near-horizon geometry

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#Contents#
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## Idea

The [[spacetime]] geometry of a [[black hole]] or [[black brane]] close to the [[horizon]] is called its _near-horizon geometry_. Accordingly, the geometry far from the horizon might be called its _far-horizon geometry_. One commonly says that the black hole/brane solution _interpolates_ between its near and far horizon geometry.

## Properties

In the following "small" and "large" [[radius]] is in [[units]] of [[Planck length]] $\ell_P$ times a root of the integer charge ("number" $N$) of the branes, i.e. being "near" to the horizon means that

$$
  r/\ell_P N^{1/k} \ll 1
$$

while "far" from the horizon means that

$$
  r/\ell_P N^{1/k} \gg 1
$$

(e.g. [AFFHS 98, (2), (7) and below (11)](#AFFHS98)).

This means that the near/far horizon limit may also be thought of as corresponding to large/small $N$, respectively.

Since the [[Planck length]] "is tiny" and due to the higher roots of $N$ appearing here, this means that $N$ must be "huge" for the near horizon limit to be visible at macroscopic scale, while, conversely, any "moderate" value of $N$ means implies that every macroscopic radius is "far" from the horizon.

### Near-horizon geometry
 {#NearHorizonGeometry}

For [[black brane|black]] [[M-branes]] ([[black branes]] in [[11-dimensional supergravity]]) of [[dimension]] $p+1$ that preserve some [[supersymmetry]], the near horizon geometry is always a [[Cartesian product]] of an [[anti-de Sitter spacetime]] $AdS_{p+2}$ with a [[compact topological space|compact]] [[Einstein manifold]] $X_{11-(p+2)}$;

$$
  Ads_{p+2} \times X_{d-(p+2)}
  \,;
$$

while for [[black brane|black]] [[D-branes]] and [[NS5-branes]] in [[type II supergravity]] the near horizon geometry is [[conformal transformation|conformal]] to a geometry of this form $AdS_{p+2} \times X_{d-(p+2)}$ (see [AFFHS 98, section 2](#AFFHS98)).

### Far-horizon geometry
 {#FarHorizonGeometry}

In contrast, the "far-horizon geometry" of all those [[black branes]] whose near horizon geometry is $AdS_{p+2} \times X_{d-(p+2)}$ (i.e. actual [[anti-de Sitter spacetime]] without conformal factor) is of the form

$$
  \mathbb{R}^{p,1} \times C\left(X_{d-(p+2)}\right)
  \,,
$$

where $\mathbb{R}^{p,1}$ is [[Minkowski spacetime]] (the brane [[worldvolume]]) and $C(X_{d-(p+2)})$ is the [[metric cone]] over $X_{d-(p+2)}$, hence an [[orbifold]] (see [AFFHS 98, section 3](#AFFHS98)). Since this "far-horizon limit" is still a solution to the [[supergravity]] [[equations of motion]] away from the tip of the cone, it may in itself be regarded as a "[[cone brane]]"-solution (see [AFFHS 98, section 3.1](#AFFHS98)).

If $X_{d-(p+2)}$ is a smooth [[quotient space]] by the [[action]] of a [[finite subgroup of SU(2)]], then the corresponding [[cone brane]] is a brane "at an [[ADE-singularity]]".

Examples and applications of such [[cone branes]], in the context of [[M-theory on G2-manifolds]], are discussed in [Atiyah-Witten 01](#AtiyahWitten01).

See also at *[[flat space holography]]*.

## Examples
 {#Examples}

The near/far horizon limits of the [[black brane|black]] [[M-branes]]:

* _[The black M2-brane](#BlackM2)_

* _[The black M5-brane](#BlackM5Brane)_

* _[The MK6-brane](#MK6Brane)_

### The black M2-brane
 {#BlackM2}

The [[black brane|black]] [[M2-brane]] is given (we follow [AFFHS98, §2.1.1](#AFFHS98)) by the [[Riemannian metric]]

\[
  \label{BlackM2BraneMetric}
  g_{M2} \;\coloneqq\;
  H^{- 2/3} g_{(\mathbb{R}^{2,1})}
  +
  H^{1/3} g_{C(X_7)}
\]

and the [[C-field]] [[field strength|strength]]

$$
  F_{M2}
     \;\coloneqq\;
  dvol_{\mathbb{R}^{2,1}} \wedge d H^{-1}
$$

where $C(X_7)$ denotes the [[metric cone]] on a [[closed manifold|closed]] 7-[[dimension|dimensional]] [[Einstein manifold]] $X_7$ for [[cosmological constant]] $\Lambda = 5$, whence

$$
  g_{C(X)}
  \;\coloneqq\;
  (d r)^2 + r^2 g_{X_7}
  \,,
$$

and

$$
  H \;\coloneqq\; 1 + \frac{\ell_{th}^6}{r^6}
  \:,
  \phantom{AAA}
  \ell_{th} \;\coloneqq\; 2^{5/6} \pi^{2/6} N^{1/6} \ell_P
$$

with $N$ the number of [[M2-branes]] and with $\ell_P$ the [[Planck length]] in 11 dimensions.

In the near-horizon/large $N$-limit $\ell_{th} \to \infty$ this becomes

$$
  \begin{aligned}
    g_{M2}
    \overset{\ell_{th} \gg 1}{\longrightarrow}
    \;\;\;
    &
    \left(
      r/\ell_{th}
    \right)^{4}
    g_{(\mathbb{R}^{2,1})}
    +
    \left( r/\ell_{th} \right)^{-2} g_{C(X_7)}
    \\
    = &
    \left(
      r/\ell_{th}
    \right)^{4}
    g_{(\mathbb{R}^{2,1})}
    +
    \left( r/\ell_{th} \right)^{-2} (d r)^2
    +
    \ell_{th}^{2}
    g_{X_7}
    \\
    = &
    \underset{
      = g_{AdS}
    }{\underbrace{
    \frac{1}{z^2}
    \left(
      2^4 g_{(\mathbb{R}^{2,1})}
      +
      (d z)^2
    \right)
    }}
    +
    \ell_{th}^{2}
    g_{X_7}
  \end{aligned}
  \,,
$$

where in the last step we set

$$
  r \;\coloneqq\; 2 \ell_{th} \frac{1}{\sqrt{z}}
$$

This reveals the first summand as being the [[metric tensor]] of [[anti-de Sitter spacetime]] of AdS radius $\ell_{th}$ in [[horospheric coordinates]], and the second summand as that of $X_{7}$ rescaled to radius $\ell_{th}$.

In contrast, in the far-horizon/small $N$-limit $\ell_{th} \to 0$ (eq:BlackM2BraneMetric) becomes

$$
  g_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\;
  g_{\mathbb{R}^{2,1}}
  +
  g_{C(X_7)}
$$

and

$$
  F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0
$$

which is the metric on a [[Cartesian product]] of flat [[Minkowski spacetime]] [[worldvolume]] of an M2-brane with the [[metric cone]] on $X_7$.

### The black M5-brane
 {#BlackM5Brane}

The [[black brane|black]] [[M5-brane]] is given (we follow [AFFHS98, §2.1.2](#AFFHS98)) by the [[Riemannian metric]]

\[
  \label{BlackM5BraneMetric}
  g_{M5}
    \;\coloneqq\;
  H^{-1/3} g_{(\mathbb{R}^{5,1})}
  +
  H^{2/3} g_{C(X_4)}
\]

and the [[C-field]] [[field strength|strength]]

$$
  F_{M2} \;\coloneqq\; \pm 3 \star_5  d H
$$

where $C(X_4)$ denotes the [[metric cone]] on a [[closed manifold|closed]] 4-[[dimension|dimensional]] [[Einstein manifold]] $X_4$ for [[cosmological constant]] $\Lambda = 3$, whence

$$
  g_{C(X)}
  \;\coloneqq\;
  (d r)^2 + r^2 g_{X_4}
  \,,
$$

and

$$
  H \;\coloneqq\; 1 + \frac{\ell_{th}^3}{r^3}
  \:,
  \phantom{AAA}
  \ell_{th} \;\coloneqq\; \pi^{1/3} N^{1/3} \ell_P
$$

with $N$ the number of [[M5-branes]] and with $\ell_P$ the [[Planck length]] in 11 dimensions.

In the near-horizon/large $N$-limit $\ell_{th} \to \infty$ this becomes

$$
  \begin{aligned}
    g_{M5}
    \overset{\ell_{th} \gg 1}{\longrightarrow}
    \;\;\;
    &
    \left(
      r/\ell_{th}
    \right)^{4}
    g_{(\mathbb{R}^{2,1})}
    +
    \left( r/\ell_{th} \right)^{-2} g_{C(X_7)}
    \\
    = &
    r/\ell_{th}
    g_{(\mathbb{R}^{2,1})}
    +
    \left( r/\ell_{th} \right)^{-2} (d r)^2
    +
    \ell_{th}^{2}
    g_{X_7}
    \\
    = &
    \underset{
      = g_{AdS}
    }{\underbrace{
    \frac{1}{z^2}
    \left(
      2 g_{(\mathbb{R}^{2,1})}
      +
      (d z)^2
    \right)
    }}
    +
    \ell_{th}^{2}
    g_{X_4}
  \end{aligned}
  \,,
$$

where in the last step we set

$$
  r \;\coloneqq\; \ell_{th}\tfrac{1}{2} \frac{1}{z^2}
$$

This reveals the first summand as being the [[metric tensor]] of [[anti-de Sitter spacetime]] of AdS radius $\ell_{th}$ in [[horospheric coordinates]], and the second summand as that of $X_{4}$ rescaled to radius $\ell_{th}$.

In contrast, in the far-horizon/small $N$-limit $\ell_{th} \to 0$ (eq:BlackM5BraneMetric) becomes

$$
  g_{M5} \;\overset{\ell_{th} \to 0}{\longrightarrow}\;
  g_{\mathbb{R}^{5,1}}
  +
  g_{C(X_4)}
$$

and

$$
  F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0
$$

which is the metric on a [[Cartesian product]] of flat [[Minkowski spacetime]] [[worldvolume]] of an M5-brane with the [[metric cone]] on $X_4$.

### The MK6-brane
 {#MK6Brane}

The [[metric tensor]] of $N$ coincident [[KK-monopoles]] in [[11-dimensional supergravity]] in the limit that $\ell_{th} \coloneqq N \ell_P \to 0$ is

\[
  \label{SmallNMK6}
  g_{MK6}
    \;=\;
  g_{\mathbb{R}^{6,1}}
  +
  (d y)^2
  +
  y^2
  \big(
    (d \theta)^2
    +
    (\sin \theta)^2 (d \varphi)^2
    +
    (\cos \theta)^2 (d \phi)^2
  \big)
\]

subject to the identification

\[
  \label{OrbifoldIdentificationForKKMonopole}
  (\varphi, \phi)
  \;\sim\;
  (\varphi, \phi) + (2\pi/N ,2\pi/N)
  \,.
\]

This is equation (47) in [IMSY 98](#IMSY98), which applies subject to the condition

$$
  U/\left(\frac{N}{g^{2/3}_{YM}}\right)
  \;=\;
  U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right)
  \;\gg\;
  1
$$

from a few lines above. Inserting this condition into the definition $y^2 \coloneqq 2 N \ell^3_P U $ right above (47) shows that

$$
  \begin{aligned}
  y^2
  & =
  2 N \ell^3_P U
  \\
  & =
  2(2\pi)^{-4/3} N^2 \ell_P^2
  \;
  \underset{
    \gg 1
  }{
  \underbrace{
     \left(U/\left(\frac{N}{ (2 \pi)^{4/3} \ell_P}\right)\right)
  }}
  \end{aligned}
$$

hence that the distance $y$ from the locus of the MK6-brane is large in units of

$$
  \ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P
  \,.
$$

The identification (eq:OrbifoldIdentificationForKKMonopole) means that this is the [[orbifold]] [[metric cone]] $\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right)$, hence an [[ADE classification|
A-type]] [[ADE-singularity]]. To make this more explicit, introduce the complex coordinates

$$
  v \;\coloneqq\; y \, e^{i \varphi} \sin \theta
  \;\;\;
  w \;\coloneqq\; y \, e^{i \phi} \cos \theta
$$

on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which (eq:SmallNMK6) becomes

$$
  g_{MK6}
  \;\coloneqq\;
  d v d \overline v
  +
  d w d \overline w
$$

and which exhibit the identification (eq:OrbifoldIdentificationForKKMonopole) as indeed that of the [[ADE classification|A-type]] $\mathbb{Z}_N$-action ([Asano 00, around (18)](#Asano00)).

[[!include KK-monopole geometries -- table]]

## Related concepts

* [[piecewise flat spacetime]]

* [[anti-de Sitter spacetime]]

* [[asymptotic boundary]]

## References

### General

See also

* Wikipedia, _[Near-horizon metric](https://en.wikipedia.org/wiki/Near-horizon_metric)_

### For extremal black holes

That the near horizon geometry of the [[extremal black hole|extremal]] [[Reissner-Nordström black hole]] in $\mathcal{N}=2$ [[4d supergravity]] is $AdS_2 \times S^2$ was observed in

* [[Gary Gibbons]], in F. del Aguila, [[J. de Azcarraga]], [[Luis Ibanez]] (eds.) _Supersymmetry, Supergravity and Related Topics_

Review:

* Hari K. Kunduri, James Lucietti, _Classification of Near-Horizon Geometries of Extremal Black Holes_ ([web](http://relativity.livingreviews.org/Articles/lrr-2013-8/title.html))

Description of the near-horizon geometry of near-extremal black holes by [[Jackiw-Teitelboim gravity]]:

* {#NSST18} Pranjal Nayak, Ashish Shukla, Ronak M Soni, [[Sandip Trivedi]], V. Vishal, _On the Dynamics of Near-Extremal Black Holes_ ([arXiv:1802.09547](https://arxiv.org/abs/1802.09547))

* {#MTV18} Upamanyu Moitra, [[Sandip Trivedi]], V. Vishal, _Near-Extremal Near-Horizons_ ([arXiv:1808.08239](https://arxiv.org/abs/1808.08239))

### For black branes

That the near horizon geometry of [[black branes]] in [[11-dimensional supergravity]] is (conformal to) [[anti de Sitter spacetime]] times some [[compact space]] is apparently due to

* {#GibbonsTownsend93} [[Gary Gibbons]], [[Paul Townsend]], _Vacuum interpolation in supergravity via super p-branes_, Phys. Rev. Lett. 71, (1993) 3754 ([arXiv:hep-th/9307049](https://arxiv.org/abs/hep-th/9307049))

* {#DuffGibbonsTownsend94} [[Mike Duff]], [[Gary Gibbons]], [[Paul Townsend]], _Macroscopic superstrings as interpolating solitons_, Phys. Lett. B. 332 (1994) 32 ([arXiv:hep-th/9405124](https://arxiv.org/abs/hep-th/9405124))

* {#GibbonsHorowitzTownend95} [[Gary Gibbons]], [[Gary Horowitz]], [[Paul Townsend]], _Higher-dimensional resolution of dilatonic black hole singularities_, Class. Quant. Grav. 12 (1995) 297 ([arXiv:hep-th/9410073](https://arxiv.org/abs/hep-th/9410073))

* [[Gary Gibbons]], Nucl. Phys. B207, (1982) 337;

* [[Renata Kallosh]], [[Amanda Peet]], Phys. Rev. B46, (1992) 5223;

* [[Sergio Ferrara]], [[Gary Gibbons]], [[Renata Kallosh]], Nucl. Phys. B500, (1997) 75;

* [[Ali Chamseddine]], [[Sergio Ferrara]], [[Gary Gibbons]], [[Renata Kallosh]], Phys. Rev. D55, (1997) 3647

The observation that the resulting [[isometry group]] is the bosonic body of one of the [[orthosymplectic super groups]] is due to

* P. Claus, [[Renata Kallosh]], J. Kumar, [[Paul Townsend]], [[Antoine Van Proeyen]], _Conformal Theory of M2, D3, M5 and D1+D5 Branes_, JHEP 9806 (1998) 004 ([arXiv:hep-th/9801206](https://arxiv.org/abs/hep-th/9801206))

A decent account is in

* {#AFFHS98} [[Bobby Acharya]], [[Jose Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_, Adv. Theor. Math. Phys. **2** (1999) 1249-1286 [[arXiv:hep-th/9808014](https://arxiv.org/abs/hep-th/9808014)]

reviewed in

* {#FigOFar98} [[Jose Figueroa-O'Farrill]], _Near-horizon geometries of supersymmetric branes_, talk at [SUSY 98](http://inspirehep.net/record/971430/) ([arXiv:hep-th/9807149](https://arxiv.org/abs/hep-th/9807149), [talk slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/SUSY98.pdf))

The near horizon geometry of coincident [[KK-monopoles]] in [[11-dimensional supergravity]] is discussed in

* {#IMSY98} [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], section 9 of _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))

* {#ItzhakiTseytlinYankielowicz98} [[Nissan Itzhaki]], [[Arkady Tseytlin]], [[Shimon Yankielowicz]], _Supergravity Solutions for Branes Localized Within Branes_, Phys.Lett.B432:298-304, 1998 ([arXiv:hep-th/9803103](https://arxiv.org/abs/hep-th/9803103))

* {#Hashimoto98} Akikazu Hashimoto, _Supergravity Solutions for Localized Intersections of Branes_, JHEP 9901 (1999) 018 ([arXiv:hep-th/9812159](https://arxiv.org/abs/hep-th/9812159))

* {#Asano00} Masako Asano, section 3 of _Compactification and Identification of Branes in the Kaluza-Klein monopole backgrounds_ ([arXiv:hep-th/0003241](https://arxiv.org/abs/hep-th/0003241))

Examples and applications of cone branes in the context of [[M-theory on G2-manifolds]] are discussed in

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

Conical [[D-branes]] are discussed in

* [[Koji Hashimoto]], Shunichiro Kinoshita, Keiju Murata, _Conic D-branes_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 8,  August 2015 ([arXiv:1505.04506](https://arxiv.org/abs/1505.04506), [slides pdf](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2015/slides/kinoshita.pdf))

Near horizon geometries of [[black branes]] which [[KK-compactification|KK-compactify]] to [[black holes]]:

* [[Mirjam Cvetič]], Paulo J. Porfírio, Alejandro Satz, _Gaussian null coordinates, near-horizon geometry and conserved charges on the horizon of extremal non-dilatonic black p-branes_ ([arXiv:2003.09304](https://arxiv.org/abs/2003.09304))

Further discussion for [[M2-branes]]:

* Harvendra Singh, _M2-branes on a resolved $\mathbb{C}^4/\mathbb{Z}_4$_, 	JHEP 0809:071, 2008 ([arXiv:0807.5016](https://arxiv.org/abs/0807.5016))

* Chethan Krishnan, Carlo Maccaferri, Harvendra Singh, _M2-brane Flows and the Chern-Simons Level_, JHEP 0905:114, 2009 ([arXiv:0902.0290](https://arxiv.org/abs/0902.0290))

Discussion for [[heterotic supergravity]]:

* Karl-Philip Gemmer, Alexander S. Haupt, Olaf Lechtenfeld, Christoph Nölle, Alexander D. Popov, _Heterotic string plus five-brane systems with asymptotic $AdS_3$_, Adv. Theor. Math. Phys. 17 (2013) 771-827 ([arXiv:1202.5046](https://arxiv.org/abs/1202.5046))

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near-ring

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# Near-rings
* table of contents
{: toc}

## Idea

A near-ring is a sort of [[ring]] in which [[addition]] need not be commutative.  However, if we simply remove the commutativity of addition from the usual definition of a ring, then nothing changes, because we can prove commutativity from the other axioms: we have
$$ (x+y)(1+1) = x(1+1) + y(1+1) = x + x + y + y $$
and also
$$ (x+y)(1+1) = (x+y)1 + (x+y)1 = x + y + x + y. $$
Canceling $x$ on the left and $y$ on the right, we have $x+y=y+x$.

Thus, in order for the notion of near-ring to be different from that of a ring, we need to relax the [[distributivity law]] as well; we impose it only on one side.

## Definition

A **near-ring** is a set $R$ equipped with

1. A [[group]] structure $(R,+,0)$,

2. A [[monoid]] structure $(R,\cdot,1)$,

3. such that for any $x,y,z\in R$ we have $(x+y)\cdot z = (x\cdot z) + (y\cdot z)$, and for any $x\in R$ we have $0\cdot x = 0$.

If $(R,+,0)$ is only a monoid or semigroup, we say instead that $R$ is a **near-rig** or a **near-semiring**.  (Of course, now it is possible to have distributivity on both sides without making addition commutative, since addition need not always cancel.)

There is also a weaker notion, of __loop near ring__, where in the
axioms above the group structure on $(R,+,0)$ is weakened to a [[loop (algebra)|loop]]
algebraic structure (i.e. left and right additive inverses no longer coincide and the associativity of addition is dropped).

## Internalization

Of course, near-rings can be defined [[internalization|internally]] to any [[cartesian monoidal category]].  More generally, they can be defined internally to a braided [[duoidal category]].

## References

* [wikipedia](en.wikipedia.org/wiki/Nearring)

For loop near rings see

* D. Ramakotaiah, C. Santhakumari. On loop near-rings.
Bull. Austral. Math. Soc. 19(3):417{435, 1978.

Loop near rings appear in topology:

* Damir Franetič, Petar Pavešić, _Loop near-rings and unique decompositions of H-spaces_, [arxiv/1511.06168](http://arxiv.org/abs/1511.06168)

category: algebra

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nearby homomorphisms from compact Lie groups are conjugate

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#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[compact Lie group]] and $\Gamma$ any [[Lie group]], every [[group homomorphisms]] $\phi \,\colon\, G \xrightarrow{\;} \Gamma$ (as [[topological groups]], [[continuous homomorphisms of Lie groups are smooth|hence]] as [[Lie groups]]) has a [[neighbourhood]] $U_\phi$ (in the homomorphism-[[subspace]] of the [[compact open topology]]) all whose elements $\phi'$ are [[conjugation|conjugate]] to $\phi$, in that

\[
  \label{ExistenceOfConjugationsForNearbyHomomorphisms}
  \underset{
    \phi' \in U_{\phi}
  }{\forall}
  \;\;\;
  \underset{
    \gamma \in \Gamma
  }{\exists}
  \;\;\;\;\;\;\;
  \phi'(-)
    \,=\,
  \gamma^{-1} \cdot \phi(-) \cdot \gamma
  \,.
\]

([Conner & Floyd 1964, Lem. 31.8](#ConnerFloyd64), [Rezk  et al. 2013](#Rezk13)).

It follows (highlighted in [Rezk 2014, Rem. 2.2.1](#Rezk14)) that the [[quotient space]] of the homomorphism space $Hom(G,\Gamma) \,\subset\, Maps(G,\Gamma)$ under the [[adjoint action]] by $\Gamma$ is [[discrete topological space|discrete]]:

\[
  \label{DiscreteQuotientSpace}
  Hom(G,\Gamma)/_{\!\!ad} \Gamma
  \;\;\;
  \in
  Set
  \xhookrightarrow{\;}
  TopSp
  \,,
\]

and that the homomorphism space itself decomposes, as a [[topological G-space|$\Gamma$-space]] via the [[adjoint action]], into the [[disjoint union]], [[indexed set|indexed]] by (eq:DiscreteQuotientSpace), of the [[coset spaces]] of $\Gamma$ by the corresponding [[stabilizer subgroups]] ("[[centralizer subgroups]]") $C_\Gamma(\phi) \subset \Gamma$:

\[
  \label{DisjointUnionOfOrbits}
  Hom(G,\Gamma)
  \;\simeq\;
  \underset{
    {[\phi] \, \in }
    \atop
    \mathclap{ Hom(G,\Gamma)/\Gamma }
  }{\bigsqcup}
  \Gamma/C_{{}_{\Gamma}}(\phi)
  \;\;\;
  \in
  \;
  \Gamma Act(TopSp)
  \,.
\]

## Examples

\begin{example}\label{Pontrjagin duality}
**([[Pontrjagin duality]])**
\linebreak
  In the special case that $\Gamma = S^1$ is the [[circle group]] and $G$ is an [[abelian group]], the homomorphism space $\widehat G \,\coloneqq\, Hom(G,S^1)$ is the [[Pontrjagin dual]] of $G$.

In this case, since $S^1$ is [[abelian group|abelian]] so that the [[conjugation action]] on $Hom(G,S^1)$ is [[trivial action|trivial]], the statement (eq:DiscreteQuotientSpace) is a special case of the classical fact that the [[Pontrjagin dual]] of a [[compact topological group]] is [[discrete group|discrete]] (see [there](Pontrjagin+dual#BasicPropertiesOfDualGroups)).
\end{example}

## Variants

### For crossed homomorphisms
 {#ForCrossedHomomorphisms}

Let

$$
  \alpha
    \;\colon\;
  G
    \xrightarrow{\;}
  Aut_{Grp}(\Gamma)
$$

be a [[group action]] by [[group homomorphism|group]] [[automorphisms]], with the special property that its restriction to the [[center of a group|center]] $C(G) \subset G$ is [[trivial action|trivial]] (for example when $G$ has [[trivial group|trivial]] center to start with):

$$
  \alpha_{\vert C(G)}
  \;=\;
  id_{\Gamma}
  \,.
$$

In that case the above statement (eq:ExistenceOfConjugationsForNearbyHomomorphisms) generalizes to say that *nearby [[crossed homomorphisms]] are [crossed conjugate](crossed+homomorphism#AdjointActionOnCrossedHomomorphisms).

More precisely, notice ([this Prop.](crossed+homomorphism#CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup)) that  a [[crossed homomorphism]] $\phi \;\colon\; G \xrightarrow{\;} \Gamma$ is equivalently
a plain [[group homomorphism]] $(\phi(-),\,(-)) \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G$ to the [[semidirect product group]], subject to the constraint that its projection to $G$ is the [[identity morphism]].
Now say that another crossed homomorphism $\phi'$ is nearby if it is so as a plain homomorphism $(\phi'(-),(-))$ to the semidirect product group (i.e. we consider [[neighbourhoods]] of crossed homomorphism in the sense of neighbourhoods of the points which they represent in $Hom(G,\Gamma \rtimes G))$.

Then the above statement (eq:ExistenceOfConjugationsForNearbyHomomorphisms) says that there is an element $(\gamma,\,h) \,\in\, \Gamma \rtimes G$ such that

\[
  \label{ConjugationOfHomomorphismsToSemidirectProductGroup}
  \underset{g \in G}{\forall}
  \;\;\;\;\;
  \big(
    \phi'(g),\,g
  \big)
  \;\;
  =
  \;\;
  \big(
    \gamma, \, h
  \big)^{-1}
  \cdot
  \big(
    \phi(g),\,g
  \big)
  \cdot
  \big(
    \gamma, \, h
  \big)
  \,.
\]

In order for such a conjugation to be a [crossed conjugation](crossed+homomorphism#AdjointActionOnCrossedHomomorphisms) of $\phi$ to $\phi'$, we need its $G$-component to be the [[neutral element]]: $h = \mathrm{e} \,\in\, G$.

Now, the further conjugation of (eq:ConjugationOfHomomorphismsToSemidirectProductGroup) by $\big(\mathrm{e},\,h^{-1}\big)$ yields:

$$
  \Big(
    \alpha(h^{-1})
    \big(
      \phi'(g)
    \big)
    ,\, g
  \Big)
  \;\;
  =
  \;\;
  \big(
    \mathrm{e},\, h
  \big)
  \cdot
  \big(
    \gamma, \, h
  \big)^{-1}
  \cdot
  \big(
    \phi(g),\,g
  \big)
  \cdot
  \big(
    \gamma, \, h
  \big)
  \cdot
  \big(
    \mathrm{e},\, h^{-1}
  \big)
  \,.
$$

But we know that $h \in C(G)$ is in the [[center of a group|center]] of $G$, since the projection of both sides of (eq:ConjugationOfHomomorphismsToSemidirectProductGroup) to $G$ must be the identity, by construction of crossed homomorphisms.

Therefore, by the assumption that the action of $G$ on $\Gamma$ restricts along the inclusion $C(G) \xhookrightarrow{\;} G$ to the [[trivial action]],
we have $\alpha(h^{-1})\big( \phi(g) \big) \,=\, \phi(g)$ and it follows that

$$
  \big(
    \gamma, \, h
  \big)
  \cdot
  \big(
    \mathrm{e},\, h^{-1}
  \big)
  \;\;
  =
  \;\;
  \big(
    \gamma, \, \mathrm{e}
  \big)
$$

exhibits the required crossed conjugation:

$$
  \phi'(-) \;=\;  \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)
  \,.
$$

### For non-Lie groups
 {#ForNonCompactGroups}

* While the [[projective unitary group]] $\Gamma \,\coloneqq\,$ [[PU(ℋ)]] on a [[separable Hilbert space]] $\mathcal{H}$ is not a (finite-dimensional) [[Lie group]] and hence outside the scope of assumption for the above general theorem, the conclusion still holds, at least for $G$ [[discrete group|discrete]], hence [[finite group|finite]]:

  The [[PU(ℋ)]][[G-space|-space]] of homomorphisms $Hom\big(G,\, PU(\mathcal{H})\big)$ is a [[disjoint union]] (eq:DisjointUnionOfOrbits) of [[orbits]] of the [[conjugation action]].

  This is established in [Uribe & Lück 2013, Sec. 15](universal+equivariant+PUH-bundle#UribeLueck13), [p. 38](https://arxiv.org/pdf/1304.4862.pdf#page=38).

## Related concepts

* [[Hilbert's fifth problem]]

## References

* {#ConnerFloyd64} [[Pierre Conner]], [[Edwin Floyd]], Ch. III, Lem. 38.1 in: _Differentiable periodic maps_, Ergebnisse der Mathematik und ihrer Grenzgebiete **33**, Springer 1964 ([doi:10.1007/978-3-662-41633-4](https://link.springer.com/book/10.1007/978-3-662-41633-4))

* {#Rezk13} [[Charles Rezk]], *Nearby homomorphisms from compact Lie groups are conjugate*, 2013 ([MO:q/123624](https://mathoverflow.net/q/123624/381))

* {#Rezk14} [[Charles Rezk]], Rem. 2.2.1 in: *[[Global Homotopy Theory and Cohesion]]*, 2014 ([pdf](http://www.math.uiuc.edu/~rezk/global-cohesion.pdf), [[Rezk14.pdf:file]])

See also:

* [[Alejandro Adem]], [[Frederick R. Cohen]], Lemma 2.5 in: *Commuting Elements and Spaces of Homomorphisms*, Math. Ann. **338** (2007) 587–626 ([arXiv:math/0603197](https://arxiv.org/abs/math/0603197), [doi:10.1007/s00208-007-0089-z](https://doi.org/10.1007/s00208-007-0089-z))

* [[Charles Rezk]], *Classifying spaces for 1-truncated compact Lie groups*, Algebr. Geom. Topol. 18 (2018) 525-546 ([arXiv:1608.02999](https://arxiv.org/abs/1608.02999), [doi:10.2140/agt.2018.18.525](https://doi.org/10.2140/agt.2018.18.525))

[[!redirects nearby homomorphisms are conjugate]]


nearly AdS2/CFT1

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The special case of [[AdS/CFT duality]] in dimensions 2/1.

## References

### General

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))

[[!include AdS2-CFT1 -- references]]

[[!redirects nearly AdS2-CFT1]]

[[!redirects AdS2/CFT1]]
[[!redirects AdS2-CFT1]]

[[!redirects AdS2-CFT1 duality]]
[[!redirects AdS2/CFT1 duality]]


necessity and possibility

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[modal logic]] and specifically in [[epistemic modal logic]], two key [[modalities]] that one wants to consider are those of _necessity_ and _possibility_.

The idea is to consider for any [[proposition]] $p$

* a proposition labeled $\Box p$ expressing the idea that "$p$ is necessarily true";

* a proposition labeled $\lozenge p$ expressing the idea that "$p$ is possibly true".

## Via S4 modal logic
 {#ViaS4ModalLogic}

A common formalization of necessity/possibility is via [[S4 modal logic|S4]] or [[S5 modal logic]], whose [[axioms]] are motivated as follows:

For every comonadic [[modality]] like necessity $\Box$, one typically demands that it preserves [[implication]], in the sense that

$$\Box(p \to q) \to (\Box p \to \Box q).$$

This preservation of implication is called the *[[axiom K (modal logic)|axiom K]]*.  In traditional non-[[categorical logic|categorical]] [[modal logic]] this is often all that is considered ("[[K modal logic]]), because the operators are considered [[de Morgan duality|de Morgan dual]]:

$$
  \label{deMorganDualityBetweenNecessiryAndPossibility}
  \Box A
  \;=\;
  \neg
  \big(
    \lozenge (\neg A)
  \big)
  \,.
$$

In an intuitionistic modal logic, one would also demand that $\lozenge$ preserves implication in a suitable sense, such as

$$\Box(p\to q) \to (\lozenge p \to \lozenge q).$$

One may also ask for a further compatibility such as $(\lozenge p \to \Box q) \to \Box(p\to q)$, as [Simpson](#Simpson) does.

Similarly, the axiom $\Box(p \to q) \to (\Box p \to \Box q)$ implies that $\Box$ preserves [[logical conjunction]], i.e. $\Box(p\wedge q) \leftrightarrow (\Box p \wedge \Box q)$.  The nullary version $\Box(\top)\leftrightarrow \top$ follows from the "necessitation rule" which says that if $p$ is provable in the empty context, then so is $\Box p$.  (This is an additional rule assumed in modal logics.)  From duality (eq:deMorganDualityBetweenNecessiryAndPossibility) it follows that $\lozenge$ preserves [[logical disjunction]] (both binary and nullary), while intuitionistically one may want to ask for this separately ([Simpson](#Simpson) does, but [Biermann & de Paiva 2000](#BiermanDePaiva00) don't).

A minimum further requirement on a formalization of $\Box$ and $\lozenge$ to have the interpretation of "necessity" and "possibility" is arguably that there are [[implications]]

* $\Box p \rightarrow p$;

* $p \rightarrow \lozenge p$,

expressing that if something is necessarily true, then that should mean that it is true in all instances, and that if something is true in one instance, then it is evidently possible for it to be true.  With these additions to the above [[K modal logic]], one speaks of *[[T modal logic]]*.

By similar plausibility arguments one often demands that

* $\Box p \to \Box \Box p$

* $\lozenge \lozenge p \to \lozenge p$

which may be read as expressing that iterating the previous reasoning does not yield any new insight.  This additional enhancement to [[T modal logic]] yields *[[S4 modal logic]]*.

\linebreak

In terms of [[categorical logic]] interpreting propositional logic into a [[Heyting algebra]], the S4 axioms just say ([Bierman & de Paiva 1996](#BiermanPaiva96), [2000](#BiermanPaiva00)) that:

* $\Box$ is a (necessarily [[idempotent comonad|idempotent]]) [[comonad]] which is [[monoidal functor|monoidal]], hence product-preserving; and

* $\lozenge$ is a (necessarily [[idempotent monad|idempotent]]) [[monad]] which is a "$\Box$-[[strong functor|strong]] functor", and (perhaps) preserves coproducts.

As usual, we can also replace the Heyting algebra by a [[cartesian closed category]], thereby obtaining a "proof-relevant" system; in this case monoidality of $\Box$ does not imply directly that it preserves products, although one might reasonably ask that it does (see [[strong monads]] [[monad in computer science|in computer science]]).
Note that if one moves from Heyting algebras to cartesian closed categories the (co)monads are not necessarily idempotent (and there are good proof-theoretical reasons why they should not be so).

The above reasoning makes plausible that any operators expressing "necessity" and "possibility" should *at least* satisfy these (co)monad axioms.  However, not every (co)monad is sensibly interpreted this way.

For example, there is

* the (idempotent) comonad $\varnothing$ which sends every proposition to [[false]] (the [[nothing|non-being]]-modality);

* the (idempotent) monad $\ast$ which sends every proposition to [[true]] (the [[being]]-modality).

These $\emptyset \dashv \ast$ satisfy all of the above axioms (as well as more axioms that are being considered, such as those called [[S5 modal logic]]) but they are not a very good interpretation of the informal concepts of "possibility" and "necessity".  Under this interpretation nothing is necessarily true, and everything is possibly true.  While this is not nonsensical, it is not very interesting and doesn't correspond well to our intuitive meanings of "necessary" and "possible".

This issue becomes more pronounced as one generalizes from the small realm of [[propositional logic]] to include both or either of:

* [[types]] more general than [[propositions]] (in [[modal type theory]]);

* [[first-order logic]] with [[quantifiers]], hence [[hyperdoctrines]]/[[dependent types]].

There are many [[modal operators]] in such contexts which are all modeled by (idempotent) (co)monads but which do not have the interpretation of expressing the modes of "necessity" or "possibility". See at _[[modal operator]]_ for some examples.

Therefore it makes sense to ask which _additional_ axioms on a modal operator make it an accurate formalization of the informal concepts of necessity and possibility.  The answer to this may depend on context and intention (after all one is trying to find a precise formulation of an a priori informal idea).

## Possible worlds via dependent types
 {#InFirstOrderLogicAndTypeTheory}

One common philosophical interpretation of "necessarily" and "possibly" is ([[Kripke semantics]]) in terms of a collection of "[[possible worlds]]" that are similar to the "real world", but not the same.  Under this interpretation, a proposition is necessarily true if it is true in all possible worlds, and possibly true if it is true in some possible world.

Now the idea of a proposition being true "necessarily in all possible cases" or "possibly at least in one case" is formally very well established in *predicate* logic: it is just the interpretation of the [[universal quantifier]] "for all" $\forall$ and of the [[existential quantifier]] "there exists" $\exists$.  In [[categorical logic]], these [[quantifiers]] (see there for details) are part of [[adjoint triple]] whose middle piece is [[context]] extension, and as such they naturally induce a [[comonad]] and a [[monad]].  Thus, if we interpret "necessarily" and "possibly" according to [[S5 modal logic]] in terms of [[possible worlds]], we can model them by this [[base change]] [[adjoint triple]] (cf. [Awodey 2006, p. 279](#Awodey06)):

### Globally
 {#Globally}

In more detail, let $W$ be the [[context]] [[type]] of [[variables]]/[[terms]] on which the propositions depend, i.e. the collection "of all [[possible worlds semantics|possible worlds]]" (in fact a [[Kripke frame]] for [[S5 modal logic]], see [there](Kripke+frame#NecessityLogic)).  Any specific choice of $W$ may be taken as specifying what is to be understood as a "possible world".

Writing $\mathbf{H}_{\ast}$ for the [[category]] of all context-free [[types]] under consideration and writing $\mathbf{H}_{/W}$ for the category of types in [[context]] "$W$", then in [[categorical logic]] (for instance $\mathbf{H}_{/(-)}$ might be a [[hyperdoctrine]] over a [[category of contexts]] containing objects $W$ and $\ast$) the [[quantifiers]] $\forall_{w\colon W}$ and $\exists_{w\colon W}$ participate in a [[base change]] [[adjoint triple]]

$$
  (\exists_W \dashv W^\ast \dashv \forall_W)
  \;\colon\;
  \mathbf{H}_{/W}
    \stackrel{\stackrel{\forall_{w \colon W}}{\longrightarrow}}{\stackrel{\stackrel{W^\ast}{\longleftarrow}}{\underset{\exists_{w\colon W}}{\longrightarrow}}}
  \mathbf{H}
  \,.
$$

In a context of pure [[logic]] this would be called

[[existential quantifier]] $\dashv$ [[context extension]] $\dashv$ [[universal quantifier]]

whereas in a context of [[dependent type theory]] this would be called

[[dependent sum]] $\dashv$ [[context extension]] $\dashv$ [[dependent product]].

In either case, under the suitable version of [[propositions as types]] (and using [[bracket types]] etc. if desired), the operations $\forall$ and $\exists$ have the usual interpretation of "for all" and "there exists".

Note, however, that these operations change the context from $W$ to $\ast$.  In other words, if a proposition $P$ depends on $W$, so that it may be true in some worlds and false in others, then $\exists_W P$ and $\forall_W P$ no longer depend on $W$.  The idea of a necessity and a possibility modality is to send a proposition in some context to a proposition in the *same* context, so that we can for instance say that $\Box P \to P$ and so on.  Thus we need to make $\exists_W P$ and $\forall_W P$ into propositions that again depend on $W$ --- even if they now depend trivially on $W$, being [[context extension|extended]] back from the absolute context $\ast$ to $W$.

This is just what is accomplished by passing from the above [[adjoint triple]] to the induced [[adjoint pair]] on $\mathbf{H}_{/W}$ by forming composites with the [[context extension]] operation $W^\ast$

\[
  \label{PossibilityNecessityAdjunctionViaBaseChange}
  (\underset{W}{\lozenge} \dashv \underset{W}{\Box})
  \coloneqq
  \left(
     \left(
     W^\ast \circ \underset{w\colon W}{\exists}
     \right)
     \dashv
     \left(
     W^\ast \circ \underset{w\colon W}{\forall}
     \right)
  \right)
  \;\colon\;
  \mathbf{H}_{/W}
    \longrightarrow
  \mathbf{H}_{/W}
  \,.
\]

With this, if $p\in \mathbf{H}_{/W}$ is a [[proposition]] about terms $w$ of $W$ (a $W$-[[dependent type]]) then

* $\underset{W}{\lozenge}(p)$ is [[true]]/[[inhabited type|inhabited]] precisely if $\underset{w \colon W}{\exists} p(w)$ is [[true]]/[[inhabited type|inhabited]], hence (that is the standard interpretation of the [[quantifier]]) if it is possible for $p(w)$ to be true for some $w$;

* $\underset{W}{\Box}(p)$ is [[true]]/[[inhabited type|inhabited]] precisely if $\underset{w \colon W}{\forall} p(w)$ is [[true]]/[[inhabited type|inhabited]], hence (that is again the standard interpretation of the quantifier) if $p(w)$ necessarily holds for all $w$.

Thus, this gives one [[syntax|syntactic]] formalization of the informal meaning of "necessity" and "possibility".  The natural [[semantics]] for these [[base change]] operations is a generalization of the simple traditional _[[possible worlds semantics]]_ of propositional necessity and possibility modalities.  (There are, however, more complicated possible worlds semantics.)

Moreover, with this formalization, the modal operator $\underset{W}{\lozenge}$ is [[left adjoint]] to $\underset{W}{\Box}$ and hence both form an [[adjoint modality]]. As discussed there, this is a formalization of [[unity of opposites|opposite]] concepts, which reflects well the opposition of necessity and possibility in their informal meaning.

Observe how the corresponding [[unit of a monad|unit]] and [[counit of a comonad|counit]] maps properly reflect the intended logic of necessity, actuality and possibility:

\[
  \label{NecessityActualityPossibility}
\]
\begin{imagefromfile}
    "file_name": "QCwthLHT-ClassicalNecessity-221003.jpg",
    "width": "900",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Some technical remarks:

1. In general, $\underset{W}{\lozenge}$ and $\underset{W}{\Box}$ as defined above are, while being a [[monad]] and [[comonad]], respectively, not an [[idempotent monad]] and [[idempotent comonad]] if generalized from [[first-order hyperdoctrines]] to more general [[dependent type theories]]. But this just reflects the usual issues with [[propositions as types]], see there for more discussion.

1. While [[base change]]-[[adjunctions]] are essentially unique and not free to choose, there is a genuine choice in the above given by the choice of [[context]] $W$. This is reflected in the subscripts of $\underset{W}{\lozenge}$ and $\underset{W}{\Box}$ above. It is the choice of this $W$ that gives different kinds of possibility and necessity. More generally there is in fact not just a choice of a context, but of a morphism of contexts, reflecting what is often called "accessibility of possible worlds". This we come to [below](#ViaBaseChangeRelatively).

### Relation to Potentiality
 {#RelationToPotentiality}

There is also an [[adjoint pair]] on the other side, $\mathbf{H}_{/\ast}$, of the [[base change]] maps, in which the [[left adjoint]] is given by [[context extension]] back to $\mathbf{H}_{/W}$ followed by $\exists_W$, and dually the [[right adjoint]] is given by $W^\ast$ followed by $\forall_W$. The former is the [[coreader comonad]], whereas the latter is the [[reader monad]] (aka [[function monad]]):

\[
  \label{ModalitiesFromDependentTypeFormers}
\]

If we think of the types $P_\bullet \,\in\, Type_B$ in the given context -- now called "$B$" -- as *[[actuality|actual types]]* (eq:NecessityActualityPossibility) then the types down in the base context $P \,\in\, Type$ should be thought of as [[potentiality|potential types]] that fail to be actual in that they are missing information about their $B$-dependency (see also at *[[nondeterministic computation]]*, [here](nondeterministic+computation#FormalizationViaIndefinitenessEffects)):

The definiteness-counit $\star P \rightarrow P$ witnesses their inhabitation as following from that of a definite type after forgetting which definite world $b \colon B$ was used to inhabit it.

{#PotentialityAsMonadicDescent} More in detail, since $(p_B)^\ast$ is a [[conservative functor]], the [[monadicity theorem]] (see [this example](monadicity+theorem#BaseChangeOfPresheavesAlongFullyFaithfulFunctor)) says

\begin{imagefromfile}
    "file_name": "PotentialTypesAsPossibilityModules-221101b.jpg",
    "width": "580",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

that the types $P \,\in\, Type$ are equivalently

1. those actual types $P_\bullet$ which carry a possibility-[[algebra over a monad|monad action]]
   $
     \lozenge_B P_\bullet
       \longrightarrow
     P_\bullet
   $

1. those actual types $P_\bullet$ which carry a necessity-[[coalgebra over a comonad|co-monad co-action]]
   $
     P_\bullet
       \longrightarrow
     \Box_B P_\bullet
   $

(The same conclusion in known in the theory of [[lenses in computer science]], see [there](lens+in+computer+science#PossibilityMonadicity).)

{#PotentialDataAsPossibilityModalData} Hence:

{#PotentialAxiomsComparedToAristotle} This formalization compares reasonably well with [[Aristotle]]'s discussion, who wrote (in the translation from [Ferroni & Gili (2016)](potentiality+and+actuality#FerroniGili16), [§44](https://www.cairn.info/revue-de-philologie-litterature-et-histoire-anciennes-2016-1-page-81.htm#pa44)) first that:

> of non-existent things some exist potentially

which is our

$$
 \underset{
   \color{blue}
   \text{non-existent thing}
 }{
  \big(
    \varnothing
    ,\,
    \lozenge_W \varnothing \xrightarrow{\exists !} \varnothing
  \big)
  }
  \underset{
    \color{blue}
    \text{is a}
  }{
    \phantom{\vert}
    \colon\;
    \phantom{\vert}
  }
  \;
  \underset{
    \color{blue}
    \text{potential thing}
  }{
    Type^{\lozenge_W}_W
  }
$$

and then:

> if they exist ... it is not possible that it is true to say that: this is possible but will not be the case.

hence (unwinding the double negation)

> if potential things exist ... and are possible then they will be the case (ie: will be actual)

which is our

$$
  \underset{
    \color{blue}\text{given a potential thing}
  }{
  (D_\bullet, \rho_\bullet)
  \,\colon\,
  Type_W^{\lozenge_W}
  }
  \;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;
  \underset{
   \color{blue} \text{which is possible}
  }{
    \lozenge_W D_\bullet
  }
  \underset{
   \color{blue}\text{then}
  }{
    \phantom{\vert}
    \xrightarrow{\;\; \rho_\bullet \;\;}
    \phantom{\vert}
  }
  \underset{
   \color{blue}\text{it is actual}
  }{
    D_\bullet
    \mathrlap{\phantom{vert_{g}}}
  }
$$

\linebreak

### Relative version
  {#ViaBaseChangeRelatively}

With this axiomatization via [[base change]], it is immediate to consider the more general relative case where instead of [[base change]] to a [[unit type]] $W \to \ast$ one considers [[base change]]

$$
  (\exists_\omega \dashv \omega^\ast \dashv \forall_\omega)
  \;\colon\;
  \mathbf{H}_{/W}
    \stackrel{\stackrel{\forall_{w \colon \omega^{-1}(-)}}{\longrightarrow}}{\stackrel{\stackrel{\omega^\ast}{\longleftarrow}}{\underset{\exists_{w\colon \omega^{-1}(-)}}{\longrightarrow}}}
  \mathbf{H}_{/W_0}
  \,.
$$

along any [[morphism]]

$$
  \omega \colon W \longrightarrow W_0
$$

and sets

$$
  (\underset{\omega}{\lozenge} \dashv \underset{\omega}{\Box})
  \coloneqq
  \left(
     \left(
     \omega^\ast \circ \underset{w\colon \omega^{-1}(-)}{\exists}
     \right)
     \dashv
     \left(
     \omega^\ast \circ \underset{w\colon \omega^{-1}(-)}{\forall}
     \right)
  \right)
  \;\colon\;
  \mathbf{H}_{/W}
    \longrightarrow
  \mathbf{H}_{/W}
  \,.
$$

If here $\omega$ is an [[effective epimorphism]] (a [[1-epimorphism]]) then it exibits an [[equivalence relation]] on $W$, where $w_1\sim w_2$ is given by $\omega(w_1) = \omega(w_2)$. In traditional [[possible worlds semantics]] such equivalence relation in a [[Kripke frame]] is called an "accessibility relation between possible worlds" for the case of [[S5 modal logic]]. Now

*  $\underset{\omega}{\lozenge} p$ is true/inhabited at $w\in W$ iff it is true/inhabited at at least one $\tilde w$ in the same equivalence class of $w$;

*  $\underset{\omega}{\Box} p$ is true/inhabited at $w\in W$ iff it is true/inhabited at all $\tilde w$ in the same equivalence classes of $w$.

For more on this see at *[[S5 modal logic]]* -- *[via dependent types](S5+modal+logic#ViaDependentTypeTheory)*.

\linebreak

### Examples
 {#Examples}

#### Historical examples

Historically, one informal example whose formalization in [[modal logic]] has been controversially discussed (see for instance [IEP "Rudolf Carnap: Modal Logic"](#IEPCarnapModal)) is the pair of informal assertions

1. "9 is necessarily greater than 7."

1. "The number of planets in the solar system is 9."

(**Remark.** It maybe adds to the joy of modal logic to notice that the second sentence, which was regarded as true in our world at the time the above example was brought up, actually [no longer is](http://www.universetoday.com/15568/how-many-planets-are-in-the-solar-system/). Of course this only highlights  that indeed this statement is not to be expected to be "necessarily true" in any sense.)

The issue with this example is that if one does not fix a decent formalization of these statements, then naively they seem to imply as correct the statement "The number of planets in the solar system is necessarily greater than 7.", which however sounds like it ought to be wrong.

We now formalize and then analyze this example with the [above](#InFirstOrderLogicAndTypeTheory) prescription.

So let $W$ be any [[type]], to be thought of as the type of [[possible worlds]]. Write

$$
  w\colon W \vdash \mathbb{N}(w) \colon Type
$$

for the $W$-dependent type that is constant on the ordinary [[type of natural numbers]], i.e.$\mathbb{N}(w) = \mathbb{N}$ for all $w\colon W$.

The terms of $\mathbb{N}$, i.e. the [[natural numbers]], canonically extend to $W$-dependent terms of this dependent type

$$
  w\colon W \vdash 9 \colon \mathbb{N}(w)
$$

namely to the constant terms, which take the same value (here: 9) for all $w \colon W$.

In contrast to this, assume now that $W$ is such there is at least one non-[[constant function]] $W \to \mathbb{N}$. In fact, in the spirit of the informal problem at hand, we require a surjective function.
This gives a non-constant term of the constantly $W$-dependent type of natural numbers, which we may just as well call

$$
  w \colon W \vdash NumberOfPlanetsInSolarSystem(w) \colon \mathbb{N}(w)
  \,.
$$

That is the formalization of the above example we consider, and it should be evident enough. Now we may step back and see what the [above formalization](#InFirstOrderLogicAndTypeTheory) produce from this.

So consider the $W$-dependent [[identity type]]

$$
  w \colon W \vdash (9 = NumberOfPlanetsInTheSolarSystem)(w) \colon Type
  \,.
$$

By the assumption that $NumberOfPlanetsInTheSolarSystem(w)$ is not constant in $W$ it follows that the [[dependent product]] over $W$ of that dependent identity type is the [[empty type]], and so the same is true for $\Box_W$ applied to it:

* $[\Box_W (9 = NumberOfPlanetsInTheSolarSystem)]$ is [[false]].

However, if we assume that there is one $w$ for which indeed $NumberOfPlanetsInTheSolarSystem(w)$ takes the value 9, then the [[dependent sum]] over the dependent identity type contains that coincidence as a term, and so $\lozenge_W$ of our dependent identity type is [[inhabited type|inhabited]]. Hence

* $[\lozenge_W (9 = NumberOfPlanetsInTheSolarSystem)]$ is [[true]].

In English words, these formal consequences are to be pronounced as:

1. "It is false that it is necessary that the number of planets in the solar system is 9."

1. "It is true that it is possible that the number of planets in the solar system is 9."

Which is just as it should be.

Similarly, the $W$-dependent type

$$
  w \colon W \vdash (NumberOfPlanetsInTheSolarSystem \gt 7)(w) \colon \mathbb{N}(w)
$$

is only going to be [[inhabited type|inhabited]] if indeed the value $NumberOfPlanetsInTheSolarSystem(w)$ is greater than 7 for all $w\colon W$. With our formalization assumption above this is not the case, and so one finds that

* $[\Box_W (NumberOfPlanetsInTheSolarSystem \gt 7)]$ is [[false]].

hence

* "It is false that it is necessary that the number of planets in the solar system is greater than 7."

\linebreak

#### Quantum modal logic
 {#ModalQuantumLogic}

> under construction

We discuss[^1] the modal logic induced from base change of [[dependent types]], as [above](#Globally), now applied to [[dependent linear types]] over [[finite sets]] ([[family of finite types]]) --- to find a natural [[modal logic|modal]] [[quantum logic]] (and in fact a natural [[quantum programming language]] for [[quantum circuits]], see at *[[quantum circuits via dependent linear types]]* for more on this).

[^1]: Following discussion at *[[schreiber:Topological Quantum Programming in Linear Homotopy Type Theory]]*

\linebreak

Given a base [[type]] (which we take to be  [[family of finite types|finite]])

\[
  \label{FiniteBaseType}
  B \;\in\; FinType
  \,,
\]

write $LinType_B$ for the $B$-[[dependent linear types]]

$$
  \mathscr{H}_\bullet \,\in\, \mathrm{LinType}_B
  \;\;\;\;\;\;\leftrightarrow\;\;\;\;\;\;
  b : B \;\;\vdash\;\; \mathscr{H}_b : \mathrm{LinType}
  \,.
$$

The intended [[categorical semantics]] is obtained by [[relation between type theory and category theory|interpreting]] $\mathscr{H}_\bullet$ as a $B$-[[indexed set]] of [[complex vector spaces]] (or rather: [[Hermitian inner product spaces]]) which the reader may want to think of as being [[finite dimensional vector space|finite dimensional]] ([hence](Hilbert+space#FiniteDimensionalInnerProductSpaces): as [[Hilbert spaces]] if positive definite), though this finite-dimensionality condition of $\mathscr{H}_b$ is not used in the following (as opposed to the finiteness condition on $B$, which is used).

We assume that our [[dependent linear type theory]] realizes [[base change]] of linear types along maps of non-linear base types in the form of [[Wirthmüller context|Wirthmüller]]-type [[yoga of six operations]]:

\begin{imagefromfile}
    "file_name": "QCwthLHT-BaseChange-221001.jpg",
    "width": "680",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Moreover, we assume that along maps with [[finite type|finite]]-[[fibers]], such as the [[terminal object|terminal]] map $B \to \ast$, the left [[base change]] agrees with the right [[base change]] ("[[ambidextrous adjunction|ambidexterity]]", so that both give the dependent *[[biproduct]]* $\otimes_{b : B}$ over $B$):

$$
  \mathrm{stability}
  \;:\;
  (p_B)_! \mathscr{H}_\bullet
  \;\;\simeq\;\;
  (p_B)_\ast \mathscr{H}_\bullet
  \;\;=:\;\;
  \underset{b : B}{\bigoplus}
  \mathscr{H}_b
  \;\;\;
  \text{(biproducts)}
$$

Curiously, in terms of [[modal logic]], this is the *[[Gell-Mann principle]]* of quantum physics, in that it means:

$$
  \array{
    \llap{\text{The}}\;\text{possible}
    &
    \text{is}
    &
    \text{necessary}
    &
    \text{and hence}
    &
    \text{actualized}
    \\
    \lozenge \mathcal{H}_\bullet
    &
    \simeq
    &
    \Box \mathcal{H}_\bullet
    &
    \xrightarrow{\phantom{--} \epsilon \phantom{--}}
    &
    \mathcal{H}_{
      \bullet
      \;
      \mathrlap{\text{with some probability}}
    }
  }
$$

Both these assumptions are verified in the [[linear homotopy type theory]] of [Riley (2022),  §2.4](dependent+linear+type+theory#Riley22Thesis), which should have [[categorical semantics]] in (a [[type-theoretic model category]] [[presentable (infinity,1)-category|presenting]]) the [[(infinity,1)-topos|$(\infty,1)$-topos]] of [[parameterized spectrum|parameterized]] [[Eilenberg-MacLane spectrum|$H\mathbb{C}$]]-[[module spectra]], among which [[complex vector spaces]] $\mathscr{H}$ [[full (infinity,1)-subcategory|embed]] as their [[Eilenberg-MacLane spectra|Eilenberg-MacLane]] [[module spectra]] $H\mathscr{H}$.

Recalling from [above](#RelationToPotentiality) the quadruple of ([[comonad|co-]])[[monad|monadic]] [[modalities]] associated with any [[base change]]
one finds that its quantum analog for [[dependent linear types]] (along [[family of finite types|finite fibers]] (eq:FiniteBaseType))

\[
  \label{ModalitiesFromDependentLineaTypeFormers}
\]

where the interpretation of the modal ([[counit of a comonad|co-]])[[unit of a monad|units]] is as follows:

\[
  \label{QuantumModalUnits}
\]
\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-230806.jpg",
    "width": "810",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In the context of linear types not just $(p_B)^\ast$ is [[monadic functor|monadic]], but also $(p_B)_!$/$(p_B)_\ast$ (for details see the *[quantum reader monad](function+monad#QuantumReaderMonad)*), so that the factorization [above](#PotentialityAsMonadicDescent) enhances to the following situation:

\[
  \label{DependentLinearTypesAsIndefiniteLinearTypes}
\]
\begin{imagefromfile}
    "file_name": "LinearTypesAsModalModules-221113.jpg",
    "width": "810",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

\[
  \label{NotationalConvention}
\text{It is convenient and meaningful to declare notational conventions to:}
  \phantom{--------------------}
\]
1. abbreviate $\mathscr{H} \,\coloneqq\, p_! \mathscr{H}_\bullet \,\simeq\, p_\ast \mathscr{H}_\bullet$

1. notationally suppress any outer application of $(p_B)^\ast \colon LinType \to LinType_B$;

Thereby, the definiteness/randomness modalities in relation to [[necessity and possibility]] of classical modal logic ([above](#ClassicalModalLogic)) may be expressed in quantum modal logic by:

\begin{imagefromfile}
    "file_name": "DefinitelyRandomlyViaPossiblyNecessarily.jpg",
    "width": "210",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Considering the basic dependent linear type

$$
  \mathbb{C}_\bullet \;\coloneqq\; (p_B)^\ast \mathbb{C}
$$
$$
  b \colon B
  \;\;\;\vdash\;\;\;
  \mathbb{C} \;\colon\; LinType
$$

(which is the [[tensor unit]] in $LinType_B$) we may understand

$$
  \mathscr{H} \;\coloneqq\; \Box_B \mathbb{C}_\bullet
$$

as the [[Hilbert space]] which is spanned by a measurement basis $\vert b \rangle$.

From all this we find, first of all, that

1. the [[necessity]]-[[counit of a comonad|counit]] exhibits the *[[collapse of the wavefunction|collapse]]* of [[quantum states]] under a [[quantum measurement]] in the $B$-basis

1. the [[possibility]]-[[unit of a monad|unit]] exhibits the condition quantum state preparation in this basis:

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumNecessity-221021.jpg",
    "width": "900",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\begin{remark}
  The [[collapse of the wavefunction|collapse]]-principle of [[quantum measurement]] accurately expressed by the above linear [[necessity]]$\to$actuality$\to$[[possibility]] map was maybe first highlighted and made explicit in [von Neumann (1932), §III.3](wave+function+collapse#vonNeumann32), at least for the special case that (in our notation here) $\mathscr{H}_\bullet \,=\, (p_B)^\ast \mathbb{C}$ (cf. [Lüders (1951)](wave+function+collapse#Lüders51)).
\end{remark}

\begin{remark}
We see that the base type $B$ (eq:FiniteBaseType) *of [[quantum measurement]] outcomes* plays the role of *both*

1. the *[[possible worlds semantics|possible worlds]]* of [[modal logic|classical modal logicians]];

1. the *[[many-worlds interpretation of quantum mechanics|many worlds]]* of [[interpretation of quantum mechanics|quantum interpretationists]].

\end{remark}

(...)

## Related concepts

* [[modal logic]], [[modal type theory]]

* [[epistemic modal logic]]

## References

Discussion of [[modal logic]] of necessity and possibility goes back to [[Aristotle]], as nicely reviewed in

* {#LemmonScott77} [[E. John Lemmon]] with [[Dana Scott]], pp 1-12 of: *An Introduction to Modal Logic -- The "Lemmon Notes"*, B. Blackwell (1977) [[ark:/13960/t3gz25k3h](https://archive.org/details/introductiontomo0000lemm/)]

In

* [[Kant]], _Critique of pure reason_ 1781

is discussion of _modality_ as one of the [[category (philosophy)|categories of pure thought]], with sub-categories _necessity_, _possibility_ and _actuality_ (Wirklichkeit).

In

* [[Georg Hegel]], [book 2, section 3](http://ncatlab.org/nlab/show/Science+of+Logic#DieWirklichkeit), especially [book 2, section 3, chapter 2](http://ncatlab.org/nlab/show/Science+of+Logic#DieWirklichkeitKapitel2) in  _[[Science of Logic]]_ (1812)

this is picked up and claimed to be refined to a [[dialectic]] system of [[unity of opposites|unities of opposites]].

Early discussion in the context of [[alethic modal logic]]:

* {#Wright51} [[Georg H. von Wright]], *An Essay in Modal Logic*, North-Holland Publishing (1951)

See also:

* {#Kripke80} [[Saul Kripke]], _[[Naming and Necessity]]_ (1980)

* Stanford Encyclopedia of Philosophy, _[Modal Logic](http://plato.stanford.edu/entries/logic-modal/#PosWorSem)_

Making explicit the [[necessity]] [[modal operator]] as a [[comonad]]:

* {#BiermanDePaiva96} [[Gavin M. Bierman]], [[Valeria de Paiva]], *Intuitionistic necessity revisited*, School of Computer Science research reports-University of Birmingham CSR (1996) [[researchgate](https://www.researchgate.net/publication/2810611_Intuitionistic_Necessity_Revisited), [[Bierman-dePaiva-NecessityRevisited.pdf:file]]]

* {#BiermanDePaiva00} [[Gavin M. Bierman]], [[Valeria de Paiva]], *On an Intuitionistic Modal Logic*, Studia Logica **65** (2000) 383–416 [[doi:10.1023/A:1005291931660](https://doi.org/10.1023/A:1005291931660), [pdf] (https://www.researchgate.net/profile/Valeria-De-Paiva/publication/226515897_On_An_Intuitionistic_Modal_Logic/links/00b4951ed416906ccc000000/On-An-Intuitionistic-Modal-Logic.pdf)]

* {#Kobayashi97} [[Satoshi Kobayashi]], *Monad as modality*, Theoretical Computer Science **175** 1 (1997) 29-74 [doi:10.1016/S0304-3975(96)00169-7]

* Natasha Alechina, Michael Mendler, [[Valeria de Paiva]], Eike Ritter: *Categorical and Kripke Semantics for Constructive S4 Modal Logic*, in *Computer Science Logic. CSL 2001* Lecture Notes in Computer Science **2142**, Springer (2001) 292 [[doi:10.1007/3-540-44802-0_21](https://doi.org/10.1007/3-540-44802-0_21)]

The understanding of [[Kripke semantics]] for [[S5 modal logic]] as [[base change]] is indicated in:

* {#Awodey06} [[Steve Awodey]], p. 279 in: *Category theory*, Oxford University Press (2006, 2010) [[doi:10.1093/acprof:oso/9780198568612.001.0001](https://doi.org/10.1093/acprof:oso/9780198568612.001.0001), [ISBN:9780199237180](https://global.oup.com/ukhe/product/category-theory-9780199237180), [pdf](http://englishonlineclub.com/pdf/Category%20Teory%20%5BEnglishOnlineClub.com%5D.pdf)]

* [[David Corfield]], Chap. 4 of: _[Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy](https://ncatlab.org/davidcorfield/show/Modal+Homotopy+Type+Theory)_ (2020)

See also:

* [[Jake Chandler]], _Modality: Necessity and Possibility_, lecture notes ([pdf](http://www.jakechandler.com/assets/2006MetaphysicsBBK/%5B6%5D%20%5BModality%5D.pdf))

* {#Simpson} [[Alex Simpson]], _The Proof Theory and Semantics of Intuitionistic Modal Logic_, Ph.D. Thesis, University of Edinburgh, 1994, [web](http://homepages.inf.ed.ac.uk/als/Research/thesis.pdf).

* {#IEPCarnapModal} Internet Encyclopedia of Philosophy, _[Rudolf Carnap: Modal Logic](http://www.iep.utm.edu/cmlogic/)_

[[!redirects necessity]]
[[!redirects possibility]]

[[!redirects necesecarily]]
[[!redirects possibly]]

[[!redirects definiteness and randomness]]

[[!redirects alethic modal logic]]
[[!redirects alethic modality]]


necklace

## Definition

A __necklace__ is a gluing of [[simplicial sets]]
$$\Delta^{n_1}\vee \Delta^{n_2}\vee\cdots\vee \Delta^{n_k}.$$

Here the final vertex of $\Delta^{n_i}$ is identified with the initial vertex of $\Delta^{n_{i+1}}$.

## Applications

Necklaces provide a way to extract a [[simplicial category]] from a [[simplicial set]] (not necessarily fibrant) in the [[Joyal model structure]].

Given such a [[simplicial set]] $X$, construct a [[simplicial category]] by taking its objects to be vertices of $X$, and for a pair of object $x$, $y$ take as the [[simplicial set]] of morphisms $x\to y$ the simplicial set whose $k$-simplices are necklaces of length $k$ in $X$ as described above, with the initial vertex of $\Delta^{n_1}$ mapping to $x$ and the final vertex of $\Delta^{n_k}$ mapping to $y$.

Dugger and Spivak prove that the resulting [[simplicial category]] is weakly equivalent to the other [[simplicial categories]] that one can extract from $X$, e.g., the left adjoint of the [[homotopy coherent nerve]].

## Related concepts

* [[homotopy coherent nerve]]

* [[simplicial category]], [[quasicategory]]

## References

* [[Daniel Dugger]], [[David I. Spivak]], _Rigidification of quasi-categories_, [arXiv:0910.0814](https://arxiv.org/abs/0910.0814).

* [[Daniel Dugger]], [[David I. Spivak]], _Mapping spaces in Quasi-categories_, [arXiv:0911.0469](https://arxiv.org/abs/0911.0469).

A similar idea is developed for [[complete Segal spaces]] in

* Shaul Barkan, Jan Steinebrunner, _Segalification and the Boardmann-Vogt tensor product_, [arXiv:2301.08650](https://arxiv.org/abs/2301.08650).


necklace Lie algebra

A necklace in a quiver is a closed oriented path in the quiver up to cyclic permutation of the arrows making up the cycle.
The space of necklaces in the path algebra of a quiver has a natural Lie algebra structure, which is a noncommutative analogue of a Poisson bracket on the sympletic reduction. In the case of one vertex and n loops the path algebra is the free associative algebra on n generators. In that case the necklace Lie algebra is introduced
in

* [[M. Kontsevich]], _Formal noncommutative sympletic geometry_, The Gelfand mathematical seminars, 1990--1992, Springer 1993, 173--187 [pdf](http://193.51.104.7/~maxim/TEXTS/Formal%20non-commutative%20symplectic%20geometry.pdf)

* related notions: [[quiver with potential]], [[formal noncommutative symplectic geometry]]
* R. Bocklandt, L. Le Bruyn, _Necklace Lie algebras and noncommutative symplectic geometry_, Math Z 240, 141–167 (2002) [doi](https://doi.org/10.1007/s002090100366)
* [[Lieven Le Bruyn]], <http://www.neverendingbooks.org/quiver-superpotentials>
* Wee Liang Gan, Travis Schedler, _The necklace Lie coalgebra and renormalization algebras_, J. Noncommut. Geom. 2 (2008) no. 2, 195--214 [doi](https://doi.org/10.4171/JNCG/19)

Quantization of the necklace Lie bialgebra

* [[Victor Ginzburg]], [[Travis Schedler]], _Moyal quantization of necklace Lie algebras_, [arXiv:0503405](https://arxiv.org/abs/math/0503405)
* T. Schedler, _A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver_, Int. Math. Res. Not. 2005 (12) (2005) 725--760

category: algebra


Neelakantan Krishnaswami

* [webpage](http://www.mpi-sws.org/~neelk/)

## Selected writings

On [[dependent linear type theory]]:

* {#KPB15} [[Neelakantan Krishnaswami]], [[Pierre Pradic]], [[Nick Benton]], *Integrating Dependent and Linear Types*,  ACM SIGPLAN Notices **50** 1 (2015)  17–30  [[pdf](https://www.cl.cam.ac.uk/~nk480/dlnl-paper.pdf),  [doi:10.1145/2775051.2676969](https://doi.org/10.1145/2775051.2676969)]

## Related entries

* [[dependent linear type theory]]

* [[Löb's theorem]]

category: people

[[!redirects Neel Krishnaswami]]


negation

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

# Contents
* table of contents
{: toc}

## Idea

In (classical) [[logic]], the __negation__ of a statement $p$ is a statement $\neg{p}$ which is true if and only if $p$ is false. Hence, viewed algebraically, the negation corresponds to the complement operator of the corresponding [[Boolean algebra]] which satisfies $a\wedge\neg a=\bot$ as well as $a\vee \neg a=\top$.

More generally, as different logics correspond to different types of lattices, one calls *negation* antitone, or polarity reversing, lattice operators that mimic or approximate the algebraic and proof-theoretic behavior of $\neg$.

## As a logic gate
 {#AsALogicGate}

As a [[logic gate]] on [[bits]] and as a [[quantum logic gate]] on [[qbits]] ($X$-[[Pauli matrix]]):

\begin{imagefromfile}
    "file_name": "NOT_LogicGate-221025.jpg",
    "width": "600",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Negation in different logics

In [[classical logic]], we have the [[double negation]] law:
$$
  \neg\neg{p}
    \;\equiv\;
  p
  \,.
$$

In [[intuitionistic logic]], we only have

$$
  \neg\neg{p}
    \;\dashv\; p
  \,,
$$

while in [[paraconsistent logic]], we instead have

$$
  \neg\neg{p}
    \;\vdash\;
  p
  \,.
$$

One may interpret intuitionistic negation as 'denial' and paraconsistent negation as 'doubt'.  So when one says that one doesn\'t deny $p$, that\'s weaker than actually asserting $p$; while when one says that one doesn\'t doubt $p$, that\'s stronger than merely asserting $p$.  Paraconsistent logic has even been applied to the theory of law: if $p$ is a judgment that normally requires only the preponderance of evidence, then $\neg\neg{p}$ is a judgment of $p$ beyond reasonable doubt.

[[linear logic|Linear logic]] features (at least) three different forms of negation, one for each of the above.  (The default meaning of the term 'negation' in linear logic, $p^\bot$, is the one that satisfies the classical double-negation law.)

Accordingly, negation mediates [[de Morgan duality]] in classical and [[linear logic]] but not in intuitionistic or paraconsistent logic.

## In type theory syntax

In usual [[type theory]] [[syntax]] negation is obtained as the [[function type]] into the [[empty type]]: $\not a = a \to \varnothing$.

Equivalently, in [[type theory]] with [[equivalence types]] but without [[function types]], negation is the [[equivalence type]] with the [[empty type]]: $\not a = a \simeq \varnothing$.

## In categorical semantics

The [[categorical semantics]] of negation is the [[internal hom]] into the [[initial object]]: $\not = [-, \emptyset]$.

In a [[topos]], the __negation__ of an object $A$ (a [[proposition]] under the [[propositions as types]]-interpretation) is the [[internal hom]] object $0^A$, where $0 = \emptyset$ denotes the [[initial object]].

This matches the [[intuitionistic mathematics|intuitionistic]] notion of negation in that there is a [[natural transformation|natural morphism]] $A \to 0^{0^A}$ but not the other way around.

## Related entries

* [[double negation]]

* [[complement]]

* [[De Morgan duality]]

* [[co-Heyting negation]]

* [[adjoint cylinder]]

[[!include logic symbols -- table]]

## References

* Y. Gauthier, _A Theory of Local Negation: The Model and some Applications_ , Arch. Math. Logik **25** (1985) pp.127-143. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002045923))

* H. Wansing, _Negation_ , pp.415-436 in Goble (ed.), _The Blackwell Guide to Philosophical Logic_ , Blackwell Oxford 2001.

[[!redirects negation]]
[[!redirects negations]]

[[!redirects not]]
[[!redirects NOT]]

[[!redirects linear negation]]


negative moment

#Contents#
* table of contents
{:toc}

##Idea

In the context of [[idempotent (∞,1)-monads]] or comonads, or _moments_, the **negative moment** is the [[homotopy fiber]] of the [[unit of an adjunction|unit]] or [[homotopy cofiber]] of the counit.

These are denoted, e.g., $\overline{\sharp}$, $\overline{\flat}$.

* The negative of $id$ (both as monad and comonad) is $\ast$.
* The negative of $\ast$ is $id$.
* The negative of $\emptyset$ is the [[maybe monad]], although not itself idempotent, so not a moment.

## Related concepts

* [[differential cohomology hexagon]]
* [[extensive quantity]]


negative thinking

<div markdown="1" class="rightHandSide toc">
[[!include higher category theory - contents]]
</div>

Negative thinking is a way of thinking about [[vertical categorification|categorification]] by considering what the original concept is a categorification of.  That is, to better understand how foos are categorified to become $2$-foos, $3$-foos, and so on, you think about how foos are themselves a categorification of $0$-foos, $(-1)$-foos, and so on.  Generally, the concept of $n$-foo stops making sense for small values of $n$ after a few steps, but it does make sense surprisingly often for at least some non-positive values.  Experienced negative thinkers can compete to see 'how low can you go'.

More generally, negative thinking can apply whenever you have a sequence of mathematical objects and ask yourself what came before the beginning?  Examples outside [[category theory]] include the $(-1)$-[[sphere]] and the $(-1)$-[[simplex]] (which are both [[empty set|empty]]), although maybe it means something that these are both from [[homotopy theory]].  [[Tim Gowers]] has called this '[generalizing backwards](http://gowers.wordpress.com/2007/09/19/one-way-of-looking-at-cauchys-theorem/)'.

For low values of $n$-[[n-category|category]], see Section 2 of [[Lectures on n-Categories and Cohomology]].  Related issues appear at [[category theory vs order theory]].  See also nearly any page here with '0' or '(-1)' in the title, such as

* [[(0,1)-category]]
* [[0-category]]
* [[(-1)-category]]
* [[(-2)-category]]
* [[0-poset]]
* [[(-1)-poset]]
* [[0-groupoid]]
* [[(-1)-groupoid]]
* [[(-2)-groupoid]]
* [[0-functor]]
* [[(-1)-functor]]
* [[(0,1)-topos]]
* [[(0,1)-site]]

[[!redirects negative thinking]]


negative type

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

# Negative types
* table of contents
{: toc}

## Idea

In [[type theory]], a **negative type** is one whose *eliminators* are regarded as primary.  Its constructors are derived from these by the rule that "to construct an element of a negative type, it is necessary and sufficient to specify how that element behaves upon applying all of the eliminators to it".

The opposite notion is a [[positive type]], forming two [[polarity in type theory|polarities]].

In [[categorical semantics]] of type theory, negative types correspond to "from the right" [[universal properties]] (i.e. for mapping in).

In [[denotational semantics]], negative types behave well with respect to "[[call-by-name]]" and other [[lazy evaluation]] strategies.

## Examples

* [[dependent product type]]

  * [[function type]]

* [[coinductive type]]

* [[unit types]], [[dependent sum types]] and cartesian [[product types]] can be presented both positively and negatively.  The two definitions are equivalent in ordinary type theory, but distinct in [[linear logic]].  The same is true of binary [[sum types]] if we allow [[sequents]] with multiple conclusions.

[[!redirects negative types]]


neighborhood

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###Context###
#### Topology
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[[!include topology - contents]]
=--
#### Geometry
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[[!include higher geometry - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea ##

In [[topology]], a _neighbourhood_ (or _neighborhood_) of a [[point]] $x$ in some [[topological space]] $X$ is a [[subset]] $U$ such that there is enough room around $x$ in $U$ to move in any direction (but perhaps not very far).  One writes $x \in U^\circ$, $U \stackrel{\circ}\ni x$, or any of the six other obvious variations to indicate that $U$ is a neighbourhood of $x$.

## Definitions ##

Let $(X,\tau)$ be a [[topological space]] and $x \in X$ a point. Then:

1. A [[subset]] $U \subset X$ is a __neighbourhood__ of $x$ if there exists an [[open subset]] $O \subset X$ such that $x \in O$ and $O \subset U$.

1. A [[subset]] $U \subset X$ is an __open neighbourhood__ of $x$ if it is both an [[open subset]] and a neighbourhood of $x$;

Beware, some authors use "neighbourhood" as a synonym for "open neighbourhood".

Similarly one says that a _closed neighbourhood_ or _compact neighbourhood_ etc. is a neighbourhood that is also a [[closed subset]] or [[compact subspace]], respectively.

## Properties ##

* When definitions of topological concepts are given in terms of neighbourhoods, it often makes no difference if the neighbourhoods are required to be open or not.  There should be some deep logical reason for this ....

* The neighbourhoods of a given point form a proper [[filter]], the __neighbourhood filter__ of that point.  A __local (sub)[[base]]__ for the topology at that point is a (sub)base for that filter.

* The concept of [[topological space]] can be defined by taking the neighbourhood relation as primitive.  One axiom is more complicated than the others; if it is dropped, then the result is the definition of [[pretopological space]].

## Related concepts

* [[neighbourhood base]]

[[!include infinitesimal and local - table]]

[[!redirects neighborhood]]
[[!redirects neighbourhood]]
[[!redirects neighborhoods]]
[[!redirects neighbourhoods]]
[[!redirects neighbourhood filter]]
[[!redirects neighborhood filter]]
[[!redirects neighbourhood filters]]
[[!redirects neighborhood filters]]
[[!redirects local base]]
[[!redirects local bases]]
[[!redirects local subbase]]
[[!redirects local subbases]]

[[!redirects open neighborhood]]
[[!redirects open neighbourhood]]

[[!redirects open neighborhoods]]
[[!redirects open neighbourhoods]]

[[!redirects closed neighborhood]]
[[!redirects closed neighbourhood]]

[[!redirects closed neighborhoods]]
[[!redirects closed neighbourhoods]]


neighborhood base

+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $(X, \tau)$ be a [[topological space]], and $x \in X$ a point. A _neighborhood base_ at $x$ is a collection $\{U_i \subset X\}_{i \in I}$ of [[neighbourhoods]] of $x$ such that every [[neighborhood]] $W$ of $x$ (which WLOG we may assume open) contains some $U_i$:

$$
  \underset{ { W \underset{\text{open}}{\subset} X } \atop { W \supset \{x\} }   }{\forall}
  \left(
    \underset{i \in I}{\exists}
    \left(
       U_i \subset W
    \right)
  \right)
  \,.
$$

## Related concepts

* [[base of a topology]]

* [[locally compact topological space]]

## References

See also

* Wikipeda, _[Neighbourhood system -- Basis](https://en.wikipedia.org/wiki/Neighbourhood_system#Basis)_

[[!redirects local base]]
[[!redirects local basis]]
[[!redirects local bases]]

[[!redirects neighborhood bases]]
[[!redirects neighborhood basis]]

[[!redirects base of neighborhoods]]
[[!redirects bases of neighborhoods]]
[[!redirects basis of neighborhoods]]

[[!redirects neighbourhood base]]
[[!redirects neighbourhood bases]]
[[!redirects neighbourhood basis]]

[[!redirects base of neighbourhoods]]
[[!redirects basis of neighbourhoods]]
[[!redirects bases of neighbourhoods]]

[[!redirects base of neighborhoods]]
[[!redirects basis of neighborhoods]]
[[!redirects bases of neighborhoods]]

[[!redirects fundamental system of neighborhoods]]
[[!redirects fundamental systems of neighborhoods]]
[[!redirects fundamental system of neighbourhoods]]
[[!redirects fundamental systems of neighbourhoods]]


Given a [[topologizing subcategory]] $\mathbb{T}$ of an abelian category $A$, the $n$-th neighborhood of $\mathbb{T}$ is the $n$-th power $\mathbb{T}^{(n)}$ of $\mathbb{T}$ with respect to the [[Gabriel multiplication]] of topologizing subcategories.

The union $\mathbb{T}^{(\infty)} := \cup_{n\gt 1} \mathbb{T}^{(n)}$ is a topologizing subcategory of $A$ closed under extensions (that is a [[thick subcategory]] in the strong sense).

A typical example is when $\mathbb{T} = \Delta_B$ is the minimal "subscheme" (= coreflective [[topologizing subcategory]]) of the category of additive endofunctors $A = End B$ containing the identity functor. See [[differential monad]].

[[!redirects n-th neighborhood of a topologizing subcategory]]
[[!redirects neighborhoods of a topologizing subcategory]]


neighborhood retract

+-- {: .rightHandSide}
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###Context###
#### Topology
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[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A topological [[subspace]] $A$ is a **neighborhood retract** of a [[topological space]] $X$ if there is a [[neighborhood]] $U\supset A$ in $X$ such that $A$ is a [[retract]] of $U$.

A [[metrisable space|metrisable]] topological space $Y$ is an _[[absolute neighborhood retract]]_ if for _any_ [[embedding of topological spaces|embedding]] $Y\subset Z$ as a [[closed subspace]] in a [[metrisable topological space]] $Z$, $Y$ is a neighborhood retract of $Z$.

A pair $(X,A)$ where $A$ is a [[closed subspace]] of $X$ is an **NDR-pair** (for 'Neighbourhood Deformation Retract') or a closed Borsuk pair if there is a function $u:X\to I=[0,1]$ and a [[homotopy]] $H:X\times I\to X$ such that $H(x,0)=x$, for all $x\in X$, $H(a,t)=a$ for all $a\in A$, $H(x,1)\in A$ for all $x\in X$ such that $u(x)\lt 1$ and $u^{-1}(0)\subset A$. (See _[[deformation retract]]_.)

## Properties

The canonical inclusion $i:A\hookrightarrow X$ corresponding to any NDR-pair $(X,A)$ is a closed [[Hurewicz cofibration]].

## Related concepts

* [[retract]], [[deformation retract]]

## References

Textbook accounts

* [[Peter May]], Section 6.4 of: _[[A concise course in algebraic topology]]_, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

[[!redirects neighborhood retracts]]

[[!redirects neighbourhood retract]]
[[!redirects neighbourhood retracts]]

[[!redirects NDR-pair]]
[[!redirects NDR-pairs]]

[[!redirects NDR pair]]
[[!redirects NDR pairs]]


Neil Ashcroft

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

* Daniel Garisto, *[Neil Ashcroft 1938-2021](https://www.aps.org/publications/apsnews/202105/ashcroft.cfm)*, APS News **30** 5 (May 2021)

## Selected writings

On [[solid state physics]]:

* [[Neil Ashcroft]], [[N. David Mermin]], *Solid State Physics*, Saunders College Publishing (1973) [[archive](https://archive.org/details/AshcroftSolidState), [Wikipedia entry](https://en.wikipedia.org/wiki/Ashcroft_and_Mermin)]

category: people


Neil Barton

* [webpage](https://neilbarton.net/)

## Selected publications

* [[Neil Barton]], [[Sy-David Friedman]], *Set Theory and Structures*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_10](https://doi.org/10.1007/978-3-030-15655-8_10), [pdf](https://philsci-archive.pitt.edu/15908/7/Set_theory_structures_2.pdf)]

category: people


Neil Constable

## Selected writings

On [[D1-D3 brane intersections]] as [[fuzzy funnels]]/[[BIons]] on [[fuzzy 2-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

* [[Neil Constable]], [[Neil Lambert]], _Calibrations, Monopoles and Fuzzy Funnels_, Phys. Rev. D66 (2002) 065016 ([arXiv:hep-th/0206243](https://arxiv.org/abs/hep-th/0206243))

On [[D1-D5 brane intersections]] as [[fuzzy funnels]] on [[fuzzy 4-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _Non-abelian Brane Intersections_, JHEP 0106:023, 2001 ([arXiv:hep-th/0102080](https://arxiv.org/abs/hep-th/0102080))

category:people


Neil Ghani

Neil Ghani is professor for [[computer science]] at University of Strathclyde in Glasgow.

* [webpage](https://personal.cis.strath.ac.uk/neil.ghani/)

## Selected writings

On [[inductive types]] and [[W-types]]:

* [[Michael Abbott]], [[Thorsten Altenkirch]], [[Neil Ghani]]: *Representing Nested Inductive Types using W-types*, in: *Automata, Languages and Programming*, ICALP 2004, Lecture Notes in Computer Science, **3142**, Springer (2004) [[doi:10.1007/978-3-540-27836-8_8](https://doi.org/10.1007/978-3-540-27836-8_8), [pdf](http://www.cs.nott.ac.uk/~psztxa/publ/icalp04.pdf)]

  exposition: _Inductive Types for Free -- Representing nested inductive types using W-types_, talk at ICALP (2004) [[pdf](http://www.cs.nott.ac.uk/~txa/talks/icalp-slides.pdf)]

## Related entries

* [[logical relation]]

* [[polymorphism]]

* [[verified software]]

category: people


Neil Lambert

Neil Lambert is professor for theoretical [[physics]] at King's College London.

* [webpage](http://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/lambertn.aspx)

## Selected writings

On obtaining [[massive type IIA supergravity]] from a form o [[D=11 supergravity]]:

* [[Paul S. Howe]], [[Neil D. Lambert]], [[Peter C. West]], *A New Massive Type IIA Supergravity From Compactification*, Phys. Lett. B **416** (1998) 303-308 [[arXiv:hep-th/9707139](https://arxiv.org/abs/hep-th/9707139), doi:10.1016/S0370-2693(97)01199-4]

On the [[3-brane in 6d]] as a [[black brane]] in an [[M5-brane]]/[[NS5-brane]] [[worldvolume]]

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Threebrane Soliton of the M-Fivebrane_, Phys. Lett. B419 (1998) 79-83 ([arXiv:hep-th/9710033](http://arxiv.org/abs/hep-th/9710033))

and of its relation to [[D=4 N=2 super Yang-Mills theory]]:

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills_, Phys. Lett. B418 (1998) 85-90 ([arXiv:hep-th/9710034](https://arxiv.org/abs/hep-th/9710034))

* [[Neil Lambert]], [[Peter West]], _Gauge Fields and M-Fivebrane Dynamics_, Nucl. Phys. B524 (1998) 141-158 ([arXiv:hep-th/9712040](https://arxiv.org/abs/hep-th/9712040))

* {#LambertWest98} [[Neil Lambert]], [[Peter West]], _$N=2$ Superfields and the M-Fivebrane_, Phys. Lett. B424 (1998) 281-287 ([arXiv:hep-th/9801104](https://arxiv.org/abs/hep-th/9801104))

* [[Neil Lambert]], [[Peter West]], _Monopole Dynamics from the M-Fivebrane_, Nucl. Phys. B556 (1999) 177-196 ([arXiv:hep-th/9811025](https://arxiv.org/abs/hep-th/9811025))

On [[Dp-D(p+2) brane intersections]] (such as [[D1-D3 brane intersections]]) as spikes/[[BIons]] from the [[M5-brane]]:

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

On multiple [[M2-branes]] and the [[BLG model]]/[[ABJM model]] of their [[worldvolume]] theory:

* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).

* [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955))

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

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

* {#Lambert12} [[Neil Lambert]], _M-Theory and Maximally Supersymmetric Gauge Theories_, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 ([arXiv:1203.4244](https://arxiv.org/abs/1203.4244), [doi:10.1146/annurev-nucl-102010-130248](https://doi.org/10.1146/annurev-nucl-102010-130248))

* {#Lambert18}  [[Neil Lambert]], _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))

and specifically in relation to [[heterotic M-theory]]:

* {#Lambert15} [[Neil Lambert]], _Heterotic M2-branes_, Physics Letters B Volume 749, 7 October 2015, Pages 363-367 ([arXiv:1507.07931](http://arxiv.org/abs/1507.07931))

On [[D1-D5-brane bound states]]:

* [[Neil Lambert]], _D-brane Bound States and the Generalised ADHM Construction_, Nucl. Phys. B519 (1998) 214-224 ([arXiv:hep-th/9707156](https://arxiv.org/abs/hep-th/9707156))

On [[M2-M5 brane bound states]]:

* {#ILPT95} [[José M. Izquierdo]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], *Dyonic Membranes*, Nucl. Phys. B **460** (1996) 560-578 [[arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177), doi:10.1016/0550-3213(95)00606-0]

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys.Lett.B384:86-92, 1996 ([arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146))

New approach to construction of candidate [[Lagrangian densities]] for [[D=6 N=(2,0) SCFTs]]:

* [[Neil Lambert]], _$(2,0)$ Lagrangian Structures_, Physics Letters B Volume 798, 10 November 2019, 134948 ([arXiv:1908.10752](https://arxiv.org/abs/1908.10752))

Description of the [[D=6 N=(1,0) SCFT]] induced form a non-[[Lorentz group|Lorentz]] invariant 5d Lagrangian:

* [[Neil Lambert]], Tristan Orchard, *Non-Lorentzian Avatars of $(1,0)$ Theories*, J. High Energ. Phys. **2021** 205 (2021) [[arXiv:2011.06968](https://arxiv.org/abs/2011.06968), doi:10.1007/JHEP02(2021)205]

reviewed in:

* [[Neil Lambert]], *A non-Lorentzian View of the M5-brane*,  talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Lambert2023)]

## Related $n$Lab entries

* [[BLG model]] ([[M2-brane]])

* [[n-Lie algebra]]

category: people

[[!redirects Neil D. Lambert]]


Neil Ross

* [personal page](https://www.mathstat.dal.ca/~neilr/)

* [institute page](https://www.dal.ca/faculty/science/math-stats/faculty-staff/our-faculty/mathematics/julien-ross.html)

## Selected writings

Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

On [[quantum compilation]] and introducing [[Proto-Quipper]]:

* [[Neil J. Ross]], *Algebraic and Logical Methods in Quantum Computation*, Ph.D. thesis, Dalhousie University (2015) [[arXiv:1510.02198](https://arxiv.org/abs/1510.02198)]

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

On [[quantum programming]] via [[dependent linear type theory]] with [[Quipper]]:

* [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 ([arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9))

On [[dependent linear type theory]] and [[categorical semantics]] for versions of "[[Quipper]]":

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

On [[quantum circuits]]:

* {#ACGMMR23} M. Amy, M. Crawford, A. N. Glaudell, M. L. Macasieb, S. S. Mendelson, [[Neil J. Ross]], *Catalytic Embeddings of Quantum Circuits* [[arXiv:2305.07720](https://browse.arxiv.org/abs/2305.07720)]

category: people

[[!redirects Neil J. Ross]]

[[!redirects Julien Ross]]


Neil S. Ostlund

## Selected writings

On [[quantum chemistry]]:

* [[Attila Szabo]], [[Neil S. Ostlund]], Sec. 2.2.3 of: *Modern Quantum Chemistry -- Introduction to Advanced Electronic Structure Theory*, Macmillan (1982), McGraw-Hill (1989), Dover (1996) $[$[pdf](https://chemistlibrary.files.wordpress.com/2015/02/modern-quantum-chemistry.pdf)$]$

## Related $n$Lab entries

* [[Slater determinant]]

category: people

[[!redirects Neil Ostlund]]


Neil Strickland

* [website](http://neil-strickland.staff.shef.ac.uk/)

## Selected writings

On [[topological spaces]] relevant in [[algebraic topology]]:

* [[Neil Strickland]], *A Bestiary of Topological Objects* \[pdf, [[Strickland-BestiaryTopological.pdf:file]]\]

On [[Hopf rings]] of [[ordinary homology]] of [[topological K-theory]]:

* [[Neil Strickland]], _Bott periodicity and Hopf rings_, 1992 ([pdf](https://neil-strickland.staff.shef.ac.uk/research/thesis.pdf), [[StricklandHopfRings.pdf:file]])

On [[stable homotopy theory]]:

* [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]],  *Axiomatic stable homotopy theory*, Memoirs Amer. Math. Soc. **610** (1997) [[ISBN:978-1-4704-0195-5](https://bookstore.ams.org/memo-128-610), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf)]

On [[elliptic genera]], the [[Witten genus]] and the [[string orientation of tmf]]:

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _The sigma orientation is an H-infinity map_, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053), [doi:10.1353/ajm.2004.0008](https://doi.org/10.1353/ajm.2004.0008))

On the [[model structure on unbounded chain complexes]]:

* [[Neil Strickland]], *The model structure for chain complexes* [[arXiv:2001.08955](https://arxiv.org/abs/2001.08955)]

## Related entries

* [[homotopy theory]]

* [[stable homotopy theory]]

category: people

[[!redirects N. P. Strickland]]


Neil Toronto

* [webpage](http://cs.umd.edu/~ntoronto/)

## related $n$Lab entries

* [[function monad]], [[random variable]]

category: people


Neil Turok

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

## Related $n$Lab entries

* [[ekpyrotic cosmology]]

category: people


Neill Graham

## Selected writings

On the [[many-worlds interpretation of quantum mechanics]]:

* [[Bryce S. DeWitt]], [[Neill Graham]] (eds.), *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press (1973, 2016) [[ISBN:9780691645926](https://press.princeton.edu/books/hardcover/9780691645926/the-many-worlds-interpretation-of-quantum-mechanics), [jstor:j.ctt13x0wwk](https://www.jstor.org/stable/j.ctt13x0wwk)]

category: people


Nekrasov function

Nekrasov functions are the coefficients of the character expansion of the matrix $\tau$-function. They were introduced in

* N.Nekrasov, _Seiberg-Witten prepotential from instanton counting_, Adv.Theor.Math.Phys. __7__ (2004) 831-864, [hep-th/0206161](http://arxiv.org/abs/hep-th/0206161)

and play the central role in the formulation of the [[AGT conjecture]].

* A. Belavin, V. Belavin, _AGT conjecture and integrable structure of conformal field theory for c=1_, [arxiv/1102.0343](http://arxiv.org/abs/1102.0343)
[[!redirects Nekrasov functions]]


Nele Callebaut

* [home page](https://www.thp.uni-koeln.de/callebaut/index.html)

## Selected writings

On [[holographic entanglement entropy]] in [[conformal field theory]]:

* [[Nele Callebaut]], *Entanglement in conformal field theory and holography* [[arXiv:2303.16827](https://arxiv.org/abs/2303.16827)]

category: people