David P. DiVincenzo

* [QuTech page](https://qutech.nl/person/david-divincenzo/)

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

* [GoogleScholar page](https://scholar.google.com/citations?user=4kTctIoAAAAJ&hl=en)

## Selected writings

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

Influential list of criteria necessary for physical realization of [[quantum computation]]:

* {#DiVincenzo00} [[David P. DiVincenzo]], *The Physical Implementation of Quantum Computation*, Fortschr. Phys. **48** 9-11 (2000) 771-783 [[arXiv:quant-ph/0002077](https://arxiv.org/abs/quant-ph/0002077), doi:10.1002/1521-3978(200009)48:9/11%3C771::AID-PROP771%3E3.0.CO;2-E]

The idea of [[qbits]] realized on [[quantum mechanics|quantum mechanical]] [[spinors]] (e.g. [[electron]]-spin or [[nucleus]]-spin, cf. [spin resonance qbits](nuclear+magnetic+resonance#SpinResonanceQBitsReferences)):

* [[Daniel Loss]], [[David P. DiVincenzo]], *Quantum Computation with Quantum Dots*, Phys. Rev. A **57** 120 (1998) [[arXiv:cond-mat/9701055](https://arxiv.org/abs/cond-mat/9701055), [doi:10.1103/PhysRevA.57.120](https://doi.org/10.1103/PhysRevA.57.120)]

On [[topological quantum computation]]:

* [[Ananda Roy]], [[David P. DiVincenzo]], *Topological Quantum Computing*, Lecture notes of the [48th IFF Spring School](https://www.fz-juelich.de/en/pgi/expertise/schools-and-courses/iff-spring-school/the-iff-spring-school-topics-and-history/springschool2017) (2017) $[$[arXiv:1701.05052](https://arxiv.org/abs/1701.05052)$]$

category:people

[[!redirects David DiVincenzo]]

[[!redirects David Di Vincenzo]]
[[!redirects David P. Di Vincenzo]]


David Penneys

* [personal page](https://people.math.osu.edu/penneys.2/8800//Math8800Spring2021.html)

## Selected writings

On [[monoidal enriched categories]]:

* [[Scott Morrison]], [[David Penneys]], *Monoidal Categories Enriched in Braided Monoidal Categories*, International Mathematics Research Notices **2019** 11 June 2019 3527–3579 [[doi:10.1093/imrn/rnx217](https://doi.org/10.1093/imrn/rnx217), [arXiv:1701.00567](https://arxiv.org/abs/1701.00567)]

On [[topological phases of matter]] (with emphasis on [[monoidal category]]-machinery such as [[unitary fusion categories]])

* [[David Penneys]], *[Topological Phases of Matter](https://people.math.osu.edu/penneys.2/8800//Math8800Spring2021.html)* (2021)

category: people


David Poland

* [webpage](http://physics.yale.edu/people/david-poland)

## Selected writings

On the [[conformal bootstrap]]:

* {#PolandSimmonsDuffin16} [[David Poland]], [[David Simmons-Duffin]], _The conformal bootstrap_, Nature Physics 535–539 (2016) [doi:10.1038/nphys3761](http://www.nature.com.edgesuite.net/nphys/journal/v12/n6/full/nphys3761.html)

* [[David Poland]], [[Slava Rychkov]], [[Alessandro Vichi]], _The Conformal Bootstrap: Numerical Techniques and Applications_, Rev. Mod. Phys. 91, 15002 (2019) ([arXiv:1805.04405](https://arxiv.org/abs/1805.04405))

For [[gauge theories]]:

* Zhijin Li, [[David Poland]], _Searching for gauge theories with the conformal bootstrap_ ([arXiv:2005.01721](https://arxiv.org/abs/2005.01721))

Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

## Related $n$Lab entries

* [[conformal bootstrap]]

category: people


David Poulin

* [CIFAR obituary](https://cifar.ca/cifarnews/2020/06/17/a-quantum-festschrift-for-david-poulin/)

## Selected writings

On the *[[DQC1]]* [[quantum complexity theory|quantum complexity]] [[complexity class|class]] ("one clean qbit"):

* [[David Poulin]], [[Raymond Laflamme]], [[Gerard J. Milburn]], [[Juan Pablo Paz]], *Testing integrability with a single bit of quantum information*, Phys. Rev. A **68** 22302 (2003) [[arXiv:quant-ph/0303042](https://arxiv.org/abs/quant-ph/0303042), [doi:10.1103/PhysRevA.68.022302](https://doi.org/10.1103/PhysRevA.68.022302)]

[[operator algebra|Operator algebraic]] formulation of [[quantum error correction]]:

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], *A Unified and Generalized Approach to Quantum Error Correction*, Phys. Rev. Lett. 94, 180501 (2005) ([arXiv:quant-ph/0412076](https://arxiv.org/abs/quant-ph/0412076))

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], [[Maia Lesosky]], *Operator quantum error correction*, Quant. Inf. & Comp., 6 (2006), 383-399 ([arXiv:quant-ph/0504189](https://arxiv.org/abs/quant-ph/0504189))

On [[tensor networks]] in [[quantum error correction]]:

* {#FerrisPoulin13} Andrew J. Ferris, [[David Poulin]], *Tensor Networks and Quantum Error Correction*, Phys. Rev. Lett. 113, 030501 (2014) ([arXiv:1312.4578](https://arxiv.org/abs/1312.4578))

category: people


David Prieto

* [InSpire page](https://inspirehep.net/authors/1808337)

## Selected writings

On [[moduli stabilization]] in [[flux compactifications]] of [[type II string theory]]:

* [[David Prieto]], *Moduli Stabilization and Stability in Type II/F-theory flux compactifications* [[arXiv:2401.13339](https://arxiv.org/abs/2401.13339)]

category: people


David R. Hilbert

(NB.  This is clearly not the same David Hilbert as [[David Hilbert]])

David R. Hilbert is a Professor of Philosophy at the University of Illinois at Chicago. His main areas of research are color, philosophy of perception, and philosophy of mind. He also has interests in philosophy of biology, early modern philosophy (especially Berkeley) and epistemology.

* [personal webpage](https://hilbert.people.uic.edu/)

category:people


David Rees

* [Wikipedia page](https://en.wikipedia.org/wiki/David_Rees_%28mathematician%29)

category: people


David Reutter

* [website](https://www.cs.ox.ac.uk/people/david.reutter/)

## Selected writings

On [[associative n-categories]] and their [[formal proof|formalization]] in [[proof assistants]] (cf.: [[Globular]], [[homotopy.io]]):

* {#ReutterVicary19} [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]

* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

Introducing a notion of [[fusion 2-categories]]:

* [[Christopher L. Douglas]], [[David J. Reutter]], *Fusion 2-categories and a state-sum invariant for 4-manifolds* [[arXiv:1812.11933](https://arxiv.org/abs/1812.11933)]

On [[weak omega-category|weak $\omega$-categories]] via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

category: people

[[!redirects David J. Reutter]]


David Ridout

* [personal page](https://researchers.ms.unimelb.edu.au/~dridout@unimelb/)

* [GoogleScholar page](https://scholar.google.com.au/citations?user=UOua7WkAAAAJ&hl=en)

## Selected writings

On [[logarithmic CFT]]:

*  [[Thomas Creutzig]], [[David Ridout]], *Logarithmic conformal field theory: beyond an introduction*,  J. Phys. A: Math. Theor. **46** (2013) 494006 ([doi:10.1088/1751-8113/46/49/494006](https://iopscience.iop.org/article/10.1088/1751-8113/46/49/494006), [arXiv:1303.0847](https://arxiv.org/abs/1303.0847))

On [[fractional-level WZW models]] as [[logarithmic CFT]]:

* [[David Ridout]], *$\widehat{\mathfrak{sl}}(2)_{-1/2}$: A Case Study*, Nucl. Phys. B **814** (2009) 485-521 $[$[arXiv:0810.3532](https://arxiv.org/abs/0810.3532), [doi:10.1016/j.nuclphysb.2009.01.008](https://doi.org/10.1016/j.nuclphysb.2009.01.008)$]$

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models I*, Nuclear Physics B **865** 1 (2012) 83-114 $[$[arXiv:1205.6513](https://arxiv.org/abs/1205.6513), [doi:10.1016/j.nuclphysb.2012.07.018](https://doi.org/10.1016/j.nuclphysb.2012.07.018)$]$

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models II*, Nuclear Physics B **875** 2 (2013) 423-458 $[$[arXiv:1306.4388](https://arxiv.org/abs/1306.4388), [doi:10.1016/j.nuclphysb.2013.07.008](https://doi.org/10.1016/j.nuclphysb.2013.07.008)$]$

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules I: rank 1 cases*, Commun. Math. Phys. **368** (2019) 627–663  $[$[arXiv:1803.01989](https://arxiv.org/abs/1803.01989), [doi:10.1007/s00220-019-03305-x](https://doi.org/10.1007/s00220-019-03305-x)$]$

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules II: classifications for affine vertex algebras*, Communications in Contemporary Mathematics, **24** 05 (2022) 2150037 $[$[arXiv:1906.02935](https://arxiv.org/abs/1906.02935), [doi:10.1142/S0219199721500371](https://doi.org/10.1142/S0219199721500371)$]$

Reviewed in:

* [[David Ridout]], *Fractional Level WZW Models as Logarithmic CFTs* (2010) $[$[pdf](https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/100225.pdf), [[Ridout-FractionalLevelWZW2010.pdf:file]]$]$

* [[David Ridout]], *Fractional-level WZW models* (2020) $[$[pdf](https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/200205.pdf), [[Ridout-FractionalLevelWZW2020.pdf:file]]$]$

On the [[Verlinde formula]] for [[logarithmic CFTs]]:

* [[David Ridout]], [[Simon Wood]], *The Verlinde formula in logarithmic CFT*,  J. Phys.: Conf. Ser. **597** 012065 $[$[arXiv:1409.0670](https://arxiv.org/abs/1409.0670), [doi:10.1088/1742-6596/597/1/012065](https://doi.org/10.1088/1742-6596/597/1/012065)$]$

category: people


David Rodríguez Entem

* [webpage](http://campus.usal.es/~dptoff/?q=node/37)

## Selected writings

On [[meson]] [[effective field theory]] ([[chiral perturbation theory]]) in [[nuclear physics]]:

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

category: people

[[!redirects David Entem]]


David Ruelle

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

## related $n$Lab entries

* [[Ruelle zeta function]]

category: people


David Rydh

I am an algebraic geometer and currently a postdoc at [UC Berkeley](http://www.math.berkeley.edu/~dary/).

At the moment I am mostly interested in algebraic stacks. A brief list of interests include:

* **Moduli** Chow and Hilbert schemes, Hilbert stacks, higher-dimensional moduli, coarse quotients, ...

* Algebraic spaces and stacks (foundational).

* Riemann-Zariski spaces, valuation theory, rigid geometry

* Two-dimensional sheaf theory.

* Log geometry

* minimal model program, multiplier ideal sheaves

* non-noetherian objects

* **Commutative algebra** (from a geometric viewpoint): divided powers, seminormality, weak normality, ideal closures such as tight closure, absolute flat (and perfect) rings

category: people


David Saunders

* [webpage](http://prf.osu.eu/kma/index.php?idc=48210)

## related $n$Lab entries

* [[jet bundle]]

category: people

[[!redirects David John Saunders]]


David Savitt

* [webpage](https://math.jhu.edu/~savitt/)

## Selected writings

* {#GHS18} [[Toby Gee]], [[Florian Herzig]], [[David Savitt]], _General Serre Weight Conjectures_, J. Eur. Math. Soc. 20 (2018), no. 12, 2859-2949 ([arXiv:1509.02527](https://arxiv.org/abs/1509.02527))

category:people


David Simmons-Duffin

* [webpage](http://sns.ias.edu/~dsd/)

## Selected writings

On the [[conformal bootstrap]]:

* {#PolandSimmonsDuffin16} [[David Poland]], [[David Simmons-Duffin]], _The conformal bootstrap_, Nature Physics 535–539 (2016) [doi:10.1038/nphys3761](http://www.nature.com.edgesuite.net/nphys/journal/v12/n6/full/nphys3761.html)

* [[David Simmons-Duffin]], _TASI Lectures on the Conformal Bootstrap_ ([arXiv:1602.07982](https://arxiv.org/abs/1602.07982))

Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

## Related $n$Lab entries

* [[conformal bootstrap]]

category: people


David Singerman


David Skinner

* [personal page](https://www.damtp.cam.ac.uk/user/dbs26/)

## Selected writings

On [[twistor string theory]]:

* {#Skinner09} [[David Skinner]], _The geometry of scattering amplitudes_, talk notes, November 2009 ([pdf](http://research.physics.unc.edu/string/transparencies/trans_20091112.pdf), [[Skinner_GeometryOfScattering.pdf:file]])

On [[scattering amplitudes]] in [[D=4 N=8 supergravity]]:

* {#CachazoSkinner12} Freddy Cachazo, [[David Skinner]], *Gravity from Rational Curves*, Phys. Rev. Lett. 110, 161301, 2012 ([arXiv:1207.0741](https://arxiv.org/abs/1207.0741), [arXiv:10.1103/PhysRevLett.110.161301](https://doi.org/10.1103/PhysRevLett.110.161301))

category: people


David Spergel

* [webpage](https://www.astro.princeton.edu/~dns/)

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

## Selected writings

Constraints from [[cosmology]] on [[MOND]] [[phenomenology]]:

* Kris Pardo, [[David Spergel]], _What is the price of abandoning dark matter? Cosmological constraints on alternative gravity theories_ ([arXiv:2007.00555](https://arxiv.org/abs/2007.00555))

category: people


David Speyer

* [University of Michigan page](http://www-personal.umich.edu/~speyer/)

category: people


David Spivak

* [website](http://www.dspivak.net/)

## Selected writings

On [[derived smooth manifolds]]:

* [[David Spivak]], _Derived smooth manifolds_, Duke Math. J. **153** 1 (2010) 22-128 [[arXiv:0810.5174](http://arxiv.org/abs/0810.5174), [doi:10.1215/00127094-2010-021](https://doi.org/10.1215/00127094-2010-021)]

On [[(∞,1)-categorical hom-spaces]] in terms of [[quasi-categories]]:

* [[Dan Dugger]], [[David Spivak]], _Mapping spaces in quasi-categories_, Algebraic & Geometric Topology 11 (2011) 263–325 ([arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263))

On [[internal categories in homotopy type theory]]:

* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], *Experience Implementing a Performant Category-Theory Library in Coq*,  In:  *Interactive Theorem Proving. ITP 2014* Lecture Notes in Computer Science, **8558** Springer (2014)  $[$[arXiv:1401.7694](http://arxiv.org/abs/1401.7694), [doi:10.1007/978-3-319-08970-6_18](https://doi.org/10.1007/978-3-319-08970-6_18)$]$

On [[hypergraph categories]]:

* {#FongSpivak} [[Brendan Fong]] and [[David Spivak]], *Hypergraph Categories* ([arxiv:1806.08304](https://arxiv.org/abs/1806.08304))

On [[category theory]]:

* {#FongSpivak18} [[Brendan Fong]], [[David Spivak]], _An invitation to applied category theory_, 2018 ([web](http://math.mit.edu/~dspivak/teaching/sp18/), [pdf](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf))

On [[lenses (in computer science)]]:

* {#SpivakACT19} [[David Spivak]], _Lenses:  applications and generalizations_, talk at [ACT 19](http://www.cs.ox.ac.uk/ACT2019/) ([slides](http://math.ucr.edu/home/baez/ACTUCR2019/ACTUCR2019_spivak.pdf), [[Spivak_Lenses.pdf:file]])

On [[polynomial functors]]:

* [[David Spivak]], _Poly: An abundant categorical setting for mode-dependent dynamics_, ([arXiv:2005.01894](https://arxiv.org/abs/2005.01894))

On [[type theory|type theoretical]] [[databases]]:

* [[Henrik Forssell]], [[Håkon Robbestad Gylterud]], [[David Spivak]], *Type theoretical databases*, Journal of Logic and Computation, Volume 30, Issue 1, January 2020, Pages 217–238, ([doi:10.1093/logcom/exaa009](https://doi.org/10.1093/logcom/exaa009), [arXiv:1406.6268](https://arxiv.org/abs/1406.6268))

On [[dynamical systems]] and their [[categorical systems theory]]:

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], *Dynamical Systems and Sheaves*, Appl Categor Struct **28** (2020) 1–57 [[arXiv:1609.08086](https://arxiv.org/abs/1609.08086), [doi:10.1007/s10485-019-09565-x](https://doi.org/10.1007/s10485-019-09565-x)]

and via [[polynomial functors]]:

* {#SpivakPoly} [[David Spivak]], [[Nelson Niu]], _Polynomial Functors: A General Theory of Interaction_ [[webpage](https://topos.site/poly-course/), [pdf](https://topos.site/poly-book.pdf)]

## Related entries

* [[derived smooth manifold]]

* [[relation between quasi-categories and simplicial categories]]

category: people

[[!redirects David I. Spivak]]


David Tong

* [webpage](http://www.damtp.cam.ac.uk/user/tong/)

## Selected writings

On [[string theory]]:

* {#Tong09} [[David Tong]], _Lectures on String Theory_ ([arXiv:0908.0333](https://arxiv.org/abs/0908.0333))

Identification of the [[Coulomb branch]] of [[D=3 N=4 super Yang-Mills theory]] with the [[moduli space of monopoles]] in [[Yang-Mills theory]]:

* N. Dorey, V. V. Khoze, M. P. Mattis, [[David Tong]], S. Vandoren, _Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold_, Nucl. Phys. B502 (1997) 59-93 ([arXiv:hep-th/9703228](https://arxiv.org/abs/hep-th/9703228))

On [[M-theory on 8-manifolds]] and [[conifold transition]]:

* [[Sergei Gukov]], [[James Sparks]], [[David Tong]], *Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds*, Class. Quant. Grav. **20** (2003) 665-706 [[arXiv:hep-th/0207244](https://arxiv.org/abs/hep-th/0207244)]

On [[vortex strings]]:

* [[Amihay Hanany]], [[David Tong]], _Vortices, Instantons and Branes_, JHEP 0307 (2003) 037 ([arXiv:hep-th/0306150](https://arxiv.org/abs/hep-th/0306150))

* [[Amihay Hanany]], [[David Tong]], _Vortex Strings and Four-Dimensional Gauge Dynamics_, JHEP 0404 (2004) 066 ([arXiv:hep-th/0403158](https://arxiv.org/abs/hep-th/0403158))

* {#Tong09} [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060))

* [[David Tong]], _Vortices, Strings, and Vortex Strings_ ([[TongVortexStrings.pdf:file]])

On [[vortices]]:

* [[David Tong]], _TASI Lectures on Solitons_ ([web](http://www.damtp.cam.ac.uk/user/tong/tasi.html)), _part 3: _Vortices_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/vortex.pdf))

On [[solitons]]:

* {#Tong05} [[David Tong]], _TASI Lectures on Solitons_ ([arXiv:hep-th/0509216](https://arxiv.org/abs/hep-th/0509216)), _Lecture 1: Instantons_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf))

On [[quantum field theory]]:

* [[David Tong]], *Lectures on Quantum Field Theory*, Cambridge (2006-7) [[pdf](https://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf), [web](https://www.damtp.cam.ac.uk/user/tong/qft.html)]

On [[fractional M2-branes]]

* [[Neil Lambert]], [[David Tong]], _Membranes on an Orbifold_, Phys.Rev.Lett.101:041602, 2008 ([arXiv:0804.1114](https://arxiv.org/abs/0804.1114))

On [[mirror symmetry]]:

* [[Mina Aganagic]], [[Kentaro Hori]], [[Andreas Karch]], [[David Tong]], _Mirror Symmetry in 2+1 and 1+1 Dimensions_, JHEP 0107:022,2001 ([arXiv:hep-th/0105075](http://arxiv.org/abs/hep-th/0105075))

On [[solid state physics]]:

* [[David Tong]], *Lectures on solid state physics* (2017) $[$[pdf](http://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf), [webpage](http://www.damtp.cam.ac.uk/user/tong/solidstate.html)$]$

On [[triality]] and its application on [[string theory]]:

* {#TongTurner19} [[David Tong]], Carl Turner, _Notes on 8 Majorana Fermions_ ([arXiv:1906.07199](https://arxiv.org/abs/1906.07199))

On the [[quantum Hall effect]] and [[Laughlin wavefunctions]]:

* {#Tong16} [[David Tong]], *The Quantum Hall Effect* (2016) $[$[course webpage](https://www.damtp.cam.ac.uk/user/tong/qhe.html), [pdf](http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf), [[Tong-QuantumHallEffect.pdf:file]]$]$

Discussion of [[anomaly cancellation]] on the [[standard model of particle physics]]:

* Nakarin Lohitsiri, [[David Tong]], _Hypercharge Quantisation and Fermat's Last Theorem_ ([arXiv:1907.00514](https://arxiv.org/abs/1907.00514))

  (relating to [[Fermat's last theorem]])

## Selected talks

On [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]:

* {#Tong17} [[David Tong]], _The Unreasonable Effectiveness of Physics in Mathematics_, [LMS Popular Lecture](https://www.lms.ac.uk/events/popular-lectures) 2017 ([webpage](https://www.maths.cam.ac.uk/features/lms-popular-lecture-unreasonable-effectiveness-physics-mathematics-david-tong), [talk recording](https://youtu.be/UVuKyZ4pBzg))

## Related $n$Lab entries

* [[string theory]]

* [[vortex]]

* [[soliton]]

* [[fractional M2-brane]]

* [[M-theory on 8-manifolds]]

* [[conifold transition]]

* [[discrete torsion]]

* [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]

category: people


David Treumann

[[!redirects Treumann]]

* [List of Publications](https://www2.bc.edu/david-treumann/materials.html)

##Related entries

* [[Carlos Simpson]]

* [[stratified space]]

category: people


David van Dantzig

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

## Selected writings

On [[pre-metric electromagnetism]]:

* {#Dantzig34} [[David van Dantzig]], *The fundamental equations of electromagnetism, independent of metrical geometry*, Mathematical Proceedings of the Cambridge Philosophical Society **30** 4  (1934) 421-427 [[doi:10.1017/S0305004100012664](https://doi.org/10.1017/S0305004100012664)]

category: people


David Vanderbilt

* [personal page](http://www.physics.rutgers.edu/~dhv/)

## Selected writings

On [[electronic band structure]] and [[Berry connection|Berry connections]] in the context of [[topological phases of matter]] ([[topological insulators]] and [[semimetals]]):

* {#Vanderbilt18} [[David Vanderbilt]],  *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))

category: people


David Vogan

* [webpage](http://math.mit.edu/~dav/)

category: people


David W. Kribs

* [Institute page](https://dkribs.uoguelph.ca/)

## Selected writings

[[operator algebra|Operator algebraic]] formulation of [[quantum error correction]]:

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], *A Unified and Generalized Approach to Quantum Error Correction*, Phys. Rev. Lett. 94, 180501 (2005) ([arXiv:quant-ph/0412076](https://arxiv.org/abs/quant-ph/0412076))

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], [[Maia Lesosky]], *Operator quantum error correction*, Quant. Inf. & Comp., 6 (2006), 383-399 ([arXiv:quant-ph/0504189](https://arxiv.org/abs/quant-ph/0504189))

on [[quantum observables]]:

* {#BenyKempfKribs06} Cédric Bény, Achim Kempf, [[David W. Kribs]], _Generalization of Quantum Error Correction via the Heisenberg Picture_, Phys. Rev. Lett. 98, 100502 – Published 7 March 2007 ([doi:10.1103/PhysRevLett.98.100502](https://doi.org/10.1103/PhysRevLett.98.100502), [arXiv:quant-ph/0608071](https://arxiv.org/abs/quant-ph/0608071))

* Cédric Bény, Achim Kempf, [[David W. Kribs]], _Quantum Error Correction of Observables_, Phys. Rev. A 76, 042303 (2007) ([arXiv:0705.1574](https://arxiv.org/abs/0705.1574))

category: people

[[!redirects David Kribs]]


David Wallace

* [webpage](http://users.ox.ac.uk/~mert0130/)

## Selected writings

On the [[interpretation of quantum mechanics]]:

* [[David Wallace]], *Philosophy of Quantum Mechanics*, Part 2 in: Dean Rickles (ed.), *The Ashgate Companion to Contemporary Philosophy of Physics* (2008) 16-98 [[ISBN:9780754655183](https://www.routledge.com/The-Ashgate-Companion-to-Contemporary-Philosophy-of-Physics/Rickles/p/book/9780754655183), [arXiv:0712.0149](http://arxiv.org/abs/0712.0149)]

## Related pages

* [[philosophy of physics]]

* [[interpretation of quantum mechanics]]

category: people


David White

* [MO page](http://mathoverflow.net/users/11540/david-white)

* [[commutative monoid in a symmetric monoidal model category]]

category: people


David Wiltshire

* [webpage](http://www2.phys.canterbury.ac.nz/~dlw24/)

category: people


David Wood

[[!redirects David W. Wood]]

* [academi.edu page](http://independent.academia.edu/DavidWWood)

## related $n$Lab entries

* [[philosophy of science]]

* [[objective idealism]]

* [[Novalis]]

category: people


David Wright

* [webpage](https://www.math.okstate.edu/~wrightd/)

## related $n$Lab entries

* [[Iwasawa-Tate theory]]

category: people


David Wärn

* [personal page](https://dwarn.se/)

* [institute page](https://www.chalmers.se/en/staff/Pages/warnd.aspx)

## Selected writings

On [[delooping]] in [[homotopy type theory]] and application to [[Eilenberg-MacLane space]] [[Eilenberg-MacLane space types|types]] and [[Steenrod operations]]:

* [[David Wärn]], *Eilenberg-MacLane spaces and stabilisation in homotopy type theory*, J. Homotopy Relat. Struct. **18**  (2023) 357–368 [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685), [doi:10.1007/s40062-023-00330-5](https://doi.org/10.1007/s40062-023-00330-5)]

category: people

[[!redirects David Warn]]


David Yetter

* [website](http://www.math.ksu.edu/~dyetter/)

## Selected writings

On [[quantales]] and [[linear logic]] as a form of [[quantum logic]]:

* {#Yetter90} [[David Yetter]], *Quantales and (noncommutative) linear logic*, Journal of Symbolic Logic **55** (1990) 41-64 [[doi:10.2307/2274953](https://doi.org/10.2307/2274953)]

On the [[Crane-Yetter model]] [[4d TQFT]]:

* {#CraneYetterKauffman94} [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997)  ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))

## Related entries

* [[topological field theory]], [[Yetter model]]

* [[quantum gravity]]

* [[quantum logic]]

category: people

[[!redirects D. Yetter]]


Davide Gaiotto

* [webpage](https://perimeterinstitute.ca/people/davide-gaiotto)

## Selected writings

On [[open/closed string duality]]:

* [[Davide Gaiotto]], [[Leonardo Rastelli]], _A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model_, JHEP 0507:053,2005 ([hep-th/0312196](https://arxiv.org/abs/hep-th/0312196))

On [[KK-compactification|dimensional reduction]] of [[black rings]] in [[D=5 gravity]] to [[black holes]] in [[D=4 gravity]]:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _5D Black Rings and 4D Black Holes_, JHEP 0602:023, 2006 ([arXiv:hep-th/0504126](https://arxiv.org/abs/hep-th/0504126))

On [[D3-D5 brane bound states]]:

* [[Davide Gaiotto]], [[Edward Witten]], _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

On [[knot invariants]] from [[Yang-Mills theory]]:

* [[Davide Gaiotto]], [[Edward Witten]], _Knot Invariants from Four-Dimensional Gauge Theory_, Advances in Theoretical and Mathematical Physics, Volume 16 (2012) Number 3 ([arxiv:1106.4789](https://arxiv.org/abs/1106.4789))

On the [[AGT correspondence]]:

* {#Gaiotto09} [[Davide Gaiotto]], _$N=2$ dualities_, JHEP08(2012)034 ([arXiv:0904.2715](http://arxiv.org/abs/0904.2715))

* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](http://arxiv.org/abs/0906.3219))

* [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation_, Advances in Mathematics __234__ (2013) 239--403 ([arXiv:0907.3987](https://arxiv.org/abs/0907.3987))

On the [[M5-brane elliptic genus]]:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _The M5-Brane Elliptic Genus: Modularity and BPS States_, JHEP 0708:070, 2007 ([hep-th/0607010](https://arxiv.org/abs/hep-th/0607010))

* [[Davide Gaiotto]], [[Xi Yin]], _Examples of M5-Brane Elliptic Genera_, JHEP 0711:004, 2007 ([arXiv:hep-th/0702012](https://arxiv.org/abs/hep-th/0702012))

On [[Coulomb branch]] [[singularities]] in [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

  (relation to [[Nahm's equations]] and the [[moduli space of monopoles]])

On the [[theta angle]] in [[QCD]]:

* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))

On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]] and [[D6-D8 brane intersections]] and their [[fuzzy funnel]] [[noncommutative geometry]]:

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP 12 (2014) 003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

On [[Kazama-Suzuki models]]:

* [[Davide Gaiotto]], _Kazama-Suzuki models and BPS domain wall junctions in $N=1$ $SU(n)$ Super Yang-Mills_ ([arxiv:1306.5661](https://arxiv.org/abs/1306.5661))

On [[generalized global symmetries]]:

* {#GKSW14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries*  J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), doi:10.1007/JHEP02(2015)172]

On [[extended TQFTs]] allegedly modelling [[topological phases of matter]] in [[solid state physics]]:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Condensations in higher categories* ([arXiv:1905.09566](https://arxiv.org/abs/1905.09566))

On the [[Witten index]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Davide Gaiotto]], Tadashi Okazaki, _Sphere correlation functions and Verma modules_ ([arXiv:1911.11126](https://arxiv.org/abs/1911.11126))

Construction of [[2d SCFTs]] from [[error-correcting codes]] and suggestion that the [[Chern-Dold character]] from [[tmf]] to [[modular forms]], sends a [[2d SCFT]] to its [[partition function]]/[[elliptic genus]]/supersymmetric index:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601  [[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)]

and discussion of the [[2d SCFTs]] (namely [[supersymmetry|supersymmetric]] [[SU(2)]]-[[WZW-models]]) conjecturally corresponding, under this conjectural identification, to the elements of
$\mathbb{Z}/24$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$
  tmf^{-3}(\ast)
    =
  \pi_3(tmf)
$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$\pi_3(\mathbb{S})$
(the [[third stable homotopy group of spheres]]):

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], [[Edward Witten]], p. 17 of: *A Note On Some Minimally Supersymmetric Models In Two Dimensions*, ([arXiv:1902.10249](https://arxiv.org/abs/1902.10249)) in S. Novikov et al. *Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry*, Proc. Symposia Pure Math., 103(2), 2021 ([ISBN: 978-1-4704-5592-7](https://bookstore.ams.org/pspum-103-2))

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Mock modularity and a secondary elliptic genus* ([arXiv:1904.05788](https://arxiv.org/abs/1904.05788))

## Related entries

* [[BPS state]], [[wall crossing]]

* [[AGT correspondence]]

category: people


Davide Rossini

* [GoogleScholar entry](https://scholar.google.it/citations?user=T1l3ZzUAAAAJ&hl=it)

## Selected writings

On [[quantum computation]] and [[quantum information]]:

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

category: people


DavidJaz

David Jaz Myers is an undergraduate, making this page as a place to semi-publicly store notes about categories.


Davorin Lešnik

* [webpage](https://www.fmf.uni-lj.si/si/imenik/3210/)

## Selected writings
 {#SelectedWritings}

On [[synthetic topology]]:

* [[Andrej Bauer]], [[Davorin Lešnik]], _Metric Spaces in Synthetic Topology_, 2010 ([pdf](http://math.andrej.com/wp-content/uploads/2010/01/csms_in_synthtop.pdf))

* [[Davorin Lešnik]], _Unified Approach to Real Numbers in Various Mathematical Settings_, 2014 ([arXiv:1402.6645](https://arxiv.org/abs/1402.6645))

* [[Davorin Lešnik]], _Synthetic Topology and Constructive Metric Spaces_, PhD, 2010 ([arXiv:2104.10399](https://arxiv.org/abs/2104.10399))

category: people


Day convolution

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[category of functors]] on a [[monoidal category]] canonically inherits itself a monoidal category structure via a [[categorification|categorified]] [[convolution product]]. This holds generally in the context of [[enriched category theory]]. This was first observed by ([Day 70](#Day70)) and accordingly these monoidal structures are called _Day convolution_ products.

In more detail, just as there is [[convolution]] of  [[functions]] $f : G \to \mathbb{C}$ whenever $G$ carries the structure of a  [[group]], or more generally just the structure of a [[monoid]], so there is convolution of  [[functors]] $f \colon  \mathcal{G} \to Set$ whenever the [[category]] $\mathcal{G}$ carries the structure of a [[monoidal category]].

This may be generalized by replacing [[Set]] with a more general [[cocomplete category|cocomplete]] [[symmetric monoidal category]] $V$. The technical condition is that $V$ is a [[monoidally cocomplete category]]: i.e. the [[tensor product]] $u \otimes v$ preserves [[colimits]] in its two arguments separately; hence that the functors $u \otimes -$ and $- \otimes v$ preserve colimits. This occurs notably when $V$ is symmetric [[closed monoidal category|closed monoidal]] (so that these functors are [[left adjoints]]).

## Definition

### For monoidal categories
 {#ForMonoidalCategories}

Let $V$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with all small [[limits]] and [[colimits]].

For $\mathcal{C}$ a  $V$-[[enriched category]], write $[\mathcal{C},V]$ for the $V$-[[enriched functor category]] to $V$, etc.

We discuss two equivalent ways of defining Day convolution

1. _[In terms of coends](#InTermsOfCoends)_

1. _[In terms of profunctors](#InTermsOfProfunctors)_

#### In terms of coends
 {#InTermsOfCoends}

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

Let $(\mathcal{C}, \otimes, 1)$ be a [[small category|small]] $V$-enriched monoidal category.

Then the **Day convolution tensor product** on $[\mathcal{C},V]$

$$
  \otimes_{Day}
    \;\colon\;
  [\mathcal{C},V] \times [\mathcal{C},V]
    \longrightarrow
  [\mathcal{C},V]
$$

is given by the following [[coend]]

$$
  X \otimes_{Day} Y
    \;\colon\;
  c
    \;\mapsto\;
  \overset{(c_1,c_2)\in \mathcal{C}\times \mathcal{C}}{\int}
    \mathcal{C}(c_1 \otimes c_2, c) \otimes_V X(c_1) \otimes_V Y(c_2)
  \,.
$$

=--

We observe now that [[Day convolution]] is equivalently a [[left Kan extension]]. This will be key for understanding [[monoids]] and [[modules]] with respect to Day convolution.

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

Let $\mathcal{C}$ be a [[small category|small]] $V$-monoidal category. Its **[[external tensor product]]** is

$$
  \overline{\otimes}
   \;\colon\;
  [\mathcal{C}, V]
    \times
  [\mathcal{C}, V]
    \longrightarrow
  [\mathcal{C}\times \mathcal{C}, V]
$$

given by

$$
  X \overline{\otimes} Y
    \;\coloneqq\;
  \otimes_V \circ (X,Y)
  \,,
$$

i.e.

$$
  (X \overline\otimes Y)(c_1,c_2)
  =
  X(c_1)\otimes_V Y(c_2)
  \,.
$$

=--

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

The [[Day convolution]] product (def. \ref{TopologicalDayConvolutionProduct}) of two functors is equivalently the [[left Kan extension]] of their external tensor product (def. \ref{ExternalTensorProduct}) along the tensor product $\otimes_{\mathcal{C}}$: there is a [[natural isomorphism]]

$$
  X \otimes_{Day} Y
  \simeq
  Lan_{\otimes_{\mathcal{C}}} (X \overline{\otimes} Y)
  \,.
$$

Hence the [[adjunction unit]] is a [[natural transformation]] of the form

$$
  \array{
    \mathcal{C} \times \mathcal{C}
    &&
      \overset{X \overline{\otimes} Y}{\longrightarrow}
    &&
      V
    \\
    & {}^{\mathllap{\otimes}}\searrow
     &\Downarrow&
    \nearrow_{\mathrlap{X \otimes_{Day} Y}}
    \\
    && \mathcal{C}
  }
  \,.
$$

=--

This perspective is highlighted in ([MMSS 00, p. 60](#MMSS00)).

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

By prop. \ref{TopologicalLeftKanExtensionBCoend} we may compute the left Kan extension as the following [[coend]]:

$$
  \begin{aligned}
     Lan_{\otimes_{\mathcal{C}}} (X\overline{\otimes} Y)(c)
     &
     \simeq
      \overset{(c_1,c_2)}{\int}
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c )
      \wedge
      (X\overline{\otimes}Y)(c_1,c_2)
     \\
    & =
    \overset{(c_1,c_2)}{\int}
    \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c)
    \wedge
     X(c_1) \otimes_V Y(c_2)
  \end{aligned}
  \,.
$$

=--

Proposition \ref{DayConvolutionViaKanExtensionOfExternalTensorAlongTensor} implies the following fact, which is the key for the identification of "[[functors with smash product]]".

+-- {: .num_cor #DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}
###### Corollary

The operation of [[Day convolution]] $\otimes_{Day}$ (def. \ref{TopologicalDayConvolutionProduct}) is universally characterized by the property that there are [[natural isomorphisms]]

$$
  [\mathcal{C}, V](X \otimes_{Day} Y, Z)
    \simeq
  [\mathcal{C}\times \mathcal{C}, V](
    X \overline{\otimes} Y,\; Z \circ \otimes_C
  )
  \,,
$$

where $\overline{\otimes}$ is the external product of def. \ref{ExternalTensorProduct}.

=--

#### In terms of profunctors
 {#InTermsOfProfunctors}

The Day convolution can also be expressed in terms of [[profunctors]].  The tensor product $\otimes :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$ induces a representable profunctor $\mathcal{C}(\otimes,1): \mathcal{C} ⇸ \mathcal{C}\otimes \mathcal{C}$.  The above definition can be interpreted to say that if $X,Y\in [\mathcal{C},V]$ are regarded as profunctors $\mathcal{C} ⇸ I$, where $I$ is the unit $V$-category, then $X\otimes_{Day} Y$ is the composite of profunctors
$$ \mathcal{C}\xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C}\xrightarrow{X\otimes Y} I\otimes I \cong I. $$
A more "global" way to say the same thing is to consider the "evaluation" functor $[\mathcal{C},V] \otimes \mathcal{C} \to V$ to be a profunctor $E:\mathcal{C}⇸[\mathcal{C},V]^{op}$.  Then the profunctor composite
$$ \mathcal{C}\xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C}\xrightarrow{E\otimes E} [\mathcal{C},V]^{op} \otimes [\mathcal{C},V]^{op}$$
is a functor $\mathcal{C}\otimes [\mathcal{C},V] \otimes [\mathcal{C},V] \to V$, which by exponential transpose gives a functor $[\mathcal{C},V] \otimes [\mathcal{C},V] \to [\mathcal{C},V]$; this is the Day convolution product.

### For promonoidal categories

The [above](#InTermsOfProfunctors) description in terms of profunctors makes it clear that the construction only depends on the representable profunctor induced by $\otimes : \mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$, i.e. on the underlying [[promonoidal category]] of $\mathcal{C}$.  In the original article ([Day 70](#Day70)), a stronger form of the convolution is discussed, in which $\mathcal{C}$ is assumed only to be a [[promonoidal category]].

Before continuing our discussion, we comment a bit on a convention adopted in [(Day's thesis)](#Day70Thesis). To define [[promonoidal structures]], Day used functors of the form $P\colon\mathcal{A}^\mathrm{op}\times\mathcal{A}^\mathrm{op}\times\mathcal{A} \to \mathcal{V}$, whereas the nLab convention is that a [[profunctor]] $P\colon\mathcal{A}\times\mathcal{A}$⇸$\mathcal{A}$ is a functor $P\colon\mathcal{A}^\mathrm{op}\times\mathcal{A}\times\mathcal{A} \to \mathcal{V}$. Following modern usage and [(Corner 2016)](#Corner2016), instead of defining Day convolution for $\mathcal{V}$-[[enriched functors]], we do so for $\mathcal{V}$-presheaves.

Let $\mathcal{V}$ be a [[Bénabou cosmos]], $(\mathcal{C},P,I,\lambda,\rho,\alpha)$ be a small $\mathcal{V}$-[[enriched category]] equipped with the structure of a [[promonoidal category]], and write $P^A_{B,C}$ for $P(A,B,C)$ (this is called the Einstein notation for profunctors; see ([Loregian 2019, Definition 5.1.10](#Fosco))).

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

The **Day convolution tensor product** on $[\mathcal{C}^\mathsf{op},\mathcal{V}]$ is the bifunctor

$$\otimes_{Day}\colon[\mathcal{C}^\mathsf{op},\mathcal{V}]\times[\mathcal{C}^\mathsf{op},\mathcal{V}]\longrightarrow[\mathcal{C}^\mathsf{op},\mathcal{V}]$$

defined on objects by the [[coend]]

$$F\otimes_{Day}G=\overset{A,B\in\mathcal{C}}{\int}F(A)\otimes_V G(B)\otimes_V P^{(-)}_{A,B}.$$

=--

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

There is an [[equivalence of categories]] between the category of [[pro-monoidal structures]] on $\mathcal{C}$ with strong pro-monoidal functors between them and the category of biclosed monoidal structures on $[\mathcal{C}^{op},V]$ with [[strong monoidal functors]] between them.

=--

This is claimed without proof in ([Day 70](#Day70)).

### Via multicategories

The monoidal structure on $[C, D]$ for $C$ promonoidal and $D$ [[monoidally cocomplete category|monoidally cocomplete]] may be seen as the multicategory structure arising from viewing $C$ and $D$ as [[multicategories]], since $C$ is [[exponentiable object|exponentiable]] in this case. See [[multicategory]] and Proposition 2.12 of [Pisani 2014](#Pisani14) for more details.

## Properties
  {#Properties}

### Closed monoidal structure
 {#ClosedMonoidalStructure}

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

For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a [[small category|small]] [[monoidal category|monoidal]] $V$-[[enriched category]],
the Day convolution product $\otimes_{Day}$ of def. \ref{DayConvolutionProduct} makes

$$
  ( [\mathcal{C}, V], \otimes_{Day}, y(I))
$$

a [[monoidal category]] with [[tensor unit]] $y(I)$ co-represented by the tensor unit $I$ of $\mathcal{C}$.

=--

This may be deduced fairly abstractly from the [above](#InTermsOfProfunctors) description of Day convolution in terms of profunctors, using the associativity of the promonoidal structure on $\mathcal{C}$. It of course also holds when $\mathcal{C}$ is only [[promonoidal category|promonoidal]], as shown in [Day70](#Day70) and [Day's thesis](#Day70Thesis).

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

For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a [[small category|small]] [[monoidal category|monoidal]] $V$-[[enriched category]], the monoidal category with Day convolution $([\mathcal{C},V], \otimes_{Day}, y(I))$ from def. \ref{DayConvolutionYieldsMonoidalCategoryStructure} is a [[closed monoidal category]]. Its [[internal hom]] $[-,-]_{Day}$ is given by the [[end]]

$$
  [X,Y]_{Day}(c)
  \simeq
   \underset{c_1,c_2}{\int}
      V\left(
        \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2),
        V(X(c_1), Y(c_2))
      \right)
  \,.
$$

or equivalently by the end

$$ [X,Y]_{Day}(c) = \int_{c_1} V\left(X(c_1),Y(c\otimes c_1)\right) $$

=--

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

First note that the equivalence between the two formulas follows from the [[Yoneda lemma]].  (We mention them both, even though the second is undoubtedly simpler, because the more general case of a promonoidal $\mathcal{C}$ this simplification is unavailable.)

In analogy to the cartesian [[closed monoidal structure on presheaves]]
we see that if the [[internal hom]] in $[\mathcal{C},V]$ exists at all,
(with $[X,-]_{Day}$ [[right adjoint]] to $(-) \otimes_{Day} X$) then by the [[enriched Yoneda lemma]] and by the [[end]]-expression for the [[hom-objects]] in the [[enriched functor category]] $[\mathcal{C},V]$ it has to be given by

$$
  \begin{aligned}
    [X,Y]_{Day}(c)
      & \simeq
    [\mathcal{C},V](y(c), [X,Y])
    \\
    &
    \simeq
      [\mathcal{C},V](y(c) \otimes_{Day} X, Y)
    \\
    & \simeq
    \underset{c_1}{\int} V((y(c) \otimes_{Day} X)(c_1), Y(c_1))
    \\
    &\simeq
    \underset{c_1}{\int}
      V\left(
         \overset{d_2}{\int}
         \overset{d_1}{\int}
         \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1)
           \otimes_V
         \mathcal{C}(c,d_1)
         \otimes_V
         X(d_2)
         ,
         Y(c_1)
      \right)
    \\
    & \simeq
    \underset{c_1}{\int}
    \underset{d_2}{\int}
      V\left(
        \underset{\simeq \mathcal{C}(c \otimes_{\mathcal{C}} d_2, c_1 )}{
          \underbrace{
            \overset{d_1}{\int}
            \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1)
              \otimes_V
            \mathcal{C}(c,d_1)
           }
         }
         \otimes_V
         X(d_2)
         ,
         Y(c_1)
      \right)
      \\
      & \simeq
      \underset{c_1,d_2}{\int}
      V\left(
        \mathcal{C}(c \otimes_{\mathcal{C}} d_2,c_1),
        V(X(d_2), Y(c_1))
      \right)
      \\
      & =
      \underset{c_1,c_2}{\int}
      V\left(
        \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2),
        V(X(c_1), Y(c_2))
      \right)
  \end{aligned}
  \,.
$$

This exists, by the assumption that $\mathcal{C}$ is [[small category|small]] and that $V$ has all small limits. Now to check that this really gives a right adjoint:

$$
  \begin{aligned}
     [\mathcal{C},V]( X, [Y,Z]_{Day} )
       & \simeq
     \underset{c}{\int} V\left(
        X(c),
        \underset{c_1,c_2}{\int}
        V\left(
          \mathcal{C}(c \otimes_{\mathcal{C}} c_1 , c_2),
          V(Y(c_1), Z(c_2))
        \right)
     \right)
     \\
     &
     \simeq
     \underset{c}{\int}
     \underset{c_1,c_2}{\int}
     V\left(
       \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2)
         \otimes_V
       X(c)
         \otimes_V
       Y(c_1)
       ,\;
       Z(c_2)
     \right)
     \\
     & \simeq
     \underset{c_2}{\int}
     V\left(
       \overset{c,c_1}{\int}
       \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2)
         \otimes_V
       X(c)
         \otimes_V
       Y(c_1)
       ,\;
       Z(c_2)
     \right)
     \\
     &\simeq
     \underset{c_2}{\int}
       V\left(
         (X \otimes_{Day} Y)(c_2),
         Z(c_2)
       \right)
     \\
     &\simeq
     [\mathcal{C},V](X \otimes_{Day} Y, Z)
  \end{aligned}
  \,.
$$

=--

While most of this page discusses the covariant Day convolution, the contravariant one interacts conveniently with the Yoneda embedding.

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

The [[Yoneda embedding]] constitutes a  [[strong monoidal functor]] $(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C}^{\mathrm{op}},V], \otimes_{Day}, y(I))$.

=--

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

The proof that the [[tensor unit]] is respected is dual to the argument in prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure}. To see that the tensor product is respected, apply the [[co-Yoneda lemma]] twice to get the following natural isomorphism

$$
  \begin{aligned}
    (y(c_1) \otimes_{Day} y(c_2))(c)
    &
    \simeq
    \underset{d_1, d_2}{\int}
      \mathcal{C}( c ,d_1 \otimes_{\mathcal{C}} d_2)
    \otimes_V
      \mathcal{C}(d_1,c_1)
    \otimes_V
      \mathcal{C}(d_2,c_2)
    \\
    & \simeq \mathcal{C}( c ,c_1\otimes_{\mathcal{C}}c_2 )
    \\
    &
    = y(c_1 \otimes_{\mathcal{C}} c_2 )(c)
  \end{aligned}
  \,.
$$

=--

### Monoids with respect to Day convolution
 {#Monoids}

Given any [[monoidal category]] then one may consider [[monoid objects]] and [[module objects]] inside it.

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

For $(\mathcal{C}, \otimes)$ a [[small category|small]] ([[symmetric monoidal category|symmetric]]) [[monoidal category|monoidal]] $V$-[[enriched category]],
then ([[commutative monoid object|commutative]]) [[monoid objects]] in the Day convolution monoidal category $([\mathcal{C},V], \otimes_{Day}, y(I))$ of prop. \ref{DayConvolutionYieldsMonoidalCategoryStructure} are equivalent to ([[symmetric monoidal functor|symmetric]]) [[lax monoidal functors]] $\mathcal{C} \to V$:

$$
  Mon([\mathcal{C},V], \otimes_{Day}, y(I))
  \simeq
  MonFunc(\mathcal{C},V)
$$

$$
  CMon([\mathcal{C},V], \otimes_{Day}, y(I))
  \simeq
  SymMonFunc(\mathcal{C},V)
  \,.
$$

In functional programming, these monoids give rise to the notion of Applicative.

Moreover, [[module objects]] over these monoid objects are equivalent to the corresponding [[modules over monoidal functors]].

=--

This is stated in some form in ([Day 70, example 3.2.2](#Day70)). It was highlighted again in ([MMSS 00, prop. 22.1](#MMSS00)). See also MO discussion [here](http://mathoverflow.net/a/130619/381).

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

A [[lax monoidal functor]] $F \colon \mathcal{C} \to V$ is given by [[natural transformations]]

$$
  I_V \longrightarrow F(I_{\mathcal{C}})
$$

$$
  F(c_1) \otimes_V F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)
$$

satisfying compatibility conditions. Under the [[natural isomorphism]] of corollary \ref{DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor} these are identified with natural transformations

$$
  y(I) \to F
$$

$$
  F \otimes_{Day} F \longrightarrow F
$$

satisfying analogous conditions. This is just the structure of a [[monoid object]] on $F$ under $\otimes_{Day}$.

Similarly for [[module objects]] and [[modules over monoidal functors]].

=--

\begin{remark}\label{rmk:promonoidal functors as monoids} This page primarily addresses the Day convolution monoidal structure on $[\mathcal{C}^{op},\mathcal{V}]$ when $\mathcal{C}$ is monoidal or promonoidal and [[enriched]] over $\mathcal{V}$. This monoidal structure "convolves" the respective structures of $\mathcal{C}$ and $\mathcal{V}$. One might ask if one can convolve the monoidal structures of two promonoidal categories in general. In other words, is it possible to equip the functor category $[\mathcal{C},\mathcal{D}]$, for promonoidal categories $\mathcal{C}$ and $\mathcal{D}$, with a convolution monoidal structure? The literature does not seem to give a definite answer to this question in general. However, in the case that $\mathcal{D}$ is [[closed monoidal category|biclosed]] and [[cocomplete]] there is a monoidal convolution structure on $[\mathcal{C},\mathcal{D}]$ as shown in Proposition 5.4 of [Day74](#Day74). In this special situation, the [[monoid]] objects of $[\mathcal{C},\mathcal{D}]$ are precisely the promonoidal functors $\mathcal{C}\to\mathcal{D}$ (Remark 2.2 of [Day77](#Day77)), where $\mathcal{D}$ obtains a promonoidal structure from its monoidal structure by Theorem 4.1 of [Day70](#Day70).
\end{remark}

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

In the case that $V$ is [[pointed topological spaces]] or pointed [[simplicial sets]] equipped with the [[smash product]] of [[pointed objects]] and that $\mathcal{C}$ is a diagram category for [[spectra]], then monoids in prop. \ref{DayMonoidsAreLaxMonoidalFunctorsOnTheSite} are known as [[ring spectra]] and the [[lax monoidal functors]] in prop. \ref{DayMonoidsAreLaxMonoidalFunctorsOnTheSite} are known as the incarnation of ring spectra as "[[functors with smash product]]".

=--

([MMSS 00, section 22](#MMSS00)).

### Modules with respect to Day convolution
 {#Modules}

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

For $(\mathcal{C},\otimes, I)$ a [[small category|small]] [[monoidal category|monoidal]] $V$-[[enriched category]], and for $R \in Mon([\mathcal{C}, V],\otimes_{Day})$ a [[monoid object]] with respect to [[Day convolution]] over $\mathcal{C}$, write

$$
  R Free_{\mathcal{C}}Mod \hookrightarrow R Mod
$$

for the [[full subcategory]] of the [[category of modules]] over $R$ on those that are [[free modules]] and moreover free on objects in $\mathcal{C}$ (under the [[Yoneda embedding]]). Hence the [[objects]] of $R Free_{\mathcal{C}}Mod$ are those of $\mathcal{C}$ and the [[hom-objects]] are

$$
    R Free_{\mathcal{C}}Mod(c_1,c_2)
      \;\coloneqq\;
    R Mod( y(c_1) \otimes_{Day} R , y(c_2) \otimes_{Day} R)
  \,.
$$

=--

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

For $(\mathcal{C},\otimes, I)$ a [[small category|small]] $V$-[[enriched category]], and for $R \in Mon([\mathcal{C}, V],\otimes_{Day})$ a [[monoid object]] with respect to [[Day convolution]] over $\mathcal{C}$, then there is an [[equivalence of categories]]

$$
  Mod_R \simeq [R Free_{\mathcal{C}}Mod^{op}, V]
$$

between the [[category of modules|category of right modules]] over $R$ and the [[enriched functor category]] out of the [[opposite category]] of that of free $R$-modules from def. \ref{FreeModulesOverAMonoidInDayConvolution}.

=--

([MMSS 00, theorem 2.2](#MMSS00))

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

Use the identification from prop. \ref{DayMonoidsAreLaxMonoidalFunctorsOnTheSite} of $R$ with a [[lax monoidal functor]] and of any $R$-[[module object]] $N$ as a functor with the structure of a [[module over a monoidal functor]], given by [[natural transformations]]

$$
  N(c_1) \otimes R(c_2) \longrightarrow N(c_1 \otimes c_2)
  \,.
$$

Notice that these transformations have just the same structure as those of the [[enriched functor|enriched functoriality]] of $N$ of the form

$$
  \mathcal{C}(c_1,c_2) \otimes N(c_1) \longrightarrow N(c_2)
  \,.
$$

Hence we may unify these two kinds of transformations into a single kind of the form

$$
   \mathcal{C}(c_1 \otimes c_4, c_2)
        \otimes
      R(c_4)
      \otimes
    N(c_1)
      \longrightarrow
    \mathcal{C}(c_1 \otimes c_4, c_2)
      \otimes
    N(c_1 \otimes c_4)
      \longrightarrow
    N(c_2)
$$

and subject to certain identifications.

Now observe that the hom-objects of $R Free_{\mathcal{C}}Mod$ (def. \ref{FreeModulesOverAMonoidInDayConvolution}) have just this structure:

$$
  \begin{aligned}
    R Free_{\mathcal{C}}Mod(c_2,c_1)
      & =
    R Mod( y(c_2) \otimes_{Day} R , y(c_1) \otimes_{Day} R)
    \\
      & \simeq
    [\mathcal{C},V](y(c_2), y(c_1) \otimes_{Day} R)
    \\
      & \simeq
    (y(c_1) \otimes_{Day} R)(c_2)
    \\
      & \simeq
     \overset{c_3,c_4}{\int}
       \mathcal{C}(c_3 \otimes c_4,c_2)
         \otimes_V
       \mathcal{C}(c_1, c_3) \otimes_V R(c_4)
     \\
     & \simeq
     \overset{c_4}{\int}
       \mathcal{C}(c_1 \otimes c_4,c_2)
       \otimes_V R(c_4)
  \end{aligned}
  \,.
$$

We claim that under this identification, composition in $R Free_{\mathcal{C}}Mod$ is given by

$$
  \begin{aligned}
    R Free_{\mathcal{C}}Mod(c_2, c_1) \otimes_V R Free_{\mathcal{C}}Mod(c_3, c_2)
    & =
    \left(
    \overset{c_4}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2)
     \otimes_V R(c_4)
    \right)
      \otimes_V
    \left(
    \overset{c_5}{\int} \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3 )
     \otimes_V R(c_5)
    \right)
    \\
    & \simeq
    \overset{c_4, c_5}{\int}
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4  , c_2  )
        \otimes_V
      \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3)
        \otimes_V
      R(c_4) \otimes_V R(c_5)
    \\
    & \longrightarrow
      \overset{c_4,c_5}{\int}
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}}  c_5 , c_2 \otimes_{\mathcal{C}} c_5  )
        \otimes_V
      \mathcal{C}(c_2 \otimes c_5, c_3)
        \otimes_V
      R(c_4 \otimes_{\mathcal{C}} c_5 )
     \\
     & \longrightarrow
      \overset{c_4, c_5}{\int}
      \mathcal{C}(c_1\otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5 , c_3)
        \otimes_V
      R(c_4 \otimes_{\mathcal{C}} c_5 )
     \\
     & \longrightarrow
      \overset{c_4}{\int}
      \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 , c_3)
        \otimes_V
      R(c_4 )
  \end{aligned}
  \,,
$$

where

1. the first morphism is, in the integrand, the tensor product of

   1. forming the tensor product of hom-objects of $\mathcal{C}$ with the identity of $c_5$

      $$
        \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \simeq
       \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes_V 1_V
       \overset{}{\longrightarrow}
       \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes \mathcal{C}(c_5,c_5)
       \longrightarrow
       \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5, c_2 \otimes_{\mathcal{C}} c_5)
      $$

   1. the monoidal functor incarnation $R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 )$ of the monoid structure on $R$;

1. the second morphism is, in the integrand, given by composition in $\mathcal{C}$;

1. the last morphism is the morphism induced on [[coends]] by regarding [[extranatural transformation|extranaturality]] in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).

It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in $\mathcal{C}$ together with the action property over $R$.

=--

## Examples

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

Let $C$ be a [[discrete category]] over a set, which is hence a [[monoid]] (for instance a [[group]]) with product $\cdot$.

Then the Day convolution product is

$$
  F \star G : e \mapsto \oplus_{c \cdot d = e} F(c) \times G(d)
  \,.
$$

Notice that if we regard the presheaves $F$ and $G$ here, assuming they take values in finite sets, as [[vertical categorification|categorifications]] of $\mathbb{N}$-valued functions $|F|, |G| : C \to \mathbb{N}$, where $|\cdot| : Set \to \mathbb{N}$ is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these $\mathbb{N}$-valued functions

$$
  \begin{aligned}
    |F \star G| : e &\mapsto
       \sum_{c,d \in C}
        |F(c)| \times |G(d)| \times \delta(e, c \otimes d)
       & =
       \sum_{c \cdot d = e} |F(c)| \cdot |F(d)|
  \end{aligned}
$$

This uses in particular that for every object $c \in C$ the functor

$$
  Hom_C(c,-) = \delta_c
$$

is in this sense the Kronecker delta-function on the set $C$ supported at $c \in C$. Precisely because by assumption $C$ has only identity morphisms.

$$
 Hom_C(c,d) = \left\{ \array{
   * & if c = d
   \\
   \emptyset & if c \neq d
 } \right.
$$

=--

Further examples:

* There is an obvious monoidal structure on the [[cube category]]. By Day convolution this induces a monoidal structure on [[cubical set|cubical sets]]. This in turn induces a monoidal structure on [[strict omega-category|strict omega-categories]].

* There is a monoidal structure on the [[augmented simplex category]] which by Day convolution induces a monoidal structure on the category of [[augmented simplicial sets]], which by restriction induces the [[join of simplicial sets|join operation]] on [[simplicial sets]].

* If $C$ is a [[large category]] in one [[universe]], then its [[universe enlargement]] to a bigger universe can be given a closed monoidal structure via Day convolution.

* The [[semantics]] of [[linear logic]] obtained from Girard's "phase spaces", or more generally from [[ternary frames]], is essentially Day convolution for [[posets]].

+-- {: .num_example #SymmetricSmashProductOfSpectra}
###### Example

The [[symmetric smash product of spectra]] on, in particular,  [[symmetric spectra]] and  [[orthogonal spectra]] is the Day convolution product for [[Top]]-[[enriched functors]] on monoidal categories of [[symmetric groups]] of [[orthogonal groups]], respectively ([MMSS 00, theorem 1.7 and section 21.](#MMSS00)).

Similarly the [[symmetric smash product of spectra]] on the [[model structure for excisive functors]] is Day convolution for [[sSet]]-[[enriched functors]] on the plain [[smash product]] of finite pointed [[simplicial sets]] ([Lydakis 98](#Lydakis98)).

See also at _[[functor with smash products]]_.

=--

## Related concepts

* [[monoidal topos]]

## References
 {#References}

The concept, and many of its basic properties, originates in several works of Brian day, including:

* {#Day70Thesis} [[Brian Day]], _Construction of Biclosed Categories_, PhD thesis. School of Mathematics of the University of New South Wales, September 1970. [Link](http://unsworks.unsw.edu.au/fapi/datastream/unsworks:58748/SOURCE01?view=true).

* {#Day70} [[Brian Day]], _On closed categories of functors_, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/DayReport.pdf)),

* {#Day74} [[Brian Day]], _On closed categories of functors II_, Category Seminar, Sydney 1972/73, Springer Lecture Notes, Vol. 420, 20-53.

* {#Day77} [[Brian Day]], _Promonoidal functor categories_, J. Austral. Math. Soc. 23 (Series A) (1977), 312-328.

(Note that some unproven statements in [(Day's report)](#Day70) are proved in [(Day's thesis)](#Day70Thesis) and vice versa.)

The universal property of the Day convolution, in the sense of free monoidal cocompletion, is discussed in

* Geun  Bin Im and G. M. Kelly, _A universal property of the convolution monoidal structure_, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.

General discussion includes

* [[Todd Trimble]] on Day convolution [here](http://golem.ph.utexas.edu/category/2008/01/the_concept_of_a_space_of_stat.html#c014365)

The application of Day convolution to the construction of [[symmetric smash products of spectra]] for [[highly structured spectra]] is due to

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [publisher](http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc))

and for [[excisive functors]] due to

* {#Lydakis98} Lydakis, _Simplicial functors and stable homotopy theory_ Preprint, available via Hopf archive, 1998 ([pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf))

(see also at [[functors with smash product]]).

Day convolution for [[monoidal bicategories]] is developed in

* {#Corner2016} [[Alexander Corner]], _Day convolution for monoidal bicategories_, School of Mathematics and Statistics of the University of Sheffield. Available through the [White Rose theses database](http://etheses.whiterose.ac.uk/16767/).

Day convolution for [[(∞,1)-categories]] is discussed in

* Saul Glasman, _Day convolution for infinity-categories_ ([arXiv:1308.4940](http://arxiv.org/abs/1308.4940))

Other references cited in this page:

* {#Fosco} [[Fosco Loregian]], _Coend calculus_ [[arXiv](http://arxiv.org/abs/1501.02503)]

On [[(infinity,1)-colimits|$\infty$-colimits]] and [[Day convolution]] in the context of [[enriched (infinity,1)-categories|enriched $\infty$-categories]]:

* [[Vladimir Hinich]], *Colimits in enriched ∞-categories and Day convolution*, Theory and Applications of Categories **39** 12 (2023) 365-422 [[tac:39-12](http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html), [arXiv:2101.09538](https://arxiv.org/abs/2101.09538)]

Various generalisations are discussed in:

* [[Brian Day]] and [[Ross Street]], _Lax monoids, pseudo-operads, and convolution_, Contemporary Mathematics 318 (2003): 75-96.

The link with multicategories is observed in:

* {#Pisani14} [[Claudio Pisani]], _Sequential multicategories_, Theory and Applications of Categories 29.19 (2014), [arXiv:1402.0253](https://arxiv.org/abs/1402.0253)

[[!redirects Day convolutions]]

[[!redirects day convolution]]

[[!redirects Day tensor product]]
[[!redirects Day tensor products]]

[[!redirects Day convolution product]]
[[!redirects Day convolution products]]


day convolution > history

< [[day convolution]]
[[!redirects day convolution -- history]]


Day tensor product

The _Day tensor product_ is the [[monoidal category|monoidal product]] on [[presheaf]] [[category|categories]] induced from [[Day convolution]].


Day's reflection theorem

\tableofcontents

## Idea

Day's reflection theorem [[Day (1972)](#Day72)] gives conditions under which a [[reflective subcategory]] of a [[symmetric monoidal closed category]] is closed under [[internal homs]].

One memorable consequence of the theorem is that the subcategory is closed under [[internal homs]] if the [[reflector]] is symmetric [[strong monoidal functor|strong monoidal]].

## Statement and Proof

\begin{theorem} (Day)
Let $R: C \to D$ be a [[fully faithful functor]] with [[left adjoint]] $L \colon D \to C$, and suppose given a symmetric monoidal closed structure on $D$ with tensor $\otimes$ and internal hom $[-, -]$. Then for objects $c$ of $C$ and $d, d'$ of $D$, the following are equivalent:

1. $u[d, R c] \colon [d, R c] \to R L[d, R c]$ is an isomorphism;

1. $[u, 1] \colon [R L d, R c] \to [d, R c]$ is an isomorphism;

1. $L(u \otimes 1) \colon L(d \otimes d') \to L(R L d \otimes d')$ is an isomorphism;

1. $L(u \otimes u) \colon L(d \otimes d') \to L(R L d \otimes R L d')$ is an isomorphism.

\end{theorem}

Some remarks before the proof:

* (a) Since $R$ is fully faithful, the [[monad]] $R L$ is [[idempotent monad|idempotent]]. Consequently, $R: C \to D$ is [[monadic functor|monadic]], and algebra structures on an object $d$ are unique when they exist. This makes algebra structure on $d$ a [[stuff, structure, property
|property]], that $d \cong R c$ for some $c$ in $C$, unique up to isomorphism. Call this property "being in $C$". Thus condition $\text{1.}$ says that $[d, d']$ is in $C$ for a particular $d' \cong R c$ in $C$. (If $D$ is [[cartesian closed category|cartesian closed]] and $[d, d'] \in C$ whenever $d' \in D$, we say $C$ is an [[exponential ideal]] in $D$.) Notice that all $D$-maps between objects in $C$ are algebra maps, by full faithfulness of $R$.

* (b) If $d$ is in $C$, then for any $d'$ the unit $u\colon d' \to R L d'$ induces an isomorphism $D(R L d', d) \cong D(d', d)$, since every $f \in D(R L d', d)$ is an algebra map $R L d' \to d$ and $R L d'$ is the [[free object|free algebra]] on $d'$.

\begin{proof}
We prove $\text{3.} \Rightarrow \text{4.} \Rightarrow \text{1.}\; \Rightarrow\; \text{3.}$, and then $\text{2.} \Leftrightarrow \text{3.}$. Some parts are merely sketched; refer to Day for full details.

$\text{3.}\; \Rightarrow\; \text{4.}$ is obvious from the symmetric structure and

\begin{center}
\begin{tikzcd}
L(d \otimes d') \ar[rr, "L(u \otimes u)"] \ar[dr, swap, "L(u \otimes 1)"]
& & L(R L d \otimes R L d') \\
& L(R L d \otimes d') \ar[ur, swap, "L(1 \otimes u)"] &
\end{tikzcd}
\end{center}

$\text{4.} \Rightarrow \text{1.}$ is proven by giving an algebra structure $R L[d, R c] \to [d, R c]$, obtained by [[currying]] the following composite:

\begin{tikzcd}
R L[d, R c] \otimes d
\ar[r, "1 \otimes u"]
& R L[d, R c] \otimes R L d
\ar[r, "u"]
& R L(R L[d, R c] \otimes R L d)
\ar[r, "R (L(u \otimes u)^{-1})"]
& R L([d, R c] \otimes d)
\ar[r, "R L eval"]
& R L R c
\ar[r, "\cong"]
& R c
\end{tikzcd}

(by idempotence of $R L$, to prove this map is an algebra structure, it suffices to show it is left inverse to the unit).

$\text{1.} \Rightarrow \text{3.}$ is proven by considering the diagram

\begin{tikzcd}
C(L(R L d \otimes d'), c)
\ar[r, "{C(L(u \otimes 1), 1)}"]
\ar[d, "\cong"]
& C(L(d \otimes d'), c)
\ar[d, "\cong"]
\\
D(R L d \otimes d', R c) \ar[r, "{D(u \otimes 1, 1)}"] \ar[d, "\cong"] & D(d \otimes d', R c) \ar[d, "\cong"] \\
D(R L d, [d', R c]) \ar[r, "{D(u, 1)}"] \ar[d, swap, "{D(1, u)}"] & D(d, [d', R c]) \ar[d, "{D(1, u)}"] \\
D(R L d, R L[d', R c]) \ar[r, "{D(u, 1)}"] & D(d, R L[d', R c])
\end{tikzcd}

which commutes by functoriality and naturality, and where the maps labeled $D(1, u)$ are isomorphisms by assuming $\text{1.}$, and the bottom map labeled $D(u, 1)$ is an isomorphism by invoking remark (b). It follows that the top horizontal map is also an isomorphism. Since this holds for all objects $c$ of $C$, the map $L(u \otimes 1)$ is an isomorphism by the Yoneda lemma, so that $\text{4.}$ holds.

Finally, $\text{2.} \Leftrightarrow \text{3.}$ is proven by considering the diagram

\begin{tikzcd}
C(L(R L d \otimes d'), c)
\ar[r, "{C(L(u \otimes 1), 1)}"]
\ar[d, "\cong"]
& C(L(d \otimes d'), c)
\ar[d, "\cong"]
\\
D(R L d \otimes d', R c) \ar[r, "{D(u \otimes 1, 1)}"] \ar[d, "\cong"] & D(d \otimes d', R c) \ar[d, "\cong"] \\
D(d', [R L d, R c]) \ar[r, "{D(1, [u, 1])}"] & D(d', [d, R c])
\end{tikzcd}

and again applying a Yoneda lemma argument. \end{proof}

## Consequences

\begin{corollary}
If $L$ is a strong [[symmetric monoidal functor]], then for any $c$ in $C$ and $d$ in $D$, the internal hom $[d, c]$ is in $C$.
\end{corollary}

\begin{proof}
Full faithfulness of $R$ is equivalent to the [[counit]] $\varepsilon \colon L R \to 1_C$ being an isomorphism. By a [[triangle identity]], this forces $L u$ to be an isomorphism for any unit $u: d \to R L d$. This in turn forces the arrow $L(u \otimes u)$ in the diagram

\begin{tikzcd}
L d \otimes L d' \ar[r, "L u \otimes L u"] \ar[d] & L R L d \otimes L R L d' \ar[d] \\
L(d \otimes d') \ar[r, "L(u \otimes u)"] & L(R L d \otimes R L d')
\end{tikzcd}
to be an isomorphism, where the vertical arrows are invertible symmetry constraints and the maps $L u$ are isomorphisms. Now apply the equivalence between $\text{4.}$ and $\text{1.}$ from the reflection theorem.
\end{proof}

\begin{corollary}
If $D$ is cartesian closed, and the reflector $L \colon D \to C$ preserves products, then $C$ is cartesian closed.
\end{corollary}

\begin{proof}
We have already seen under the hypotheses that if $c, c'$ are in $C$, then the exponential $c^{c'}$ as calculated in $D$ is in $C$. Furthermore, $C$ inherits products from $D$, because $R \colon C \to D$ is monadic and monadic functors reflect limits. The adjunction $C(c'' \times c', c) \cong C(c'', c^{c'})$ holds because it holds when interpreted in $D$ and $C$ is fully embedded in $D$.
\end{proof}

## Reference

* {#Day72} [[Brian Day]], *A reflection theorem for closed categories*, Journal of Pure and Applied Algebra **2** 1 (1972), 1-11. [doi:10.1016/0022-4049(72)90021-7]


DBI-action higher curvature corrections -- references

### Higher curvature corrections to the DBI-action for D-branes
 {#ReferencesDBIHigherCurvatureCorrections}

On [[higher curvature corrections]] to the (abelian) [[DBI-action]] for (single) [[D-branes]]:

* {#AndreevTseytlin88} Oleg Andreev, [[Arkady Tseytlin]], _Partition-function representation for the open superstring effective action:: Cancellation of Möbius infinites and derivative corrections to Born-Infeld lagrangian_, Nuclear Physics B Volume 311, Issue 1, 19 December 1988, Pages 205-252 (doi:10.1016/0550-3213(88)90148-4)

* {#BachasBainGreen99} [[Constantin Bachas]], P. Bain, [[Michael Green]], _Curvature terms in D-brane actions and their M-theory origin_, JHEP 9905:011, 1999 ([arXiv:hep-th/9903210](https://arxiv.org/abs/hep-th/9903210))

* Niclas Wyllard, _Derivative corrections to D-brane actions with constant background fields_, Nucl. Phys. B598 (2001) 247-275 ([arXiv:hep-th/0008125](https://arxiv.org/abs/hep-th/0008125))

* Oleg Andreev, _More About Partition Function of Open Bosonic String in Background Fields and String Theory Effective Action_, Phys. Lett. B513:207-212, 2001 ([arXiv:hep-th/0104061](https://arxiv.org/abs/hep-th/0104061))

* Niclas Wyllard, _Derivative corrections to the D-brane Born-Infeld action: non-geodesic embeddings and the Seiberg-Witten map_, JHEP 0108 (2001) 027 ([arXiv:hep-th/0107185](https://arxiv.org/abs/hep-th/0107185))

* [[Mohammad Garousi]], _T-duality of curvature terms in D-brane actions_, JHEP 1002:002, 2010 ([arXiv:0911.0255](https://arxiv.org/abs/0911.0255))

* [[Mohammad Garousi]], _S-duality of D-brane action at order $O(\alpha'{}^2)$_, Phys. Lett. B701:465-470, 2011 ([arXiv:1103.3121](https://arxiv.org/abs/1103.3121))

* Ali Jalali, [[Mohammad Garousi]], _On D-brane action at order $\alpha'{}^2$_, Phys. Rev. D 92, 106004 (2015) ([arXiv:1506.02130](https://arxiv.org/abs/1506.02130))

* {#Garousi15} [[Mohammad Garousi]], _An off-shell D-brane action at order $\alpha'{}^2$ in flat spacetime_, Phys. Rev. D 93, 066014 (2016) ([arXiv:1511.01676](https://arxiv.org/abs/1511.01676))

* Komeil Babaei Velni, Ali Jalali, _Higher derivative corrections to DBI action at $\alpha'{}^2$ order_, Phys. Rev. D 95, 086010 (2017) ([arXiv:1612.05898](https://arxiv.org/abs/1612.05898))


dcpo

> This entry is about domains in [[domain theory]]. For other uses, see at _[[domain]]_.

***

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#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
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=--

# DCPOs
* table of contents
{: toc}

## Idea

A *directed-complete partial order* (DCPO), is a [[poset]] with all [[directed joins]].  Often a DCPO is required to have a [[bottom element]] $\bot$; then it is called a *[[pointed object|pointed]] DCPO* or a *CPO* (but this term is ambiguous).

The [[morphisms]] between DCPOs preserve the directed joins; equivalently, they are [[Scott continuity|Scott-continuous]].  Morphisms between pointed DCPOs may or may not be required to preserve $\bot$, depending on the application.

In [[domain theory]], a DCPO $P$ is interpreted as a [[type]] (in the sense of [[programming languages]], cf. *[[data type]]*), and its [[elements]] are possible partial (in the sense of a [[partial function]]) results of a [[computation]].  The [[bottom element]] (if there is one) indicates that no result has been obtained; if $x \leq y$ in $P$, then $x$ consists of part of the information in $y$.  A [[directed set|directed subset]] $D$ of $P$ indicates a collection of partial results which are mutually consistent, since for any two results $x, y \in D$, there is a partial result that subsumes them both.  The required join of $D$ is then a partial result encoding the same information as $D$ itself.

## Definitions

Recall that a [[poset]] $P$ consists of a collection of [[elements]] equipped with a [[binary relation]] $\leq$ such that $x \leq x$ always and $x \leq z$ whenever $x \leq y$ and $y \leq z$; we also consider $x$ and $y$ to be equal whenever $x \leq y$ and $y \leq x$.

Recall that a [[subset]] $D$ of $P$ is [[directed set|semidirected]] iff, whenever $x, y \in D$, there is some $z \in D$ such that $x \leq z$ and $y \leq z$.  Then $D$ is directed iff it is semidirected and [[inhabited subset|inhabited]].

Recall that an [[upper bound]] of a subset $D$ of $P$ is an element $x$ such that $y \leq x$ whenever $y \in D$, and a [[join]] of $D$ is an upper bound $x$ such that $x \leq y$ whenever $y$ is an upper bound of $D$.  The join of $D$, if one exists, is unique, and we write it $\bigvee D$ (or even put a little arrow on the right flank of the symbol when $D$ is directed).  A [[bottom element]] is a join of the [[empty subset]].

\begin{definition}
\label{TheDefinition}
A **directed-complete partial order** (*DCPO*) or _predomain_ is a [[poset]] in which every [[directed set|directed]] [[subset]] has a [[join]].

A _pointed DCPO_ or **complete partial order** (*CPO*) (or _inductive partial order_ (*IPO*) or _semidirected-complete partial order_ or _domain_ or *lift algebra*) is a DCPO with a [[bottom element]], equivalently a poset in which every semidirected subset has a join.
\end{definition}

## Properties

### The Scott topology

Every poset $P$ becomes a [[topological space]] under the [[Scott topology]], but this is particularly nice for DCPOs, so we review it.

A [[subset]] $A$ of a DCPO is Scott-[[open subset|open]] iff it's an [[upper subset]] and any directed subset of $P$ whose join belongs to $A$ must meet $A$; it\'s Scott-[[closed subset|closed]] iff it is a [[lower subset]] that is directed-complete in its own right.

The [[specialisation order]] of the Scott topology of a DCPO is its original order (but the Scott topology is finer than the [[specialisation topology]]).

## Related entries

* [[omega-complete poset]]

* [[domain theory]]

* [[continuous lattice]]

## References ##

See the references at *[[domain theory]]*.

See also:

* Wikipedia, *[Scott domain](https://en.wikipedia.org/wiki/Scott_domain)*

* Wikipeida, *[Domain theory](https://en.wikipedia.org/wiki/Domain_theory)*

* D.J. Lehmann, M.B., Smyth, *Algebraic specification of data types: A synthetic approach*,  Math. Systems Theory **14** (1981) 97–139 [[doi:10.1007/BF01752392](https://doi.org/10.1007/BF01752392)]

Discussion in [[homotopy type theory]]/[[univalent foundations]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *The Univalence Principle* ([abs:2102.06275](https://arxiv.org/abs/2102.06275))

Generalization to the context of [[quantum computation]] (via [[quantum sets]] carrying [[quantum relations]] providing [[categorical semantics]] for [[quantum programming languages]] like [[Quipper]] equipped with type recursion):

* [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

[[!redirects dcpos]]
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[[!redirects directed-complete partial order]]
[[!redirects directed-complete partial orders]]
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[[!redirects directed complete partial orders]]

[[!redirects directed-complete poset]]
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[[!redirects directed complete poset]]
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[[!redirects pointed dcpo]]
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[[!redirects pointed directed complete partial order]]
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[[!redirects pointed directed-complete poset]]
[[!redirects pointed directed-complete posets]]
[[!redirects pointed directed complete poset]]
[[!redirects pointed directed complete posetss]]

[[!redirects cpo]]
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[[!redirects complete partial orders]]

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dd^c-lemma

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

## References

* [[Pierre Deligne]], [[Phillip Griffiths]], J. Morgan, [[Dennis Sullivan]], _Real homotopy theory of Kaehler manifolds_,. Invent. Math., 29:245–274, 1975.

* {#Cavalcanti03} [[Gil Cavalcanti]], _New aspects of the $d d^c$-lemma_ ([arXiv:math/0501406](https://arxiv.org/abs/math/0501406))

[[!redirects ddc-lemma]]


de Broglie wavelength

## References

* Wikipedia, _[Matter wave](https://en.wikipedia.org/wiki/Matter_wave)_

[[!redirects de Broglie wavelengths]]


de Bruijn indices

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

# Context
* table of contents
{: toc}

## Idea

De Bruijn indices are used to represent [[terms]] in [[lambda calculus]] and [[type theory]] without naming [[bound variables]].

Usually in informal mathematical language, an expression involving a lambda is written out as $\lambda x.\phi(x)$. However, this involves the use of [[bound variables]], which are hard to deal with when formalizing [[substitution]] of variables; in addition, the statements $\lambda x.\phi(x)$ and $\lambda y.\phi(y)$ refer to the same thing. De Bruijn indices involve the use [[natural numbers]] instead of bound variables; both $\lambda x.\phi(x)$ and $\lambda y.\phi(y)$ would be written as $\lambda \phi(\underline{0})$ when using De Bruijn indices, and $\lambda x.\lambda y.\phi(x, y)$ would be written as $\lambda \lambda \phi(\underline{1}, \underline{0})$. A statement like $x \lambda x.\phi(x)$ where $x$ is both a bound variable and a free variable would be written as $\underline{1} \lambda \phi(\underline{0})$ using de Bruijn indices.

## See also

* [[explicit substitution]]

## References

de Bruijn indices are discussed in section 9.1 of:

* [[Théo Winterhalter]], *Formalisation and Meta-Theory of Type Theory*, Nantes (2020) [[pdf](https://github.com/TheoWinterhalter/phd-thesis/releases/download/v1.2.1/TheoWinterhalter-PhD-v1.2.1.pdf), [github](https://github.com/TheoWinterhalter/phd-thesis)]

The notion is due to:

* [[Nicolaas de Bruijn]], *Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem*, Indagationes Mathematicae (Proceedings) **75** 5 (1972) 381-392 [doi:10.1016/1385-7258(72)90034-0, [pdf](http://alexandria.tue.nl/repository/freearticles/597619.pdf)]

See also:

* Wikipedia, [De Bruijn index](https://en.wikipedia.org/wiki/De_Bruijn_index)

[[!redirects de Bruijn index]]


de Donder-Weyl field theory

See *[[De Donder-Weyl-Hamilton equation]]*.


De Donder-Weyl-Hamilton equation

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###Context###
#### Variational calculus
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[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The  _De Donder-Weyl-Hamilton equation_ is a refinement of [[Hamilton's equation]] from a single [[variational derivative|variational]] direction to many.

Where [[Hamilton's equation]] appears as a natural condition in [[symplectic geometry]], so the De Donder-Weyl-Hamilton equation appears as the analogous condition in [[multisymplectic geometry]]/[[n-plectic geometry]].

Where [[Hamilton's equation]] appears as the [[equations of motion]] of a [[mechanical system]] whose [[dynamics]] is described by evolution along a single parameter ([[time]]), so the _De Donder-Weyl Hamilton equation_ appears as the [[equations of motion]] of a higher dimensional [[field theory]] given by a [[local Lagrangian]] where "evolution" is along more parameters ([[spacetime]]).

## Definition

Let $\Sigma$ be a [[smooth manifold]] to be regarded as [[worldvolume]]/[[spacetime]] and let $F \to \Sigma$ be a smooth [[bundle]] to be thought of as the [[field bundle]].

For simplicity of exposition we first consider local patches and take without essential restriction $\Sigma$ to be a [[Cartesian space]] and the [[field bundle]] to be a trivial [[vector bundle]] $X \times \Sigma$. In all of the following the [[summation convention]] for summation over repeated induces is understood.

We then denote the canonical [[coordinates]] on $\Sigma$ by $\{\sigma^\mu\}$ and those on $X$ by $\{\phi^i\}$.

Write $(J^1 E^)^\ast$ for the dualized first [[jet bundle]] of $E$. Under the above assumptions this has canonical local coordinates $\{\sigma^\mu, \phi^i, \pi_i^\mu\}$. Here we call $\pi^\mu_i$ the "$\mu$-th [[momentum]] of the $i$-th field".

Consider a [[local Lagrangian]] $L$ which is _of first order_, hence which is a function on the first [[jet bundle]] $J^1 E$ of the [[field bundle]].

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

Its [[Euler-Lagrange equations]] are

$$
  \partial_\mu
  \frac{\delta L }{\delta (\partial_\mu \phi^i)}
   =
  \frac{\delta L}{ \delta \phi^i}
  \,.
$$

=--

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

The _De Donder-Weyl-Hamiltonian_ of $L$ is its generalized [[Legendre transform]] (if it exists), the function

$$
  H \coloneqq \pi^\mu_i \partial_\mu \phi^i - L
$$

on the dual jet bundle.

=--

+-- {: .num_prop #DWHEquations}
###### Proposition/Definition

In terms of the De Donder-Weyl Hamiltonian $H$, def. \ref{dWHamiltonianByLegendre},  the [[Euler-Lagrange equations of motion]], def. \ref{ELEquations}, are equivalent to the [[differential equations]]

$$
  \partial_\mu \phi^i
    =
   \frac{\partial H}{\partial \pi_i^\mu}
  \;\;\,,
  \;\;\;\;
  \partial_\mu \pi^\mu_i
   =
  -
  \frac{\partial H}{ \partial \phi^i}
  \,.
$$

These are called the **De Donder-Weyl-Hamilton equations**.

=--

## Properties

### As a multisymplectic/$n$-plectic Hamiltonian equation
 {#AsnplecticHamiltonEquation}

For $(X,\omega)$ a [[symplectic manifold]] and $H \in C^\infty(X)$ a [[smooth function]] regarded as a [[Hamiltonian]], then [[Hamilton's equations]] are equivalent to

$$
  \iota_v \omega = \mathbf{d}H
$$

for $v$ a [[tangent vector]] to a [[trajectory]] in [[phase space]]. This may be referred to as the "non-relativistic" form of the symplectic version of Hamilton's equations, as the "[[time]]"-parameter is not part of the [[phase space]].

By passing from plain phase space to the corresponding [[multisymplectic geometry|dual jet bundle]], hence by adjoining a [[worldline]] coordinate $t$, this is equivalent to

$$
  \iota_v \Omega = 0
$$

where now

$$
  \Omega = \omega + \mathbf{d}H \wedge \mathbf{d}t
  \,.
$$

This may accordingly be thought of as the [[relativity|relativistic]] version of Hamilton's equations.

We now discuss the analog of both the "non-relativistic" and of this "[[relativity|relativistic]] version" of [[Hamilton's equation]] for the de Donder-Weyl-Hamilton equation.

In both cases, the $n$-tuple of [[tangent vectors]] to a [[section]] which satisfies the [[equations of motion]] is characterized as a _[[Hamiltonian n-vector field]]_. See there for more discussion.

#### Relativistic form

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

Given a De Donder-Weyl-Hamiltonian $H$, def. \ref{dWHamiltonianByLegendre}, define a [[differential form]]

$$
  \Omega \in \Omega^{n+1}_{cl}((J^1 E)^\ast)
$$

on the dual jet bundle (the _[[multisymplectic form]]_ or _[[n-plectic form|pre-(n+1)-plectic form]]_) by

$$
  \Omega
   \;\coloneqq \;
    \mathbf{d} \phi^i \wedge \mathbf{d} \pi^\mu_i
   \wedge (\iota_{\partial_\mu} vol_\Sigma)
    +
    \mathbf{d} H \wedge vol_\Sigma
  \,.
$$

=--

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

A [[section]] $(\phi^i, \pi^\mu_i) \colon \Sigma \to J^1 E^\ast$ of the dualized first [[jet bundle]]  satisfies the [[equations of motion]], prop. \ref{DWHEquations}, precisely if its [[tangent vectors]]

$$
  v_\mu
    \coloneqq \partial_{\mu}
  +
    \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i}
  +
  \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i}
$$

jointly annihilate the pre-$(n+1)$-plectic form of def. \ref{PrePlecticForm} in that the [[equation]]:

$$
  (\iota_{v_1} \cdots \iota_{v_n}) \Omega
  = 0
$$

holds.

=--

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

First, the component of this equation which does not contain any $\mathbf{d}\sigma^\mu$ is

$$
  \iota_{v_i}
  \;
  \mathbf{d}\phi^i \wedge \mathbf{d}\pi^i_\mu
  =
  \mathbf{d}H
  \,.
$$

This is already equivalent to the DWH equation, prop. \ref{DWHEquations}.

Second, this already implies that the components of the equation that are proportional to $\mathbf{d}\sigma^\mu$ are automatically satisfied; because these components are

$$
  \frac{\partial \phi^i}{\partial \sigma^\mu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\nu}
  \mathbf{d}\sigma^\nu
  -
  \frac{\partial \phi^i}{\partial \sigma^\nu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\mu}
  \mathbf{d}\sigma^\nu
  =
  \frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  +
  \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
$$

and inserting the DWH equations on the left makes the left side identically equal to the right hand side.

=--

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

Proposition \ref{DWHInRelativisticPlecticForm} is indeed true in the general case where $H$ may be [[spacetime]] dependent (depend nontrivially on the $\sigma^\mu$).

=--

#### Non-relativistic form

To obtain the "non-relativistic" form of the $(n+1)$-plectic
form of the DWH equation, consider the _affine_ dual first jet bundle with canonical coordinates $\{\sigma^a , \phi^i, \pi^a_i, e \}$.

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

The canonical [[n-plectic form|pre-(n+1)-plectic form]] on
the affine dual first jet bundle is

$$
  \omega
   \coloneqq
  \mathbf{d}\phi^i \wedge \mathbf{d}\phi^\mu_a
  \iota_{\partial_{\sigma^\mu}} vol_\Sigma
  +
  \mathbf{d}e \wedge vol_\Sigma
  \,.
$$

=--

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

Vector fields tangent to a [[section]] of the affine dual first jet bundle are of the form

$$
  v_\mu
  \coloneqq
    \partial_{\mu}
  +
    \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i}
  +
    \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i}
  +
    \frac{\partial e}{\partial \sigma^\mu} \partial_e
  \,.
$$

=--

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

On the affine dual first jet bundle the de Donder-Weyl-Hamilton equation characterizes those [[sections]] whose [[tangent vectors]] as above satisfy

$$
  (\iota_{v_1} \cdots \iota_{v_n})
  \omega
  =
  \mathbf{d}(H + e)
$$

=--

This has been pointed out in ([Hélein 02, around equation (4)](#Helein02)).

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

This is a slight variant of the proof of prop. \ref{DWHInRelativisticPlecticForm}.
First, the component of the equation independend of $\mathbf{d}\sigma^\nu$ is

$$
  \iota_{v_i}
  \;
  \mathbf{d}\pi^i_\mu \wedge \mathbf{d}\phi^i
  +
  \mathbf{d}e
  =
  \mathbf{d}H
  +
  \mathbf{d}e
  \,.
$$

Here the term $\mathbf{d}e$ cancels on both sides and leaves the equation equivalent to the DWH equation as in the first step of the proof of prop. \ref{DWHInRelativisticPlecticForm}.

Second, the component of the claimed equation proportional to $\mathbf{d}\sigma^\mu$ is now

$$
  \frac{\partial \phi^i}{\partial \sigma^\mu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\nu}
  \mathbf{d}\sigma^\nu
  -
  \frac{\partial \phi^i}{\partial \sigma^\nu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\mu}
  \mathbf{d}\sigma^\nu
  +
  \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  = 0
  \,.
$$

Inserting into this the DWH equation makes it equivalent to

$$
  \frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  +
  \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  +
  \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  = 0
$$

This has a unique solution, up to a global constant, given by

$$
  e(\{ \sigma^\nu \})
   =
  - H(
    \{
      \phi^i(\{ \sigma^\nu \}),
      \pi^\mu_i (\{ \sigma^\nu \})
    \}
    )
  \,.
$$

=--

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

By the proof of prop. \ref{DWHInNonRelativisticPlecticForm} we have _on shell_ that $H + e = 0$ and that $\omega = \Omega$.

=--

## Related concepts

* [[Hamiltonian n-vector field]]

## References

Maybe the first example of what is now called De Donder-Weyl theory appeared in

* [[Constantin Carathéodory]], _Über die Extremalen und geodätischen Felder in der Variationsrechnung der mehrfachen Integrale_, Acta Sci. Math. (Szeged) 4 (1929) 193-216

Then Weyl and de Donder independently published

* [[Hermann Weyl]], _Geodesic fields in the calculus of variations_, Ann. Math. (2) 36 (1935) 607-629.

* [[Théophile De Donder]], _Théorie invariante du calcul des variations_, Nuov. éd, Gauthiers–Villars, Paris 1935

Reviews:

* Wikipedia, _[De Donder-Weyl theory](http://en.wikipedia.org/wiki/De_Donder%E2%80%93Weyl_theory)_

* [[Narciso Román-Roy]], _Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories_, SIGMA 5 (2009), 100 ([journal](http://www.emis.de/journals/SIGMA/2009/100/), [arXiv:math-ph/0506022](http://arxiv.org/abs/math-ph/0506022))

* [[Frédéric Hélein]], _Hamiltonian formalisms for
multidimensional calculus of variations and perturbation theory_, ([arXiv:math-ph/0212036](http://arxiv.org/abs/math-ph/0212036))
 {#Helein02}

See also

* [[Igor V. Kanatchikov]] *Geometric (pre)quantization in the polysymplectic approach to field theory*, in *Differential Geometry and Its Applications*, Proc. Conf., Opava (August 2001) [[arXiv:hep-th/0112263](https://arxiv.org/abs/hep-th/0112263), [pdf](http://conferences.math.slu.cz/8icdga/PDF/309-322.pdf)]

* [[Igor V. Kanatchikov]], _De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory_, Rept. Math. Phys. **43** (1999) 157-170 [[arXiv:hep-th/9810165](http://arxiv.org/abs/hep-th/9810165), doi:10.1016/S0034-4877(99)80024-X]

For [[teleparallel gravity]]:

* [[Igor V. Kanatchikov]]. *The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization* (2023) [[arXiv:2308.10052](https://arxiv.org/abs/2308.10052)]

For more see the references at _[[multisymplectic geometry]]_, at _[[n-plectic geometry]]_ and at _[[Hamiltonian n-vector field]]_.

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de Finetti's theorem

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### Context
#### Measure and probability theory
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Motivation

(...)

## In measure-theoretic probability

(...)

## In Markov categories

(...)

## Quantum versions

(For now, see the references.)

## See also

* [[probability theory]], [[categorical probability]]
* [[infinitary tensor product]], [[Kolmogorov extension theorem]]
* [[Giry monad]], [[probability monad]], [[Markov category]]
* [[exchangeability]], [[limit]]

## References

* {#definetti_limit} [[Bart Jacobs]], [[Sam Staton]], _De Finetti's construction as a categorical limit_, Proceedings of CMCS, 2021. ([arXiv:2003.01964](https://arxiv.org/abs/2003.01964))

* {#fritz_definetti} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], _De Finetti's theorem in categorical probability_. Journal of Stochastic Analysis, 2021. ([arXiv:2105.02639](https://arxiv.org/abs/2105.02639))

* {#det_submonad} Sean Moss, [[Paolo Perrone]], _Probability monads with submonads of deterministic states_, LICS 2022. ([arXiv:2204.07003](https://arxiv.org/abs/2204.07003))

* {#fritz_representable} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], Eigil Fjeldgren Rischel, _Representable Markov categories and comparison of statistical experiments in categorical probability_, Theoretical Computer Science 961, 2023. ([arXiv:2010.07416](https://arxiv.org/abs/2010.07416))

* {#ergodic_dagger} Noé Ensarguet, [[Paolo Perrone]], Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, [arXiv:2310.04267](https://arxiv.org/abs/2310.04267)

For the quantum case:

* {#definetti_ned} [[Sam Staton]] and Ned Summers, _Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of $C^\ast$-algebras_, Proceedings of Quantum Physics and Logic, 2022. ([arXiv](https://arxiv.org/abs/2207.05832))

* {#quantum_definetti} [[Tobias Fritz]] and Antonio Lorenzin, _Involutive Markov categories and the quantum de Finetti theorem_, 2023. ([arXiv](https://arxiv.org/abs/2312.09666))

category: probability

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