Joël Scherk

Joël Scherk (1946-1980)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Jo%C3%ABl_Scherk)

* [van Nieuwenhuizen 1981](Peter+van+Nieuwenhuizen#Nieuwenhuizen81):

  > **In memoriam Joël Scherk**

  > Joël Scherk died on May 16, 1980. Although 33 years old, he had made many very important discoveries in dual [[string theory]] and [[supergravity]], as will be manifest in this review. It is striking to note how logical and sustained the development of his thinking was.

  > His first important contribution was the [[renormalization]] of the [[one-loop]] [[open string|open]] dual [[string scattering|string amplitude]]. Then he developed the idea of the zero-slope limit when all masses except the lightest tend to infinity, and found that this leads to [[quantum field theories]]. In particular the [[closed string]] sector yielded [[general relativity]], and from then on he was fascinated by [[gravitation]]. The [[one-loop]] contributions of the [[open string|open string model]] in $d = 26$ dimensions [[Goddard-Thorn no-ghost theorem|were shown]] to correspond to a new ghost-free set of states, corresponding to closed Strings and interpreted as Pomeron states. One of these states was described by an [[B-field|antisymmetric tensor gauge field]], which is nowadays of interest. Then he started wondering what happens with the extra dimensions in our four-dimensional world. This led him to the idea of spontaneous compactification of space-time, extending [[Kaluza-Klein compactification|ideas of Kalusza and Klein]].

  > In [[supergravity]] he started by solving the matter [[interaction|coupling]] problem and worked out the super-[[Higgs effect]]. Then he constructed the [[D=4 N=3 supergravity|$N = 3$ supergravity]], and both the $SO(4)$ and the $SU(4)$ versions of [[D=4 N=4 supergravity|$N=4$ extended supergravity]]. The latter model was suggested by taking the zero slope limit of the [[superstring theory|$d = 10$ dimensional closed string dual model with fermions]]. The $d = 10$ open string model yielded in the zero slope limit [[D=4 N=4 SYM|$N = 4$ supersymmetric Yang—Mills theory]]. In the $SU(4)$ model he found a noncompact $SU(1, 1)$ [[global symmetry]], and in the $N = 2, 3, 4$ models he gave a complete treatment of the duality and chiral symmetries. He then formulated [[D=11 N=1 supergravity|$N = 1 d = 11 supergravity$]], which yields the [[D=4 N=8 supergravity|$N=8$ $d=4$ model]] upon [[Kaluza-Klein reduction|dimensional reduction]]. He also developed the idea of [[spontaneous symmetry breaking]] by means of dimensional reduction, and worked out the particle spectrum. There he found a massless vector boson and proposed the idea that it yields the (testable) force of antigravity.

  > Despite this impressive record, Joel was a relaxed and very friendly person. He was among the best and most pedagogical speakers and his articles were crystal clear. He was always open to discussion and had a very developed sense of humour. He also contributed actively to the popularization of [[theoretical physics]] by radio talks, lectures for lay-men and articles in popular scientific magazines.

## Selected writings

On the early theory of [[relativistic strings]] ([[Nambu-Goto action]]) then motivated a explanations of the "dual resonance model" for [[hadrons]] (cf. *[[Polyakov gauge-string duality]]*):

* [[Joël Scherk]], *An introduction to the theory of dual models and strings*, Rev. Mod. Phys. **47** 123 (1975) [[doi:10.1103/RevModPhys.47.123](https://doi.org/10.1103/RevModPhys.47.123)]

On the [[GSO projection]] for the [[spinning string]]/[[superstring]]:

* {#GSO77} F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supersymmetry, supergravity theories and the dual spinor model_, Nucl. Phys, B122, 253 (1977)

* {#GSO77} F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supersymmetry, Supergravity Theories and the Dual Spinor Model_, Nucl. Phys. B 122 (1977), 253 ([spire](http://inspirehep.net/record/111434))

Introducing [[D=11 supergravity]]:

* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. B **76** (1978) 409 [doi:10.1016/0370-2693(78)90894-8]

On the [[Scherk-Schwarz mechanism]]:

* {#ScherkSchwarz79} [[Joël Scherk]], [[John Schwarz]], _Spontaneous breaking of supersymmetry through dimensional reduction_, Physics Letters B Volume 82, Issue 1, 12 March 1979, Pages 60-64 (doi:10.1016/0370-2693(79)90425-8)

* {#ScherkSchwarz89} [[Joël Scherk]], [[John Schwarz]], _How to get masses from extra dimensions_, in: Supergravities in Diverse Dimensions, pp. 1282-1309 (1989)  ([doi:10.1142/9789814542340_0083](https://doi.org/10.1142/9789814542340_0083))

[[!redirects Joel Scherk]]

category: people


Juan Carlos Rocha Barriga

## Selected writings

On [[twisted differential K-theory]] and its application to the [[K-theory classification of D-brane charge]]:

* [[Fabio Ferrari Ruffino]], [[Juan Carlos Rocha Barriga]], _Twisted differential K-characters and D-branes_, Nuclear Physics B Volume 960, November 2020, 115169 ([arXiv:2009.04223](https://arxiv.org/abs/2009.04223). [doi:10.1016/j.nuclphysb.2020.115169](https://doi.org/10.1016/j.nuclphysb.2020.115169))

category: people

[[!redirects Juan Barriga]]


Juan González-Meneses

* [personal page](https://personal.us.es/meneses/)

* [GoogleScholar entry](https://scholar.google.com/citations?user=ELNo00oAAAAJ&hl=es)

## Selected writings

On [[braid groups]]:

* {#González-Meneses11} [[Juan González-Meneses]], *Basic results on braid groups*, Annales Mathématiques Blaise Pascal, Tome 18 1 (2011) 15-59 [[ambp:AMBP_2011__18_1_15_0](https://ambp.centre-mersenne.org/item/?id=AMBP_2011__18_1_15_0)]

category: people

[[!redirects Juan Gonzalez-Meneses]]


Juan José Manjarín

* [Mathematics Genealogy page](http://www.genealogy.ams.org/id.php?id=144487)

## related $n$Lab entries

* [[D-brane]], [[D6-brane]]

category: people

[[!redirects Juan Jose Manjarin]]

[[!redirects Juan Manjarín]]
[[!redirects Juan Manjarin]]


Juan Maldacena

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

* [wikipedia entry](http://en.wikipedia.org/wiki/Juan_Mart%C3%ADn_Maldacena)

## Selected writings

On microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:

* [[Curtis Callan]], [[Juan Maldacena]], _D-brane Approach to Black Hole Quantum Mechanics_, Nucl. Phys. B472:591-610, 1996 ([arXiv:hep-th/9602043](http://arxiv.org/abs/hep-th/9602043))

Discussion of [[black hole entropy]] of [[D2-D6 brane bound states]] as [[black holes in string theory]]:

* [[Juan Maldacena]], [[Andrew Strominger]], _Statistical Entropy of Four-Dimensional Extremal Black Holes_, Phys. Rev. Lett. 77:428-429, 1996 ([arXiv:hep-th/9603060](https://arxiv.org/abs/hep-th/9603060))

On [[D1-D3 brane intersections]] as spikes/[[BIons]] in the [[D3-brane]] [[DBI action|DBI-theory]]:

* [[Curtis Callan]], [[Juan Maldacena]], _Brane Dynamics From the Born-Infeld Action_, Nucl. Phys. B513 (1998) 198-212 ([arXiv:hep-th/9708147](https://arxiv.org/abs/hep-th/9708147))

On the [[AdS-CFT correspondence]]:

* {#Maldacena97a} [[Juan Maldacena]], _The Large N limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998 ([hep-th/9711200](http://arxiv.org/abs/hep-th/9711200))

* {#Maldacena97b} [[Juan Maldacena]], _Wilson loops in Large $N$ field theories_, Phys. Rev. Lett. __80__ (1998) 4859 ([hep-th/9803002](http://arxiv.org/abs/hep-th/9803002))

* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys.Rept.323:183-386,2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

On [[supergravity]], [[D-branes]] and the [[large N limit]] of [[super Yang-Mills theories]]:

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

On [[AdS3-CFT2]] for [[D1/D5 brane bound states]] and [[black hole entropy]] [[black holes in string theory|in string theory]]:

* [[Robbert Dijkgraaf]], [[Juan Maldacena]], [[Gregory Moore]], [[Erik Verlinde]], _A Black Hole Farey Tail_ ([arXiv:hep-th/0005003](https://arxiv.org/abs/hep-th/0005003), [spire:526744](http://inspirehep.net/record/526744))

On [[Inönü-Wigner contraction]] of [[anti de Sitter spacetimes]] to [[pp-wave spacetimes]], [[single trace operators]] in the [[AdS/CFT correspondence]], and introducing the [[BMN matrix model]]:

* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], Section 2 of: _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP 0204 (2002) 013 ([arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021))

On [[boundary conditions]] ([[BCFT]]/[[D-branes]]) for the [[gauged WZW model]] via [[parafermions]]:

* [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], *Geometrical interpretation of D-branes in gauged WZW models*, JHEP 0107:046 (2001) $[$[arXiv:hep-th/0105038](https://arxiv.org/abs/hep-th/0105038), [doi:10.1088/1126-6708/2001/07/046](https://doi.org/10.1088/1126-6708/2001/07/046)$]$

On [[M5-branes]] in the [[BMN matrix model]]:

* [[Juan Maldacena]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Transverse Fivebranes in Matrix Theory_, JHEP 0301:038, 2003 ([arXiv:hep-th/0211139](https://arxiv.org/abs/hep-th/0211139))

and a limit where aspects of [[little string theory]] become visible:

* [[Hai Lin]], [[Juan Maldacena]], _Fivebranes from gauge theory_, Phys. Rev. D74:084014, 2006 ([arXiv:hep-th/0509235](https://arxiv.org/abs/hep-th/0509235))

Proposing [[ER = EPR]]:

 * {#MaldacenaSusskind13} [[Juan Maldacena]], [[Leonard Susskind]], _Cool horizons for entangled black holes_, Fortsch. Phys. 61 (2013) 781-811, 2013 ([arXiv:1306.0533](http://arxiv.org/abs/1306.0533), [doi:10.1002/prop.201300020](https://doi.org/10.1002/prop.201300020))

General aspects of [[nearly AdS2-CFT1]] ([[JT-gravity]]/matrix models):

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ([arXiv:1606.01857](https://arxiv.org/abs/1606.01857))

Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]

On  [[AdS2/CFT1]] with the [[BFSS matrix model]] on the CFT side and [[black hole in string theory|black hole-like solutions]] in [[type IIA supergravity]] on the AdS side:

* [[Juan Maldacena]], Alexey Milekhin, _To gauge or not to gauge?_, JHEP 04 (2018) 084 ([arxiv:1802.00428](https://arxiv.org/abs/1802.00428))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020)  \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\]

Review:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]

On [[magnetic black holes]]:

* [[Juan Maldacena]], *Comments on magnetic black holes*,  J. High Energ. Phys. **2021** 79 (2021) [[arXiv:2004.06084](https://arxiv.org/abs/2004.06084), doi:10.1007/JHEP04(2021)079]

Exposition of the [[BFSS matrix model]] in view of the [[holographic principle]]:

* [[Juan Maldacena]], *A simple quantum system that describes a black hole* [[arXiv:2303.11534](https://arxiv.org/abs/2303.11534)]

On [[graviton]] [[scattering amplitudes]] in [[D=11 supergravity]] computed via the [[BFSS matrix model]]:

* [[Aidan Herderschee]], [[Juan Maldacena]], *Three Point Amplitudes in Matrix Theory* [[arXiv:2312.12592](https://arxiv.org/abs/2312.12592)]

and the [[soft graviton theorem]]:

* [[Aidan Herderschee]], [[Juan Maldacena]], *Soft Theorems in Matrix Theory* [[arXiv:2312.15111](https://arxiv.org/abs/2312.15111)]

## Related entries

* [[AdS/CFT correspondence]]

* [[holographic principle]]

category: people


Juan Orendain

* [institute page](https://www.qiss.fr/juan-orendain/)

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

## Selected writings

On [[higher gauge theory|higher]] [[lattice gauge theory]]:

* [[Juan Orendain]], [[José A. Zapata]], *Higher homotopy and lattice gauge fields* [[arXiv:2311.02363](https://arxiv.org/abs/2311.02363)]

category: people


Juan Pablo Paz

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

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

## Selected writings

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

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

category: people

[[!redirects Juan P. Paz]]
[[!redirects Juan Paz]]


Juan Polo Gomez

Juan Polo is postdoc at the [Technology Innovation Institute](https://www.tii.ae/) (TII) in Abu Dhabi

* [webpage](https://www.tii.ae/team/dr-juan-polo)

## Selected writings

On [[atomtronics]]:

* [[Luigi Amico]], [[Malcolm G. Boshier]], [[Juan Polo Gomez]] et al.: *Roadmap on Atomtronics: State of the art and perspective*, AVS Quantum Sci. **3** 039201 (2021) [[arXiv:2008.04439](https://arxiv.org/abs/2008.04439), [doi:10.1116/5.0026178](https://doi.org/10.1116/5.0026178)]

[[!redirects Juan Polo]]


Jucys-Murphy element

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

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$, consider the [[complex numbers|complex]] [[group algebra]] $\mathbb{C}[Sym(n)]$ of the [[symmetric group]] on $n$ elements.

\begin{defn}\label{JucycMurphyElements}

The *Jucys-Murphy elements*

$$
  J_k \;\in\; \mathbb{C}[Sym(n)] \;\;\;\; k \in \{1, \cdots, n\}
$$

are the following [[sums]] of [[transpositions]] $(i,j) \in Sym(n) \subset \mathbb{C}[Sym(n)]$ in this [[group algebra]]:

\[
  \label{J1IsZero}
  J_1 \;\coloneqq\; 0
\]

\[
  \label{JkAsSum}
  J_k \;\coloneqq\; \underoverset{i = 1}{k-1}{\sum} (i,k)
\]

\end{defn}

([Murphy 81, p. 287 (1 of 11)](#Murphy81), [Murphy 83, p. 259 (2 of 8)](#Murphy83))

Notice that $J_k$ may be understood as the sum of all transpositions in $Sym(k)$ that are not in the [[subgroup]] $Sym(k-1) \subset Sym(k)$.

## Properties

### Eigenvalues

Write $sYTableaux_n$ for the [[set]] of [[standard Young tableaux]] with $n$ boxes, and write

\[
  \label{VectorsInSeminormalRepresentations}
  \big\{ v_{T,k} \in \mathbb{C}[Sym(n)]\big\}_{
    { T \in sYTableaux_n}
    \atop
    { 1 \leq m \leq dim(S^{(\left\vert T\right\vert)})  }
  }
\]

for the [[Gelfand-Tsetlin basis]] (Young's [[seminormal representation]]).

We will notationally suppress the multiplicity index $m$ in the following.

\begin{prop}\label{EigenvaluesOfJMElements}

  The $v_T$ (eq:VectorsInSeminormalRepresentations) are joint [[eigenvectors]] of the Jucys-Murphy elements (Def. \ref{JucycMurphyElements}) for [[eigenvalues]] the "content" $j - i$ of the box $(i,j)$ that contains the number $k$ in the standard Young tableau
$$
  J_k v_T \;=\; (j - i) v_T
  \,,
  \;\;\;
  T_{i,j} = k
  \,.
$$

\end{prop}

In particular, the Jucys-Murphy elements all commute with each other.

This is due to [Jucys 71](#Jucys71), recalled as [Jucys 74 (12)](#Jucys74), and independently due to [Murphy 81 (3.8)](#Murphy81).

### Factorization of Cayley-Distance kernel

\begin{prop}\label{FactorizationOfCayleyDistanceKernelAsCharPolynomialOfJMElements}
 The [[characteristic polynomial]] of the Jucys-Murphy elements is proportional to the [[Cayley distance kernel]]:
$$
  \big(
    t + J_1
  \big)
  \big(
    t + J_2
  \big)
  \cdots
  \big(
    t + J_n
  \big)
  \;=\;
  \underset{
    \sigma \in Sym(n)
  }{\sum}
  e^{ ln(t) \cdot \# cycles(\sigma) }
  \, \sigma
  \;\;\;\;\;
  \in
  \mathbb{C}[Sym(n)][t]
$$

\end{prop}

On the [Wikipedia page](https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element) this is attributed to Jucys, but without explicit reference. It might be in [Jucys 71](#Jucys71) (but is not mentioned in the review in [Jucys 74](#Jucys74)). However, it is not much of a theorem, anyways:
\begin{proof}
By [[induction]]:

For $n = 1$ the claim reduces to

$$
  \big(
    t + J_1
  \big)
  \;=\;
  e^{ ln(t) \cdot 1 }
  \,,
$$

which holds by (eq:J1IsZero).

Now assume that the statement is true for $n \in \mathbb{N}$.

Observe that every permutation $\sigma \in Sym(n+1)$ may be written as product

$$
  \sigma
  \;=\;
  c_1(\sigma)
  \cdot
  c_2(\sigma)
  \cdots
  c_{\# cycles(\sigma)}(\sigma)
$$

of products of transpositions of the form

$$
  c_i(\sigma)
  \;\;
  \;=\;
   (i_{\ell_i-1},i_{\ell_i-2})
   \circ
   \cdots
   \circ
   (i_3,i_2)
   \circ
   (i_2,i_1)
   \circ
   (i_1,i_{\ell_i})
  \,,
$$

where $\ell_i$ is the length of the $i$th [[permutation cycle]].

Here we may assume without restriction that

$$
  i_j
    \lt
  i_{\ell_i}
  \,,
  {\phantom{AAA}}
  \text{and}
  {\phantom{AAA}}
  i \lt j
   \;\;
   \Rightarrow
   \;\;
   i_{\ell_i} \lt j_{\ell_j}
$$

because if not then we may (1) rearrange the $c_i(\sigma)$ within their product (these commute with each other, since they contain permutations among elements within distinct cycles) and (2) cyclically rearrange the factors withing each $c_i(\sigma)$.

Once all permutations are uniquely represented as products this way, the representative of any $\sigma \in Sym(n+1)$ is uniquely obtained from the representative of an element in $Sym(n)$, by either:

1. multiplying from the right with a transposition $(k, n+1)$ for $k \leq n$, in which case the $n+1$st element will be part of a cycle that was already present;

1. by retaining the previous product as is, in which case the $n+1$st element will be its own new cycle.

This means that

$$
    \underset{\sigma \in Sym(n+1)}{\sum}
    t^{ \# cycles(\sigma) }
    \,
    \sigma
    \;
    =
    \;
    \underset{\sigma' \in Sym(n)}{\sum}
    t^{ \# cycles(\sigma') }
    \,
    \sigma' (t + J_{n+1})
$$
and hence the claim follows by the induction assumption.
\end{proof}

Combining
Prop. \ref{EigenvaluesOfJMElements} with
Prop. \ref{FactorizationOfCayleyDistanceKernelAsCharPolynomialOfJMElements}
yields:

\begin{corollary}

  The [[eigenvalues]] of the [[Cayley distance kernel]]
  $$
    e^{ - ln(t) \cdot d_C }
    \;=\;
    e^{- ln(t) \cdot n}
    \underset{
      \sigma \in Sym(n)
    }{\sum}
    e^{ ln(t) \cdot \# cycles(\sigma) }
    \, \sigma \cdot
  $$
  (regarded on the right as the linear map given by right multiplication in the [[group algebra]]) are [[indexed set|indexed]] by [[Young diagrams]] $\lambda$ and are given by
  $$
    EigVals[e^{- ln(t) \cdot d_C}]_\lambda
    \;=\;
    e^{ - ln(t) \cdot  n}
    \underset{
      { 1 \leq i \leq rows(\lambda) }
      \atop
      { 1 \leq j \leq \lambda_i }
    }{\prod}
    \big(
      t - j + i
    \big)
  $$

\end{corollary}

An alternative derivation of this statement, using the formula for [[Cayley graph spectra]] and the [[hook length formula]]/[[hook-content formula]] is [this Prop.](Cayley+distance+kernel#EigenvaluesInTermsOfJustBoxContent) at *[[Cayley distance kernel]]* ([CSS 21](Cayley+distance+kernel#CSS21)).

### Projectors
 {#Projectors}

[Murphy 81, p. 291 (5 of 11)](#Murphy81)

[Murphy 83, p. 259 (2 of 8)](#Murphy83)

[MathOverflow:a/151375](https://mathoverflow.net/a/151375/381)

## References

The Jucys-Murphy elements are independently due to:

* {#Jucys71} [[Algimantas Adolfas Jucys]], *Factorization of Young projection operators for the symmetric group*, Lietuvos Fizikos Rinkinys, **11** (1) 9 (1971) [[journal content](http://www.itpa.lt/%7Elfd/Lfz/Turiniai/Turi1971.html#), [[Jucys-1971.pdf:file]]]

* {#Jucys74} [[Algimantas Adolfas Jucys]], *Symmetric polynomials and the center of the symmetric group ring*, Reports on Mathematical Physics, Volume 5, Issue 1, February 1974, Pages 107-112 (doi:10.1016/0034-4877(74)90019-6)

and

* {#Murphy81} [[G. E. Murphy]], *A new construction of Young's seminormal representation of the symmetric groups*, Journal of Algebra Volume 69, Issue 2, April 1981, Pages 287-297 (doi:10.1016/0021-8693(81)90205-2)

* {#Murphy83} [[G. E. Murphy]], *The idempotents of the symmetric group and Nakayama's conjecture*, Journal of Algebra Volume 81, Issue 1, March 1983, Pages 258-265 ([pdf](https://core.ac.uk/download/pdf/82692868.pdf))

* {#Murphy92} [[G. E. Murphy]], *On the representation theory of the symmetric groups and associated Hecke algebras*, J. Algebra 152 (1992) 492–513 ([pdf](https://core.ac.uk/download/pdf/82422274.pdf), doi:10.1016/0021-8693(92)90045-N))

(According to [Vershik-Okounkov 04, Footnote 2](representation+theory+of+the+symmetric+group#VershikOkounkov04), [Jucys 74](#Jucys74) "remained unnoticed" until [Murphy 81](#Murphy81) first re-discovered the results, and then Jucys' paper. In fact [Jucys 71](#Jucys71), which must have the actual proofs, seems to remain electronically unavailable.)

Review:

* Wikipedia, *[Jucys-Murphy element](https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element)*

Further discussion:

* {#Chan18} Kelvin Tian Yi Chan, *Induction Relations in the Symmetric Groups and Jucys-Murphy Elements*, 2018 ([hdl:10012/13601](http://hdl.handle.net/10012/13601), [pdf](https://uwspace.uwaterloo.ca/bitstream/handle/10012/13601/Chan_Kelvin.pdf?sequence=3&isAllowed=y))

[[!redirects Jucys-Murphy elements]]


Judea Pearl

Judea Pearl is a recipient of the Turing prize with major work on causality. He works in the Computer Science Department
Cognitive Systems Lab. at UCLA.

* [Webpage](http://bayes.cs.ucla.edu/jp_home.html)

##Key publications:

* (with Dana MacKenzie), _The Book of Why, The New Science of Cause and Effect_.

* _Causal inference in statistics: An overview_, Statistics Surveys __3__ (2009) 96–146[doi](https://doi.org/10.1214/09-SS057) [pdf](https://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf)

with a book review of it

* _Causality: models, reasoning, and inference_, Cambridge University Press 2000 (book review [pdf](http://bayes.cs.ucla.edu/BOOK-2K/neuberg-review.pdf))

category: people


judgements for types and terms - table

| [[type theory]] | [[category theory]] |
|--|--|
| [[syntax]] | [[semantics]] |
| **[[judgment]]** | **[[diagram]]** |
| [[type]] | [[object]] in [[category]] |
| $\vdash\; A \; \mathrm{type}$ | $A \in \mathcal{C}$ |
| [[term]] | [[element]] |
| $\vdash\; a \colon A$ | $* \stackrel{a}{\to} A$ |
| [[dependent type]] |  [[object]] in [[slice category]] |
| $x \colon X \;\vdash\; A(x) \; \mathrm{type}$ | $\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}$ |
| [[term in context]] | [[generalized elements]]/[[element]] in [[slice category]] |
| $x \colon X \;\vdash \; a(x)\colon A(x)$ | $\array{X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{}} \\ && X}$ |
| $x \colon X \;\vdash \; a(x)\colon A$ | $\array{X &&\stackrel{(id_X,a)}{\to}&& X \times A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_1}} \\ && X}$ |


judgment

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Judgments
* table of contents
{: toc}

## Idea
 {#Idea}

In [[mathematics]] and especially in [[formal logic]], by a *judgement* [[Frege (1879, §2)](#Frege1879)] one means the *assertion* [[Russell & Whitehead (1910, p. xviii)](#RussellWhitehead1910)] or *validation* [[Kochen (1961, p. 2)](#Kochen61)] of a [[proposition]] $P$, hence the claim (or even the specification) of a [[proof]] of $P$ [[Church (1940, §5)](#Church40)].

To bring out the subtle (cf. [jstor:43154266](https://www.jstor.org/stable/43154266)) but crucial distinction --- eloquently amplified by [Martin-Löf (1996, lectures 1-2)](#Martin-Löf96) --- between the notion of a [[proposition]] $P$ as such and the judgement/assertion/validation/proof of its [[truth]], it is tradition to prefix the latter with the symbol "$\vdash$", known as the *Urteilsstrich* (German: "judgement stroke") of [[Frege (1879, §2)](#Frege1879)] (see the historical pointers [below](#History)), as:

\[
  \label{JudgementInIntroduction}
  \vdash \; P
  \,.
\]

For example, the [[Riemann hypothesis]] certainly is a [[proposition]], but it remains open whether we may truthfully assert or validate it, hence we cannot yet *judge* that it holds true: No [[proof]] is known, even though it seems likely to be in fact [[true]].

In other words, the judgement symbols "$\vdash \, P$ " convey that *we know* (a [[proof]] of, hence the [[truth]] of) $P$ [[Martin-Löf (1996, lectures 1-2)](#Martin-Löf96)].

As such, judgements may be formalized as "meta-proposition", namely [[propositions]] but in an ambient [[meta-language]] (the [[deductive system]] or [[logical framework]]) rather than in the [[object language]] in which the given $P$ itself is formulated. (More specifically, any [[deductive system]] includes, as part of its specification, which strings of symbols are to be regarded as the _judgments_.  Some of these symbols may themselves express a [[proposition]] in the object language, but this is not necessarily the case.)

More generally, one considers *hypothetical judgements* (maybe first made explicit in [[Church (1940, §5)](#Church40)]) which are such assertions/validations but conditioned on premises, hence claims/specifications of [[proofs]] based on assumptions $A$: Writing

\[
 \label{HypotheticalJudgementInIntroduction}
  A \;\;\vdash\;\; P
\]

is to express the judgement that *given* any assertion/validation/proof of $A$ we know how to produce (from that) a validation/proof of $P$.

In systems of ([[dependent type theory|dependent]]) [[type theory]] (where such notation seems to go back to [Hofmann (1995, p. 31)](#Hofmann95), cf. [below](#MLDidNot)) this is quite literally the case in that the [[categorical semantics]] of such a hypothetical judgement is a given by a [[function]] (and typically a [[computable function]], hence an [[algorithm]]) which [[map|maps]] proofs of $A$ to proofs of $P$.

More generally yet, in [[type theory]] the [[propositions]] $A$ and $P$ appearing here are allowed to be any ([[dependent type|dependent]]) [[types]] (see at *[[propositions as types]]*). In this generality, the [[denotational semantics]] of hypothetical judgements (eq:HypotheticalJudgementInIntroduction) is given by (computable) [[functions]]/[[morphisms]] with [[domain]] $A$ and [[codomain]] $P$. In [[programming languages]] based on such [[type theories]] this translates to the denotation of an actual [[program]] which produces [[data]] of [[data type]] $P$ when run on input [[data]] of [[data type]] $A$.

Judgements are themselves subject to a calculus of [[natural deduction]] by which further judgements may be inferred from given ones. For example the notation

$$
  \frac{
    A \;\;\vdash\;\; P_1
    ,\,
    \;\;\;\;\;\;\;
    A \;\;\vdash\;\; P_2
  }{
    A
      \;\;\vdash\;\;
    P_1 \times P_2
  }
$$

traditionally expresses that whenever we can assert both $P_1$ as well as $P_2$ assuming a validation of $A$ (hence: given a [[pair]] of [[functions]] with [[domain]] [[type]] $A$), then with that same assumption we find assertion/validation/proof of their [[logical conjunction]] $P_1 \times P_2$ (namely a [[function]] with [[codomain]] to their [[product type]]).

Finally, to make explicit which particular assumptions a hypothetical judgement really depends on, it is common to adjoin a generic [[context]] symbol $\Gamma$ to the hypothesis on the left hand side of the *Urteilsstrich*: Notation such as

$$
  \Gamma
  ,\,\;\;
  a \colon A
  \;\;\;
  \vdash
  \;\;\;
  p_a \,\colon\, P
$$

expresses the judgement that $P$ may be asserted given validation of $\Gamma \times A$, but that in doing so only the validation of $A$ is actually made use of, while $\Gamma$ may be any [[context]] of assumptions (whose validation might however be used in previous/further [[natural deduction|deduction]] steps).

## Examples

### In first-order logic

In [[first-order logic]], a paradigmatic example of a judgement is the judgement that a certain string of symbols is a well-formed [[proposition]].  This is often written as "$P \;prop$", where $P$ is a [[metavariable]] standing for a string of symbols that denotes a proposition.

Another example of a judgement is the judgement that these symbols form a proposition [[proof|proved]] to be [[true]].  This judgment is often written as "$P\;true$".

Neither of these judgements is the same thing as the proposition $P$ itself.  In particular, the proposition is a statement *in* the logic, while the judgement that the proposition is a proposition, or is provably true, is a statement *about* the logic.  However, often people abuse notation and conflate a proposition with the judgment that it is true, writing $P$ instead of $P\;true$.

### In type theory

The distinction between judgements and [[propositions]] is particularly important in [[intensional type theory]].

The paradigmatic example of a judgment in [[type theory]] is a *typing judgment*.  The assertion that a [[term]] $t$ has [[type]] $A$ (written "$t:A$") is not a statement *in* the type theory (that is, not something which one could apply logical operators to in the type-theoretic system) but a statement *about* the type theory.

Often, type theories include only a particular small set of judgments, such as:

* typing judgments (written $t:A$, as above)

* judgments of typehood (usually written $A \;type$)

* judgments of [[equality]] between typed terms (written say $(t=t'):A$)

(In a type theory with a [[type of types]], judgments of typehood can sometimes be incorporated as a special case of typing judgments, writing $A:Type$ instead of $A\;type$.)

These limited sets of judgments are often defined [[inductive definition|inductively]] by giving [[type formation]]/[[term introduction]]/[[term elimination]]- and [[computation rules]] (see [[natural deduction]]) that specify under what hypotheses one is allowed to conclude the given judgment.

These inductive definitions can be formalized by choosing a particular [[type theory]] to be the meta-language; usually a very simple type theory suffices (such as a [[dependent type theory]] with only [[dependent product types]]).  Such a meta-type-theory is often called a [[logical framework]].

## Hypothetical and generic judgments

It may happen that a judgment $J$ is only derivable under the assumptions of certain other judgments $J_1,\dots, J_2$.  In this case one writes
$$ J_1,\dots,J_n \;\vdash J.$$
Often, however, it is convenient to incorporate hypotheticality into judgments themselves, so that $ J_1,\dots,J_n \;\vdash J $ becomes a single _[[hypothetical judgment]]_.  It can then be a consequence of other judgments, or (more importantly) a hypothesis used in concluding other judgments.  For instance, in order to conclude the truth of an [[implication]] $\phi\Rightarrow\psi$, we must conclude $\psi$ *assuming* $\phi$; thus the [[introduction rule]] for implication is
$$ \frac{\phi \;\vdash\; \psi}{\vdash\; \phi\Rightarrow\psi}$$
with a hypothetical judgment as its hypothesis.  See [[natural deduction]] for a more extensive discussion.

In a [[type theory]], we may also consider the case where the hypotheses $J_1$ are typing judgments of the form $x:A$, where $x$ is a [[variable]], and in which the conclusion judgment $J$ involves these variables as [[free variables]].  For instance, $J$ could be $\phi\;prop$, where $\phi$ is a valid (well-formed) proposition only when $x$ has a specific [[type]] $X$.  In this case we have a _[[generic judgement]]_, written
$$ (x \colon X) \;\vdash\; (\phi \; prop). $$
which expresses that _assuming the hypothesis or [[antecedent]] judgement_ that $x$ is of type $X$, as a consequence we have the [[succedent]] judgement that $\phi$ is a proposition. If on the right here we have a typing judgment

$$
  (x \colon X) \;\vdash\; (t \colon A)
$$

we have a _[[term in context]]_.

For more about the precise relationship between the various meanings of $\vdash$ here, see [[natural deduction]] and [[logical framework]].

While this may seem to be a very basic form of (hypothetical/generic) judgement only, in systems such as [[dependent type theory]] or [[homotopy type theory]], all of [[logic]] and a good bit more is all based on just this.

## History of the notion and notation
 {#History}

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The terminology of *judgement* in [[formal logic]], and the notation

$$
  \vdash A
$$

for the *judgement* of *content* $A$ is due to [Frege (1879, §2)](#Frege1879) (Frege's *das Urtheil* -- or *das Urteil* in modern German spelling -- directly translates to: *the judgement*).

{#RusselWhiteheadAssertion} This notion and notation (often "Frege's Urteilsstrich" -- "judgement stroke") was adopted by [Russell & Whitehead (1910, p. xviii)](#RussellWhitehead1910), who however speak of "assertion" instead of "judgement"  --- see [Martin-Löf (1996, p. 6)](#Martin-Löf96) for speculation that these authors may have wanted "to avoid any association with Kantian philosophy" (cf. [below](#KantTheoryOfJudgement)):

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\end{imagefromfile}

and then by [Church (1940, §5)](#Church40) (without any attribution) --- now allowing hypotheses/assumptions on the left of the *Urteilsstrich* --- to mean *provable* under these assumptions:

\begin{imagefromfile}
    "file_name": "Church-ProvableUnderAssumptions.jpg",
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    "margin": {
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        "bottom": 20,
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        "left": 20
    }
\end{imagefromfile}

Similarly, [Kochen (1961, p. 2)](#Kochen61) introduced (in a context of [[model theory]]) the notation "$\mathfrak{A} \;\vDash\; P$" (apparently not actively remembering its historical origin, cf. [Kochen (2017)](#Kochen17)) for a notion in [[model theory]] pronounced  "$\mathfrak{A}$ *validates* $P$" --- which is at least closely related to these hypothetical judgements (and often used synonymously, e.g. [here](https://open.conted.ox.ac.uk/sites/open.conted.ox.ac.uk/files/resources/Create%20Document/Turnstiles.pdf)):

\begin{imagefromfile}
    "file_name": "Kochen-Validation.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 10,
        "left": 20
    }
\end{imagefromfile}

The need (which had been contested by some logicians such as [[Ludwig Wittgenstein|Wittgenstein]], cf. [jstor:43154266](https://www.jstor.org/stable/43154266)) for Frege's distinction between a proposition "$A$", as such, and its judgement/assertion/proof/validation "$\vdash A$" was eloquently argued for by [Martin-Löf (1984, pp. 2)](#Martin-Löf84), [(1987)](#Martin-Löf87) and in [(1996, lectures 1-2)](#Martin-Löf96), together with a re-promotion of Frege's *Urteilsstrich* notation "$\vdash$" ([ML84, p. 2](#Martin-Löf84),  [ML96, p. 2, 6](#Martin-Löf96)):

\begin{imagefromfile}
    "file_name": "Martin-Loef--Judgement.jpg",
    "width": 620,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 10,
        "left": 20
    }
\end{imagefromfile}

The notation for *hypothetical* judgements

\[
  \label{GenericHypotheticalJudgement}
  context \;\vdash\; judgement
\]

was then (without attribution to Church or Martin-Löf) adopted for [[dependent type theory]] in [Hofmann (1995, p. 31)](#Hofmann95), later published as [Hofmann (1997, p. 82)](#Hofmann97):

\begin{imagefromfile}
    "file_name": "Hofmann-judgements.jpg",
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    "margin": {
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\end{imagefromfile}

This eventually became the trademark of the practice of [[dependent type theory]] (though mostly used without any attribution), e.g.: [Jacobs (1998, p. 2, 121, 586)](dependent+type+theory#Jacobs98); [Bauer, Haselwarter & Lumsdaine (2020)](#BauerHaselwarterLumsdaine20).

{#MLDidNot} (Curiously, apparently [[Per Martin-Löf|Martin-Löf]] did not actually use the general notation (eq:GenericHypotheticalJudgement) in publications: When in [1996, p. 29](#Martin-Löf96) it comes to hypothetical judgements, he switched to using just a vertical "$\vert$" instead of "$\vdash$".)

{#HistorySummary} In summary:

| terminology | author |
| ------------|--------|
| *judgement* | [Frege (1879)](#Frege1879) |
| *assertion* | [Russell & Whitehead (1910)](#RussellWhitehead1910) |
| *proof*     | [Church (1940)](#Church40) |
| *validation* | [Kochen (1961)](#Kochen61) |
| *judgement* | [Martin-Löf (1984](#Martin-Löf84), [1996)](#Martin-Löf96) <br/> [Hofmann (1995](#Hofmann95), [1997)](#Hofmann97) |

\linebreak

## Related concepts

* [[infinite judgment]]

* [[natural deduction]], [[sequent calculus]]

* [[context]], [[context extension]]

* [[type theory]], [[dependent type theory]]

[[!include mathematical statements --- contents]]

## References
 {#References}

{#PreHistory} Pre-history of the notion of judgement (*Urteil*) in (informal) logic:

* {#KantTheoryOfJudgement} SEP: *[Kant’s Theory of Judgment](https://plato.stanford.edu/entries/kant-judgment/)*

* [[Georg Hegel]], first section, second chapter of *[[Science of Logic]]* (1813)

* [[Franz Brentano]], *Psychologie vom empirischen Standpunkte* (1874) [[web](https://archive.org/details/psychologievome00kraugoog/page/n8/mode/2up)]

  see: SEP: *[Brentano’s Theory of Judgement](https://plato.stanford.edu/entries/brentano-judgement/)*

The origin of the notion of "judgement" (and of the notation "$\vdash$" for it) in [[formal logic]]:

* {#Frege1879} [[Gottlob Frege]], *Begriffsschrift* (1879) [[web scan](https://gdz.sub.uni-goettingen.de/id/PPN538957069?tify=%7B%22pages%22%3A%5B4%5D%2C%22view%22%3A%22info%22%7D), [Wikipedia entry](https://en.wikipedia.org/wiki/Begriffsschrift)]

and under the name "assertion" in:

* {#RussellWhitehead1910} [[Bertrand Russell]], [[Alfred Whitehead]], p. xviii in: *[[Principia Mathematica]]* (1910)

Keeping the symbol "$\vdash$", now allowing hypothesis on the left, but pronouning it as "[[proof]] under assumptions":

* {#Church40} [[Alonzo Church]], §5 of: *A Formulation of the Simple Theory of Types*, The Journal of Symbolic Logic **5** 2 (1940) 56-68  [[doi:10.2307/2266170](https://doi.org/10.2307/2266170)]

  > (see also at *[[simple type theory]]*)

Discussion in the context of what became known as [[Martin-Löf dependent type theory]]:

* {#Martin-Löf84} [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), p. 2-4 in: _Intuitionistic type theory_, Lecture in Padua 1984, Bibliopolis (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

* {#Martin-Löf87} [[Per Martin-Löf]], *Truth of a proposition, evidence of a judgement, validity of a proof*, Synthese **73** (1987) 407–420 [[doi:10.1007/BF00484985](https://doi.org/10.1007/BF00484985), [pdf](https://archive-pml.github.io/martin-lof/pdfs/Truth-of-a-Proposition-Evidence-of-a-Judgment-1987.pdf)]

* {#MartinLoef90} [[Per Martin-Löf]], _A path from logic to metaphysics_, talk at _Nuovi problemi della logica e della filosofia della scienza_, Jan 1990 ([pdf](https://github.com/michaelt/martin-lof/raw/master/pdfs/A-path-from-logic-to-metaphysics-1991.pdf))

* {#Martin-Löf96} [[Per Martin-Löf]], *On the Meanings of the Logical Constants and the Justifications of the Logical Laws*, Nordic Journal of Philosophical Logic, **1** 1 (1996) 11-60 [[pdf](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf), [[MartinLofOnTheMeaning96.pdf:file]]]

* {#Hofmann95} [[Martin Hofmann]], §2.1.2 in: *Syntax and semantics of dependent types*, Chapter 2 in: _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

  also published as:

* {#Hofmann97} [[Martin Hofmann]], p. 82 of: *Syntax and semantics of dependent types*, in *Semantics and logics of computation*, Publ. Newton Inst. **14**, Cambridge Univ. Press (1997) 79-130 [[doi:10.1017/CBO9780511526619.004](https://doi.org/10.1017/CBO9780511526619.004)]

Reviewed in:

* {#PfenningDavies01} [[Frank Pfenning]], Rowan Davies, §2, §3 of: _A judgemental reconstruction of modal logic_, Mathematical Structures in Computer Science **11** 4 (2001) 511-540 [[doi:10.1017/S0960129501003322](https://doi.org/10.1017/S0960129501003322), [pdf](http://www.cs.cmu.edu/~fp/papers/mscs00.pdf)]

Further discussion in

* Greta Coraglia, [[Ivan Di Liberti]], *Context, Judgement, Deduction* [[arXiv:2111.09438](https://arxiv.org/abs/2111.09438)]

Textbook acccount in a context of [[programming languages]]:

* [[Robert Harper]], section I.3 of  _[[Practical Foundations for Programming Languages]]_ (2015)

The alternative notation "$\vDash$" for *satisfaction* with closely related (if different) usage [seems](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/) to originate in

* {#Kochen61} [[Simon Kochen]], p. 223 in: *Ultraproducts in the Theory of Models*, Ann. Math.  **74** 2 (1961) 221-261 [[doi:10.2307/1970235](https://doi.org/10.2307/1970235)]

whose author

> cannot recollect whether [he] originated it or took it from some other paper.

according to:

* {#Kochen17} [[Simon Kochen]], *Origins of the double turnstile*, FOM mailing list posting (Feb 2017) [[web](https://cs.nyu.edu/pipermail/fom/2017-February/020265.html), [screenshot](/nlab/files/Kochen-OriginOfDoubleTurnstile.jpg)]

See also:

* Formal Logic Wikibook: *[Satisfaction](https://en.wikibooks.org/wiki/Formal_Logic/Predicate_Logic/Satisfaction)*

[[!redirects judgment]]
[[!redirects judgments]]
[[!redirects judgement]]
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[[!redirects hypothetical judgment]]
[[!redirects hypothetical judgments]]
[[!redirects hypothetical judgement]]
[[!redirects hypothetical judgements]]
[[!redirects generic judgment]]
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[[!redirects assertion]]
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[[!redirects validation]]
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judgmental equality

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

\tableofcontents

## Idea

In any [[type theory]], judgmental equality is the notion of [[equality]] which is defined to be a [[judgment]]. Judgmental equality is most commonly used in single-level type theories like [[Martin-Löf type theory]] or [[higher observational type theory]] for making [[inductive definitions]], but it is also used in [[cubical type theory]] and [[simplicial type theory]] to define probe shapes for [[(infinity,1)-category theory|(infinity,1)-categorical]] types which could not be coherently defined in vanilla dependent type theory.

There are two different kinds of judgmental equalities

* Judgmental equality of terms

* Judgmental equality of types, in [[dependent type theories]] with a separate [[type]] [[judgment]].

Judgmental equality of types is not necessary for [[dependent type theory]] with a separate type judgment. It behaves similarly to the [[equality]] between [[sets]] in [[structural set theory]], and the equality between sets is not necessary for structural set theory since one could simply work with [[bijections]] or [[one-to-one correspondences]] between sets. Similarly, in [[dependent type theory]], one could just work with [[strict equivalence of types]] or some notion of [[weak equivalence of types]] instead of judgmental equality of types.

## Judgmental equality of terms

Judgmental equality of terms is given by the following judgment:

* $\Gamma \vdash a \equiv a' : A$ - $a$ and $a'$ are judgmentally equal well-typed terms of type $A$ in context $\Gamma$.

There are two different notions of judgmental equality of terms which could be distinguished:

* Weak judgmental equality of terms is just a shorthand for the [[identity type]] as [[typal equality]]

* Strict judgmental equality is additional structure on types which gives every type the structure of a [[set]] in addition to the [[infinity-groupoid|$\infty$-groupoidal]] structure on a type from the [[identity type]].

Strict judgmental equalities of terms are used in most [[dependent type theories]]. Weak judgmental equalities of terms can be used in [[weak type theories]], where a direct translation of the [[inference rules]] of the types in [[Martin-Löf type theory]] results in a weak version of Martin-Löf type theory.

Judgmental equality of terms can be contrasted with [[propositional equality]] of terms, where equality is a [[proposition]] in the sense of [[first-order logic]], and [[typal equality]] of terms, where equality is a [[type]].

### Weak judgmental equality

Weak judgmental equality of terms is simply given by a reflection rule into the [[identity type]]:

$$\frac{\Gamma \vdash a \equiv a':A}{\Gamma \vdash \delta_{a, a'}:a =_A a'}$$

### Strict judgmental equality

Strict judgmental equality is an equivalence relation:

* Reflexivity of judgmental equality

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv a:A}$$

* Symmetry of judgmental equality
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b \equiv a:A}$$

* Transitivity of judgmental equality
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A \quad b \equiv c:A }{\Gamma \vdash a \equiv c:A}$$

In addition, strict judgmental equality of terms has congruence rules for substitution, the [[principle of substitution]]:

* Principle of substitution for judgmentally equal terms:
$$\frac{\Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B}{\Gamma, \Delta(a) \vdash c(a) \equiv c(b): B}$$

If there is a separate [[type]] [[judgment]], then there is also a separate rule for the principle of substitution into type families.

If one has judgmental equality of types, then the principle of substitution into type families is given by

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, \Delta(a) \vdash B(a) \equiv B(b) \; \mathrm{type}}$$

This implies the reflection rule of weak judgmental equalities because one could derive the following rule:

$$\frac{\Gamma \vdash a \equiv b:A}{\Gamma \vdash \mathrm{refl}_A(a):a =_A b}$$

Otherwise, the principle of substitution into type families is given by [[strict transport]] across judgmental equality as [[explicit conversion]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}:B(a) \simeq B(b)}$$

where $A \simeq B$ is the [[type of strict equivalences]] defined using [[natural deduction]] [[inference rules]]. If one doesn't have a type of strict equivalences, one could define it by components

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}(y):B(b)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a: A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{a \equiv a}(y) \equiv y:B(a)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{b \equiv a}(\mathrm{tr}_{B(-)}^{a \equiv b}(y)) \equiv y:B(a)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma \vdash b \equiv c : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, y:B(a), \Delta(a) \vdash \mathrm{tr}_{B(-)}^{b \equiv c}(\mathrm{tr}_{B(-)}^{a \equiv b}(y)) \equiv \mathrm{tr}_{B(-)}^{a \equiv c}(y):B(c)}$$

This shows that transport across judgmental equality forms a [[groupoid]].

Either way, this also implies the reflection rule of weak judgmental equalities because one could derive the following rule

$$\frac{\Gamma \vdash a \equiv b:A}{\Gamma \vdash \mathrm{tr}_{a =_A (-)}^{a \equiv b}(\mathrm{refl}_A(a)):a =_A b}$$

Similarly, for a term $c(x):B(x)$ dependent upon $x:A$, if one has judgmental equality of types, then the principle of substitution across $c(x)$ is given by the rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B(x)}{\Gamma, \Delta(b) \vdash c(a) \equiv c(b):B(b)}$$

Otherwise, it is given by a judgmental version of [[function application to identifications]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B(x)}{\Gamma, \Delta(b) \vdash \mathrm{tr}_{B(-)}^{a \equiv b}(c(a)) \equiv c(b):B(b)}$$

### In computation and uniqueness rules

Judgmental equality of terms can be used in the [[computation rules]] and [[uniqueness rules]] of types:

* Computation rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) \equiv b(a):B(a)}$$

* Uniqueness rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}$$

* Computation rules for negative dependent sum types:

$$\frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \pi_2(a, b) \equiv b:B(a)}$$

If one does not have judgmental equality of types, then one would have to use transport across judgmental equality for the second computation rule:

$$\frac{\Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B(a)}{\Gamma \vdash \mathrm{tr}_{B(-)}^{\pi_1(a, b) \equiv a}(\pi_2(a, b)) \equiv b:B(a)}$$

* Uniqueness rules for negative dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}$$

* Computation rules for identity types:

$$\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C(a, b, p) \; \mathrm{type} \quad \Gamma \vdash t:\prod_{c:A} C(c, c, \mathrm{refl}_A(c))}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv t:C(c, c, \mathrm{refl}_A(c))}$$

## Judgmental equality of types

In [[dependent type theory]] with a separate [[type]] [[judgment]], judgmental equality of types is given by the following judgment:

* $\Gamma \vdash A \equiv A' \; \mathrm{type}$ - $A$ and $A'$ are judgmentally equal well-typed types in context $\Gamma$.

There are two different notions of judgmental equality of types which could be distinguished:

* Weak judgmental equality of types is just a shorthand for equivalence of types

* Strict judgmental equality of types could be thought of as making explicit the implicit [[coercion]] of [[equivalence of types|equivalent types]] as [[subtypes]], and is preserved throughout the type theory as [[congruences]].

In either case, judgmental equality of types is primarily used for [[definitional equality]] of types.

### Weak judgmental equality

Weak judgmental equality of types is given by one of the two sets of structural rules:

* The variable conversion rule for judgmentally equal types:
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

or

* Rules for isomorphisms between judgmentally equal types:

$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}(x):B} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}^{-1}(x):A}$$

$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}^{-1}(\delta_{A, B}(x)) \equiv x:A} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}(\delta_{A, B}^{-1}(y)) \equiv y:B}$$

In the first case, one could construct isomorphisms from the variable conversion rule, other structural rules, and the rules for function types:

From the generic term rule and the variable conversion rule for judgmentally equal types $A \equiv A'$ we have $x:A' \vdash x:A$ and $x:A \vdash x:A'$, whereby from the introduction and computation rules for function types we have functions $\lambda x:A'.x:A' \to A$ and $\lambda x:A.x:A \to A'$ such that
$$(\lambda x:A'.x)((\lambda x:A.x)(x)) \equiv (\lambda x:A'.x)(x) \equiv x:A$$
$$(\lambda x:A.x)((\lambda x:A'.x)(x)) \equiv (\lambda x:A.x)(x) \equiv x:A'$$
making both functions $\lambda x:A'.x$ and $\lambda x:A.x$ isomorphisms.

### Strict judgmental equality

In addition to the variable conversion rule, there are reflexivity, symmetry, and transitivity rules making strict judgmental equality for types an [[equivalence relation]]:

* Reflexivity of judgmental equality

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{type}}$$

* Symmetry of judgmental equality
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{type}}$$

* Transitivity of judgmental equality
$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma \vdash B \equiv C \; \mathrm{type} }{\Gamma \vdash A \equiv C \; \mathrm{type}}$$

### Congruence rules for judgmental equality of types

In addition, strict judgmental equalities have [[congruence rules]] for every type in the type theory.

* Congruence rules for dependent function types

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \prod_{x:A} B(x) \equiv \prod_{x:A'} B'(x)\; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\
	\Gamma, x:A \vdash b(x) \equiv b'(x):B(x)
\end{array}
}{\Gamma \vdash \lambda x:A.b(x) \equiv \lambda x:A.b'(x):\prod_{x:A}.B(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \quad f':\prod_{x:A} B(x) \\
	\Gamma \vdash f \equiv f':\prod_{x:A} B(x)
\end{array}
}{\Gamma, x:A \vdash f(x) \equiv f'(x):B(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\
	\Gamma, x:A \vdash b(x) \equiv b'(x):B(x)
\end{array}
}{\Gamma \vdash \beta_{\prod}^{A, B} x:A.b(x) \equiv \beta_{\prod}^{A, B} x:A.b'(x):\prod_{x:A} b(x) =_{B(x)} (\lambda x:A.b(x))(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash B'(x) \; \mathrm{type} \\
	\Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \eta_{\prod}^{A, B} \equiv \eta_{\prod}^{A, B'}:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)}$$

* Congruence rules for dependent pair types:

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \sum_{x:A} B(x) \equiv \sum_{x:A'} B'(x)\; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma, x:A, y:B(x) \vdash \mathrm{pair}_{\sum}^{A, B} \equiv \mathrm{pair}_{\sum}^{A', B'}:\sum_{x:A} B(x)}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \mathrm{ind}_{\sum}^{A, B, C} \equiv \mathrm{ind}_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{z:\sum_{x:A} B(x)} C(z)}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \beta_{\sum}^{A, B, C} \equiv \beta_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{x:A} \prod_{y:B(x)} \mathrm{ind}_{\sum}^{A, B, C}(g, \mathrm{pair}_{\sum}^{A, B}(x, y)) =_{C(\mathrm{pair}_{\sum}^{A, B}(x, y))} g(x, y)}$$

* Congruence rules for identity types:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma, x:A, y:A \vdash x =_A y \equiv x =_{A'} y}$$

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma \vdash \mathrm{refl}_A \equiv \mathrm{refl}_{A'}:\prod_{x:A} x =_A x}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \mathrm{ind}_{=}^{A, C} \equiv \mathrm{ind}_{=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} C(x, y, p)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \beta_{ \mathrm{ind}_=}^{A, C} \equiv \beta_{\mathrm{ind}_=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \mathrm{ind}_{=}^{A, C}(t, x, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t(x)}$$

* Congruence rules for the empty type:

$$\frac{\Gamma, x:\emptyset \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\emptyset^C \equiv \mathrm{ind}_\emptyset^{C'}:\prod_{x:\emptyset} C(x) \; \mathrm{type}}$$

* Congruence rules for the type of booleans:

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{2}^C \equiv \mathrm{ind}_\mathbb{2}^{C'}:\prod_{a:C(0)} \prod_{b:C(1)} \prod_{x:\mathbb{2}} C(x)}$$

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{0, C} \equiv \beta_\mathbb{2}^{0, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 0) =_{C(0)} a}$$

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{1, C} \equiv \beta_\mathbb{2}^{1, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 1) =_{C(1)} b}$$

* Congruence rules for the natural numbers type:

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C \equiv \mathrm{ind}_\mathbb{N}^{C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N} C(x)}}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{0, C} \equiv \beta_\mathbb{N}^{0, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0) =_{C(0)} c_0}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{s, C} \equiv \beta_\mathbb{N}^{s, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N}} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(x)) =_{C(s(x))} c_s(x)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, x))}$$

Similarly, we have congruence rules for every [[axiom]] in the dependent type theory, such as

* congruence rule for [[function extensionality]]:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{funext}_{A, B} \equiv \mathrm{funext}_{A', B'}:\prod_{f;\prod_{x:A} B(x)} \prod_{g:\prod_{x:A} B(x)} (f =_{\prod_{x:A} B(x)} g) \simeq \prod_{x:A} f(x) =_{B(x)} g(x)}$$

* congruence rule for the [[axiom of choice]]:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash C(x, y) \equiv C'(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{choice}_{A, B, C} \equiv \mathrm{choice}_{A', B', C'}:\left(\mathrm{isSet}(A) \times \prod_{x:A} \mathrm{isSet}(B(x))\right) \to \forall x:A.\exists y:B(x).C(x, y) \to \exists g:\prod_{x:A} B(x).\forall x:A.C(x, g(x))}$$

## Judgmental equality of contexts

In some [[dependent type theories]], there is also judgmental equality of [[contexts]], which is given by the following [[judgment]]:

* $\Gamma \equiv \Gamma' \; \mathrm{ctx}$ - $\Gamma$ and $\Gamma'$ are judgmentally equal contexts.

In addition to the variable conversion rule, there are reflexivity, symmetry, and transitivity rules making judgmental equality for contexts an [[equivalence relation]]:

* Reflexivity of judgmental equality

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \equiv \Gamma \; \mathrm{ctx}}$$

* Symmetry of judgmental equality
$$\frac{\Gamma \equiv \Delta \; \mathrm{ctx}}{\Delta \equiv \Gamma \; \mathrm{ctx}}$$

* Transitivity of judgmental equality
$$\frac{\Gamma \equiv \Delta \; \mathrm{ctx} \quad \Delta \equiv \Xi \; \mathrm{ctx}}{\Gamma \equiv \Xi \; \mathrm{ctx}}$$

## See also

* [[equality]]

* [[propositional equality]], [[typal equality]]

* [[coercion]]

## References

* Robin Adams, _Pure type systems with judgemental equality_, Journal of Functional Programming, Volume 16 Issue 2(2006)   ([web](http://dl.acm.org/citation.cfm?id=1114675))

* Vincent Siles, Hugo Herbelin, _Equality is typable in semi-full pure type systems_ ([pdf](http://pauillac.inria.fr/~herbelin/publis/lics-SilHer10-pts-typed-conv.pdf))

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf)) (478 pages)

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Jules Hedges

* [personal website](https://julesh.com/)

## Selected writings

On [[categorical systems theory]] ([[cybernetics]]):

* [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], [[Eigil F. Rischel]], *Towards Foundations of Categorical Cybernetics*, EPTCS **372** (2022) 235-248 [[arXiv:2105.06332](https://arxiv.org/abs/2105.06332), [doi:10.4204/EPTCS.372.17](https://doi.org/10.4204/EPTCS.372.17)]

## Related Pages

* [[open game]]

* [[lens (in computer science)]]

* [[applied category theory]]

category: people


Julia Kempe

* [institute page](https://cims.nyu.edu/~kempe/)

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

## Selected writings

On [[adiabatic quantum computation]]:

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

category: people


Julia set

## Related entries

* [[Mandelbrot set]]

## References

* Wikipedia, _[Julia set](https://en.wikipedia.org/wiki/Julia_set)_

[[!redirects Julia sets]]


Julian Cole

* [[Cole's theory of spectrum]]

* [[Julian Cole]], _The Bicategory of Topoi, and Spectra_ , ms. ([pdf](http://www.oliviacaramello.com/Unification/ColeBicategoryTopoiSpectra.pdf))

category: people


Julian Holstein

Julian Holstein  is Junior Professor in the Algebra and Number Theory group at the Department of Mathematics, of Universität Hamburg. His research interests lie 'at the intersection of algebraic geometry and algebraic topology with higher category theory and homotopical algebra'.

* [webpage](https://www.math.uni-hamburg.de/home/holstein/h)

* [Mathematical Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=186736)

## References

On [[Koszul duality]] duality:

* [[Julian Holstein]], [[Andrey Lazarev]], *Categorical Koszul duality*, Advances in Mathematics **409** B (2022) 108644 [[arXiv:Koszul duality2006.01706](https://arxiv.org/pdf/2006.01705.pdf), [doi:10.1016/j.aim.2022.108644](https://doi.org/10.1016/j.aim.2022.108644)]

On the [[Riemann-Hilbert correspondence]]:

* [[Joseph Chuang]], [[Julian Holstein]], [[Andrey Lazarev]] _Maurer–Cartan moduli and theorems of Riemann–Hilbert type_, Applied Categorical Structures **29** (2021) 685–728 [[doi:10.1007/s10485-021-09631-3](https://doi.org/10.1007/s10485-021-09631-3)]

category:people

[[!redirects J. Holstein]]


Julian Rachman


Julian Rachman > history

< [[Julian Rachman]]


Julian Schwinger

* [webpage](https://en.wikipedia.org/wiki/Julian_Schwinger)

* Steve Nadis, _Julian Schwinger: A Centennial Celebration at Harvard_ ([arXiv:1812.04123](https://arxiv.org/abs/1812.04123))

## Selected writings

Introducing [[adiabatic switching]]:

* {#LippmannSchwinger50} B. A. Lippmann, [[Julian Schwinger]], *Variational Principles for Scattering Processes. I*,  Phys. Rev. 79, 469 (1950) ([doi:10.1103/PhysRev.79.469](https://doi.org/10.1103/PhysRev.79.469))

Introducing the [[Schwinger effect]] of [[vacuum polarization]]:

* {#Schwinger51} [[Julian Schwinger]], _On Gauge Invariance and Vacuum Polarization_, Phys. Rev. 82, 664 (1951) ([doi:10.1103/PhysRev.82.664](https://doi.org/10.1103/PhysRev.82.664))

On [[effective field theory]]-foundations of [[nuclear physics]] (introducing, among other things, the concept of the [[sigma-meson]]):

* [[Julian Schwinger]], _A Theory of the Fundamental Interactions_, Annals Phys. 2 (1957) 407-434 ([spire:12286](http://inspirehep.net/record/12286/))

On [[Dirac charge quantization]]:

* [[Julian Schwinger]], *Magnetic Charge and Quantum Field Theory*, Phys. Rev. **144** (1966) 1087 [[doi:10.1103/PhysRev.144.1087](https://doi.org/10.1103/PhysRev.144.1087)]

On [[classical field theory|classical]] [[electrodynamics]]:

* [[Julian Schwinger]] et al., *Classical Electrodynamics*, based on Schwinger's 1976 lecture, Routledge (1999) [[ISBN:9780738200569](https://www.routledge.com/Classical-Electrodynamics/Schwinger-Jr-Milton-Tsai/p/book/9780738200569), [pdf](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/schwinger_EM.pdf)]

## Related entries

* [[perturbative quantum field theory]]

* [[Schwinger-Tomonaga-Feynman-Dyson]]

* [[quantum electrodynamics]]

* [[adiabatic switching]]

* [[Tomonaga-Schwinger equation]]

* [[Schwinger-Keldysh formalism]]

category: people

[[!redirects Schwinger]]


Julian Sochocki

_Julian Karol Sochocki_ was a Russian-Polish mathematician, born 1842 in Warsaw and deceased 1927 in Leningrad. This is his native Polish spelling. Sometimes it is differently transliterated from Russian, e.g. Sokhotsky, Sokhotskii and Sokhotski.

He is famous for Sochocki-Plemelj formulae involving the principal value of $1/x$.

* wikipedia: [Julian Sochocki](https://en.wikipedia.org/wiki/Julian_Sochocki)

category: people

[[!redirects Yulian Vasilievich Sokhotski]]
[[!redirects Yulian Vasilievich Sokhotskii]]
[[!redirects Yulian Vasilievich Sokhotsky]]
[[!redirects Yulian Sokhotski]]
[[!redirects Julian Karol Sochocki]]


Julian Sonner

* [homepage](http://www.sonnerphysics.net/)

## Selected writings

On [[AdS-CFT in condensed matter physics]]:

* [[Hong Liu]], [[Julian Sonner]], _Quantum many-body physics from a gravitational lens_, Nature Rev. Phys. **2** (2020)  615-633 [[arXiv:2004.06159](https://arxiv.org/abs/2004.06159), [doi:10.1038/s42254-020-0225-1](https://doi.org/10.1038/s42254-020-0225-1)]

category: people


Julie Bergner

* [website](http://www.people.virginia.edu/~jeb2md/)

## Selected writings

Introducing the [[model structure on sSet-categories]]:

* {#Bergner04} [[Julie Bergner]], _A model category structure on the category of simplicial categories_,  Trans. Amer. Math. Soc. 359 (2007), 2043-2058  ([arXiv:0406507](http://arxiv.org/abs/math/0406507), [doi:10.1090/S0002-9947-06-03987-0](https://doi.org/10.1090/S0002-9947-06-03987-0))

On the [[homotopy theory of homotopy theories]] (the [[(infinity,1)-category of (infinity,1)-categories]]):

* {#Bergner07} [[Julia Bergner]], _Three models for the homotopy theory of homotopy theories_, Topology Volume 46, Issue 4, September 2007, Pages 397-436 ([arXiv:math/0504334](https://arxiv.org/abs/math/0504334), [doi:10.1016/j.top.2007.03.002](https://doi.org/10.1016/j.top.2007.03.002))

On comparison of [[model categories]] of [[(∞,1)-categories]]:

* [[Julie Bergner]], _A survey of $(\infty,1)$-categories_,  In: [[John Baez]], [[Peter May]] (eds.),  _[[Towards Higher Categories]]_ The IMA Volumes in Mathematics and its Applications, vol 152, Springer 2007 ([arXiv:math/0610239](http://arxiv.org/abs/math/0610239), [doi:10.1007/978-1-4419-1524-5_2](https://doi.org/10.1007/978-1-4419-1524-5_2))

* [[Julia Bergner]], _Equivalence of models for equivariant $(\infty,1)$-categories_, Glasgow Mathematical Journal, Volume 59, Issue 1 (2016) ([arXiv:1408.0038](https://arxiv.org/abs/1408.0038), [doi:10.1017/S0017089516000136](https://doi.org/10.1017/S0017089516000136))

On [[model categories]] presenting [[(infinity,0)-categories|$(\infty,0)$-categories]], namely models for [[infinity-groupoids|$\infty$-groupoids]] (such as the [[model structure on simplicial groupoids]]) akin to corresponding models for [[(infinity,1)-categories|$(\infty,1)$-categories]] (such as the [[model structure on simplicial categories]]):

* [[Julia E. Bergner]], *Adding inverses to diagrams II: Invertible homotopy theories are spaces*, Homology, Homotopy Appl. **10** 2 (2008) 175-193 [[doi:10.4310/HHA.2008.v10.n2.a9](https://dx.doi.org/10.4310/HHA.2008.v10.n2.a9), [doi:0710.2254](https://arxiv.org/abs/0710.2254), erratum:[doi:10.4310/HHA.2012.v14.n1.a15](https://dx.doi.org/10.4310/HHA.2012.v14.n1.a15)]

## Related entries

* [[model category]]

* [[(∞,1)-category]]

* [[simplicially enriched category]]

* [[complete Segal space]]

* [[(∞,n)-category]]

* [[cobordism hypothesis]]

category: people

[[!redirects Julia Bergner]]
[[!redirects Julia E. Bergner]]


Julien Mairal

* [personal page](https://thoth.inrialpes.fr/people/mairal/)

## Selected writings

On [[kernel methods]] in [[machine learning]]:

* [[Julien Mairal]], [[Jean-Philippe Vert]], *Machine Learning with Kernel Methods*, 2017 ([pdf](http://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/slides/master2017/master2017.pdf), [[MarailVertKernelMethods.pdf:file]])

category: people


Julien Narboux

* [webpage](http://dpt-info.u-strasbg.fr/~narboux/)

## related $n$Lab entries

* [[Euclidean geometry]], [[synthetic geometry]]

* [[formal proof]], [[Coq]]

category: people


Julio Rubio

* [GoogleScholar page](https://scholar.google.com/citations?user=kSyA2sAAAAAJ&hl=en)

## Selected writings

On [[constructive mathematics|constructive]] methods in [[algebraic topology]] ([[constructive algebraic topology]]):

* [[Julio Rubio]], [[Francis Sergeraert]], *Constructive Algebraic Topology*, Bulletin des Sciences Mathématiques
**126** 5 (2002) 389-412 [[arXiv:math/0111243](https://arxiv.org/abs/math/0111243), doi:10.1016/S0007-4497(02)01119-3]

On the [[Kenzo]] software for computations in [[constructive algebraic topology]]:

* Jonathan Heras, Vico Pascual, Ana Romero, [[Julio Rubio]], *Integrating multiple sources to answer questions in Algebraic Topology*, Lectures Notes in Artificial Intelligence **6167** (2010) [[arXiv:1005.0749](https://arxiv.org/abs/1005.0749)]

* [[Julio Rubio]], *Formalization of Mathematics: why Algebraic Topology?*, MAP Spring School 2012 [[pdf](http://www-sop.inria.fr/manifestations/MapSpringSchool/contents/JulioRubio.pdf)]

On [[homological algebra]] in [[constructive mathematics]]:

* [[Julio Rubio]], [[Francis Sergeraert]], _Constructive homological algebra and applications_ [[arXiv:1208.3816](http://arxiv.org/abs/1208.3816)]

category: people


Julius Wess

_Julius Wess_ was a theoretical physicist, one of the discoverers of the [[supersymmetry]] as a possible symmetry in physical models (with Bruno Zumino, and independently from Akulov and Volkov). With Zumino he introduced Wess-Zumino sigma model. From around 1990, he was working also on quantum groups and noncommutative geometry with a view toward physics on noncommutative space-times.

* [wikipedia](http://en.wikipedia.org/wiki/Julius_Wess)

# Selected writings

On [[chiral perturbation theory]]:

* [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. I_, Phys. Rev. 177, 2239 (1969) ([doi:10.1103/PhysRev.177.2239](https://doi.org/10.1103/PhysRev.177.2239))

* [[Curtis Callan]], Jr., [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. II_, Phys. Rev. 177, 2247 (1969) ([doi:10.1103/PhysRev.177.2247](https://doi.org/10.1103/PhysRev.177.2247))

Introducing the [[WZW term]] of [[chiral perturbation theory]]/[[quantum hadrodynamics]] which reproduces the [[chiral anomaly]] of [[QCD]] in the [[effective field theory]] of [[mesons]] and [[Skyrmions]]:

* [[Julius Wess]], [[Bruno Zumino]], _Consequences of anomalous Ward identities_, Phys. Lett. B 37 (1971) 95-97 ([spire:67330](https://inspirehep.net/literature/67330), doi:10.1016/0370-2693(71)90582-X)

Introducing [[supersymmetry]] for [[quantum field theory]] on 4d [[spacetime]]:

* {#WessZumino74} [[Julius Wess]], [[Bruno Zumino]], _Supergauge transformations in four dimensions_, Nucl. Phys. B70 (1974) 39; Phys. Letters 49B (1974) 52 ([doi:10.1142/9789814542340_0002](https://doi.org/10.1142/9789814542340_0002))

category: people


Jun John Sakurai

* [Wikipedia entry](https://en.wikipedia.org/wiki/J._J._Sakurai)

## Selected writings

Origin of [[effective field theory]] of [[mesons]] in [[nuclear physics]], via [[hidden local gauge symmetry]]:

* [[Jun John Sakurai]], _Theory of strong interactions_, Annals Phys. 11 (1960) 1-48 ([spire:3218](http://inspirehep.net/record/3218), doi:10.1016/0003-4916(60)90126-3)

On [[meson]]-[[effective field theory]], introducing [[vector meson dominance]]:

* [[Jun John Sakurai]], _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))

On [[quantum mechanics]]:

* [[Jun John Sakurai]], Jim Napolitano, *Modern Quantum Mechanics*, Cambridge University Press (1985, 1994, 2020) [[doi:10.1017/9781108587280](https://doi.org/10.1017/9781108587280), [Wikipedia](https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics)]

category: people

[[!redirects J. J. Sakurai]]

[[!redirects Jun Sakurai]]


Jun Murakami

* [webpage](http://www.f.waseda.jp/murakami/)

## Selected writings

On [[universal Vassiliev invariants]]:

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _The universal Vassiliev-Kontsevich invariant for framed oriented links_, Compositio Math. 102 (1996), 42–64 ([arXiv:hep-th/9401016](https://arxiv.org/abs/hep-th/9401016))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _Representation of the category of tangles by Kontsevich's iterated integral_, ([euclid.cmp/1104272488](https://projecteuclid.org/euclid.cmp/1104272488))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], Tomotada Ohtsuki, _On a universal perturbative invariant of 3-manifolds_, Topology Volume 37, Issue 3, May 1998, Pages 539-574 (https://doi.org/10.1016/S0040-9383(97)00035-9)

On the [[colored Jones polynomial]] and the [[volume conjecture]]:

* {#MurakamiMurakami01} [[Hitoshi Murakami]], [[Jun Murakami]], _The Colored Jones Polynomial And The Simplicial Volume Of A Knot_, Acta Math. 186 (2001) 85-104 ([euclid.acta/1485891370](https://projecteuclid.org/euclid.acta/1485891370))

* {#MMOTY02} [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, _Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links_, Experiment. Math. 11 (2002) 427-435 ([arXiv:math/0203119](https://arxiv.org/abs/math/0203119))

## Related $n$Lab entries

* [[volume conjecture]]

category: people


Jun Zhang

Jun Zhang is a mathematician with interests in algebraic topology (esp. [[persistent homology]]), symplectic geometry, microlocal analysis.

* Paul Biran, Octav Cornea, Jun Zhang, _Triangulation and persistence: Algebra 101_, [arXiv:2104.12258](https://arxiv.org/abs/2104.12258)

> This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of objects of some triangulated categories. We also provide examples of TPCs coming from algebra, algebraic topology, microlocal sheaf theory and symplectic topology.

* Jun Zhang, _Quantitative Tamarkin theory_, book, CRM Short Courses, [arXiv:1807.09878](https://arxiv.org/abs/1807.09878)
[doi](https://doi.org/10.1007/978-3-030-37888-2)

> ...we try to explain how standard symplectic techniques, for instance, generating function, capacities, symplectic homology, etc., are elegantly packaged in the language of sheaves as well as related intriguing sheaf operators. In addition, many concepts developed in Tamarkin category theory are natural generalizations of [[persistent homology]] theory...

category: people


June Huh

* [website](https://web.math.princeton.edu/~huh/)

## Selected writings

On [[Lorentzian polynomials]]:

* [[Petter Brändén]], [[June Huh]], *Lorentzian polynomials*, Annals of Mathematics, **192** 3 (2020) [[arXiv:1902.03719](https://arxiv.org/abs/1902.03719), [doi:10.4007/annals.2020.192.3.4](https://doi.org/10.4007/annals.2020.192.3.4)]

and on [[intersection cohomology]] of [[matroids]]:

* Tom Braden, [[June Huh]], Jacob P. Matherne, [[Nicholas Proudfoot]], Botong Wang, *Singular Hodge theory for combinatorial geometries* [[arXiv:2010.06088](https://arxiv.org/abs/2010.06088)]

A survey of the use of [[Hodge theory]] in [[combinatorics]]:

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]

[[Schur polynomials]] as Lorentzian:

* [[June Huh]], Jacob P. Matherne, Karola Mészáros, Avery St. Dizier, _Logarithmic concavity of Schur and related polynomials_ ([arXiv:1906.09633](https://arxiv.org/abs/1906.09633))

* Petter Brändén, [[June Huh]], _Hodge-Riemann relations for Potts model partition functions_ ([arXiv:1811.01696](https://arxiv.org/abs/1811.01696))

## Related $n$Lab entries

* [[hard Lefschetz theorem]]
* [[Hodge theory]]
* [[Lorentzian polynomial]]

category: people


Junya Yagi

* [webpage](https://sites.google.com/site/junyagi/)

## Selected writings

* {#Yagi10} [[Junya Yagi]], _Vanishing Chiral Algebras and Höhn-Stolz Conjecture_ ([arXiv:1002.0028](http://arxiv.org/abs/1002.0028))

  (on the [[Stolz conjecture]] in terms of [[2d (2,0)-superconformal QFT]])

* [[Junya Yagi]], _3d TQFT from 6d SCFT_, JHEP08(2013)017 ([arXiv:1305.0291](https://arxiv.org/abs/1305.0291))

  (on [[3d-3d correspondence]])

category: people


Jure Kališnik

* [webpage](http://www.fmf.uni-lj.si/si/imenik/3121/)

category: people

[[!redirects Jure Kalisnik]]
[[!redirects J.  Kališnik]]
[[!redirects J. Kalisnik]]


Jussi Kalkkinen

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=133091)

## Selected writings

On [[D6-D8 brane bound states]] with emphasis on the [[charge quantization]] of the [[RR-fields]] via [[bundle gerbes]]:

* [[Jussi Kalkkinen]], _Gerbes and Massive Type II Configurations_, JHEP 9907  :002, 1999 ([arXiv:hep-th/9905018](https://arxiv.org/abs/hep-th/9905018))

On the [[Hopf-Wess-Zumino term]] of the [[M5-brane]]:

* {#KalkkinenStelle02} [[Jussi Kalkkinen]], [[Kellogg Stelle]], Section 3.2 of: _Large Gauge Transformations in M-theory_, J. Geom. Phys. 48 (2003) 100-132 ([arXiv:hep-th/0212081](https://arxiv.org/abs/hep-th/0212081))

On [[M5-branes]] and [[Spin(5)]]-[[spin geometry|geometry]]:

* {#Kalkkinen06} [[Jussi Kalkkinen]], _Global Spinors and Orientable Five-Branes_, JHEP 0609: 028, 2006 ([arXiv:hep-th/0604081](https://arxiv.org/abs/hep-th/0604081))

category: people


Justin Curry

* [webpage](http://www.math.upenn.edu/~jucurry/)

## related $n$Lab entries

* [[Verdier duality]]

category: people


Justin Greenough

* [personal page](https://www.math.umb.edu/people/faculty_homepage.php?id=50)

## Selected writings

On [[bimodule categories]]:

* [[Justin Greenough]], *Monoidal 2-structure of Bimodule Categories*, Journal of Algebra **324** 8 (2010) 1818-1859 [[arXiv:0911.4979](https://arxiv.org/abs/0911.4979), [doi:10.1016/j.jalgebra.2010.06.018](https://doi.org/10.1016/j.jalgebra.2010.06.018)]

category: people


Justin Hsu

* [personal page](https://www.justinhsu.net/)

* [institute page](https://www.engineering.cornell.edu/faculty-directory/justin-hsu)

## Selected writings

Understanding the [[linear logic]]-sector of [[bunched logic]] as a kind of [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

category: people


Justin Kaidi

* [Institute page](https://phys.washington.edu/people/justin-kaidi)

* [GoogleScholar page](https://scholar.google.com/citations?user=nilwx_0AAAAJ&hl=en)

## Selected writings

On [[modular forms]] in relation to [[string theory]]:

* [[Eric D'Hoker]], [[Justin Kaidi]], *Lectures on modular forms and strings* [[arXiv:2208.07242](https://arxiv.org/abs/2208.07242)]

category: people


Justin Khoury

* [webpage](https://www.physics.upenn.edu/people/standing-faculty/justin-khoury)

## Selected writings

On [[open/closed string duality]]:

* [[Justin Khoury]], [[Herman Verlinde]], _On Open/Closed String Duality_ ([arXiv:hep-th/0001056](http://arxiv.org/abs/hep-th/0001056))

On [[fuzzy dark matter]] and its [[MOND]]-like [[phenomenology]]:

* {#Khoury16} [[Justin Khoury]], _Another Path for the Emergence of Modified Galactic Dynamics from Dark Matter Superfluidity_, Phys. Rev. D 93, 103533 (2016) ([arXiv:1602.05961](https://arxiv.org/abs/1602.05961))

* [[Justin Khoury]], *Dark Matter Superfluidity* ([arXiv:2109.10928](https://arxiv.org/abs/2109.10928))

## Related $n$Lab entries

* [[MOND]], [[fuzzy dark matter]]

category: people


Justin Noel

* [webpage](http://www.nullplug.org)

category: people


Justin Read

* [webpage](https://www.surrey.ac.uk/people/justin-read)

## Selected writings

On possible resolution of the [[cusp-core problem]] in [[dark matter]] [[model (in theoretical physics)|models]] via star-formation feedback in the [[structure formation]] of [[galaxies]]:

* [[Justin Read]], O. Agertz, M. L. M. Collins, _Dark matter cores all the way down_, Monthly Notices of the Royal Astronomical Society, Volume 459, Issue 3, 1 July 2016 ([arXiv:1508.04143](https://arxiv.org/abs/1508.04143))

* {#ReadWalkerSteger18} [[Justin Read]], M. G. Walker, P. Steger, _Dark matter heats up in dwarf galaxies_, Monthly Notices of the Royal Astronomical Society  ([arXiv:1808.06634](https://arxiv.org/abs/1808.06634), [doi:10.1093/mnras/sty3404](https://doi.org/10.1093/mnras/sty3404),  [talk recording](http://online.kitp.ucsb.edu/online/cdm-c18/read/), [press release](https://www.surrey.ac.uk/news/dark-matter-move))

* {#Read19} [[Justin Read]], _[Dark matter heats up in dwarf galaxies](https://www.simonsfoundation.org/event/dark-matter-heats-up-in-dwarf-galaxies/)_, Simons Foundation Lecture 2019

## Related $n$Lab entries

* [[dark matter]], [[cusp-core problem]]

category:people


Justin Roberts

* [webpage](http://www.math.ucsd.edu/~justin/)

## Selected writings

On [[Rozansky-Witten theory]]:

* {#Roberts01} [[Justin Roberts]], _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))

* [[Justin Roberts]], [[Justin Sawon]], _Generalisations of Rozansky-Witten invariants_, Geom. Topol. Monogr. 4 (2002) 263-279 ([arXiv:math/0112210](https://arxiv.org/abs/math/0112210))

On [[Rozansky-Witten weight systems]] as [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]], and unified [[Wheels theorem]]:

* {#RobertsWillerton} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

category: people


Justin Sawon

* [webpage](http://sawon.web.unc.edu)

## Selected writings

On [[Rozansky-Witten theory]]:

* [[Justin Roberts]], [[Justin Sawon]], _Generalisations of Rozansky-Witten invariants_, Geom. Topol. Monogr. 4 (2002) 263-279 ([arXiv:math/0112210](https://arxiv.org/abs/math/0112210))

* [[Justin Sawon]], _Rozansky-Witten invariants of hyperkähler manifolds_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

category: people


Juven Wang

* [personal page](http://www.sns.ias.edu/~juven/)

* [GoogleScholar page](https://scholar.google.ca/citations?user=eVd7eJsAAAAJ&hl=en)

* [InSpire page](https://inspirehep.net/authors/1268730)

## Selected writings

On [[symmetry protected topological phases]]:

* [[Juven Wang]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_, Phys. Rev. Lett. **114** (2015) 031601 [[arxiv:1405.7689](https://arxiv.org/abs/1405.7689), [doi:10.1103/PhysRevLett.114.031601](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)]

On [[CPT symmetry]]:

* [[Juven Wang]], *C-P-T Fractionalization*, Phys. Rev. D **106** (2022) 105009 [[arXiv:2109.15320](https://arxiv.org/abs/2109.15320), [doi:10.1103/PhysRevD.106.105009](https://doi.org/10.1103/PhysRevD.106.105009)]

On [[quantum anomalies]] related to [[spin^c structure]] and [[spin^h structure]]:

* {#WWW19} [[Juven Wang]], [[Xiao-Gang Wen]], [[Edward Witten]], _A New SU(2) Anomaly_, Journal of Mathematical Physics **60**  (2019) 052301 [[arXiv:1810.00844](https://arxiv.org/abs/1810.00844), [doi:10.1063/1.5082852](https://doi.org/10.1063/1.5082852)]

Speculation of [[topological phases of matter]] relevant in the [[standard model of particle physics]]:

* [[Juven Wang]], *Ultra Unification*, Phys. Rev. D **103**  (2021) 105024 [[arXiv:2012.15860](https://arxiv.org/abs/2012.15860), [doi:10.1103/PhysRevD.103.105024](https://doi.org/10.1103/PhysRevD.103.105024)]

On stability of the [[proton]]:

* [[Juven Wang]], Zheyan Wan, Yi-Zhuang You, *Proton Stability: From the Standard Model to Beyond Grand Unification*, Phys. Rev. D **106** (2022) 025016 [[arXiv:2204.08393](https://arxiv.org/abs/2204.08393), [doi:10.1103/PhysRevD.106.025016](https://doi.org/10.1103/PhysRevD.106.025016)]

On [[generalized global symmetries]] hypothesized to play a role in the [[standard model of particle physics]]:

* {#PutrovWang23} [[Pavel Putrov]], [[Juven Wang]], *Categorical Symmetry of the Standard Model from Gravitational Anomaly* [[arXiv:2302.14862](https://arxiv.org/abs/2302.14862)]

category: people


jxb

Just a philosopher.

Some of my (hopefully more relevant) interests:

Duality in (and between) philosophy and mathematics

Philosophical interpretations of adjointness (and adjointness in philosophical concepts!)

Dialectic (esp. Platonic dialectic) and formalization (formalization of dialectic, dialectic of formalization)

Reflexivity, reflection, self-reference, and indexicality in philosophy and mathematics

Fixed point theorems and reflection principles

The ontology of algebraic structures, especially groups

Intensionality and extensionality

Equivalences among the different ways that first-order theories can be exceeded, extended, and broken

Metalogic as ethics

[my academia page](https://villanova.academia.edu/JohnBova)


Ján Pulmann

* [institute page](https://www.maths.ed.ac.uk/school-of-mathematics/people/a-z?person=1043)

* [homepage](https://jpulmann.github.io/)

## Selected writings

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Martin Doubek]], [[Branislav Jurčo]], [[Ján Pulmann]], *Quantum L∞ Algebras and the Homological Perturbation Lemma*  Comm. Math. Phys. **367** (2019) 215–240 [[arXiv:1712.02696](https://arxiv.org/abs/1712.02696), [doi:10.1007/s00220-019-03375-x](https://doi.org/10.1007/s00220-019-03375-x)]

On [[duality in physics|duality]] for [[higher gauge theories]], subsuming [[non-abelian T-duality]] and [[electric-magnetic duality]], via [[boundary conditions]] for [[AKSZ sigma-models]]:

* [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], _A non-abelian duality for (higher) gauge theories_ ([arXiv:1909.06151](https://arxiv.org/abs/1909.06151))

category: people

[[!redirects Jan Pulmann]]


János Kollár

* [personal page](https://web.math.princeton.edu/~kollar/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/J%C3%A1nos_Koll%C3%A1r)
## Selected writings

On [[Hilbert schemes]] of [[rational curves]] on [[algebraic varieties]]:

* [[János Kollár]], *Rational Curves on Algebraic Varieties*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **32**, Springer 1996 ([doi:10.1007/978-3-662-03276-3](https://doi.org/10.1007/978-3-662-03276-3))

On [[birational geometry]]:

* [[János Kollár]], [[Shigefumi Mori]], _Birational geometry of algebraic varieties_, Cambridge Tracts in Mathematics __134__, Cambridge University Press 2010 ([doi:10.1017/CBO9780511662560](https://doi.org/10.1017/CBO9780511662560))

On [[algebraic hypersurfaces]]:

* [[János Kollár]], *Algebraic hypersurfaces*, Bull. Amer. Math. Soc. 56 (2019), 543-568  ([arXiv:1810.02861](https://arxiv.org/abs/1810.02861), [doi:10.1090/bull/1663](https://doi.org/10.1090/bull/1663))

category: people

[[!redirects Janos Kollar]]


Jérôme Cayssol

* [Institute page](https://www.loma.cnrs.fr/jerome-cayssol/)

## Selected writings

On [[electronic band structure]] with focus on [[topological phases of matter]]:

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

category: people

[[!redirects Jerome Cayssol]]


Jérôme Gaillard

## Selected writings

On [[G-structure]] in [[KK-compactification]] of [[supergravity]] and application to [[holographic QCD]]:

* [[Jérôme Gaillard]], _On $G$-structures in gauge/string duality_, 2011 ([cronfa:42569](https://cronfa.swan.ac.uk/Record/cronfa42569) [spire:1340775](http://inspirehep.net/record/1340775), [[GaillardGStructure.pdf:file]])

On [[Seiberg duality]]:

* Eduardo Conde, [[Jérôme Gaillard]], [[Carlos Núñez]], Maurizio Piai, Alfonso V. Ramallo, _Towards the string dual of tumbling and cascading gauge theories_ ([arXiv:1112.3346](http://arxiv.org/abs/1112.3346))

On [[D6-branes]] lifting to [[Kaluza-Klein monopoles]] in [[M-theory]]:

* {#GaillardSchmude09} [[Jérôme Gaillard]], [[Johannes Schmude]], _The lift of type IIA supergravity with D6 sources: M-theory with torsion_, JHEP 1002:032,2010 ([arXiv:0908.0305](http://arxiv.org/abs/0908.0305))

category: people

[[!redirects Jerome Gaillard]]


Jérôme Poineau

* [webpage](http://www-irma.u-strasbg.fr/~poineau/)

[[!redirects Jerome Poineau]]