D=6 supergravity

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## Idea

[[supergravity]] in [[dimension]] 6

## Properties

### U-duality

[[!include U-duality -- table]]

## Related concepts

* [[D=6 N=(1,0) SCFT]]

* [[D=6 N=(2,0) SCFT]]

* [[self-dual string in 6d]]

* [[3-brane in 6d]]

## References

### General

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter III.7 in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific, 1991

  (in [[D'Auria-Fré formulation of supergravity]])

* [[Sergio Ferrara]], [[Fabio Riccioni]], [[Augusto Sagnotti]], _Tensor and Vector Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. B519 (1998) 115-140 ([arXiv:hep-th/9711059](https://arxiv.org/abs/hep-th/9711059))

* [[Fabio Riccioni]], [[Augusto Sagnotti]], _Some Properties of Tensor Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. Proc. Suppl. 67 (1998) 68-73 ([arXiv:hep-th/9711077](https://arxiv.org/abs/hep-th/9711077))

* [[Fabio Riccioni]], _All Couplings of Minimal Six-dimensional Supergravity_, Nucl. Phys. B605 (2001) 245-265 ([arXiv:hep-th/0101074](https://arxiv.org/abs/hep-th/0101074))

* Vijay Kumar, Daniel S. Park, [[Washington Taylor]], _6D supergravity without tensor multiplets_, JHEP 1104:080,2011 ([arXiv:1011.0726](http://arxiv.org/abs/1011.0726))

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity*, Journal of Geometry and Physics **158** (2020) [[arXiv:1912.08723](https://arxiv.org/abs/1912.08723), [doi:10.1016/j.geomphys.2020.103868](https://doi.org/10.1016/j.geomphys.2020.103868)]

  > (relation to [[contact geometry]])

* Edoardo Lauria, [[Antoine Van Proeyen]], _$\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions_ ([arXiv:2004.11433](https://arxiv.org/abs/2004.11433))

{#ReferencesFourZero} For "exotic" $\mathcal{N} = (4,0)$-supersymmetry:

* [[Chris Hull]], *Strongly Coupled Gravity and Duality*, Nucl.Phys. B **583** (2000) 237-259 [[arXiv:hep-th/0004195](https://arxiv.org/abs/hep-th/0004195), doi:10.1016/S0550-3213(00)00323-0]

* [[Chris Hull]], *Symmetries and Compactifications of $(4,0)$ Conformal Gravity*, JHEP 0012: 007 (2000) [[arXiv:hep-th/0011215](https://arxiv.org/abs/hep-th/0011215), [doi:10.1088/1126-6708/2000/12/007](https://doi.org/10.1088/1126-6708/2000/12/007)]

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Toward Exotic 6D Supergravities*, Phys. Rev. D **103** (2021) 046002 [[arXiv:2007.11644](https://arxiv.org/abs/2007.11644), [doi:10.1103/PhysRevD.103.046002](https://doi.org/10.1103/PhysRevD.103.046002)]

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Supersymmetric action for 6D $(4,0)$ supergravity*, JHEP (2022) [[arXiv:2206.04100](https://arxiv.org/abs/2206.04100)]

* [[Chris Hull]], *Gravity, Duality and Conformal Symmetry*, in *Quantum gravity, branes and M-theory* [[arXi:2209.11716](https://arxiv.org/abs/2209.11716)]

### Killing spinors

Discussion of [[Killing spinors]]/enhancement of [[number of supersymmetries]] in [[D=6 supergravity]]:

* Mehmet Akyol, [[George Papadopoulos]], _Spinorial geometry and Killing spinor equations of 6-D supergravity_, Class. Quant. Grav. 28:105001, 2011 ([arXiv:1010.2632](https://arxiv.org/abs/1010.2632))

* Mehmet Akyol, [[George Papadopoulos]], _$(1,0)$ superconformal theories in six dimensions and Killing spinor equations_, JHEP 07 (2012) 070 ([arXiv:1204.2167](https://arxiv.org/abs/1204.2167))

### KK-reduction  to JT gravity

Realization of [[JT-gravity]] as [[Kaluza-Klein reduction]] of [[D=6 supergravity]] on the [[worldvolume]] of [[D1-D5 brane bound states]] or [[M2-M5 brane bound states]]:

* Yue-Zhou Li, Shou-Long Li, H. Lu, _Exact Embeddings of JT Gravity in Strings and M-theory_, Eur. Phys. J. C (2018) 78: 791 ([arXiv:1804.09742](https://arxiv.org/abs/1804.09742))

* [[Iosif Bena]], Pierre Heidmann, David Turton, _$AdS_2$ Holography: Mind the Cap_, JHEP 1812 (2018) 028 ([arXiv:1806.02834](https://arxiv.org/abs/1806.02834))

[[!redirects 6d supergravity]]

[[!redirects 6-dimensional supergravity]]


D=7 super Yang-Mills theory

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## Idea

[[super Yang-Mills theory]] in [[dimension]] $7 = 6+1$.

The [[worldvolume]] theory of [[Chan-Paton gauge fields]] on [[D6-branes]].

## References

* Christoph Lüdeling, _Seven-Dimensional Super-Yang-Mills Theory in $N=1$ Superfields_ ([arXiv:1102.0285](http://arxiv.org/abs/1102.0285))

From [[M-theory on G2-manifolds]]:

* {#Acharya02} [[Bobby Acharya]], section 3.2.1 of _M Theory, $G_2$-manifolds and Four Dimensional Physics_,  Classical and Quantum Gravity Volume 19 Number 22, 2002  ([pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))

On 3-[[Sasakian manifolds]]:

* Nikolaos Iakovidis, Jian Qiu, Andreas Rocén, Maxim Zabzine, _7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds_ ([arXiv:2003.12461](https://arxiv.org/abs/2003.12461))

[[!redirects 7d super Yang-Mills theory]]


D=7 supergravity

[[!redirects 7-dimensional supergravity]]

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## Idea

[[Kaluza-Klein mechanism|Reductions]] of [[11-dimensional supergravity]] on a 4-[[dimension]]al space to a 7-dimensional theory.

## Properties

### U-duality

Maximal 7d supergravity has $SL(5,\mathbb{Z})$-[[U-duality]]
([Kumar-Vafa 96, section 3.4](KumarVafa96), [Rozali 97](#Rozali97)).

### Holography

The [[AdS/CFT|AdS7/CFT6-correspondence]] relates
7-dimensional supergravity on asymptocially 7d [[anti de Sitter spacetimes]] to the [[6d (2,0)-superconformal QFT]].

## Related concepts

* [[11-dimensional supergravity]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* **7-dimensional supergravity**

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

## References

### General

The maximally [[number of supersymmetries|N=4]] supersymmetric $SO(5)$-[[gauged supergravity]] in 7-dimensions was first constructed in

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], _Gauged Maximally Extended Supergravity in Seven Dimensions_ , Phys. Lett. 143B (1984) 103 ([spire:15064](https://inspirehep.net/literature/15064), doi:10.1016/0370-2693(84)90813-X)

There are also gaugings for non-compact groups

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], [[Nicholas Warner]], _Noncompact Gauging and Critical Points of Maximal Supergravity in Seven Dimensions_, Nucl. Phys. B249 (1985) 381 (doi:10.1016/0550-3213(85)90046-X>doi:10.1016/0550-3213(85)90046-X)

The [[number of supersymmetries|N=2]] [[Sp(1)]]-gauged theory was described in

* [[Paul Townsend]], [[Peter van Nieuwenhuizen]], _Gauged Seven-Dimensional Supergravity_ , Phys. Lett. 125B (1983) 41 ([spire:189055](http://inspirehep.net/record/189055), doi:10.1016/0370-2693(83)91230-3)

See also

* [[Eric Bergshoeff]], D C Jong and [[Ergin Sezgin]], _Noncompact gaugings, chiral reduction and dual sigma models in supergravity_ Class. Quantum Grav. 23 (2006) 2803–2831 ([arXiv:hep-th/0509203](https://arxiv.org/abs/hep-th/0509203), [pdf](http://iopscience.iop.org/0264-9381/23/9/003/pdf/0264-9381_23_9_003.pdf))

* Der-Chyn Jong, _Aspects of 7D and 6D gauged supergravities_ PhD thesis (2007) ([spire:776484](https://inspirehep.net/literature/776484), [web](https://oaktrust.library.tamu.edu/handle/1969.1/ETD-TAMU-2110))

Discussion of [[AdS spacetime|AdS vacua]]:

* Nikolay Bobev, P. Marcos Crichigno, _Universal RG Flows Across Dimensions and Holography_, J. High Energ. Phys. 2017, 65 (2017) ([arXiv:1708.05052](https://arxiv.org/abs/1708.05052))

* Parinya Karndumri, Patharadanai Nuchino, _Twisted compactifications of 6D field theories from maximal 7D gauged supergravity_, Eur. Phys. J. C 80, 201 (2020) ([arXiv:1912.04807](https://arxiv.org/abs/1912.04807))

### Heterotic 7d supergravity

* [[Paul Townsend]], _String-Membrane Duality in Seven Dimensions_,  	Phys. Lett. B354:247-255, 1995 ([arXiv:hep-th/9504095](https://arxiv.org/abs/hep-th/9504095))

* {#GherghettaKehagias02} Tony Gherghetta, [[Alex Kehagias]], _Anomaly Cancellation in Seven-Dimensional Supergravity with a Boundary_, Phys.Rev. **D68** (2003), 065019, ([arXiv:hep-th/0212060](http://arxiv.org/abs/hep-th/0212060))

* Spyros D. Avramis, [[Alex Kehagias]], _Gauged $D=7$ Supergravity on the $S^1/\mathbb{Z}_2$ Orbifold ([arXiv:hep-th/0407221](https://arxiv.org/abs/hep-th/0407221))

* T.G. Pugh, [[Ergin Sezgin]], [[Kellogg Stelle]], _$D=7$ / $D=6$ Heterotic Supergravity with Gauged R-Symmetry_ ([arXiv:1008.0726](https://arxiv.org/abs/1008.0726))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _BPS objects in $D=7$ supergravity and their M-theory origin_, JHEP 12 (2017) 041 ([arXiv:1707.06152](https://arxiv.org/abs/1707.06152))

* Bernardo Fraiman, Héctor Parra De Freitas, *Symmetry Enhancements in 7d Heterotic Strings* ([arXiv:2106.08189](https://arxiv.org/abs/2106.08189))

### By reduction from 10d/11d supergravity

The following references discuss construction of full 7d sugra theories or of their solutions by [[Kaluza-Klein mechanism|KK-reduction]] from [[11-dimensional supergravity]] or from 10d [[type II supergravity]].

The construction of 7d supergravity by [[KK-reduction]] from [[11-dimensional supergravity]] was first indicated in

* [[Peter Freund]], M. A. Rubin, _Dynamics of dimensional reduction_  Phys. Lett. 97B (1980) 233.

via what is now called _[[Freund-Rubin compactification]]_.

That their Ansatz is indeed consistent has been shown in

* [[Horatiu Nastase]], Diana Vaman, [[Peter van Nieuwenhuizen]], _Consistent nonlinear KK reduction of 11d supergravity on $AdS_7\times S_4$ and self-duality in odd dimensions_ Phys.Lett. B469 (1999) 96-102 ([arXiv:hep-th/9905075](http://arxiv.org/abs/hep-th/9905075))

* [[Ali Chamseddine]], W.A. Sabra, _$D=7$ $SU(2)$ gauged supergravity from $D=10$ supergravity_ Physics Letters B
Volume 476, Issues 3-4, 16 (2000), Pages 415-419

The $SL(5,\mathbb{Z})$-[[U-duality]] in 7d is discussed in

* {#KumarVafa96} Alok Kumar, [[Cumrun Vafa]], section 3.4 of _U-Manifolds_, Phys.Lett. B396 (1997) 85-90 ([arXiv:hep-th/9611007](http://arxiv.org/abs/hep-th/9611007))

* {#Rozali97} [[Moshe Rozali]], _Matrix Theory and U-Duality in Seven Dimensions_, Phys.Lett. B400 (1997) 260-264 ([arXiv:hep-th/9702136](http://arxiv.org/abs/hep-th/9702136))

Explicit compactifications are discussed in

* [[Peter van Nieuwenhuizen]], N. P. Warner, _New Compactificafions of Ten and Eleven Dimensional Supergravity on Manifolds Which Are Not Direct Products_ Commun. Math. Phys. 99, 141-151 (1985)

* [[Jerome Gauntlett]], Nakwoo Kim, Stathis Pakis, [[Daniel Waldram]], _M-Theory solutions with AdS factors_ ([arXiv:hep-th/0202184](http://arxiv.org/abs/hep-th/0202184))

See also

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _BPS objects in D=7 supergravity and their M-theory origin_, J. High Energ. Phys. (2017) 2017: 41 ([arxiv:1707.06152](https://arxiv.org/abs/1707.06152))

Relation to [[massive type IIA supergravity]] and [[bound states]] of [[D-branes]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_, PoS CORFU2018 (2019) 142 ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455), [doi:10.22323/1.347.0142](https://doi.org/10.22323/1.347.0142))

### D'Auria-Fré formulation of supergravity

Discussion in the [[D'Auria-Fré formulation of supergravity]] is in

* [[Pietro Fré]], [[Pietro Antonio Grassi]], [[Lucrezia Ravera]], M. Trigiante, _Minimal $D=7$ Supergravity and the supersymmetry of Arnold-Beltrami Flux branes_, JHEP 1606 (2016) 018 ([arXiv:1511.06245](https://arxiv.org/abs/1511.06245))

and the 7d analog of the [[DF-algebra]] (the [[Chevalley-Eilenberg algebra]]  of a spinorial extension of the [[M-theory super Lie algebra]]) is due to

* [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_, JHEP 1608 (2016) 095 ([arXiv:1606.07328](https://arxiv.org/abs/1606.07328))

* [[Lucrezia Ravera]], _Group Theoretical Hidden Structure of Supergravity Theories in Higher Dimensions_ ([arXiv:1802.06602](https://arxiv.org/abs/1802.06602))

[[!redirects 7d supergravity]]


D=8 supergravity

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## Idea

[[supergravity]] in [[dimension]] 8

## Properties

### U-duality

The [[U-duality]] group of maximax 8d SuGra is $SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$ (e.g. [Liu-Minasian 97](#LiuMinasian97))

[[!include U-duality -- table]]

## Related concepts

* [[supergravity]]

* [[11-dimensional supergravity]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* **8-dimensional supergravity**

* [[7-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

## References

*  [[Abdus Salam]], Ergin Sezgin, *d=8 supergravity*, Nuclear Physics B **258** (1985) 284-304 [doi:10.1016/0550-3213(85)90613-3]

* M. Awada, [[Paul Townsend]], _d = 8 Maxwell/Einstein supergravity_, Physics Letters B Volume 156, Issues 1–2, 13 June 1985

* [[Mees de Roo]], [[Giuseppe Dibitetto]], Yihao Yin, _Critical points of maximal $D=8$ gauged supergravities_, JHEP 1201:029 (2012) [[arXiv:1110.2886](http://arxiv.org/abs/1110.2886)]

Discussion of [[U-duality]] in 8d:

* {#LiuMinasian97} [[James Liu]], [[Ruben Minasian]], _U-branes and $T^3$ fibrations_, Nucl.Phys. B510 (1998) 538-554 ([arXiv:hep-th/9707125](http://arxiv.org/abs/hep-th/9707125))

Further discussion:

* Bing-Xin Lao, [[Ruben Minasian]], *Consistency of eight-dimensional supergravities: Anomalies, Lattices and Counterterms* [[arXiv:2306.07766](https://arxiv.org/abs/2306.07766)]

[[!redirects 8d supergravity]]
[[!redirects 8-dimensional supergravity]]


Daciberg Lima Gonçalves

* [Webpage](https://www.ime.usp.br/~dlgoncal/)

## Selected writing

On [[group cohomology]] of [[cyclic groups]] with [[coefficients]] in cyclic groups:

* {#GolasińskiGonçalves11} [[Marek Golasiński]], [[Daciberg Lima Gonçalves]], *On cohomologies and extensions of cyclic groups*, Topology and its Applications **158** (2011) 1858–1865 [[doi:10.1016/j.topol.2011.06.022](https://doi.org/10.1016/j.topol.2011.06.022)]

category:people


dagger 2-poset

[[!redirects dagger 2-posets]]
[[!redirects dagger 2-preorder]]
[[!redirects dagger 2-preorders]]
[[!redirects dagger 2-proset]]
[[!redirects dagger 2-prosets]]
[[!redirects univalent dagger 2-poset]]
[[!redirects univalent dagger 2-posets]]

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## Contents ##
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## Definition ##

A dagger 2-poset is a [[dagger category]] $C$ such that

1. for each object $A \in Ob(C)$ and $B \in Ob(C)$ and morphisms $R \in Hom(A, B)$ and $S \in Hom(A, B)$ there is a [[binary relation]] $R \leq_{A, B} S$
2. for each object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $R \in Hom(A, B)$, $R \leq_{A, B} R$.
3. for each object $A \in Ob(C)$ and $B \in Ob(C)$ and morphisms $R \in Hom(A, B)$ and $S \in Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} R$ implies that $R = S$.
4. for each object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $R \in Hom(A, B)$, $S \in Hom(A, B)$, and $T \in Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} T$ implies that $R \leq_{A, B} T$.
5. for each object $A \in Ob(C)$ and $B \in Ob(C)$ and morphisms $R \in Hom(A, B)$ and $S \in Hom(A, B)$, $R \leq_{A, B} S$ implies $R^\dagger \leq_{B, A} S^\dagger$.

A dagger 2-poset which only satisfies 1., 2., 4., and 5. is called a **dagger 2-preorder** or a **dagger 2-proset**.

A dagger 2-poset is a **univalent dagger 2-poset** if the underlying dagger-category is a [[univalent dagger category]].

## Morphisms ##

* [[onto morphism in a dagger 2-poset]]
* [[functional morphism in a dagger 2-poset]]
* [[entire morphism in a dagger 2-poset]]
* [[injective morphism in a dagger 2-poset]]
* [[map in a dagger 2-poset]]
* [[category of maps]]
* [[monic map in a dagger 2-poset]]
* [[epic map in a dagger 2-poset]]
* [[unitary isomorphism]]

## Examples ##

* [[semiadditive dagger 2-poset]]
* [[unital dagger 2-poset]]
* [[tabular dagger 2-poset]]
* [[division dagger 2-poset]]
* [[power dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]
* [[W-topical dagger 2-poset]]
* [[well-pointed topical dagger 2-poset]]
* [[coherent dagger 2-poset]]
* [[geometric dagger 2-poset]]
* [[Heyting dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[Boolean topical dagger 2-poset]]
* [[allegory]]
* [[groupoid]]
* [[ETCR]]

## See also ##

* [[dagger category]]
* [[2-poset]]


dagger category

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# $\dagger$ categories
* tic
{: toc}

## Idea

The definition of a [[category]] effectively enforces an ordering on the "0-faces" -- the source and target [[object]]s -- of every 1-cell (every [[morphism]]). In many cases this is essential, in that there is no way to regard the generic morphism $a \stackrel{f}{\to} b$ in the category as a morphism from $b$ to $a$ instead.

But there are many categories for which this is not the case, where every morphism naturally only comes with the information of an unordered pair $\{a,b \}$ of objects, without any prejudice on which is to be regarded as source and which as target. An important general example is:

* the category $Spans(C)$ of [[span|spans]] in a category $C$ with pullbacks, or [[duality|dually]], the category $CoSpans(C)$ of [[cospan|cospans]] in a category $C$ with pushouts.

More concrete examples are:

* categories of [[cobordism]]s (but notice that cobordisms are naturally regarded as [[cospan]]s which makes this a special case of the above example);

* the category [[Hilb]] of Hilbert spaces, where for every linear map $f : H_1 \to H_2$ we also have the adjoint map (in the sense of Hilbert spaces, not in the categorical sense) $f^\dagger : H_2 \to H_1$ (but notice that according to [[groupoidification]] this is also essentially to be regarded as a special case of categories of spans).

A _dagger structure_ on a category is extra structure which encodes the idea of _removing_ the ordering information on the 0-faces of 1-cells in a category: it is a contravariant functor which sends every morphism $f : a \to b$ to a morphism going the other way, $f^\dagger : b \to a$.

The notation and terminology here is motivated from the example [[Hilb]] of Hilbert spaces, where $f^\dagger$ is traditionally the notion for the adjoint of a linear map $f$.  The canonical †-structure on [[Hilb]] and on [[nCob]] is crucial in [[FQFT|quantum field theory]] where it is used to encode the idea of **unitarity**:

a _unitary_ [[FQFT|functorial QFT]] of dimension $n$ is supposed to be a functor $n Cob \to Hilb$ which respects the †-structure on both sides.

## Terminology

In Wikipedia a [dagger category](http://en.wikipedia.org/wiki/Dagger_category) is said to be the same as _involutive category_ or _category with involution_.  However, [Springer's Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Category_with_involution) requires that a "category with involution" is also compatibly [[enriched category|enriched]] over [[posets]].

In enriched category theory, involutive categories have also been called **symmetric categories**. Sometimes the involution is required to be strict in the sense that the dagger is an *equality* rather than an *isomorphism*: in line with [[horizontal categorification]], one could argue these should be called **commutative categories**, since a one-object category with such an involution is a [[commutative monoid]].

## Definition

### Dagger categories

#### With a family of functions

\begin{definition}
A **dagger category** or $\dagger$-category $C$ is a [[category]] with a function $(-)^\dagger: Hom_C(A,B) \to Hom_C(B,A)$ for every object $A,B \in Ob(C)$, such that

* for every $A \in Ob(C)$, $(1_A)^\dagger = 1_A$,
* for every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$ and $g \in Hom_C(B,C)$, $(g \circ f)^\dagger = f^\dagger \circ g^\dagger$,
* and for every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$, $((f)^\dagger)^\dagger = f$.

\end{definition}

#### With a function

\begin{definition}
Given a category $C$ with a type or class of objects $Ob(C)$ and a set of morphisms $Mor(C)$ with source and target functions $s:Mor(C) \to Ob(C)$ and $t:Mor(C) \to Ob(C)$, $C$ is a **dagger category** if it has a function $(-)^\dagger:Mor(C) \to Mor(C)$ such that

* for every term $f:Mor(C)$, $s(f) = t(f^\dagger)$
* for every term $f:Mor(C)$, $t(f) = s(f^\dagger)$
* for every term $f:Mor(C)$, $(f^\dagger)^\dagger = f$
* for every term $a:Ob(C)$, $id(a)^\dagger = id(a)$
* for every term $f:Mor(C)$ and $g:Mor(C)$ such that $t(f) =_D s(g)$, $(g \circ f)^\dagger = f^\dagger \circ g^\dagger$.

\end{definition}

#### With a functor

\begin{definition}
A **dagger category** is a [[category]] $C$ equipped with a [[contravariant functor|contravariant endofunctor]], hence an ordinary [[functor]] from the [[opposite category]] $C^{op}$ of $C$ to $C$ itself

$$
  \dagger : C^{op} \to C
$$

which

1. is the [[identity-on-objects]],

1. is an [[involution]] $\dagger^{op} \circ \dagger = \mathrm{id}_C$.

\end{definition}

This definition, which is perhaps the most concise, does rely on a notion of [[identity-on-objects functor]].  This is no problem in most foundations for mathematics, although it violates the [[principle of equivalence]] (except for [[strict categories]]).  In [[homotopy type theory]] (or more generally [[intensional type theory]]) it doesn't make sense to talk about "being the identity on objects" as a property of a given functor, but one can still define an "identity-on-objects endofunctor" (covariant or contravariant) as a basic notion; see [[identity-on-objects functor]] for details.  When unwound for $\dagger$-categories, this yields the above "family of functions" definition.

### Special morphisms

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

A morphism $f$ in a †-category is called a **[[unitary morphism]]** if its †-adjoint equals its [[inverse]]:

$$
  f^\dagger = f^{-1}
  \,.
$$

=--

For the purpose of considering what makes two objects of a $\dagger$-category [[equivalence|equivalent]], one should not consider all [[isomorphism]]s (invertible morphisms) but rather all unitary isomorphisms.

The unitary isomorphisms form a [[groupoid]], which may be regarded as the _dagger-[[core]]_ of the $\dagger$-category.

For example, in [[Hilb]], there are many invertible linear operators, but only those of norm $1$ (the invertible isometries) are unitary.

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

A morphism $f$ in a †-category is called a **[[self-adjoint morphism]]** if it equals its †-adjoint

$$
  f^\dagger = f
  \,.
$$

=--

### The category of †-categories {#CatOfDagCats}

+-- {: .un_defn}
###### Definition
Given two $\dagger$-categories $A$ and $B$, a [[dagger functor|$\dagger$-functor]] $F : A \to B$ consists of a function $F_0 : Ob(A) \to Ob(B)$ with a function $F_{a,b}:Hom_A(a,b) \to Hom_B(F a,F b)$ for every object $a,b:Ob(A)$, where $F_{a,b}$ is generally also denoted as $F$, such that

* for every object $a \in Ob(A)$, $F(1_a)=1_{F a}$,
* for every object $a,b,c \in Ob(A)$ and morphisms $f \in Hom_A(a,b)$ and $g \in Hom_A(b,c)$, $F(g \circ_A f) = F g \circ_B F f$
* for every object $a,b \in Ob(A)$ and morphism $f \in Hom_A(a,b)$, $F(f^{\dagger_A}) = (F f)^{\dagger_B}$.
=--

More concisely, one can say that a $\dagger$-functor $(A,\dagger) \to (B,\ddagger)$ is a [[functor]] $F : A \to B$ of the underlying categories that commutes with the $\dagger$-structures in that $F \circ \dagger = \ddagger \circ F^{op}$.  This appears to violate the [[principle of equivalence]], since it is a strict equality of functors; however, once $\dagger$ and $\ddagger$ are known to be the identity on objects (whatever that means), the two functors $F\circ \dagger$ and $\ddagger \circ F^{op}$ automatically agree on objects.

A [[natural transformation]] between $\dagger$-functors is just a natural transformation of the underlying functors.

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

The †-adjoint $\eta^*$ of a natural transformation

$$
  \eta : F \to G
$$

between two †-functors $F, G : (C,\dagger) \to (D,\ddagger)$ is given by the componentwise $\ddagger$-adjoint:

$$
  (\eta^*)_a := (\eta_a)^\ddagger
  \,.
$$

=--

To check that $\eta^*$ is indeed a natural transformation $\eta^* : G \to F$
consider $f : a \to b$ any morphism in $C$ and $f^\dagger : b \to a$ its $\dagger$-adjoint and let

$$
  \array{
    F(a) &\stackrel{\eta_a}{\to}& G(a)
    \\
    \uparrow^{\mathrlap{F(f^\dagger)}} && \uparrow^{\mathrlap{G(f^\dagger)}}
    \\
    F(b) &\stackrel{\eta_b}{\to}& G(b)
  }
$$

be the corresponding naturality square of $\eta$. Taking the $\ddagger$-adjoint of the entire diagram yields

$$
  \array{
    F(a) &\stackrel{\eta_a^\ddagger}{\leftarrow}& G(a)
    \\
    \downarrow^{\mathrlap{F(f^\dagger)^{\ddagger}}}
    &&
    \downarrow^{\mathrlap{G(f^\dagger)^{\ddagger}}}
    \\
    F(b) &\stackrel{\eta_b^{\ddagger}}{\leftarrow}& G(b)
  }
  \;\;\;
  =
  \;\;\;
  \array{
    F(a) &\stackrel{\eta_a^\ddagger}{\leftarrow}& G(a)
    \\
    \downarrow^{\mathrlap{F(f)}}
    &&
    \downarrow^{\mathrlap{G(f)}}
    \\
    F(b) &\stackrel{\eta_b^{\ddagger}}{\leftarrow}& G(b)
  }
$$

by the fact that $F$ and $G$ are †-functors. This is the naturality square over $f$ of $\eta^* : G \to F$.

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

Write $DagCat$ for the [[category]] whose objects are †-categories and whose morphisms are †-functors.

For $(C,\dagger)$ and $(D,\dagger)$ two †-categories, write
$([(C,\dagger),(D,\ddagger)]_{dag}, \star) \in DagCat$ for the †-category whose objects are †-functors, whose morphisms are natural transformations, with the †-operation $\star : \eta \mapsto \eta^*$ as above.

=--

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

The assignment $((C,\dagger),(D,\ddagger)) \mapsto [(C,\dagger),(D,\ddagger)]_{dag}, \star)$ extends to an [[internal hom]]-functor

$$
  [-,-] : DagCat^{op} \times DagCat \to DagCat
$$

that makes $DagCat$ into a [[cartesian closed category]].

=--

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

This follows step-by-step the standard proof that [[Cat]] is cartesian closed, while observing that each step respects the respect for †-structures.

To indicate the main point, let $C, D$ and $E$ be †-categories and consider a functor $F : C \times D \to E$. For $(f : c_1 \to c_2) \in C$ and $(g : d_1 \to d_2) \in D$ we have natural assignments

$$
  \array{
    (c_1, d_1) &\stackrel{(Id,g)}{\to}& (c_1, d_2)
    \\
    \downarrow^{\mathrlap{(f,Id)}} &\searrow^{(f,g)}& \downarrow^{\mathrlap{(f,Id)}}
    \\
    (c_2, d_1)
    &\stackrel{(Id,g)}{\to}&
    (c_2, d_2)
  }
  \;\;\;\;\;
  \mapsto
  \;\;\;\;\;
  \array{
    F(c_1, d_1) &\stackrel{F(Id,g)}{\to}& F(c_1, d_2)
    \\
    \downarrow^{\mathrlap{F(f,Id)}} && \downarrow^{\mathrlap{F(f,Id)}}
    \\
    F(c_2, d_1)
    &\stackrel{F(Id,g)}{\to}&
    f(c_2, d_2)
  }
$$

that respect daggering all morphisms, in the evident way.

Keeping $d_1$ and $d_2$ fixed, respectively this makes $F(-,d_1), F(-,d_2) : C \to E$ †-functors. We see from the diagrams that $F(-,(d_1 \stackrel{g}{\to}) d_2)$ is a natural transformation between these †-functors, and the fact that $F$ intertwines the dagger operation of $D$ with that of $E$ means $F$ regarded as a functor $D \to [C,E]$ intertwines the †-structures of $D$ and $[D,E]_{dag}$, by the above definition.

=--

## Examples

* The category [[Rel]] of sets and [[relations]] is a †-category, taking dagger as relational converse.

* More generally, let $C$ be a category with [[pullbacks]] and let $Span_1(C)$ be the 1-category of [[spans]] up to isomorphism: its morphisms are spans with one leg labeled as source, the other labeled as target. Then the functor $\dagger : Span_1(C)^{op} \to Span_1(C)$ which just exchanges this labeling is a †-structure on $Span_1(C)$.

* $\mathcal{R}(G)$, the category of unitary [[representation]]s of a (discrete) [[group]] $G$ and intertwining maps, is a †-category. For an intertwiner $\phi : R \rightarrow S$, let $\phi^\dagger : S \rightarrow R$ be the adjoint of $\phi$ in [[Hilb]].

* The [[category of couplings]] between [[probability spaces]].

* Every [[symmetric proset]] is a [[thin category|thin]] †-category.

## Variants

* [[strict dagger category]]

* [[univalent dagger category]]

### Monoidal †-categories

* [[monoidal dagger category]]

  * [[braided monoidal dagger category]]

  * [[symmetric monoidal dagger category]]

  * [[compact closed dagger category]]

  * [[cartesian monoidal dagger category]]

  * [[cocartesian monoidal dagger category]]

  * [[semiadditive dagger category]]

### $\dagger$-2-posets

A [[dagger 2-poset|$\dagger$-2-poset]] is a $\dagger$-category that is also a [[2-poset]]. Examples of $\dagger$-2-posets include [[allegories]] and [[bicategories of relations]].

### Model structure on †-categories {#ModelStructure}

> the following is based on a remark by [[Andre Joyal]], [posted](https://github.com/punkdit/categories/blob/master/gmane/science/mathematics/categories/5477) to the CategoryTheory mailing list on Jan 5, 2010, with a [follow-up](https://github.com/punkdit/categories/blob/master/gmane/science/mathematics/categories/5487) on Jan 6.

Consider †-categories from the point of view of [[homotopy theory]].

Recall that the category [[Cat]] of [[small category|small categories]] naturally admits the [[model category]] structure called the [[folk model structure on Cat]].

The category of small †-categories $DCat$ also admits
a "natural" [[model category]] structure:

* †-functor $f:A \to B$ is a weak equivalence iff it is

  * [[full and faithful functor|full and faithful]];

  * and unitary surjective, meaning that every object of $B$ is unitary isomorphic to an object in the image of the functor $f$;

* the cofibrations and the trivial fibrations are as in [[folk model structure on Cat|Cat]];

* fibrations are the unitary isofibration: maps having the [[right lifting property]] for unitary isomorphisms.

The [[forgetful functor]] $DCat \to Cat$ is a [[right adjoint]]
but it is not a [[Quillen adjunction|right Quillen functor]] with respect to the natural model structures on these categories.

Moreover, a forgetful functor $XStruc \to Cat$ should reflect
weak equivalences in addition to preserving them.
The forgetful functor $DCat\to Cat$ preserves weak equivalences
but it does not reflect them. Because two objects in a †-category can be isomorphic without been unitary isomorphic.

In other words the forgetful functor $DCat\to Cat$ is wrong.
This may explains why a †-category cannot be
regarded as a category equipped a homotopy invariant structure, as discussed in more detail in the example sections of the entry _[[principle of equivalence]]_.

But the notion of †-category is perfectly reasonable
from an homotopy theoretic point of view. This is because the model category $DCat$ is a [[combinatorial model category]].
It follows, by a general result, that the notion of †-category is homotopy essentially algebraic.
There is a homotopy limit sketch whose category of models (in spaces) is [[Quillen equivalence|Quillen equivalent]] to the model category $DCat$. This is true also for the model category Cat.

### $\dagger$-simplicial set

> the following is based on a remark by [[Andre Joyal]], posted to the CategoryTheory mailing list on Jan 6, 2010

A †-[[simplicial set]] can be defined to be a simplicial set $X$ equipped with an involutive [[isomorphism]] $\dagger :X\to X^{op}$ which is the identity on 0-cells.
The category of †-simplicial sets (and dagger preserving maps)
is the category of [[presheaf|presheaves]] on the category whose objects are the ordinals $[n]$, but where the maps $[m]\to [n]$ are order reversing or preserving.

#### $\dagger$-Graphs

* [[dagger-graph]]

###  $(\infty,1)$-†-categories {##oo1Version}

> the following is based on a remark by [[Andre Joyal]], posted to the CategoryTheory mailing list on Jan 6, 2010

There should be a notion of †-[[quasi-category]] based on $\dagger$-simplicial sets as ordinary quasi-categories are based on ordinary simplicial sets.

(...)

## Properties

### Relation to anti-involutive monoids

The [[horizontal categorification]] of an [[anti-involution|anti-involutive]] [[monoid]] is a $\dagger$-category

### Relation to symmetric prosets

Every [[symmetric proset]] is a [[thin category|thin]] $\dagger$-category, and the [[groupoidal categorification]] of a symmetric proset is a $\dagger$-category, with the $\dagger$ operation being the groupoidal categorification of the symmetric property.

### Relation to star-algebras

The [[category convolution algebra]] of a dagger category is naturally a [[star-algebra]]. The star-involution is given by [[pullback of functions]] along the $\dagger$-functor.

## Applications

### Quantum mechanics in terms of $\dagger$-compact categories

The [[finite-dimensional Hilbert spaces]] of interest in [[quantum information theory]] and [[quantum computation]] form $\dagger$-categories that are
also [[compact closed categories]] ([[dagger compact categories]]), a fact that allow a [[string diagram]]-calculus for [[quantum mechanics in terms of †-compact categories]] which motivates much of the contemporary discussion of dagger-categories, starting with [Abramsky & Coecke 2004](#AbramskyCoecke04).

## Related concepts

* [[pseudograph]]

* [[star-category]], [[C-star-category]]

* [[dagger functor]]

* [[dagger category in homotopy type theory]]

* [[symmetric bicategory]]

\linebreak

[[!include oidification - table]]

\linebreak

## References
 {#References}

### General

Motivated by the example of [[finite-dimensional Hilbert spaces]], the concept of $\dagger$-categories appears in the form of *strongly compact closed categories* in

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], around Prop. 7.3 *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04), IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130)]

then named *[[dagger compact closed categories]]* in

* {#Selinger07} [[Peter Selinger]], *Dagger compact closed categories and completely positive maps*, Electronic Notes in Theoretical Computer Science **170** (2007) 139-163 [[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018), [web](http://www.mscs.dal.ca/~selinger/papers.html#dagger), [pdf](https://www.mscs.dal.ca/~selinger/papers/dagger.pdf)]

and as such became the foundation of *[[quantum information theory via dagger-compact categories]]*.

Exposition in this context of [[quantum physics]]:

* [[John Baez]], *The $\star$-category of Hilbert spaces*, Section 3 in: *Quantum quandaries: a category-theoretic perspective*, in _Structural Foundations of Quantum Gravity_, Oxford U. Press (2006) 240-265 [[web](http://math.ucr.edu/home/baez/quantum/node3.html), [arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040)]

and of [[quantum information theory via dagger-compact categories]]:

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], §2.3 in: *Categories for Quantum Theory*, Oxford University Press 2019 [[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)]

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], §2.3 in: *Lectures on categorical quantum mechanics* (2012) [[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]]

But the notion of plain dagger categories has been considered much earlier, reinvented many times under different names (see [Karvonen 2018 §1.2](#Karvonen18) for more historical references) such as under the name "categories with involution":

* [[Saunders Mac Lane]], pp. 1045 of: *An Algebra of Additive Relations*, Proceedings of the National Academy of Sciences of the United States of America **47** 7 (1961) 1043-1051 [[jstor:71127](https://www.jstor.org/stable/71127)]

* [[Dieter Puppe]], §1 in *Korrespondenzen in abelschen Kategorien*, Mathematische Annalen **148** (1962) 1–30 [[doi:10.1007/BF01438388](https://doi.org/10.1007/BF01438388)]

* {#Burgin70} [[Mark S. Burgin]], *Categories with involution, and correspondences in $\gamma$-categories*, Trudy Moskovskogo Matematicheskogo Obshchestva **22** (1970) 161–228 [[mathnet:mmo235](https://www.mathnet.ru/eng/mmo235)]

Vaguely related is the notion of "symmetric category" is (analogous to [[symmetric proset]] or [[symmetric bicategory]])

* {#Walters81} [[Robert Walters]], *Sheaves and Cauchy complete categories*, Cahiers Top. Geom. Diff. Cat. **22** 3 (1981) 283-286 [[numdam:CTGDC_1981__22_3_283_0](http://www.numdam.org/item?id=CTGDC_1981__22_3_283_0)]

Still, dedicated discussion of these notions only appears later under the name of dagger-categories:

* {#Karvonen18} [[Martti Karvonen]], *The Way of the Dagger*, Edinburgh (2018) [[arXiv:1904.10805](https://arxiv.org/abs/1904.10805), [pdf](https://mysite.science.uottawa.ca/mkarvone/papers/Karvonen%20-%20Thesis.pdf)]

concerning [[limits]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Limits in dagger categories*, Theory and Applications of Categories **34** 18 (2019) 468-513 [[arXiv:1803.06651](https://arxiv.org/abs/1803.06651), [34-18](http://www.tac.mta.ca/tac/volumes/34/18/34-18abs.html)]

concerning [[monads]] (such as the [[Frobenius monad|Frobenius]] [[quantum reader monad]]):

* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [tac:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

concerning abstract characterization:

* [[Luuk Stehouwer]], [[Jan Steinebrunner]], *Dagger categories via anti-involutions and positivity* [[arXiv:2304.02928](https://arxiv.org/abs/2304.02928)]

Certain specially nice $\dagger$-categories, such as [[C-star-category|$C^*$-categories]] and [[modular tensor categories]], play an important role in [[topological quantum field theory]] and the theory of [[quantum groups]], see the references there for more.

### Higher dagger categories
 {#ReferencesHigherDagger}

On dagger [[higher categories]]:

Three definitions of dagger [[(infinity,1)-categories|$(\infty,1)$-categories]] have been proposed by:

* [[Simon Henry]], *$(n,1)$-dagger categories* [[MO:a/427322](https://mathoverflow.net/a/427322/381)]

Generalization to dagger [[(infinity,n)-categories|$(\infty,n)$-categories]]:

* [[Theo Johnson-Freyd]], *Dagger Higher Categories*, workshop (2023) [[web](http://categorified.net/dagger2023.html)]

* [[Theo Johnson-Freyd]], *Higher Dagger Categories*, [talk at](Center+for+Quantum+and+Topological+Systems#JohnsonFreydNov2023) [[CQTS]] (08 Nov 2023) [slides:[pdf](http://categorified.net/NYUADtalk.pdf), [[JohnsonFreyd-HigherDaggerNYUAD.pdf:file]], video:[YT](https://youtu.be/zvtziTpl2T0)]

[[!redirects dagger-category]]
[[!redirects dagger-categories]]
[[!redirects dagger categories]]
[[!redirects †-category]]
[[!redirects †-categories]]

[[!redirects symmetric category]]
[[!redirects symmetric categories]]

[[!redirects category with involution]]
[[!redirects categories with involution]]
[[!redirects categories with involutions]]

[[!redirects dagger higher category]]
[[!redirects dagger higher categories]]

[[!redirects higher dagger category]]
[[!redirects higher dagger categories]]


dagger epimorphism

[[!redirects dagger epimorphisms]]

# Contents
* automatic table of contents goes here
{: toc}

## Definition ##

A morphism $f:hom_A(a,b)$ of a [[dagger category]] $A$ is a **dagger epimorphism** if $f \circ f^\dagger=1_a$.

## See also ##

* [[self-adjoint morphism]]

* [[unitary morphism]]

* [[dagger monomorphism]]

* **dagger epimorphism**

## References ##
* Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))


dagger functor

[[!redirects dagger functors]]

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

# $\dagger$ functors
* tic
{: toc}

## Idea ##

A map between two [[dagger categories]] that preserves the dagger category structure.

## Definition ##

+-- {: .un_defn}
###### Definition
Given two $\dagger$-categories $A$ and $B$, a **$\dagger$-functor** $F : A \to B$ consists of a function $F_0 : Ob(A) \to Ob(B)$ with a function $F_{a,b}:Hom_A(a,b) \to Hom_B(F a,F b)$ for every object $a,b:Ob(A)$, where $F_{a,b}$ is generally also denoted as $F$, such that

* for every object $a \in Ob(A)$, $F(1_a)=1_{F a}$,
* for every object $a,b,c \in Ob(A)$ and morphisms $f \in Hom_A(a,b)$ and $g \in Hom_A(b,c)$, $F(g \circ_A f) = F g \circ_B F f$
* for every object $a,b \in Ob(A)$ and morphism $f \in Hom_A(a,b)$, $F(f^{\dagger_A}) = (F f)^{\dagger_B}$.
=--

There is another definition which violates the [[principle of equivalence]], since the definition of the dagger in a dagger category in this case is a functor that imposes equations on objects: A $\dagger$-functor $F : (C, \dagger) \to (D, \ddagger)$ between two $\dagger$-categories $(C, \dagger)$ and $(D, \ddagger)$ is a [[functor]] $F : C \to D$ of the underlying categories, which commutes with the $\dagger$-structures in that $F \circ \dagger = \ddagger \circ F^{op}$.

## See also ##

* [[dagger category]]

* [[monoidal dagger category]]

* [[unitary functorial field theory]]

[[!redirects dagger functors]]

[[!redirects dagger-functor]]
[[!redirects dagger-functors]]


dagger monomorphism

[[!redirects dagger monomorphisms]]

# Contents
* automatic table of contents goes here
{: toc}

## Definition ##

A morphism $f:hom_A(a,b)$ of a [[dagger category]] $A$ is a **dagger monomorphism** if $f^\dagger \circ f=1_a$.

## See also ##

* [[self-adjoint morphism]]

* [[unitary morphism]]

* **dagger monomorphism**

* [[dagger epimorphism]]

## References ##
* Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))


dagger-compact category

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

# Contents
* table of contents
{: toc}

## Idea

A _$\dagger$-compact category_ is a category which is a

* [[compact closed category]]

and a

* [[symmetric monoidal †-category]]

in a compatible way.  So, notably, it is a [[monoidal category]] in which

* every object has a dual;

* every morphism has an $\dagger$-adjoint.

## Definition

A category $C$ that is equipped with the structure of a [[symmetric monoidal †-category]] and is [[compact closed category|compact closed]] is **$\dagger$-compact** if the dagger-operation takes [[units]] of [[dual objects]] to [[counits]] in that for every object $A$ of $C$ we have

$$
  \array{
    && A \otimes A^*
    \\
    & {}^{\epsilon_A^\dagger}\nearrow
    \\
    I && \downarrow^{\mathrlap{\sigma_{A \times A^*}}}
    \\
    & {}_{\eta_A}\searrow
    \\
    && A^* \otimes A
  }
  \,.
$$

## Examples
 {#Examples}

\begin{example}
**(finite-dimensional Hilbert spaces)**
\linebreak
The category of [[Hilbert spaces]] (over the [[complex numbers]]) with [[finite number|finite]] [[dimension]] is a standard example of a $\dagger$-compact category. This example is [[completeness theorem|complete]] for [[equation]]s in the language of $\dagger$-compact categories; see [Selinger 2012](#Selinger2012).

The finite parts of [[quantum mechanics]] ([[quantum information theory]] and [[quantum computation]]) are naturally formulated as the theory of $\dagger$-compact categories. For more on this see at _[[finite quantum mechanics in terms of †-compact categories]]_.
\end{example}

\begin{example}
**(spans)**
\linebreak
For $C$ a category with [[finite limits]]  the category $Span_1(C)$ whose morphisms are [[span|spans]] in $C$ is $\dagger$-compact.  The $\dagger$ operation is that of relabeling the legs of a span as source and target.  The tensor product is defined using the cartesian product in $C$.   Every object $X$ is dual to itself with the unit and counit given by the span
$ I \stackrel{!}{\leftarrow} X \stackrel{Id \times Id}{\to} X \times X$.
See [Baez 2007](#Baez07).
\end{example}

## Properties

### Relation to self-duality

If each [[object]] $X$ of a [[compact closed category]] is equipped with a [[self-duality]] structure $X \simeq X^\ast$, then sending morphisms to their [[dual morphisms]] but with these identifications pre- and postcomposed

$$
  (-)^\dagger \;\colon\;
  (X \stackrel{f}{\longrightarrow} Y)
  \mapsto
  (Y \stackrel{\simeq}{\to} Y^\ast \stackrel{f^\ast}{\longrightarrow} X^\ast \stackrel{\simeq}{\to} X)
$$

constitutes a dagger-compact category structure.

See for instance ([Selinger, remark 4.5](#Selinger)).

Applied for instance to the category of finite-dimensional [[inner product spaces]] this dagger-operation sends [[matrices]] to their [[transpose]].

A good example of a $\dagger$-compact category where most objects are *not* isomorphic to their duals is the category of continuous unitary representations of [[U(n)]] on finite-dimensional complex Hilbert spaces.

## Related concepts

* [[bra-ket notation]]

* [[quantum information theory via dagger-compact categories]]

## References

The concept was introduced in

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], _A categorical semantics of quantum protocols_, in _Proceedings of the 19th IEEE conference on Logic in Computer Science_ (LiCS'04), IEEE Computer Science Press, 2004. ([arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130))

with an expanded version in

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) [[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/)]

under the name "strongly compact" and used for [[finite quantum mechanics in terms of dagger-compact categories]].
The topic was taken up

* [[Peter Selinger]], _Dagger compact closed categories and completely positive maps_, in _Proceedings of the 3rd International Workshop on Quantum Programming Languages_ (QPL 2005), ENTCS 170 (2007), 139--163.
([web](http://www.mscs.dal.ca/~selinger/papers.html#dagger), [pdf](http://www.mscs.dal.ca/~selinger/papers/dagger.pdf))

where the alternative terminology "dagger-compact" was proposed, and used for the abstract characterization of [[quantum operations]] ([[completely positive maps]] on [[Bloch regions]] of [[density matrices]]).

The examples induced from [[self-duality]]-structure are discussed abstractly in

* {#Selinger} [[Peter Selinger]], _Autonomous categories in which $A \simeq A^\ast$_, talk at QPL 2010 ([[SelingerSelfDual.pdf:file]])

That finite-dimensional Hilbert spaces are "complete for dagger-compactness" is shown in

* {#Selinger2012} [[Peter Selinger]] (2012), _Finite dimensional Hilbert spaces are complete for dagger compact closed categories_, ([arXiv:1207.6972](http://arxiv.org/abs/1207.6972)).

The example of [[spans]]:

* {#Baez07} [[John Baez]], _Spans in quantum theory_ ([web](http://math.ucr.edu/home/baez/span/), [pdf](http://math.ucr.edu/home/baez/span/span.pdf), [blog](http://golem.ph.utexas.edu/category/2007/10/spans_in_quantum_theory.html))

[[!redirects dagger compact category]]
[[!redirects dagger compact categories]]
[[!redirects dagger-compact category]]
[[!redirects dagger-compact categories]]
[[!redirects † compact category]]
[[!redirects † compact categories]]
[[!redirects †-compact category]]
[[!redirects †-compact categories]]

[[!redirects dagger compact closed category]]
[[!redirects dagger compact closed categories]]
[[!redirects dagger-compact closed category]]
[[!redirects dagger-compact closed categories]]
[[!redirects † compact closed category]]
[[!redirects † compact closed categories]]
[[!redirects †-compact closed category]]
[[!redirects †-compact closed categories]]


dagger-graph

#Contents#
* table of contents
{:toc}

## Idea

In view of the relation between [[categories]] with their [[underlying]] ([[reflexive graph|reflexive]]) [[directed graphs]] and in view of the notion of *[[dagger-categories]]* it makes sense to say that:

A _$\dagger$-graph_ is a ([[reflexive graph|reflexive]]) [[directed graph]] equipped for each [[pair]] of vertices $x,y$ with an [[involution]] $\sigma_{x,y} \colon V(x,y) \to V(y,x)$ on sets of [[edges]] between these vertices, changing the direction.

This is one way to think of _undirected_ graphs as [[directed graphs]] with [[extra structure]]. In fact, some authors define [[pseudographs]] to be directed graphs equipped with such an involution.

The point of the terminology here is that the [[underlying]] graph of any [[dagger-category]] is canonically a dagger-graph, in the above sense.

## Related concepts

* [[pseudograph]]

[[!redirects dagger graph]]
[[!redirects dagger graphs]]

[[!redirects dagger-graphs]]

[[!redirects †-graph]]
[[!redirects †-graphs]]


Dai Tamaki

* [webpage](http://pantodon.shinshu-u.ac.jp/)

* [University webpage](http://soar-rd.shinshu-u.ac.jp/profile/en.OafeOUkh.html#research_career)

On [[algebraic topology]] with focus on [[complex oriented cohomology|complex oriented]] [[Whitehead-generalized cohomology]]:

* [[Dai Tamaki]], [[Akira Kono]], _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

On [[homotopy theory]] motivated from [[classifying spaces]] for [[principal bundles]]/[[fiber bundles]]:

* [[Dai Tamaki]], *Fiber Bundles and Homotopy*, World Scientific 2021 ([doi:10.1142/12308](https://doi.org/10.1142/12308))

## Related $n$Lab entries

* [[generalized (Eilenberg-Steenrod) cohomology]]

  * [[Brown representability theorem]]

  * [[multiplicative cohomology theory]]

category: people


Daisuke Bekki

Daisuke Bekki is a theoretical linguist employing [[dependent type theory]] in a framework, known as [[Dependent Type Semantics]], to provide a natural language semantics.

* [website](http://www.is.ocha.ac.jp/~bekki/publications.html)

category: people


Daisuke Tambara

* [webpage](https://researchmap.jp/read0167804?lang=en)

### Related entries

* [[Tambara functor]]

* [[Tambara module]]

### Selected writings

* Daisuke Tambara, _The coendomorphism bialgebra of an algebra_, J. Fac. Sci. Univ. Tokyo Sect. IA Math, __37__, 425-456, 1990 [pdf](https://repository.dl.itc.u-tokyo.ac.jp/record/39399/files/jfs370210.pdf)

* {#Tam93} [[Daisuke Tambara]], _On multiplicative transfer_, Comm. Algebra 21 (1993), no. 4, 1393–1420 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/tambara.pdf)).

* [[Daisuke Tambara]], _Distributors on a tensor category_, Hokkaido mathematical journal, 35(2):379–425, 2006. ([project euclid](https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-35/issue-2/Distributors-on-a-tensor-category/10.14492/hokmj/1285766362.full))

On what came to be called *[[Tambara-Yamagami categories]]*:

* {#Tambara1998tensor} [[Daisuke Tambara]], [[Shigeru Yamagami]], *Tensor categories with fusion rules of self-duality for finite abelian groups*, Journal of Algebra **209**  (1998)  692-707 [[doi:10.1006/jabr.1998.7558](https://doi.org/10.1006/jabr.1998.7558)]

category: people


Dale H. Peterson

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=62288)

## Selected writings

On [[quantum cohomology]] of [[flag varieties]]:

* [[Dale H. Peterson]] (notes by [[Arun Ram]]), *Quantum Cohomology of $G/H$*, MIT (1997) [[web](http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html), [pdf](http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf), [[Peterson-QuantumCohomology.pdf:file]]]

On [[affine Lie algebras]] in relation to [[modular forms]]:

* [[Victor G. Kac]], [[Dale H. Peterson]], *Affine Lie algebras and Hecke modular forms*,  Bull. Amer. Math. Soc. (N.S.) **3** 3 (1980) 1057-1061 [[bams:1183547694](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-3/Affine-Lie-algebras-and-Hecke-modular-forms/bams/1183547694.full)]

* [[Victor G. Kac]], [[Dale H. Peterson]], *Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics **53** 2 (1984) 125-264 [doi:10.1016/0001-8708(84)90032-X]

category: people

[[!redirects Dale Peterson]]


Dale Husemöller

* [wikipedia entry](http://de.wikipedia.org/wiki/Dale_Husem%C3%B6ller)

## Selected writings

On [[fiber bundles]] ([[principal bundles]], [[associated bundles]], [[vector bundles]]):

* [[Dale Husemoeller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) ([gBooks](http://books.google.hr/books?id=DPr_BSH89cAC), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller))

and with an eye towards [[topological K-theory]]:

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,  Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

On [[bilinear forms]]:

* [[John Milnor]], [[Dale Husemöller]], *Symmetric Bilinear Forms*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **73**, Springer (1973) [[doi:10.1007/978-3-642-88330-9](https://doi.org/10.1007/978-3-642-88330-9)]

## Related entries

* [[Milnor construction]]

* [[fiber bundle]]

  * [[principal bundle]], [[associated bundle]], [[vector bundle]]

* [[Thom space]]

* [[Hopf invariant]]

* [[Hopf invariant one]]

* [[higher category theory and physics]]

category: people

[[!redirects Dale Husemoeller]]


Dale Rolfsen

* [personal page](https://personal.math.ubc.ca/~rolfsen/)

* [MathGenealogy entry](https://www.mathgenealogy.org/id.php?id=760)

## Selected writings

On [[knots]] and [[links]]:

* [[Dale Rolfsen]], _Knots and links_, AMS Chelsea __346__ (2003) [[ams:chel-346-h](https://bookstore.ams.org/chel-346-h), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rolfsen.pdf)]

category: people


Dalimil Mazáč

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

## Selected writings

On using the [[conformal bootstrap]] to re-prove the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24, and interpretation of the result, via [[AdS/CFT]] in terms of the [[weak gravity conjecture]]:

* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))

category: people

[[!redirects Dalimil Mazac]]


Dalitz decay

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

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum hadrodynamics]] a _Dalitz decay_ is the [[decay]] of an electrically neutral [[pseudoscalar meson]] into a [[dilepton]] pair and a either a [[photon]] (this is the original sense of [Dalitz 51](#Dalitz51), a [[radiative leptonic decay]]) or a [[vector meson]] (a [[semileptonic decay]]).

### Radiative leptonic Dalitz decay
 {#RadiativeLeptonicDalitzDecay}

In [[quantum hadrodynamics]] a _Dalitz decay_ in the original sense ([Dalitz 51](#Dalitz51)) is the [[radiative leptonic decay]] of an [[electromagnetism|electromagnetically]] neutral [[pseudoscalar meson]] $P$ into a [[photon]] and a [[dilepton]]:

$$
  P \,\to\, \gamma \,+\, l^+ \,+\, l^-
  \,.
$$

The dominant contribution to this decay is by a purely radiative decay $P \to \gamma+ \gamma$ followed by an [[electron-photon interaction]] $\gamma + e^+ + e^-$ ("Dalitz pair") of one of the two resulting photons (which thus participates as a [[virtual particle|virtual]] [[photon]] $\gamma^\ast$ with [[spacelike]] [[wave vector]] $q^2 = m^2_{e^+ e^-}$):

\begin{imagefromfile}
  "file_name": "DalitzDecayPseudoscalarMesonToPhoton.jpg",
  "width": 320,
  "caption": "from [Kunkel 12, p. 9](#Kunkel12)",
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  }
\end{imagefromfile}

The original and archetypical example is the [[decay]] of the neutral [[pion]] into an [[electron]]/[[positron]]-[[pair]] and a [[photon]]:

$$
  \pi^0 \,\to\, e^+ \,+\, e^- \,+\, \gamma
  \,.
$$

The [[Feynman amplitude]] of the process is proportional to

\[
  \label{FeynmanAmplitudedRadiativeDalitzDecay}
  \epsilon^{\mu_1 \mu_2 \nu \kappa}
  \underset{
    photon
  }{
    \underbrace{
      q_{\mu_1}
      \epsilon_{\mu_2}
    }
  }
  \underset{
    {scalar}
    \atop
    {meson}
  }{
    \underbrace{
      p_\nu
    }
  }
  \underset{
    dilepton
  }{
    \underbrace{
      \overline{L} \!\cdot\! \gamma_\kappa  \!\!\cdot\! L
    }
  }
  \underset{
    {virtual}
    \atop
    {photon}
  }{
    \underbrace{
      \tfrac{1}{(p_1 + p_2)^2}
    }
  }
\]

where

* $\epsilon^{\mu_1 \mu_2 \nu \kappa}$ is the [[Levi-Civita symbol]],

* $q_{\mu_1}$ is the [[wave vector]] and $\epsilon_{\mu_2}$ the [[polarization]] of the [[photon]],

* $p_\mu$ is the [[momentum]] of the (pseudo-)[[scalar meson]],

* $\overline{L} \!\cdot\! \gamma_\kappa  \!\!\cdot\! L$ is the [[lepton current]]

* $p_1, p_2$ are the momenta of the two [[leptons]], hence $(p_1 + p_2)$ is that of the [[virtual particle|virtual]] [[photon]] mediating the decay.

(see [Yia-Sang 09, (9)](#YiaSang09) (symbol definition on p. 12), also e.g. [KKN 02, p. 2](#KKN02), [KKN06, p. 6](#KKN06) ).

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

The corresponding [[interaction]] [[Lagrangian density]] for the [[Feynman amplitude]] (eq:FeynmanAmplitudedRadiativeDalitzDecay) (hence essentially its [[Fourier transform]] times the [[volume form]]) is, at fixed  $(p_1 + p_2)^2$ (all on 4d [[Minkowski spacetime]]):

\[
  \label{LagrangianDensityForRadiativeDalitzDecay}
  \mathbf{L}_{rad}
  \;\sim\;
  d A \wedge d \pi^0 \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu
  \,,
\]

where $A = A_\mu d x^\mu$ is the [[differential 1-form]] defined by the [[photon]] [[field (physics)|field]] (the [[gauge symmetry]]-[[connection on a principal bundle|connection]]-form), $\pi^0$ is the function ([[scalar field]]) which is the neutral component of the [[pion]] [[field (physics)|field]] and $d$ denotes is the [[de Rham differential]].

=--

### Semileptonic Dalitz decay

More generally, one speaks of (generalized) Dalitz decays for the analogous process as [above](#RadiativeLeptonicDalitzDecay) with the [[photon]] replaced by some [[vector meson]], hence for [[semileptonic decays]] $A \to B + l^+ + l^-$ for $A$ and $B$ a [[pseudoscalar meson]] and [[vector meson]], respectively, with dominant decay mode given by this [[Feynman diagram]]:

\begin{imagefromfile}
  "file_name": "DalitzDecayPseudoscalarMesonToVectorMeson.jpg",
  "width": 320,
  "caption": "from [Kunkel 12, p. 9](#Kunkel12)",
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  }
\end{imagefromfile}

The [[Feynman amplitude]] of the process is, directly analogous to (eq:FeynmanAmplitudedRadiativeDalitzDecay), proportional to

\[
  \label{FeynmanAmplitudedSemileptonicDalitzDecay}
  \epsilon^{\mu_1 \mu_2 \nu \kappa}
  \underset{
    {vector}
    \atop
    {meson}
  }{
    \underbrace{
      q_{\mu_1}
      \epsilon_{\mu_2}
    }
  }
  \underset{
    {scalar}
    \atop
    {meson}
  }{
    \underbrace{
      p_\nu
    }
  }
  \underset{
    dilepton
  }{
    \underbrace{
      \overline{L} \!\cdot\! \gamma_\kappa  \!\!\cdot\! L
    }
  }
  \underset{
    {virtual}
    \atop
    {photon}
  }{
    \underbrace{
      \tfrac{1}{(p_1 + p_2)^2}
    }
  }
\]

where

* $\epsilon^{\mu_1 \mu_2 \nu \kappa}$ is the [[Levi-Civita symbol]],

* $q_{\mu_1}$ is the [[wave vector]] and $\epsilon_{\mu_2}$ the [[polarization]] of the [[vector meson]],

* $p_\mu$ is the [[momentum]] of the (pseudo-)[[scalar meson]],

* $\overline{L} \!\cdot\! \gamma_\kappa  \!\!\cdot\! L$ is the [[lepton current]]

* $p_1, p_2$ are the momenta of the two [[leptons]], hence $(p_1 + p_2)$ is that of the [[virtual particle|virtual]] [[photon]] mediating the decay.

(e.g. [Wachs 00, (2.60)](#Wachs00),  [FLQY 12](#FLQY12), [GLMY 19, (2)](#GLMY19))

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

The corresponding [[interaction]] [[Lagrangian density]] for the [[Feynman amplitude]] (eq:FeynmanAmplitudedSemileptonicDalitzDecay) (hence essentially its [[Fourier transform]] times the [[volume form]]) is, at fixed $(p_1 + p_2)^2$:

\[
  \label{LagrangianDensityForSemileptonicDalitzDecay}
  \mathbf{L}_{sem}
  \;\sim\;
  d V \wedge d \pi \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\!\cdot\! L) d x^\mu
  \,,
\]

where $V = V_\mu d x^\mu$ is the [[differential 1-form]] defined by the [[vector meson]] [[field (physics)|field]] (the [[hidden local symmetry]]-[[connection on a principal bundle|connection]]-form), $P$ is the [[scalar field]] function which is the neutral component of the [[scalar meson]] [[field (physics)|field]] and $d$ denotes is the [[de Rham differential]].

=--

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

Up to possibly a global sign, it does not matter whether $q_\mu$ in (eq:FeynmanAmplitudedRadiativeDalitzDecay) or (eq:FeynmanAmplitudedSemileptonicDalitzDecay) is taken to be the wave vector of the external photon/vector meson (as stated for instance in [Yia-Sang 09, (9)](#YiaSang09)) or of the interval virtual photon, equivalently of the dilepton pair (as stated for instance in [FLQY 12](#FLQY12), [GLMY 19, (2)](#GLMY19)): Due to momentum conservation at the first interaction vertex their difference is proportional to the momentum $p_\mu$ of the scalar meson, and due to anti-symmetry of the [[Levi-Civita symbol]] this difference vanishes inside the above expression.

Equivalently, in terms of the [[Lagrangian density]] (eq:LagrangianDensityForSemileptonicDalitzDecay), this equivalence is reflected by Lagrangian densities defined only up to [[total derivative]]:

$$
  \begin{aligned}
  &
  d V \wedge d \pi \wedge (\overline{L} \!\cdot\!\gamma_\mu \!\cdot\! L) d x^\mu
  \\
  & \sim\;
  V \wedge d \pi \wedge d (\overline{L} \!\cdot\!\gamma_\mu \!\cdot\! L) \wedge d x^\mu
  \end{aligned}
$$

=--

## References

The original article:

* {#Dalitz51} [[Richard Dalitz]], _On an alternative decay process for the neutral $\pi$-meson_, Proceedings of the Physical Society. Section A 64 (7), 667, 1951 ([doi:10.1088/0370-1298/64/7/115](https://doi.org/10.1088/0370-1298/64/7/115))

Survey:

* {#Kunkel12} M. Kunkel, _Dalitz Decays of Pseudo-Scalar Mesons_, talk at [Light Meson Decays Workshop August 5, 2012](https://www.jlab.org/conferences/lmd2012/) ([[KunkelDalitzDecay.pdf:file]])

Further discussion of the Dalitz decay of [[pions]]:

* {#KKN02} Karol Kampf, Marc Knecht, Jiri Novotny, _Some aspects of Dalitz decay $\pi^0 \to e^+ e^- \gamma$_,  presented at Int. Conf. Hadron Structure '02, September 2002, Slovakia ([arXiv:hep-ph/0212243](https://arxiv.org/abs/hep-ph/0212243))

* {#KKN06} Karol Kampf, Marc Knecht, Jiri Novotny, _The Dalitz decay $\pi^0 \to e^+ e^- \gamma$ revisited_, Eur. Phys. J. C46:191-217, 2006 ([arXiv:hep-ph/0510021](https://arxiv.org/abs/hep-ph/0510021))

* Esther Weil, Gernot Eichmann, Christian S. Fischer, Richard Williams, section III.A of: _Electromagnetic decays of the neutral pion_, Phys. Rev. D 96, 014021 (2017) ([arXiv:1704.06046](https://arxiv.org/abs/1704.06046))

Dalitz decay of/into [[omega-mesons]]:

* {#Wachs00} Mirko Wachs, _Die Selbstenergie des Omega-Mesons_, 2000 ([epda:000050](http://elib.tu-darmstadt.de/diss/000050))

Of [[pions]] and [[eta-mesons]]:

* Sergi González-Solís, _Single and double Dalitz decays of $\pi^0$, $\eta$ and $\eta'$ mesons_, Nuclear and Particle Physics Proceedings Volumes 258–259, January–February 2015, Pages 94-97 ([doi:10.1016/j.nuclphysbps.2015.01.021](https://doi.org/10.1016/j.nuclphysbps.2015.01.021))

Of [[pions]], [[eta-mesons]] and [[omega-mesons]]:

* {#Berghauser10} Henning Berghäuser, _Investigation of the Dalitz decays and the electromagnetic form factors of the $\eta$ and $\pi^0$-meson_, 2010 ([spire:1358057](https://inspirehep.net/literature/1358057))

Discussion of Dalitz decay of [[quarkonium]]:

* {#YiaSang09} Yu Jia, Wen-Long Sang, _Observation prospects of leptonic and Dalitz decays of pseudoscalar quarkonia_, JHEP 0910:090, 2009 ([arXiv:0906.4782](https://arxiv.org/abs/0906.4782))

Of [[charmonium]]:

* {#FLQY12} Jinlin Fu, Hai-Bo Li, Xiaoshuai Qin, Mao-Zhi Yang, _Study of the electromagnetic transitions $J/\psi \to P l^+ l^-$ and probe dark photon_,  Modern Physics Letters A Vol. 27, No. 38, 1250223 (2012) ([arXiv:1111.4055](https://arxiv.org/abs/1111.4055) [doi:10.1142/S0217732312502239](https://doi.org/10.1142/S0217732312502239))

* _Study of the Dalitz decay $J/\psi \to e^+e^- \eta$_, Phys. Rev. D 99, 012006 (2019) ([arXiv:1810.03091](https://arxiv.org/abs/1810.03091))

Of [[upsilon-mesons]]:

* {#GLMY19} Li-Min Gu, Hai-Bo Li, Xin-Xin Ma, Mao-Zhi Yang, _Study of the electromagnetic Dalitz decays $\psi(\Upsilon) \to \eta_c(\eta_b) l^+ l^-$_, Phys. Rev. D 100, 016018 (2019) ([arXiv:1904.06085](https://arxiv.org/abs/1904.06085))

Of [[baryons]]:

* G. Ramalho, _A covariant model for the decuplet to octet Dalitz decays_ ([arXiv:2002.07280](https://arxiv.org/abs/2002.07280))

[[!redirects Dalitz decays]]

[[!redirects Dalitz pair]]
[[!redirects Dalitz pairs]]

[[!redirects radiative Dalitz decay]]
[[!redirects radiative Dalitz decays]]

[[!redirects semileptonic Dalitz decay]]
[[!redirects semileptonic Dalitz decays]]


Damien Calaque

*  [webpage](https://imag.umontpellier.fr/~calaque/)

## Selected writings

On [[tensor categories]]:

* [[Damien Calaque]], [[Pavel Etingof]], _Lectures on tensor categories_, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) ([arXiv:math/0401246](https://arxiv.org/abs/math/0401246))

On the [[Duflo isomorphism]]:

* [[Damien Calaque]], [[Carlo A. Rossi]], _Lectures on Duflo isomorphisms in Lie algebra and complex geometry_, European Math. Soc. (2011) [[doi:10.4171/096](https://doi.org/10.4171/096), [pdf](https://imag.umontpellier.fr/~calaque/lecturenotes-LecturesDufloETH)]

On [[mapping stacks]] and [[TFT]]:

* [[Damien Calaque]], _Lagrangian structures on mapping stacks and semi-classical TFTs_, 2013, [arXiv:1306.3235](https://arxiv.org/abs/1306.3235).

On [[derived geometry|derived]] [[symplectic geometry]] ([[shifted symplectic structures]]):

* [[Damien Calaque]], _Three lectures on derived symplectic geometry and topological field theories_, 2014,
[pdf](https://imag.umontpellier.fr/~calaque/lecturenotes-DerivedTFT) - see also [these notes](https://imag.umontpellier.fr/~calaque/lecturenotes-LagStr)

* [[Damien Calaque]], [[Tony Pantev]], [[Bertrand Toen]], [[Michel Vaquie]], [[Gabriele Vezzosi]], _Shifted Poisson structures and deformation quantization_, Journal of topology 10.2 (2017) 483--584 [arXiv:1506.03699](http://arxiv.org/abs/1506.03699)

See also

* Tristan Bozec, [[Damien Calaque]], Sarah Scherotzke, _Calabi-Yau structures for multiplicative preprojective algebras_, [arXiv:2102.12336](https://arxiv.org/abs/2102.12336)

Review of [[shifted symplectic structures]]:

* [[Damien Calaque]], *Derived symplectic geometry*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2308.04210](https://arxiv.org/abs/2308.04210)]

category: people


Damjan Kobal

* [institute page](https://www.fmf.uni-lj.si/sl/imenik/104/kobal-damjan/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=3116)

## Selected writings

On [[Hermitian K-theory]]:

* [[Damjan Kobal]], *K-Theory, Hermitian K-Theory and the Karoubi Tower*, K-Theory **17** 113–140 (1999) [[doi:10.1023/A:1007799508729](https://sci-hub.ru/10.1023/A:1007799508729)]

category: people


Damjan Pištalo

* [institute page](https://wwwen.uni.lu/research/fstm/dmath/people/damjan_pistalo)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=233720)

## Selected writings

On [[homotopy limits]], [[homotopy pullbacks]] and their [[pasting law for pullbacks|pasting law]] in/via [[model categories]]:

* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], *Indeterminacies and models of homotopy limits* [[arXiv:2109.12395](https://arxiv.org/abs/2109.12395)]

category: people

[[!redirects Damjan Pistalo]]


Dan Abramovich

* [webpage](http://www.math.brown.edu/~abrmovic/)

## Selected writings

On [[t-structures]] on [[derived categories of coherent sheaves]]:

* [[Dan Abramovich]], [[Alexander Polishchuk]], *Sheaves of t-structures and valuative criteria for stable complexes*, J. reine angew. Math. **590** (2006) 89-130 [[arXiv:math/0309435](https://arxiv.org/abs/math/0309435), [doi:10.1515/CRELLE.2006.005](https://doi.org/10.1515/CRELLE.2006.005)]

category: people


Dan Burghelea

__Dan Burghelea__ is a topologist at Ohio State University

* [gScholar](https://scholar.google.com/citations?user=yKc9M2gAAAAJ)

## Selected works

* _The cyclic homology of the group rings_, Comm. Math. Helv. 60 (1985) 354-365 [doi](https://doi.org/10.1007/BF02567420) [pdf](https://www.researchgate.net/profile/Dan-Burghelea/publication/227109904_The_cyclic_homology_of_the_group_rings/links/5537dec70cf226723ab5bf97/The-cyclic-homology-of-the-group-rings.pdf)

* _"Barcodes" for continuous maps and a brief introduction to Alternative Morse Theory_, [arXiv:2305.19828](https://arxiv.org/abs/2305.19828)

category: people


Dan E. Browne

* [personal page](https://sites.google.com/site/danbrowneucl/)

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

## Selected writings

On [[measurement-based quantum computation]]:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

category: people

[[!redirects Dan Browne]]


Dan Frumin

## Selected writings

On [[finite sets]] in [[homotopy type theory]]/[[univalent foundations of mathematics]]:

* [[Dan Frumin]], [[Herman Geuvers]], [[Léon Gondelman]], [[Niels van der Weide]], _Finite Sets in Homotopy Type Theory_, in *CPP 2018: Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs* (2018) 201–214 [[doi:10.1145/3167085](https://doi.org/10.1145/3167085), [pdf](http://cs.ru.nl/~nweide/FiniteSetsInHoTT.pdf)]

On [[univalent bicategory|univalent]] [[bicategories]] in [[homotopy type theory]]:

* [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], _Bicategories in Univalent Foundations_, Mathematical Structures in Computer Science **31** 10 (2022) 1232-1269 [[arXiv:1903.01152](https://arxiv.org/abs/1903.01152), [doi:10.1017/S0960129522000032](https://doi.org/10.1017/S0960129522000032)&rbrack

category: people


Dan Ghica

* [website](https://www.cs.bham.ac.uk/~drg/)

### Selected writings

* [[Dan Ghica]], [[Fabio Zanasi]], _Hierarchical string diagrams and applications_ ([arXiv:2305.18945](https://arxiv.org/abs/2305.18945))

category: people

[[!redirects Dan R. Ghica]]


Dan Isaksen > history

< [[Dan Isaksen]]
[[!redirects Dan Isaksen>History]]
[[!redirects Dan Isaksen>history]]


Dan Israel

* [personal page](https://www.lpthe.jussieu.fr/~israel/)

## Selected writings

On [[super-orbifolds]] as [[target spaces]] in [[superstring theory]] ([[orbispace|orbi-]][[superspace]]):

* [[Matthias Gaberdiel]], [[Dan Israel]], [[Eliezer Rabinovici]], _D-branes at multicritical points_, JHEP 0804:086, 2008 [[arXiv:0803.0291](https://arxiv.org/abs/0803.0291)]


Dan Margalit

* [personal page](https://dmargalit7.math.gatech.edu/index.shtml)

## Selected wriings

On [[finitely presented group|finite presentations]] of the [[pure braid group]]:

* [[Dan Margalit]], [[Jon McCammond]], *Geometric presentations for the pure braid group*,  Journal of Knot Theory and Its Ramifications **18** 01 (2009) 1-20 [[arXiv:math/0603204](https://arxiv.org/abs/math/0603204), [doi:10.1142/S0218216509006859](https://doi.org/10.1142/S0218216509006859)]

On [[Joan Birman]]:

* [[Dan Margalit]], *The Mathematics of Joan Birman*, Notices of the AMS **66** 3 (2019) [[pdf](https://www.ams.org/journals/notices/201903/rnoti-p341.pdf), [pdf](https://www.ams.org/journals/notices/201903/201903FullIssue.pdf)]

On the relation between [[mapping class groups]] and [[braid groups]]:

* {#MargalitWinarski21} [[Dan Margalit]], [[Rebecca R. Winarski]], *Braid groups and mapping class groups: The Birman–Hilden theory*, Bull. London Math. Soc. **53** 3 (2021) 643-659 [[arXiv:1703.03448](https://arxiv.org/abs/1703.03448), [doi:10.1112/blms.12456](https://doi.org/10.1112/blms.12456)]

category: people


Dan Marsden

* [website](http://www.cs.ox.ac.uk/people/dan.marsden/)

category: people


Dan Petersen

* [webpage](https://www.math.ku.dk/english/about/news/dan_petersen/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=174794)

## Selected writings

Relating [[singular cohomology]] to [[sheaf cohomology]] via [[hypercovers]]:

* {#Petersen21} [[Dan Petersen]], _A remark on singular cohomology and sheaf cohomology_ ([arXiv:2102.06927](https://arxiv.org/abs/2102.06927))

category: people


Dan Rockmore

* [personal page](https://www.cs.dartmouth.edu/rockmore/)

* [institute page](https://faculty-directory.dartmouth.edu/daniel-rockmore)

## Selected writings

On [[Cayley graph spectra]]:

* {#RockmoreKostelecHordijkStadler02} [[Dan Rockmore]], [[Peter Kostelec]], [[Wim Hordijk]], [[Peter F. Stadler]], *Fast Fourier Transform for Fitness Landscapes*, Applied and Computational Harmonic Analysis Volume 12, Issue 1, January 2002, Pages 57-76 ([doi:10.1006/acha.2001.0346](https://doi.org/10.1006/acha.2001.0346))

category: people

[[!redirects Daniel Rockmore]]


Dan-Virgil Voiculescu

_Dan-Virgil Voiculescu_ is a mathematician at University of Berkeley, specialized in [[operator algebra]]s, and the founder of [[free probability]] theory.

* [webpage](https://math.berkeley.edu/people/faculty/dan-virgil-voiculescu) at U. Berkeley.

* [[Dan-Virgil Voiculescu]], _Free probability theory: random matrices and von Neumann algebras_, Proc. ICM 1994 [pdf]()
* D.V. Voiculescu, K.J. Dykema, A. Nica, _Free random variables_, CRM Monograph Series, vol. 1, Amer. Math. Soc. 1992.
* D.V. Voiculescu, _Limit laws for random matrices and free products_, Inventiones Mathematicae __104__:201--220, 1991 [MR1094052](https://mathscinet.ams.org/mathscinet-getitem?mr=1094052)
* D.V. Voiculescu, _Symmetries of some reduced free product $C^\ast$-algebras_, in Operator algebras and their connections with topology and ergodic theory (Bucsteni, 1983), Lecture Notes in Math. 1132, 556--588. Springer 1985.

A noncommutative analogue of [[Fisher metric]]

* D.-V. Voiculescu, _The analogues of entropy and of Fisher’s information measure in free probability theory. V: Noncommutative Hilbert transforms_, Invent. Math. 132:1 (1998) 189--227.

category: people, operator algebras


Dana May Latch

Dana May Latch (1943-2015) was an American algebraic topologist and category theorist who worked in North Carolina State University.

##Selected works

* with [[Rudolf Fritsch]], _Homotopy inverses for nerve_, Bull. Amer. Math. Soc. (N.S.), Volume 1, Number 1 (1979), 258-262.

* with [[Robert Thomason]] and [[W. Stephen Wilson]], _Simplicial sets from categories_, Mathematische Zeitschrift 164(3):195-211, (1979).

category: people


Dana Scott

* [Homepage](http://www.cs.cmu.edu/~scott/)

* [Wikipedia page](http://en.wikipedia.org/wiki/Dana_Scott)

* [Scans of selected papers](https://github.com/CMU-HoTT/scott)

* [[Peripatetic Seminar on Sheaves and Logic]] (History)

## Selected writings

On the theory of [[computation]] and introducing the notion of [[denotational semantics]] of [[programming languages]] by what came to be called *[[domain theory]]*:

* [[Dana S. Scott]], *Outline of a mathematical theory of computation*, in: Proceedings of the *Fourth Annual Princeton Conference on Information Sciences and Systems* (1970) 169–176. [[pdf](https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf), [[Scott-TheoryOfComputation.pdf:file]]]

* [[Dana S. Scott]], [[Christopher Strachey]], *Toward a Mathematical Semantics for Computer Languages*, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [[pdf](https://www.cs.ox.ac.uk/files/3228/PRG06.pdf), [[ScottStrachey-MathematicalSemantics.pdf:file]]]

* {#Scott76} [[Dana Scott]], *Data types as lattices*. SIAM Journal of Computing **5** 3 (1976) 522--587 [[doi:10.1137/0205037](https://doi.org/10.1137/0205037), [pdf](https://www.cs.ox.ac.uk/files/3287/PRG05.pdf)]

* {#GierzHofmannKeimelLawsonLisloveScott03} G. Gierz, [[Karl H. Hofmann]], K. Keimel, J. D. Lawson, [[Michael W. Mislove]], [[Dana S. Scott]], *Continuous Lattices and Domains* in *Encyclopedia of Mathematics and its Applications* 93, Cambridge University Press (2003)  [[doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725)]

On [[modal logic]]:

* [[E. John Lemmon]] with [[Dana Scott]], *An Introduction to Modal Logic -- The "Lemmon Notes"*, B. Blackwell (1977) [[ark:/13960/t3gz25k3h](https://archive.org/details/introductiontomo0000lemm/)]

## Related entries

* [[domain theory]]

* [[Scott's trick]]

* [[equilogical space]]

* [[Scott topology]]

* [[lambda-calculus]]

* [[reflexive object]]

category: people

[[!redirects Dana S. Scott]]


Danel Ahman

* [personal page](https://danel.ahman.ee/)

## Selected writings

On ([[delta lens|delta]]-)[[lenses in computer science]] and [[monads in computer science|(co)monadic effects]]:

* {#AhmanUustalu17} [[Danel Ahman]], [[Tarmo Uustalu]], *Taking updates seriously*, Proceedings of the Sixth International Workshop on [Bidirectional Transformations (Bx 2017)](https://ceur-ws.org/Vol-1827/), CEUR Workshop Proceedings **1827** (2017) [[pdf](http://ceur-ws.org/Vol-1827/paper11.pdf)]

On ([[comonad|co-]])[[monads in computer science]] (such as the [[coreader comonad]]:

* {#AhmanUustalu19} [[Danel Ahman]], [[Tarmo Uustalu]], p. 3 of: *Decomposing Comonad Morphisms*, CALCO 2019, Leibniz International Proceedings in Informatics (LIPIcs) **139** (2019) [[doi:10.4230/LIPIcs.CALCO.2019.14](https://doi.org/10.4230/LIPIcs.CALCO.2019.14)]

## Related entries

* [[cowriter comonad]]

category: people


Daniel Allcock

* [personal page](https://web.ma.utexas.edu/users/allcock/)

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

## Selected writings

On the classification of [[spherical space forms]] (i.e. [[free action|free]] [[group actions on n-spheres]] of [[finite groups]] through [[isometries]]):

* {#Allcock15} [[Daniel Allcock]], _Spherical space forms revisited_, Trans. Amer. Math. Soc. 370 (2018), 5561-5582  ([arXiv:1509.00906](https://arxiv.org/abs/1509.00906), [doi:10.1090/tran/7167](https://doi.org/10.1090/tran/7167))

category: people


Daniel Altschuler

Daniel Altschuler is a theoretical physicist.

## Selected works

On ([[universal Vassiliev invariant|universal]]) [[Vassiliev invariants]] and relation to [perturbative](quantization+of+3d+Chern-Simons+theory#ReferencesPerturbativeCSTheory) [[Chern-Simons theory]]

* [[Daniel Altschuler]], [[Laurent Freidel]], _On Universal Vassiliev invariants_, Commun. Math. Phys. **170** (1995) 41-62 [[arXiv:hep-th/9403053](https://arxiv.org/abs/hep-th/9403053), [doi:10.1007/BF02099438](https://doi.org/10.1007/BF02099438)]

* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. **187** (1997) 261-287 [[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)]

Relations of knot invariants to quasi-Hopf algebras

* Daniel Altschuler, Antoine Coste, _Invariants de rubans et algèbres quasi Hopf_, Les rencontres physiciens-mathématiciens de Strasbourg - RCP25, 1994, tome 46
« Conférences de A. Beauville, A. Coste, J. Dubochet, A. Gramain, M. Holschneider, C.
Itzykson, P. Le Tallec, V. Rivasseau, C. Weber », exp. no 1, p. 1-42
[numdam](http://www.numdam.org/item?id=RCP25_1994__46__1_0)
* D. Altschuler, A. Coste, _Quasi-quantum groups, knots, three-manifolds, and topological field theory_Commun.Math. Phys. 150 (1992) 83--107 [doi](https://doi.org/10.1007/BF02096567)

Other works

* Daniel Altschuler, _Quantum equivalence of coset space models_, Nuclear Physics B __313__:2 (1989) 293-307 doi
* Daniel Altschuler, [[Claude Itzykson]], _Remarks on integration over Lie algebras_,  Annales de l’I. H. P., section A, tome 54, no 1 (1991), p. 1-8 [numdam](http://www.numdam.org/item?id=AIHPA_1991__54_1_1_0)
* Daniel Altschuler, Brain Davis, _Quantum loop models and quantum spin chains_, Int. J. Mod. Phys. A __09__:22 (1994) 3925-3958 [doi](https://doi.org/10.1142/S0217751X94001588)

category: people


Daniel Baumann

[Institute page](http://www.uva.nl/en/profile/b/a/d.d.baumann/d.d.baumann.html)

## Selected writing

On [[cosmic structure formation]]:

* [[Daniel Baumann]], _The big bang and the origin of structure_, 2019 ([web](https://www.dropbox.com/s/as3vb4jpv65822k/BigHistory-2019.pdf?dl=0))

On [[cosmic inflation]] in [[string theory]]:

* [[Daniel Baumann]], [[Liam McAllister]], _Inflation and String Theory_, Cambridge University Press (2015) [[arXiv:1404.2601](https://arxiv.org/abs/1404.2601), [doi:10.1017/CBO9781316105733]( https://doi.org/10.1017/CBO9781316105733)]

On [[cosmology]] and the [[cosmic microwave background]]:

* [[Daniel Baumann]], _TASI Lectures on Primordial Cosmology_ ([arXiv:1807.03098](https://arxiv.org/abs/1807.03098))

Reviewing the classification of possible long-range [[forces]], hence of [[scattering]] processes of [[mass|massless]] [[field (physics)|fields]], by classification of suitably factorizing and decaying [[Poincaré group|Poincaré]]-[[invariant]] [[S-matrices]] depending on particle spin, leading to uniqueness statements about [[Maxwell theory|Maxwell]]/[[photon]]-, [[Yang-Mills theory|Yang-Mills]]/[[gluon]]-, [[gravity]]/[[graviton]]- and [[supergravity]]/[[gravitino]]-[[interactions]]:

* [[Daniel Baumann]], _What long-range forces are allowed?_, 2019 ([[BaumannLongRange19.pdf:file]])

On [[general relativity]]:

* [[Daniel Baumann]], *[General Relativity](http://cosmology.amsterdam/general-relativity/)*, 2021 ([pdf](http://cosmology.amsterdam/wp-content/uploads/2021/10/GR-Oct19.pdf))

## Related $n$Lab entries

* [[cosmology]]

* [[cosmic structure formation]]

* [[cosmic microwave background]]

* [[cosmic inflation]], [[string theory]]

category: people


Daniel Bennequin

Daniel Bennequin is a French mathematician who gave the first example of an exotic [[contact structure]] on $\mathbb{R}^3$ in his dissertation. He was a [member](https://books.google.com.au/books?id=-CXn6y_1nJ8C&pg=PA17) of [[Bourbaki]].

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

##Selected writings

On contact geometry and [[links]]:

* [[Daniel Bennequin]], _Entrelacements et equations de Pfaff_, Astérique, 107-108 (1983), 87-161. ([pdf](http://www.numdam.org/article/AST_1983__107-108__87_0.pdf))

On a cohomological interpretation of [[entropy]]:

* Pierre Baudot, [[Daniel Bennequin]], _The homological nature of entropy_, Entropy, 17 (2015), 3253-3318. ([doi](https://doi.org/10.3390/e17053253))

##Related $n$Lab pages

* [[contact manifold]]

* [[entropy]]


Daniel Berwick-Evans

* [webpage](http://math.illinois.edu/~danbe/index.html)

## Selected writings

On identifying the [[shape modality]] on the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]] with the [[path ∞-groupoid]]-construction ([[shape via cohesive path ∞-groupoid]]):

* {#BEBP} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

Discussion of [[topological modular forms]] via [[(2,1)-dimensional Euclidean field theory]]:

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?* $[$[arXiv:2303.09138](https://arxiv.org/abs/2303.09138)$]$

Review:

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23), video:[YT](https://www.youtube.com/watch?v=fw7yxFsDmjs)]

category: people


Daniel C. Cabra

* [institute page](https://www.iflp.unlp.edu.ar/sft_cabra/sft_cabra-15280)

## Selected writings

On the [[CS-WZW correspondence]] (with motivation by [[Laughlin wavefunctions]] for [[anyons]]):

* [[Daniel C. Cabra]], [[Gerardo L. Rossini]], *Explicit connection between conformal field theory and 2+1 Chern-Simons theory*, Mod. Phys. Lett. A **12** (1997) 1687-1697 [[arXiv:hep-th/9506054](https://arxiv.org/abs/hep-th/9506054), [doi:10.1142/S0217732397001722](https://doi.org/10.1142/S0217732397001722)]

Relating [[parafermions]]  to the [[affine Lie algebra|affine]] [[su(2)]]-[[current algebra]]/[[WZW model]] ([[su(2)-anyons]]):

* [[Daniel Cabra]], *Spinons and parafermions in fermion cosets*, talk at International Seminar dedicated to the memory of D. V. Volkov, Kharkov (January 5-7, 1997) [[arXiv:hep-th/9704015](https://arxiv.org/abs/hep-th/9704015)]

* {#CFRS00} [[Daniel C. Cabra]], [[Eduardo Fradkin]], G. L. Rossini, F. A. Schaposnik, Section 4 of: *Non-Abelian fractional quantum Hall states and chiral coset conformal field theories*, International Journal of Modern Physics A **15** 30 (2000) 4857-4870 [[doi:10.1142/S0217751X00002354](https://doi.org/10.1142/S0217751X00002354), [arXiv:cond-mat/9905192](https://arxiv.org/abs/cond-mat/9905192)]

category: people

[[!redirects Daniel Cabra]]


Daniel C. Cohen

* [personal page](https://www.math.lsu.edu/~cohen/)

## Selected writings

On [[local system]]-[[twisted de Rham cohomology]] over [[configuration spaces of points]] (such as for the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]):

* [[Daniel C. Cohen]], [[Peter Orlik]], *Arrangements and local systems*, Math. Res. Lett. **7** (2000) 299-316 $[$[arXiv:math/9907117](https://arxiv.org/abs/math/9907117), [doi:10.4310/MRL.2000.v7.n3.a5](https://dx.doi.org/10.4310/MRL.2000.v7.n3.a5)$]$

On [[Gauss-Manin connections]] over [[configuration spaces of points]]:

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin Connections for Arrangements, I Eigenvalues*, Compositio Math. **136** (2003) 299-316 $[$[arXiiv:math/0105063](https://arxiv.org/abs/math/0105063), [doi:10.1023/A:1023262022279](https://doi.org/10.1023/A:1023262022279)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, II Nonresonant weights*, Amer. J. Math. **127** (2005) 569-594 $[$[arXiv:math/0207114](https://arxiv.org/abs/math/0207114), [jstor:40067930](https://www.jstor.org/stable/40067930)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, III Formal connections*, Trans. Amer. Math. Soc. **357** (2005) 3031-3050 $[$[arXiv:math/0307210](https://arxiv.org/abs/math/0307210), [doi:10.1090/S0002-9947-04-03621-9](https://doi.org/10.1090/S0002-9947-04-03621-9)$]$

On [[Milnor fibration]] and [[local system]]s for arrangements of complex hyperplanes

* [[Daniel C. Cohen]], Alexander I. Suciu, _On Milnor ﬁbrations of arrangements_, J. London Math. Soc. 51 (1995) 105--119; MR96e:32034 [doi](https://doi.org/10.1112/jlms/51.1.105)

On [[braid group representations]] seen in the [[topological K-theory]] of the [[classifying space]] of the [[braid group]]:

* [[Alejandro Adem]], [[Daniel C. Cohen]], [[Frederick R. Cohen]], *On representations and K-theory of the braid groups*, Math. Ann. **326** (2003) 515-542 ([arXiv:math/0110138](https://arxiv.org/abs/math/0110138), [doi:10.1007/s00208-003-0435-8](https://doi.org/10.1007/s00208-003-0435-8))

On [[topological complexity]] of [[connected sums]]:

* {#CohenVandembroucq18} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Motion planning in connected sums of real projective spaces_ (2018), [arXiv:1807.09947](https://arxiv.org/abs/1807.09947)

On [[topological complexity]] of the [[Klein bottle]]:

* {#CohenVandembroucq16} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Topological Complexity of the Klein bottle_ (2018), [arXiv:1612.03133v3](https://arxiv.org/abs/1612.03133v3)

category: people

[[!redirects Daniel Cohen]]


Daniel C. Thompson

* [institute page](https://www.swansea.ac.uk/staff/d.c.thompson/)

* [InSpire page](https://inspirehep.net/authors/1048758)

## Selected writings

On [[boundary conditions]] for interacting [[M2-branes]] in regarding through the [[ABJM model]]:

* {#BPST09} [[David Berman]], [[Malcolm J. Perry]], [[Ergin Sezgin]], [[Daniel C. Thompson]], *Boundary Conditions for Interacting Membranes*, JHEP 1004:025, 2010 [[arXiv:0912.3504](https://arxiv.org/abs/0912.3504), doi:10.1007/JHEP04(2010)025]

On [[U-duality]] and [[diffeomorphisms]] in [[exceptional generalized geometry]]:

* [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], [[Daniel C. Thompson]], _The gauge structure of generalised diffeomorphisms_, High Energ. Phys. 2013, 64 (2013) [[arXiv:1208.5884](http://arxiv.org/abs/1208.5884)]

Generalization of [[Poisson-Lie T-duality]] to [[U-duality]] in [[exceptional generalized geometry]]:

* [[Emanuel Malek]], [[Daniel C. Thompson]], _Energy Physics - Theory Poisson-Lie U-duality in Exceptional Field Theory_, JHEP 0420 (2020) 058 [[arxiv:1911.07833](https://arxiv.org/abs/1911.07833)]

* [[Chris D. A. Blair]], [[Daniel C. Thompson]], Sofia Zhidkova, _Exploring Exceptional Drinfeld Geometries_, JHEP 09 (2020) 151 [[arxiv:2006.12452](https://arxiv.org/abs/2006.12452)]

category: people

[[!redirects Daniel Thompson]]


Daniel Carranza

* [personal page](https://daniel-carranza.github.io/)

## Selected writings

On a [[calculus of fractions]] generalized from categories to [[quasi-categories]] ("$(\infty,1)$-calculus of fractions", for *[[localization of an (infinity,1)-category|localization of $(\infty,1)$-categories]]*):

* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]

exposition:

* [[Daniel Carranza]], *Calculus of Fractions for Quasicategories (Part II)*, [talk at](CQTS#CarranzaOct2023) [[CQTS]] (25 Oct 2023) [video:[YT](https://youtu.be/Z41YDb99cZk)]

category: people


Daniel Conduché

[[!redirects D. Conduché]]
Daniel Conduché is a French mathematician who has worked at Rennes.

His work includes the definition and exploration of the notion of [[2-crossed module]] and  the [[decomposition theorem for simplicial groups]].

####Selected publications

For his work on 2-crossed modules, and simplicial groups,

* [[D. Conduché]], _Modules croisés généralisés de longueur 2_ , J. Pure Appl. Alg., 34, (1984), 155--178.

In his work on homotopy $n$-types, [[Loday]] gave a construction of what he called a 'mapping cone' for a [[crossed square]]. Conduch\'{e} later noticed that this naturally had the structure of a [[2-crossed module]],

* D. Conduché, Simplicial Crossed Modules and Mapping Cones, Georgian Math. J., 10, (2003), 623–636.

category:people


Daniel Dugger

Daniel K. Dugger is a professor of mathematics at the University of Oregon.
He got his PhD in 1999 at MIT, advised by [[Michael J. Hopkins]].

* [website](https://pages.uoregon.edu/ddugger/)

## Selected writings

On [[model structures on simplicial presheaves]] and the $\mathbb{R}^1$-[[Bousfield localization of model categories|localization]] of that on the site of [[smooth manifolds]] (later: "[[smooth infinity-groupoids]]"):

* [[Daniel Dugger]], _Sheaves and homotopy theory_ (1998) [[pdf](http://math.mit.edu/~dspivak/files/cech.pdf), [[DuggerSheavesAndHomotopyTheory.pdf:file]]]

On [[hypercovers]] in [[simplicial presheaves]]:

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. ([arXiv:0205027](http://arxiv.org/abs/math/0205027), [K-theory archive](http://www.math.uiuc.edu/K-theory/0563/))

On [[combinatorial model categories]]:

* _[[Universal Homotopy Theories]]_

* _[[Combinatorial model categories have presentations]]_

On [[Delta-generated spaces]]:

* [[Daniel Dugger]], _Notes on Delta-generated spaces_, 2003 ([web](https://pages.uoregon.edu/ddugger/delta.html), [[DuggerDeltaGenerated.pdf:file]])

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 [[Cayley-Dickson algebras]]:

* Daniel K. Biss, [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras_,
 Communications in Algebra 36 (2), 632-664, 2008 ([arxiv:math/0511691](https://arxiv.org/abs/math/0511691))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras II_, 	Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 ([arxiv:math/0702075](https://arxiv.org/abs/math/0702075))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Eigentheory of Cayley-Dickson algebras_,
Forum Mathematicum 21(5) (2009), 833-851 ([arxiv:0905.2987](https://arxiv.org/abs/0905.2987))

On [[model categories of spectra]]:

* [[Daniel Dugger]], *Stable categories and spectra via model categories* ([pdf](https://pages.uoregon.edu/ddugger/models_chapter.pdf)), in:  [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press

On [[Kronholm's freeness theorem]] in [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]

* [[Daniel Dugger]], [[Christy Hazel]], [[Clover May]], *Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$*, Journal of Pure and Applied Algebra **228** 3 (2024) 107473 [[arXiv:2203.05287](https://arxiv.org/abs/2203.05287), [doi:10.1016/j.jpaa.2023.107473](https://doi.org/10.1016/j.jpaa.2023.107473)]

## Related entries

* [[model structure on simplicial presheaves]]

* [[descent for simplicial presheaves]]

* [[combinatorial model category]]

[[!redirects Dan Dugger]]

category: people


Daniel F. Styer

* [personal page](https://www2.oberlin.edu/physics/dstyer/)

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

## Selected writings

On [[path integral quantization]]:

* [[Daniel F. Styer]], *The Errors of Feynman and Hibbs* [[pdf](https://www2.oberlin.edu/physics/dstyer/FeynmanHibbs/ErrorsOfFeynmanHibbsX.pdf)]

* [[Daniel F. Styer]], [[Richard Feynman]], [[Albert R. Hibbs]], *Quantum Mechanics and Path Integrals: Emended Edition*, Dover (2010) [[ISBN:0486477223](https://store.doverpublications.com/0486477223.html), [pdf](http://www-f1.ijs.si/~ramsak/km1/FeynmanHibbs.pdf)]

category: people

[[!redirects Daniel Styer]]