Donald W. Anderson

Donald Werner Anderson was a professor of mathematics at UCSD.

He got his PhD degree from Berkeley in 1964, advised by [[Emery Thomas]].

Among other things, he is known for [[Anderson duality]].

His PhD students include [[Chris Reedy]] of [[Reedy categories]].

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=12015)

## Selected writings

Introducing [[Anderson duality]]:

* {#Anderson69} [[Donald W. Anderson]], _Universal coefficient theorems for K-theory_, mimeographed notes, Univ. California, Berkeley, Calif. (1969) [[pdf](https://faculty.tcu.edu/gfriedman/notes/Anderson-UCT.pdf)]

On [[homotopy pullbacks]] under [[geometric realization of simplicial topological spaces]] and stating (though not proving) the [[canonical model structure on groupoids]]:

* [[Donald Werner Anderson]], *Fibrations and geometric realization*,  Bull. Amer. Math. Soc. **84** 5 (1978) 765-788 [[euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139)]

category: people

[[!redirects Donald Werner Anderson]]
[[!redirects D. W. Anderson]]


Donald Yau

* [webpage](https://math.osu.edu/people/yau.22)

## Selected writings

On [[2-categories]] and [[2-category theory]]:

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

On bimonoidal or [[rig categories]]

* [[Niles Johnson]], [[Donald Yau]], _Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory_ ([three volume book](https://nilesjohnson.net/En-monoidal.html); abridged version [arXiv:2107.10526](https://arxiv.org/abs/2107.10526)).

On [[Mackey functors]] [[enriched]] over [[closed category|closed]] [[multicategories]]:

* [[Niles Johnson]], [[Donald Yau]], _Homotopy Theory of Enriched Mackey Functors_ ([arXiv:2212.04276](https://arxiv.org/abs/2212.04276))

## Related $n$Lab entries

* [[homotopical algebraic quantum field theory]]

category: people


Donaldson theory

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## Idea

(...)

## Properties

A [[topological field theory|topological]] [[FQFT]]-formulation of Donaldson theory is supposed to be given as a [[functor]] from a suitable [[symplectic category]] of [[symplectic manifolds]] with [[Lagrangian correspondences]] between them which sends a [[symplectic manifold]] to its [[Fukaya category]]. For more on this see at _[[Lagrangian correspondences and category-valued TFT]]_.

## Related concepts

* [[topologically twisted D=4 super Yang-Mills theory]]

* [[Seiberg-Witten theory]]

* [[Gromov-Witten theory]]

## References

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

The relation to the [[topologically twisted N=2 D=4 super Yang-Mills theory]] is due to

* [[Edward Witten]], _Topological quantum field theory_, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 ([Euclid](http://projecteuclid.org/euclid.cmp/1104161738))

Review:

* [[Jan Manschot]], *Four-Manifold Invariants and Donaldson-Witten Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14709](https://arxiv.org/abs/2312.14709)]

[[!redirects Donaldson-Witten theory]]


Donaldson-Thomas invariant

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## Idea

### In mathematics

Consider a complex projective [[Calabi-Yau manifold|Calabi-Yau 3-manifold]] $X$ with volume form $vol_X$. R. Thomas considered in his 1997 thesis a holomorphic version of the [[Casson invariant]] which may be defined using the holomorphic [[Chern-Simons theory|Chern-Simons functional]].

For a holomorphic connection $A = A_0 +\alpha$, the holomorphic Chern-Simons functional is given by

$$CS(A) = \int_X Tr(\bar\nabla_{A_0} \alpha\wedge\alpha +\frac{1}{2}\alpha\wedge\alpha\wedge\alpha) vol_X $$

Its critical points are holomorphically flat connections: $F^{0,2}_A = 0$. One would like to count the critical points in appropriate sense, which means the integration over the suitable compactified moduli space of solutions. These solutions may be viewed as Hermitean Yang-Mills connections or as [[BPS state]]s in physical interpretation. The issues of compactification involve stability conditions which depend on the underlying [[Kähler manifold|Kähler form]]; as the Kähler form varies there are discontinuous jumps at the places of [[wall crossing]].

Under the [[mirror symmetry]], the holomorphic bundles correspond to the [[Lagrangian submanifold]]s in the mirror, and the stability condition restricts the attention to the [[special Lagrangian submanifold]]s in the mirror.

### In string theory

... in [[heterotic string theory]]: [He-Lee 12](#HeLee12)...

### Motivic DT invariants {#Motivic}

A more general setup of [[motive|motivic]] Donaldson-Thomas invariants is given by [[Dominic Joyce]] and by [[Maxim Kontsevich]] and [[Yan Soibelman]], see the references below.

## Related concepts

* [[Gromov-Witten/Donaldson-Thomas correspondence]]

## References

### General

* [[Simon Donaldson]], ...

* Young-Hoon Kiem, Jun Li, _Categorification of Donaldson-Thomas invariants via Perverse Sheaves_, [arxiv/1212.6444](http://arxiv.org/abs/1212.6444)

### Motivic Donaldson-Thomas invariants

The original articles are

* [[Dominic Joyce]], Yinan Song, _A theory of generalized Donaldson-Thomas invariants_ ([arxiv/0810.5645](http://arxiv.org/abs/0810.5645))

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_, ([arXiv:0811.2435](http://arxiv.org/abs/0811.2435));

summarized in

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Motivic Donaldson-Thomas invariants: summary of results_, ([0910.4315](http://arxiv.org/abs/0910.4315));

and with lecture notes in

* [[Yan Soibelman]], _Motivic Donaldson-Thomas invariants and wall-crossing formulas_,  [Chern-Simons Research Lectures](http://math.berkeley.edu/~reshetik/CSR-Lectures.html) 2010 ([pdf](http://math.berkeley.edu/~reshetik/CSRL/Yan-Berkeley-2010-2.pdf))

See also

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Cohomological [[Hall algebra]], exponential Hodge structures and motivic Donaldson-Thomas invariants_ ([arxiv/1006.2706](http://arxiv.org/abs/1006.2706))
* D.-E. Diaconescu, Z. Hua, Y. Soibelman, _HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants_, [arxiv/1202.4651](http://arxiv.org/abs/1202.4651)
* Tudor Dimofte, Sergei Gukov, _Refined, Motivic, and Quantum_, [arxiv/0904.1420](http://arxiv.org/abs/0904.1420)
* Vittoria Bussi, Shoji Yokura, _Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems_, [arxiv/1306.4725](http://arxiv.org/abs/1306.4725)
* Andrew Morrison, Sergey Mozgovoy, Kentaro Nagao, Balazs Szendroi, _Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex_, [arxiv/1107.5017](http://arxiv.org/abs/1107.5017)
* Markus Reineke, _Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers_, [arxiv/1102.3978](http://arxiv.org/abs/1102.3978)

### Relation to wall crossing

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry_, ([arxiv/1303.3253](http://arxiv.org/abs/1303.3253))

* [[Sergio Cecotti]], [[Cumrun Vafa]], *BPS wall crossing and topological strings*  [[arXiv/0910.2615](http://arxiv.org/abs/0910.2615)]

* [[Davide Gaiotto]], [[Greg Moore|Gregory W. Moore]], [[Andrew Neitzke]], _[[wall crossing|Wall-crossing]], [[Hitchin system]]s, and the [[WKB approximation]], [arxiv/0907.3987](http://arxiv.org/abs/0907.3987)

* sbseminar blog: [Hall algebras and Donaldson-Thomas invariants-i](http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i)

### Relation to string theory
 {#RelationToStringTheory}

In [[heterotic string theory]]:

* {#HeLee12} [[Yang-Hui He]], Seung-Joo Lee, _Quiver Structure of Heterotic Moduli_, J. High Energ. Phys. (2012) 2012: 119 ([arXiv:1208.3004](https://arxiv.org/abs/1208.3004))

Relation to [[Hilbert schemes]]:

* Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], _Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory_, Nucl. Phys. B809: 452-518, 2009 ([arXiv:0803.4188](https://arxiv.org/abs/0803.4188))

* Artan Sheshmani, _Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories_, Notices of the International Congress  of Chines Mathematics (2019) ([j.mp:2U7qd01](http://j.mp/2U7qd01), [pdf](https://scholar.harvard.edu/files/artan/files/iccm_07_02_a03.pdf))

[[!redirects Donaldson-Thomas invariants]]

[[!redirects motivic Donaldson-Thomas invariant]]

[[!redirects Donaldson-Thomas theory]]


Donaldson-Uhlenbeck-Yau theorem

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## Idea

A series of results on [[moduli spaces of connections]] in the presence of [[complex structure]] due to ([Donaldson 85](#Donaldson85), [Uhlenbeck-Yau 86](#UhlenbeckYau86)), the most famous of which says that on a [[compact topological space|compact]] [[Kähler manifold]] any [[stable vector bundle|semistable]] [[holomorphic vector bundle]] with trivial [[determinant line bundle]] admits a [[Hermite-Einstein connection]].

The theorem is recalled for instance as ([Scheinost-Schottenloher 96, theorem 1.15](#ScheinostSchottenloher96)). This result is a key step in the construction of the [[Kähler polarization]] on a [[moduli space of flat connections]] via [[symplectic reduction]] ([Scheinost-Schottenloher 96, corollary 1.16](#ScheinostSchottenloher96)), a [[non-abelian cohomology|non-abelian]] version of the Griffiths [[intermediate Jacobian]] $J^1$ (see there at _[Examples -- Picard variety](intermediate+Jacobian#ExamplePicard)_).

The correspondence between [[stable vector bundle|semi-stable vector bundles]] and [[Hermite-Einstein connections]] holds more generally over [[complex manifolds]], where it is known as the _[[Kobayashi-Hitchin correspondence]]_. The special case of that over [[Riemann surfaces]] in turn is essentially the _[[Narasimhan-Seshadri theorem]]_.

## Related concepts

* [[Narasimhan-Seshadri theorem]]

## References

The original articles are

* {#Donaldson85} [[Simon Donaldson]], _Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles_, Proc. LMS 50 (1985) 1-26

* {#UhlenbeckYau86} [[Karen Uhlenbeck]], [[Shing-Tung Yau]], _On the existence of Hermitean Yang-Mills-connections on stable bundles over Kähler manifolds_, Comm. Pure Appl. Math. 39 (1986)  257-293

Reviews in the context of discussion of [[Kähler polarization]] of [[moduli spaces of connections]] is in

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))


Dongho Lee

* [personal page](https://lmf.cnrs.fr/Perso/Dongho_Lee)

## Selected writings

On [[software verification|verification]] of [[quantum circuits]] with the [[QWIRE]]-[[quantum programming language]]:

* [[Robert Rand]], [[Jennifer Paykin]], [[Dong-Ho Lee]], [[Steve Zdancewic]], *ReQWIRE: Reasoning about Reversible Quantum Circuits*, EPTCS **287** (2019) 299-312 [[arXiv:1901.10118](https://arxiv.org/abs/1901.10118), [doi:10.4204/EPTCS.287.17](https://doi.org/10.4204/EPTCS.287.17)]

On [[dynamic lifting]] (of [[quantum measurement]]-results) in [[quantum programming]]/[[quantum computing]] such as [[Quipper]]:

* [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]

* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]

* [[Dongho Lee]], *Formal Methods for Quantum Programming Languages*, Paris Saclay (Dec 2022) [[hal:tel-03895847](https://hal.science/LMF/tel-03895847)]

category: people

[[!redirects Dong-Ho Lee]]


Dongmin Gang

## Selected writings

On [[3d-3d correspondence]]:

* [[Dongmin Gang]], Eunkyung Koh, Sangmin Lee, [[Jaemo Park]], _Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus_, JHEP01(2014)063 ([arXiv:1305.0937](https://arxiv.org/abs/1305.0937))

* {#GangKimLee14a} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of Wrapped M5-branes and Chern-Simons theory_, Physics Letters B
Volume 733, 2 June 2014, Pages 316-319 ([arXiv:1401.3595](https://arxiv.org/abs/1401.3595))

* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

* Jin-Beom Bae, [[Dongmin Gang]], Jaehoon Lee, _3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes_, JHEP 08 (2017) 118 ([arXiv:1610.09259](https://arxiv.org/abs/1610.09259))

* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))

* {#GangKimPandoZayas19} [[Dongmin Gang]], [[Nakwoo Kim]], Leopoldo A. Pando Zayas, _Precision Microstate Counting for the Entropy of Wrapped M5-branes_ ([arXiv:1905.01559](https://arxiv.org/abs/1905.01559))

Discussion of the [[volume conjecture]] by combining the [[3d-3d correspondence]] with [[AdS/CFT]] for background of [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

* [Gang-Kim-Lee 14b, Section 3.2](#GangKimLee14b)

* [Gang-Kim 18, around (21)](#GangKim18)

Enhanced to a [[defect field theory]]:

* [[Dongmin Gang]], [[Nakwoo Kim]], Mauricio Romo, Masahito Yamazaki, _Aspects of Defects in 3d-3d Correspondence_, J. High Energ. Phys. (2016) ([arXiv:1510.05011](https://arxiv.org/abs/1510.05011))

relation to [[Reidemeister torsion]]:

* [[Dongmin Gang]], Seonhwa Kim, Seokbeom Yoon,  _Adjoint Reidemeister torsions from wrapped M5-branes_ ([arXiv:1911.10718](https://arxiv.org/abs/1911.10718))

category: people


Donovan van Osdol

[[!redirects Don van Osdol]]
Donovan H. Van Osdol is a mathematician who is Emeritus professor at the University of New Hampshire. He worked with [[Mike Barr]] and [[Jack Duskin]], on category theory and homological algebra.

A minimalist Faculty  webpage is [here](https://ceps.unh.edu/person/donovan-vanosdol)

## Selected writings

Introducing [[Barr-exact categories]] and [[regular categories]]:

* {#BarrGrilletvonOsdool} [[Michael Barr]],  [[Pierre Grillet]], [[Donovan van Osdol]], *Exact Categories and Categories of Sheaves*,  Lec. Notes in Math. __236__, Springer 1971, ([doi:10.1007/BFb0058579](https://link.springer.com/book/10.1007/BFb0058579))

On [[cohomology]] induced from a [[monad]]/[[comonad]]-pair and early discussion of [[distributive laws]] of monads over comonads:

* {#vanOsdol73} [[Donovan van Osdol]], *Bicohomology Theory*, Transactions of the American Mathematical Society **183** (1973) 449-476 [[jstor:1996479](https://www.jstor.org/stable/1996479)]

See also:

* [[D. van Osdol]], _Long exact sequences in the first variable for algebraic cohomology theories,_ J. Pure Appl. Alg., 23, (1982), 271–309.

category:people
[[!redirects Donovan H. van Osdol]]
[[!redirects D. van Osdol]]


Donu Arapura

__Donu Arapura__ is an [[algebraic geometer]], a Professor at the Purdue University. His main specialty is [[Hodge theory]].

* [webpage](http://www.math.purdue.edu/~dvb)

category: people


Doosung Park

* [webpage](http://math.berkeley.edu/people/grad/doosung-park)

## related $n$Lab entries

* [[Berkovich analytic space]]

category: people


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## Idea

The *Doplicher-Roberts reconstruction theorem* allows one to [[reconstruction theorem|reconstruct]] a [[compact topological group]] or "[[supergroup]]" (here essentially: $\mathbb{Z}/2$-graded group) from its [[category of representations]], given enough [[extra structure]] on this [[category]].  Unlike some other versions of [[Tannaka duality]], such as [[Deligne's theorem on tensor categories]], it does not require knowledge of the [[forgetful functor]] ("[[fiber functor]]") from the category of representations to [[VectorSpaces]].

This result is named after [[Sergio Doplicher]] and [[John Roberts]], who conceived of it in the context of [[algebraic quantum field theory]] as an intrinsic characterization of the [[superselection sectors]] of a [[quantum field theory]]. See at _[[DHR superselection theory]]_ for more on this aspect.
A simplified and self-contained proof was given in [Müger 06](#Mueger06).

In brief, the DR reconstruction theorem says that every [[symmetric monoidal category|symmetric]] [[tensor category|tensor]] [[star-category]] with conjugates, [[direct sums]], [[subobjects]] and [[endomorphism ring]] of the [[tensor unit]] [[isomorphism|isomorphic]] to the [[complex numbers]], $End(1)= \mathbb{C}$,  is [[equivalence of categories|equivalent]] to the category of [[finite dimensional vector space|finite dimensional]] [[unitary group|unitary]] [[linear representations]] of a [[compact topological group|compact]] [[supergroup]], which is unique up to [[isomorphism]].  For definitions of these terms, see [Müger 06](#Mueger06).

Beware that there are non-isomorphic finite groups whose [[categories of representations]] are equivalent considered merely as [[tensor categories]], ignoring the [[symmetric monoidal category|symmetric]] [[mathematical structure|structure]] (e.g. [Etingof-Gelaki 00](#EtingofGelaki00)).

## Related concepts

* [[Deligne's theorem on tensor categories]]

* [[Wigner classification]]

  * [[unitary representation of the Poincaré group]]

  * [[unitary representation of the super Poincaré group]]

## References

Original statement and proof in the context of [[AQFT]]:

* [[Sergio Doplicher]], [[John Roberts]], ...

Generalization to supergroups and streamlined proof:

* {#Mueger06} [[Michael Müger]], _Abstract duality theory for symmetric tensor $\ast$-categories_ appendix ([pdf](http://www.math.ru.nl/~mueger/PDF/16.pdf)), in [[Hans Halvorson]],  _Algebraic quantum field theory_ ([arXiv:math-ph/0602036](http://arxiv.org/abs/math-ph/0602036)), in J. Butterfield & J. Earman (eds.) _Handbook of Philosophy and Physics_

See also:

* {#EtingofGelaki00} [[Pavel Etingof]], [[Shlomo Gelaki]], _Isocategorical groups_ ([arXiv:math/0007196](https://arxiv.org/abs/math/0007196))

[[!redirects Doplicher-Roberts reconstruction]]
[[!redirects Doplicher-Roberts duality]]


Dorette Pronk

Dorette Pronk is a category theorist at Dalhousie University, Canada.

* [webpage](http://www.mscs.dal.ca/Faculty/pronk.htm)

## Selected writings

On [[localizations of 2-categories]] via [[bicategories of fractions]]:

* [[Dorette A. Pronk]], _Etendues and stacks as bicategories of fractions_, Comp. Math. **102** 3 (1996) pp.243-303. ([numdam:CM_1996__102_3_243_0](http://www.numdam.org/item/?id=CM_1996__102_3_243_0))

On [[model structures on DblCat]]:

* {#FPPModel} [[Thomas M. Fiore]], [[Simona Paoli]], [[Dorette A. Pronk]], *Model structures on the category of small double categories*, _Algebraic & Geometric Topology_ **8** (2008) 1855–1959 [[doi:10.2140/agt.2008.8.1855](https://doi.org/10.2140/agt.2008.8.1855)]

On [[orbifold cohomology]] (such as [[orbifold K-theory]]):

* {#PronkScull07} [[Dorette Pronk]], [[Laura Scull]], _Translation Groupoids and Orbifold Bredon Cohomology_, Canad. J. Math. 62 (2010), 614-645 ([arXiv:0705.3249](https://arxiv.org/abs/0705.3249), [doi:10.4153/CJM-2010-024-1](https://doi.org/10.4153/CJM-2010-024-1))

On [[exponential objects]] among [[internal groupoids]] in [[Top]] ([[topological groupoids]]):

* [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

On the [[equivariant fundamental groupoid]] of an [[orbifold]]:

* {#PronkScull21} [[Dorette Pronk]], [[Laura Scull]], *The Equivariant Fundamental Groupoid as an Orbifold Invariant*, Homology, Homotopy and Applications **23** 1 (2021) ([arXiv:1908.01201](https://arxiv.org/abs/1908.01201), [doi:10.4310/HHA.2021.v23.n1.a3](https://dx.doi.org/10.4310/HHA.2021.v23.n1.a3))

category: people

[[!redirects D. Pronk]]
[[!redirects Dorette A. Pronk]]
[[!redirects D. A. Pronk]]


Dorian Goldfeld

* [webpage](http://www.math.columbia.edu/~goldfeld/)

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

## related $n$Lab entries

* [[L-function]]

* [[automorphic representation]]

* [[adelic integration]]

category: people


Dorit Aharonov

* [personal page](https://www.cs.huji.ac.il/w~doria/)

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

* [GoogleScholar page](https://scholar.google.com/citations?user=hTBcSYQAAAAJ)

## Selected writings

On [[quantum circuits]] with [[mixed quantum states]]/[[density matrices]]:

* [[Dorit Aharonov]], [[Alexei Kitaev]], [[Noam Nisan]], *Quantum Circuits with Mixed States*, *Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC)* (1998) 20-30 [[arXiv:quant-ph/9806029](https://arxiv.org/abs/quant-ph/9806029), [doi:10.1145/276698.276708](https://doi.org/10.1145/276698.276708)]

Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:

* [[Dorit Aharonov]], [[Michael Ben-Or]], *Fault-Tolerant Quantum Computation With Constant Error Rate*, SIAM J. Comput., **38** 4 (2008) 1207–1282 [[arXiv:quant-ph/9906129](https://arxiv.org/abs/quant-ph/9906129), [doi:10.1007/978-3-642-32512-0_31](https://doi.org/10.1007/978-3-642-32512-0_31)]

On [[adiabatic quantum computation]]:

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

category: people


Dorje C. Brody

* [institute page](https://www.surrey.ac.uk/people/dorje-c-brody)

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

## Selected writings

On the [[symplectic formulation of quantum mechanics]]:

* {#BrodyHughston01} [[Dorje C. Brody]], [[Lane P. Hughston]], *Geometric Quantum Mechanics*, J. Geom. Phys. **38** (2001) 19-53 [[arXiv:quant-ph/9906086](https://arxiv.org/abs/quant-ph/9906086), doi:10.1016/S0393-0440(00)00052-8]

category: people

[[!redirects Dorje Brody]]


Doron Gepner

* [webpage](http://www.weizmann.ac.il/physics/staff/Gepner.htm)

## related $n$Lab entries

* [[WZW model]]

* [[Gepner model]]

category: people


Doron Zeilberger

__Doron Zeilberger__ is a mathematician at Rutgers University, mainly working in [[discrete mathematics]].

* [web](http://www.math.rutgers.edu/~zeilberg)

category: people

[[!redirects Doron Zeilberger]]
[[!redirects D. Zeilberger]]
[[!redirects D Zeilberger]]


double affine Hecke algebra

## Idea

Double Hecke algebras or __double affine Hecke algebras__ (DAHA) or __Cherednik algebra__s are a particular class of multiparametric families of associative algebras; one parameter is $q$ and then there are 1 or more parameters $t_\alpha$ where labels $\alpha$ depend on a root system. While Iwahori-Hecke algebra is a deformation of the group algebra of a Weyl group, DAHA are flat deformations of certain crossed product algebras (or twisted group algebras) involving Coxeter groups. Affine denotes the relation to affine Weyl group in DAHA case.
[[Ivan Cherednik]] introduced DAHA in proving [[Macdonald conjecture]]s about orthogonal polynomials attached to reduced root systems.

Notice that [[rational Cherednik algebra]]s defined by Etingof and Ginzburg are not literally DAHAs, but certain degenerations of those and are a special case of [[symplectic reflection algebra]]s.

## Related concepts

* [[Hecke algebra]]

* [[Calogero-Moser system]], [[Dunkl operator]]
* [[symplectic reflection algebra]]
* [[rational Cherednik algebra]]

## References

* Wikipedia, _[Double affine Hecke algebra](https://en.wikipedia.org/wiki/Double_affine_Hecke_algebra)_

Monograph:

* I. Cherednik, _Double affine Hecke algebras_, London Math. Soc. Lecture Note Series 319, Cambridge Univ. Press 2005

It has been introduced in a series of works in early 1990s:

* I. Cherednik, _Double affine Hecke algebras. Knizhnik-Zamolodchikov equations, and Macdonald operators_, IMRN (Duke Math. J.) 9 (1992), 171--180 MR1185831 (94b:17040), _A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras_, Invent. math. __106__ (1991) 411--431 MR1128220 (93b:17040); _Double affine Hecke algebras and Macdonald’s conjectures_, Ann. of Math.(2) 141:1 (1995) 191--216.

An introduction into the monograph above is the preprin

* Ivan Cherednik, _Introduction to double Hecke algebras_, [math.QA/0404307](https://arxiv.org/abs/math/0404307)
* I. Cherednik, Y. Markov, _Hankel transform via double Hecke algebra_, math.QA/0004116

* [[Mark Haiman]], _Cherednik algebras, Macdonald polynomials and combinatorics_, Proc. ICM, Madrid 2006, Vol. III, 843--872, [djvu scan](http://www.mathunion.org/ICM/ICM2006.3/Main/icm2006.3.0843.0872.ocr.djvu); author's [pdf](https://math.berkeley.edu/~mhaiman/ftp/icm-2006/comb-mac-chered.pdf)

* [[Pavel Etingof]], _Lectures on Calogero-Moser systems_, [pdf](http://www-math.mit.edu/~etingof/zlecnew.pdf)
* Wee Liang Gan, [[Victor Ginzburg]], _Almost-commuting variety, $\mathcal{D}$-modules, and Cherednik algebras_, With an appendix by Ginzburg. IMRP Int. Math. Res. Pap. 2006, 26439, 1--54, [MR2008c:14062](https://www.ams.org/mathscinet-getitem?mr=2210660)

A construction via [[factorization homology]] of [[quantum group]] [[representation categories]] is indicated in

* [[David Ben-Zvi]], [[Adrien Brochier]], [[David Jordan]], _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_ ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))

category: algebra

[[!redirects Cherednik algebra]]
[[!redirects double Hecke algebra]]
[[!redirects double affine Hecke algebras]]
[[!redirects DAHA]]


double affine Hecke algebra > history

[[!redirects double affine Hecke algebra -- history]]

< [[double affine Hecke algebra]]


double bicategory

##Idea

A **double bicategory** is a structure similar to a [[double category]], but where composition of both vertical and horizontal arrows is only weakly associative and unital.

To define this notion, we need to include extra shapes of 2-dimensional cells in addition to the squares that appear in a double category: we also need vertical and horizontal [[globe|globes]].  The reason is that if we have associators for horizontal morphisms given
by squares, it appears to be impossible to formulate the pentagon identity for these squares unless either 1) composition of vertical morphisms is strict, or 2) we introduce cells with new shapes.  In case 1) --- that is, if associativity and the unit laws hold strictly in one direction --- we have a 'pseudo double category', as studied by Grandis, Paré and Fiore.  (See [[double category]] for more on this concept.)  In case 2) we have a double bicategory.

The concept and terminology here were introduced by Verity in his thesis ([Verity 1992](#Verity92)). The definition can also be found in ([Morton 2009](#Morton09)) (especially [Section 3: Double bicategories](http://arxiv.org/PS_cache/math/pdf/0611/0611930v3.pdf#page=9) (direct pdf link)) and ([Morton 2007](#Morton07)).

The term 'double bicategory' may be confusing, since while a double category is a category internal to $Cat$, a double bicategory is not the fully general sort of bicategory internal to $Bicat$.  This issue is addressed in Morton's work.

A double bicategory can also be regarded as a special sort of [[intercategory]].

## References

* {#Verity92} [[Dominic Verity]], _Enriched categories, internal categories and change of base_ Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html))

* {#Morton09} [[Jeffrey C. Morton]], _Double bicategories and double cospans_,  Journal of Homotopy and Related Structures, Vol. 4 (2009), No. 1, pp. 389-428,
[journal abstract page](https://www.emis.de/journals/JHRS/volumes/2009/n1a19/abstract.htm), arXiv:[math/0611930](http://arxiv.org/abs/math/0611930),

* {#Morton07} [[Jeffrey C. Morton]], _Extended TQFT's and Quantum Gravity_, PhD thesis University of California, Riverside (2007), arXiv:[0710.0032](http://arxiv.org/abs/0710.0032).

[[!redirects double bicategories]]


double category

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#### 2-Category theory
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# Contents
* table of contents
{:toc}

## Definition ##

A **double category** $D$ is an [[internal category]] in [[Cat]].   Similarly, a **double groupoid** is an internal [[groupoid]] in [[Grpd]].

However, these definitions obscure the essential symmetry of the concepts.  We think of a double category $D_1 \stackrel{\to}{\to} D_0$ as having

* *[[objects]]*: the objects  of $D_0$
* *[[vertical morphisms]]*: the morphisms of $D_0$
* *[[horizontal morphisms]]*: the objects of $D_1$
* *[[2-morphisms]]* or *squares* or *2-cells*: the morphisms of $D_1$.

We may picture a 2-cell in a double category as a square:

$$
  \begin{matrix}
    x_0
    &\stackrel{f}{\to} &
    x_1
    \\
    {}^{\mathllap{\alpha_0}}\downarrow
    &\Downarrow^{\mathrlap{\phi}}&
    \downarrow^{\mathrlap{\alpha_1}}
    \\
    y_0
    &\underset{g}{\to} &
    y_1
 \end{matrix}
$$

Here $x_i,y_i$ are objects, $f$ and $g$ are horizontal arrows, $\alpha_i$ are vertical arrows and $\phi$ is the 2-cell itself.  This makes it clear why $\phi$ is called a 'square'.

Alternatively one can apply [[Poincaré duality]] to obtain a [[string diagram]] representation for 2-cells, similar to the one obtained for [[2-category|2-categories]] ([Myers 16](#Myers16)):

\begin{centre}
    \begin{tikzpicture}[scale=2]
        \usetikzlibrary{positioning,calc,decorations.pathmorphing,decorations.pathreplacing,decorations.markings,patterns}

        \tikzset{fwd/.style={decoration={markings,mark=at position 0.7 with {\arrow{>}}},postaction={decorate}}}
        \tikzset{bwd/.style={decoration={markings,mark=at position 0.5 with {\arrow{<}}},postaction={decorate}}}

        \path[fill={rgb,255:red,212; green,166; blue,204}] (0,0) rectangle (.5,.5);
        \path[fill={rgb,255:red,152; green,209; blue,162}] (.5,0) rectangle (1,.5);
        \path[fill={rgb,255:red,248; green,168; blue,156}] (0,.5) rectangle (.5,1);
        \path[fill={rgb,255:red,243; green,237; blue,155}] (.5,.5) rectangle (1,1);
        \draw[gray] (0,0) rectangle (1,1);
        \node[circle,draw,inner sep=1.5pt,fill=white] at (.5,.5) (alpha) {$\phi$};
        \draw[fwd] (0,.5) -- (alpha);
        \draw[fwd] (alpha) -- (1,.5);
        \draw[bwd] (.5,0) -- (alpha);
        \draw[bwd] (alpha) -- (.5,1);
        \node at (-.2,.5) {$\alpha_0$};
        \node at (1.2,.5) {$\alpha_1$};
        \node at (.5,1.2) {$f$};
        \node at (.5,-.2) {$g$};
        \node at (.25,.75) {$x_0$};
        \node at (.75,.75) {$x_1$};
        \node at (.25,.25) {$y_0$};
        \node at (.75,.25) {$y_1$};
    \end{tikzpicture}
\end{centre}

The vertical and horizontal arrows form categories (called **edge categories**), and the squares have two category structures which respect the edge category structures.

Vertical composition is given by the composition in the ordinary [[category|categories]] $D_0$ and $D_1$, while horizontal composition is given by the composition operation specified on $D_1 \stackrel{\to}{\to} D_0$ by virtue of it being a [[category]] [[internalization|internal to]] [[Cat]].

In particular, the **transpose** of a double category, which switches the vertical and horizontal arrows, is again a double category.

A double category is an important special case of an [[n-fold category]], namely the case where $n = 2$. There is more on $n$-fold categories in the paper by Brown and Higgins referenced below.

## Examples ##

* If $C$ is a [[2-category]], there is its _[[double category of squares]]_ $Sq(C)$ whose objects are those of $C$, both of whose types of morphisms are the morphisms in $C$, and whose squares are [[2-morphisms]] in $C$ with their source and target both decomposed as a composite of two morphisms.  (These squares are sometimes called *quintets* $(\alpha,f,g,h,k)$ where $\alpha\colon f g \to h k$, and so this double category is said to be a [[quintet construction]].)

  (In this example, the two edge categories coincide.  Double categories with this property are called **edge-symmetric**.)

* Since any 1-category can be regarded as a 2-category with only identity 2-cells, for any 1-category $C$ we have a double category $Sq(C)$ whose squares are the commutative squares in $C$.  We can also restrict the commutative squares considered, such as taking only [[pullback]] squares.

* Any 2-category can also be made into a double category in two more ways, by defining the vertical or horizontal morphisms to consist only of identities.  In this way 2-categories can be considered as a special case of double categories.

  In the other direction, any double category has two underlying 2-categories, consisting of the objects, the vertical (resp. horizontal) arrows, and the squares whose horizontal-arrow (resp. vertical-arrow) source and target are identities.  We call these its **vertical 2-category** and **horizontal 2-category**.

* Finally, we can also make a 2-category $C$ into a double category with the same objects, whose horizontal arrows are the morphisms of $C$, whose vertical arrows are the [[adjunctions]] in $C$, and whose 2-cells are [[mate]]-pairs of 2-cells in $C$.  Naturality properties of the mate correspondence are concisely expressed by the existence of this double category.

* There is a double category $Prof$ whose objects are (small) categories, whose vertical arrows are functors, whose horizontal arrows are [[profunctors]], and whose squares are natural transformations.  This double category is in fact an [[equipment]], as are many other similar ones (such as for [[enriched categories]]).

* Any topological space has a "homotopy double groupoid" whose objects are points, whose morphisms of both types are paths (so it is edge-symmetric), and whose 2-cells are homotopies.

* There is a double category $MonCat$ whose objects are [[monoidal categories]], whose horizontal arrows are [[lax monoidal functor]]s, whose vertical arrows are [[colax monoidal functor]]s, and whose 2-cells are generalized [[monoidal natural transformation]]s.  An analogous double category can be constructed involving the algebras for any [[2-monad]]; see [[double category of algebras]].

* There is a [[double category of model categories]] whose [[objects]] are [[model categories]], whose horizontal arrows are the right [[Quillen functor]]s, vertical arrows the left Quillen functors, and whose 2-cells are arbitrary natural transformations.  Passage to [[derived functors]] is a functor on this double category.  More generally, we can define a double category of [[homotopical categories]] and "left derivable" and "right derivable" functors.

## Weakenings ##

An internal category in the 1-category $Cat$ might more properly be called a *strict* double category, since all its composition operations are strictly associative and unital.  Since a double category is a 2-dimensional structure, it makes sense to allow these compositions to be weak as well.

### Pseudo double categories

A **pseudo double category** is a weakly internal category in the 2-category $Cat$.  Here "weakly internal category" in a 2-category is interpreted as being associative and unital up to coherent isomorphism, just as a [[bicategory]] is a "weakly enriched category."  This makes the composition in one direction weak, but the composition in the other direction remains strict (it is the composition in the objects of $Cat$ that make up the pseudo double category).  Many naturally occurring examples, such as $Prof$, are pseudo double categories.

Luckily, every pseudo double category is equivalent to a double category of the usual sort, where composition of arrows in both directions is strictly associative.  This is Theorem 7.5 of Grandis and Paré's paper [Limits in double categories](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1999__40_3/CTGDC_1999__40_3_162_0/CTGDC_1999__40_3_162_0.pdf).

### Double bicategories

One way to define a double category which is "weak in both directions" is a [[double bicategory]].  Double bicategories were defined in [[Dominic Verity]]'s thesis. The definition can also be found in Jeffrey Morton's paper [Double bicategories and double cospans](http://arxiv.org/abs/math/0611930).

The definition of double bicategory takes as given two ordinary [[bicategories]], representing the vertical and horizontal bicategories, together with a set of squares which is "acted on" by both of these.  In the basic definition, there is no requirement that the 2-cells in the vertical (resp. horizontal) bicategory be exactly those squares whose vertical (resp. horizontal) 1-cell boundaries are identities, but such an axiom can easily be added.

The reason for stipulating the vertical and horizontal bicategories as a basic part of the structure is that if identities are strict in neither direction, then the vertically (resp. horizontally) "globular" 2-cells (those whose vertical or horizontal boundaries are identities) can't seemingly be composed vertically (resp. horizontally), so it's hard to state coherence axioms for the associativity and unit constraints.

### Strictly unital weak double categories

Another way around this difficulty is to require the *units* to be strict, but not the associativity.  This is not such a terrible thing, since units can usually be strictified much more easily than associativity.  Thus, in this approach we assume as given

* a set of objects,
* two sets of vertical and horizontal arrows, equipped with identities and strictly unital (but not associative) composition operations,
* a set of squares, equipped with identities and strictly unital composition in both directions, and
* two sets of associativity squares, one with identity vertical boundary and one with identity horizontal boundary, satisfying the usual axioms, which are possible to state since identities are strict.

### Cubical bicategories

Yet another approach to doubly-weak double categories was proposed by [[Richard Garner]] in an email to the categories list on 5 Mar 2010, in response to a query of [[Ronnie Brown]].  This approach avoids the coherence problems by being completely "unbiased."

> A **cubical bicategory** is given by sets of objects, of vertical arrows, of horizontal arrows and of squares, satisfying the obvious source and target criteria, together with operations of identity and binary composition for vertical and horizontal arrows, satisfying no laws at all; and finally, for every $n \times m$ grid of squares (where possibly $n$ or $m$ are zero), and every way of composing up the horizontal and vertical boundaries using the nullary and binary compositions, a composite square with those boundaries. The coherence axioms which this structure must satisfy say that any two ways of composing up a diagram of squares must give the same answer.

He then commented:

> I would be very interested to know if anyone can extract from this definition a finite collection of composition operations on squares, and a finite collection of equations between them, which together generate all the others. The key obstable seems to be problem that identity 1-cells are not strict in either direction.

It also seems that a "cubical bicategory" in this sense need not have underlying horizontal or vertical bicategories, for the same reason.  This presents something of an obstacle to practical applications.

## Unbiased composition and associativity {#Unbiased}

It is natural to ask what the "unbiased" notion of composition and associativity is for squares in a double category.  The natural thing to take as input for an "unbiased composition" is a dissection of a rectangle into smaller rectangles, the smaller rectangles representing squares that we want to "compose up" in a double category.  As usual, we can try to build up such a composite by means of the "biased" binary composites that are given in the definition of double category.

It can be shown that if a given such dissection can be composed up in *some* way using binary composites, then any two ways to compose it up must give the same result.  See

* Dawson and Paré, _General associativity and general composition for double categories_ , [link](http://www.numdam.org/item/CTGDC_1993__34_1_57_0/)

Also of interest is "Characterizing tileorders" by the same authors, in which they characterize such rectangle-dissections, called [[tileorders]], in purely combinatorial terms.

However, there are dissections which admit *no* composition in a general double category, foremost among which is the "pinwheel:"

[[double-category-pinwheel.png:pic]]

It is shown in the paper

* Dawson, _A forbidden-suborder characterization of binarily-composable diagrams in double categories_ , [TAC](http://www.tac.mta.ca/tac/volumes/1995/n7/1-07abs.html)

that this is the _only obstacle_ in the same sense that $K_5$ and $K_{3,3}$ are the only obstacles to [[planar graph|planarity]] of a [[graph]].  Namely, if a diagram is not composable in a double category, then every "attempt" to compose it by a sequence of binary compositions will eventually result in a diagram which "contains" a pinwheel in a suitable sense (specifically, as a sub-[[tileorder]]).

In many double categories, however, arbitrary arrangements of squares, even pinwheels, can be composed.  By this we mean that given a pinwheel diagram in such a double category (for example), there is a natural square with the same boundary which it makes sense to regard as "the composite" of that pinwheel.  For example:

* In the double category $Sq(C)$ of squares/quintets in a 2-category, the general pasting theorem for 2-categories shows that any arrangement of squares can be composed.  In particular, this applies to commutative squares in a 1-category.  The same is true for double categories of adjunctions.

* If all the squares in a pinwheel are pullback squares in a 1-category, so is the whole thing.  The same is true for more general arrangements in a double category of pullback squares.

* Arbitrary arrangements can also be composed in $Model$ and its variants, again by the pasting theorem for 2-categories, since its 2-cells are just ordinary natural transformations.

* Pinwheels and more general arrangements can also be composed in the double category $MonCat$, along with its relatives $T Alg$.  The proof of this is not quite as trivial, but fairly straightforward.

* In any [[fibrant double category]], all arrangements can be composed.

This last example is due to a factorization property: if we can factor the two long vertical rectangles in the pinwheel into two squares, then we can compose it.  The cartesian cells in a fibrant double category certainly allow such a factorization.  One can also axiomatize more precisely what factorization properties are necessary; this is done in the above paper of Dawson and Paré.

In fact, it seems hard to find a naturally occurring double category in which pinwheels and more general arranguments *cannot* be composed, although such composability definitely does not follow from the double-category axioms.  This suggests that perhaps the usual definition of double category is too naive, and we should actually require all "unbiased" composites to exist, including the pinwheel.  There is probably a monad of some sort on a suitable category whose algebras are "double categories with generalized composites" in this sense.

(It is tempting to try to axiomatize such a structure finitely by adding a basic "pinwheel-composition" operation, but although every non-composable arrangement reduces to one "containing" a pinwheel, the containment is not necessarily of a sort to which such an operation could be applied.  For example, here is a configuration that cannot be composed even with a "pinwheel-composition" operation:

\begin{tikzpicture}[scale=0.7]
      \draw (0,0) rectangle (2,1);
      \draw (1,1) rectangle (2,2);
      \draw (0,1) rectangle (1,3);
      \draw (1,2) rectangle (4,3);
      \draw (3,1) rectangle (4,2);
      \draw (2,0) rectangle (3,2);
      \draw (3,0) rectangle (5,1);
      \draw (4,1) rectangle (5,3);
\end{tikzpicture}

Thus, it is not clear whether there is any finitary axiomatization of such "augmented" double categories.)

## Higher categories of double categories

There are notions of [[double functor|functor]] and of [[vertical transformation|transformation]] between double categories, which make them into a [[2-category]].  In fact, due to the vertical-horizontal symmetry, there are two such 2-categories, depending on the direction we choose the transformations to go in.  Moreover, we can also take the functors to be strict, pseudo, lax, or colax in either direction, so there are many possible 2-categories of double categories (although for some choices one has to worry about compatibility).

Probably the most useful such 2-categories are where we choose one direction to be the "arrow" direction and the other the "proarrow" direction, we take functors which are lax, colax, or pseudo in the proarrow direction and strict in the arrow direction, and transformations that go in the arrow direction.  This is natural when considering double categories such as [[proarrow equipments]].  If we allow the functors and transformations to be only pseudo in the arrow direction as well, and add in [[modifications]] in the arrow direction, we obtain a [[3-category]] instead.  In some situations, such as the study of [[generalized multicategories]], this 3-category is a natural place in which to work (or an enlargement of it to include [[virtual equipments]] as well).

We can also combine the horizontally lax and colax functors, together with a general notion of vertical transformation, into a large double category of double categories.  This is a special case of the double category of $T$-algebras for a particular 2-monad $T$ mentioned above; see [[double category of algebras]].

In some contexts, a natural observation is that the 1-category $DblCat$ is [[cartesian closed category|cartesian closed]], and thus enriched over itself--thus it forms a [[locally double bicategory]], i.e. a category enriched over double categories.  This can be regaded as a structure like a [[3-category]], except that it has two different kinds of 2-cells (in this case, the vertical and horizontal transformations).

In any of these higher categories, we can apply the notions and methods of higher category theory.  For instance, we automatically obtain a notion of [[monoidal double category]], namely a [[pseudomonoid]] in $DblCat$.  Likewise we have *cartesian double categories*, which are [[cartesian objects]] in $DblCat$.  If the double category in question is [[fibrant double category|fibrant]], then monoidal structures can be lifted to the horizontal bicategory; many naturally occurring [[cartesian bicategories]] can be obtained in this way.

Finally, if we want to discuss [[weighted limits]] and colimits in double categories, or construct [[generalized multicategories]] based on double categories, we may want to [[2-category equipped with proarrows|equip DblCat with proarrows]].  There is a notion of [[double profunctor]], but they do not compose associatively, and hence do not form a proarrow equipment in the usual sense; but they do form a [[virtual equipment]], which is sufficient for many purposes.

## Model structures on the category of double categories

The category of double categories admits a plethora of [[model category|Quillen model structures]], some of which are described in papers by [Fiore, Paoli, and Pronk](#FPPModel), [Fiore and Paoli](#FPThomason), [Moser, Sarazola, and Verdugo I](#MSV2Cat) and [Moser, Sarazola, and Verdugo II](#MSVHInvar) and [Campbell](#CampbellGreg).

For more see *[[model structure on DblCat]]*.

### Gregarious model structure

The __gregarious model structure__ on the category of double categories is defined uniquely by the following two properties:

* Every double category is fibrant.
* A double functor is a trivial fibration if it is surjective on objects, full on
horizontal morphisms, full on vertical morphisms, and fully faithful on double cells.

Weak equivalences in this model structure can be characterized as follows.
A double functor $F\colon A\to B$ is a weak equivalence in the gregarious model
structure for double categories if and only if it satisfies the following four conditions.

* $F$ is surjective on objects up to gregarious equivalence, meaning that for every object $b\in B$ there is an object $a\in A$ and a gregarious equivalence $Fa\to b$ in $B$;
* $F$ is full on horizontal morphisms up to globular isomorphism;
* $F$ is full on vertical morphisms up to globular isomorphism;
* $F$ is fully faithful on double cells.

Here a _globular isomorphism_ is an invertible 2-cell in the horizontal respectively vertical [[2-category]] associated to the given double category $A$.

A __gregarious equivalence__ is a [[companion pair]] $(f,u)$ of morphisms in $A$
such that $f$ is an equivalence in the horizontal [[2-category]] of $A$
and $u$ is an equivalence in the vertical [[2-category]] of $A$.

## Related pages

* [[2-category equipped with proarrows]], [[framed bicategory]]

* [[double functor]], [[double pseudofunctor]]

* [[vertical transformation]], [[horizontal transformation]]

* [[double profunctor]]

* [[double limit]]

* [[monoidal double category]], [[locally double bicategory]]

* [[triple category]], [[intercategory]], and [[multiple category]]

* [[applications of double category theory]]

* [[right-connected double category]]

* [[double fibration]]

## References ##

The concept goes back to:

* [[Charles Ehresmann]], *Catégories double et catégories structurées*, C.R. Acad. Paris 256 (1963) 1198-1201 [[[Ehresmann-CategoriesDoubles.pdf:file]], [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246)]

* {#Ehresmann63} [[Charles Ehresmann]], Déf. 10, p. 389 in: *Catégories structurées*, Annales scientifiques de l'École Normale Supérieure. Vol. 80. No. 4. Elsevier, 1963 [[eudml:81794](https://eudml.org/doc/urn:eudml:doc:81794)]

For a comprehensive treatment, it is probably best to consult the series of papers by Grandis and Paré:

* Marco Grandis and Robert Paré, [Limits in double categories](http://www.numdam.org/article/CTGDC_1999__40_3_162_0.pdf), _Cahiers de Topologie et Géométrie Différentielle Catégoriques_ **40** (1999), 162--220.

* Marco Grandis and Robert Paré, [Adjoints for double categories](http://www.numdam.org/article/CTGDC_2004__45_3_193_0.pdf), _Cahiers de Topologie et Géométrie Différentielle Catégoriques_ **45** (2004), 193--240.

* Marco Grandis and Robert Paré, [Kan extensions in double categories (On weak double categories, Part III)](http://www.tac.mta.ca/tac/volumes/20/8/20-08abs.html), Theory and Applications of Categories, Vol. 20, 2008, No. 8, pp 152-185.

* Marco Grandis and Robert Paré, [Lax Kan extensions for double categories (On weak double categories, Part IV)](http://www.numdam.org/article/CTGDC_2007__48_3_163_0.pdf) Cahiers de topologie et géométrie différentielle catégoriques, tome 48, no 3 (2007), p. 163-199

For basic theory of the (virtual) double category of presheaves on a double category, the Yoneda embedding, and the density theorem

* Robert Paré [Yoneda theory for double categories](http://www.tac.mta.ca/tac/volumes/25/17/25-17abs.html), Theory and Applications of Categories, Vol. 25, 2011, No. 17, pp 436-489.

* Robert Paré [Composition of Modules for Lax Functors](http://www.tac.mta.ca/tac/volumes/27/16/27-16abs.html), Theory and Applications of Categories, Vol. 27, 2013, No. 16, pp 393-444.

Textbook accounts:

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 12.3 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

* {#Grandis19} [[Marco Grandis]], _Higher dimensional categories: from double to multiple categories_, World Scientific, 2019. ([doi:10.1142/11406](https://doi.org/10.1142/11406))

See also:

* [[The Catsters]], Double Categories ([YouTube](http://www.youtube.com/watch?v=kiCZiSA2W3Q&feature=channel_page)).

* Tom Fiore, [Double categories and pseudo algebras](http://www-personal.umd.umich.edu/~tmfiore/1/CT2006Fioreslides.pdf).

* Jeffrey C. Morton, [Double bicategories and double cospans](http://arxiv.org/abs/math/0611930).

* [[Ronnie Brown]] and C.B. Spencer, [Double groupoids and crossed modules](http://www.numdam.org/item/CTGDC_1976__17_4_343_0), _Cahiers de Topologie et Géométrie Différentielle Catégoriques_ **17** (1976), 343--362.

* [[Ronnie Brown]]  and [[Philip Higgins]], [The equivalence of $\infty$-groupoids and crossed  complexes,](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1981__22_4/CTGDC_1981__22_4_371_0/CTGDC_1981__22_4_371_0.pdf)  _Cah. Top. Géom. Diff_. 22 (1981) 371-386.

* [[Mike Shulman]], [Comparing composites of left and right derived functors](http://nyjm.albany.edu/j/2011/17-5v.pdf), _New York J. Math._ **17** (2011) 75--125. [arXiv:0706.2868](http://arxiv.org/abs/0706.2868)

* {#Myers16} [[David Jaz Myers]], _String diagrams for double categories and (virtual) equipments_ ([arXiv:1612.02762](https://arxiv.org/abs/1612.02762))

* Kenny Courser, _Open systems: a double categorical perspective_, PhD thesis, [arxiv/2008.02394](https://arxiv.org/abs/2008.02394)

* Antonin Delpeuch, _The word problem for double categories_, [arxiv](https://arxiv.org/abs/1907.09927)

* Virtual Double Categories Workshop 2022 ([webpage](https://bryceclarke.github.io/virtual-double-categories-workshop/)).

On [[model structures on DblCat]]:

* {#FPPModel} [[Thomas M. Fiore]], [[Simona Paoli]], [[Dorette A. Pronk]], *Model structures on the category of small double categories*, _Algebraic & Geometric Topology_ **8** (2008) 1855–1959 [[doi:10.2140/agt.2008.8.1855](https://doi.org/10.2140/agt.2008.8.1855)]

* {#FPThomason} [[Thomas Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*_Algebraic & Geometric Topology_ **10** (2010) 1933–2008 [[doi:10.2140/agt.2010.10.1933](http://doi.org/10.2140/agt.2010.10.1933)]

* {#MSV2Cat} [[Lyne Moser]], Maru Sarazola, Paula Verdugo, *A $2Cat$-inspired model structure for double categories*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **LXIII**2 (2022) [[arxiv:2004.14233](https://arxiv.org/abs/2004.14233), journal:[pdf](http://cahierstgdc.com/wp-content/uploads/2022/04/MoserSarazolaVerdugo-LXIII-2.pdf)]

* {#CampbellGreg} [[Alexander Campbell]], *The gregarious model structure for double categories* (2020) [[talk slides pdf](https://acmbl.github.io/greg_slides.pdf)]

* {#MSVHInvar} [[Lyne Moser]], Maru Sarazola, Paula Verdugo,  *A model structure for weakly horizontally invariant double categories*, Algebraic & Geometric Topology **23** 4 (2023) 1725-1786 [[arxiv:2007.00588](https://arxiv.org/abs/2007.00588), journal:[pdf](https://msp.org/agt/2023/23-4/agt-v23-n4-p08-p.pdf)]

An explicit description of what it means to be a one-object double category (i.e. filling in the gap in "a delooping of a ___ is a double category") is given in [this Math.StackExchange answer](https://math.stackexchange.com/a/3790232).

A [[symmetric multicategory]] structure on the [[category]] of pseudo double categories is described in the appendix of:

* {#Campbell18} [[Alexander Campbell]], _How strict is strictification?_, [arxiv](https://arxiv.org/abs/1802.07538)

For [[univalent double categories]] in [[dependent type theory]]:

* [[Niels van der Weide]], [[Nima Rasekh]], [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent Double Categories* [[arXiv:2310.09220](https://arxiv.org/abs/2310.09220)]

[[!redirects double category]]
[[!redirects double categories]]
[[!redirects double groupoid]]
[[!redirects double groupoids]]
[[!redirects vertical 2-category]]
[[!redirects horizontal 2-category]]
[[!redirects pseudo double category]]
[[!redirects pseudo double categories]]
[[!redirects strict double category]]
[[!redirects strict double categories]]


double category of algebras

# Double category of algebras

* table of contents
{: toc}

## Idea

For a [[2-monad]] $T$, there are several naturally defined [[2-categories]] of $T$-[[algebra over a monad|algebras]], depending on whether we take the [[morphisms]] to be [[lax morphism|lax]], colax (oplax), strong/pseudo, or strict.  Of these, the lax and colax cases $T Alg_l$ and $T Alg_c$ are the most general, since a strict morphism is also strong, and a strong morphism can be regarded as either lax or colax (using the structure morphisms or their inverses).

There is no natural 2-category that contains *both* lax and colax morphisms, since a lax morphism cannot be composed with a colax morphism and obtain any sort of $T$-morphism.  However, lax and colax $T$-morphisms can be combined as the horizontal and vertical morphisms in a [[double category]].  This double category allows a characterization of strong morphisms as [[companion pairs]], of [[doctrinal adjunctions]] as [[conjoint pairs]], and its 2-cells are relevant for defining [[oplax/lax comma objects]].

## Definition

Let $T$ be a [[2-monad]] on a [[2-category]] $K$.  The objects of the double category $T \mathbf{Alg}$ are the $T$-algebras, the horizontal morphisms are the [[lax morphism|lax]] $T$-morphisms, and the vertical morphisms are the colax $T$-morphisms.  The 2-cells are 2-cells in $K$ between composites of underlying morphisms, such that a certain cube of structure 2-cells commutes: consider a square

\begin{tikzcd}
A \ar[r, "f"]\ar[d,"h"'] & B\ar[d, "g"] \\
C \ar[r,"k"']& D
\end{tikzcd}

where horizontal arrows are lax $T$-morphism, and vertical ones are colax. A 2-cell $\alpha : g f \Rightarrow k h$ fills the square and it is such that the equality of pasting diagrams

\begin{tikzcd}[row sep=5mm]
& TB \ar[shorten <=12mm,shorten >=12mm, Rightarrow, dddl, "\bar f"]\arrow[rd, "Tg"] \arrow[dd] & & & & TB\ar[shorten <=4mm,shorten >=4mm, Rightarrow, dd, "T\alpha"] \arrow[rd, "Tg"] & \\
TA \arrow[ru, "Tf"] \arrow[dd,"a"'] & & TD\arrow[dd, "d"] & & TA \arrow[rd, "Th"] \arrow[ru, "Tf"] \arrow[dd, "a"'] & & TD \ar[shorten <=12mm,shorten >=12mm, Rightarrow, dddl, "\bar k"] \arrow[dd,"d"] \\
& B\ar[shorten <=4mm,shorten >=4mm, Rightarrow, ur, "\tilde g"] \ar[shorten <=4mm,shorten >=4mm, Rightarrow, dd, "\alpha"]\arrow[rd,"g"] & & = & & TC \arrow[ru, "Tk"] \arrow[dd] & \\
A \arrow[ru, "f"] \arrow[rd, "h"'] & & D & & A \arrow[rd, "h"'] \ar[shorten <=4mm,shorten >=4mm, Rightarrow, ur, "\tilde h"] && D \\
 & C \arrow[ru,"k"'] & & & & C \arrow[ru,"k"'] &
\end{tikzcd}

holds. This means that a certain diagram of 2-cells, that can be obtained translating the above equality into a commutative hexagon, is commutative.

For a concrete example in the case of monoidal categories, take [[lax monoidal functors]] $F:A\to B$ and $G:C\to D$ and colax monoidal functors $H:B\to D$ and $K:A\to C$, this cube becomes a commutative hexagon:

$$\array{
H(F x \otimes F y) & \to & H F (x\otimes y) & \to & G K (x\otimes y) \\
\downarrow &&&& \downarrow \\
H F x \otimes H F x & \to & G K x \otimes G K x & \to & G (K x \otimes K y).
}$$

If $K$ is a [[strict 2-category]], then $T \mathbf{Alg}$ is a strict double category; otherwise it is "pseudo in both directions" in some sense (such double categories can be defined, but are tricky; see [[double category]]).  As the objects we can use either strict $T$-algebras (if $T$ is a strict 2-monad) or pseudo algebras (if $T$ is either a strict or a pseudo 2-monad).

## Properties

* The horizontal 2-category of $T \mathbf{Alg}$ is the 2-category $T Alg_l$ of lax $T$-morphisms, and its vertical 2-category is $T Alg_c$.

* There is a forgetful double functor from $T \mathbf{Alg}$ to the double category $\mathbf{Q}(K)$ of [[quintets]] in $K$.  In fact, $T \mathbf{Alg}$ can be enhanced to a [[triple category]] whose "transversal morphisms" are *strict* $T$-morphisms (or pseudo ones, if $T$ is only pseudo, in which case we have to deal with "triply pseudo triple categories"), and the induced forgetful functor from this triple category to the triple category of "quintets and commutative cubes" in $K$ is [[monadic functor|monadic]] in an appropriate sense.

* A [[companion pair]] in $T \mathbf{Alg}$ consists of two isomorphic morphisms in $K$ between a pair of $T$-algebras along with a pseudo $T$-morphism structure on one (hence both).  The category of companion pairs between two $T$-algebras is thus equivalent (though not isomorphic) to the category of pseudo $T$-morphisms.

* A [[conjoint pair]] in $T \mathbf{Alg}$ is precisely a "colax/lax" or [[doctrinal adjunction]], and the theorems about [[doctrinal adjunction]] can naturally be expressed as lifting properties of the forgetful functor from $T \mathbf{Alg}$ to $\mathbf{Q}(K)$.

* The universal 2-cell of an [[oplax/lax comma object]] has the structure of a square in $T \mathbf{Alg}$.  Its universal property ought to be related to $T \mathbf{Alg}$ as well, but this is still unclear.

[[!redirects double category of algebras]]
[[!redirects double category of T-algebras]]
[[!redirects double category of algebras over a 2-monad]]
[[!redirects double category of algebras for a 2-monad]]

[[!redirects double categories of algebras]]
[[!redirects double categories of T-algebras]]
[[!redirects double categories of algebras over a 2-monad]]
[[!redirects double categories of algebras for a 2-monad]]


double category of model categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #DoubleCategoryOfModelCategories}
###### Definition
**([[double category]] of [[model categories]])**

The ([[very large category|very large]]) _[[double category]] of [[model categories]]_ $ModCat_{dbl}$ has

1. as [[objects]]: [[model categories]] $\mathcal{C}$;

1. as [[vertical morphisms]]: [[left Quillen functors]] $\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E}$;

1. as [[horizontal morphisms]]: [[right Quillen functors]] $\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D}$;

1. as [[2-morphisms]] [[natural transformations]] between the [[composition|composites]] of [[underlying]] [[functors]].

   $$
     L_2\circ R_1
       \overset{\phi}{\Rightarrow}
     R_2\circ L_1
     \phantom{AAAAA}
     \array{
       \mathcal{C}
         &\overset{\phantom{AA}R_1\phantom{AA}}{\longrightarrow}&
       \mathcal{D}
       \\
       {}^{\mathllap{L_1}}\Big\downarrow
       &{}^{\mathllap{ \phi }}\swArrow&
       \Big\downarrow{}^{\mathrlap{L_2}}
       \\
       \mathcal{C}
         &\underset{\phantom{AA}R_2\phantom{AA}}{\longrightarrow}&
       \mathcal{D}
     }
   $$

and [[composition]] is given by ordinary [[composition]] of [[functors]], horizontally and vertically, and by [[whiskering]]-composition of [[natural transformations]].

=--

([Shulman 07, Example 4.6](#Shulman07))

There is hence a [[forgetful functor|forgetful]] [[double functor]]

$$
  F \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)
$$

to the [[double category of squares]] in the [[2-category of categories]], which forgets the [[model category]]-[[structure]] and the [[Quillen functor]]-[[property]].

There is also another [[double pseudofunctor]] to $Sq(Cat)$ of interest, this is Prop. \ref{HomotopyDoublePseudofunctor} below.

## Properties

+-- {: .num_prop #HomotopyDoublePseudofunctor}
###### Proposition
**(homotopy [[double pseudofunctor]] on the [[double category of model categories]])**

There is a [[double pseudofunctor]]

$$
  Ho(-) \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)
$$

from the double category of model categories (Def. \ref{DoubleCategoryOfModelCategories}) to the [[double category of squares]] in the [[2-category]] [[Cat]], which sends

1. a [[model category]] $\mathcal{C}$ to its [[homotopy category of a model category]];

1. a [[left Quillen functor]] to its [[left derived functor]];

1. a [[right Quillen functor]] to its [[right derived functor]];

1. a [[natural transformation]]

   $$
     \array{
       \mathcal{C}
       &\overset{R_1}{\longrightarrow}&
       \mathcal{D}
       \\
       {}^{\mathllap{L_1}}\Big\downarrow
         &{}^{\mathllap{ \phi }}\swArrow&
       \Big\downarrow{}^{\mathrlap{L_2}}
       \\
       \mathcal{E}
       &\underset{R_2}{\longrightarrow}&
       \mathcal{F}
     }
   $$

   to the "[[derived natural transformation]]"

   $$
     \array{
       Ho(\mathcal{C})
       &\overset{\mathbb{R}R_1}{\longrightarrow}&
       Ho(\mathcal{D})
       \\
       {}^{\mathllap{\mathbb{L}L_1}}\Big\downarrow
         &\overset{Ho(\phi)}{\swArrow}&
       \Big\downarrow{}^{\mathrlap{\mathbb{L}L_2}}
       \\
       Ho(\mathcal{E})
       &\underset{\mathbb{R}R_2}{\longrightarrow}&
       Ho(\mathcal{F})
     }
   $$

   given by the [[zig-zag]]

   \[
     \label{DerivedNaturalTransformation}
     Ho(\phi)
     \;\colon\;
     L_2 Q R_1 P
     \overset{}{\longleftarrow}
     L_2 Q R_1 Q P
     \longrightarrow
     L_2 R_1 Q P
     \overset{\phi}{\longrightarrow}
     R_2 L_1 Q P
     \longrightarrow
     R_2 P L1 Q P
     \longleftarrow
     R_2 R L_1 Q
     \,,
   \]

    where the unlabeled morphisms are induced by [[fibrant resolution]] $c \to P c$ and [[cofibrant resolution]] $Q c \to c$, respectively.

=--

([Shulman 07, Theorem 7.6](#Shulman07))

+-- {: .num_prop #DerivedNaturalTransformationUpToIsos}
###### Proposition
**(recognizing derived natural isomorphisms)**

For the [[derived natural transformation]] $Ho(\phi)$ in (eq:DerivedNaturalTransformation) to be invertible in the [[homotopy category of a model category|homotopy category]], it is sufficient that for every [[object]] $c \in \mathcal{C}$ which is both [[fibrant object|fibrant]] and [[cofibrant object|cofibrant]] the following [[natural transformation]]

$$
  R_2 Q L_1 c
    \overset{ R_2 p_{L_1 c} }{\longrightarrow}
  R_2 L_1 c
   \overset{\phi}{\longrightarrow}
  L_2 R_1 c
    \overset{ L_2 j_{R_1 c} }{\longrightarrow}
  L_2 P R_1 c
$$

is invertible in the homotopy category, hence that the composite is a [[weak equivalences]] (by [this Prop.](geometry+of+physics+--+categories+and+toposes#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen)).

=--

([Shulman 07, Remark 7.2](#Shulman07))

## Examples

+-- {: .num_example #DerivedFunctorOfLeftRightQuillenFunctor}
###### Example
**([[derived functor]] of left-right [[Quillen functor]])**

Let $\mathcal{C}$, $\mathcal{D}$ be [[model categories]], and let

$$
  \mathcal{C} \overset{\phantom{A}F\phantom{A}}{\longrightarrow} \mathcal{C}
$$

be a functor that is both a [[left Quillen functor]] as well as a [[right Quillen functor]]. This means equivalently that there is a [[2-morphism]] in the [[double category of model categories]] (Def. \ref{DoubleCategoryOfModelCategories}) of the form

\[
  \label{DoubleMorphismExhibitingLeftRightQuillenFunctor}
  \array{
    \mathcal{C}
      &\overset{\phantom{AA}F\phantom{AA}}{\longrightarrow}&
    \mathcal{D}
    \\
    {}^{\mathllap{F}}\Big\downarrow
      &{}^{id}\swArrow&
    \Big\downarrow{}^{\mathrlap{id}}
    \\
    \mathcal{D}
      &\underset{\phantom{A}id\phantom{A}}{\longrightarrow}&
    \mathcal{D}
  }
\]

It follows that the [[left derived functor]] $\mathbb{L}F$ and [[right derived functor]] $\mathbb{R}F$ of $F$ are [[natural isomorphism|naturally isomorphic]]:

$$
  Ho(\mathcal{C})
    \overset{ \mathbb{L}F \simeq \mathbb{R}F }{\longrightarrow}
  Ho(\mathcal{D})
  \,.
$$

=--

([Shulman 07, corollary 7.8](#Shulman07))

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

To see the [[natural isomorphism]] $\mathbb{L}F \simeq \mathbb{R}F$: By Prop. \ref{HomotopyDoublePseudofunctor} this is implied once the [[derived natural transformation]] $Ho(id)$ of (eq:DoubleMorphismExhibitingLeftRightQuillenFunctor) is a [[natural isomorphism]]. By Prop. \ref{DerivedNaturalTransformationUpToIsos} this is the case, in the present situation, if the composition of

$$
  Q F c
    \overset{ p_{F c} }{\longrightarrow}
  F c
    \overset{  j_{F c} }{\longrightarrow}
  P F c
$$

is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of [[fibrant resolution|fibrant/cofibrant resolution]].

=--

## Related concepts

* [[2-category of model categories]]

* [[2Ho(CombModCat)]]

## References

* {#Shulman07} [[Michael Shulman]], _Comparing composites of left and right derived functors_, New York Journal of Mathematics 17 (2011), 75-125 ([arXiv:0706.2868](https://arxiv.org/abs/0706.2868))

[[!redirects double categories of model categories]]

[[!redirects derived natural transformation]]
[[!redirects derived natural transformations]]


double category of squares > history

> see [[quintet construction]]


double Chu construction

# Double Chu and 2-Chu constructions

## Definition

The [[Chu construction]] can be expressed as a [[functor]] from the category of co-
subunary (symmetric) [[polycategories]] to the category of symmetric polycategories, which is right adjoint to the forgetful functor.  Since it is a right adjoint, it preserves limits, and therefore also [[internal categories]].

Any [[2-polycategory]] $C$ has a double polycategory $\mathbb{Q}(C)$ of [[quintets]], in which the horizontal arrows are those of $C$ and the vertical arrows are the unary co-unary arrows in $C$, with the 2-cells filled by "square" 2-cells in $C$.  (Of course, an ordinary polycategory can be regarded as a locally discrete 2-polycategory, and in this case the 2-cells in $\mathbb{Q}(C)$ are *commutative* squares in $C$.)  If $C$ is a co-subunary 2-polycategory, then $Chu(\mathbb{Q}(C))$ is a [[double polycategory]] called its **double Chu construction** $\mathbb{C}hu(C)$.

Discarding the nonidentity vertical arrows in $\mathbb{C}hu(C)$ yields a 2-polycategory called the **2-Chu construction**.

## Related pages

* [[double gluing]]

## References

* [[Mike Shulman]], The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions. [arxiv:1806.06082](https://arxiv.org/abs/1806.06082)

[[!redirects 2-Chu construction]]


double comma object

# Double comma objects
* table of contents
{: toc}

##Definition#

The **double comma object** of three morphisms $f:A\to D$, $g:B\to D$, and $h:C\to D$ in a [[2-category]] can be defined as
$$(f/g/h) = (f/g)\times_B (g/h)$$
where $(f/g)$ and $(g/h)$ are the ordinary [[comma object]]s.  It can also be characterized as a [[2-limit]] in its own right.

## Examples

### in Cat

A double comma category is among other things the strict [[pullback]]

$$
  \array{
     (f/g/h) &\to& [I^{\vee 2}, D]
     \\
     \downarrow && \downarrow^{d_0 \times d_1 \times d_2}
     \\
     A \times B \times C &\stackrel{f\times g \times h}{\to}&
     D \times D \times D
  }
  \,,
$$

where $I^{\vee 2} = \{a \to b \to c\}$ is the category freely generated from a [[composable pair]] of morphisms (the linear [[quiver]] of length 2), obtained from the standard [[interval object]] in [[Cat]] by gluing it to itself. $[I^{\vee 2],D]$ is the [[functor category]], i.e. the category of composable pairs of morphisms in $D$.

* If $A=C=1$ are the [[terminal category]] in [[Cat]] and $g$ is the identity functor, then $f=x$ and $h=y$ are objects of $D$ and $(f/g/h) = (x/D/y)$ is sometimes called the **[[over category|over]]-[[under category|under]]-category**.

* If $f,g,h$ are all the identity functor of $A$, then $(f/g/h)$ is the [[power]] $A^{(\to\to)}$, the "object of composable pairs in $A$."

[[!redirects double comma objects]]
[[!redirects double comma category]]
[[!redirects double comma categories]]


double complex

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

#Contents#
* table of contents
{:toc}

## Idea

A **double complex** or **bicomplex** is a [[diagram]] of shape $\mathbb{Z}_{\leq} \times \mathbb{Z}_{\leq}$ (in some [[additive category]]):

$$
  \array{
   && \vdots && \vdots
   \\
   & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}}
   \\
    \cdots &\to &
    X_{n,m} &\stackrel{\partial^h}{\to}& X_{n-1,m}
    & \to & \cdots
   \\
   & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}}
   \\
    \cdots &\to &
    X_{n,m-1} &\stackrel{\partial^h}{\to}& X_{n-1,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{\mathrlap{\partial^v}} && \downarrow^{\mathrlap{\partial^v}}
   \\
   && \vdots && \vdots
  }
$$

such that each row and each column is a [[complex]] (the [[differentials]] square to 0: $\partial^v \circ \partial^v = 0$ and $\partial^h \circ \partial^h = 0$) and such that all the squares [[commuting diagram| commute]].

This means that a **double complex** is a [[complex]] in a category of [[complexes]].

Accordingly, a **double chain complex** is a [[chain complex]] in a [[category of chain complexes]].

$$
  X_{\bullet, \bullet} =
  \left[
      \cdots \to X_{n,\bullet} \stackrel{\partial^h}{\to} X_{n-1,\bullet} \to \cdots
  \right]
  \,.
$$

In the presence of [[direct sums]], there is a _[[total complex]]_ $(Tot X)_\bullet$ associated to a double complex, which in degree $n$ is the direct sum of all terms of _total degree_ $n$

$$
  (Tot X)_n \coloneqq \oplus_{k+l = n} X_{k,l}.
$$

Often it is such total complexes that are of interest.

The [[differential]] of the total complex is the sum of the horizontal and the vertical differential _made anti-commutative by adjusting signs_. There is a second convention in which one often sees the double complex defined as above from a complex of complexes, but then with the differentials in every second row (or every second column) multiplied by $(-1)$. This is just a different way of sign-bookkeeping, a detailed discussion of this is below. Which convention to use is sometimes influenced by the context, the traditions of the sources in that application of double complexes, and largely a question of taste, that is which one the writer is used to.

Double chain complexes often arise from the application of bifunctors -- [[additive functors]] of two variables -- of [[additive category|additive categories]] $C_1, C_2, C_3$

$$
  F :  C_1 \times C_2 \to C_3
$$

to complexes in their two arguments. Combining this with the formation of [[total complexes]] then yields bifunctors from  categories of complexes to categories of complexes.

$$
 \tilde F : Ch(C_1) \times Ch(C_2) \to Ch(C_3)
 \,.
$$

The most important examples of this are induced by the [[hom-functor]] and the [[tensor product]] functor together with their [[derived functors]], [[Ext]] and [[Tor]].

Notice that under the [[Dold-Kan correspondence]] and with sufficient [[resolutions]], such $\tilde F$ can be understood as the internal hom or tensor products, etc., between [[infinity-groupoid|higher groupoids]].

Although we suggest (and prefer) the 'complex of complexes' definition, as above, rather than the equivalent  _anticommutiing diagram_ one, we give both and will discuss how to go between them in some detail.

## Definition

### With commuting differentials

A double complex, $X$, is a commutative diagram in an additive category in which  the objects are bi-indexed by the integers,

$$\{ X_{p,q} \mid p,q\in \mathbb{Z} \}$$

and with two classes of '[[differentials]]' or 'boundary morphisms':

* $d_X^v: X_{p,q}\to X_{p,q-1}$ called the 'vertical boundary morphisms' or 'vertical differentials', with $d_X^v d_X^v = 0$;

and

* $d_X^h: X_{p,q}\to X_{p-1,q}$ called the 'horizontal boundary morphisms' or 'horizontal differentials', with  $d_X^h d_X^h = 0$;

such that all [[commuting diagram|squares commute]]

* $d_X^h \circ d_X^v = d_X^v \circ d_X^h$.

(To state the obvious, this means $d_X^h d_X^v - d_X^v d_X^h=0$, in contrast to the formula in the second version [below](#AntiCommutingDifferentials).)

This gives a commutative diagram:

$$
  \array{
   && \vdots && \vdots
   \\
   & & \downarrow^{d_X^v} && \downarrow^{d_X^v}
   \\
    \cdots &\to &
    X_{n,m} &\stackrel{d_X^h}{\to}& X_{n,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{d_X^v} && \downarrow^{d_X^v}
   \\
    \cdots &\to &
    X_{n-1,m} &\stackrel{d_X^h}{\to}& X_{n-1,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{d_X^v} && \downarrow^{d_X^v}
   \\
   && \vdots && \vdots
  }
$$

### With anti-commuting differentials
 {#AntiCommutingDifferentials}

A double complex, $X$, is an anticommutative diagram in an additive category in which the objects are bi-indexed by the integers,

$$\{ X_{p,q} \mid p,q\in \mathbb{Z} \}$$

and with two classes of 'differentials' or 'boundary morphisms':

* $d_X^v: X_{p,q}\to X_{p,q-1}$ called the 'vertical boundary morphisms' or 'vertical differentials', with $d_X^v d_X^v = 0$;

and

* $\bar{d}_X^h: X_{p,q}\to X_{p-1,q}$ called the 'horizontal boundary morphisms' or 'horizontal differentials', with  $\bar{d}_X^h \bar{d}_X^h = 0$;

in addition $\bar{d}_X^h d_X^v + d_X^v \bar{d}_X^h = 0$.

This gives an anticommutative diagram:
$$
  \array{
   && \vdots && \vdots
   \\
   & & \downarrow^{d_X^v} && \downarrow^{d_X^v}
   \\
    \cdots &\to &
    X_{n,m} &\stackrel{\bar{d}_X^h}{\to}& X_{n,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{d_X^v} & \swArr_{-1} & \downarrow^{d_X^v}
   \\
    \cdots &\to &
    X_{n-1,m} &\stackrel{\bar{d}_X^h}{\to}& X_{n-1,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{d_X^v} && \downarrow^{d_X^v}
   \\
   && \vdots && \vdots
  }
$$

### Equivalence of the two definitions
 {#EquivalenceOfTheTwoDefinitions}

Which definition is 'better'? 'Commuting squares', i.e., the first version, is convenient if you want to define a double complex as a chain complex in the category of chain complexes. On the other hand, 'anticommuting squares' and version 2 is sometimes convenient for defining the total complex (for computing total homology). Does it matter which you use? The following says they are just two views of the same situation.

One makes a double complex $X$ with commutative squares into a double complex with anticommutative squares by using the same vertical differential $d^v$ but taking $\bar{d}^h : X_{p,q} \to X_{p,q-1}$ to be $(-1)^p d^h$. The same trick can, of course, be used to make a double complex with anticommutative squares into a double complex with commutative squares.

## Properties

### Total complex of a double complex

The **[[total complex]]** of a double complex (under the convention that squares commute) is

$$
  tot_{\oplus}^k = \bigoplus_{m+n=k} X_{n,m}
$$

$$
  d^k_{tot_\oplus}|_{X_{n,m}} =
  d^v_X + (-1)^\bullet d_X^h
$$

Similarly, one can define the product total complex as

$$
  tot_{\prod}^k = \prod_{m+n=k} X_{n,m}
$$
$$
  d^k_{tot_\prod}|_{X_{n,m}} =
  d^v_X + (-1)^\bullet d_X^h
$$

Note that these two coincide when the set of non-zero objects $X_{n,m}$ such that $n + m = k$ is finite, for example, when $X$ is a first quadrant double complex.

### Fundamental lemmas

There is series of basic [[lemmas]] in [[homological algebra]]
which determine the horizontal/vertical [[homology groups]] of a double complex in some row or column from exactness information in other columns.
The most fundamental of these is maybe the

* [[salamander lemma]]

from which a series of others follow:

* [[snake lemma]] (see also [[connecting homomorphism]] and [[long exact sequence in homology]])

* [[3x3 lemma]]

* [[5-lemma]]

* [[horseshoe lemma]] .

## Relation to homotopy colimits

The total complex of the double complex induced by a [[chain map]]
is a model for the [[mapping cone]] of that map, see at _[mapping cone -- via double complexes](mapping+cone#ConeViaDoubleComplex) for more.

[[!redirects double complexes]]

[[!redirects bicomplex]]
[[!redirects bicomplexes]]
[[!redirects bi-complex]]
[[!redirects bi-complexes]]

[[!redirects double chain complex]]
[[!redirects double chain complexes]]


double coset

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## Idea

Given a [[group]] $G$ and two [[subgroups]] $H$ and $K$, then the set of **double cosets**, $H \backslash G/K$, is composed of [[equivalence classes]] of elements of $G$, where two elements are regarded as equivalent if they differ by left multiplication with an element in $H$ and right multiplication with an element of $K$.

This construction generalises to [[topological groups]] and [[Lie groups]], where it is called a **double coset space**. The double coset space for $G$ a [[compact Lie group]] and $H$ and $K$ [[closed subspace|closed]] [[connected topological space|connected]] [[subgroups]], with $H$ [[action|acting]] [[free action|freely]] on the singe [[coset space]] $G/K$, is also known as a _[[biquotient space]]_.

From the [[nPOV]], a weak form of the double coset construction is often more natural, $H \backslash \backslash G//K$, and can be defined as the [[homotopy pullback]] of maps of [[looping and delooping|delooped]] groups, $\mathbf{B} H \to \mathbf{B} G$ and $\mathbf{B} K \to \mathbf{B} G$. This holds more generally for any three [[∞-groups]], with any homomorphisms $H \to G$ and $K \to G$.

## Properties

### Mackey's formula
 {#MackeyFormula}

> the following needs to state the assumption on the kind of group, e.g. [[finite group]], [[compact Lie groups]], ...

Double coset decompositions are useful in [[representation theory]], for example in [[George Mackey]]'s formula for the [[restricted representation|restriction]] back to $H$ of a module [[induced representation|induced]] from $K$. Let $W$ be a [[linear representation]] of $K$. Then

$$
  Res^G_H Ind^G_K W
   \cong
  \underset{[g] \in H \backslash G/K} \bigoplus Ind^H_{K_g} W_g
  \,,
$$

where $K_g \coloneqq H \intersection g K g^{-1}$, and $W_g$ is the representation of $K_g$ which is $W$ as a vector space, but with action $x \cdot w = (g^{-1} x g) w$ where now $g^{-1} x g \in K$. This formula, sometimes called _Mackey's decomposition theorem_, amounts to the [[Beck-Chevalley condition]] for the homotopy pullback square. It appears, as the _Mackey axiom_, in one of the original definitions of [[Mackey functor]] ([Bouc 02](#Bouc02)).

## Examples

### Gromoll-Meyer sphere

The [[Gromoll-Meyer sphere]] -- an [[exotic 7-sphere]] -- arises as the [[biquotient]] of [[Sp(2)]] by two copies of [[Sp(1)]].

## Related concepts

* [[biquotient space]]

* [[Clifford-Klein form]]

* [[locally Klein geometry]]

* [[Hecke algebra]]

* [[Bruhat decomposition]]

##References

* {#Bouc02} Serge Bouc, _Mackey functors_, ([pdf](http://www.lamfa.u-picardie.fr/bouc/mackey.pdf))

* Burt Totaro, _Cheeger manifolds and the classification of biquotients_ ([arXiv:math/0210247](https://arxiv.org/abs/math/0210247))

[[!redirects double cosets]]

[[!redirects double coset space]]
[[!redirects double coset spaces]]


double cover

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{:toc}

## Definition

A _double cover_ is equivalently

* a $\mathbb{Z}_2$-[[principal bundle]];

* an [[etale space]] with local [[sections]] the 2-element set.

## Examples

### Orientation double cover

For $X$ a [[manifold]], not necessarily [[orientation|oriented]] or even orientable, write

$$
  \array{
    && B O
    \\
    & {}^{\mathllap{\hat T X}}\nearrow & \downarrow
    \\
    X &\stackrel{T X}{\to}& B GL
  }
$$

for any choice of [[orthogonal structure]]. The **orientation double cover** or **orientation bundle** of $X$ is the $\mathbb{Z}_2$-[[principal bundle]] classified by the first [[Stiefel-Whitney class]] (of the [[tangent bundle]]) of $X$

$$
  w_1(\hat T X) :  X \stackrel{\hat T X}{\to} B O \stackrel{w_1}{\to} B \mathbb{Z}_2
  \,.
$$

One may identify this with the bundle that over each [[neighbourhood]] $x \in U \subset X$ of a point $x$ has as fibers the two different choices of [[volume forms]] up to positive rescaling (the two different choices of orientation).

More generally, for $E \to X$ any [[orthogonal group]]-[[principal bundle]] classified by a morphism $E : X \to \mathbf{B} O$, the corresponding orientation double cover is the $\mathbb{Z}_2$-bundle classified by

$$
  w_1(E) : X \stackrel{E}{\to} \mathbf{B} O \stackrel{w_1}{\to} \mathbf{B} \mathbb{Z}_2
  \,.
$$

### Real Hopf fibration

The [[real Hopf fibration]] is the non-trivial double cover of the [[circle]] by itself.

### Spin double cover

* [[spin group]]

## Related concepts

* [[orientifold]], [[Jandl gerbe]]

* [[orientation]]

## References

An exposition in a broader context is in the section _[higher spin structures](#)_ at

* _[[geometry of physics]]_.

[[!redirects double covers]]

[[!redirects double covering]]
[[!redirects double coverings]]

[[!redirects orientation double cover]]
[[!redirects orientation double covers]]

[[!redirects orientation bundle]]
[[!redirects orientation bundles]]


double derivation

## Overview

Given a _commutative_ [[ring]] $k$ and an associative  $k$-algebra $A$ over $k$, the [[tensor product]] $A\otimes_k A$ is equipped with two [[bimodule]] structures, "outer" and "inner". For the outer structure $a\cdot_o(b\otimes c)\cdot_o d = a b\otimes c d$ and for the inner $a\cdot_i(b\otimes c)\cdot_i d = b d\otimes a c$. The two bimodule structures mutually commute. A $k$-linear map $\alpha\in Hom_k(A,A\otimes A)$ is called a __double derivation__ if it is also a map of $A$-bimodules with respect to the _outer_ bimodule structure ($\alpha\in A Mod A({}_A A_A,{}_A A\otimes_k A_A)$); thus the $k$-module $Der(A,A\otimes A)$ of all double derivations becomes an $A$-bimodule with respect to the _inner_ $A$-bimodule structure.

The tensor algebra $T_A Der(A,A\otimes A)$ of the $A$-bimodule $Der(A,A\otimes A)$ (which is the [[free monoid]] on $Der(A,A\otimes A)$ in the monoidal category of $A$-bimodules) is a step in the definition of the deformed [[preprojective algebra]]s of [[Bill Crawley-Boevey]]. A theorem of [[Michel Van den Bergh]] says that for any associative $A$ the tensor algebra $T_A Der(A,A\otimes A)$ has a canonical [[double Poisson bracket]].

## Literature

* [[Michel Van den Bergh]], _Double Poisson algebras_, Trans. Amer. Math. Soc. __360__ (2008) 5711--5769, [arXiv:math.AG/0410528](https://arXiv.org/abs/math/0410528)
* Anne Pichereau, Geert Van de Weyer, _Double Poisson cohomology of path algebras of quivers_, J. Alg. __319__, 5 (2008), 2166--2208 [doi](https://doi.org/10.1016/j.jalgebra.2007.09.021)
* Jorge A. Guccione, Juan J. Guccione, _A characterization of quiver algebras based on double derivations_, [arXiv:0807.1148](https://arxiv.org/abs/0807.1148)
* [[William Crawley-Boevey]], [[Pavel Etingof]], [[Victor Ginzburg]], _Noncommutative geometry and quiver algebras_,  Adv. Math. __209__:1 (2007) 274-336 [doi](https://doi.org/10.1016/j.aim.2006.05.004)
* [[Victor Ginzburg]], [[Travis Schedler]], _Differential operators and BV structures in noncommutative geometry_, Sel. Math. New Ser. 16, 673–730 (2010) [doi](https://doi.org/10.1007/s00029-010-0029-8)
* [[Victor Ginzburg]], [[Travis Schedler]], [[Boris Tsygan]], _Free products, cyclic homology, and the Gauss–Manin connection_, Advances in Mathematics __231__:3-4 (2012) 2352--2389 [doi](https://doi.org/10.1016/j.aim.2012.06.016)

category: algebra