D-brane charge quantization in topological K-theory -- references

### D-brane charge quantization in topological K-theory
 {#DBraneChargeQuantizationInTopologicalKTheory}

On the conjectural [[D-brane charge quantization in topological K-theory]]:

#### Origin and basics

The idea that [[D-branes]] have [[Dirac charge quantization]] in [[topological K-theory]] originates with the observation that their charge expressed in [[RR-field]] [[flux densities]] resembles the image of a [[Chern character]]:

* {#GreenHarveyMoore97} [[Michael Green]], [[Jeffrey A. Harvey]], [[Gregory Moore]], *I-Brane Inflow and Anomalous Couplings on D-Branes*, Class. Quant. Grav. **14** (1997) 47-52 $[$[arXiv:hep-th/9605033](https://arxiv.org/abs/hep-th/9605033), [doi:10.1088/0264-9381/14/1/008](https://doi.org/10.1088/0264-9381/14/1/008)$]$

* {#MinasianMoore97} [[Ruben Minasian]], [[Gregory Moore]], *K-theory and Ramond-Ramond charge*, JHEP 9711:002 (1997) $[$[arXiv:hep-th/9710230](http://arxiv.org/abs/hep-th/9710230), [doi:10.1088/1126-6708/1997/11/002](https://doi.org/10.1088/1126-6708/1997/11/002)$]$

Further early discussion:

* {#Witten98} [[Edward Witten]], _D-Branes And K-Theory_, JHEP 9812:019 (1998) \[arXiv:hep-th/9810188, [doi:10.1088/1126-6708/1998/12/019](https://doi.org/10.1088/1126-6708/1998/12/019)\]

* {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], *On Ramond-Ramond fields and K-theory*, JHEP 0005 (2000) 044 $[$[arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027), [doi:10.1088/1126-6708/2000/05/044](https://doi.org/10.1088/1126-6708/2000/05/044)$]$

and with emphasis on the full picture of [[twisted differential K-theory|twisted]] [[differential K-theory]] in:

* {#Freed00} [[Daniel Freed]], *[[Dirac charge quantization and generalized differential cohomology]]*, Surveys in Differential Geometry,  Int. Press, Somerville, MA, 2000, pp. 129–194 ([arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392))

{#BasisOfTheConjecture} Here:

* [Green, Harvey & Moore (1997)](#GreenHarveyMoore97), [Minasian & Moore (1997)](#MinasianMoore97) observe that [[RR-field]] [[flux]] [[differential form|form]]-expressions for [[D-brane charge]] look like images of K-theory classes under the [[Chern character]];

\begin{imagefromfile}
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    "caption": "From [Sch 18](https://ncatlab.org/schreiber/show/Equivariant+Stable+Cohomotopy+and+Branes)"
\end{imagefromfile}

* [Witten 98, Section 3](#Witten98) adds the observation that the [[tachyon condensation]] -- which is expected (this is _[[Sen's conjecture]]_ from [Sen 98](Sen's+conjecture#ReferencesForSuperstrings)) for [[open strings]] between [[D-brane]]/[[anti-D-branes]] -- plausibly implements on their [[Chan-Paton gauge field|Chan-Paton]] [[vector bundles]] the defining [[equivalence relation]] ([here](topological+K-theory#eq:DefiningEquivalenceRelation)) of [[topological K-theory]].

Expression of these D-brane K-theory classes via the [[Atiyah-Hirzebruch spectral sequence]]:

* {#MaldacenaMooreSeiberg01} [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], _D-Brane Instantons and K-Theory Charges_, JHEP 0111:062,2001 ([arXiv:hep-th/0108100](http://arxiv.org/abs/hep-th/0108100))

* {#EvslinSati06} [[Jarah Evslin]], [[Hisham Sati]], _Can D-Branes Wrap Nonrepresentable Cycles?_, JHEP0610:050,2006 ([arXiv:hep-th/0607045](http://arxiv.org/abs/hep-th/0607045))

Specifically for D-branes in [[WZW models]] see

* [[Peter Bouwknegt]], _A note on equality of algebraic and geometric D-brane charges in WZW models_ ([pdf](http://people.physics.anu.edu.au/~drt105/papers/BR0312259.pdf))

Understanding the [[solitonic brane|solitonic]] (non-singular) D-branes and their [[T-duality]] in K-theory:

* {#BergmanGimonHořava99} [[Oren Bergman]], [[Eric G. Gimon]], [[Petr Hořava]], *Brane Transfer Operations and T-Duality of Non-BPS States*, JHEP 9904 (1999) 010 \[doi:10.1088/1126-6708/1999/04/010, [arXiv:hep-th/9902160](https://arxiv.org/abs/hep-th/9902160)\]

Towards a [[BFSS matrix model|matrix model]] taking these K-theoretic effects into account ([[K-matrix model]]):

* {#AsakawaSugimotoTerashima01} [[Tsuguhiko Asakawa]], [[Shigeki Sugimoto]], [[Seiji Terashima]], *D-branes, Matrix Theory and K-homology*, JHEP 0203 (2002) 034 $[$[arXiv:hep-th/0108085](https://arxiv.org/abs/hep-th/0108085), [doi:10.1088/1126-6708/2002/03/034](https://doi.org/10.1088/1126-6708/2002/03/034)$]$

#### Twisted, equivariant and differential refinement

Discussion of charge quantization in [[twisted K-theory]] for the case of non-vanishing [[B-field]]:

* [Witten 98, Sec. 5.3](#Witten98) (for [[torsion subgroup|torsion]] twists)

* [[Peter Bouwknegt]], [[Varghese Mathai]], *D-branes, B-fields and twisted K-theory*, Int. J. Mod. Phys. A **16** (2001) 693-706 $[$[arXiv:hep-th/0002023](https://arxiv.org/abs/hep-th/0002023), [doi:10.1088/1126-6708/2000/03/007](https://doi.org/10.1088/1126-6708/2000/03/007)$]$

An elaborate proposal for the correct flavour of [[equivariant K-theory|equivariant]] [[KR-theory]] needed for [[orientifolds]] is sketched in:

* {#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 full-blown [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]]

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], *Ramond-Ramond fields and twisted differential K-theory*, Advances in Theoretical and Mathematical Physics **26** 5 (2022)  $[$[doi:10.4310/ATMP.2022.v26.n5.a2](https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a2), [arXiv:1903.08843](https://arxiv.org/abs/1903.08843)$]$

Discussion of full-blown [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

#### Reviews

* {#OlsenSzabo00} Kasper Olsen, [[Richard Szabo]], _Brane Descent Relations in K-theory_, Nucl.Phys. B566 (2000) 562-598 ([arXiv:hep-th/9904153](https://arxiv.org/abs/hep-th/9904153))

* Kasper Olsen, [[Richard Szabo]], _Constructing D-Branes from K-Theory_, Adv. Theor. Math. Phys. 3 (1999) 889-1025 ([arXiv:hep-th/9907140](https://arxiv.org/abs/hep-th/9907140))

* [[John Schwarz]], _TASI Lectures on Non-BPS D-Brane Systems_ ([arXiv:hep-th/9908144](https://arxiv.org/abs/hep-th/9908144))

* {#Witten00} [[Edward Witten]], _Overview Of K-Theory Applied To Strings_, Int. J. Mod. Phys. A16:693-706, 2001 ([arXiv:hep-th/0007175](https://arxiv.org/abs/hep-th/0007175))

* {#Moore02} [[Greg Moore]], _K-Theory from a physical perspective_, in: [[Ulrike Tillmann]] (ed.) _Topology, Geometry and Quantum Field Theory_, Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, Cambridge University Press (2004)  ([arXiv:hep-th/0304018](http://arxiv.org/abs/hep-th/0304018), [doi:10.1017/CBO9780511526398.011](https://doi.org/10.1017/CBO9780511526398.011))

* {#Manjarin04} [[Juan José Manjarín]], _Topics on D-brane charges with B-fields_, Int. J. Geom. Meth. Mod. Phys. 1 (2004) ([arXiv:hep-th/0405074](http://arxiv.org/abs/hep-th/0405074))

* {#Evslin06} [[Jarah Evslin]], _What Does(n't) K-theory Classify?_, Modave Summer School in Mathematical Physics ([arXiv:hep-th/0610328](https://arxiv.org/abs/hep-th/0610328), [spire:730502](https://inspirehep.net/literature/730502))

* {#Fredenhagen08} [[Stefan Fredenhagen]], _Physical Background to the K-Theory Classification of D-Branes: Introduction and References_ ([doi:10.1007/978-3-540-74956-1_1](https://doi.org/10.1007/978-3-540-74956-1_1)), chapter in: [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]],  _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Lecture Notes in Physics, Springer (2008) 1-9 $[$[doi:10.1007/978-3-540-74956-1](https://doi.org/10.1007/978-3-540-74956-1), [pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf)$]$

* [[Fabio Ruffino]], _Topics on topology and superstring theory_ ([arXiv:0910.4524](http://arxiv.org/abs/0910.4524))

Amplification of [[torsion subgroup|torsion]]-charges implied by charge quantization in Ktheory

* [[Volker Braun]], *K-Theory Torsion* [arXiv:hep-th/0005103]

* [[Ilka Brunner]], [[Jacques Distler]], *Torsion D-Branes in Nongeometrical Phases*, Adv. Theor. Math. Phys. **5** (2002) 265-309 \[doi:10.4310/ATMP.2001.v5.n2.a3, [arXiv:hep-th/0102018](https://arxiv.org/abs/hep-th/0102018)\]

* [[Ilka Brunner]], [[Jacques Distler]], Rahul Mahajan, *Return of the Torsion D-Branes*, Adv. Theor. Math. Phys. **5** (2002) 311-352 \[doi:10.4310/ATMP.2001.v5.n2.a4, [arXiv:hep-th/0106262](https://arxiv.org/abs/hep-th/0106262)\]

Review of D-branes charge seen in [[KK-theory]]:

* {#Szabo} [[Richard Szabo]], _D-branes and bivariant K-theory_, Noncommutative Geometry and Physics 3 1 (2013): 131. ([arXiv:0809.3029](http://arxiv.org/abs/0809.3029))

based on

* {#ReisSzabo05} Rui Reis, [[Richard Szabo]], _Geometric K-Homology of Flat D-Branes_ , Commun. Math. Phys. **266** (2006) 71-122 [[arXiv:hep-th/0507043](https://arxiv.org/abs/hep-th/0507043)]

* {#BrodzkiMathaiRosenbergSzabo06} [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_, Commun. Math. Phys. **277** (2008) 643-706 $[$[arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020), [doi:10.1007/s00220-007-0396-y](https://doi.org/10.1007/s00220-007-0396-y)$]$

* {#BMRS2} [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]],  _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys. 13:497-552, 2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))

* [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]],  _D-branes, KK-theory and duality on noncommutative spaces_, J. Phys. Conf. Ser. **103** 012004 (2008) $[$[arXiv:0709.2128](http://arxiv.org/abs/0709.2128), [doi:10.1088/1742-6596/103/1/012004](https://doi.org/10.1088/1742-6596/103/1/012004)$]$

In particular ([BMRS2](#BMRS2)) discusses the definition and construction of D-brane charge as a generalized [[index]] in [[KK-theory]]. The discussion there focuses on the untwisted case. Comments on the generalization of this to topologicall non-trivial [[B-field]] and hence [[twisted K-theory]] is in

* [[Richard Szabo]], _D-Branes, Tachyons and K-Homology_,  	Mod. Phys. Lett. A17 (2002) 2297-2316 ([arXiv:hep-th/0209210](http://arxiv.org/abs/hep-th/0209210))

#### Conceptual problems
 {#DBraneChargeInKTheoryReferencesConceptualProblems}

{#KTheorySubtleties} But there remain conceptual issues with the proposal that D-brane  charge is in K-theory, as highlighted in

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 4.5.2 and 4.6.5 of  *Triples, Fluxes, and Strings*, Adv. Theor. Math. Phys. **4** (2002) 995-1186 \[arXiv:hep-th/0103170, [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2000/0004/0005/ATMP-2000-0004-0005-a001.pdf)\]

* {#Evslin06} [[Jarah Evslin]], section 8 of: *What Does(n't) K-theory Classify?*, Second Modave Summer School in Mathematical Physics \[arXiv:hep-th/0610328\]

In particular, actual checks of the proposal that D-brane charge is given by K-theory, via concrete computation in [[boundary conformal field theory]], have revealed some subtleties:

* {#FredenhagenQuella05} [[Stefan Fredenhagen]], [[Thomas Quella]], *Generalised permutation branes*, JHEP 0511:004 (2005) \[arXiv:hep-th/0509153, doi:10.1088/1126-6708/2005/11/004\]

  > It might surprise that despite all the progress that has been made in understanding branes on group manifolds, there are usually not enough D-branes known to explain the whole charge group predicted by (twisted) K-theory. \[...\]  it is fair to say that a satisfactory answer is still missing.

The closest available towards an actual check of the argument for K-theory via [[Sen's conjecture|open superstring tachyon condensation]] ([Witten 98, Section 3](#Witten98)) seems to be

* {#Erler13} [[Theodore Erler]], _Analytic Solution for Tachyon Condensation in Berkovits' Open Superstring Field Theory_, JHEP 1311 (2013) 007 \[doi:10.1007/JHEP11(2013)007, [arXiv:1308.4400](https://arxiv.org/abs/1308.4400)\]

which, however, concludes (on [p. 32](https://arxiv.org/pdf/1308.4400.pdf#page=33)) with:

> It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge $[$[75, 76, 77](D-brane#DBraneChargeQuantizationInTopologicalKTheory)$]$. We leave these questions for future work.

See also

* {#Erler19} [[Theodore Erler]], _Four Lectures on Analytic Solutions in Open String Field Theory_ ([arXiv:1912.00521](https://arxiv.org/abs/1912.00521), [spire:1768105](https://inspirehep.net/literature/1768105))

which still lists (on [p. 112](https://arxiv.org/pdf/1912.00521.pdf#page=113)) among open problems of [[string field theory]]:

> "Are there topological invariants of the open string star algebra representing D-brane charges?"

#### For orbifolds in equivariant K-theory

{#ChargeInEquivariantKTheory} The proposal that D-brane charge on [[orbifolds]] is measured in [[equivariant K-theory]] ([[orbifold K-theory]]) goes back to

* [Witten 98, section 5.1](#Witten98)

It was pointed out that only a subgroup of equivariant K-theory can be physically relevant in

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], around (137) of:  _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

Further discussion of [[equivariant K-theory]] for D-branes on [[orbifolds]] includes the following:

* Hugo García-Compeán, _D-branes in orbifold singularities and equivariant K-theory_, Nucl.Phys. B557 (1999) 480-504 ([arXiv:hep-th/9812226](https://arxiv.org/abs/hep-th/9812226))

* [[Matthias Gaberdiel]], [[Bogdan Stefanski]], _Dirichlet Branes on Orbifolds_, Nucl.Phys.B578:58-84, 2000 ([arXiv:hep-th/9910109](https://arxiv.org/abs/hep-th/9910109))

* [[Igor Kriz]], Leopoldo A. Pando Zayas, Norma Quiroz, _Comments on D-branes on Orbifolds and K-theory_, Int.J.Mod.Phys.A23:933-974, 2008 ([arXiv:hep-th/0703122](https://arxiv.org/abs/hep-th/0703122))

* [[Richard Szabo]], [[Alessandro Valentino]], _Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory_, Commun.Math.Phys.294:647-702, 2010 ([arXiv:0710.2773](https://arxiv.org/abs/0710.2773))

Discussion of [[real K-theory]] for D-branes on [[orientifolds]] includes the following:

The original observation that [[D-brane charge]] for [[orientifolds]] should be in [[KR-theory]] is due to

* [Witten 98, section 5](#Witten98)

and was then re-amplified in

* {#Gukov99} [[Sergei Gukov]], _K-Theory, Reality, and Orientifolds_, Commun.Math.Phys. 210 (2000) 621-639 ([arXiv:hep-th/9901042](https://arxiv.org/abs/hep-th/9901042))

* {#BergmanGimonSugimoto01} [[Oren Bergman]], E. Gimon, [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

With further developments in

* [[Varghese Mathai]], [[Michael Murray]], [[Daniel Stevenson]], _Type I D-branes in an H-flux and twisted KO-theory_, JHEP 0311 (2003) 053 ([arXiv:hep-th/0310164](https://arxiv.org/abs/hep-th/0310164))

Discussion of orbi-orienti-folds using [[equivariant K-theory|equivariant]] [[KO-theory]] is in

* N. Quiroz, [[Bogdan Stefanski]], _Dirichlet Branes on Orientifolds_, Phys.Rev. D66 (2002) 026002 ([arXiv:hep-th/0110041](https://arxiv.org/abs/hep-th/0110041))

* [[Volker Braun]], [[Bogdan Stefanski]], _Orientifolds and K-theory_ ([arXiv:hep-th/0206158](https://arxiv.org/abs/hep-th/0206158))

* H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, _D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory_, JHEP 0812:007, 2008 ([arXiv:0809.4238](https://arxiv.org/abs/0809.4238))

Discussion of the alleged K-theory classification of D-brane charge in relation to the [[M-theory]] [[supergravity C-field|C-field]] is in

* {#DMW00} [[Duiliu-Emanuel Diaconescu]], [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134,2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))

See also

* {#GarciaUranga05} Inaki Garcia-Etxebarria, [[Angel Uranga]], _From F/M-theory to K-theory and back_, JHEP 0602:008,2006 ([arXiv:hep-th/0510073](https://arxiv.org/abs/hep-th/0510073))

More complete discussion of [[double dimensional reduction]] of the [[supergravity C-field]] in 11d to the expected [[B-field]] and [[RR-field]] [[flux]]  forms in 10d:

* {#MathaiSati03} [[Varghese Mathai]], [[Hisham Sati]], _Some Relations between Twisted K-theory and $E_8$ Gauge Theory_, JHEP0403:016,2004 ([arXiv:hep-th/0312033](http://arxiv.org/abs/hep-th/0312033))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Rational sphere valued supercocycles in M-theory and type IIA string theory]]_, Journal of Geometry and Physics, Volume 114, Pages 91-108 April 2017 ([arXiv:1606.03206](http://arxiv.org/abs/1606.03206), [doi:10.1016/j.geomphys.2016.11.024](http://dx.doi.org/10.1016/j.geomphys.2016.11.024))


D-brane geometry

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## Idea

Regarding the [[string theory|quantum string]] as a [[2-spectral triple]], it defines a [[spectral geometry]] (typically but not necessarily a [[noncommutative geometry]]) which is the [[effective field theory|effective]] [[spacetime]] as seen by this quantum string (as read of from, notably, its [[energy]] [[spectrum of a linear operator|spectrum]]). For the [[open string]] the most prominent aspect of its [[2d SCFT]] [[worldsheet]] theory are its [[boundary conditions]]. In the spectral interpretation these correspond to the presence of [[D-branes]] in the effective [[target spacetime]]. Much geometric information is contained in these D-brane states, and the resulting concept of (noncommutative) geometry has accordingly been called _D-brane geometry_ or _D-geometry_ for short.

## Examples

### Fuzzy spheres

The [[fuzzy spheres]] appear in [[D-brane geometry]]:

1. the [[fuzzy funnels]] of [[Dp-D(p+2)-brane intersections]] have [[fuzzy 2-sphere]] slices

1. the [[fuzzy funnels]] of [[Dp-D(p+4)-brane intersections]] have [[fuzzy 4-sphere]] slides

1. the [[supersymmetry|supersymmetric]] [[classical field theory|classical]] solutions of the [[BMN matrix model]] are precisely [[fuzzy 2-sphere]] configurations ([BMN 02 (5.4)](BMN+matrix+model#BerensteinMaldacenaNastase02)).

## References

* [[Michael Douglas]], _Two Lectures on D-Geometry and Noncommutative Geometry_ ([arxiv:hep-th/9901146](http://arxiv.org/abs/hep-th/9901146))

* [[Liang Kong]], _Conformal field theory and a new geometry_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ ([arXiv:1107.3649](http://arxiv.org/abs/1107.3649))

* Chien-Hao Liu, _Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes_ ([arXiv:1112.4317](http://arxiv.org/abs/1112.4317))

With emphasis on the [[IKKT matrix model]] and the [[noncommutative geometry]] of [[fuzzy spheres]]:

* {#Steinacker10} [[Harold Steinacker]], _Emergent Geometry and Gravity from Matrix Models: an Introduction_, Class. Quant. Grav.27: 133001, 2010 ([arXiv:1003.4134](https://arxiv.org/abs/1003.4134))

* [[Harold Steinacker]], _Non-commutative geometry and matrix models_, PoS QGQGS2011 (2011) 004 ([arXiv:1109.5521](https://arxiv.org/abs/1109.5521))

* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))


D-branes > history


D-geometry

> This entry is about the concept in the [[algebraic geometry]] of [[jet bundles]] and [[differential equations]]. For the concept of geometry as seen by [[D-branes]] see at _[[D-brane geometry]]_.

***

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## Idea

The term _$\mathcal{D}$-geometry_ refers to a [[synthetic differential geometry|synthetic]] formulation of [[differential geometry]] with a focus on the geometry of [[de Rham spaces]]. A [[quasicoherent sheaf of modules]] over a de Rham space is called a _[[D-module]]_, short for _[[module]] over the algebra of [[differential operators]]_, which is where the term "D-geometry" derives from.

## Related entries

* [[regular differential operator]], [[ordinary differential equation]]

* [[connection]], [[flat connection]], [[local system]], [[D-module]], [[perverse sheaf]]

* [[holonomic D-module]], [[meromorphic connection]], [[characteristic variety]]

* [[diffiety]], [[Grothendieck connection]], [[crystal]], [[de Rham space]]

* [[synthetic differential geometry]], [[infinitesimal object]]

* [[Kähler differential]], [[tangent category]], [[tangent (∞,1)-category]], [[cotangent complex]], [[deformation theory]], [[formally smooth morphism]], [[jet bundle]], [[jet space]]

* [[Stokes phenomenon]], [[wall crossing]]

* [[microlocal analysis]], [[algebraic microlocalization]]

* [[Riemann-Hilbert problem]], [[Fuchsian equation]]

* [[algebraic approaches to differential calculus]]

* [[regular differential operator in noncommutative geometry]]

* [[holonomic quantum field]], [[Hodge theory]], [[algebraic analysis]]

## References

* [[Alexander Beilinson]] and [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

  _chapter 2, Geometry of D-schemes_ ([pdf](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch2.pdf))

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]


d-invariant

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#Contents#
* table of contents
{:toc}

## Idea

Let

\[
  \label{TheMap}
  \big[
    X \overset{f}{\longrightarrow} Y
  \big]
\]

be a [[morphism]] in the [[stable homotopy category]] (for instance the [[stabilization]] of a morphism in the [[classical homotopy category]]) and let $E$ be a [[Whitehead-generalized cohomology theory]] (usually assumed to be at least [[multiplicative cohomology theory|multiplicative]]).

Then the _$d$-invariant of $f$ with respect to $E$_ ([Adams 66, Sec. 3, p. 26](#Adams66), short for "degree invariant", see Example \ref{DInvariantSpecializesToDegreeOfAContinuousFunction} below)  is the class of the corresponding [[pullback in cohomology|pullback]] morphism $f^\ast_E$, regarded as an element of the 0th [[Ext-group]] (i.e. the plain [[hom-object]]) between the $E$-cohomologies of $X$ and $Y$:

\[
  \label{ThePullbackInECohomology}
  d(f)
  \;\coloneqq\;
  f^\ast_E
  \;\in\;
  \big(
    E^\bullet(Y);
    \,
    E^\bullet(X)
  \big)
  \;=\;
  Ext^0
  \big(
    E^\bullet(Y);
    \,
    E^\bullet(X)
  \big)
  \,.
\]

The fancy name is justified by the fact that this is beginning of a hierarchy of more interesting invariants of stable maps seen in $E$-cohomology, each of which defined when the previous one vanishes: The next is the _[[e-invariant]]_, then comes the _[[f-invariant]]_ (e.g. [BL 09, Section 2](#BL09)). These are the elements that appear in the first lines on the second page of the $E$-[[Adams spectral sequence]] for $[X,Y]_\bullet$.

## Properties

### Relation to the Hopf degree
  {#RelationToHopfDegree}

+-- {: .num_example #DInvariantSpecializesToDegreeOfAContinuousFunction}
###### Example
**([[d-invariant]] specializes to [[degree of a continuous function]])**

In the special case that

* $E = H\mathbb{Z}$ is [[integral cohomology|integral]] [[ordinary cohomology]];

* $X$ and $Y$ are [[connected topological space|connected]] [[closed manifold|closed]] [[orientation|oriented]] [[topological manifolds]] of the same [[dimension]] $n = dim(X) = dim(Y)$

then the $n$-component of the $d$-invariant (eq:ThePullbackInECohomology)  is the _[[degree of a continuous function|degree]] of $f$_

\[
  \label{DegreeOfAContinuousFunction}
  d^n(f)
  =
  deg(f)
  \;\in\;
  Hom_{Ab}
  \big(
    H^n(Y;\mathbb{Z}),
    \,
    H^n(YX;\mathbb{Z})
  \big)
  \;\simeq\;
  \mathbb{Z}
  \,.
\]

If moreover $Y = S^n$ is the [[n-sphere]], so that $f$ represents a class in the $n$th [[Cohomotopy]] of $X$, then $\widetilde H^\bullet(Y; \mathbb{Z})$ is concentrated on $\mathbb{Z}$ in degree $n$ and hence the $n$ component $d^n(f)$ is the full information in the d-invariant.

In this case, the [[Hopf degree theorem]] says that the degree of $f$, and hence the d-invariant of $f$ in integral cohomology, completely characterizes the homotopy class $[f]$.

=--

### Trivializations of the d-invariant
  {#TrivializationsOfThedInvariant}

When the d-invariant vanishes, then the [[e-invariant]] exists. But in fact, classical constructions of the e-invariant proceed via a _choice of a trivialization_ of the d-invariant and conclude by showing the result to be independent of this choice (this is meant to be brought out at _[Adams e-invariant -- Construction via unit cofiber theories](Adams+e-invariant#ConstructionViaUnitCofiberTheories)_).

Here we discuss the space of choices of trivializations of the d-invariant
(following [[schreiber:M-Theory as Mf-Theory|SS 21]]):

\begin{definition}[Set of trivializations of the d-invariant]
  \label{SetOfTrivializationsOfTheDInvariant}
  For $E$ a [[multiplicative cohomology theory]]
  and $n, d \in \mathbb{N}$,
  write
  \begin{equation}
    \label{SetOfTrivializationsOfdInvariant}
    \begin{aligned}
    H^E_{n-1}\!Fluxes\!\big( S^{n+d-1} \big)
    &
    \;\coloneqq\;
    \underset{
      [c] \in \mathbb{S}_{d-1}
    }{\sqcup}
    \pi_0
    \mathrm{Paths}_0^{c^\ast (1^E)}
    \Big(
    \mathrm{Maps}^{\ast/}\!
      \big(
        S^{n+d-1}
        \,,\,
        E^{n}
      \big)
    \Big)
    \end{aligned}
  \end{equation}
  for the [[set]] of [[tuples]] consisting of the
  [[stable Cohomotopy]] class
  $\big[ G^{\mathbb{S}}_{n}\!(c) \big]$
  of a [[map]] $S^{n+d-1} \xrightarrow{\;c\;}S^{n}$
  and the [[2-homotopy]] class
  $\big[ H^E_{n-1}\!(c) \big]$
  of a trivialization (if any) of its
  [[d-invariant]] $c^\ast(1^E)$ in $E$-cohomology.

So an element of $H^E_{n-1}\!Fluxes\!\big( S^{n+d-1} \big)$ (eq:SetOfTrivializationsOfdInvariant) is the [[2-homotopy]] class relative the boundary of a [[homotopy coherent diagram]]of the following form:

\begin{xymatrix@C=75pt@R=36pt}
      S^{n+d-1}
      \ar[rr]_>>{\ }="s"
      \ar[d]
        |-{
          \mathclap{\phantom{\vert^{\vert}}}
          c
          \mathclap{\phantom{\vert_{\vert}}}
        }
      \ar[dr]|-{
        G^{\mathbb{S}}_{n}\!(c)
      }
      ^>>>>>{\ }="t"
      &&
      \ast
      \ar[d]
        |-{
          \mathclap{\phantom{\vert^{\vert}}}
          0
          \mathclap{\phantom{\vert_{\vert}}}
        }
      \\
      S^{n}
      \ar@/_1.3pc/[rr]|-{ \;\Sigma^{n}(1^E)\; }
      \ar[r]|-{
        \;
        \Sigma^{n}
        (1^{\mathbb{S}})
        \;
      }
      &
      \mathbb{S}^{n}
      \ar[r]|-{
        \;
        (e^E)^{n}
        \;
      }
      &
      E^{n}
      %
      \ar@{==>}
        |-{
          \;
          \color{orange}
          H^E_{n-1}\!(c)
          \;
        }
        "s"; "t"
\end{xymatrix}

\end{definition}

\begin{definition}[Unit cofiber cohomology theory]
  \label{UnitCofiberCohomologyTheory}

For $E$ a [[multiplicative cohomology theory]] with unit map $\mathbb{S} \overset{ e^E }{\longrightarrow} E$ we denote the corresponding [[homotopy cofiber]]-theory as follows:

\begin{xymatrix@C=50pt@R=12pt}

    \mathbb{S}
    \ar[dd]
      ^>>{\ }="t"
    \ar[rr]
      ^-{
        \;1^E\;
      }
      _>>{\ }="s"
    &&
    \overset{
      \mathclap{
      \raisebox{4pt}{
        \tiny
        \color{blue}
        \bf
        \begin{tabular}{c}
          $E$-cohomology
        \end{tabular}
      }
      }
    }{
      E
    }
    \ar[dd]
      _-{
        \mathclap{\phantom{\vert^{\vert^{\vert}}}}
        i^E
        \mathclap{\phantom{\vert_{\vert}}}
      }
      ^>>{\ }="t2"
    \ar[rr]
      ^-{
      }
      _>>{\ }="s2"
    &&
    \ast
    \ar[dd]
    \\
    {\phantom{A}}
    \\
    \ast
    \ar[rr]
    &&
    \underset{
      \mathclap{
      \raisebox{-4pt}{
        \tiny
        \color{blue}
        \bf
        \begin{tabular}{c}
          $E$-cofiber
          \\
          cohomology
          \\
          theory
        \end{tabular}
      }
      }
    }{
      E/\mathbb{S}
    }
    \ar[rr]
      |-{ \;\partial^E\; }
      _-{
        \mbox{
          \tiny
          \color{green}
          \bf
          \begin{tabular}{c}
            \phantom{-}
            \\
            boundary
            \\
            operation
          \end{tabular}
        }
      }
    &&
    \underset{
      \mathclap{
      \raisebox{-6pt}{
        \tiny
        \color{blue}
        \bf
        \begin{tabular}{c}
          shifted stable
          \\
          Cohomotopy
        \end{tabular}
      }
      }
    }{
      \Sigma \mathbb{S}
    }
    \,,
    &
    {\phantom{AAA}}
    %
    \ar@{=>}^-{ \mbox{\tiny\rm(po)} }
      "s"; "t"
    \ar@{=>}^-{ \mbox{\tiny\rm(po)} }
      "s2"; "t2"

\end{xymatrix}

\end{definition}

\begin{proposition}[Trivializations of d-invariant are classes in cofiber theory]
  \label{EFluxesAreCocycleInCofiberTheory}
  There is a map  between the set
  (eq:SetOfTrivializationsOfdInvariant)
  of trivializations of the d-invariant
  and the [[cohomology group]] of the unit cofiber theory
  $E\!/\mathbb{S}$ (Def. \ref{UnitCofiberCohomologyTheory}):

\begin{xymatrix@=18pt}
     \Big(
       \big[
         G^{\mathbb{S}}_n\!(c)
       \big]
       \,,\,
       \big[
         H^E_{n-1}\!(c)
       \big]
      \Big)
    \ar@{|->}[dd]
    &
     \overset{ \mathllap{
        \mbox{
          \tiny
          \bf
          \begin{tabular}{c}
          \end{tabular}
        }
      }}{
      H^E_{n-1}\mbox{\rm{Fluxes}}\!
      \big(
        S^{n + d - 1}
      \big)
      }
      \ar[rr]^-{ \simeq }
      \ar[dd]
      &&
      \big(
        E \!/\mathbb{S}
      \big)_{d}
      \ar[dd]^{\partial}
      \\
      \\
        \big[
          G^{\mathbb{S}}_n\!(c)
        \big]
      &
      G^{\mathbb{S}}_n\mathrm{Fluxes}
      \big(
        S^{n+d-1}
      \big)
      \ar@{=}[rr]
      &&
      \mathbb{S}_{d-1}
\end{xymatrix}
  compatible with the fibrations
  of both over the underlying stable Cohomotopy classes
  $
    G^{\mathbb{S}}_n Fluxes
    \big(
      S^{n+d-1}
    \big)
    \;\coloneqq\;
    \widetilde {\mathbb{S}}{}^d
    \big( S^{n + d - 1}\big)
    \,\simeq\,
    \mathbb{S}_{d-1}
  $.
\end{proposition}

We give two proofs: A quick abstract one and a more explicit one that proceeds via classes on the cofiber space of $c$. The latter is close to the old argument of [Conner-Floyd 66, Thm. 16.2](MUFr#ConnerFloyd66) (there for $E/\mathbb{S} = $ [[MUFr]], see [this section](MUFr#RelationToMUAndFr) for more).

\begin{proof}\label{QuickAbstractProof}[quick abstract]

By Definition \ref{SetOfTrivializationsOfTheDInvariant}, an element in $H_{n-1}Fluxes^E\big( S^{n+d-1} \big)$ is equivalently the class of a homotopy [[cone]] with tip $\Sigma^{n+d-1} \mathbb{S}$ over the [[cospan]] formed by the ring spectrum unit $e^E$ and the [[zero morphism]]:

\begin{xymatrix@=26pt}
    \Sigma^{n+d-1}\mathbb{S}
    \ar[rr]_>>>{\ }="s"
    \ar[dd]^>>>{\ }="t"
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        c
        \mathclap{\phantom{\vert_{\vert}}}
      }
    &&
    \ast
    \ar[dd]
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        0
        \mathclap{\phantom{\vert_{\vert}}}
      }
    \\
    \\
    \Sigma^n \mathbb{S}
    \ar[rr]
      |-{
        \; \Sigma^n(e^E) \;
      }
    &&
    \Sigma^n E
    %
    \ar@{=>}
      |-{
        \mathclap{\phantom{\vert^{\vert^{\vert}}}}
        {
          \color{orange}
          H^E_{n-1}\!(c)
        }
        \mathclap{\phantom{\vert_{\vert_{\vert}}}}
      }
      "s"; "t"
\end{xymatrix}

But in [[Spectra]] [[homotopy cofiber sequences]] are [[homotopy fiber sequences]] (by [this Prop.](Spectrum#InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences)), so that by the [[universal property]] of [[homotopy fibers]] the class of the above diagram is equivalently the class of a map from $\Sigma^{n+d-1}\mathbb{S}$ to  $fib\big( \Sigma^{n} e^E \big) \,\simeq\, \Sigma^{n-1} (E/\mathbb{S}$):

\begin{xymatrix@=26pt}
    \Sigma^{n+d-1}\mathbb{S}
    \ar@/_1pc/[dddr]
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        c
        \mathclap{\phantom{\vert_{\vert_{\vert}}}}
      }
    \ar@/^1pc/[rrrd]
    \ar@{-->}[dr]
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        \vdash
        {
          \color{orange}
          H^E_{n-1}\!(c)
        }
        \mathclap{\phantom{\vert_{\vert}}}
      }
    \\
    &
    \Sigma^{n-1}
    (\mathbb{S}/E)
    \ar[rr]_>>>{\ }="s"
    \ar[dd]^>>>{\ }="t"
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        \partial
        \mathclap{\phantom{\vert_{\vert}}}
      }
    &&
    \ast
    \ar[dd]
      |-{
        \mathclap{\phantom{\vert^{\vert}}}
        0
        \mathclap{\phantom{\vert_{\vert}}}
      }
    \\
    \\
    &
    \Sigma^n \mathbb{S}
    \ar[rr]
      |-{
        \;\Sigma^n(e^E)\;
      }
    &&
    \Sigma^n E
    %
    \ar@{=>}
      ^-{ \mbox{\tiny\color{orange}(pb)} }
    "s"; "t"
\end{xymatrix}

This implies the claim, by

$$
  \pi_0
  Maps
  \Big(
    \Sigma^{n+d - 1}\mathbb{S}
    \,,\,
    \Sigma^{n-1} (E/\mathbb{S})
  \Big)
  \;\simeq\;
  (E/\mathbb{S})_{d}
  \,.
$$

\end{proof}

\begin{proof}[more explicit, Conner-Floyd-style]

Let
$\big[ S^{n + d - 1} \overset{c}{\longrightarrow} S^{n} \big] \;\in\; \pi^n\big(S^{n+d-1}\big)$ be a given class in Cohomotopy.
We need to produce a map of the form

\begin{xymatrix@R=10pt}
    {
      \color{orange}
      H^E_{n-1}\!(c)
    }
    \ar@{}[rr]|-{\longmapsto}
    &&
    M^d
    \\
    H_{n-1}\mbox{Fluxes}(X)_c
    \ar[rr]
    \ar[dd]
    &&
    \big(
      E\!/\mathbb{S}
    \big)_d
    \ar[dd]^-{ \partial }
    \\
    {\phantom{A}}
    \\
    \big\{
      [
        \Sigma^\infty c
      ]
    \big\}
    \;
    \ar@{^{(}->}[rr]
    &&
    \;
    \mathbb{S}_{d-1}
\end{xymatrix}

and show that it is a bijection onto this fiber, hence that the square is [[cartesian square|cartesian]]. To this end, we discuss the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]], all of whose cells are homotopy cartesian:

\begin{xymatrix@R=15pt@C=60pt}
    \Sigma^\infty S^{n + d - 1}
    \ar[dd]_-{
      \Sigma^\infty
      {\color{green}
        c
      }
    }
    \ar[rr]
    &&
    \ast
    \ar[dd]
    \\
    \\
    \Sigma^\infty S^{n}
    \ar[dd]
    \ar[rr]|-{
      %\;\Sigma^\infty q_c\;
    }
    \ar@/_1.4pc/[rrr]
      |<<<<<<<<<<<<<<<<<<<{
        \; \Sigma^{n}(e^{E}) \;
      }
    &&
    \Sigma^\infty C_c
    \ar[dd]|<<<<{ \phantom{A} }
    \ar@{-->}[r]
    \ar@{}[r]
    |-{
      \scalebox{.7}{$
        \begin{array}{c}
          \vdash
          {\color{orange}
            H^E_{n-1}\!(c)
          }
          \\
          {\phantom{a}}
        \end{array}
      $}
    }
    \ar@/^2pc/[rr]
    &
    \Sigma^n E
    \ar[r]
    \ar[dd]
    &
    \ast
    \ar[dd]
    \\
    \\
    \ast
    \ar[rr]
    &&
    \Sigma^{\infty\scalebox{.6}{$+1$}} S^{n + d - 1}
    \ar[r]^-{
      \color{magenta}
      M^{d}
    }
    \ar@/_1.2pc/[rr]|-{
      \;
      \Sigma^{\infty\scalebox{.5}{$+1$}}
      {\color{green}
        c
      }
      \;
    }
    &
    \Sigma^{n}(E\!/\mathbb{S})
    \ar[r]^{ \partial }
    &
    \Sigma^{\infty\scalebox{.6}{$+1$}} S^{n}
\end{xymatrix}

For given $H^E_{n-1}\!(c)$, this diagram is constructed as follows
(where we say "square" for any _single_ cell and "rectangle" for the pasting composite of any adjacent _pair_ of them):

* The two squares on the left are the stabilization of the homotopy pushout squares defining the cofiber space $C_c$ and the suspension of $S^{n + d - 1}$

* The bottom left rectangle (with $\Sigma^n(e^E)$ at its top)
is the homotopy pushout defining $\Sigma^n(E\!/\mathbb{S})$.

* The classifying map for the given $(n-1)$-flux, shown as a dashed arrow,
completes a co-cone under the bottom left square. Thus the map ${\color{magenta}M^d}$ forming the bottom middle square is uniquely implied by the homotopy pushout property of the bottom left square.
Moreover, the [[pasting law]] implies that this bottom middle square is itself homotopy cartesian.

* The bottom right square is the homotopy pushout defining $\partial$.

* By the [[pasting law]] it follows that also the bottom right rectangle is homotopy cartesian, hence that, after the two squares on the left, it exhibits the third step in the long homotopy cofiber sequence of $\Sigma^\infty c$. This means that its total bottom morphism is $\Sigma^{\infty + 1} c$, and hence that $\partial \big[ M^d \big] = [c]$.

In conclusion, these construction steps yield a map map $H^E_{n-1}\!(c) \mapsto M^d$ which is as required.

\end{proof}

## Related concepts

* [[degree of a continuous function]], [[Hopf degree theorem]]

* [[e-invariant]]

* [[f-invariant]]

[[!include notions of pullback -- contents]]

## References

The notion was introduced, together with that of the [[e-invariant]], in:

* {#Adams66} [[John Adams]], Section 3 of: _On the groups $J(X)$ IV_, Topology 5: 21 (1966)   ([pdf](http://math.unice.fr/~cazanave/Gdt/ImJ/J-IV.pdf), doi:10.1016/0040-9383(66)90004-8)

Discussion in the broader context including also the [[f-invariant]]:

* {#BL09} Mark Behrens, Gerd Laures, _$\beta$-Family congruences and the $f$-invariant_, Geometry & Topology Monographs 16 (2009) 9–29 ([arXiv:0809.1125](https://arxiv.org/abs/0809.1125), [doi: 10.2140/gtm.2009.16.9](https://msp.org/gtm/2009/16/p002.xhtml))

[[!redirects d-invariants]]

[[!redirects Adams d-invariant]]
[[!redirects Adams d-invariants]]


D-meson

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _D-meson_ is a light [[meson]] including a [[charm quark]], hence a bound state of a [[charm quark]] with either of an [[up quark]], [[down quark]] or [[strange quark]].

The class of D-mesons goes alongside that of [[K-mesons]] ([[light mesons]] with a [[strange quark]]) and [[B-mesons]] (mesons with a [[bottom quark]]),

[[!include flavors of fundamental particles -- table]]

## References

### General

On [[D-meson]] [[decays]]:

* Marina Artuso, Brian Meadows, [[Alexey Petrov]], _Charm Meson Decays_, Ann. Rev. Nucl. Part. Sci.58:249-291, 2008 ([arXiv:0802.2934](https://arxiv.org/abs/0802.2934))

See also:

* Wikipedia, _[D-meson](https://en.wikipedia.org/wiki/D_meson)_

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!redirects D-mesons]]

[[!redirects D meson]]
[[!redirects D mesons]]


D-module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### In terms of differential operators

A _D-module_ (introduced by [[Mikio Sato]]) is a [[abelian sheaf|sheaf]] of [[modules]] over the [[sheaf]] $D_X$ of [[regular differential operators]] on a 'variety' $X$ (the latter notion depends on whether we work over a [[scheme]], [[manifold]], analytic complex manifold etc.), which is quasicoherent as $O_X$-module. As $O_X$ is a subsheaf of $D_X$ consisting of the zeroth-order differential operators (multiplications by the sections of structure sheaf), every $D_X$-module is an $O_X$-module.  Moreover, the (quasi)coherence of $D_X$-modules implies the (quasi)coherence of a $D_X$-module regarded as an $O_X$-module (but not vice versa).

### In terms of sheaves on the deRham space

The category of $\mathcal{D}$-modules on a smooth scheme $X$ may equivalently be identified with the category of [[quasicoherent sheaf|quasicoherent sheaves]] on its [[deRham space]] $dR(X)$ (in non-smooth case one needs to work in derived setting, with de Rham stack instead).

([Lurie, above theorem 0.4](#LurieCrystal), [Gaitsgory-Rozenblyum 11, 2.1.1](#GaitsgoryRozenblyum11))

Remembering, from this discussion there, that

* the deRham space is the decategorification of the [[infinitesimal shape modality|infinitesimal path groupoid]] $\Pi_{inf}(X)$ of $X$;

* a quasicoherent sheaf on $dR(X)$ is a generalization of a [[vector bundle]] on $X$;

* a vector bundle with a flat [[connection on a bundle|connection]] is an [[equivariant cohomology|equivariant]] vector bundle on the infinitesimal path $\infty$-groupoid $\Pi^{inf}$ of $X$

this shows pretty manifestly how D-modules are "sheaves of modules with flat connection", as described more below.

## Meaning and usage

$D$-modules are useful as a means of applying the methods of [[homological algebra]] and [[Categories and Sheaves|sheaf theory]] to the study of analytic systems of partial differential equations.

Insofar as an $O$-module on a [[ringed site]] $(X, O)$ can be interpreted as a generalization of the [[sheaf]] of sections of a vector bundle on $X$, a D$-$module can be interpreted as a generalization of the [[sheaf]] of sections of a vector bundle on $X$ _with flat [[connection]]_ $\nabla$.  The idea is that the action of the differential operation given by a vector field $v$ on $X$ on a section $\sigma$ of the sheaf (over some patch $U$) is to be thought of as the covariant derivative $\sigma \mapsto \nabla_v \sigma$ with respect to the flat connection $\nabla$.

In fact when $X$ is a complex analytic manifold, any $D_X$-module which is coherent as $O_X$-module is isomorphic to the sheaf of sections of some holomorphic vector bundle with flat connection. Furthermore, the subcategory of nonsingular $D_X$-modules coherent as $D_X$-modules is equivalent to the category of [[local systems]].

If $X$ is a variety over a field of positive characteristic $p$, the terms "$O_X$-coherent coherent $D_X$-module" and "vector bundle with flat connection" are not interchangeable, since $D_X$  no longer is the enveloping algebra of $O_X$ and $\text{Der}_X(O_X,O_X)$. A theorem by Katz states that for smooth $X$ the category of $O_X$-coherent $D_X$-modules is equivalent to the category with objects sequences $(E_0, E_1,\ldots)$ of locally free $O_X$-modules together with $O_X$-isomorphisms $\sigma_i: E_i\rightarrow F^* E_{i+1}$, where $F$ is the Frobenius endomorphism of $X$.

+-- {: .query}

[[John Baez]]: it would be nice to have a little more explanation about how not every $D$-module that is coherent as an $O$-module is coherent as a $D$-module.  If I understand correctly, this may be the same question as how not every holomorphic vector bundle with flat connection is a local system.  Perhaps the answer can be found under [[local system]]?  Apparently not.  Perhaps the point is that not every flat connection on a holomorphic vector bundle is locally holomorphically trivializable?  If so, this is different than how it works in the $C^\infty$ category, which might explain my puzzlement.

=--

## Properties

### Six operations yoga
 {#SixOperationsYoga}

Discussion of [[six operations yoga]] for pull-push of ([[coherent D-module|coherent]], [[holonomic D-module|holonomic]]) D-modules is in ([Bernstein, around p. 18](#Bernstein)). This is  reviewed for instance in ([Etingof](#Etingof), [Ben-Zvi & Nadler 09](#BenZviNadler09)).

The most efficient and intuitive way to define the [[six operations]]
on D-modules is to transfer them from [[Ω-modules]]
(i.e., modules over the differential graded algebra
of differential forms)
using [[Koszul duality]].
The [[six operations]] on [[Ω-modules]] can be defined
in the standard way using the fact that differential
forms can be pulled back, unlike differential operators.
See the article [[Koszul duality]] for more information.

### Relation to geometric representation theory
 {#RelationToGeometricRepresentationTheory}

For the moment see at _[[Harish Chandra transform]]_.

## Related concepts

* [[coherent D-module]], [[holonomic D-module]]

* [[arithmetic D-module]]

* [[linear differential equation]]

* [[D-geometry]]

* [[D-scheme]]

* [[D-algebra]]

* [[Riemann-Hilbert correspondence]]

* [[crystalline cohomology]]

* [[Weyl algebra]], [[regular differential operator]], [[local system]], [[differential bimodule]], [[Grothendieck connection]], [[crystal]], [[algebraic analysis]].

* [[Hecke category]], [[Harish Chandra transform]]

## References

A comprehensive account is in chapter 2 of

* [[Alexander Beilinson]] and [[Vladimir Drinfeld]], chapter 2 of _[[Chiral Algebras]]_

* [[Armand Borel]] et al., _Algebraic D-modules_, Perspectives in Mathematics, Academic Press, 1987 ([djvu](http://www.math.columbia.edu/~scautis/dmodules/boreletal.djvu))

* R. Hotta, K. Takeuchi, T. Tanisaki, _D-modules, perverse sheaves, and representation theory_, Progress in Mathematics __236__, Birkhäuser ([pdf](http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf))

Discussion in [[derived algebraic geometry]] is in

* {#GaitsgoryRozenblyum11} [[Dennis Gaitsgory]], [[Nick Rozenblyum]], _Crystals and D-modules_,  Pure and Applied Mathematics Quarterly Volume 10 (2014) Number 1 ([arXiv:1111.2087](http://arxiv.org/abs/1111.2087), [publisher](http://www.intlpress.com/site/pub/pages/journals/items/pamq/content/vols/0010/0001/a002/index.html))

Lecture notes:

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

* [[Severino C. Coutinho]], *A primer of algebraic D-modules*, London Math. Soc. Stud. Texts **33**, Cambridge University Press (1995) [[doi:10.1017/CBO9780511623653](https://doi.org/10.1017/CBO9780511623653)]

* {#Bernstein} [[Joseph Bernstein]], _Algebraic theory of D-modules_ ([[BernsteinDModule.pdf:file]], [ps](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps), [dvi](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.dvi))

* Peter Schneiders' [notes](http://www.analg.ulg.ac.be/jps/rec/idm.pdf),

* Dragan Miličić's [notes](http://www.math.utah.edu/~milicic/Eprints/dmodules.pdf), , [Localization and representation theory of reductive Lie groups](http://www.math.utah.edu/~milicic/Eprints/book.pdf);

* [[Victor Ginzburg]]'s 1998 Chicago notes [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf); A.

* Braverman-T. Chmutova, _Lectures on algebraic D-modules_, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf)

* R. Bezrukavnikov, MIT course notes, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/bezr_notes.pdf)

* Notes in Gaitsgory's seminar [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf)
* A. [[Beilinson|Beĭlinson]], J. Bernstein, _A proof of Jantzen's conjectures_, I. M. Gelʹfand Seminar,  1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf)

See also

* {#Saito89} [[Morihiko Saito]], _Induced D-modules and differential complexes_, Bull. Soc. Math. France 117 (1989), 361–387, [pdf](http://smf4.emath.fr/Publications/Bulletin/117/pdf/smf_bull_117_361-387.pdf)

* D. Gieseker, _Flat vector bundles and the fundamental group in non-zero characteristics_,  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)  2  (1975), no. 1, 1--31.

* J.-E. Björk, _Rings of differential operators_, North-Holland Math. Library 21. North-Holland Publ. 1979. xvii+374 pp.

* [[Masaki Kashiwara|M. Kashiwara]], W.Schmid, _Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups_, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488

* [[M. Kashiwara]], _D-modules and representation theory of Lie groups_,
Annales de l'institut Fourier, 43 no. 5 (1993), p. 1597-1618, [article](http://aif.cedram.org/item?id=AIF_1993__43_5_1597_0), [MR95b:22033](http://www.ams.org/mathscinet-getitem?mr=95b:22033)

* P. Maisonobe, C. Sabbah, _D-modules cohérents et holonomes_, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)

* [[Donu Arapura]], _Notes on D-modules and connection with Hodge theory_, [pdf](http://www.math.purdue.edu/~dvb/preprints/dmod.pdf)

* Nero Budur, _On the V-filtration of D-modules_, [math.AG/0409123](http://arxiv.org/abs/math/0409123), in "Geometric methods in algebra and number theory" Proc. 2003 conf. Univ. of Miami, edited by F. Bogomolov, Yu. Tschinkel

Review of [[six operations yoga]] for D-modules is in

* {#Etingof} [[Pavel Etingof]], _Formalism of six functors on all (coherent) D-modules_ ([pdf](http://www-math.mit.edu/~etingof/dmodfactsheet.pdf))

* {#BenZviNadler09} [[David Ben-Zvi]], [[David Nadler]], section 3 of _The Character Theory of a Complex Group_ ([arXiv:0904.1247](http://arxiv.org/abs/0904.1247))

See also

* A. Beilinson, I. N. Bernstein, _A proof of Jantzen conjecture_, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022

* Secret Blogging Seminar [Musings on D-modules](http://sbseminar.wordpress.com/2007/07/07/musings-on-d-modules/), [Musings on D-modules, part 2](http://sbseminar.wordpress.com/2007/07/14/musings-on-d-modules-part-2/)

* The Everything Seminar [D-module Basics I](http://cornellmath.wordpress.com/2007/09/06/d-module-basics-i/), [D-Module Basics II](http://cornellmath.wordpress.com/2007/09/09/d-module-basics-ii/).

[[!redirects D-modules]]

[[!redirects twisted D-module]]
[[!redirects twisted D-modules]]


D-scheme

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

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[scheme]], analogous to how an $X$-scheme is a [[scheme]] $E \to X$ over $X$, a _$\mathcal{D}_X$-scheme_ is a scheme over the [[de Rham space]] $\mathbf{\Pi}_{inf}(X)$ of $X$.

See also [[diffiety]].

## Definition

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

For $X$ a [[scheme]], a _$\mathcal{D}_X$-scheme_ is a [[scheme]] $E \to \mathbf{\Pi}_{inf}(X)$ over the [[de Rham space]] $\mathbf{\Pi}_{inf}(X)$ of $X$.

=--

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

This definition makes sense in much greater generality: in any context of [[differential cohesion]].

=--

## Properties

### Relation to D-modules

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

In the [[sheaf topos]] over [[affine scheme]]s, an
$X$-affine $\mathcal{D}_X$-scheme is a [[commutative monoid]] object in the [[monoidal category]] of [[quasicoherent sheaves]] $QC(\mathbf{\Pi}_{inf}(X))$, which is equivalently the category of [[D-module]]s over $X$:

$$
  Aff \mathcal{D}_X Scheme \simeq CMon(\mathcal{D}Mod(X))
  \,.
$$

=--

This is ([BeilinsonDrinfeld, section 2.3](#BeilinsonDrinfeld)).

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

This is indeed equivalent to the above abstract definition

=--

This appears as ([Lurie, theorem, 0.6 and below remark 0.7](#Lurie))

### Relation to jet schemes

The [[free functor|free]] $\mathcal{D}_X$-scheme on a given $X$-scheme $E \to X$ is the [[jet bundle]] of $E$.

This is ([BeilinsonDrinfeld, 2.3.2](#BeilinsonDrinfeld)).

This fact makes $\mathcal{D}$-geometry a natural home for [[variational calculus]].

## Related concepts

* [[scheme]]

* [[D-module]]

* [[de Rham space]]

* [[diffiety]]

## References

The definition in terms of monoids in [[D-module]]s is in section 2.3 in

* {#BeilinsonDrinfeld} [[Alexander Beilinson]] and [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

  _chapter 2, Geometry of D-schemes_ ([pdf](http://www.math.uchicago.edu/~mitya/langlands/chiral/cha_ch2.pdf))

The observation that this is equivalent to the abstract definition given above appears in pages 5 and 6 of

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

[[!redirects D-schemes]]


d-space > history

< [[d-space]]
[[!redirects d-space -- history]]


D-term

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum hadrodynamics]], the *D-term* is a subtle [[conserved charge]] of [[hadrons]] (alongside the well-known charges of [[mass]], [[spin]] and [[electric charge]]), which may be understood as a [[limit of a sequence|limit]] of a [[gravity|gravitational]] [[form factor]]. Accordingly, the most natural way to [[measurement|measure]] the D-term charge for any particle is (or would be) via [[scattering]] with [[gravitons]] and, as a result, there is currently no [[experiment|experimental]] determination of the D-term of any particle.

But more indirect measurements of the D-term may be possible (cf. [BEG18](#BEG18)), such as at a future electron-ion collider, the vague prospect of which recently led to renewed attention of the problem (see [FHSU22, p. 1](#FHSU22)).

Generally, it is expected that the D-term must be [[negative number|negative]] for otherwise [[hadrons]] would seem to be unstable.

A computation of the D-term via [[holographic QCD]] is claimed in [FHSU22](#FHSU22), whose authors indeed find a (small and) negative value.

## References

Original articles:

* I Yu Kobzarev, L B Okun, *On gravitational interaction of fermions*, Eksp. Teor. Fiz. **43** (1962) 1904-1909 $[$[osti:4744739](https://www.osti.gov/biblio/4744739)$]$ (in Russian)

* Heinz Pagels, *Energy-Momentum Structure Form Factors of Particles*,
Phys. Rev. **144** (1966) 1250 $[$[doi:10.1103/PhysRev.144.1250](https://doi.org/10.1103/PhysRev.144.1250)$]$

* [[Maxim V. Polyakov]], C. Weiss, *Skewed and double distributions in pion and nucleon* $[$[arXiv:hep-ph/9902451](https://arxiv.org/abs/hep-ph/9902451)$]$

* [[Maxim V. Polyakov]], [[Peter Schweitzer]], *D-term, strong forces in the nucleon, and their applications* $[$[arXiv:1801.05858](https://arxiv.org/abs/1801.05858), [spire:1648787](https://inspirehep.net/literature/1648787)$]$

Review:

* {#PolyakovSchweitzer18} [[Maxim V. Polyakov]], [[Peter Schweitzer]], *Forces inside hadrons: pressure, surface tension, mechanical radius, and all that*, Int. J. Mod. Phys. A **33** (2018) 1830025 $[$[doi:10.1142/S0217751X18300259](https://doi.org/10.1142/S0217751X18300259), [arXiv:1805.06596](https://arxiv.org/abs/1805.06596)$]$

Experimental measurement:

* {#BEG18} V. D. Burkert, L. Elouadrhiri, F. X. Girod, *The pressure distribution inside the proton*, Nature **557** (2018) 396–399 $[$[doi:10.1038/s41586-018-0060-z](https://doi.org/10.1038/s41586-018-0060-z)$]$

Computation/prediction of the D-term for hadrons via [[holographic QCD]], specifically via [[D4-D8-brane bound states]]:

* {#FHSU22} Mitsutoshi Fujita, Yoshitaka Hatta, [[Shigeki Sugimoto]], Takahiro Ueda, *Nucleon D-term in holographic QCD* $[$[arXiv:2206.06578](https://arxiv.org/abs/2206.06578)$]$

[[!redirects D-terms]]


D-topological infinity-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A _D-topological $\infty$-groupoid_ is an [[∞-groupoid]] equipped with  [[cohesive (∞,1)-topos|cohesion]] in the form [[D-topology]], as in [[D-topological spaces]]/[[Delta-generated topological spaces]]. The latter are among the [[concrete object|concrete]] [[0-truncated object in an (infinity,1)-category|0-truncated]] D-topological $\infty$-groupoids, containing, in particular, the [[topological manifolds]].

Examples of [[1-truncated object in an (infinity,1)-category|1-truncated objects]] in D-topological $\infty$-groupoids are [[topological groupoids]]/[[topological stacks]] which are presented by [[internal groupoids]] in [[D-topological spaces]] (hence degree-wise [[concrete object|concrete]]).

More generally, every [[simplicial topological space]] whose topology is degreewise [[D-topological space|D-topological]] canonically presents a D-topological $\infty$-groupoid. Various constructions with simplicial toppological spaces find their natural home in this [[(∞,1)-topos]]. For instance:

* [[geometric realization of simplicial topological spaces|geometric realization of simplicial topological manifolds]] is equivalently the image $\Pi(X)$ of the corresponding Euclidean-topological $\infty$-groupoid $X$ under the canonical [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]].

* topological [[simplicial principal bundle]]s over topological [[simplicial group]]s are the corresponding [[principal ∞-bundle]]s in $ETop\infty Grpd$ classified by its [[cohomology|internal cohomology]].

## Definition

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

Let [[CartSp]]${}_{top}$ be the [[site]] whose underlying [[category]] has as [[objects]] the [[Cartesian space]]s $\mathbb{R}^n$, $n \in \mathbb{N}$ equipped with the [[Euclidean topology]] and as [[morphism]]s the [[continuous maps]] between them; and whose [[coverage]] is given by [[good open cover]]s.

=--

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

Define

$$
  ETop \infty Grpd := (\infty,1)Sh(CartSp_{top})
$$

to be the [[(∞,1)-category of (∞,1)-sheaves]] on $CartSp_{top}$.
=--

## Properties

### General

+-- {: .un_prop #ETopInfGrpdAsCohesiveTopos}
###### Proposition

The [[(∞,1)-topos]] $ETop \infty Grpd$ is a [[cohesive (∞,1)-topos]].

=--

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

The site [[CartSp]]${}_{top}$ an  [[∞-cohesive site]].  See there for details.

=--

For completeness we record  general properties of [[cohesive (∞,1)-topos]]es implied by this.

+-- {: .un_prop #PointLikePropertiesOfETopInfGrpd}
###### Corollary

$ETop\infty Grpd$ is

* [[hypercomplete (∞,1)-topos|hypercomplete]];

* of [[cohomological dimension]] 0;

* of [[homotopy dimension]] 0;

* of the [[shape of an (∞,1)-topos]] of the point.

=--

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

We say that $ETop \infty Grpd$ defines **Euclidean-topological cohesion**. An object in $ETop \infty Grpd$ we call a **Euclidean-topological $\infty$-groupoid**.

=--

+-- {: .un_prop #ToposOverTopMfd}
###### Proposition

Write [[TopMfd]] for the category of [[topological manifold]]s. This becomes a [[large site]] with the [[open cover]] [[coverage]]. We have an [[equivalence of (∞,1)-categories]]

$$
  ETop\infty Grpd
  \simeq
  \hat Sh_{(\infty,1)}(TopMfd)
$$

with the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]] on [[TopMfd]].

=--

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

Since every [[topological manifold]] admits an [[open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]] it follows that [[CartSp]]${}_{top}$ is a [[dense sub-site]] of $TopMfd$. Accordingly the [[categories of sheaves]] are [[equivalence of categories|equivalent]]

$$
  Sh(CartSp_{top}) \simeq Sh(TopMfd)
  \,.
$$

By the discussion at [[model structure on simplicial sheaves]] it follows that the [[hypercomplete (∞,1)-topos]]es  over these sites are [[equivalence of (∞,1)-categories|equivalent]]

$$
  \hat Sh_{(\infty,1)}(CartSp_{top})
  \simeq
  \hat Sh_{(\infty,1)}(TopMfd)
 \,.
$$

But by the [above proposition](#ETopInfGrpdAsCohesiveTopos) we have that before [[hypercompletion]] $Sh_{(\infty,1)}(CartSp_{top})$ is [[cohesive (∞,1)-topos|cohesive]]. This means that it is in particular a [[local (∞,1)-topos]]. By the discussion there, this means that it already coincides with its [[hypercompletion]], $Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(CartSp_{top})$.

=--

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

Write $Top_1$ for the 1-[[category]] of [[Hausdorff space|Hausdorff]] [[topological spaces]] and continuous maps. There is a canonical functor

$$
  j : Top_1 \to \tau_{\leq 0}ETop\infty Grpd \hookrightarrow  ETop\infty Grpd
$$

given by sending a topological space $X$ to the 0-[[truncated]]
[[(∞,1)-sheaf]] (=  [[sheaf]]) on [[CartSp]]${}_{top}$ externally represented by $X$ under the embedding $CartSp_{top} \hookrightarrow Top$:

$$
  j(X) : (U \in CartSp_{top}) \mapsto Hom_{Top}(U,X)
  \in Set \hookrightarrow \infty Grpd
  \,.
$$
=--

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

The functor $j$ exhibits [[TopMfd]] as a full [[sub-(∞,1)-category]] of $ETop\infty Grpd$

$$
  TopMfd \hookrightarrow ETop\infty Grpd
  \,.
$$

=--

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

With the [above proposition](#ToposOverTopMfd) this follows directly by the [[(∞,1)-Yoneda lemma]].

=--

### Model category presentation {#ModelCategoryPresentation}

We dicuss some aspects of the [[presentable (∞,1)-category|presentation]] of $ETop \infty Grpd$ by [[model category]] structures.

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

Let $[CartSp_{top}^{op}, sSet]_{proj,loc}$ be the Cech-local projective [[model structure on simplicial presheaves]]. This is a [[presentable (∞,1)-category|presentation]] of $ETop \infty Grpd$

$$
  ([CartSp_{top}^{op}, sSet]_{proj,loc})^\circ
    \simeq
   ETop \infty Grpd
  \,.
$$

Also the [[model structure on simplicial sheaves]] $sSh(CartSp_{top})_{loc}$ is a presentation

$$
  (sSh(CartSp_{top})_{loc})^\circ
    \simeq
   ETop \infty Grpd
  \,.
$$

=--

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

The first statement is a special case of the general discussion at [[model structure on simplicial presheaves]]. Similarly, by the general discussion at [[model structure on simplicial sheaves]] we have that this presents the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]]. But by [the above](#PointLikePropertiesOfETopInfGrpd) $ETop\infty Grpd$ already is hypercomplete.

=--

Moreover:

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

$ETop\infty Grpd$ is also the [[hypercompletion]] of the [[(∞,1)-topos]] presented by the local [[model structure on simplicial presheaves]] over all of [[Mfd]] (or over any [[small category|small]] [[dense sub-site]] such as for instance the full sub-category of manifolds bounded in size by some [[regular cardinal]]).

$$
  \hat{}([Mfd^{op}, sSet]_{proj,loc})^\circ
    \simeq
   ETop \infty Grpd
  \,.
$$

=--

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

By the [above proposition](#ToposOverTopMfd).

=--

While the model structures on simplicial presheaves over both sites present the same [[(∞,1)-category]], they lend themselves to different computations:

the model structure over $CartSp_{top}$ has more fibrant objects and hence fewer cofibrant objects, while the model structure over $Mfd$ has more cofibrant objects and fewer fibrant objects. More specifically:

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

Let $X \in [Mfd^{op}, sSet]$ be an object that is _globally fibrant_ , _separated_ and _locally trivial_, meaning that

1. $X(U)$ is an [[inhabited]] [[Kan complex]] for all $U \in Mfd$;

1. for every [[covering]] $\{U_i \to U\}$ in [[Mfd]] the [[descent]] comparison morphism
   $X(U) \to [Mfd^{op}, sSet](C(\{U_i\}), X)$ is a [[full and faithful (∞,1)-functor]];

1. for [[contractible]] $U$ we have $\pi_0[Mfd^{op}, sSet](C(\{U_i\}), X) \simeq *$.

Then the restriction of $X$ along $CartSp_{top} \hookrightarrow Mfd$ is a fibrant object in the local model structure $[CartSp_{top}^{op}, sSet]_{proj,loc}$.

=--

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

The fibrant objects in the local model structure are precisely those that are Kan complexes over every object and for which the descent morphism is an equivalence for all covers.

The first condition is given by the first assumption. The second and third assumptions imply the second condition over contractible manifolds, such as the [[Cartesian space]]s.

=--

+-- {: .un_example}
###### Example

Let $G$ be a [[topological group]], regarded as the presheaf over [[Mfd]] that it [[representable functor|represents]]. Write $\bar W G$ (see the notation at [[simplicial group]]) for the simplicial presheaf on $Mfd$ given by the [[nerve]] of the [[topological groupoid]] $(G \stackrel{\to}{\to} *)$. (This is a presentation of the [[delooping]] of the [[0-truncated]] [[∞-group]] $G \in ETop\infty Grpd$, see the discussion [below](#Groups). )

The fibrant [[resolution]] of $\bar W G$ in $[Mfd^{op}, sSet]_{proj,loc}$ is (the rectification of) its [[stackification]]: the [[stack]] $G Bund$ of topological $G$-[[principal bundle]]s. But the canonical morphism

$$
  \bar W G \to G Bund
$$

is a [[full and faithful functor]] (over each object $U \in Mfd$): it includes the single object of $\bar W G$ as the trivial $G$-[[principal bundle]]. The automorphism of the single object in $\bar W G$ over $U$ are $G$-valued [[continuous function]]s on $U$, which are precisely the automorphisms of the trivial $G$-bundle. Therefore this inclusion is full and faithful, the presheaf $\bar W G$ is a [[separated prestack]].

Moreover, it is locally trivial: every [[Cech cohomology|Cech cocycle]] for a $G$-bundle over a [[Cartesian space]] is equivalent to the trivial one. Equivalently, also $\pi_0 G Bund(\mathbb{R}^n) \simeq *$.

Therefore $\bar W G$, when restricted $CartSp_{top}$, does become a fibrant object in $[CartSp_{top}^{op}, sSet]_{proj,loc}$.

On the other hand, let $X \in Mfd$ be any non-contractible manifold. Since in the projective [[model structure on simplicial presheaves]] every representable is cofibrant, this is a cofibrant object in $[Mfd^{op}, sSet]_{proj,loc}$. However, it fails to be cofibrant in $[CartSp_{top}^{op}, sSet]_{proj,loc}$. Instead, there a cofibrant replacement is given by the [[Cech nerve]] $C(\{U_i\})$ of any [[good open cover]] $\{U_i \to X\}$.

This yields two different ways to compute the first [[nonabelian cohomology]]

$$
  H^1_{ETop}(X,G) := \pi_0 ETop\infty Grpd (X, \mathbf{B}G)
$$

in $ETop\infty Grpd$ on $X$ with coefficients in $G$, as

1. $\cdots \simeq \pi_0 [Mfd^{op}, sSet](X, G Bund) \simeq \pi_0 G Bund(X)$;

1. $\cdots \simeq \pi_0 [CartSp_{top}^{op}, sSet](C(\{U_i\}), \bar W G) \simeq H^1_{Ch}(X,G)$.

In the first case we need to construct the fibrant replacement $G Bund$. This amounts to computing $G$-cocycles = $G$-bundles over _all_ manifolds and then evaluate on the given one, $X$, by the [[2-Yoneda lemma]].

In the second case however we cofibrantly replace $X$ by a good open cover, and then find the [[Cech cohomology|Cech cocycles]] with coefficients in $G$ on that.

For ordinary $G$-bundles the difference between the two computations may be irrelevant in practice, because ordinary $G$-bundles are very well understood. However for more general coefficient objects, for instance general topological [[simplicial group]]s $G$, the first approach requires to find the full [[∞-stackification]] to the [[∞-stack]] of all [[principal ∞-bundle]]s, while the second approach requires only to compute specific coycles over one specific base object. In practice the latter is often all that one needs.

=--

## Structures in the cohesive $(\infty,1)$-topos $ETop \infty Grpd$

We discuss what some of the general abstract Structures in a cohesive (∞,1)-topos look like in the model $ETop \infty Grpd$.

As usual, write

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) :
  ETop \infty Grpd
    \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
$$

for the defining quadruple of [[adjoint (∞,1)-functor]]s that refine the [[global section]] [[(∞,1)-geometric morphism]] to [[∞Grpd]].

### Cohesive $\infty$-groups {#Groups}

By the general properties of [[cohesive (∞,1)-topos]]es with an [[∞-cohesive site]] of definition, every [[∞-group]] object is presented by a presheaf of [[simplicial group]]s. For  $ETop\infty Grpd$ among these are the [[simplicial topological group]]s. See there for more details.

### Geometric homotopy and Galois theory
 {#GeometricHomotopy}
 {#GeometricRealization}

We discuss the realization of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] in $ETop \inft Grpd$.

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

Let $X$ be a [[paracompact topological space]] such that that $X$ admits a [[good open cover]] by [[open ball]]s (for instance a [[paracompact manifold]]).

Then $\Pi(X) := \Pi(i(X)) \in \infty Grpd$ is equivalent to the standard [[fundamental ∞-groupoid]] of a [[topological space]] that is presented by the [[singular simplicial complex]] $Sing X$

$$
  \Pi(X) \simeq Sing X
  \,.
$$

Equivalently, under [[geometric realization]] $\mathbb{L}|-| : \infty Grpd \to Top$ we have that there is a [[weak homotopy equivalence]]

$$
  X \simeq |\Pi(X)|
  \,.
$$
=--

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

By the discussion at [[∞-cohesive site]] we have an equivalence $\Pi(-) \simeq \mathbb{L} \lim_\to$ to the [[derived functor]] of the [[sSet]]-[[colimit]] functor $\lim_\to : [CartSp^{op}, sSet]_{proj,loc} \to sSet_{Quillen}$.

To compute this derived functor, let $\{U_i \to X\}$ be a [[good open cover]] by [[open balls]], hence [[homeomorphic]]ally by [[Cartesian space]]. By goodness of the cover the [[Cech nerve]] $C(\coprod_i U_i \to X) \in [CartSp^{op}, sSet]$ is degreewise a [[coproduct]] of [[representable functor|representable]]s, hence a [[split hypercover]]. By the discussion at [[model structure on simplicial presheaves]] we have that in this case the canonical morphism

$$
  C(\coprod_i U_i \to X) \to X
$$

is a cofibrant [[resolution]] of $X$ in $[CartSp^{op}, sSet]_{proj,loc}$. Accordingly we have

$$
  \Pi(X) \simeq (\mathbb{L} \lim_\to) (X) \simeq \lim_\to C(\coprod_i U_i \to X)
  \,.
$$

Using the [[equivalence of categories]] $[CartSp^{op}, sSet] \simeq [\Delta^{op}, [CartSp^{op}, Set]]$ and that [[colimit]]s in [[presheaf categories]] are computed objectwise and finally using that the colimit of a [[representable functor]] is the point (an incarnation of the [[Yoneda lemma]]) we have that $\Pi(X)$ is presented by the [[Kan complex]] that is obtained by contracting in the [[Cech nerve]] $C(\coprod_i U_i)$ each open subset to a point.

The classical [[nerve theorem]] asserts that this implies the claim.
=--

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

We may regard [[Top]] itself as a [[cohesive (∞,1)-topos]]. $(\Pi_{Top}\dashv Disc_{Top} \dashv \Gamma_{Top} \dashv coDisc_{Top}) Top \stackrel{\simeq}{\to} \infty Grpd$. This is discussed at [[discrete ∞-groupoid]].

Using this the above proposition may be stated as saying that for $X$ a [[paracompact topological space]] that admits a [[good open cover]] we have

$$
  \Pi_{ETop\infty Grpd}(X) \simeq \Pi_{Top}(X)
  \,.
$$

=--

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

Let $X_\bullet$ be a [[good simplicial topological space]] that is degreewise [[paracompact topological space|paracompact]] and degreewise admits a [[good open cover]], regarded naturally as an object $X_\bullet \in Top^{\Delta^{op}} \to ETop \infty Grpd$.

We have that the intrinsic $\Pi(X_\bullet) \in \infty Grpd$ coincides under [[geometric realization]] $\mathbb{L}|-| : \infty Grpd \stackrel{\simeq}{\to} Top$ with the ordinary [[geometric realization of simplicial topological spaces]] $|X_\bullet|_{Top^{\Delta^{op}}}$

$$
  |\Pi(X_\bullet)| \simeq |X_\bullet|_{Top^{\Delta^{op}}}
  \,.
$$
=--

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

Write $Q$ for Dugger's cofibrant replacement functor on $[CartSp^{op}, sSet]_{proj,loc}$ (discussed at [[model structure on simplicial presheaves]]). On a simplicially constant simplicial presheaf $X$ it is given by

$$
 Q X := \int^{[n] \in \Delta} \Delta[n] \cdot
    \left(
      \coprod_{U_0 \to \cdots \to U_n \to X} U_0
    \right)
   \,,
$$

where the [[coproduct]] in the integrand of the [[coend]] is over all sequences of morphisms from [[representable functor|representable]]s $U_i$ to $X$ as indicated. On a general [[simplicial presheaf]] $X_\bullet$ it is given by

$$
  Q X_\bullet := \int^{[k] \in \Delta} \Delta[k] \cdot Q X_k
  \,,
$$

which is the simplicial presheaf that over any $\mathbb{R}^n \in CartSp$ takes as value the [[diagonal]] of the [[bisimplicial set]] whose $(n,r)$-entry is
$\coprod_{U_0 \to \cdots \to U_n \to X_k} CartSp_{top}(\mathbb{R}^n,U_0)$.

Since [[coend]]s are special [[colimit]]s, the colimit functor itself commutes with them and we find

$$
  \begin{aligned}
     \Pi(X_\bullet)
     & \simeq
      (\mathbb{L} \lim_\to) X_\bullet
     \\
     & \simeq \lim_\to Q X_\bullet
     \\
      & \simeq \int^{[n] \in \Delta} \Delta[k] \cdot \lim_\to (Q X_k)
    \,.
  \end{aligned}
$$

By the discussion at [[Reedy model structure]] this [[coend]] is a [[homotopy colimit]] over the simplicial [[diagram]] $\lim_\to Q X_\bullet : \Delta \to sSet_{Quillen}$

$$
  \cdots \simeq hocolim_\Delta \lim_\to Q X_\bullet
  \,.
$$

By the [above proposition](#FundGroupoidOfParacompact) we have
for each $k \in \mathbb{N}$ weak equivalences $\lim_\to Q X_k \simeq (\mathbb{L} \lim_\to) X_k \simeq Sing X_k$, so that

$$
  \begin{aligned}
    \cdots &\simeq hocolim_\Delta Sing X_k
    \\
     & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot Sing X_k
    \\
     & \simeq diag Sing(X_\bullet)_\bullet
  \end{aligned}
  \,.
$$

By the discussion at [[geometric realization of simplicial topological spaces]], this maps to the [[homotopy colimit]] of the simplicial topological space $X_\bullet$, which is just its geometric realizaiton if it is proper.
=--

### Paths and geometric Postnikov towers
 {#PresentationOfPathGroupoid}

We discuss the notion of _[geometric path  ∞-groupoids](http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#Paths)_ realized in $ETop\infty Grpd$.

In the [above](#GeometricRealization) constructions of $\Pi(X)$ the actual paths are not explicit. We discuss here presentations of $\mathbf{\Pi}(X)$ in terms of actual paths.

By prop. \ref{FundGroupoidOfParacompact} we have

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

Let $X$ be a a [[paracompact topological space]], regarded as an object of $ETop\infty Grpd$. Then $\mathbf{\Pi}(X)$ is presented by the constant [[simplicial presheaf]]

$$
  Disc Sing(X)
   \,:\,
  (U,[k])
    \mapsto
  Hom_{Top}(\Delta^k, X)
  \,.
$$

=--

Possibly more natural would seem to look at the topological Kan complex that remembers the topology on the spaces of paths:

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

For $X$ a paracompact topological space, define the simplicial presheaf

$$
  \mathbf{Sing} X
  :
  (U,[k])
  \mapsto
  Hom_{Top}(U \times \Delta^k, X)
  \,.
$$

=--

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

Also $\mathbf{Sing} X$ is a presentation of $\mathbf{\Pi}(X)$

=--

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

For each fixed $U \in CartSp$ the inclusion of simplicial sets

$$
  Sing X \to \mathbf{Sing}(X)(U)
$$

is a [[weak homotopy equivalence]], since $U \in CartSp$ is [[contractible space|contractible]].

Therefore the inclusion of simplicial presheaves

$$
  Disc Sing X \to \mathbf{Sing} X
$$

is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$. This implies the claim with prop. \ref{DiscretePathGroupoid}.

=--

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

Typically one is interested in mapping out of $\mathbf{\Pi}(X)$. While it is clear that $Disc Sing X$ is cofibrant in $[CartSp^{op}, sSet]_{proj,loc}$, it is harder to determine the necessary [[resolution]]s of $\mathbf{Sing}X$.

=--

### Cohomology and principal $\infty$-bundles
{#Cohomology}

We dicuss aspects of the intrinsic [[cohomology]] of $E Top \infty Grpd$ and of the [[principal ∞-bundle]]s that it classifies.

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

Let $A \in $ [[∞Grpd]] be any [[discrete ∞-groupoid]]. Write $|A| \in $ [[Top]] for its [[geometric realization]]. For $X$ any [[topological space]], the [[nonabelian cohomology]] of $X$ with coefficients in $A$ is the set of [[homotopy]] classes of maps $X \to |A|$

$$
  H_{Top}(X,A) := \pi_0 Top(X,|A|)
  \,.
$$

We say $Top(X,|A|)$ itself is the [[cocycle]] [[∞-groupoid]] for $A$-valued [[nonabelian cohomology]] on $X$.

Similarly, for $X, \mathbf{A} \in ETop \infty Grpd$ two e-topological $\infty$-groupoids, write

$$
  H_{ETop}(X,\mathbf{A}) := \pi_0 ETop\infty Grpd(X,\mathbf{A})
$$

for the intrinsic [[cohomology]] of $ETop \infty Grpd$ on $X$ with coefficients in $\mathbf{A}$.

=--

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

Let $A \in $ [[∞Grpd]], write $Disc A \in ETop \infty Grpd$ for the corresponding [[discrete ∞-groupoid|discrete topological ∞-groupoid]]. Let $X \in Top_1 \stackrel{i}{\hookrightarrow} ETop \infty Grpd$ be a [[paracompact topological space]] regarded as a 0-[[truncated]]
Euclidean-topological $\infty$-groupoid.

We have an [[isomorphism]] of cohomology sets

$$
  H_{Top}(X,A) \simeq H_{ETop}(X,Disc A)
$$

and in fact an [[equivalence in an (∞,1)-category|equivalence]] of [[cocycle]] [[∞-groupoid]]s

$$
  Top(X,|A|) \simeq ETop\infty Grpd(X, Disc A)
  \,.
$$

=--

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

By the $(\Pi \dashv Disc)$-[[adjunction]] of the [[locally ∞-connected (∞,1)-topos]] $ETop \infty Grpd$ we have

$$
  ETop\infty Grpd(X, Disc A) \simeq \infty Grpd(\Pi(X), A)
  \underoverset{\simeq}{|-|}{\to}
  Top(|\Pi X|, |A|)
  \,.
$$

From this the claim follows by the [above proposition](#FundGroupoidOfParacompact).
=--

+-- {: .un_prop #GeometricRealizationOfHomotopyFibers}
###### Proposition

 Let $G$ be a [[well-pointed simplicial topological group]] degreewise in [[TopMfd]].
Then the $(\infty,1)$-functor
$\Pi : \mathrm{ETop}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd}$
 preserves [[homotopy fiber]]s of all morphisms of the form
$X \to \mathbf{B}G$
that are presented in $[\mathrm{CartSp}_{\mathrm{top}}^{\mathrm{op}}, \mathrm{sSet}]_{proj}$
by morphism of the form $X \to \bar W G$ with $X$ fibrant.

=--

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

Notice that since [[(∞,1)-sheafification]] preserves
finite [[(∞,1)-limit]]s  we may
indeed discuss the homotopy fiber in the global model structure on
simplicial presheaves.

Write $Q X \stackrel{\simeq}{\to} X$ for the global cofibrant resolution
given by $Q X : [n] \mapsto \coprod_{\{U_{i_0} \to \cdots \to U_{i_n} \to X_n\}} U_{i_0}$, where the $U_{i_k}$ range over $\mathrm{CartSp}_{\mathrm{top}}$ .
(Discussed at model structure on simplicial presheaves -- cofibrant replacement. )
This has degeneracies splitting off as direct summands, and hence is a
[[good simplicial topological space]] that is degreewise in
[[TopMfd]].  Consider then the pasting of two [[pullback]] diagrams of
simplicial presheaves

$$
  \array{
    P' &\stackrel{\simeq}{\to}&  P &\to& W G
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Q X &\stackrel{\simeq}{\to}& X &\to & \bar W G
  }
  \,.
$$

By the discussion at [[geometric realization of simplicial topological spaces]] we have that the rightmost vertical morphism is a fibration in $[CartSp_{top}^{op}, sSet]_{proj}$. Since fibrations are stable under pullback, the middle vertical morphism is also a fibration (as is the leftmost one). Since the global model structure is a right [[proper model category]] it follows then that also the top left horizontal morphism is a weak

Since the square on the right is a pullback of fibrant objects with one morphism being a  fibration, $P$ is a presentation of the [[homotopy fiber]] of $X \to \bar W G$. Hence so is $P'$, which is moreover the pullback of a diagram of good simplicial spaces.

By prop. \ref{FundGroupoidOfSimplicialParacompact} we have that on the outer
diagram $\Pi$ is presented by
[[geometric realization of simplicial topological spaces]] $|-|$.
By the discussion of realization of [[simplicial principal bundle]]s there,
we have a pullback in $\mathrm{Top}_{\mathrm{Quillen}}$

$$
  \array{
    {|P|} &\to&  {|W G|}
    \\
    \downarrow && \downarrow
    \\
    {|Q X|} & \to & {|\bar W G|}
  }
$$

which exhibits $|P|$ as the [[homotopy fiber]] of $|Q X| \to |\bar W G|$. But this is a model for $|\Pi(X \to \bar W  G)|$.

=--

### Twisted cohomology

See _[[twisted bundle]]_ .

### Universal coverings and geometric Whitehead towers {#WhiteheadTowers}

We discuss geometric Whitehead towers in $ETop\infty Grpd$.

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

Let $X$ be a [[pointed object|pointed] [[paracompact topological space]] that admits a [[good open cover]].  Then its ordinary [[Whitehead tower]]
$* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X$ in [[Top]] coincides with the image under the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|intrinsic fundamental ∞-groupoid functor]] $|\Pi(-)|$ of its [[Whitehead tower in an (∞,1)-topos|geometric Whitehead tower]]
$X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X$ in $ETop \infty Grpd$:

$$
  \begin{aligned}
     |\Pi(-)| & :
   (X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X) \in ETop\infty Grpd
    \\
     & \mapsto
      (* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X)
       \in Top
  \end{aligned}
  \,.
$$

=--

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

By the general discussion at [[Whitehead tower in an (∞,1)-topos]] the geometric Whitehead tower is characterized for each $n$ by the [[fiber sequence]]

$$
  X^{\mathbf{(n)}} \to X^{\mathbf{(n-1)}} \to
  \mathbf{B}^n \mathbf{\pi}_n(X)
  \to
  \mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{(n-1)}(X)
  \,.
$$

By the above [proposition on the fundamental ∞-groupoid](#FundGroupoidOfParacompact) we have that $\mathbf{\Pi}_n(X) \simeq Disc Sing X$. Since $Disc$ is [[right adjoint]] and hence preserves [[homotopy fiber]]s this implies that
$\mathbf{B} \mathbf{\pi}_n(X) \simeq \mathbf{B}^n Disc \pi_n(X)$, where $\pi_n(X)$ is the ordinary $n$th [[homotopy group]] of the pointed topological space $X$.

Then by the above [proposition on geometric realization of homotopy fibers](#GeometricRealizationOfHomotopyFibers) we have that under $|\Pi(-)|$ the space $X^{\mathbf{(n)}}$ maps to the homotopy fiber of $|\Pi(X^{\mathbf{(n-1)}})| \to B^n |Disc \pi_n(X)| = B^n \pi_n(X)$.

By induction over $n$ this implies the claim.

=--

### Path $\infty$-groupoid and geometric Postnikov towers
{#PathInftinityGroupoids}

Let $C$ be an [[∞-connected site]]. We give an explicit presentation of the _constant path inclusion_ $X \to \mathbf{\Pi}(X)$ in the [[locally ∞-connected (∞,1)-topos]] over $C$ such that the component maps are cofibrations.

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

The projective [[model structure on simplicial presheaves]]
$[C^{op}, sSet]_{proj}$ has a set of generating cofibrations

$$
  I =
  \{
    U \cdot \partial \Delta[n] \hookrightarrow U \cdot \Delta[n] | U \in C, n \in \mathbb{N})
  \}
  \,.
$$
=--

See [[model structure on functors]] for details.

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

Write

$$
  \mathbf{Sing} : C \to [C^{op}, sSet]
$$

for the functor given by applying the [[small object argument]] to this set $I$ to obtain a functorial factorization of the [[terminal object|terminal morphisms]] $U \to *$ into a cofibration followed by an acyclic fibration

$$
  U \hookrightarrow \mathbf{Sing} U \stackrel{\simeq}{\to} *
  \,.
$$

Let

$$
  \mathbf{Sing} : [C^{op}, sSet] \to [C^{op}, sSet]
$$

be the [[Yoneda extension]] (left [[Kan extension]] through the [[Yoneda embedding]]) of this functor to all of $[C^{op}, sSet]$.
=--

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

For $U \in C$ the [[simplicial presheaf]] $\mathbf{Sing}U$ is a [[resolution]] of the ([[nerve]] of the) [[fundamental groupoid]] $\Pi_1(U)$:

the non-degenerate components of $\mathbf{Sing}U$ at the first stage of the [[small object argument]] are such that a map out of them into a simplicial presheaf $A$ are given by commuting diagrams

$$
  \array{
    U_0 \coprod U_0 &\stackrel{(s,t)}{\to}& U
    \\
    \downarrow && \downarrow
    \\
    U_0 \times \Delta[1] &\to& A
  }
  \,.
$$

This is a $U$-parameterized family of objects of $A$ together with a $U_0$-parameterized family of morphisms of $A$ associated to the pairs of points $(s,t) \in U$, hence to the "straight paths" from $s$ to $t$. At the next stage for every triangle of such straight path a 2-morphism is thrown in, and so on. So $\mathbf{Sing}U$ indeed is an $\infty$-groupoid of paths in $U$.
=--

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

The functor $\mathbf{Sing}$ is the [[left adjoint]] of a [[Quillen adjunction]]

$$
  (\mathbf{Sing} \dashv R) :
    [C^{op}, sSet]_{proj, loc} \to [C^{op}, sSet]_{proj, loc}
  \,.
$$

Its left [[derived functor]] is equivalent to the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]

$$
  \mathbb{L}\mathbf{Sing}(-) \simeq \Pi(-)
$$

and the _constant path inclusion_ $Id \to \Pi$ is presented by the canonical [[natural transformation]] $Id \to \mathbf{Sing}$.
=--

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

On an arbitrary simplicial presheaf $X$ the functor $\mathbf{Sing}$ is given by the [[coend]]

$$
  \mathbf{Sing} : X \mapsto
   \int^{U \in C} X(U) \cdot \mathbf{Sing}U
  \,.
$$

By construction this preserves all colimits. Hence by the [[adjoint functor theorem]] (using that domain and codomain are [[presheaf categories]]) we have that $\mathbf{Sing}$ is a [[left adjoint]]. Explicitly, the [[right adjoint]] is given by

$$
  R X : U \mapsto [C^{op}, sSet](\mathbf{Sing}U, X)
  \,.
$$

We check that $\mathbf{Sing}$ is also a [[Quillen adjunction|left Quillen functor]] first for the [[model structure on simplicial presheaves|global projective model structure]].
For that, notice that the above expression is the evaluation of the left [[Quillen bifunctor]] (see the examples-section there for details)

$$
  \int^C (-) \cdot (-) :
    [C^{op}, sSet]_{proj} \times [C, [C^{op}, sSet]_{proj}]_{inj}
  \to
   [C^{op}, sSet]_{proj}
  \,.
$$

Since every representable $U$ is cofibrant in $[C^{op}, sSet]_{proj}$ and since $U \to \mathbf{Sing}U$ is a cofibration by the [[small object argument]], we have that $\mathbf{Sing}U$ is cofibrant in $[C^{op}, sSet]_{proj}$ for all $U$. This means that also $\mathbf{Sing}(-)$ is cofibrant in $[C, [C^{op}, sSet]_{pro}]_{inj}$. Since $\int^C (-) \cdot (-)$ is a left Quillen bifunctor it follows that $\int^C (-)\cdot \mathbf{Sing}$ is a left Quillen functor. Hence it preserves cofibrations and acyclic cofibrations.

This establishes that $\mathbf{Sing}$ is a left simplicial Quillen functor on $[C^{op}, sSet]_{proj}$.

Since this is a [[left proper model category]] we have by the discussion at [[simplicial Quillen adjunction]] that for showing that this does descend to the local model structure it is sufficient to check that the right adjoint preserves local fibrant objects. Which, in turn, is implied if $\mathbf{Sing}$ send covering [[Cech nerve]]s to weak equivalences.

Let therefore $C(\coprod_i U_i \to U)$ be the Cech nerve of a covering family in the [[site]] $C$. We may write this as the [[coend]]

$$
  C(\coprod_i U_i) = \int^{[k] \in \Delta} \Delta[k] \cdot
  \left(
    \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n}
  \right)
  \,,
$$

where by assumption on the [[∞-connected site]] $C$ all the $U_{i_0, \cdots, i_n}$ are representable. By precomposing the projection $C(\coprod_i U_i) \to X$ with the objectwise [[Bousfield-Kan map]] that replaces the simplices with the [[fat simplex]] $\mathbf{\Delta} : \Delta \to sSet$, we get the morphisms

$$
  C(\coprod_i U_i) = \int^{[k] \in \Delta}
   \mathbf{\Delta}[k] \cdot
  \left(
    \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n}
  \right)
   \stackrel{\simeq}{\to}
   C(\coprod_i U_i)
   \to U
  \,.
$$

Here the first map is an objectwise weak equivalence by Bousfield-Kan (see the examples at [[Reedy model structure]] for details). Hence by 2-out-of-3 we may equivalently check that $\mathbf{Sing}$ sends these morphisms to weak equivalences in $[C^{op}, sSet]_{proj}$.

Since $\mathbf{Sing}$ commutes with all colimits and hence coends the result of applying it to this morphism is

$$
  \int^{[k] \in \Delta}
   \mathbf{\Delta}[k] \cdot
  \left(
    \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n}
  \right)
   \to \mathbf{Sing}U
  \,.
$$

Since the [[fat simplex]] is cofibrant in $[\Delta, sSet_{Quillen}]_{proj}$ and since the above is an evaluation of the left [[Quillen bifunctor]]

$$
  \int^\Delta (-) \cdot (-) :
   [\Delta, sSet_{Quillen}]_{proj} \times
   [\Delta^{op}, [C^{op}, sSet]_{proj}]_{inj}
  \to
   [C^{op}, sSet]_{proj}
$$

the functor $\int^\Delta \mathbf{\Delta} \cdot (-)$ is left Quillen and hence preserves weak equivalences between cofibrant objects (by the [[factorization lemma]]), such as the morphisms $\mathbf{Sing}U \stackrel{\simeq}{\to} *$. Therefore we have a commuting diagram

$$
  \array{
    \int^{[k] \in \Delta}
     \mathbf{\Delta}[k] \cdot
    \left(
      \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n}
    \right)
      &\stackrel{\simeq}{\to}&
    \int^{[k] \in \Delta}
     \mathbf{\Delta}[k] \cdot
    \left(
      \coprod_{i_0, \cdots, i_n} *
    \right)
    \\
    \downarrow && \downarrow^{\simeq}
    \\
    \mathbf{Sing}U &\stackrel{\simeq}{\to}& *
   }
  \,,
$$

with weak equivalences in $[C^{op}, sSet]_{proj}$ as indicated: the top morphism is a weak equivalence by the argument just given, the bottom one by the [[small object argument]]-construction of $\mathbf{Sing}$ and the right vertical morphism is a weak equivalence by the assumption on an [[∞-connected site]]. It follows by 2-out-of-3 that also the left vertical morphism is a weak equivalence.

This establishes the fact that $\mathbf{Sing}$ is left Quillen on the local model structure on simplicial presheaves. By the discussion at [[simplicial Quillen adjunction]] this implies that its left [[derived functor]] is a [[left adjoint|left]] [[adjoint (∞,1)-functor]]. Hence it preserves [[(∞,1)-colimit]]s and so is determined on representatives. There $\mathbf{Sing} U \simeq *$ does coindice with $\Pi(U) \simeq *$, hence both [[(∞,1)-functor]]s are equivalent.
=--

+-- {: .un_corollary}
###### Corollary

For all cofibrant $X \in [C^{op}, sSet]_{proj,loc}$, the de Rham coefficient object $\mathbf{\Pi}_{dR} X$ is presented by the ordinary [[pushout]]

$$
  \array{
     X &\to& *
     \\
     \downarrow && \downarrow
     \\
     \mathbf{Sing}X &\to& \mathbf{\Pi}_{dR} X
  }
$$

in $[C^{op}, sSet]$.
=--

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

By definition we have that $\mathbf{\Pi}_{dR}$ is the [[(∞,1)-pushout]] $\mathbf{\Pi}(X) \coprod_X *$ in $Sh_{(\infty,1)}(C)$. By the above proposition we have a cofibrant presentation of the pushout diagram as indicated (all three objects cofibrant, at least one of the two morphisms a cofibration). By the general discussion at [[homotopy colimit]] the ordinary pushout of that diagram does compute the [[(∞,1)-colimit]].
=--

## Homotopy localization

We discuss that the [[homotopy localization]] of topological $\infty$-groupoids reproduces [[Top]] $\simeq$ [[∞Grpd]], following ([Dugger](#Dugger)).

### Idea

A central result about the [[(∞,1)-topos]] $Sh_{(\infty,1)}(Top)$ of [[∞-stack]]s on [[Top]] is that the [[homotopy localization]] is equivalent to [[Top]] itself

$$
  Sh_{(\infty,1)}(Top)^I \simeq Top
  \,.
$$

A discussion of this is in (the nice but not quite finished) ([Dugger](#Dugger)).

In fact, this is true even for [[Lie ∞-groupoid]]s, i.e. [[∞-stack]]s on [[Diff]]: the homotopy invariant ones model plain [[topological space]]s.

This provides a useful resolution of [[topological space]]s that often helps to disentangle the two different roles played by a topological space: on the one hand as a model for an [[∞-groupoid]], in the other as a [[locale]].

### Details

Let $SPSh(Diff)^{loc}$ be the local [[model structure on simplicial presheaves]] obtained by left [[Bousfield localization of model categories|Bousfield localization]] at the [[Cech nerve]]s of
[[Cech cover]]s with respect to the standard [[Grothendieck topology]] on [[Diff]]. This is a [[models for ∞-stack (∞,1)-toposes|model for ∞-stacks]] on [[Diff]].

Let $SPSh(Diff)^{loc}_I$ be furthermore the left [[Bousfield localization of model categories|Bousfield localization]]
at the set of projection morphisms out of [[product]]s of the form $X \times \mathbb{R} \to X$ for all $X \in Diff$. The $\infty$-stacks that are [[local object]]s with respect
to these morphisms are the _homotopy invariant_ $\infty$-stacks, so this localization models the [[(∞,1)-topos]] of homotopy invariant
$\infty$-stacks on $Diff$.

There is a [[adjunction]]

$$
  L : SSet \stackrel{\leftarrow}{\to} SPSh(Diff) : R
$$

where $L$ sends a simplicial set to the simplicial presheaf constant on that simplicial
set, and where evaluates a simplicial presheaf on the manifold that is the [[point]].

+-- {: .un_theorem}
###### Theorem (Dugger)

This adjunction $(L \dashv R)$ is a [[Quillen equivalence]] with respect to the  standard [[model structure on simplicial sets]] on the left and the above model structure $SPSh(Diff)_{loc}^I$ on the right.
=--

## Related concepts

* [[D-topological space space]], [[diffeological space]]

* [[cohesive (∞,1)-topos]]

  * [[discrete ∞-groupoid]]

  * [[topological infinity-groupoid]]

    * **Euclidean-topological ∞-groupoid**

    * [[locally contractible topological infinity-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

  * [[super ∞-groupoid]]

  * [[smooth super ∞-groupoid]]

  * [[synthetic differential super ∞-groupoid]]

## References

Section 3.2 in

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

Some discussion of the $(\infty,1)$-category of $(\infty,1)$-sheaves on the category of manifolds and its restriction to open balls and a discussion of its homotopy localization is in:

* {#Dugger} [[Dan Dugger]], _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [dvi](http://www.uoregon.edu/~ddugger/cech.dvi), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

Discussion of [[geometric realization of simplicial topological spaces|geometric realization of simplicial topological]] principal bundles and of their [[classifying spaces]] is in

* {#RobertsStevenson} [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundle in parameterized spaces_ ([arXiv:1203.2460](http://arxiv.org/abs/1203.2460))

* {#Stevenson} [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](http://arxiv.org/abs/1203.2461))

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