noncommutative Hodge structure

#Contents#
* table of contents
{:toc}

## Idea

Noncommutative Hodge theory is an extension of the classical [[Hodge theory]]. The basic notion is a **noncommutative Hodge structure**, the generalization of _[[Hodge structure]]_, which is formulated in the language of meromorphic connections.

Unlike classical Hodge theory for complex varieties, noncommutative Hodge structures can be attached to a wider class of [[noncommutative algebraic geometry|noncommutative spaces]]. In the framework of Katzarkov-Kontsevich-Pantev, noncommutative spaces are represented by [[dg-categories]], or more generally, A-infinity categories. In particular, they are interested in the dg-categories which arise in [[homological mirror symmetry]]: for example, [[Fukaya category|Fukaya categories]], (dg enhanced) [[derived categories]] of (quasi-)coherent sheaves, [[matrix factorization]] categories, [[Fukaya-Seidel categories]].

The noncommutative analogue of [[Dolbeault cohomology]] is the [[Hochschild homology]] of the category. The analogue of [[de Rham cohomology]] is the periodic [[cyclic cohomology|cyclic homology]] of the category. The analogue of the Hodge-de Rham spectral sequence is the Hochschild-cyclic spectral sequence. There is work of Weibel which makes this analogy precise.

There is a conjecture of Kontsevich that the Hochschild-cyclic spectral sequence degenerates for smooth and proper noncommutative spaces. (This is the analogue of the Hodge-de Rham degeneration for smooth and proper varieties.) Kontsevich's conjecture has been proven in some cases by [[Dmitri Kaledin]], who adapts Deligne-Illusie's proof of Hodge-de Rham degeneration (using reduction mod p) to the noncommutative setting. Kontsevich's conjecture is known as the "degeneration conjecture".

## References

Noncommutative Hodge theory is being developed in

* [[Ludmil Katzarkov]], [[Maxim Kontsevich]], [[Tony Pantev]], _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)

* [[D. Kaledin]], _Cartier isomorphism and Hodge theory in the non-commutative case_,  Arithmetic geometry,  537--562, Clay Math. Proc. __8__, Amer. Math. Soc. 2009, [arxiv/0708.1574](http://arxiv.org/abs/0708.1574)

* D. Kaledin, _Tokyo lectures "Homological methods in non-commutative geometry"_, [pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex)

* Claus Hertling, Christian Sevenheck, _Twistor structures, $tt^*$-geometry and singularity theory_, [arxiv/0807.2199](http://arxiv.org/abs/0807.2199)

* C. Hertling, C. Sabbah, _Examples of non-commutative Hodge structure_ (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures),
J. Inst. Math. Jussieu __10__, Spec. Issue 3: in honour of L. Boutet de Monvel & P. Schapira (2011) 635 - 674 [arxiv/0912.2754](http://arxiv.org/abs/0912.2754) [doi](https://doi.org/10.1017/S147474801100003X)

* C. Sabbah, _Non-commutative Hodge structures_,  [pdf](http://www.math.polytechnique.fr/~sabbah/sabbah_grenoble10.pdf)

* [[Dmytro Shklyarov]], _Non-commutative Hodge structures: Towards matching categorical and geometric examples_, Trans. Amer. Math. Soc. 366 (2014), 2923-2974 [arxiv/1107.3156](http://arxiv.org/abs/1107.3156) [doi](https://doi.org/10.1090/S0002-9947-2014-05913-8)

[[!redirects noncommutative Hodge theory]]
[[!redirects Noncommutative Hodge theory]]


noncommutative integrable system

Noncommutative integrable systems pertain to analogues of [[integrable systems]] in which the coordinates and momenta are formal noncommuting  variables.

* [[Pavel Etingof]], [[Israel Gelfand]], Vladimir Retakh, _Nonabelian integrable systems, quasideterminants, and Marchenko lemma_,  Math. Res. Lett. __5__  (1998), no. 1-2, 1--12, [MR99h:58083](http://www.ams.org/mathscinet-getitem?mr=99h:58083)
* [[Vladimir Retakh]], [[Vladimir Rubtsov]], _Noncommutative Toda chains, Hankel quasideterminants and [[Painleve transcendent|Painlevé]] II equation_, [arxiv/1007.4168](http://arxiv.org/abs/1007.4168)
* Semeon Arthamonov, _Noncommutative inverse scattering method for the Kontsevich system_, [arxiv/1410.5493](http://arxiv.org/abs/1410.5493)
* Philippe Di Francesco, Rinat Kedem, _Noncommutative integrability, paths and quasi-determinants_, [http://arxiv.org/abs/1006.4774](http://arxiv.org/abs/1006.4774)
* Philippe Di Francesco, _Discrete integrable systems, positivity, and continued fraction rearrangements_ [arxiv.org/abs/1009.1911](http://arxiv.org/abs/1009.1911)

[[!redirects noncommutative integrable systems]]


noncommutative localization

The [[noncommutative localization]] is a common term for the localization in noncommutative algebra, that is the localizations of noncommutative algebraic structures, most often noncommutative rings and associative algebras, as well as localization functors on various categories of modules (and bimodules) over possibly noncommutative algebras. Most often this term is used for several kinds of [[localization]] which occur in noncommutative [[ring theory]] and in particular the (left or right) [[Ore localization]], [[Gabriel localization]] and Cohn [[universal localization]] of rings and of their associated [[Mod|categories of modules]]. Many notion here have straightforward extension to general [[Grothendieck categories]].

Gabriel's locazalization is usually stated in terms of [[Gabriel filter]]s, but it fits into a more general abelian [[localization of abelian categories]] on [[thick subcategories]].

* [[Pierre Gabriel]], [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448, [numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0)

An alternative, but equivalent, approach to Gabriel localization, via [[kernel functor]]s is introduced in

* O. Goldman, _Rings and modules of quotients_, J. Algebra __13__, 1969 10--47, [MR245608](http://www.ams.org/mathscinet-getitem?mr=245608), [doi](http://dx.doi.org/10.1016/0021-8693%2869%2990004-0)

The language of [[torsion theory|torsion theories]] was originally developed also in this context, but in fact it can pertain to a more general situation than to the categories of modules over rings. Hereditary torsion theories on the categories of 1-sided modules correspond to Gabriel lcoalization and the localization functors are flat. Cohn [[universal localization]]s of rings corresponds at the level of categories of modules to certain nonhereditary torsion theories.

See also [[torsion theory]], [[localization]], [[Q-category]], [[thick subcategory]], [[topologizing subcategory]], [[Gabriel filter]]

* Bo Stenström, _Rings and modules of quotients_. Lecture Notes in Mathematics __237__, Springer-Verlag 1971. vii+136 pp. [MR0325663](http://www.ams.org/mathscinet-getitem?mr=325663)
* Nicolae Popescu, _An introduction to Abelian categories with applications to rings and modules_, Acad. Press 1973
* A. W. Goldie, _Torsion-free modules and rings_, J. Algebra __1__ 1964, 268–287 [MR164991](http://www.ams.org/mathscinet-getitem?mr=164991) doi
* [[Zoran Škoda]], _Noncommutative localization in noncommutative geometry_, London Math. Society Lecture Note Series __330__, ed. [[A. Ranicki]]; pp. 220–313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276)

category: algebraic geometry


noncommutative measure theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

According to the [[duality between geometry and algebra]],
various ([[opposite category|opposite]]) [[categories]] of [[commutative algebras]] and similar objects
are [[equivalence of categories|equivalent]] to certain corresponding categories of [[spaces]].

For example, [[Gelfand duality]] establishes
a [[opposite category|contravariant]] [[equivalence of categories]]
between the category of [[commutative algebra|commutative]] [[associative unital algebra|unital]] [[C*-algebras]]
and the category of [[compact Hausdorff topological spaces]].

Thus, one can drop the commutativity condition and argue
that studying arbitrary [[C*-algebras]] amounts
to studying [[noncommutative topology|noncommutative general topology]],
a part of [[noncommutative geometry]].

Analogously, in [[measure theory]] one works with [[commutative von Neumann algebras]] instead, which are equivalent to several other categories:
[[localizable measurable space|compact strictly localizable]] [[enhanced measurable spaces]],
[[hyperstonean topological spaces]] and [[open maps]],
[[hyperstonean locales]] and [[open maps]],
[[measurable locales]] (and arbitrary morphisms of [[locales]]).

Thus, dropping the commutativity condition
results in the [[category of von Neumann algebras]]
(with ultraweakly continuous $*$-homomorphisms),
and we can view this [[category]]
as the [[category]] of __noncommutative measurable spaces__.

Thus, noncommutative measure theory is more-or-less the study
of [[von Neumann algebras]], more precisely those aspects of them
that reproduce known phenomena in [[measure theory]] when specialized
to the commutative case.

## Examples

The notion of a [[spatial product]] of (compact strictly localizable enhanced) [[measurable spaces]]
can be expressed algebraically as the [[spatial tensor product]]
of [[von Neumann algebras]].
This is quite different from the categorical [[product]]
of measurable spaces, which corresponds to the Dauns [[tensor product]]
of [[von Neumann algebras]].

As another example, the theory of $L^p$-[[Lebesgue space|spaces]] generalizes
to [[von Neumann algebras]].
This is a work of many authors, including [[von Neumann]],
[[Edward Nelson]], [[Irving Segal]], and [[Uffe Haagerup]].
It plays a crucial role in [[Tomita-Takesaki modular theory]],
which is essentially nothing else than the theory of $L^p$-spaces
where $p$ is a purely [[imaginary number]].
In particular, it allows one to classify [[type III factors]].

The notion of a [[measure]] is replaced by that of a normal [[weight]] on a [[von Neumann algebra]].
A [[trace]] on a [[von Neumann algebra]] is a special type of a [[weight]].

The notion of a [[measurable field of Hilbert spaces]]
can be expressed using two equivalent categories:
the category of [[W*-modules]] over [[von Neumann algebras]],
and the category of [[representations of von Neumann algebras]]
on a [[Hilbert space]].

## Related concepts

* [[quantum probability theory]]

* [[noncommutative topology]], [[noncommutative geometry]]

* [Locales -- Examples -- Measure theory](locale#ExamplesMeasureTheory)

* [[measure theory]], [[measurable space]], [[measurable function]]

## References

* [[Masamichi Takesaki]], _Theory of Operator Algebras I, II, III_, Springer, I: 1979. vii+415 pp. ISBN: 0-387-90391-7; II: 2003. xxii+518 pp. ISBN: 3-540-42914-X; III: 2003. xxii+548 pp. ISBN: 3-540-42913-1.

* {#WeaverKuperberg11} [[Nik Weaver]], [[Greg Kuperberg]], *A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations*, Memoirs of the AMS **215** 1010 (2011) [[ams:memo-215-1010](https://bookstore.ams.org/memo-215-1010)]

[[!redirects noncommutative measure theory]]
[[!redirects non-commutative measure theory]]

[[!redirects noncommutative measurable space]]
[[!redirects noncommutative measurable spaces]]
[[!redirects non-commutative measurable space]]
[[!redirects non-commutative measurable spaces]]


noncommutative motive

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[motives]] in algebraic geometry can be adapted to derived [[noncommutative geometry]]. The idea and the first version has been developed by [[Maxim Kontsevich]]. There is a remarkable observation that the [[category of Chow motives]] (after localizing at the [[Lefschetz motive]]) can be embedded into the category of Kontsevich's noncommutative motives. More recently this direction has been systematically studied by Cisinski and Tabuada.

A second approach is due to [[Bertrand Toën]], [[Michel Vaquié]], [[Gabriele Vezzosi]].  They construct a [[motivic stable homotopy theory]] for [[noncommutative geometry|noncommutative spaces]] (in the sense of [[Kontsevich]]).

There is another approach by [[Arne Ostvaer]].

In [[noncommutative geometry]] à la [[Alain Connes]], Connes and [[Matilde Marcolli]] have also introduced some motivic ideas. Marcolli also has a  recent collaboration with Tabuada on the algebraic side, see her [[Matilde Marcolli|webpage]].

## Definition
 {#Definition}

### As the universal additive/localizing invariant
 {#AsUniversalAditiveInvariant}

The definition in ([Blumberg-Gepner-Tabuada 10](#BlumbergGepnerTabuada10)) is the following.

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

Write

* (stable) $Cat_\infty^{stab}$ for the [[(∞,1)-category]] of [[stable (∞,1)-categories]];

* (small with linear maps) $Cat_\infty^{ex}$ for the $(\infty,1)$-category of _[[small (infinity,1)-category|small]]_ stable $\infty$-categories with [[finite (infinity,1)-limit]]/colimit-preserving functors between them;

* (idem-complete) $Cat_\infty^{perf} \hookrightarrow Cat_\infty^{ex}$ for the further [[full sub-(infinity,1)-category]] of the small stable $\infty$-categories on the [[idempotent complete (infinity,1)-categories]].

=--

([Blumberg-Gepner-Tabuada 10, def. 2.12 and above def 2.14](#BlumbergGepnerTabuada10))

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

The inclusion from def. \ref{Catperf} is a [[reflective sub-(infinity,1)-category]]

$$
  Cat^{perf} \stackrel{\overset{Idem}{\longleftarrow}}{\hookrightarrow} Cat^{ex}
$$

the reflector $Idem$ being [[idempotent completion]].

=--

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

Passing to [[ind-objects]] yields an [[equivalence of (infinity,1)-categories]]

$$
  Ind \colon
  Cat_\infty^{per}
  \stackrel{\simeq}{\longrightarrow}
  Pr_{St,cg}^{L}
  \hookrightarrow
  Pr_{St}^{L}
$$

with the [[stable (infinity,1)-category|stable]] [[compactly generated (∞,1)-categories]].

=--

([[Higher Topos Theory|HTT, 5.5.7]])

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

Say that a morphism in $Cat_\infty^{ex}$ is a [[Morita equivalence]] if it is an $Idem$-equivalence, hence if it becomes an [[equivalence of (∞,1)-categories]] under idempotent completion, prop. \ref{IdempotentCompletion}.

=--

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

Say that a sequence in $Cat_\infty^{ex}$ is (split-)exact if it is an exact sequence (...see section 5...) under idempotent completion, prop. \ref{IdempotentCompletion}

=--

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

A functor $Cat_\infty^{ex} \to \mathcal{D}$ to a stable [[presentable (∞,1)-category]] is called a **localizing invariant** (**additive invariant**) if it

  1. inverts Morita equivalences, def. \ref{MotitaEquivalence};

  1. preserves [[filtered (∞,1)-colimits]];

  1. sends (split-) exact sequences, def. \ref{ExactSequences}, to (split) cofiber sequences (...see section 5...).

=--

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

The $(\infty,1)$-category $Mot_{add}$ or $Mot_{loc}$, respectively, of **noncommutative motives** is the [[universal construction|universal]] localizing/additive invariant, def. \ref{additiveinvariant}

$$
  \mathcal{U}_{loc} \colon Cat_\infty^{ex} \to Mot_{loc}
  \,.
$$

$$
  \mathcal{U}_{add} \colon Cat_\infty^{ex} \to Mot_{add}
  \,.
$$

=--

([Blumberg-Gepner-Tabuada 10, theorem 1.1, section 8](#BlumbergGepnerTabuada10))

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

The localization property here (be additive, invert Morita, preserve split sequences) is of the same form as that which defines the localization of [[C*-algebras]] to [[KK-theory]] in [[noncommutative stable homotopy theory]]. See at _[KK-theory -- Universal characterization](KK-theory#UniversalCharacterization)_. See also ([Blumberg-Gepner-Tabuada 10, paragraph 1.5](#BlumbergGepnerTabuada10)).

=--

## Properties

### Relation to algebraic K-theory
 {#RelationToAlgebraicKTheory}

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

For $\mathcal{A}, \mathcal{B} \in Cat_\infty^{stab}$ with $\mathcal{B}$ smooth and proper, hence a [[compact object in an (infinity,1)-category|compact object]], then the hom-[[spectrum]] in $Mot_{loc}$ between $\mathcal{A}$ and $\mathcal{B}$ is given by the [[non-connective algebraic K-theory]] $\mathbb{K}$ of the tensor product in that there is a [[natural equivalence]]

$$
  Hom_{Mot_{loc}}(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A}))
  \simeq
  \mathbb{K}(\mathcal{B}^{op}\widehat \otimes \mathcal{A})
  \,.
$$

=--

([Blumberg-Gepner-Tabuada 10](#BlumbergGepnerTabuada10), theorem 9.36)

### Relation to correspondences equipped with cocycles
 {#RelationToCorrespondences}

By ([Blumberg-Gepner-Tabuada 10, theorem 9.36](#BlumbergGepnerTabuada10)), the morphisms of noncommutative motives from $\mathcal{A}$ to $\mathcal{B}$ for $\mathcal{B}$ suitably dualizable/compact are given by

$$
  Maps(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A}))
  \simeq
   \mathbb{K}(\mathcal{B}^{op}\otimes\mathcal{A})
  \,,
$$

hence by the [[non-connective algebraic K-theory]] of the [[Deligne tensor product]] of the two categories.

Thinking of these as categories of [[quasicoherent sheaves]] on some spaces (by definition in [[noncommutative algebraic geometry]]), this are $\mathbb{K}$-cocycles on the product [[correspondence]] space.

(...)

### Relation to Chow motives
 {#RelationToChowMotives}

The category of ordinary [[Chow motives]], after factorizing out the action of the [[Tate motive]] essentially sits inside that of noncommutative Chow motives. This is recalled as  ([Tabuada 11, theorem 4.6](#Tabuada11)). For more see ([Tabuada 11 ChowNCG](#Tabuada11Chow)).

This relation is best understood as being exhibited by [[K-motives]], see there.

### Relation to KK-theory

Noncommutative motives receive a universal functor from [[KK-theory]]

$$
  KK \longrightarrow NCC_{dg}
$$

which is given by sending a [[C*-algebra]] to the [[dg-category]] of [[perfect complexes]] over (the [[unitalization]] of) its underlying [[associative algebra]] ([Mahanta 13](#Mahanta13)).

## Applications

* [[Tabuada]] has used noncommutative motives to compute the [[cyclic homology]] of [[twisted projective homogeneous varieties]].  Also, he showed that the noncommutative motive of such a variety is trivial if and only if the [[Brauer class|Brauer classes]] of the associated [[central simple algebras]] are trivial.  See ([Tabuada 13](#Tabuada13Twisted)).

* Bernardara and Tabuada have used noncommutative motives to compute the rational [[Chow groups]] of certain complete [[intersections]] of [[curves]].  See ([Bernardara-Tabuada 13](#BerTab13)).

## Related concepts

* [[bivariant cohomology]], [[bivariant algebraic K-theory]]

* [[motivic symplectic category]]

* [[motive]]

  * [[pure motive]]

    * [[Chow motive]], [[numerical motive]]

  * [[mixed motive]]

* [[motivic cohomology]]

[[!include noncommutative motives - table]]

## References
 {#References}

A survey is in

* [[Goncalo Tabuada]], _A guided tour through the garden of noncommutative motives_, in Guillermo Cortinas, _Topics in Noncommutative Geometry_ Clay Mathematics Proceedings Vol 16, 2012 ([arxiv1108.3787](http://arxiv.org/abs/1108.3787));
 {#Tabuada11}

Discussion of [[Maxim Kontsevich]]'s definition of
noncommutative motives include

* [[Maxim Kontsevich]], _Noncommutative motives_, talk at the conference on Pierre Deligne's 61st birthday (2005) (pdf of part of the talk, notes by [[Zoran Skoda]])

* [[Maxim Kontsevich]], _Geometry and Arithmetic - Non-commutative motives_, talk at  Institute for Advanced Study October 20, 2005 ([video](http://video.ias.edu/Geometry-and-Arithmetic-Kontsevich))

The following article has the treatment of $A_\infty$-categories representing smooth, proper, separated etc. noncommutative varieties, notions which are used in Kontsevich's approach to motives in the above talks.

* [[Ludmil Katzarkov]], [[Maxim Kontsevich]], [[Tony Pantev]], _Hodge theoretic aspects of mirror symmetry_, in [[Ron Donagi]] and [[Katrin Wendland]] (eds.) _From Hodge theory to integrability  and TQFT: $tt^\ast$-geometry_, Proceedings of Symposia in Pure Mathematics vol. 78 (2008), 87-174 ([arXiv:0806.0107](http://arxiv.org/abs/0806.0107))

An abstract characterization of noncommutative motives in [[dg-category]] theory and higher [[algebraic K-theory]] is in

* [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], _Non connective K-theory via universal invariants_. Compositio
Mathematica 147 (2011), 1281–1320 ([arXiv:0903.3717](http://arxiv.org/abs/0903.3717))

* [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], _Symmetric monoidal structure on Non-commutative motives_, ([arxiv/1001.0228](http://arxiv.org/abs/1001.0228))

* [[Gonçalo Tabuada]], _K-theory via universal invariants_, Duke Math. J. 145 (2008), no.1, 121–206.

and a further lift of this to [[(∞,1)-category theory]] is in

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_,  Geometry and Topology 17 (2013) 733–838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))

with discussion of the corresponding [[cyclotomic trace]] in

* [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _Uniqueness of the multiplicative cyclotomic trace_,  Advances in Mathematics ([arXiv:1103.3923](http://arxiv.org/abs/1103.3923))

See also

* [[Goncalo Tabuada]],

  _Bivariant cyclic cohomology and Connes' bilinear pairings in Non-commutative motives_, [arxiv/1005.2336](http://arxiv.org/abs/1005.2336);

  _Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives_, [arxiv/1101.0731](http://arxiv.org/abs/1101.0731);

* {#Tabuada11Chow} [[Goncalo Tabuada]], _Chow motives versus non-commutative motives_ ([arxiv/1103.0200](http://arxiv.org/abs/1103.0200)

* [[Goncalo Tabuada]],  _Galois descent of additive invariants_, [arxiv/1301.1928](http://arxiv.org/abs/1301.1928)

* [[Matilde Marcolli]], [[Goncalo Tabuada]], _Kontsevich's noncommutative numerical motives_, [arxiv/1108.3785](http://arxiv.org/abs/1108.3785); _Noncommutative motives, numerical equivalence, and semi-simplicity_, [arxiv/1105.2950](http://arxiv.org/abs/1105.2950); _Noncommutative numerical motives, Tannakian structures, and motivic Galois groups_, [arxiv/1110.2438](http://arxiv.org/abs/1110.2438)

* [[Ivo Dell'Ambrogio]], [[Gonçalo Tabuada]], _Tensor triangular geometry of non-commutative motives_, [arxiv/1104.2761](http://arxiv.org/abs/1104.2761)

For the approach of [[Bertrand Toën]]-[[Michel Vaquié]]-[[Gabriele Vezzosi]], see

* [[Michel Vaquié]], _A new approach on non commutative motives_, lecture at [JAMI 2011](http://www.mathematics.jhu.edu/new/jami2011/noncommgeo.htm).  [video](http://streams1.nts.jhu.edu/mathematics/jami2011/), [notes](http://streams1.nts.jhu.edu/mathematics/jami2011/jami2011pdf/Vaquie.pdf)

and the Ph.D. thesis of [[Marco Robalo]], under the supervision of [[Bertrand Toën]]:

* [[Marco Robalo]], _Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes_, June 2012 ([arxiv:1206.3645](http://arxiv.org/abs/1206.3645))

* [[Marco Robalo]], _Noncommutative Motives II: K-Theory and Noncommutative Motives_, June 2013, ([arxiv:1306.3795](http://arxiv.org/abs/1306.3795))

Also the lectures notes:

* [[Marco Robalo]], _Noncommutative motives and K-theory_, talk at [Higher Categories and Topological Quantum Field Theories,
Vienna, 2013](http://www.mat.univie.ac.at/~favero/Workshops/Higher.html), [notes](http://www.mat.univie.ac.at/~favero/Workshops/here/NotesRobalo.pdf)

Another survey article is

* [[Snigdhayan Mahanta]], Noncommutative geometry in the framework of differential graded categories, [arXiv:0805.1628](http://arxiv.org/abs/0805.1628).

Discussion of how the [[triangulated categories of sheaves|derived category]] of a [[scheme]] determines its commutative and [[noncommutative motive|noncommutative]] [[Chow motive]] is in

* [[Adeel Khan]], _On derived categories and noncommutative motives of varieties_, [arXiv](http://arxiv.org/abs/1401.7222).

In

* {#Mahanta13} [[Snigdhayan Mahanta]], _Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality_, ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf), [talk notes](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf))

it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C*-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].

* {#Tabuada13Twisted} [[Goncalo Tabuada]], _Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives_, ([arXiv](http://arxiv.org/abs/1310.4063)).

* [[Marcello Bernardara]], [[Goncalo Tabuada]].  _Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives._  ([arXiv](http://arxiv.org/abs/1310.6020))
 {#BerTab13}

category: algebraic geometry, noncommutative geometry

[[!redirects noncommutative motif]]
[[!redirects noncommutative motive]]
[[!redirects noncommutative motives]]
[[!redirects noncommutative motives]]

[[!redirects noncommutative Chow motive]]
[[!redirects noncommutative Chow motives]]

[[!redirects (infinity,1)-category of non-commutative motives]]
[[!redirects (∞,1)-category of non-commutative motives]]

[[!redirects (infinity,1)-category of noncommutative motives]]
[[!redirects (∞,1)-category of noncommutative motives]]

[[!redirects (infinity,1)-categories of non-commutative motives]]
[[!redirects (∞,1)-categories of non-commutative motives]]

[[!redirects (infinity,1)-categories of noncommutative motives]]
[[!redirects (∞,1)-categories of noncommutative motives]]

[[!redirects non-commutative motive]]
[[!redirects non-commutative motives]]


noncommutative motives - table

| geometric context | universal additive [[bivariant cohomology theory|bivariant]] (preserves [[split exact sequences]]) | universal localizing [[bivariant cohomology|bivariant]] (preserves all [[exact sequences]] in the middle) | universal additive invariant | universal localizing invariant |
|---|---|---|--|--|
| [[noncommutative algebraic geometry]] | [[noncommutative motives]] $Mot_{add}$ | [[noncommutative motives]] $Mot_{loc}$ | [[algebraic K-theory]] | [[non-connective algebraic K-theory]] |
| [[noncommutative topology]] | [[KK-theory]] | [[E-theory]] | [[operator K-theory]] | ... |


noncommutative principal bundle

There are many approaches to the generalizations of [[principal bundles]] to various flavours of [[noncommutative geometry]]. Not only that the base and total space of a principal bundle are replaced by noncommutative spaces, but various frameworks of noncommutative geometry also allow that the structure group be replaced by some analogue or generalization, something like [[quantum group]]. Some people hence talk about "quantum principal bundles".

In [[noncommutative algebraic geometry]], the most studied is the case in which the base and total space are affine, i.e. each represented by a single algebra, say base by $U$ and the total space by $E$. If the structure group is a [[Hopf algebra]], then the standard requirement is that $E$ is a right $H$-comodule algebra which is a [[Hopf-Galois extension]] of $U$ (for algebraic story, see the references there). An affine case of connections on principal bundles is considered in

* [[Tomasz Brzeziński]], [[Shahn Majid]], _Quantum group gauge theory on quantum spaces_, Commun.Math. Phys. __157__, 591–638 (1993) [doi](https://doi.org/10.1007/BF02096884)

One generalization of this picture are the "coalgebra bundles"

* [[T. Brzeziński]], [[Shahn Majid|S. Majid]], _Coalgebra bundles_, Comm. Math. Phys.  191  (1998),  no. 2, 467--492 ([arXiv version](http://arxiv.org/abs/q-alg/9602022)).

In that case, the Hopf algebra is replaced by a [[coalgebra]] $C$, the total space by an algebra which is a $C$-comodule, bu then an entwining structure is needed as an additional structure, and again a version of a Galois condition is required. The entwining which is a mixed [[distributive law]] is in fact in a role of lifting certain induced action of a monoidal category associated to $C$ from the base ground scheme cf.

* [[Z. Škoda]], _Distributive laws for actions of monoidal categories_, [math.CT/0406310](http://arxiv.org/abs/math.CT/0406310).

The liftings can be defined more generally in nonaffine situations leading to the concept of [[geometrically admissible action]]s as in

* [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770).

The Galois condition can then be defined locally on some compatible cover. A generalized version of the Schneider's descent theorem hold in this generality.

As a particular case, this allows the $H$-comodule algebras which are not Hopf-Galois extension of their coinvariants, but became so when localizing on some affine coaction compatible cover by affine "biflat" localizations (see [[zoranskoda:gluing categories from localizations|here]]). In other words the localized algebras are Hopf-Galois extensions (for example [[crossed product algebra]]s) of the [[localized coinvariant]]s as in

* [[Z. Škoda]], _Localizations for construction of quantum coset spaces_, [math.QA/0301090](http://front.math.ucdavis.edu/math.QA/0301090),  Banach Center Publ. __61__, pp. 265--298, Warszawa 2003;

* Z. Škoda, _Coherent states for Hopf algebras_, Letters in Mathematical Physics __81__, N.1, pp. 1-17, July 2007. (earlier arXiv version: [math.QA/0303357](http://arxiv.org/abs/math.QA/0303357)).

A principal bundle is called __locally trivial__ if there is a cover on which the Hopf-Galois extensions are in fact [[crossed product algebra|Hopf smash products]]. In the commutative case, and with affine algebraic group as a structure group, this is the same as the local triviality in [[fpqc topology]].

Another point of view to generalized Galois conditions in noncommutative algebraic geometry based on spaces represented by monoidal categories can be found in

* [[Tomasz Maszczyk]], _Noncommutative geometry through monoidal categories_, [math.QA/0611806] (http://arxiv.org/abs/math.QA/0611806)

As [[Hopf algebroid]]s generalize (function algebras on) groupoids, there is a well motivated study of Galois conditions (hence torsors) in the world of Hopf algebroids:

*  Gabriella Böhm, _Galois theory for Hopf algebroids_, Annali dell'Universita di Ferrara, Sez VII, Sci. Mat., Vol. LI (2005) 233--262 [math.QA/0409513](http://arxiv.org/abs/math.QA/0409513)

See also

* [[Tomasz Brzeziński]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arxiv:0912.0213](http://uk.arxiv.org/abs/0912.0213); _Quantum group differentials, bundles and gauge theory_, Encyclopedia of Mathematical Physics, Acad. Press. 2006, pp. 236–244
[doi](http://dx.doi.org/10.1016/B0-12-512666-2/00050-X)
* Francesco D'Andrea, Alessandro De Paris, _On noncommutative equivariant bundles_,  Commun. Alg. __47__:12 (2019) 5443-5461 [arxiv/1606.09130](https://arxiv.org/abs/1606.09130) [doi](https://doi.org/10.1080/00927872.2019.1631320)

For principal bundles in the context of operator algebras, see

* D. A. Ellwood, A new characterisation of principal actions, J. Funct. Anal. __173__(2000) 49–60, Zbl 0960.46049 [MR1760277](https://mathscinet.ams.org/mathscinet-getitem?mr=1760277)
* Kenny De Commer, Makoto Yamashita, _A construction of finite index $C^*$-algebra inclusions from free actions of compact quantum groups_, Publ. Res. Inst. Math. Sci. __49__ (2013) 709-735 [doi](https://dx.doi.org/10.4171/PRIMS/117) [arxiv/1201.4022](https://arxiv.org/abs/1201.4022)

In framework of [[spectral triple]]s, see

* Branimir Ćaćić, Bram Mesland, _Gauge theory on noncommutative Riemannian principal bundles_, [arxiv/1912.04179](https://arxiv.org/abs/1912.04179)

See also specifically the theory of [[connections on a noncommutative bundle]].

category: noncommutative geometry
[[!redirects noncommutative principal bundles]]
[[!redirects noncommutative torsor]]
[[!redirects quantum principal bundle]]
[[!redirects quantum principal bundles]]


noncommutative projective geometry

The __noncommutative projective geometry__ is a name of several directions in [[noncommutative algebraic geometry]]. The noncommutative projective schemes are supposed to be represented with noncommutative $\mathbb{N}$-[[graded algebra]]s; as in commutative case, different graded algebras may give rise to the isomorphic schemes. By definition the category $Qcoh Proj A$ of quasicoherent sheaves over the $Proj$ of a noncommutative graded algebra $A$ is the full category of (say left) modules $grMod_A$ modulo the torsion subcategory of the category of modules of finite length.

Most often one limits to noncommutative algebras which are close to commutative in certain sense. One very usable axiomatics is due Artin and Zhang and has been widely used:

* [[Michael Artin|M. Artin]], J. J. Zhang, _Noncommutative projective schemes_, Adv. Math. __109__ (1994), no. 2, 228-287, [MR96a:14004](http://www.ams.org/mathscinet-getitem?mr=1304753), [doi](http://dx.doi.org/10.1006/aima.1994.1087)

category: algebraic geometry, noncommutative geometry


noncommutative rational function

__Noncommutative rational identities__ are formally studied in

* S. A. Amitsur, _Rational identities and application to
algebra and geometry_, J. Alg. __3__ (1966), 304--359.
* P. M. Cohn, _Free rings and their relations_, Academic Press 1971.

As explained in the Cohn's book, one can define the division ring of __noncommutative rational functions__ on a given alphabet.

Given a commutative field $k$
(_field of constants_), and a finite set $X = \{x_1,\ldots,x_n\}$,
consider the free algebra on $X$ (in the sense of universal algebra)
of signature $\{ +_2, \cdot_2, (\,)^{-1}_1, -_1\}$
and with constants in $k$ and denote it by $\mathcal{R}(X,k)$.
Expressions like $0^{-1}$ and $(x-x)^{-1}$ are in $\mathcal{R}(X,k)$
as no relations are imposed (one starts with terms which are elements
of $X\cup k$ and continues by nesting sequence of algebraic
operations eventually connecting all terms).
Let $R$ be an associative $k$-algebra, and $\phi : X\to R$
a map of sets. Then there is
a subset $\mathcal{R}_\phi \subset \mathcal{R}(X,k)$
and a map $\phi_* : \mathcal{R}_\phi\to R$ uniquely determined by the rules

* (constants evaluate) If $c \in k$ then $c \in \mathcal{R}_\phi$
and $\phi_*(c) = c$.

* (variables evaluate) If $x \in X$ then $x \in \mathcal{R}_\phi$
and $\phi_*(x) = \phi(x)$.

* (sums, products and negatives of evaluables evaluate)
If $f, g \in\mathcal{R}_\phi$, then $f+g, f\cdot g, -f \in\mathcal{R}_\phi$,
$\phi_*(f+g) = \phi_*(f) + \phi_*(g)$, $\phi_*(f\cdot g) =
\phi_*(f)\cdot\phi_*(g)$ and $\phi_*(-f) = -\phi_*(f)$.

* If $g \in \mathcal{R}_\phi$, and $\phi_*(g)$ is invertible in $R$,
then $g^{-1}\in \mathcal{R}_\phi$ and $\phi_*(g^{-1}) = \phi_*(g)^{-1}$.

For every $f \in \mathcal{R}(X,k)$ define $\mathrm{Dom}_\phi\,f$ to be the
set of all $|X|$-tuples $\vec{r} = (r_1,\ldots,r_{|X|}) \in R^{|X|}$
such that $f \in \mathcal{R}_\phi$ where $\phi = \phi_{\vec{r}}$ satisfies
$\phi(x_i) = r_i$ for $i = 1,\ldots, |X|$. Those $f$ for which
$\mathrm{Dom}_\phi f \neq \emptyset$ are called nondegenerate. It is clear
that $f \notin\mathcal{R}_\phi$ iff there is a subexpression in $f$ of the form
$(f_0)^{-1}$ where $f_0 \in \mathcal{R}_\phi$ and $\phi_*(f_0)$ is not
invertible in $R$.

* [[I. M. Gel'fand]], [[V. Retakh|V. S. Retakh]], _Determinants
of matrices over noncommutative rings_, Funct. Anal. Appl. __25__ (1991), no.2, pp. 91--102; engl. transl. __21__ (1991), pp. 51--58; _A theory of noncommutative
determinants and characteristic functions of graphs_,
Funct. Anal. Appl. __26__ (1992), no.4, pp. 231--246.
* [[Zoran Škoda]], _Universal noncommutative flag variety_, preprint (the exposition above is mainly taken from here)

It is used in the study of [[skewfield]]s, [[Cohn localization]], [[quasideterminant]]s, noncommutative integrable systems and so on.

* Natalia Iyudu, Stanislav Shkarin, _A proof of the Kontsevich periodicity conjecture_, [arxiv/1305.1965](http://arxiv.org/abs/1305.1965)

* [[M. Kontsevich]], _Noncommutative identities_, writeup of the 2011 MPIM Bonn Arbeitstagung talk, [arxiv/1109.2469](http://arxiv.org/abs/1109.2469)

category: algebra
[[!redirects noncommutative rational functions]]
[[!redirects noncommutative rational identity]]
[[!redirects noncommutative rational identities]]


noncommutative residue

#Contents#
* table of contents
{:toc}

## Idea

The *noncommutative residue* or *Wodzicki residue* is (up to a constant multiple) the only nontrivial trace functional $Res$ on the algebra of [[pseudodifferential operators]] (of arbitrary order) on a compact smooth manifold $M$. For a $\psi DO$ $P$ it can be computed as the coefficient in front of $log t$ in the asymptotic expansion of $Tr (P e^{-t \Delta})$, where $\Delta$ is the Laplacian, or equivalently, in terms of the usual residue, $res_{s=0} Tr(P \Delta^{-s})$.

If $M$ is Riemannian, $P$ an elliptic operator of the negative integer order equal $-dim M$ then, by a result of Connes, the Wodzicki residue coincides with the [[Dixmier trace]].

## References

* [[Mariusz Wodzicki]], *Spectral asymmetry and noncommutative residue*, PhD thesis, Steklov Institute (1984)

* [[Mariusz Wodzicki]], *Noncommutative residue. I. Fundamentals*, in: *K-theory, arithmetic and geometry*, Lecture Notes in Math. __1289__, Springer (1987) 320-399 [[doi:10.1007%2FBFb0078372](http://dx.doi.org/10.1007%2FBFb0078372), [MR923140](http://www.ams.org/mathscinet-getitem?mr=923140)]

* [[eom]]: [Wodzicki residue](http://www.encyclopediaofmath.org/index.php/Wodzicki_residue)

* wikipedia [noncommutative residue](http://en.wikipedia.org/wiki/Noncommutative_residue)

* Nigel Kalton, Steven Lord, Denis Potapov, Fedor Sukochev, _Traces of compact operators and the noncommutative residue_, Adv. Math. __235__ (2013) 1-55, [arxiv/1210.3423](http://arxiv.org/abs/1210.3423) [doi](10.1016/j.aim.2012.11.007)

* [[Alain Connes]], __; _Noncommutative geometry_, Academic Press 1984.

category: analysis, noncommutative geometry

[[!redirects Wodzicki residue]]


noncommutative scheme

# Noncommutative schemes
* table of contents
{: toc}

## Idea

One would like to have a proper extension of a notion of [[scheme]] in [[noncommutative algebraic geometry]] which should include also noncommutative spaces represented by noncommutative rings ("noncommutative affine schemes") and "noncommutative projective schemes" represented by graded rings, but *not* including commutative spaces which are not schemes ([[stacks]], [[algebraic space]]s, etc.). To this aim, one notices that schemes are glued from affine schemes in the [[Zariski topology]], i.e. they are a subclass of [[locally affine space]]s where locally pertains to the [[Zariski site]]. At the level of the category $QCoh_X$ of [[quasicoherent sheaf|quasicoherent sheaves]] on a scheme $X$, affine subschemes $Y\subset X$ correspond to especially nice [[localization]] functors $Q^*_Y$: first of all they are affine localizations (i.e. have a [[right adjoint]] which is [[fully faithful functor|fully faithful]] and which has its own right adjoint), second they are exact ([[flat functor|flat]]) and their
right adjoint is also flat. Second the localizations pertaining to Zariski topology are of finite type (a fortiori, Zariski topology on an affine scheme is generated by the principal open subsets $D_f$). The fact that a family of affine subschemes covers $X$ is expressed via the property that the family of localizations $Q_Y$ for $Y$ in the cover is a (jointly) conservative family (i.e. a morphism is invertible iff it is invertible in each element of a cover). Using the idea of Grothendieck which says, according to Manin, that _to do a geometry you do not need a space you only need an algebra of functions on this would-be space_, we express all the properties of gluing affine schemes via catgories of quasicoherent sheaves. This is justified by various evidence including Serre's theorem on $Proj$, the observation that important notions in noncommutative geometry are Morita-invariant, and the reconstruction theorems of Gabriel--Rosenberg and, in the monoidal and monoidal triangulated case, by Balmer and by Garkusha, that a scheme $X$ can be reconstructed up to isomorphism from $QCoh_X$.

## Formal frameworks

The above ideas are sometimes loosely expressed by practioners to take an [[abelian category]] as a noncommutative space (Smith calls it a _quasischeme_ in that case). Some prefer to have a pair of an abelian category and a distinguished object in it. The work of Kontsevich--Rosenberg on noncommutative smooth spaces notices that the main noncommutative examples (those close to [[free object|free]] algebras, rather than almost commutative quantum, PI and similar examples) like noncommutative projective spaces and noncommutative grassmanians (unlike quantum projective spaces and quantum flag varieties) are *not* locally affine with respect to the covers by biflat affine localizations, but rather locally affine in certain noncommutative smooth topologies (covers by Cohn localizations, which are not biflat, also appear occasionally). For the projective case, F. van Oystaeyen prefers to work with Ore localizations only and defines an interesting class of so-called __schematic algebras__ and axiomatizes a more general property of torsion lattices which can play a role in an extension of quasicoherent sheaf theory to similar situations ("noncommutative topology"). T. Maszczyk prefers to generalize categories of quasicoherent sheaves to monoidal abelian categories and requires that the corresponding direct or inverse image functors respect the monoidal structure in a lax or colax way. That means that the commutative modules are viewed as [[central bimodule]]s. Bimodules can be related to correspondences ([[spans]]), that is in one-sided module point of view they correspond to functors: however Maszczyk argues that this point of view is only partial and that monoidal categories should be put in the fundamentals.

Rosenberg notices that the considerations from the Idea section force one to consider not noncommutative schemes in an absolute sense, but rather noncommutative schemes relative to the ground category $C = QCoh_{Spec k}$. Suppose all categories involved are abelian and functors [[additive functor|additive]] (there is a more subtle variant in the nonabelian case). Then the general nonsense basically forces one to a unique notion of a noncommutative scheme.

## Rosenberg's definition

(A. Rosenberg, _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125, [doi](http://dx.doi.org/10.1023/A:1000479824211))

A quasicompact __relative noncommutative scheme__ over a category $C$ is a category $A$ with an [[adjoint functor|adjoint pair]] of functors (direct and inverse image)  $g_* : A\to C : q^*$, such that there exists a finite conservative family of affine localizations $\{Q^*_\lambda : A\to A_\lambda\}_{\lambda\in\Lambda}$ whose direct image $Q_{\lambda *}$ and inverse image $Q_\lambda^*$ parts are exact and such that the composition of the direct image functor of each localization $Q_{\lambda*} : A_\lambda\to A$ followed by the direct image functor $g_* : A\to C$ is also affine.

(A pair of adjoint functors, inverse and direct image represent an affine morphism if the direct image is faithtful and it has its own right adjoint, so we have an adjoint triple with middle term, that is the direct image, faithful. For the affine localizations, the direct image is in addition full.)

## Properties

The word 'localization' here is in the sense of Gabriel--Zisman. Notice that if the ground category $C$ has a distinguished object (say the generator in the category of vector spaces over a field) then its direct image functor is a distinguished object in $A$. In general this is not required. The exactness properties required above guarentee that the quasicoherent sheaves can be glued from the restrictions on the covers, and moreover that the derived functors of the direct image like functors can be computed using standard resolutions. If the cover is semiseparated (a notion from Thomason--Trobaugh), that is the localization functors for the cover mutually commute, then the standard resolutions can be replaced by alternating Čech resolutions.

See also [[zoranskoda:gluing categories from localizations]].

## Examples

Quantum projective spaces, quantum flag varieties, commutative quasicompact schemes, the $D$-schemes of Beilinson, the almost schemes of Gabber etc.

## Nonexamples

Noncommutative projective spaces, noncommutative grassmannians, universal noncommutative flag variety.

## Literature

* [[Tomasz Maszczyk]], _Noncommutative geometry through monoidal categories_, [math.QA/0611806] (http://arxiv.org/abs/math.QA/0611806)

* A. Rosenberg, _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125, [doi](http://dx.doi.org/10.1023/A:1000479824211)

* P. Gabriel, M. Zisman, _Calculus of Fractions and Homotopy Theory_, Springer 1967.

* S. Paul Smith, _Subspaces of non-commutative spaces_, Trans. Amer. Math. Soc. __354__(2002), 2131--2171

* V. A. Lunts, A. L. Rosenberg, _Localization for quantum groups_, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123--159.

* Z. Škoda, _Some equivariant constructions in noncommutative geometry_, Georgian Math. J. __16__ (2009) 1; 183--202 ([arXiv:0811.4770](http://front.math.ucdavis.edu/0811.4770))

* F. van Oystaeyen, L. Willaert, _Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras_, J. Pure Appl. Alg. __104__(1995), p. 109--122

* M. Artin, J. Zhang, _Noncommutative projective schemes_, Adv. Math. 109, 228--287 (1994).

For a derived analogue see

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, [arXiv](http://arxiv.org/abs/1402.7364v1); _Derived noncommutative schemes, geometric realizations, and finite dimensional algebras_, [arxiv/1808.02287](https://arxiv.org/abs/1808.02287)

category: algebraic geometry, noncommutative geometry
[[!redirects noncommutative scheme]]
[[!redirects noncommutative schemes]]
[[!redirects non-commutative scheme]]
[[!redirects non-commutative schemes]]


noncommutative Schur function

These are analogues of [[Schur function]]s within the frameworks of [[noncommutative symmetric function]]s and quasisymmetric functions.

* S. Fomin, C. Greene, _Noncommutative Schur functions and their applications_, Discrete Math. __193__ (1998), pp. 179–200
* J. Haglund, K. Luoto, S. Mason, S. van Willigenburg,
_Quasisymmetric Schur functions_, J. Combin. Theory Ser. A, 118 (2011), pp. 463–490.
* K. Luoto, S. Mykytiuk, S. van Willigenburg, _An introduction to quasisymmetric Schur functions_, in: Hopf algebras, quasisymmetric functions, and Young composition tableaux, Springer Briefs in Math., Springer 2013
* Vasu V. Tewari,  _A Murnaghan-Nakayama rule for noncommutative Schur functions_, [arxiv/1403.0607](http://arxiv.org/abs/1403.0607)

[[!redirects noncommutative Schur functions]]
[[!redirects quasisymmetric Schur function]]


noncommutative sheaf

The term __'noncommutative [[sheaf]]'__ is an informal term which for some people denotes either

* a sheaf of [[noncommutative algebras]] on a space or on a site, or

* a sheaf (of anything, say, of sets or abelian groups) on a [[noncommutative space]].

The latter concept of a sheaf is in a more general sense than sheaf on a [[site]]; see

* [[sheaf on a noncommutative space]].

category: disambiguation, noncommutative geometry

[[!redirects noncommutative sheaf]]
[[!redirects noncommutative sheafs]]
[[!redirects noncommutative sheaves]]
[[!redirects non-commutative sheaf]]
[[!redirects non-commutative sheafs]]
[[!redirects non-commutative sheaves]]


noncommutative space as a cover

Noncommutative algebraic geometry studies objects much more general than a single algebra: one wants to glue algebras to more general "systems". This idea has been around from 1980-s. Of course, one needs a good equivalence among systems. One idea is to replace the algebra by a module and glue categories of modules and then look if the categories are equivalent. But the categories of modules loose the track of the Morita equivalence class, which one would like to keep. For example, one wants to define morphisms between such systems as some class of geometric morphisms, but when one restricts to algebras one would like to have only morphisms of algebras, not the bimodules corresponding to nonaffine morphisms.

This is nicely achieved in a formalism which significantly refines the standard techniques of gluing via corings sketched in chapter 2 of

* [[Alexander Rosenberg]], [[Maxim Kontsevich]], _Noncommutative smooth spaces_, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Seminars, [gBooks](http://books.google.hr/books?id=1xI6j05m7NUC&lpg=PA85&ots=FEtnAU-ImI&dq=alexander%20rosenberg%20gelfand%20seminar%20kontsevich&hl=en&pg=PA85#v=onepage&q&f=false), [math.AG/9812158](http://arxiv.org/abs/math/9812158)

#### STEP 1: Category of Covers

Given a $k$-algebra $R$, a $R$-[[coring]] $M = (M,\delta^M,\epsilon^M)$ is a comonoid in the category of $R$-bimodules (i.e. $R\otimes R^{op}$-modules). One defines a category of covers as the category of pairs $(R,M)$ where $R$ is a $k$-algebra and $M$ is an $R$-coring; one requires that $M$ is faithfully flat as the right  $R$-module i.e. tensoring $M\otimes_R$ is exact and faithful functor. The morphisms $(R_1,M_1)\to (R_2,M_2)$ are given by pairs of algebraic maps in opposite direction, i.e. an algebra map $f_R : R_2\to R_1$ and a bimodule map $f_M:M_2\to M_1$ satisfying the obvious compatibility conditions with the rest of the structure. This defines the category of covers, $Cover_k$.

One also defines the category of quasicoherent modules $QCoh(R,M)$ as the category of left $M$-comodules, i.e. left $R$-modules $X$ together with the coaction $X\to M\otimes_R X$.

#### STEP 2: Introducing structure morphism $s_C$

The objects of the category of "space covers" $Cover_k^{sp}$ is the category of pairs $(C,s_C)$ where $C = (R,M) = (R,M,\delta^M,\epsilon^M)$ is a cover
and $s^C: R\otimes_k R\to M$ is a surjective homomorphism of coalgebras in the category of $R$-bimodules. The morphisms in $Cover_k^{sp}$ are the morphisms in $Cover_k$ compatible with "structure morphisms" $s^C$.

#### STEP 3: Equivalence relation for the morphisms

Still some morphisms induce the same inverse image functors for $QCoh$ categories. To eliminate this, one calls two morphisms $f,g:(C,s_C)\to (C',s_{C'})$ equivalent if for any element $u = \sum x_\alpha\otimes y_\alpha$ in $Ker(s_C)$ the tensor product $(f\otimes g)(u) = 0$ and $(g\otimes f)(u)=0$. After moding out this equivalence we obtain $\widetilde{Cover}^{sp}_k$

#### STEP 4: Localizing on refinements

Any cover $(R,M)$ in $Cover_k$ together with a monomorphism of $k$-algebras $R\hookrightarrow S$ induces a cover $(S,S\otimes_R M\otimes_R S)$ together with a canonical morphism $(S,S\otimes_R M\otimes S)\to (R,M)$ in $Cover_k$. Morphisms of this type are called refinement morphisms; they are closed under composition and pullback. Thus one can define the localized category $Cover_k[Ref^{-1}]$; moreover, $(S,S\otimes_R M\otimes S)$ can be canonically equipped with a structure morphism and we can in this vain obtain the category of noncommutative spaces $NSpace_k := \tilde{Cover}^{sp}_k[Ref^{-1}]$.

### Properties

Theorem. (AR/MK, sec.2.2) The category of separated quasi-compact schemes over $k$ is equivalent to a full subcategory of $NSpace_k$. The category $Alg^{op}_k$ is also equivalent to a full subcategory of
$NSpace_k$. Category $NSpace_k$ has finite limits.

Theorem. (AR/MK, sec.2.2) Construction $(B,M)\to QCoh(B,M)$ extends to a functor from the category $NSpace_k$ to the category of abelian $k$-linear categories. For separated quasi-compact schemes, it gives usual quasi-coherent sheaves of commutative algebraic geometry. For associative algebras, it gives categories of left modules.

[[!redirects noncommutative spaces as covers]]


noncommutative stable homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By _noncommutative stable homotopy theory_ is meant an analog of classical _[[stable homotopy theory]]_ obtained as one passes from [[topology]] to [[noncommutative topology]], hence as one generalizes [[topological spaces]] to non-neccesarily commutative (formal duals of) [[C*-algebras]].

The concrete implementation of such _noncommutative stable homotopy theory_ is usually taken to be _[[KK-theory]]_ or rather _[[E-theory]]_. See there for details.

## Related concepts

* [[homotopical structure on C*-algebras]]

* [[index theory]]

## References

A reference that mentions the term "noncommutative stable homotopy theory" explicitly is

* Martin Grensing, _Noncommutative stable homotopy theory_ ([arXiv:1302.4751](http://arxiv.org/abs/1302.4751))

But see at _[[KK-theory]]_ and _[[E-theory]]_ for more references.


noncommutative symmetric function

# Noncommutative symmetric functions
* table of contents
{: toc}

## Idea

Noncommutative symmetric functions are a generalisation of [[symmetric functions]].  Many concepts and ideas extend from symmetric functions to noncommutative symmetric functions and the way in which they extend sheds light on their behaviour for ordinary symmetric functions.

Noncommutative symmetric functions also arise in their own right as interesting objects of study.

## Definition

+--{: .num_defn #nsymm}
###### Definition
The graded [[Hopf algebra]] of **noncommutative symmetric functions**, $\NSymm$, is defined in the following way.

1. As an algebra, it is $\mathbb{Z}\langle Z_1, Z_2, \dots \rangle$, the free algebra in countably many indeterminants over $\mathbb{Z}$.
1. The comultiplication is given by $\Delta Z_n = \sum_{i + j = n} Z_i \otimes Z_j$, where $Z_0 = 1$.
1. The counit is $\epsilon(Z_n) = 0$ for $n \ge 1$.
1. The antipode is $\iota(Z_n) = \sum_{wt(\alpha) = n} (-1)^{length(\alpha)} Z_\alpha$, where $\alpha = [\alpha_1, \cdots, \alpha_k]$ is a word in $\{1,2,\dots\}$ with $length(\alpha) = k$ and $wt(\alpha) = \sum \alpha_i$.
1. The degree of $Z_n$ is $n$.
=--

## Properties

1. The object $\NSymm$ represents a functor from not-necessarily-commutative rings to groups given by sending $B$ to formal power series in $B$ with leading term $1$, $B \mapsto 1 + t B[\![t]\!]$.

1. It is dual to the Hopf algebra of [[quasi-symmetric functions]].

1. The Hopf algebra of [[symmetric functions]] is a quotient of $\NSymm$.  The quotient mapping is given by sending $Z_i$ to the $i$th symmetric function.

## References

### Full research articles

* G. Duchamp, F. Hivert, J.-Y. Thibon, _Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras_, Internat. J. Alg. Comput. 12 (2002), 671–717.
* [[I. M. Gelfand]], D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, _Noncommutative symmetric functions_, Adv. in Math. __112__ (1995), 218–348, [hep-th/9407124](http://arxiv.org/abs/hep-th/9407124)
* Jean-Christophe Novelli, Jean-Yves Thibon, _Noncommutative symmetric functions and Lagrange inversion_, [math.CO/0512570](http://arxiv.org/abs/math/0512570); _Noncommutative symmetric functions and an amazing matrix_ [arxiv/1109.1184](http://arxiv.org/abs/1109.1184)
* Lenny Tevlin, _Noncommutative Monomial Symmetric Functions_, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings [pdf](http://www.fpsac.org/FPSAC07/SITE07/PDF-Proceedings/Talks/83.pdf)
* D. Krob, J.-Y. Thibon, _Noncommutative symmetric functions IV:
Quantum linear groups and Hecke algebras at $q = 0$_, [pdf](http://hal.inria.fr/docs/00/05/79/10/PDF/ncsf4.pdf)

### Long surveys and lecture notes

* Michael Hazewinkel, _Symmetric functions, noncommutative symmetric functions and quasisymmetric functions_, [pdf](http://oai.cwi.nl/oai/asset/10344/10344B.pdf)
* [[V. Retakh]] and R. Wilson, Advanced Course on Quasideterminants and Universal Localization: [pdf](http://castellet.cat/Publications/quaderns/Quadern41.pdf) (see the part _Factorization of Noncommutative Polynomials
and Noncommutative Symmetric Functions_)

### Expositions/short summaries

* Mike Zabrocki, _Non-commutative symmetric functions II:
Combinatorics and coinvariants_, slides from a talk [pdf](http://garsia.math.yorku.ca/~zabrocki/talks/coinvariants.pdf), III: A representation theoretical approach [pdf](http://garsia.math.yorku.ca/~zabrocki/talks/reptheory.pdf)
* Lenny Tevlin, _Introduction to quasisymmetric and noncommutative
symmetric functions_, slides, Fields Institute 2010  [pdf](http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/Noncommutative-Symmetric-and-Quasi-Symmetric-Functions-Fields-2010.pdf)

[[!redirects noncommutative symmetric function]]
[[!redirects noncommutative symmetric functions]]


noncommutative thin scheme

Given a commutative [[field]] $k$, a **noncommutative thin $k$-scheme** (in the terminology of [[Maxim Kontsevich]]) is a [[left exact functor]] from the category $Alg^{fd}_k$ of finite dimensional $k$-algebras to [[Set]], or equivalently by duality, from $Coalg^{fd}_k^{op}$ to $Set$. Every such functor is [[representable functor|representable]] by a $k$-[[coalgebra]].

* L. le Bruyn, _Noncommutative geometry and dual coalgebras_, [arXiv:0805.2377](http://arxiv.org/abs/0805.2377)

* M. Kontsevich, Y. Soibelman, _Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I_, [math.RA/0606241](http://arxiv.org/abs/math/0606241)

* [blog entry](http://matrix.cmi.ua.ac.be/fun/index.php/noncommutative-f_un-geometry-2.html)

category: algebraic geometry, noncommutative geometry


noncommutative topology

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#### Topology
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#### Higher geometry
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=--
#### Higher algebra
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[[!include higher algebra - contents]]
=--
#### Index theory
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[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By [[Gelfand duality]], suitable [[topological spaces]] are [[contravariant functor|contravariantly]] [[equivalence of categories|equivalent]] to [[commutative C-star-algebra|commutative C*-algebras]]. Therefore conversely, non-commutative $C^\ast$-algebras may be thought as the formal [[duality|duals]] of generalized topological spaces, "noncommutative topological spaces". Therefore the study of [[operator algebra]] and [[C-star-algebra]] theory is sometimes called _noncommutative topology_. This is a special case of the general idea of [[noncommutative geometry]].

(Not to be confused with _[[algebraic topology]]_, which is instead the study of ordinary [[topology]] and of its [[homotopy theory]] by [[algebra|algebraic]] tools.)

## Related concepts

* [[noncommutative stable homotopy theory]]

  * [[homotopical structure on C*-algebras]]

* [[noncommutative topology of quasiperiodicity]]

* [[noncommutative geometry]]

* [[noncommutative measure theory]]

[[!include Isbell duality - table]]

$\,$

[[!include noncommutative motives - table]]

## References

* [[Alain Connes]], _[[Noncommutative Geometry]]_

[[!redirects non-commutative topology]]


noncommutative topology of quasiperiodicity

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=--
#### Geometry
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=--
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#Contents#
* table of contents
{:toc}

## Idea

Quasi-periodic geometries such as [[Penrose tilings]] and [[quasicrystals]] tend to admit useful descriptions by [[noncommutative topology]]/[[noncommutative geometry]].

## References

### For Penrose tilings

Noncommutative topology of [[Penrose tilings]]:

* [[Alain Connes]], Section 2.3 of: *[[Noncommutative Geometry]]*, Academic Press, San Diego, CA, 1994 ([ISBN:9780080571751](https://www.elsevier.com/books/noncommutative-geometry/connes/978-0-08-057175-1), [pdf](http://www.alainconnes.org/docs/book94bigpdf.pdf))

* Hervé Oyono-Oyonoa, Samuel Petite, *$C^\ast$-algebras of Penrose hyperbolic tilings*, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 ([doi:10.1016/j.geomphys.2010.09.019](https://doi.org/10.1016/j.geomphys.2010.09.019))

### For quasicrystals

Discussion of the [[noncommutative topology]]/[[KK-theory]] of the [[Brillouin zone]] of [[quasi-crystals]] in the spirit of the [[K-theory classification of topological phases of matter]]:

* {#Bellissard03} Jean Bellissard, *The Noncommutative Geometry of Aperiodic Solids*, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) ([pdf](http://people.math.gatech.edu/~jeanbel/Publi/ncg02.pdf), [doi:10.1142/9789812705068_0002](https://doi.org/10.1142/9789812705068_0002))

* {#Ypma05} Fonger Ypma, *Quasicrystals, $C^\ast$-algebras and K-theory*, 2005 ([pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.4414&rep=rep1&type=pdf))

* {#Putnam10} Ian F. Putnam, *Non-commutative methods for the K-theory of $C^\ast$-algebras of aperiodic patterns from cut-and-project systems*,  Commun. Math. Phys. 294, 703–729 (2010) ([pdf](https://www.math.uvic.ca/faculty/putnam/r/0802_CMP.pdf), [doi:10.1007/s00220-009-0968-0](https://doi.org/10.1007/s00220-009-0968-0))


noncommutative toric variety

* [[Ludmil Katzarkov]], [[Ernesto Lupercio]], Laurent Meersseman, Alberto Verjovsky, _Noncommutative toric varieties_, [arxiv/1308.2774](http://arxiv.org/abs/1308.2774)

related: [[convex geometry]], [[toric geometry]], [[tropical geometry]]

applications: [[homological mirror symmetry]], [[topological insulator]]

category: algebraic geometry, noncommutative geometry


noncommutative torus

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#Contents#
* table of contents
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## Idea

A [[quotient]] of the [[torus]], taken in the sense of [[noncommutative geometry]].

## Definition

### As a Lie groupoid

The __noncommutative torus__ with parameter $\theta \in \mathrm{U}(1)$ away from [[roots of unity]] is the [[groupoid quotient]] of the [[circle group]] $\mathrm{U}(1)$ by the [[action]] of the [[discrete group]] $\mathbf{Z}$ of [[integers]], where the element $1 \in \mathbf{Z}$ acts by [[multiplication]] by $\theta$.

Here we can take $\theta = \exp(2\pi i \hbar)$, for $\hbar$ an [[irrational number]] (which, if we think of the non-commutative algebra here as arising from [[quantization]], may be thought of as [[Planck's constant]]).

The following [[Lie groupoid]] is [[Morita equivalent]] to the above
[[Lie groupoid]]:

take the [[groupoid quotient]] of $\mathrm{U}(1)^2$ (i.e., a [[torus]])
by the action of the [[Lie group]] $\mathbf{R}$ of [[real numbers]]
such that

$$
  a \cdot (u,v)
  \;=\;
  \big(
    u \exp(2\pi i a), v \exp(2\pi i a \hbar)
  \big)
$$

### As an algebra

Taking the [[groupoid algebra]] of the [[Lie groupoid]] constructed above,
we obtain a [[noncommutative space]] in the sense of [[Connes noncommutative geometry]].

For instance, in the realm of [[C*-algebras]]
we get the [[C*-algebra]] generated by two [[unitary]] operators
$U$ and $V$ subject to the relation
$$U V=\theta V U.$$

## Classification

The following classification is due to [Rieffel 81](#Rieffel81).

If $\theta$ is a [[root of unity]], then the resulting [[Lie groupoid]] is [[Morita equivalent]] to the [[smooth manifold]] $\mathrm{U}(1)$, i.e., the  [[circle]].

Otherwise, the noncommutative torus with parameter $\theta$ is not
[[Morita equivalent]] to the circle (or any [[smooth manifold]]).
Furthermore, two noncommutative tori with paremeters $\theta_1$ and $\theta_2$
are [[Morita equivalent]] if and only if $\theta_1$ and $\theta_2$
are in the same orbit with respect to the action of $\mathrm{GL}(2,\mathbf{Z})$
on $\mathrm{U}(1)$ defined as follows:

$$
  \left[
    \array{
      a & b
      \\
      c & d
    }
  \right]
  \cdot
  \exp(2\pi i \alpha)
  \;\coloneqq\;
  \exp
  \left(
    2\pi i
    \frac
      {a\alpha + b}
      {c\alpha + d}
  \right)
  \,.
$$

## Related concepts

* [[fuzzy sphere]]

* [[fuzzy funnel]]

## References

Original articles:

* {#Rieffel81} [[Marc A. Rieffel]], _$C^\ast$-algebras associated with irrational rotations_, Pacific Journal of Mathematics 93:2 (1981), 415–429 ([pdf](https://msp.org/pjm/1981/93-2/pjm-v93-n2-p12-s.pdf), [doi:10.2140/pjm.1981.93.415](http://dx.doi.org/10.2140/pjm.1981.93.415), [euclid:pjm/1102736269](https://projecteuclid.org/euclid.pjm/1102736269))

* [[Marc Rieffel]], [[Albert Schwarz]], _Morita equivalence of multidimensional noncommutative tori_, Int. J. Math. 10 (1999) 289-299 ([arXiv:math/9803057](https://arxiv.org/abs/math/9803057))

* George A. Elliott and Hanfeng Li, _Morita equivalence of smooth noncommutative tori_, Acta Math. Volume 199, Number 1 (2007), 1-27 ([euclid:acta/1485891908](https://projecteuclid.org/euclid.acta/1485891908))

Review:

* [[Alain Connes]], p. 55, p. 217, p. 356 of: _[[Noncommutative Geometry]]_, Academic Press, San Diego, CA, 1994 ([ISBN:9780080571751](https://www.elsevier.com/books/noncommutative-geometry/connes/978-0-08-057175-1), [pdf](http://www.alainconnes.org/docs/book94bigpdf.pdf))

* Wikipedia, _[Noncommutative torus](https://en.wikipedia.org/wiki/Noncommutative_torus)_

* [[John Baez]], _[Noncommutative tori](https://math.ucr.edu/home/baez/braids/node8.html)_

Non-commutative tori understood as [[strict deformation quantizations]]:

* [[Marc A. Rieffel]], *Deformation quantization of Heisenberg manifolds*, Communications in Mathematical Physics **122** (1989) 531–562 [[doi:10.1007/BF01256492](https://doi.org/10.1007/BF01256492)]

[[!redirects noncommutative tori]]

[[!redirects non-commutative torus]]
[[!redirects non-commutative tori]]


noncommutative vector bundle

# Noncommutative vector bundles
* table of contents
{: toc}

## Scope

There are at least 3 groups of ideas to define noncommutative vector bundles

* taking Serre--Swan theorem as a definition in the noncommutative case; i.e. a finite-dimensional bundle is simply a projective module over the algebra of functions. Of course, in the algebraic case, this is satisfactory at best for the affine case, when possibly some Noetherianess assumptions are needed, even in the commutative case. Not good for e.g. for (nonaffine) [[noncommutative scheme]]s.

* as a locally free sheaf of vector spaces; this requires some notion of topology or covers on the noncommutative base space. There are candidates of noncommutative topologies, cf. [[descent in noncommutative algebraic geometry]]. Problem: In commutative case, SGA gives a construction which as an input takes a quasicoherent module and as an output its "underlying space". There is no satisfactory construction of this form in noncommutative algebraic geometry, so the locally free sheaves do not have naturally defined total space in the category of noncommutative spaces.

* as an associated bundle to some sort of [[noncommutative principal bundle]]s, typically with a Hopf algebra in the place of a structure group. Typically, the associated bundle is defined as a dual to some cotensor product locally, and in the affine case this is pretty standard. This has a problem as the locally free sheaf: if one wants to make a total space a space, one does a cotensor product with an algebra of functions on the space, not with a vector space itself, as the latter will just give the global sections or sheaf of sections at best. In fact one wants consistently both: to get the global sections locally as a cotensor product and to get the bundle of noncommutative spaces from a cotensor product at algebra level. The sections (in some category of noncommutative space) of the inclusion of algebras should reproduce the space of global sections, but good theorems of that kind are not known. See [[noncommutative associated bundle]].

## References

* [[S. Majid]], [[T. Brzeziński]], _Quantum group gauge theory on quantum spaces_, Commun. Math. Phys. __157__, 591--638 (1993); Erratum 167, 235 (1995) [MR94g:58015](http://www.ams.org/mathscinet-getitem?mr=1243712), [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104254023)

* [[Tomasz Brzeziński]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arxiv:0912.0213](http://uk.arxiv.org/abs/0912.0213).

* T. Brzeziński, _On modules associated to coalgebra Galois extensions_, J. Algebra __215__, 290–317 (1999) [MR2000c:16047](http://www.ams.org/mathscinet-getitem?mr=1684158), [doi](http://dx.doi.org/10.1006/jabr.1998.7738)

* Zoran Škoda, _Localizations for construction of quantum coset spaces_, Banach Center Publications __61__, 265–298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090)

* Zoran Škoda, _Coherent states for Hopf algebras_, Letters in Mathematical Physics __81__, N.1, pp. 1-17, July 2007, [pdf](http://www.irb.hr/korisnici/zskoda/skodacohstates.pdf), (earlier different arXiv version: [math.QA/0303357](http://arxiv.org/abs/math.QA/0303357))

* [[José Gracia-Bondia]], [[Joseph Varilly]], Héctor Figueroa, _Elements of noncommutative geometry_, Birkhäuser 2001. xviii+685 pp. ([doi:10.1007/978-1-4612-0005-5](https://link.springer.com/book/10.1007/978-1-4612-0005-5), [gBooks](http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false))

* R. Coquereaux, A. O. Garcí, R. Trinchero, _Associated quantum vector bundles and symplectic structure on a quantum plane_, [arXiv:math-ph/9908007](http://arxiv.org/pdf/math-ph/9908007)

* Guanglian Zhang, R.B. Zhang, _Equivariant vector bundles on quantum homogeneous space_, [pdf](http://www.maths.usyd.edu.au/u/rzhang/MathResLett.pdf); R. B. Zhang, _Quantum group equivariant homogeneous
vector bundles_, [pdf slides](http://www.newton.ac.uk/programmes/NCG/seminars/122114001.pdf)

The following article is working with local triviality in the sense of ideals instead of localizations, what unfortunately corresponds to the covers by closed sets, not open:

* Piotr M. Hajac, Rainer Matthes, Wojciech Szymanski, _A locally trivial quantum Hopf fibration_, Algebr. Represent. Theory __9__ (2006), no. 2, 121–146.

category: noncommutative geometry
[[!redirects noncommutative vector bundle]]
[[!redirects noncommutative vector bundles]]
[[!redirects non-commutative vector bundle]]
[[!redirects non-commutative vector bundles]]
[[!redirects quantum vector bundle]]
[[!redirects quantum vector bundles]]


nondeterministic automaton

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### Context
#### Computing
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[[!include constructivism - contents]]
=--
=--
=--

# Non-deterministic automata
* table of contents
{:toc}

## Definitions

(For the moment, mostly as an example of a [[coalgebra for an endofunctor]] and to start discussion of models for non-determinism.)

This will give the classical definition as a non-deterministic state-based system and then show how to turn that form into the coalgebraic form.

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

A **non-deterministic [[automaton]]** consists of the following data:

* a [[set]] of _states_, $Q$;

* a set, $\Sigma$, of _input symbols_ ;

* a function, $\delta : Q\times \Sigma \to \mathcal{P}(Q)$, called the _next state relation_,

and

* a [[predicate]], $final : Q \to bool = \{\bot, \top\}$, the [[boolean domain]].
=--

In the usual interpretation if the automaton is in state $q$, and is 'given' the input $\sigma$, then it changes to being in one of the states in the subset, $\delta(q,\sigma)$, of $Q$.

The 'final' predicate, of course, returns $\top$ if a state is a _final state_ and $\bot$ otherwise. (There will, in general, be a set of final or 'accept' states.)

(For the moment we will not look at the links between automata and languages.)

## Properties

### Coalgebraic description

The first step in transforming the above definition to the coalgebraic form is to [[currying|curry]] $\delta$, so as to obtain it in the form $\delta : Q\to \mathcal{P}(Q)^\Sigma$. We thus have for a state, $q\in Q$, $\delta(q,) : \Sigma \to \mathcal{P}(Q)$.  We then also have a product function

$$\alpha = \delta \times (final) : Q \to  \mathcal{P}(Q)^\Sigma\times bool.$$

If we now write $HQ = \mathcal{P}(Q)^\Sigma\times bool$, we get a functor (for you to check) $H : Set \to Set$ and the non-deterministic automaton corresponded precisely to a [[coalgebra for an endofunctor|coalgebra]], $(Q,\alpha)$, for $H$.

## Related concepts

* [[automaton]]

* [[non-deterministic computation]]

## References

Monographs:

*  M. V. Lawson, _Finite Automata_, [Google books](http://books.google.co.uk/books?id=MDQ_K7-z2AMC&dq=automata+theory+lawson&source=bl&ots=0w-Xp8AaDa&sig=6dXKjG0tVY6aUlvlAc5UaT-ObX8&hl=en&ei=Ywy6TPmtCJSOjAfzgYXYDg&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CBoQ6AEwAA)

See also:

* Wikipedia, *[Automata theory](http://en.wikipedia.org/wiki/Automata_theory)*

Discussion via [[coalgebra]]:

*  [[Alexander Kurz]]:  _Coalgebras and Modal Logic._ Course Notes for ESSLLI 2001, Version of October 2001. Appeared on the CD-Rom ESSLLI'01, Department of Philosophy, University of Helsinki, Finland, available from [site](http://www.cs.le.ac.uk/people/akurz/works.html).

Discussion of [[non-deterministic automata]] as 1-dimensional [[defect QFT|defect]] [[TQFTs]]:

* Paul Gustafson, [[Mee Seong Im]], Remy Kaldawy, [[Mikhail Khovanov]], Zachary Lihn, *Automata and one-dimensional TQFTs with defects* [[arXiv:2301.00700](https://arxiv.org/abs/2301.00700)]

Review:

* [[Mee Seong Im]], *Correspondence between automata and one-dimensional Boolean topological theories and TQFTs*, 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#ImMar2023), video:[YT](https://www.youtube.com/watch?v=q5kynYxwTmk)]

[[!redirects nondeterministic automaton]]
[[!redirects nondeterministic automata]]
[[!redirects non-deterministic automaton]]
[[!redirects non-deterministic automata]]


nondeterministic computation

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#### Computation
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Broadly speaking *nondeterministic* [[computations]]/[[algorithms]] are those whose outcome is [indefinite](necessity+and+possibility#RelationToPotentiality), or else dependent on parameters that are not known or specified when the computation/algorithm is laid out (they may become known at "runtime").

Beware that in [[complexity theory]], at least, by a "non-determinstic algorithm" one really refers instead (or more specifically) to the *shortest path* that a non-deterministic machine *could* potentially take to arrive at a desired result (cf. the [NIST entry](#NIST)). This is the sense in which "non-determinism" appears for instance in the name of the [[complexity class]] "[[NP]]" .

(...)

## Formalizations

### Via indefiniteness-effects
 {#FormalizationViaIndefinitenessEffects}

A basic [[programming language]]-model of non-determinism is via dependence of a [[program]] on an unspecified parameter $b \colon B$, whose value is indefinite but imagined to become known at execution time -- say via user input, or by [[observation|observing]] the actual value of a [[random variable]] in an [[experiment]] and/or by making a [[measurement]] of a [[stochastic process]] or a [[quantum measurement]] of a [[quantum state]] (see [below](#ExamplesQuantumComputation)).

In this model, a non-deterministic program taking input [[data]] of [[data type]] $D$ and producing output data of *nominal* type $D'$ would be an actual (determinstic) program of the form

$$
  prog \;\colon\; D \longrightarrow \big( B \to D' \big)
  \,,
$$

whose *de facto* output [[data type]] is not $D'$ but the [[function type]] $B \to D'$. This means that the $D'$-result of computing $prog(d)$ for any $d \colon D$ remains "un-determined" or "indefinite" unless and until a parameter $b \colon B$ is somehow chosen (say "randomly", "experimentally" or in whichever way), in which case only we obtain definite data $prog(d)(b) \;\colon\; D'$.

It makes sense to say that the data type

$$
  \bigcirc_B D' \;\coloneqq\; \big( B \to D'\big)
$$

exhibits a "$B$-indefinitess effect" that goes along with the execution of the program, making its output indefinite. Such *side effects* in computing may be modeled by [[monads in computer science]] (see [there](monad+in+computer+science#BasicIdea) for more). Indeed $\bigcirc_B$ is a classical example of such a monad, often called the *$B$-[[reader monad]]*, reflecting the idea that it postpones returning definite output data until it has "read in" an otherwise unspecified parameter.

> Incidentally, in terms of [[modal logic]] the parameter type $B$ plays the role of the set of "[[possible worlds]]" in which the given non-deterministic computation may take definite values. For more on this modal logic perspective see at *[[necessity and possibility]]* the section *[relation to potentiality](necessity+and+possibility#RelationToPotentiality)*.

Now the [[monad]]-[[structure]] on $\bigcirc_B$ allows to *carry along* this *indefiniteness effect* so that followup programs, nominally waiting for input data in $D'$, are consistently treated as non-deterministic, dependent on specification of $b \colon B$, too.

Concretely, if

$$
  prog \;\colon\; D \to \bigcirc_B D'
$$

and

$$
  prog' \;\colon\; D' \to \bigcirc_B D''
$$

are two such non-deterministic programs but conditioned on the *same* global random variable $b \colon B$, then their consistent [[composition]], as such, is their [[Kleisli composition]] (which involves the categorical [[multiplication]]-operation in the [[reader monad]], see there for details):

$$
  prog
    \;\;\text{>>=}\;\;
  prog'
  \;\;\;\;
  \equiv
  \;\;\;\;
  bind^{\bigcirc_B} prog' \;\circ\; prog
  \;\colon\;
  D \to \bigcirc_B D''
  \,.
$$

Of course, in general the non-determinism of a program need not be parameterized over a fixed parameter type $B$. If the sequence $(B_i)_{i \colon I}$ of parameters is known beforehand, one can form their [[product type]] $B \equiv \prod_i B_i$ and proceed as above, but in general even that sequence may not be pre-specified and further effect handling besides a fixed [[reader monad]] needs to be considered (...).

## Examples

### Quantum computation
 {#ExamplesQuantumComputation}

A fundamental example of non-deterministic computation is [[quantum computation]], where the process of [[quantum measurement]] produces outcomes that -- to the best knowledge of the current state of [[fundamental physics]] -- are fundamentally non-deterministic in that there is not even in principle a way to predict a given outcome with certainty (only their [[probability distribution]] is fixed by [[quantum theory]]).

The possible results of a [[quantum measurement]] are indexed by a "measurement basis" which is exactly a ([[finite set|finite]]) [[set]] $B$ serving as the parameter type in the [[modal logic]]-discussion [above](#FormalizationViaIndefinitenessEffects).
The only difference for quantum computation is that quantum data types are [[linear types]], so that the relevant [[reader monad]] here is a linear version of the classical reader monad -- see at *[[quantum reader monad]]* for details.

A formulation of (non-deterministic) [[quantum computation]] in terms of the resulting $\bigcirc_B$-[[modal type theory|modal]] [[linear type theory]] is discussed at *[[quantum circuits via dependent linear types]]*.

But in fact the [[quantum reader monad]] is (for [[finite set|finite]] $B$) a [[Frobenius monad]] and as such [[isomorphism|isomorphic]] (see [here](quantum+reader+monad#QuantumReaderMonadIsSpecialFrobeniusWriterMonad)) to the "classical structures" Frobenius monad has long been proposed to model [[quantum measurement]] in [[quantum information theory via dagger-compact categories]] ([Coecke & Pavlović (2008)](quantum+information+theory+via+dagger-compact+categories#CoeckePavlović08))

### Differential linear logic
 {#ExamplesDifferentialLinearLogic}

Something reminiscent of --  but different from -- non-deterministic computation appears (see the quotes [below](#QuotesForDifferentialLinearLogic)) in [[differential linear logic]], where the [[cut-elimination]] procedure has been argued to "intuitively correspond" to non-determinism:

The differential linear cut rule receives as input a [[proof]] of type $\Gamma \vdash \Delta$ and returns as output a [[formal sum]] of proofs of type $\Gamma \vdash \Delta$. During the procedure, several proofs can be formed to eliminate cuts from the proof in the previous step, these different possibilities are summed and thus the algorithm returns a [[formal sum]] of proofs. The nondetermism is here realized by the [[addition]] of proofs as formal sums of proofs that can be understood as a [[disjunction]] but in not exactly the same way that the [[linear disjunction]] because it's a disjunction of proofs and not of formulas. An algorithm which returns $\pi_{1} + \pi_{2}$ where $\pi_{1},\pi_{2}$ are two proofs of the same type $\Gamma \vdash \Delta$ returns in fact the disjunction of $\pi_{1}$ and $\pi_{2}$ because it can't chose between the two possibilities. On the level of the syntactic proof theory, one cannot code this sum only by the intermediate of a [[biproduct]] in a [[semiadditive category]], because it would imply that the algorithm would introduce [[cuts]] that will not be eliminated: in a semiadditive category, the sum is defined by combining the structural morphisms of the products and coproducts, which are then equal, through composition, ie. through the [[cut rule]]. One thus works in a [[CMon-enriched symmetric monoidal category]] where we have access to a sum of morphism without necessarily biproducts. In the case where one considers the additive connectors in the logic, the [[additive disjunction]] and the [[additive conjunction]]  are then equated under the presence of formal sums and one can reobtain this formal sum through the biproduct in the semantics. But the semantics is not able to explain the syntactic subtleties related to cut elimination and the difference between a formal sum of two proofs and a sum obtained through cuts and biproducts.

In contradistinction, the cut elimination of [[linear logic]] is a deterministic algorithm and doesn't imply formal sums of proofs. Thus, in linear logic, if one includes the [[additive disjunction]] and the [[additive conjunction]], they are not necessarily equated forming a [[biproduct]] in the syntax, as in differential linear logic.

{#QuotesForDifferentialLinearLogic} From [Ehrhard (2008), Introduction](#Ehrhard08):

> "when one linearizes a proof obtained by contracting two linear inputs of a proof, one has to choose between these two inputs, and there is no canonical way of doing so: we take the non-deterministic superposition of the two possibilities. Syntactically, this means that one must accept the possibility of adding proofs of the same formula, which is standard in mathematics, but hard to accept as a primitive logical operation on proofs."

However, [Erhardt & Regnier (2009), p. 3](differential+linear+logic#EhrhardtRegnier09) admit that:

> "Notice that in our axiomatization of sums in the differential lambda-calculus, the non-deterministic reduction rule $s + s' \to s$ will not be valid, but sums will nevertheless intuitively correspond to a version of non-deterministic choice where all the actual choice operations are postponed: the result of a term reduction will in general be a large formal sum of terms, and we can reduce the really "non-deterministic" part of the computation to a unique ultimate step consisting in choosing one term of that sum. This step is however a fiction and will not appear as a reduction rule of our calculus"

## Related concepts

* [[computation]], [[quantum computation]]

* [[non-deterministic automaton]]

* [[repeat-until-success computing]]

* [[reversible computation]]

* [[parallel computing]]

## References

* {#Sipser2012} [[Michael Sipser]], §1.2 in: *Introduction to the Theory Of Computation*, 3rd ed: Cengage Learning (2012) [ISBN:978-1-133-18779-0,[pdf](https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf), [pdf](https://fuuu.be/polytech/INFOF408/Introduction-To-The-Theory-Of-Computation-Michael-Sipser.pdf)]

See also:

* Wikipedia, *[Nondeterministic algorithm](https://en.wikipedia.org/wiki/Nondeterministic_algorithm)*

Nondeterministic Turing machines:

* [Sipser (2012), §3.2](#Sipser2012)

* Wikipedia, *[Nondeterministic Turing machine](https://en.wikipedia.org/wiki/Nondeterministic_Turing_machine)*

The notion in [[complexity theory]]:

* {#NIST} NIST, *[nondeterministic algorithm](https://xlinux.nist.gov/dads/HTML/nondetermAlgo.html)*

The above treatment of non-deterministic [[quantum computation]] via the [[quantum reader monad]] is made explicit in

* {#CQTS22} [[CQTS]], *[Quantum Data Types via Linear HoTT](https://ncatlab.org/schreiber/show/QDataInLHoTT#QTML2022)*, talk at *[Workshop on Quantum Software](https://quasar.unina.it/qtml2022/workshop.php)* @ *[QTML 2022](https://quasar.unina.it/qtml2022.html)* (Nov 2022) [[pdf](/schreiber/files/QuantumDataInLHoTT-221117.pdf)]

Concerning the quasi-example of cut elimination in differential linear logic:

* {#Ehrhard08}[[Thomas Ehrhard]], *An introduction to Differential Linear Logic: proof-nets, models and antiderivatives*, Mathematical Structure in Computer Science **28** 7 (2018) 995-1060 ([arXiv:1606.01642](https://arxiv.org/abs/1606.01642), [doi:10.1017/S0960129516000372](https://doi.org/10.1017/S0960129516000372))

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