [pdf](http://www.ihes.fr/~maxim/TEXTS/determinants_elliptic_11.pdf); _Geometry of determinants of elliptic operators_, Functional Analysis on the Eve of the 21-st Century, vol. I, Ed. S.Gindikin, J.Lepowsky, R.Wilson,  Progress in Mathematics 131, Birkhauser 1995, 173-197, [pdf](http://www.ihes.fr/~maxim/TEXTS/geometry_determinants_12.pdf)

* Matthias Lesch, Carolina Neira Jiménez, _Classification of traces and hypertraces on spaces of classical pseudodifferential operators_, J. Noncommut. Geom. 7 (2013), 457-498, [arxiv/1011.3238](http://arxiv.org/abs/1011.3238) [doi](http://dx.do.org/10.4171/JNCG/123)

* Cyril Levy, Carolina Neira Jiménez, Sylvie Paycha, _The canonical trace and the noncommutative residue on the noncommutative torus_, [arxiv/1303.0241](http://arxiv.org/abs/1303.0241)

* Tobias Hartung, Simon Scott, _A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $\zeta$-functions_, [arxiv/1510.07324](http://arxiv.org/abs/1510.07324)

category:analysis

[[!redirects Kontsevich-Vishik determinant]]


Koopman operator

[[!redirects koopman operator]]
The ''Koopman operator'' $C_\phi$ of a morphism $\phi$ is defined by pre-composing with $\phi$. i.e., $C_\phi(f) := f \circ \phi.$

* Wikipedia entry on [Koopman operator](https://en.wikipedia.org/wiki/Koopman_operator)


Kornél Szlachányi

[[!redirects Kornel Szlachanyi]]
__Kornél Szlachányi__ is a Hungarian mathematical physicist working in Budapest and teaching also in Pécs.

* Pécs: [web](http://physics.ttk.pte.hu/tanszekek/szlachanyi.shtml)

* category list question on flat functors and Grothendieck topologies with Johnstone's partial answer: [gmane](http://article.gmane.org/gmane.science.mathematics.categories/5052)

* _The monoidal Eilenberg-Moore construction and bialgebroids_, J. Pure Appl. Algebra 182, 287-315 (2003)

* _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)

[[!redirects K. Szlachányi]]


Korteweg de Vries equation

#Contents#
* table of contents
{:toc}

## Idea

The _Korteweg de Vries equation_ is a [[differential equation]], a particular nonlinear variant of a [[wave equation]]:

$$
  u_t = (3u^2 + u_{xx})_x
$$

This is one of the main examples in the study of [[integrable systems]].

## Related entries

* [[Witten conjecture]]

## References

* M. Adler, _On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg de Vries type equations_, Invent. Math. __50__  (1978/79), no. 3, 219--248, [MR520927](http://www.ams.org/mathscinet-getitem?mr=520927)

[[!redirects KdV equation]]

[[!redirects Korteweg-de Vries integrable hierarchy]]


Kosta Došen

Kosta Došen (1954-2017)

* [personal page](http://www.mi.sanu.ac.rs/novi_sajt/biography/kosta/index.html)

* Peter Schroeder-Heister, *Kosta Došen (1954-2017)*, Studia Logica: An International Journal for Symbolic Logic **107** 1, General Proof Theory (February 2019) 7-9 [[jstor:45096838](https://www.jstor.org/stable/45096838)]

## Selected writings

On [[substructural logic]]:

* [[Kosta Došen]], Peter Schroder-Heister, *Substructural Logics*, Clarendon Press (1993) [[ISBN:9780198537779](https://global.oup.com/academic/product/substructural-logics-9780198537779?cc=de&lang=en&)]

On [[proof theory]] and [[category theory]]:

* [[Kosta Došen]], *Cut Elimination in Categories*, Trends in Logic **6** [[doi:10.1007/978-94-017-1207-1](https://doi.org/10.1007/978-94-017-1207-1)]

category: people

[[!redirects Kosta Dosen]]


Kostant-Souriau extension

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## Idea

### Traditional notion

Given a [[presymplectic manifold]] $(X,\omega)$ there is the [[Poisson bracket]] [[Lie algebra]] $\mathfrak{Poisson}(X.\omega)$ of [[Hamiltonians]] and their [[Hamiltonian vector fields]]. This is a [[extension of Lie algebras]] of the Lie algebra of just Hamiltonian vecotor fields. Over a [[connected topological space|connected]] manifold it is an extension by $\mathbb{R}$. The [[Lie integration]] of this extension is the [[quantomorphism group]]-extension of the group of [[Hamiltonian symplectomorphisms]].

These extensions are called **Kostant-Souriau extensions** after [[Bertram Kostant]] and [[Jean-Marie Souriau]]. They play a central role in [[geometric quantization]].

### Refinement to higher geometry

For $(X,\omega)$ a [[pre-n-plectic manifold]] there is an [[L-∞ algebra]] extension that generalizes the Kostant Souriau extension: the [[Poisson bracket L-∞ algebra]] ([Rogers](#Rogers)).

These in turn are special cases of [[L-∞ algebra cohomology|L-∞ algebra extension]]/[[∞-group extensions]] which are variants of the [[higher Atiyah groupoid]]-extensions that exist in general in [[cohesive]] [[higher geometry]] ([Hgpt](#Hgpt)), as indicated in the following table:

[[!include geometric quantization extensions - table]]

## References

The traditional Kostant-Souriau extension appears for instance prop. 2.3.9 in

* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_, Birkhäuser (1993)

The refinement to [[n-plectic geometry]] is due to

* [[Chris Rogers]], _$L_\infty$ algebras from multisymplectic geometry_ ,
Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50  ([arXiv:1005.2230](http://arxiv.org/abs/1005.2230), [journal](http://link.springer.com/article/10.1007%2Fs11005-011-0493-x)).
 {#Rogers}

A general characterization in [[higher geometry]] is in

* _[[schreiber:Higher geometric prequantum theory]]_

[[!redirects Kostant-Souriau extensions]]

[[!redirects Kostant-Souriau central extension]]
[[!redirects Kostant-Souriau central extensions]]

[[!redirects Kostant Souriau extension]]
[[!redirects Kostant Souriau extensions]]


Kostas Skenderis

* [webpage](http://www.southampton.ac.uk/maths/about/staff/ks8e11.page)

## Selected writings

On [[Dp-D(p+4) brane bound states]] as [[black holes in string theory]] and their [[black hole entropy]]:

* [[Konstadinos Sfetsos]], [[Kostas Skenderis]], _Microscopic derivation of the Bekenstein-Hawking entropy formula for non-extremal black holes_, Nucl. Phys. B517 (1998) 179-204 ([arXiv:hep-th/9711138](https://arxiv.org/abs/hep-th/9711138))

## Related $n$Lab entries

* [[AdS-CFT correspondence]]

* [[Fefferman-Graham ambient construction]]

category: people


Koszul > history

see _[[Jean-Louis Koszul]]_


Koszul algebra

## Definition

A positively [[graded algebra]] $A$ is called **Koszul** (after [[Jean-Louis Koszul]]) if every graded [[simple object]] has a [[projective resolution]] where the $i$th term is generated in grade $i$.  Equivalently, this holds if the internal and homological gradings on the [[Ext]] $A^!=\Ext_A(A_0,A_0)$ coincide.

The algebra $(A^!)^{\op}$ is itself Koszul and is called the _[[Koszul dual]]_ of $A$.

[[!redirects Koszul algebras]]


Koszul complex

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## Idea

The _Koszul complex_ of a sequence of elements $(x_1, \cdots, x_d)$ in a [[commutative ring]] $R$ (or more generally of [[center|central]] elements in a [[non-commutative ring]]) is a [[cochain complex]] whose entry in degree $-n$ is the [[exterior power]] $\wedge^n R^d$ of the [[free module]] $R^d = R^{\oplus_d}$ over $R$ of [[rank]] $d$, and whose [[differential]] is given in each degree on the $k$th summand by multiplication with $x_k$.

The key property of the Koszul complex is that in good cases (namely if the sequence $(x_1, \cdots, x_d)$ is a [[regular sequence]] in $R$), it constitutes is a [[free resolution|free]] [[homological resolution]] of the [[quotient]] $R/(x_1, \cdots, x_d)$ of $R$ by the [[ideal]] generated by these elements (see prop. \ref{KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing} below).

In cases where the Koszul complex fails to be a [[homological resolution]] of the [[quotient ring]], it may be augmented by further generators to yield a resolution after all then called a _[[Koszul-Tate resolution]]_.

From the perspective of [[derived algebraic geometry]] the Koszul complex may be interpreted as the [[formal duality|formal dual]] of the [[derived critical locus]] of the elements  $(x_1, \cdots, x_d)$, regarded as functions on the [[spectrum of a ring|spectrum]] $Spec(R)$.

In this guise the Koszul complex appears prominently in [[Lagrangian field theory]], under the name _[[BV-complex]]_, as a potential [[homological resolution]] of the _[[shell]]_ (the solution locus of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]). In this case the obstruction to the Koszul complex providing a resolution of the shell is its cochain cohomology in degree -1 (via prop. \ref{KoszulResolutionForNoetherianRngAndElementsInJacobson} below) which has the interpretation as the [[infinitesimal gauge symmetries]] of the [[Lagrangian density]] that have not been made explicit. Making them explicit by promoting them to elements in the [[Chevalley-Eilenberg algebra]] of the corresponding [[action Lie algebroid]] yields what is called the _[[BRST complex]]_ of the theory, and _its_ Koszul complex then yields the respective [[Koszul-Tate resolution]], now called the _[[BV-BRST complex]]_ of the theory.

## Definition

Let $R$ be a [[unital ring]].

Consider also a finite [[sequence]] $(x_1,\ldots,x_r)$ of [[elements]] $x_i \in R$.

Given any [[center|central element]] $x\in Z(R)$, one can define a two term [[cochain complex]]

$$ K(x) \coloneqq (0\to R\stackrel{x}\to R\to 0) $$

concentrated in degrees $0$ and $1$, where the map (the [[differential]]) is the left multiplication by $x$. Given a sequence $(x_1,\ldots,x_r)$ of central elements in $R$ one can define the tensor product

$$
  K(x_1,\ldots,x_r)
   \coloneqq
  K(x_1)\otimes_R K(x_2)\otimes_R\cdots \otimes_R K(x_r) $$

of complexes of left $R$-[[modules]]. The degree $p$ part of $K(x_1,\ldots,x_r)$ equals the [[exterior power]] $\Lambda^{p+1}R^r$. Consider the usual [[basis]] elements $e_{i_0}\wedge \cdots \wedge e_{i_p}$ of $\Lambda^{p+1}R^r$, where $1\leq i_0\lt i_1\lt\cdots\lt i_p\leq r$. Then the differential is given by

$$
d(e_{i_0}\wedge \cdots \wedge e_{i_p}) = \sum_{k = 0}^{p}(-1)^{k+1} x_{i_k} e_{i_0}\wedge \cdots\wedge \hat{e}_{i_k} \wedge \cdots\wedge e_{i_r}
$$

The differential can be obtained from the faces of the obvious Koszul [[semi-simplicial object|semi-simplicial]] $R$-[[module]] and the cochain complex above is obtained by the usual alternating sum rule.

Now let $A$ be a finitely generated left $R$-module. Then the abelian [[chain homology]] [[abelian group|groups]]

$$ H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A), $$

$$ H^q(x_1,\ldots,x_r;A) = H^q(Hom_R(K(x_1,\ldots,x_r),A)), $$

together with [[connecting homomorphisms]], form a homological and cohomological [[delta-functor]] (in the sense of [[nlab:Tohoku]]) respectively, deriving the zero parts

$$ H_0 = A/(x_1,\ldots,x_r)A$$

$$ H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)$$

where $(x_1,\ldots,x_r)A$ is the left $R$-submodule generated by $x_1,\ldots,x_r$. A Poincare-like duality holds: $H_p(x_1,\ldots,x_r;A) = H^{r-p}(x_1,\ldots,x_r;A)$.

The sequence $\mathbf{x} = (x_1,\ldots,x_r)$ is called __$A$-regular__ (or regular on $A$) if for all $i$ the image of $x_i$ in $A/(x_1,\ldots,x_{i-1})A$ annihilates only zero. This terminology is in accord with calling a non-zero divisor in a ring a "regular element" (and is in accord with the terminology regular local rings).

If $\mathbf{x}$ is a regular sequence on/in $R$ then $K(\mathbf{x},R)$ is a free resolution of the module $R/(x_1,\ldots,x_r)R$ and the cohomology $H^q(x_1,\ldots,x_r;A) = Ext^q_R(R/(x_1,\ldots,x_r)R,A)$ while [[Koszul homology]] is $H_q(x_1,\ldots,x_r;A) = Tor_q^R(R/(x_1,\ldots,x_r)R,A)$.

The resolution of $R/(x_1,\ldots,x_r)R$ can be written

$$
0 \to \Lambda^r(R^r)\to \cdots \to \Lambda^2(R^r)\to R^r \to R \to R/(x_1,\ldots,x_r)R\to 0$$

and the $R$-linear map $R^r\to R$ is given by the row vector $(x_1,\ldots,x_r)$.

## Properties
 {#Properties}

+-- {: .num_prop #KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing}
###### Proposition
**(Koszul complex of [[regular sequence]] is [[free resolution]] of [[quotient ring]])**

Let $R$ be a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[regular sequence]] of elements in $R$. Then the Koszul complex $K(x_1,\cdots, x_d)$ is a [[free resolution]] of the [[quotient ring]] $R/(x_1, \cdots, x_d)$.

=--

+-- {: .num_prop #KoszulResolutionForNoetherianRngAndElementsInJacobson}
###### Proposition
**(Koszul resolution detected in degree (-1))**

Let $R$ be a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[sequence]] of elements in $R$, such that

1. $R$ is [[Noetherian ring|Noetherian]];

1. each $x_i$ is contained in the [[Jacobson  radical]] of $R$

then the following are equivalent:

1. the [[cochain cohomology]] of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;

1. the Koszul complex $K(x_1, \cdots, x_d)$ is a [[free resolution]] of the [[quotient ring]] $R/(x_1, \cdots, x_d)$, hence its [[cochain cohomology]] vanishes in all degrees  $\leq -1$;

1. the sequence $(x_1, \cdots, x_d)$ is a [[regular sequence]].

=--

A **proof** is spelled out on [Wikipedia - Properties of Koszul homology](https://en.wikipedia.org/wiki/Koszul_complex#Properties_of_a_Koszul_homology)

## Examples

+-- {: .num_example #KoszulComplexForFormalPowerSeriesAlgebras}
###### Example
**(Koszul complex for formal power series algebras)**

Let $k$ be a [[field]], let $R = k[ [ X_1,\cdots, X_n ] ]$ be a [[formal power series algebra]] over $k$ in $n$ [[variables]], and let $(f_1, \cdots, f_r)$ be formal power series whose constant term vanishes. Then the Koszul complex is a homological resolution precisely already if its cohomology in degree -1 vanishes

$$
  \left(
    H^{-1}(K(f_1, \cdots, f_n))
    = 0
  \right)
   \;\Leftrightarrow\;
  \left(
    K(f_1, \cdots, f_n)
      \overset{\simeq_{qi}}{\longrightarrow}
    k[ [X_1, \cdots X_n] ]/(f_1, \cdots, f_r)
  \right)
  \,.
$$

This is because the assumptions of prop. \ref{KoszulResolutionForNoetherianRngAndElementsInJacobson} are met: A formal power series ring over a field is Noetherian ([this example](noetherian+ring#PolynomialAlgebraOverNoetherianRingIsNoetherian)) and an element of a formal power series algebra is in the Jacobson radical precisely if its constant term vanishes ([this example](Jacobson+radical#JacobsonRadicalOfFormalPowerSeriesAlgebra)).

=--

## Related concepts

* [[Koszul-Tate resolution]]

* [[syzygy]]

* [[BV-formalism]]

## References

The original reference is

* [[Jean-Louis Koszul]], _Homologie et cohomologie des algèbres de Lie_ , Bulletin de la Société Mathématique de France, 78, 1950, pp 65-127.

A standard textbook reference is

* [[Charles Weibel]], section 4.5 of _Homological algebra_ ([pdf](http://www.math.unam.mx/javier/weibel.pdf))

A generalization of Koszul complexes to (appropriate resolutions of) [[algebras over operads]] is in

* Joan Millès, _The Koszul complex is the cotangent complex_, MPIM2010-32, [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=4143)

See also

* {#Wikipedia} Wikipedia, _[Koszul complex](https://en.wikipedia.org/wiki/Koszul_complex)_

[[!redirects Koszul complexes]]


Koszul duality

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# Koszul duality
* table of contents
{: toc}

## Idea

__Koszul duality__ (named after [[Jean-Louis Koszul]]) is a duality and phenomenon generalizing the duality between the symmetric and exterior algebra of a vector space to so-called quadratic [[differential graded algebra]]s (which can be obtained as a free dga module an ideal of relations which live in degree 2). For a pair of Koszul dual algebras, there is a correspondence between certain parts of their derived categories (precise formulation involves some finiteness conditions). In a setup in which one of the algebras is replaced by a cocomplete dg coalgebra, there is a formulation free of finiteness conditions, but involving [[twisting cochain]] (see that entry).

## On operads

Koszul duality is a duality between [[quadratic operad]]s, predicted in

* [[Maxim Kontsevich|M. Kontsevich]], _Formal (non)commutative symplectic geometry_,  The Gelʹfand Mathematical Seminars, 1990--1992,  173--187, Birkhäuser, Boston, MA, 1993 (cf. [[formal noncommutative symplectic geometry]]).

and developed in

* [[Victor Ginzburg]], [[Mikhail Kapranov]], _Koszul duality for operads_, Duke Math. J. __76__ (1994),  no. 1, 203--272; reprint [arxiv/0709.1228](http://arxiv.org/abs/0709.1228); *Erratum to: Koszul duality for operads*, Duke Math. J. __80__ (1995),  no. 1, 293.

## For associative and dg-algebras

From Ben Webster on [mathoverflow](https://mathoverflow.net/a/394/2362):

There are a lot of algebras whose derived categories are equivalent in surprising ways.  [[Morita equivalences]] are pretty simple, especially for finite dimensional algebras; essentially the only free parameter is the dimension of the object.

Namely, if $A Mod$ and $B Mod$ are equivalent, then the image of $A$ as a module over itself is a [[projective object|projective]] [[generator]] of $B Mod$, and for a finite-dimensional algebra, essentially the only thing you can do is take several copies of the [[indecomposable object|indecomposible]] projectives of $B$.

On the other hand, if you take the derived category of [[differential graded module|dg-modules]] over $A$ (the dg part of this is not a huge deal; it's just that they're very close to, but a bit better behaved than, actual derived/[[triangulated categories]], which are just crude truncations of truly functorial dg/$A_\infty$ versions), this is equivalent to the category of dg-modules over the [[endomorphism algebra]] (this is in the dg sense, so it's a dg-algebra whose cohomology is the $Ext$ algebra) of *any* generating object.  There are a *lot* more generating objects than projective generators, so there are a lot of derived equivalences.

In particular, let $A$ be a finite dimensional algebra $A$, then an  obvious not-very-projective generating object is the sum of all the [[simple modules]], say $L$.  As mentioned above, there's an equivalence $A dg Mod = \mathrm{Ext}(L,L) dg Mod$, just given by taking $\mathrm{Ext}(L,-)$.

In general, $\mathrm{Ext}(L,L)$ is a very complicated object (for example, often for [[group algebras]] over [[finite field]]s), but sometimes it turns out to be nice.  For example, if $A$ is an  [[exterior algebra]], you'll get a [[polynomial ring]] on the dual vector space.  Another (closely related) example is that the cohomology of a [[reductive group]] (over $\mathbb{C}$) is Koszul dual to the cohomology of its classifying space.

One way to ensure that $\mathrm{Ext}(L,L)$ is nice is if the algebra $A$ is [[graded algebra|graded]]. Then $\mathrm{Ext}(L,L)$ inherits an "internal" grading in addition to its homological one.  If these coincide, then $A$ is called _Koszul_.

In this case, $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting $A_\infty$ operations, they would break the grading), so you're dealing with a derived equivalence between actual algebras, though you have to be a bit careful about the dg-issues.  Thus the derived category of usual modules over $A$ is equivalent to dg-modules over $B$ (with its unique grading) and vice versa.  This can be fixed by taking graded modules on both sides.

## Examples

* The most famous example of Koszul dual algebras are the [[exterior algebra]] $Alt^{\bullet}V[-1]$ and the [[polynomial algebra]] $Sym^\bullet V^*[-2]$.

* A regular block of [[category O]] for any [[semisimple Lie algebra]] is a self-Koszul dual.

  * More generally, a singular block of [[parabolic category O]] is dual to a different singular block of parabolic category O where the combinatorial data determining the central character and finiteness conditions switch.

* Braden, Licata, Proudfoot and Webster gave a combinatorial construction of a large family of Koszul dual algebras in [Gale duality and Koszul duality](http://arxiv.org/abs/0806.3256).

## Koszul duality between D-modules and Ω-modules

An important special case of Koszul duality
establishes a Quillen equivalence between model categories of D-modules
and Ω-modules.
This was first observed by Kapranov.

Here $D$ is the sheaf of differential operators
on a smooth manifold or a smooth variety
and $\Omega$ is the sheaf of differential forms on the same
manifold or variety.
Both $D$ and $\Omega$ are equipped with their canonical filtrations
(differential operators of order at most~$k$
respectively differential forms of degree at least~$k$)
and all constructions below work with sheaves
of filtered chain complexes.

The equivalence is implemented by tensoring with a certain filtered
$\Omega$-$D$-bimodule $DR$.
If we discard the differential, $DR=\Omega\otimes D$ as a filtered graded bimodule.
The differential is canonically determined by its degree 0
component, where we take the coevaluation map $O\to \Omega^1\otimes D^1_0$,
where $D^1_0$ denotes differential operators of order at most~1
with vanishing constant term, i.e., vector fields.

This equivalence allows one to define [[six operations]]
on D-modules by transfering them from Ω-modules,
where they can be defined in the usual manner,
since differential forms can be pulled back along maps,
unlike differential operators.

This fully explains the somewhat unintuitive explicit
formulas for the [[six operations]] on D-modules.

## Linear Koszul duality

Linear Koszul duality is an equivalence between the symmetric algebra of a dg vector bundle and the symmetric algebra of the shifted dual dg vector bundle.

* [[Ivan Mirković]], Simon Riche, _Linear Koszul duality_, Compos. Math. __146__:1 (2010) 233-258

## Categorical Koszul duality

A form of Koszul duality with dg algebras replaced by dg categories has been developed by Holstein and Lazarev. In this, the Koszul dual of a dg category $\mathcal{D}$, given by a bar construction, $B\mathcal{D}$ is a pointed curved coalgebra; see

 * [[Julian Holstein]], A. Lazarev, _Categorical Koszul duality_ [arXiv:2006.01706](https://arxiv.org/pdf/2006.01705.pdf)

## Global Koszul duality

Another variant of Koszul duality defined in terms of model structures on curved dg coalgebras.

* Matt Booth, Andrey Lazarev, _Global Koszul duality_, [arXiv:2304.08409](https://arxiv.org/abs/2304.08409)

## Related concepts

* [[bar and cobar construction]]

* [[holography as Koszul duality]]

* [[symplectic duality]]

## References

Other historical references on Koszul duality include

* A. A. Beilinson, V. A. Ginsburg, V. V. Schechtman, _Koszul duality_,  J. Geom. Phys.  5  (1988),  no. 3, 317--350 [MR1048505](http://www.ams.org/mathscinet-getitem?mr=1048505) doi:10.1016/0393-0440(88)90028-9

* [[Alexander Beilinson]], [[Victor Ginzburg]], [[Wolfgang Soergel]], _Koszul duality patterns in representation theory_,  J. Amer. Math. Soc.  9  (1996),  no. 2, 473--527 [corrigenda](http://home.mathematik.uni-freiburg.de/soergel/PReprints/KorrBGS.pdf)

Koszul duality is also discussed in

* [[John Baez]], [TWF Week 238](http://math.ucr.edu/home/baez/week238.html), [Week 239](http://math.ucr.edu/home/baez/week239.html)

* _The Everything Seminar_ , _[koszul-duality-and-lie-algebroids](http://cornellmath.wordpress.com/2008/03/25/koszul-duality-and-lie-algebroids)_

* MathOverflow: [What is Koszul duality?](http://mathoverflow.net/questions/329/what-is-koszul-duality), [Beilinson-Bernstein and Koszul duality](http://mathoverflow.net/questions/309/beilinson-bernstein-and-koszul-duality), [what-extra-conditions-are-necessary-for-the-following-version-of-koszul-duality](http://mathoverflow.net/questions/67909/what-extra-conditions-are-necessary-for-the-following-version-of-koszul-duality)
* David Eisenbud, Gunnar Fløystad, Frank-Olaf Schreyer, _Sheaf cohomology and free resolutions over
exterior algebras_,  Trans. Amer. Math. Soc. 355 (2003), 4397-4426 [MR1990756](http://www.ams.org/mathscinet-getitem?mr=1990756) [doi](http://dx.doi.org/10.1090/S0002-9947-03-03291-4)
* Tyler Foster, Po Hu, Igor Kriz, _D-structures and derived Koszul duality for unital operad algebras_, [arxiv./1507.07151](http://arxiv.org/abs/1507.07151)

A "curved" generalization is discussed in

* Joseph Hirsh, Joan Millès, _Curved Koszul duality theory_, Max Planck preprint MPIM2010-104, [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=4217)
* Gunnar Fløystad, _Koszul duality and equivalences of categories_,  Trans. Amer. Math. Soc. __358__ (2006), 2373-2398 [math.AG/0012264](http://arxiv.org/abs/math/0012264)
[MR2204036](http://www.ams.org/mathscinet-getitem?mr=2204036)
[doi](http://dx.doi.org/10.1090/S0002-9947-05-04035-3)

[[Bernhard Keller]] and his student Lefèvre-Hasegawa described rather general framework for Koszul duality using dg-(co)algebras and [[twisting cochain]]s:

* [[Bernhard Keller]], _Koszul duality and coderived categories (after Lefèvre-Hasegawa)_ (2003) [abstract](http://www.math.jussieu.fr/~keller/publ/kdcabs.html) [dvi](http://www.math.jussieu.fr/~keller/publ/kdc.dvi) [pdf](http://www.math.jussieu.fr/~keller/publ/kdc.pdf) [ps](http://www.math.jussieu.fr/~keller/publ/kdc.ps)

See also:

* Kenji Lefèvre-Hasegawa, _Sur les A-infini catégories_, [pdf](http://people.math.jussieu.fr/~keller/lefevre/TheseFinale/tel-00007761.pdf) [math/0310337](http://arxiv.org/abs/math.CT/0310337)
* Leonid Positselski, _Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence_, [arXiv/0905.2621](http://arxiv.org/abs/0905.2621)

* [[Dev Sinha]], _Koszul duality in algebraic topology - an historical perspective_, J. Homotopy Relat. Struct. (2013) 8: 1 ([arXiv:1001.2032](https://arxiv.org/abs/1001.2032))

* Aaron M Royer, _Generalized string topology and derived Koszul duality_, [arXiv/1306.6708](https://arxiv.org/abs/1306.6708)

* [[Mikhail Kapranov]], _On DG-modules over the de Rham complex and the vanishing cycles functor_, Algebraic Geometry, Springer Lecture Notes in Mathematics __1479__ (1991) 57-86

* [[Theo Johnson-Freyd]], _Exact triangles, Koszul duality, and coisotropic boundary conditions_, [arXiv/1608.08598](https://arxiv.org/abs/1608.08598)

* [[Jonathan Beardsley]], [[Maximilien Péroux]], _Koszul Duality in Higher Topoi_ ([arXiv:1909.11724](https://arxiv.org/abs/1909.11724))

[[!redirects Koszul dual]]


Koszul homology

[[!redirects Kozsul homology]]

## Idea

The _Koszul homology_ of an [[ideal]] in some algebra is the [[chain homology]] of its [[Koszul complex]].

The linear dual of the Koszul homology of the symmetric algebra over the tangent space of a manifold is known as [[Spencer cohomology]].

## References

* Lecture notes on Koszul homology [pdf](http://www.math.lsa.umich.edu/~hochster/711F07/L10.03.pdf)


Koszul-Malgrange theorem

+-- {: .rightHandSide}
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### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Statement

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

[[holomorphic vector bundle|Holomorphic vector bundles]] over a ([[compact topological space|compact]]) [[complex manifold]] $X$ are equivalently [[complex vector bundles]] $E$ which are equipped with a holomorphically [[flat connection]] $\nabla$, hence with a [[covariant derivative]]

$$
  \bar \partial_E \colon \Gamma(E)\longrightarrow \Gamma(E) \otimes_{C^\infty(X, \mathbb{C})}\Omega^{0,1}(X)
$$

such that $\bar \partial_E \circ \bar \partial_E = 0$, hence with a compatible [[Dolbeault operator]]

The analogous statement is true for generalization of vector bundles to [[chain complexes]] of [[module sheaves]] with [[coherent cohomology]].

=--

For [[complex vector bundles]] over [[complex varieties]] this statement is due to [[Alexander Grothendieck]] and ([Koszul-Malgrange 58](#KoszulMalgrange58)), recalled for instance as ([Pali 06, theorem 1](#Pali06)). It may be understood as a special case of the [[Newlander-Nirenberg theorem]], see ([Delzant-Py 10, section 6](#DelzantPy10)), which also generalises the proof to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles. Over [[Riemann surfaces]], see [below](#OverRiemannSurfaces), the statement was highlighted in ([Atiyah-Bott 83](#AtiyahBott83)) in the context of the [[Narasimhan-Seshadri theorem]].

As an [[equivalence of categories]] and with [[vector bundles]] generalized to [[coherent sheaves]] the statement appears in ([Pali 06](#Pali06)). This is generalized to an equivalence of [[homotopy categories]] of [[categories of chain complexes]] of coherent sheaves, hence to the [[derived category]] $D_{coh}^b(X)$ of $\mathcal{O}_X$-[[modules]] with [[coherent cohomology]], in ([Block 05, theorem 4.1.3](#Block05)).

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

The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of [[differential geometry]]/[[differential cohomology]] with constructions in [[algebraic geometry]]. For instance [[intermediate Jacobians]] arise in [[differential geometry]] and [[quantum field theory]] as [[moduli spaces of flat connections]] equipped with [[symplectic structure]] and [[Kähler polarization]], all of which in terms of [[algebraic geometry]] directly comes down [[moduli spaces]] of [[abelian sheaf cohomology]] with [[coefficients]] in the [[structure sheaf]] (and/or some variants of that, under the exponential exact sequence).

=--

We state the derived version in more detail:

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

For $X$ a [[complex manifold]], write

$$
  CE(T_{hol} X) \coloneqq (\Omega^{0,\bullet}(X), \bar \partial)
$$

for its [[Dolbeault complex]] regarded as a [[differential graded algebra]] (the [[Chevalley-Eilenberg algebra]] of its holomorphic [[tangent Lie algebroid]]). Define a [[dg-category]] $Rep(T_{hol} X)$ whose

* [[objects]] are [[finite type]] [[∞-Lie algebroid representations]] of $T_{hol} X$, hence [[cochain complexes]] of [[complex vector bundles]] $E^{\bullet}$ over $X$ equipped with a $\mathbb{C}$-linear map

  $$
    \bar \partial_E
    \;\colon\;
    \Gamma(E)^\bullet
    \longrightarrow
    \Gamma(E)^\bullet \otimes_{C^\infty(X,\mathbb{C})} CE(T_{hol}X)
  $$

  that extend to a graded [[derivation]] on $\Gamma(E)^\bullet \otimes_{C^\infty(X,\mathbb{C})} CE(T_{hol}X)$;

* [[hom-objects]] are the evident [[chain complexes]] (...).

=--

([Block 05, def. 2.3.2 in theorem 4.1.3](#Block05)).

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

For $X$ a [[compact topological space|compact]] [[complex manifold]], there is an [[equivalence of categories]]

$$
  Ho Rep(T_{hol}X) \simeq D^b_{coh}(X)
$$

between the [[homotopy category]] of the [[dg-category]] of complexes of holomorphic flat connections, def. \ref{ChainComplexesOfHolomorphicFlatConnections}, and the bounded [[derived category]] of $\mathcal{O}_X$-[[module sheaves]] with [[coherent cohomology]].

=--

([Block 05, theorem 4.1.3](#Block05)).

## Related theorems

* [[Narasimhan-Seshadri theorem]]

* [[Donaldson-Uhlenbeck-Yau theorem]]

* [[Kobayashi-Hitchin correspondence]]

## References

The classical statement of theorem \ref{KoszulMalgrangeTheorem} is due (according to [Pali 06](#Pali06)) to [[Alexander Grothendieck]] and

* {#KoszulMalgrange58} [[Jean-Louis Koszul]], [[Bernard Malgrange]], _Sur certaine structures fibrées complexes_, arch. mat, vol IX, 1958

Over [[Riemann surfaces]] and in the context of the [[moduli space of flat connections]]:

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

* {#Evans} [[Jonathan Evans]], _[Aspects of Yang-Mills theory](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm)_, lecture notes, ([lecture 10](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture10.pdf), [lecture 11](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture11.pdf), [lecture 12](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture12.pdf))

Generalization to [[coherent sheaves]] is due to

* {#Pali06} N. Pali, _Faisceaux $\bar \partial$-cohrents sur les variété complexes_ ( _$\bar \partial$-Coherent sheaves on complex manifolds) Math.
Ann. 336 (2006), no. 3, 571–615 ([arXiv:math/0305422](http://arxiv.org/abs/math/0305422))

Further Generalization to [[chain complexes]] of holomorphic vector bundles is discussed in

* {#Block05} [[Jonathan Block]], _Duality and equivalence of module categories in noncommutative geometry I_ ([arXiv:0509284](http://arxiv.org/abs/math/0509284))

in terms of [[Lie infinity-algebroid representations]] of the holomorphic [[tangent Lie algebroid]].

Generalization to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles is in

* {#DelzantPy10} Thomas Delzant, Pierre Py, _Kähler groups, real hyperbolic spaces and the Cremona group_, Compositio Math. 148, no. 1 (2012), 153--184 ([arXiv:1012.1585](http://arxiv.org/abs/1012.1585))


Koszul-Tate resolution

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#### Homological algebra
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[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $A$ an [[associative algebra|algebra]] and $I \subset A$ an [[ideal]], a **Koszul-Tate resolution** is a [[resolution]] of the [[quotient]] $A/I$ by a cochain [[dg-algebra]] in non-positive degree that is degreewise free/[[projective object|projective]].

It is a refinement of a [[Koszul complex]] or rather an extension.

## Applications

* A Koszul-Tate resolution is one part of the [[BRST-BV complex]].

## References

* [[Jean-Louis Koszul]], _Homologie et cohomologie des algèbres de Lie_ , Bulletin de la Société Mathématique de France, 78, 1950, pp 65-127.

* [[John Tate]], _Homology of Noetherian rings and local rings_ , Illinois Journal of Mathematics, 1, 1957, pp. 14-27

[[!redirects Koszul-Tate resolutions]]

[[!redirects Koszul resolution]]
[[!redirects Koszul resolutions]]

[[!redirects Tate complex]]
[[!redirects Tate complexes]]

[[!redirects Koszul complex]]
[[!redirects Koszul complexes]]

[[!redirects Koszul-Tate complex]]
[[!redirects Koszul-Tate complexes]]


Kouyemon Iriye

* [Institute page](Kouyemon Iriye)

## Selected writings

Discussion of $\mathbb{Z}_2$-[[equivariant stable homotopy groups]]/[[equivariant stable cohomotopy]] of [[spheres]]:

* {#ArakiIriye82} [[Shôrô Araki]], [[Kouyemon Iriye]], _Equivariant stable homotopy groups of spheres with involutions. I_, Osaka J. Math. Volume 19, Number 1 (1982), 1-55. ([Euclid:1200774828](http://projecteuclid.org/euclid.ojm/1200774828))

* {#Iriye82} [[Kouyemon Iriye]], _Equivariant stable homotopy groups of spheres with involutions. II_, Osaka J. Math. Volume 19, Number 4 (1982), 733-743 ([euclid:1200775536](https://projecteuclid.org/euclid.ojm/1200775536))

On (among other things) [[coset space]]-realizationa of [[complex projective 3-space]]:

* {#Iriye90} [[Kouyemon Iriye]], _Manifolds which have two projective space bundle structures from the homotopical point of view_, J. Math. Soc. Japan Volume 42, Number 4 (1990), 639-658 ([euclid:jmsj/1227108441](https://projecteuclid.org/euclid.jmsj/1227108441))

Discussion of fiberwise [[H-space]]-structure on [[spheres]], [[quaternionic Hopf fibration]], [[octonionic Hopf fibration]], see also at [[Hopf construction]]):

* {#Iriye95} [[Kouyemon Iriye]], _Equivariant Hopf structures on a sphere_, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 ([euclid:kjm/1250518704](http://projecteuclid.org/euclid.kjm/1250518704))

category: people


KR cohomology theory

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#### Cohomology
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[[!include cohomology - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _KR-theory_ ([Atiyah 66](#Atiyah66)) is variant of [[topological K-theory]] on [[spaces]] equipped with a $\mathbb{Z}_2$-[[action]] (by [[homeomorphism]], hence equipped with one [[involution|involutive]] homeomorphism -- a "[[real space]]").

In terms of [[cocycle]] models, classes of KR-theory are represented by [[complex vector bundles]] over $X$ on which the involution on their base space lifts to an _anti-linear_ involution of the total space. Over manifolds with trivial involution these are precisely the [[complexification]] of [[real vector bundles]] and hence over such spaces $KR$-theory reduces to [[KO]]-[[cohomology theory|theory]]. Conversely, over two copies $X \cup X$ of $X$ equipped with the involution that interchanges the two, $KR$-theory reduces to [[KU]]-[[cohomology theory|theory]]. Finally over $X \times S^1$ with the involution the antipodal identification on the second ([[circle]]) factor , $KR$-theory reduces to the self-conjugate [[KSC-theory]] ([Anderson 64](#Anderson64)). So in general $KR$-theory interpolates between all these cases. For instance on $X \times S^1$ with the reflection-involution on the circle (the [[real space]] denoted $S^{1,1}$, the non-trivial $\mathbb{Z}_2$-[[representation sphere]]) it behaves like $KO$-theory at the two involution fixed points (the two [[O-planes]]) and like $KU$ in their complement (a model that makes this very explicit is given in [DMR 13, section 4](#DMR13)), schematically:

$$
  KR(S^{1,1}) = ( KO --- KU --- KO )
$$

More abstractly, complex conjugation of complex vector bundles induces on the [[complex K-theory]] [[spectrum]] [[KU]] an involutive [[automorphism]]. $KR$ is the corresponding $\mathbb{Z}_2$-[[equivariant spectrum]], and $KR$-theory the corresponding $\mathbb{Z}_2$-[[equivariant cohomology]] theory.  In particular, the [[homotopy fixed point]] of [[KU]] under this automorphism is [[KO]]

$$
  KO \simeq (KU)^{\mathbb{Z}/2}
$$

(e.g.[Karoubi 01](#Karoubi01), [Dugger 03, corollary 7.6](#Dugger03), [Hill-Hopkins-Ravenel, section 7.3](#HillHopkinsRavenel)) and this way where in complex K-theory one has [[KU]]-[[modules]] ([[∞-modules]]), so in KR-theory one has $KO$-modules.

KR is an example of a [[real-oriented cohomology theory]], together with for instance [[MR-theory]] and [[BPR-theory]].

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

An [[involution]] on a space by a [[homeomorphism]] (or [[diffeomorphism]]) as they appear in KR theory may be thought of as a "non-linear [[real structure]]", and therefore spaces equipped with such involutions are called "[[real spaces]]". Following this, $KR$-theory is usually pronounced "real K-theory". But **beware** that this terminology easily conflicts with or is confused with [[KO]]-theory. For disambiguation the latter might better be called "orthogonal K-theory".

=--

## Definition

### As the Grothendieck group of complex vector bundles with real structure

...([Atiyah 66](#Atiyah66))...

### As a genuine $\mathbb{Z}_2$-Spectrum

The following gives $KR$ as a [[genuine G-spectrum]] for $G = \mathbb{Z}_2$.

Using that every orthogonal representation of $\mathbb{Z}_2$ is contained in an $\mathbb{C}^n$ with its [[complex conjugation]] action, one can restrict attention to these. Write

$$
  \mathbb{C}P^1 = S^{2,1} = S^{\mathbb{C}}
  \,.
$$

The reduced [[canonical line bundle]] over this (the [[Hopf fibration]]) is classified by a map

$$
  S^{2,1}= \mathbb{C}P^1 \to \mathbb{Z}\times BU
$$

to the [[classifying space]] for [[topological K-theory]]. The homotopy-associative  multiplication on this space then yields the structure map of a $\mathbb{Z}_2$-spectrum

$$
  S^{2,1} \wedge (\mathbb{Z} \times BU)\to \mathbb{Z}\times BU
  \,.
$$

This is in fact an [[Omega spectrum]], by equivariant complex [[Bott periodicity]] (for instance in [Dugger 03, p. 2-3](#Dugger03)).

## Properties

### Bigrading

As any genuine [[equivariant cohomology theory]] $KR$-theory is naturally graded over the [[representation ring]] $RO(\mathbb{Z}_2)$. Write $\mathbb{R}$ for the trivial 1-dimensional representation and $\mathbb{R}_-$ for that given by the sign involution. Then the general orthogonalrepresentation decomposes as a [[direct sum]]

$$
  V = \mathbb{R}^+\oplus \mathbb{R}_-^q
  \,.
$$

The corresponding [[representation sphere]] is

$$
  S^V = (some\; convention)
  \,.
$$

### As induced from the derived moduli stack of tori
 {#InducedFromModuliStackOfTori}

The relation between $KU$, $KO$ and $KR$ naturally arises in [[chromatic homotopy theory]] as follows.

Inside the [[moduli stack of formal group laws]] sits the _[[moduli stack of tori|moduli stack of one dimensional tori]]_ $\mathcal{M}_{\mathbb{G}_m}$ ([Lawson-Naumann 12, def. A.1, A.3](#LawsonNaumann12)). This is equivalent to the [[quotient stack]] of the point by the [[group of order 2]]

$$
  \mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2
$$

([Lawson-Naumann 12, prop. A.4](#LawsonNaumann12)). Here the $\mathbb{Z}_2$-action is the inversion involution on [[abelian groups]].

Using the [[Goerss-Hopkins-Miller theorem]] this stack carries an [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{top}$ ([Lawson-Naumann 12, theorem A.5](#LawsonNaumann12)); and by the above equivalence this is a single [[E-∞ ring]] equipped with a $\mathbb{Z}_2$-[[∞-action]]. This is [[KU]] with its involution induced by [[complex conjugation]], hence essentially is $KR$.

Accordingly, the [[global sections]] of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-[[homotopy fixed points]] of this action, hence is $KO$. This is further amplified in ([Mathew 13, section 3](#Mathew13)) and ([Mathew, section 2](#Mathew)).

As suggested there and by the main ([Lawson-Naumann 12, theorem 1.2](#LawsonNaumann12)) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of [[tmf]] built as the global sections of the similarly derived [[moduli stack of elliptic curves]].

## Related concepts

* [[real-oriented cohomology theory]]

  * [[MR-theory]]

  * [[BPR-theory]]

* [[real algebraic K-theory]]

* [[orientifold]]

* [[KO-dimension]]

[[!include chromatic tower examples - table]]

[[!include string theory and cohomology theory -- table]]

[[!include moduli stack of curves -- table]]

## References

KR theory was introduced in:

* {#Atiyah66} [[Michael Atiyah]], *K-theory and reality*, The Quarterly Journal of Mathematics. Oxford. Second Series **17** 1 (1966)  367-386 [[doi:10.1093/qmath/17.1.367](https://doi.org/10.1093/qmath/17.1.367), [[AtiyahKReal.pdf:file]], ISSN:0033-5606]

The version of $KSC$-theory was introduced in

* {#Anderson64} D. W. Anderson, _The real K-theory of classifying spaces_ Proc. Nat. Acad. Sci. U. S. A., 51(4):634–636, 1964.

Early discussion in the broader context of [[real-oriented cohomology theory]] and [[Real cobordism cohomology]]:

* [[Shôrô Araki]], *$\tau$-Cohomology Theories*, Japanese Journal of Mathematics **4** 2 (1978) [[doi:10.4099/math1924.4.363](https://doi.org/10.4099/math1924.4.363)]

The dual concept of KR-homology:

* [[Gennady Kasparov]], _The operator K-functor and extensions of $C^\ast$-algebras,_ Izv. Akad. Nauk. SSSR Ser. Mat. 44, 571-636 (1980)

Discussion for [[algebraic varieties]]:

* [[Max Karoubi]], [[Charles Weibel]], *Algebraic and Real K-theory of Algebraic varieties*, Topology **42** (2003) 715-742 [[arXiv:math/0509412](https://arxiv.org/abs/math/0509412), doi:10.1016/S0040-9383(02)00069-1]

Discussion in [[equivariant homotopy theory]]:

* Paolo Masulli, _Equivariant homotopy: $KR$-theory_, Master thesis (2011) ([pdf](http://www.math.ku.dk/~jg/students/masulli.msthesis.2011.pdf), [[MasulliKRTheory2011.pdf:file]])

On [[equivariant K-theory]] and [[KR-theory]] for [[finite group|finite]] [[equivariance groups]]:

* {#BrunerGreenlees10} [[Robert Bruner]], [[John Greenlees]], *Connective Real K-Theory of Finite Groups*, Mathematical Surveys and Monographs **169** AMS 2010 ([ISBN:978-0-8218-5189-0](https://bookstore.ams.org/surv-169))

Computations over [[compact Lie groups]] in the context of [[twisted ad-equivariant K-theory]]:

* [[Chi-Kwong Fok]], *The Real K-theory of compact Lie groups*, SIGMA 10 (2014), 022, ([arXiv:1308.3871](https://arxiv.org/abs/1308.3871), [doi:10.3842/SIGMA.2014.022](https://doi.org/10.3842/SIGMA.2014.022))

* [[Chi-Kwong Fok]], *Equivariant twisted Real K-theory of compact Lie groups*,   Journal of Geometry and Physics 124 (2018) 325-349 ([arXiv:1503.00957](https://arxiv.org/abs/1503.00957), [doi:10.1016/j.geomphys.2017.11.013](https://doi.org/10.1016/j.geomphys.2017.11.013))

Discussion in the general context of [[real oriented cohomology theory]] is in

* {#Kriz01} [[Po Hu]], [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

Further discussion includes

* [[Johan Dupont]], _Symplectic bundles and $KR$-theory_ (1967) ([pdf](http://www.mscand.dk/article.php?id=1908))

Reviews include

* Wikipedia, _[KR-theory](http://en.wikipedia.org/wiki/KR-theory)_

Relation to [[topological Hermitian K-theory]] via the [[hyperbolic functor]]:

* {#Crabb80} [[Michael C. Crabb]], §9 of: *$\mathbb{Z}/2$-Homotopy Theory*, Lond. Math. Soc. Lecture Notes **44**, Cambridge University Press (1980) [[ark:/13960/t3jx7bg4w](https://archive.org/details/zz2homotopytheor0000crab), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabb.pdf)]

* [[Po Hu]], [[Igor Kriz]], Appendix of: *Topological Hermitian cobordism*, Journal of Homotopy and Related Structures **11** (2016) 173–197 [[doi:10.1007/s40062-014-0100-9](https://doi.org/10.1007/s40062-014-0100-9)]

* [[Max Karoubi]], [[Charles Weibel]], Thm. 3.5 in: *Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups*, Journal of Pure and Applied Algebra **221** 7 (2017) 1629-1640 [[doi:10.1016/j.jpaa.2016.12.020](https://doi.org/10.1016/j.jpaa.2016.12.020)]

Remarks on homotopy-theoretic KR in the context of [[algebraic K-theory]] are in

* [[Vigleik Angeltveit]], [[Andrew Blumberg]], [[Teena Gerhardt]], [[Michael Hill]], _Algebraic K-theory and equivariant homotopy theory_ (2012) [[pdf](http://www.birs.ca/workshops/2012/12w5116/report12w5116.pdf)]

Discussion of [[equivariant K-theory|equivariant]] and [[twisted K-theory|twisted]] versions of KR-theory

* [[El-kaïoum M. Moutuou]], _Twistings of KR for Real groupoids_ ([arXiv:1110.6836](http://arxiv.org/abs/1110.6836))

* [[El-kaïoum M. Moutuou]], _Graded Brauer groups of a groupoid with involution_, J. Funct. Anal. 266 (2014), no.5 ([arXiv:1202.2057](https://arxiv.org/abs/1202.2057))

* {#Freed12} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]])

* [[Daniel Freed]], [[Gregory Moore]], Section 7 of: _Twisted equivariant matter_,  Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007%2Fs00023-013-0236-x](https://link.springer.com/article/10.1007%2Fs00023-013-0236-x))

* {#Gomi17} [[Kiyonori Gomi]], _Freed-Moore K-theory_ ([arXiv:1705.09134](https://arxiv.org/abs/1705.09134), [spire:1601772](http://inspirehep.net/record/1601772))

This is with motivation from _[[orientifolds]]_, see the references given there for more. A long list of computations of twisted $KR$-classes on tori with applications to [[T-duality]] on [[orientifolds]]/[[O-planes]] is in

* {#Gukov00} [[Sergei Gukov]], _K-Theory, Reality, and Orientifolds_, Commun.Math.Phys. 210 (2000) 621-639 ([arXiv:hep-th/9901042](http://arxiv.org/abs/hep-th/9901042))

* {#DMR13} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For Orientifolds and Twisted KR-theory_ ([arXiv:1306.1779](http://arxiv.org/abs/1306.1779))

  (but see [HMSV 19, p.5 footnote 1](#HMSV19))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](http://arxiv.org/abs/1402.4885))

A general proposal for [[differential K-theory|differential]] [[equivariant K-theory|equivariant]] KR-theory of [[orientifolds]] and [[O-plane charge]]

* {#DistlerFreedMoore09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

Discussion of $KO$ as the $\mathbb{Z}_2$-[[homotopy fixed points]] of $KU$ (or $KR$) is in

* {#Karoubi01} [[Max Karoubi]], _A descent theorem in topological K-theory_, K-theory 24 (2001), no. 2, 109–114 ([arXiv:math/0509396](http://arxiv.org/abs/math/0509396))

* {#Dugger03} [[Daniel Dugger]], _An Atiyah-Hirzebruch spectral sequence for $KR$-theory_, Ktheory
35 (2005), 213–256. ([arXiv:0304099](http://arxiv.org/abs/math/0304099))

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], section 7.3 of _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_ ([[HHRKervaire.pdf:file]])

Discussion of $KU$ with its $\mathbb{Z}_2$-action as the [[E-∞ ring]]-valued [[structure sheaf]] of the moduli stack of tori is due to

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], appendix of _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_ ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

which is reviewed and amplified further in

* {#Mathew13} [[Akhil Mathew]], section 3 of _The homology of $tmf$_ ([arXiv:1305.6100](http://arxiv.org/abs/1305.6100))

* {#Mathew} Akhil Mathew, section 2 of _The homotopy groups of $TMF$_, talk notes ([pdf](http://math.mit.edu/~sglasman/tmfhomotopy.pdf))

Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Adv. Theor. Math. Phys. 23 (2019) 2093-2159, [doi:10.4310/ATMP.2019.v23.n8.a5](https://doi.org/10.4310/ATMP.2019.v23.n8.a5) ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466))

* {#HMSV19} [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_, Commun. Math. Phys. 378, 1843–1873 (2020) ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

[[!redirects KR-homology]]

[[!redirects KR-theory]]

[[!redirects KR]]

[[!redirects Real K-theory]]
[[!redirects real K-theory]]


Kramers-Wannier duality

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#Contents#
* table of contents
{:toc}

## Idea

Kramers-Wannier duality is a [[duality in physics]] of the [[Ising model]] [[2d CFT]].

## Related concepts

* [[Euclidean field theory]]

## References

Discussion by [[defect QFT]] in the context of the [[FRS-formalism]] [[QFT with defects|with defects]]:

* {#FFRS04} [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _Kramers-Wannier duality from conformal defects_, Phys.Rev.Lett. 93 (2004) 070601 ([arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051))

Here in [[FRS-formalism]] a [[2d CFT]] is formalized as a [[boundary field theory]] to a 3d [[TQFT]] based on the [[Turaev-Viro model]], similar to the [[CS-WZW correspondence]].

Another discussion involving just these keywords is in

* {#FreedTeleman18} [[Daniel Freed]], [[Constantin Teleman]], _Topological dualities in the Ising model_ ([arXiv:1806.00008](https://arxiv.org/abs/1806.00008))

Realization in [[lattice field theory]]:

* David Aasen, Roger S. K. Mong, Paul Fendley, *Topological Defects on the Lattice I: The Ising model*, J. Phys. A: Math. Theor. **49** (2016) 354001 [[arXiv:1601.07185](https://arxiv.org/abs/1601.07185), [doi:10.1088/1751-8113/49/35/354001](https://doi.org/10.1088/1751-8113/49/35/354001)]

* David Aasen, Paul Fendley, Roger S. K. Mong, *Topological Defects on the Lattice: Dualities and Degeneracies* [[arXiv:2008.08598](https://arxiv.org/abs/2008.08598), [inspire:1812559](https://inspirehep.net/literature/1812559)]

* Mao Tian Tan, Yifan Wang, Aditi Mitra, *Topological Defects in Floquet Circuits* [[arXiv:2206.06272](https://arxiv.org/abs/2206.06272)]

and simulation on a [[quantum computer]]:

* Sutapa Samanta, Derek S. Wang, Armin Rahmani, Aditi Mitra, *Isolated Majorana mode in a quantum computer from a duality twist* [[arXiv:2308.02387](https://arxiv.org/abs/2308.02387)]


Kriegl and Michor's cartesian closed category of manifolds

# Kriegl and Michor's Cartesian Closed Category of Smooth Manifolds # {: #title}

+-- {: .num_section #sectiona }
=--

## Introduction ## {: .num_section}

In ([Michor 1984a](#MR764972)) and ([Michor 1984b](#MR767683)), Michor gives a description of a category of "smooth manifolds" due to himself and Kriegl which is [[cartesian closed]]. The key to achieving this is to replace charts and atlases by notions based on smooth curves. The resulting objects have a considerable amount of structure built in, but nonetheless replicate ordinary [[smooth manifolds]] in the finite dimensional situation. This is also true in the [[Banach space]] situation; which is not stated in the paper, but follows readily from the finite dimensional situation.

+-- {: .num_section #sectionb }
=--
## Definition ## {: .num_section}

The final data of a "smooth manifold" is the following:

1. Two sets, $M$ and $T M$, and a mapping $\pi _{M} \colon T M \to M$ such that each fibre is a [[locally convex space]] of a certain type.

1. A set $\mathcal{S} (\mathbb{R},M)$ of curves in $M$, closed under $C^\infty$--reparametrisations and containing all constants.

1. For each $t \in \mathbb{R} $, a mapping $\delta _{t} \colon \mathcal{S} (\mathbb{R},M) \to T M$ such that:

   1. $\pi _{M} \circ \delta _{t} = \ev_{t}$,

   1. $\delta _{t} (c \circ f) = f'(t) \delta _{f(t)} c$,

   1. $c$ is constant if for all $t$ then $\delta _{t} c = 0$.

1. A mapping $\Pt^{T M} = \Pt\colon \mathcal{S} (\mathbb{R},M) \times \mathbb{R} \to \mathcal{L} (T M, T M)$ such that:

   1. $\Pt(c,t) \colon T_{c(0)} M \to T_{c(t)} M$ is linear and continuous,

   1. $\Pt(c,0) = \Id$,

   1. $\Pt(c,f(t)) = Pt(c \circ f,t) \Pt(c,f(0))$.

1. $t \mapsto \Pt(c,t)^{-1} (\delta _{t} c)$ is a $C^\infty$--curve in the locally convex space $T_{c(0)} M$.

1. A mapping $\Geo^{M} = \Geo\colon T M \to \mathcal{S} (\mathbb{R},M)$ such that:

   1. $\Geo(t \cdot v)(s) = \Geo(v)(t s)$,

   1. $\delta _{t}( \Geov) = \Pt(\Geo(v),t) \cdot v$,

   1. $\Geo(\delta _{t} \Geo(v))(s) = \Geo(v)(t + s)$.

1. $\Pt\colon \mathcal{S} (\mathbb{R},M) \times \mathbb{R} \to L(T M, T M)$ is smooth.

1. $\Geo\colon T M \to \mathcal{S} (\mathbb{R},M)$ is smooth.

There is a notion of a *smooth map* between such objects (and in which the last two conditions must be interpreted). The resulting category is cartesian closed.

+-- {: .num_section #sectionc }
=--
## In Details ## {: .num_section}

The actual definition is built up in stages. The first definition given is that of a *pre-manifold*. This consists of the first six conditions in the above, to wit:

+-- {: .num_defn #defna .thdefn }
###### Definition ######
A *pre-manifold* consists of the following data.

1. Two sets, $M$ and $T M$, and a mapping $\pi _{M} \colon T M \to M$ such that each fibre is a [[locally convex space]] of a certain type.

1. A set $\mathcal{S} (\mathbb{R},M)$ of curves in $M$, closed under $C^{\infty } $--reparametrisations and containing all constants.

1. For each $t \in \mathbb{R} $, a mapping $\delta _{t} \colon \mathcal{S} (\mathbb{R},M) \to T M$ such that:

   1. $\pi _{M} \circ \delta _{t} = \ev_{t}$,

   1. $\delta _{t} (c \circ f) = f'(t) \delta _{f(t)} c$,

   1. $c$ is constant if for all $t$ then $\delta _{t} c = 0$.

1. A mapping $\Pt^{T M} = \Pt\colon \mathcal{S} (\mathbb{R},M) \times \mathbb{R} \to \mathcal{L} (T M, T M)$ such that:

   1. $\Pt(c,t) \colon T_{c(0)} M \to T_{c(t)} M$ is linear and continuous,

   1. $\Pt(c,0) = \Id$,

   1. $\Pt(c,f(t)) = Pt(c \circ f,t) \Pt(c,f(0))$.

1. $t \mapsto \Pt(c,t)^{-1} (\delta _{t} c)$ is a $C^{\infty } $--curve in the locally convex space $T_{c(0)} M$.

1. A mapping $\Geo^{M} = \Geo\colon T M \to \mathcal{S} (\mathbb{R},M)$ such that:

   1. $\Geo(t \cdot v)(s) = \Geo(v)(t s)$,

   1. $\delta _{t}( \Geov) = \Pt(\Geo(v),t) \cdot v$,

   1. $\Geo(\delta _{t} \Geo(v))(s) = \Geo(v)(t + s)$.
=--

There is an auxiliary concept of a *pre-vector bundle* over a pre-manifold which is also introduced. A pre-manifold defines a pre-vector bundle in a natural way, modelling the fact that the tangent bundle of an ordinary manifold is a vector bundle. The initial reason for introducing pre-vector bundles is that it can be shown that the total space of a pre-vector bundle is again a pre-manifold, and thus if $M$ is a pre-manifold then we can define an associated pre-manifold $T M$ and so on.

The next important concept is that of a *smooth map*, and this leads to a category of pre-manifolds. In fact, two definitions of morphisms between pre-manifolds are given in the two papers. The main one is that of a smooth map. The second, called $S^{1}$ in the papers, is a truncated version.

+-- {: .num_defn #defnb .thdefn }
###### Definition ######
Let $M$ and $N$ be pre-manifolds. A set map $f \colon M \to N$ is smooth if there is a sequence of maps $T^{n} f \colon T^{n} M \to T^{n} N$ with the property that for each $n$, $(T^{n} f)_{*} \colon \mathcal{S} (\mathbb{R},M) \to \mathcal{S} (\mathbb{R},N)$ makes sense and satisfies $(T^{n} f)_{*} \delta _{0} = \delta _{0} (T^{n-1} f)_{*}$.
=--

An $S^{1}$ map is simply a map $f \colon M \to N$ for which $f_{*} \colon \mathcal{S} (\mathbb{R},M) \to \mathcal{S} (\mathbb{R},N)$ is defined.

+-- {: .num_section #sectiond }
=--
## Relationship to Other Theories ## {: .num_section}

Underneath the structure of a pre-manifold is a notion of a [[generalised smooth space|generalized smooth space]] which fits in with the scheme defined in ([Stacey 2011](#MR2805746)). It is formed by taking the category of test spaces to be the one-object category associated to the monoid $C^{\infty } (\mathbb{R},\mathbb{R})$, the underlying category to be $\Set$, the input forcing condition is the input terminal condition (all constant maps are smooth), and the output forcing condition is saturation (whence output functions can be safely ignored).

+-- {: .num_defn #defnc .thdefn }
###### Definition ######
Let $\mathcal{KM} $ be the category of generalised smooth spaces so described.
=--

The input forcing condition is extremely weak. This means that it sits far to the left in the diagram at [[generalised smooth spaces|generalized smooth spaces]]. However, the fact that the category of test objects is extremely small means that there is less potential for variation than with those categories where the test category is somewhat larger.

In the literature, Chen's first (1973) definition comes closest to this in terms of forcing condition but the category of [[Frölicher spaces]] is closest in terms of underlying category and test category.

There is an inclusion functor $\mathcal{Fr} \to \mathcal{KM} $ and this has a left adjoint which is found by saturating the smooth curves with respect to the smooth functions to $\mathbb{R} $.

For [[Chen spaces]] and [[diffeological spaces]], the story is similar. Each category has a functor to $\mathcal{KM} $ but it is no longer an inclusion (for example, $\mathbb{R} ^{2}$ with the standard diffeology and with the wire diffeology give the same object in $\mathcal{KM} $). These functors have left adjoints where a plot is smooth if it locally factors through one of the original smooth curves. Note that this means that for $\mathbb{R} ^{2}$ we recover the wire diffeology and not the standard one.

## References ##

* {: #MR764972 } [[Peter Michor]], (1984). *A convenient setting for differential geometry and global analysis*. _Cahiers Topologie Géom. Différentielle_, _25_, 63–109.

* {: #MR767683 } [[Peter Michor]], (1984). *A convenient setting for differential geometry and global analysis. II*. _Cahiers Topologie Géom. Différentielle_, _25_, 113–178.

* [[Andreas Kriegl]], [[Peter Michor]], [[The Convenient Setting of Global Analysis]], Mathematical Surveys and Monographs, 53 AMS (1997) [pdf](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)

* {: #MR2805746 } [[Andrew Stacey]] (2011). Comparative smootheology. _Theory Appl. Categ._, _25_, No. 4, 64–117.


Kripke frame

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

> This entry is about [[models]] of [[modal logic]]; for *[[frames]]* in the sense of [[geometric logic]] see instead there.

***

# Contents
* table of contents
{:toc}

## Idea

In [[formal logic]], *Kripke frames*, after [Kripke (1959)](#Kripke59), [(1963)](#Kripke63), serve as the [[context|contextual]] substrate of [[set theory|set-theoretic]] [[semantics]]/[[models]] for [[modal logics]] (called *[[geometric model for modal logics|geometric models]]* as opposed to *[[algebraic model for modal logics|algebraic models]]*).

In itself, a Kripke frame is just

* an [[inhabited set|inhabited]] [[set]] $W$

  often called the set of *[[possible worlds]]* (or the set of *nodes* or *states* or *points*)

* equipped with an [[indexed set]] of binary [[relations]] $R_i : W \times W \to Prop$, one for each [[modal operator]] $\Box_i$ in the given [[modal logic]]

  called the *accessibility* or *transition* relations between worlds, or similar, where $R_i (w,w')$ is often interpreted as (in the epistemic case): "To agent $i$ in world $w$, world $w'$ appears (just as well) possible."

But note that, as there are many flavors of modal logic (epistemic, aletheic, deontic, temporal, dynamic, et cetera), there are many different interpretations of the accessibility relation in the corresponding Kripke frame.

Hence a Kripke frame is a [[concept with an attitude]], namely the attitude to serve as a model for a [[modal logic]] -- the idea is to interpret:

1. the [[elements]] of $W$ as the "[[possible worlds]]" about which the [[propositions]] in the modal logic make statements,

2. the [[relation]] $R_i(w,w')$ as asserting that it makes sense to reason about world $w'$ at world $w$.

Namely a [[geometric model for modal logics|model]] of the modal logic based on such a Kripke frame will interpret

* each [[proposition]] $P$ of the logic as a $W$-[[dependent type|dependent]] proposition $P \colon W \to Prop$,

* each [[modal operator|modal]] proposition $\Box_i P$ as the $W$-[[dependent type|dependent]] proposition which at $w \in W$ asserts that "$P$ holds at all those $w'$ that are in $i$th relation to $W$", hence "$P$ holds in all worlds $w'$ that $i$ deems to be just as possible as $w$"; formally (eg. [Blackburn & van Benthem (2007) p. 10](#BlackburnVanBenthem07)):

> {#SubtypeNotation} (Here we are writing --- following notation for *[[subtypes]]* --- $P \,\colon\,W \to Prop$ for a $W$-[[dependent type|dependent]] [[proposition]], equivalently a proposition about the elements of $W$, equivalently the [[subset]] of $W$ where the proposition holds.)

## Definition

A (uni-modal) Kripke frame is a set $W$ and a binary $R$ relation on $W$. A multimodal Kripke frame for modal operators $\{\Box_i\}_{i \in I}$ is a set $W$ together with a binary relations $R_i$ for each $i \in I$. Equivalently, a uni-modal Kripke frame is a set $W$ and a [[coalgebra for an endofunctor|coalgebra]] for the powerset endofunctor, $f_R: W \to \mathscr{P}(W)$. A multimodal Kripke frame is, equivalently, a set $X$ and a coalgebra for the endofunctor $\mathscr{P}(I \times -)$.

A morphism of Kripke frames, called a bounded morphism or p-morphism (short for pseudo-epimorphism) is a homomorphism of the coalgebras.

We call the category of unimodal Kripke frames $\mathbf{Kr}$. $\mathbf{Kr}$ is dually equivalent to the category of complete, atomic, *completely additive* [[algebraic model for modal logics|boolean algebras with operators]]. This is the analogue of the result that $\mathbf{Set}$ is dually equivalent to the category of complete, atomic boolean algebras.

A *Kripke model* for a propositional language $L$ is a Kripke frame $\langle{W, f_R\rangle}$ together with a *valuation* $V: L \to \mathbb{2}^W$ which preserves the appropriate logical structure. The non-modal connectives are handled in the usual way with their set-theoretic analogues (intersection for $\land$, union for $\lor$, complement for negation, et cetera). For the modal operator $\Box$, the truth clause is $V(\Box p)(w) = 1$ if and only if $f_R(w) \subseteq V(p)$.

We often consider particular classes of Kripke frames, identified by common features of the accessibility relations: e.g. reflexivity, transitivity, symmetry Euclidean, density, et cetera. These correspond to the validity of certain logical principles, and so, different modal logics. However, Kripke frames are not sufficient to classify all (normal) modal logics as detailed in [Holliday and Litak, 2019](#HollidayLitak19). Although every class of Kripke frames defines a normal modal logic for which the class is complete, some normal modal logics are not complete for any class of Kripke frames, a phenomenon known as "Kripke incompleteness." By duality, this means there are normal modal logics which are not complete with respect to any class of complete, atomic, completely additive boolean algebras with operators .

Below we detail some classes of Kripke frames.

## Some classes of Kripke frames

**The transparent case of S5 logic.** Specifically, for the case of [[S5 modal logic]] with a single [[comonad|comonadic]] [[modal operator]] $\Box$, a Kripke frame (originally considered in [Kripke 1963](#Kripke63)) is an

* [[inhabited set]]$\;$ $W$

equipped with a single

* [[equivalence relation]]$\;$ $R$

  (hence a relation which is [[reflexive relation|reflexive]], [[transitive relation|transitive]] and [[symmetric relation|symmetric]])

which means that $W$ decomposes as the [[disjoint union]] of its [[equivalence classes]], these being the [[fibers]] of the [[quotient]] [[coprojection]]

\[
  \label{QuotientCoprojectionInCaseOfS5}
  W \xrightarrow{\;\; p_W \;\;} W/R
  \,.
\]

{#NecessityLogic} **The simple case necessity logic.** In the further special case there is just a single such class (hence that "all possible worlds are related", originally considered by [Kripke 1959](#Kripke59)) this becomes

\[
  \label{QuotientCoprojectionInCaseOfGlobalS5}
  W \xrightarrow{\;\; p_W \;\;} \ast
\]

so that in [[geometric model for modal logics|models]] based on such a frame the interpretation of the [[modal operator]] $\Box$ is that of *[[necessity]]*: $\Box P$ now asserts, independently of the world that it is evaluated on, that $P$ holds at *all* $w \in W$, hence that $P$ holds *necessarily* in the sense of: no matter which world.

The general case of Kripke frames is really just this basic idea with some variations added. For instance the general case (eq:QuotientCoprojectionInCaseOfS5) of S5 modal logic is just a $W/R$-indexed [[disjoint union]] of this basic situation.

**Perspective of dependent type theory.**
As discussed in detail at *[Possible worlds via Dependent types](necessity+and+possibility#InFirstOrderLogicAndTypeTheory)*,
the Kripke model for the modal operator $\Box$ in the case (eq:QuotientCoprojectionInCaseOfGlobalS5) of [[S5 modal logic]] with a global accessibility relation is just the [[base change]] [[comonad]] along the projection (eq:QuotientCoprojectionInCaseOfGlobalS5), and the same argument immediately applies to the general S5-case (eq:QuotientCoprojectionInCaseOfS5):

\[
  \label{BaseChange}
\]
\begin{tikzcd}
    W \ar[rr, "{ p_{{}_W} }"] && W/R
    \\
    \mathrm{Prop}_W
    \ar[out=180-42, in=180+42,
      <-,
      looseness=4.5,
      shorten=0pt,
      shift right=3pt,
      "\scalebox{1.1}{${
          \hspace{1pt}
          \mathclap{
            {
              \Box
            }
          }
          \hspace{4pt}
        }$}"{
          pos=.5,
          swap
        }
    ]
    \ar[
      from=rr,
      shift right=7pt,
      "{ (p_{{}_W})^\ast }"{swap}
    ]
    \ar[
      rr,
      shift right=7pt,
      "{ (p_{{}_W})_\ast }"{swap}
    ]
    \ar[
      rr,
      phantom,
      "{ \scalebox{.7}{$\bot$} }"
    ]
    &&
    \mathrm{Prop}_{W/R}
  \end{tikzcd}

**The case of S4 logic.**
If the [[relation]] $R$ in a Kripke frame is ([[reflexive relation|reflexive]] and) [[transitive relation|transitive]] but not necessarily [[symmetric relation|symmetric]], then it still serves to provide models of [[S4 modal logic]] (cf. [Kripke 1959, §2](#Kripke59)).

In this case the relation no longer defines a [[groupoid]] ("[[setoid]]") as it does for the S5-case, but it defines a [[(0,1)-category]] with [[objects]] the elements of $W$.

As such, to retain the type-theoretic perspective (eq:BaseChange) in this case will require some form of *[[directed type theory]]*.

\linebreak

**General case**

[[!include flavors of modal logics and relations -- table]]

## Related entries

* [[possible worlds]]

* [[necessity and possibility]]

* [[epistemic modal logic]]

## References

The original articles:

* {#Kripke59} [[Saul A. Kripke]], *A Completeness Theorem in Modal Logic*, The Journal of Symbolic Logic **24** 1 (1959) 1-14 [[doi:10.2307/2964568](https://doi.org/10.2307/2964568), [jstor:2964568](https://www.jstor.org/stable/2964568), [pdf](https://www.filosoficas.unam.mx/~morado/Cursos/17Modal/Kripke1959.pdf)]

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

* [[Saul A. Kripke]], *Semantical Considerations on Modal Logic*, Acta Philosophical Fennica **16** (1963) 83-94 [[pdf](http://saulkripkecenter.org/wp-content/uploads/2019/03/Semantical-Considerations-on-Modal-Logic-PUBLIC.pdf)]

* {#Kripke65} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi*, in *The Theory of Models* (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [[doi:10.1016/B978-0-7204-2233-7.50026-5](https://doi.org/10.1016/B978-0-7204-2233-7.50026-5)]

Modern exposition

* {#BlackburnVanBenthem07} [[Patrick Blackburn]], [[Johan van Benthem]], *Modal logic: a semantic perspective*, Ch 1 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) [doi:10.1016/S1570-2464(07)80004-8, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)]

Textbook accounts;

* {#BlackburnDeRijkeVenema01} [[Patrick Blackburn]], [[Maarten de Rijke]], [[Yde Venema]], §1.3 in: *Modal Logic*, Cambridge Tracts in Theoretical Computer Science **53**, Cambridge University Press (2001) [[doi:10.1017/CBO9781107050884](https://doi.org/10.1017/CBO9781107050884)]

* [[Valentin Goranko]], [[Martin Otto]], §1.2 in: *Model Theory of Modal Logic*, in Section 5 in: *Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 249-329 [[pdf](https://www2.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf),  [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

* {#GasquetHerzigSaidSchwarzentruber13} Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber,  *Kripke's Worlds: An Introduction to Modal Logics via Tableaux*, Studies in Universal Logic, Springer (2014) [[doi:10.1007/978-3-7643-8504-0](https://doi.org/10.1007/978-3-7643-8504-0), [ISBN:978-3764385033]()]

Generalization of Kripke frames:

* [[Dov Samet]], §6 in: *S5 knowledge without partitions*, Synthese **172** (2010) 145–155 [[doi:10.1007/s11229-009-9469-0](https://doi.org/10.1007/s11229-009-9469-0)]

On Kripke-frame incompleteness:

* {#HollidayLitak19} Holliday, W., Litak, T. (2019). Complete Additivity and Modal Incompleteness. The Review of Symbolic Logic, 12(3), 487-535. [[doi:10.1017/S1755020317000259](https://doi:10.1017/S1755020317000259)]

[[!redirects Kripke frame]]
[[!redirects Kripke frames]]
[[!redirects frame (Kripke)]]
[[!redirects frames (Kripke)]]
[[!redirects frame (modal logic)]]
[[!redirects frames (modal logic)]]


Kripke's schema

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

#Contents#
* table of contents
{:toc}

## Idea

Kripke's schema is a principle first found implicitly in the [[intuitionistic mathematics|school of Brouwer]], going beyond base [[constructive logic]]. It ties [[propositions]] (or, in some formulations, membership of [[subsets]] of the [[natural numbers|naturals]]) to the values of infinite [[sequences]] in $\{0, 1\}$.

## Formulation
__KS__:
For every statement $\phi$ holds: There exists a binary sequence $a$, such that $\phi$ holds iff there is some $n$ such that $a_n=1$.

In the formalization of evolving [[choice sequences]] by [[Saul Kripke]], this may be motivated by the idea that, roughly speaking, for any proposition there would be a sequence capturing its proof search, so that, thus, if and only if the proposition is provable, that search eventually terminates.

## Weaker variants
KS implies, for example, the following Weak Principle of Finite Possibility, which can be stated just in terms of sequences:

__WPFP__:
For every binary sequence $b$, there exists a binary sequence $a$, such that $b$ is the zero sequence iff $a$ is not the zero sequence.

This is to say, to any sequence there is a certain sort of value-inverted sequence. Classically, of course, the value $1 - {\mathrm{max}}({\mathrm{range}}(b))$ exactly determines whether $b$ is the zero sequence and thus the constant sequence $a$ with that value does the job. But constructively one generally cannot inspect all, infinitely many return values of a sequence $b$.

## Implications
Binary sequences are nicely behaved objects and, in turn, KS can be shown to be equivalent to the claim that every inhabited subset of the naturals is countable.

The principles also has implications for [[metric spaces]].

It can be shown that (WPFP + [[Markov's principle | MP]]) iff [[principle of omniscience | LPO]]. Those principles are thus not provable in Russian constructivism (adopting MP but not LPO). Indeed, formulations of KS contradict the so called Constructive Church's Thesis.

## Related concepts

* [[intuitionism]]

* [[Markov's principle]], MP

* [[principle of omniscience]], LPO

## References

* D. Van Dalen, _The Use of Kripke's Schema as a Reduction Principle_, The Journal of Symbolic Logic Vol. 42, No. 2 (1977) ([euclid:jsl/1183739949](https://projecteuclid.org/euclid.jsl/1183739949))

* Mark van Atten, _The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs_, Indagationes Mathematicae
Volume 29, Issue 6, 2018
([arXiv:1805.00404](https://arxiv.org/abs/1805.00404), [doi:10.1016/j.indag.2018.06.005](https://doi.org/10.1016/j.indag.2018.06.005))

[[!redirects Kripke schema]]