regular theory

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## Definition

A **regular theory** is a [[theory]] in [[regular logic]].

## Related concepts

* [[finitely complete category]], [[cartesian functor]], [[cartesian logic]], [[cartesian theory]]

* [[regular category]], [[regular functor]], [[regular logic]], **regular theory**, [[regular coverage]], [[regular topos]]

* [[coherent category]], [[coherent functor]], [[coherent logic]], [[coherent theory]], [[coherent coverage]], [[coherent topos]]

* [[geometric category]], [[geometric functor]], [[geometric logic]], [[geometric theory]]

## References

Section D1.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects regular theories]]


regular topos

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#### Topos Theory
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## Definition

A **regular topos** is a [[sheaf topos]] with a [[site]] of definition that is a [[regular category]] with [[regular coverage]].

Notice that the terminology is slightly confusing: every topos is a [[regular category]].

## Examples

* The [[classifying topos]] of a [[regular theory]] is a regular topos.

## Related concepts

[[!include categories and logic - table]]

[[!redirects regular toposes]]
[[!redirects regular topoi]]


regular value

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## Idea

A __regular value__ of a [[map]] $f$ between [[smooth manifolds]] is an [[element]] of the [[codomain]] of $f$ that is not a __critical value__, i.e., not the [[image]] of a [[critical point]] of $f$.

## Definition

+-- {: .num_defn #RegularValue}
###### Definition
**([[regular value]])**

For
$$
  f
    \;\colon\;
  X
    \longrightarrow
  Y
$$
a [[differentiable function]] between [[differentiable manifolds]] (e.g. a [[smooth function]] between [[smooth manifolds]]) a [[point]] $q \in f(X) \subset Y$ in the [[image]] of $f$ is called a _regular value_ of $f$ if at all points $p \in f^{-1}(\{q\})$ in its [[preimage]], the [[differential]]
$$
  d f_p \;\colon\; T_p X \longrightarrow T_{f(x)} Y = T_{q}Y
$$
is a [[surjective function]] between the corresponding [[tangent spaces]].

A function all whose values are regular values is called a _[[submersion]]_.

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(e.g. [Kosinski 93, II (2.4)](#Kosinski93))

+-- {: .num_remark #RelationToTransversality}
###### Remark
**(relation to [[transversal map|transversality]])**

That $q \in Y$ is a [[regular value]] (Def. \ref{RegularValue}) of $f \colon X \to Y$ means equivalently that $f$ is a [[transverse map]] to the [[submanifold]]-inclusion $\ast \overset{q}{\hookrightarrow} Y$.

In this sense [[transversal map|transversality]] generalizes the concept of [[regular values]].

=--

## Properties

### Inverse image

The [[inverse function theorem]] implies that:

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

The [[inverse image]] $f^{-1}(q) \subset X$ of a [[smooth function]] $f \colon X \to Y$ at a [[regular value]] $q \in Y$ is a [[smooth manifold]] of $X$.

=--

Together with [[Thom's transversality theorem]], this is the key to the [[proof]] of the [[Pontryagin-Thom isomorphism]].

## Related concepts

* [[immersion]], [[submersion]]

* [[formally smooth morphism]]

* [[transverse map]]

* [[Thom's transversality theorem]]

* [[Pontryagin-Thom isomorphism]]

## References

* {#Pontrjagin55} [[Lev Pontrjagin]], p. 5 of: _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

* {#Thom54} [[René Thom]], p. 18 of: _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

* {#Kosinski93} [[Antoni Kosinski]], _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

[[!redirects regular values]]

[[!redirects regular point]]
[[!redirects regular points]]

[[!redirects critical value]]
[[!redirects critical values]]


regularity

## Definition

In [[cubical type theory]], **regularity** is the condition where [[composition]] along a [[degenerate]] open box is the [[identity]].

## References

* [[Andrew Swan]], *Separating Path and Identity Types in Presheaf Models of Univalent Type Theory*, ([arXiv:1808.00920v3](https://arxiv.org/abs/1808.00920v3))


regularization (physics)

e.g.

* [[zeta function regularization]]

## Related concepts

* [[renormalization]]

* [[Feynman propagator]]

## References

* Wikipedia, _[Regularization (physics)](http://en.wikipedia.org/wiki/Regularization_(physics)_


regulator of a number field

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## Idea

Given a _[[number field]]_ $K$, with [[ring of integers]] $\mathcal{O}_K$, then the _regulator_ $Reg_K$ is a number which measures the size of the [[group of units]] $GL_1(\mathcal{O}_K)$.

Since the [[determinant]] constitutes a group [[homomorphism]] $K_1 \to GL_1$ from first [[algebraic K-theory]] to the [[group of units]], the regulator may also be thought of as extracting information on the first [[algebraic K-theory]] group. This is the perspective taken in the generalization to _[[higher regulators]]_ (Beilinson regulators) which are effectively [[Chern characters]] for algebraic K-theory.

## Properties

### Relation to zeta function and class number formula

The [[Dedekind zeta function]] $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The _[[class number formula]]_ says that its [[residue]] there is proportional to the product of the regulator and the [[class number]] of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

## Related concepts

* [[Dirichlet's unit theorem]]

* [[higher regulator]]

## References

* Wikipedia, _[Dirichlet's unit theorem](http://en.wikipedia.org/wiki/Regulator_of_a_number_field)_ -- _[The regulator](http://en.wikipedia.org/wiki/Regulator_of_a_number_field#The_regulator)_

[[!redirects regulators of a number field]]
[[!redirects regulators of number fields]]


Rehren duality

## Idea

What is sometimes called __Rehren duality__ is a relation between [[local nets of observables]] and their restriction to a boundary. It was originally claimed to be a formalization of a toy model of [[AdS/CFT]] duality, but in that context field theories are not simply restricted to their boundaries (instead [[state]]s are mapped to [[correlator]]s).

## References

* Karl-Henning Rehren, _QFT lectures on AdS-CFT_, [hep-th/0411086](http://arxiv.org/abs/hep-th/0411086), in B. Dragovich et al.. Proc. 3rd Summer School in Modern Math. Physics, Zlatibor 2004. Belgrade: Institute of Physics. pp. 95–118.

* comments on [[Jacques Distler]]'s [blog](http://golem.ph.utexas.edu/~distler/blog/archives/000987.html)

* [n-cafe](http://golem.ph.utexas.edu/category/2007/07/making_adscft_precise.html), 2007

* wikipedia: [AdS/CFT correspondence](http://en.wikipedia.org/wiki/AdS/CFT_correspondence)


Reid William Barton

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Reid_W._Barton)

## Selected writings

On [[premodel categories]]:

* [[Reid William Barton]], _A model 2-category of enriched combinatorial premodel categories_ ([arXiv:2004.12937](https://arxiv.org/abs/2004.12937))

On [[o-minimal structure|o-minimal geometry]]:

* [[Reid Barton]], Johan Commelin, _Model categories for o-minimal geometry_ ([arXiv:2108.11952](https://arxiv.org/abs/2108.11952))

category: people

[[!redirects Reid Barton]]


Reidemeister move

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# Contents
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## Idea

The _Reidemeister moves_ are local changes to [[link diagrams]] which can be realised as [[isotopy|isotopies]] between the link diagrams from which they arise. Because of Reidemeister's theorem, Theorem \ref{TheoremReidemeister}, they are of fundamental importance in knot theory, allowing knots and links to be treated purely combinatorially and diagrammatically.

The moves were introduced in the book [Reidemeister](#Reidemeister).

## Definition

The convention below is that part of a [[link diagram]] is shown with the 'move' indicated, but that outside that, the diagram of the link is unchanged. As usual in knot theory, everything is up to [[planar isotopy]].

\begin{defn} The _first Reidemeister move_, or _R1 move_, allows either of the following replacements of a fragment of a link diagram to be made.

[[!include Reidemeister move 1 - SVG]]

\end{defn}

\begin{defn} The _second Reidemeister move_, or _R2 move_, allows either of the following replacements of a fragment of a link diagram to be made.

[[!include Reidemeister move 2 - SVG]]

\end{defn}

\begin{defn} The _third Reidemeister move_, or _R3 move_, allows the following replacement of a fragment of a link diagram to be made.

[[!include Reidemeister move 3 - SVG]]

\end{defn}

\begin{defn} A pair of link diagrams $D$ and $D'$ are  _isotopic_, or _isotopic by moves_, or _equivalent_ if there is a finite sequence of the moves R1, R2 and R3 taking $D$ to $D'$ up to planar isotopy. \end{defn}

\begin{defn} A pair of link diagrams $D$ and $D'$ are  _regularly isotopic_ if there is a finite sequence of the moves R2 and R3 taking $D$ to $D'$ up to [[planar isotopy]]. \end{defn}

\begin{rmk} Both isotopy and and regular isotopy define equivalence relations on link diagrams. \end{rmk}

\begin{rmk} There are a number of variants of the R3 move. All of them can be obtained as a finite sequence of the Reidemeister moves as defined above. \end{rmk}

##Reidemeister's theorem

The following crucial theorem establishes that the Reidemeister moves are of central significance in knot theory.

\begin{thm} \label{TheoremReidemeister} Let $L_{1}$ and $L_{2}$ be [[links]], and let $D_{1}$ and $D_{2}$ be [[link diagrams]] of $L_{1}$ and $L_{2}$ respectively. Then $D_{1}$ and $D_{2}$ are equivalent if and only if $L_{1}$ is [[isotopy|isotopic]] to $L_{2}$. \end{thm}

The key idea of the proof is that of subdivision. Working piecewise-linearly, one can reduce to the consideration of a few 'minimal' moves on knot diagrams, which one shows can be obtained as a finite sequence of Reidemeister moves. See for example section 2.1 of [Kauffman](#KauffmanKnotDiagrammatics).

##Consequences of Reidemeister's theorem

Because of Theorem \ref{TheoremReidemeister}, we can use the Reidemeister moves to verify invariance of a potential [[link invariant]]. For instance, as none of the moves alters the number of components of a link diagram, the number of components is an isotopy invariant (which is not at all surprising!). We can conclude that the [[Hopf link]] and the [[Borromean rings]] are not isotopic.

A slightly deeper observation is that the [[linking number]] is a link invariant, for which see that entry.

[[colourable knot|3-colourability]] is a [[knot invariant]] as is easy to check. This shows, for instance, that the trefoil knot is not equivalent to the unknot, and hence shows, very simply, that non-trivial knots exist. It also shows that the figure eight knot is not equivalent to the trefoil.

## References
The original source is the following.

* [[K. Reidemeister]], _Knotentheorie,_ Ergebnisse der Mathematik (1932)
{#Reidemeister}

Simple examples of invariants under the Reidemeister Moves are given in, for instance, the following.

* [[Nick Gilbert]], [[Tim Porter]], *Knots and surfaces*, Oxford University Press (1994) [[ISBN:9780198514909](https://global.oup.com/academic/product/knots-and-surfaces-9780198514909?cc=de&lang=en&)]

* {#KauffmanKnotDiagrammatics}
 [[Louis Kauffman | Louis H. Kauffman]], _Knot diagrammatics_, Handbook of knot theory, Elsevier, 233 - 318, 2005 [arXiv:math/0410329](https://arxiv.org/abs/math/0410329)

Most texts on [[Knot Theory]] contain discussions of the Reidemeister Moves.

category: knot theory

[[!redirects Reidemeister moves]]
[[!redirects Reidemeister's theorem]]
[[!redirects first Reidemeister move]]
[[!redirects R1 move]]
[[!redirects second Reidemeister move]]
[[!redirects R2 move]]
[[!redirects third Reidemeister move]]
[[!redirects R3 move]]


Reidemeister move 1 - SVG

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category: svg


Reidemeister move 2 - SVG

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category: svg


Reidemeister move 3 - SVG

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Reidemeister torsion

## Idea

(...)

## Properties

### In perturbative Chern-Simons theory

appears in [[perturbative quantization of 3d Chern-Simons theory]]

## Related concepts

* [[analytic torsion]]

## References

An introduction and survey is in

* L. Nicolaescu, _Notes on Reidemeister torsion_ ([pdf](http://www.nd.edu/~lnicolae/Torsion.pdf))

The notion of Reidemeister torsion originates in

* [[Kurt Reidemeister]], _Automorphismen von Homotopiekettenringen_, Math. Ann. 112 (1936), 586–593.

and was then extended in

* W. Franz, _Über die Torsion einer Überdeckung_, J. Reine Angew. Math. 173 (1935), 245–254.

and in

* [[Georges de Rham]], _Complexes á automorphismes et homéomorphie différentiable_, Ann. Inst. Fourier (Grenoble) 2 (1950), 51–67.

Reidemeister torsion was identified with the [[Alexander polynomial]] in

* [[John Milnor]], _A duality theorem for the Reidemeister torsion_, Ann. of Math. 76 (1962),
137–147.

Then

* [[Jeff Cheeger]], _Analytic torsion and the heat equation_, Ann. of Math. 109 (1979), 259–322.

and

* W. Müller, _Analytic torsion and R-torsion of Riemannian manifolds_, Adv. in Math.
28 (1978), 233–305.

proved that on compact [[Riemannian manifolds]] it coincides with [[analytic torsion]].

Relation to the [[volume]] is discussed in

* Pere Menal-Ferrer, Joan Porti, _Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds_ ([arXiv:1110.3718](http://arxiv.org/abs/1110.3718))

Relation to [[3d-3d correspondence]]:

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


Reidemeister trace

# Contents
* table of contents
{: toc}

## Idea

The Reidemeister trace, developed by Reidemeister and Wecken, is an algebraic invariant of a self-map of a "finite" topological space.  It gives information about the existence or nonexistence of fixed points, and refines both the [[Lefschetz number]] and [[Nielsen number]].

## Definition

Suppose $M$ is a [[closed manifold]] and $f\colon M\to M$ a [[self-map]].  Deform $f$ so that it has isolated fixed points.  We say that two fixed points $x$ and $y$ are in the same [[fixed-point class]] if there is a path $\gamma$ from $x$ to $y$ such that $f(\gamma)$ is [[homotopy|homotopic]] to $\gamma$ rel the endpoints ($x$ and $y$).  Let $\mathbb{Z}[\pi_1(M)_f]$ denote the free abelian group on the set of fixed-point classes.  Then the **Reidemeister trace** of $f$ is the formal sum

$$ R(f) \coloneqq \sum_{f(x)=x} ind_f(x) \cdot [x] \in \mathbb{Z}[\pi_1(M)_f] $$

where $ind_f(x)$ is the [[index]] of the fixed point $x$ of $f$.  This definition is homotopy invariant.

An equivalent definition can be obtained algebraically, or category-theoretically using the [[bicategorical trace]].

## Properties

* The sum of all the coefficients in the Reidemeister trace is the [[Lefschetz number]] $L(f)$.

* The number of nonzero coefficients in the Reidemeister trace is the [[Nielsen number]] $N(f)$.

* If $M$ is a closed manifold of dimension at least 3, and $R(f)=0$, then $f$ is homotopic to a map without fixed points.  Thus, the Reidemeister trace supports a converse to the [[Lefschetz fixed-point theorem]].

## References

The Reidemeister trace was introduced in

* [[Kurt Reidemeister]], _Automorphismen von Homotopiekettenringen_, Mathematische
Annalen, 112:586–593 (1936)

A modern treatment is in

* S. Husseini, _Generalized Lefschetz numbers_, Transactions of the American
Mathematical Society, 272:247–274 (1982)

See also

* Peter Staecker, _The Reidemeister trace: computation by nilpotentization and extension to coincidence theory_ (PhD thesis)

* Peter Staecker, _Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds_, ([arXiv:0704.1891v2](http://arxiv.org/abs/0704.1891v2))

A reformulation of the Reidemeister trace in terms of [[bicategorical trace]] is in

* [[Kate Ponto]], [[Michael Shulman]], _The multiplicativity of fixed point invariants_ ([arXiv:1203.0950](http://arxiv.org/abs/1203.0950))


Reidemeister-Schreier theorem

Reidemeister-Schreier theorem/method/algorithm/rewriting process

Reidemeister 1927, [[Otto Schreier]] 1927

* A. N. Parshin, Igorʹ Rostislavovich Shafarevich, _Algebra VII: combinatorial group theory : applications to geometry_

* A. J. Weir, _The Reidemeister-Schreier and Kuroš subgroup theorems_, Mathematica (1956), 3: 47-55, [doi](http://dx.doi.org/10.1112/S0025579300000887)

* René Hartung, _A Reidemeister-Schreier theorem for finitely $L$-presented_, [arxiv/1108.2403](http://arxiv.org/abs/1108.2403)

* _Reidemeister-Schreier type rewriting for semigroups_, Semigroup Forum __51__, Number 1, 47-62[doi](http://dx.doi.org/10.1007/BF02573619)

* MO: [Reidemeister-Schreier Method for Finding Stabilizer of an Element in a Group Action](http://mathoverflow.net/questions/64572/reidemeister-schreier-method-for-finding-stabilizer-of-an-element-in-a-group-acti)

[[!redirects Reidemeister-Schreier method]]


Reiko Miyaoka

* [webpage](http://www.math.tohoku.ac.jp/people/miyaoka/index-ev3.html)

## related $n$Lab entries

* [[quaternionic Hopf fibration]]

* [[G2]]

category: people


Reinhard Börger

* [home page](https://www.fernuni-hagen.de/MATHEMATIK/ALGGEO/Boerger/boerger.html)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=26756)

## Selected writings

On [[superconvex spaces]]:

* {#BörgerKemper94a} [[Reinhard Börger]], [[Ralf Kemper]], *There is no cogenerator for total convex spaces*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome **35** 4 (1994) 335-338 [[numdam:CTGDC_1994__35_4_335_0](http://www.numdam.org/item/CTGDC_1994__35_4_335_0)]

* {#BörgerKemper94b} [[Reinhard Börger]], [[Ralf Kemper]], *Cogenerators for convex spaces*, Applied Categorical Structures **2** (1994) 1-11 [[doi:10.1007/BF00878499](https://doi.org/10.1007/BF00878499)]

See also:

* [[Reinhard Börger]], [[Walter Tholen]], _Strong regular and dense generators_,  [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **32** 3 (1991) 257-276 [[numdam:CTGDC_1991__32_3_257_0](http://www.numdam.org/item?id=CTGDC_1991__32_3_257_0), [MR1158111](http://www.ams.org/mathscinet-getitem?mr=1158111)]

category: people


Reinhard Diestel

* [home page](https://www.math.uni-hamburg.de/home/diestel/)

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

## Selected writings
A standard text on [[graph theory]]:

* Reinhard Diestel, *Graph Theory*, Graduate Texts in Mathematics **173** 5th edition (2017) [[website](https://diestel-graph-theory.com/), [doi:10.1007/978-3-662-53622-3](https://doi.org/10.1007/978-3-662-53622-3)]

On the [[axiomatization]] of [[infinite set|infinite]] [[matroids]]:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

category: people


Reinhard Honegger

* [InSpire page](https://inspirehep.net/authors/2270461)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=65265)

## Selected writings

On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

category: people


Reinhold Baer

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

category: people


Reinhold Heckmann

## Selected writings

On [[valuation (measure theory)|valuations]]:

* {#Heckmann96} [[Reinhold Heckmann]], *Spaces of valuations*, Annals of the New York Academy of Sciences, **806** 1 *Papers on General Topology and Applications*, (1996) $[$[doi:10.1111/j.1749-6632.1996.tb49168.x](https://doi.org/10.1111/j.1749-6632.1996.tb49168.x), [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf)$]$

On the [[exponential law for spaces]]:

* [[Martín Escardó]], [[Reinhold Heckmann]], *Topologies of spaces of continuous functions* (2001) $[$[pdf](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf), [[EscardoHeckmann-TopologiesOnFunctionSpaces.pdf:file]]$]$

category: people


Reinhold Remmert

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

## Writings

* _[[Non-Archimedean Analysis]]_

## Related $n$Lab entries

* [[analytic space]], [[complex analytic space]], [[Stein manifold]]

category: people


Reissner-Nordström spacetime

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The Reissner-Nordström solution to the [[equations of motion]] of [[Einstein-Maxwell theory]] describes the [[spacetime]] outside of a [[electric charge|electrically charged]] (and/or [[magnetic charge|magnetically charged]]) point [[mass]], hence describes a _charged_ [[black hole]].

## Details

### Metric

For [[mass]] $M$, [[electric charge]] $Q$ and [[magnetic charge]] $P$ the [[pseudo-Riemannian metric]] for the Reissner-Nordström spacetime is, in [[polar coordinates]] $(t,r,\phi,\theta)$ on $\mathbb{R}^4 - \{0\} = \mathbb{R} \times \mathbb{R}_{\gt 0} \times S^2$, given by

$$
  ds^2_{M,Q,P} \coloneqq - H d t^2 + H^{-1} d r^2 + r^2 ds_{S^2}^2
$$

where (in natural units)

$$
  H \coloneqq 1 - \frac{M}{r} + \frac{Q^2 + P^2}{r^2}
  \,.
$$

The corresponding [[Faraday tensor]]/electromagnetic [[field strength]] [[curvature|curvature 2-form]] is

$$
  F = \frac{Q}{r^2} d t \wedge d r + P \sin(\theta) d\theta \wedge d\phi
  \,.
$$

> (check sign)

### Horizons

The function $H$ above vanishes at

$$
  r_{\pm}
    \coloneqq
  \tfrac{1}{2}\left(
    M \pm \sqrt{M^2 - 4 (P^2 + Q^2)}
  \right)
  \,.
$$

The larger of the two solutions is an [[event horizon]], the other is a [[Cauchy horizon]]. In the case that they coincide, for $M = 2\sqrt{Q^2 + P^2}$, one speaks of an [[extremal black hole]] (at least when also $P = 0$).

## Properties

### Penrose diagrams
 {#PenroseDiagrams}

The [[Penrose diagram]] for maximally extended RN black holes (from [HE 73, Section 5.5](#HE73)):

\linebreak

\begin{imagefromfile}
    "file_name": "RNBHPenroseDiagramHEFig25.jpg",
    "width": 550,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [HE 73, Fig. 25](#HE73)"
\end{imagefromfile}

\linebreak

\linebreak

\begin{imagefromfile}
    "file_name": "RNBHPenroseDiagramHEFig26.jpg",
    "width": 550,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [HE 73, Fig. 26](#HE73)"
\end{imagefromfile}

### Interior solutions
 {#InteriorSolutions}

Instead of having a mass and charge of singular support at the singular locus of the RN-spacetime, one may consider a finite mass and charge distribution, i.e. a "charged star". Outside of this mass/charge distribution then the metric is the RN-spacetime as above, but in the interior it becomes smoothed out to a non-singular interior solution.

Several attempts have been made to round out the Reissner metric with an interior. [Tiwari-Rao-Kanakamedala 85](#TiwariRaoKanakamelda85)  found an interior with the condition $g_{11} g_{44} = 1$, which obviously holds for the exterior Reissner, but does not match the interior Schwarzschild solution for e = 0. [Kyle-Martin 67](#KyleMartin67) found a rather complicated solution. They discussed the self-energy of the fields of charged matter. [Wilson 69](#Wilson69) modified this solution assuming a different value for the total charge. [Cohen-Cohen 69](#CohenCohen69) applied this solution to the special case of a charged thin shell and showed that the energy density of the electromagnetic field contributes to the mass. [Boulware 73](#Boulware73) studied the time development of thin shells. [Graves-Brill 60](#GravesBrill60) considered a possible oscillatory character of the Reissner-Nordström metric by examining the metric by Kruskal-like coordinates [Bekenstein 71](#Bekenstein71) studied another ansatz for the stress-energy-tensor of charged matter. [Krori-Jayantimala 75](#KroriJayantimala75) made use of the same ansatz, and he found a solution free of singularities. [Gautreau-Hoffman 73](#GautreauHoffman73) studied the sources of Weyl-type electrovac fields. They obtained the parameters for the source with the junction condition for the exterior solution. [Efinger 64](#Efinger64) deduced the stability of a charged particle from the self-energy of the gravitation field. In [Burghardt 09](#Burghardt09) is constructed another interior solution with the help of embedding the geometry in a 5-dimensional flat space.

(Literature summary taken from [Burghardt 09](#Burghardt09))

See also ([Mehra 82](#Mehra82), [WX 87](#WX87), [Gupta-Kumar-Pratibha 12](#GuptaKumarPratibha12)).

## Related concepts

* [[no-hair theorem]]

[[!include charged and rotating black holes -- table]]

## References

Original articles:

* [[Hans Reissner]], _Über die Eigengravitation des elektrischen Feldes nach der Einstein'schen Theorie_, Annalen der Physik 50 pp. 106–120, 1916 ([doi:10.1002/andp.19163550905]( https://doi.org/10.1002/andp.19163550905), [pdf](http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Reissner.pdf),  [atticus](https://www.atticusrarebooks.com/pages/books/1026/hans-reissner-jacob-also-jacob-johannes-reissner/uber-die-eigengravitation-des-elektrischen-feldes-nach-der-einstein-schen-theorie-offprint-from))

Review:

* Jonatan Nordebo, _The Reissner-Nordström metric_, 2016 ([pdf](https://www.diva-portal.org/smash/get/diva2:912393/FULLTEXT01.pdf), [diva2:912393](http://umu.diva-portal.org/smash/record.jsf?pid=diva2%3A912393&dswid=7103))

* Wikipedia, _Reissner-Nordström metric_

That the [[near horizon geometry]] of the [[extremal black hole|extremal]] Reissner-Nordström black hole in 4d ins 2d [[anti de Sitter spacetime]] times the [[2-sphere]], $AdS_2 \times S^2$, was observed in

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

   (see section 11.1 here: [pdf](http://www.hartmanhep.net/topics2015/11-nearhorizon.pdf) [[ExtremalReissnerNordstrom.pdf:file]])

Discussion of interior solutions:

* {#GravesBrill60} Graves J. C., Brill D. R., _Oscillatory character of Reissner-Nordström metric for an ideal charged wormehole._ Phys. Rev. 120, 1507, 1960

* {#Efinger64} Efinger H. J., _Der kugelsymmetrische Fall einer statischen Ladungsverteilung bei stark komprimierter Materie in der allgemeinen Relativitätstheorie. Acta Phys. Austr. 17, 347, 1964.

  _Über die Selbstenergie und Ladung eines durch Gravitationswirkung stabilisierten Teilchens in einem gekrümmten Raum_, Acta Phys. Austr. 188, 31, 1965

* {#KyleMartin67} Kyle C. F., Martin A. W., _Self-energy considerations in general relativity and the exact fields of charge and mass distribution_, Nuov. Cim 50, 883, 1967

* {#CohenCohen69} Cohen J. M., Cohen M. D., _Exact fields of charge and mass distributions in general relativity_. Nuov. Cim. 60, 241, 1969

* {#Wilson69} Wilson, S. J., _Exact solution of a static charged sphere in general relativity_. Can. Journ. Phys. 47, 2401, 1969

* {#Bekenstein71} [[Jacob Bekenstein]], _Hydrostatic equilibrium and gravitational collapse of relativistic charged fluid balls_, Phys. Rev. D 4, 2185, 1971

* {#HE73} S. W. Hawking, G. F. R. Ellis, Section 5.5 of: _The Large Scale Structure of Space-Time_, Cambridge University Press (1973) ([doi:10.1017/CBO9780511524646](https://doi.org/10.1017/CBO9780511524646))

* {#Boulware73} Boulware, D. G., _Naked singularities, thin shells, and the Reissner-Nordström metric_, Phys. Rev. D 8, 2363, 1973

* {#GautreauHoffman73} Gautreau R., Hoffman R. B., The structure of Weyl-type electrovac fields in general relativity. Nouv. Cim. 16 B, 162, 1973

* {#KroriJayantimala75} Krori K. D., Jayantimala B., _A singularity-free solution for a charged fluid in general relativity_, J. Phys. A 8, 508, 1975

* {#Mehra82} A.L. Mehra, _An interior solution for a charged sphere in general relativity_,  Physics Letters A Volume 88, Issue 4, 8 March 1982, Pages 159-161 ([publisher](http://www.sciencedirect.com/science/article/pii/0375960182905515))

* {#TiwariRaoKanakamelda85} Tiwari R. N., Rao J. R., Kanakamedala R. R., _Electromagnetic mass models in general relativity_, Phys. Rev. D 30, 489, 1984

* {#WX87} Wang Xingxiang, _Exact Solution of a Static Charged Sphere in
General Relativity_, General Relativity and Gravitation, Vol. 19, No. 7, 1987

* {#Burghardt09} Rainer Burghardt, _Reissner exterior and interior_ [arXiv:0902.0918](https://arxiv.org/abs/0902.0918)

* {#GuptaKumarPratibha12} Gupta, Y. K.; Kumar, Jitendra; Pratibha, _A Class of Well Behaved Charged Analogues of Schwarzchild's Interior Solution_, International Journal of Theoretical Physics October 2012, Volume 51, Issue 10, pp 3290–3302 ([publisher](http://link.springer.com/article/10.1007%2Fs10773-012-1209-4))

Discussion of magnetic black holes (similar to [[magnetic monopoles]]):

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

[[!redirects Reissner-Nordström spacetimes]]

[[!redirects Reissner-Nordstrom spacetime]]
[[!redirects Reissner-Nordstrom spacetimes]]

[[!redirects Reissner-Nordström black hole]]
[[!redirects Reissner-Nordström black holes]]

[[!redirects Reissner-Nordstrom black hole]]
[[!redirects Reissner-Nordstrom black holes]]

[[!redirects charged black hole]]
[[!redirects charged black holes]]

[[!redirects magnetic black hole]]
[[!redirects magnetic black holes]]


Rel

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

# $Rel$
* table of contents
{: toc}

## Idea

$Rel$ is the [[category]] whose [[objects]] are [[sets]] and whose [[morphisms]] are (binary) [[relations]] between sets.   It becomes a [[2-category]] $\mathbf{Rel}$ (in fact, a [[2-poset]]) by taking [[2-morphisms]] to be inclusions of relations.

## Definition

$Rel$ is a category whose  [[objects]] or $0$-cells are [[sets]], whose [[morphisms]] or $1$-cells $X \to Y$ are [[relations]] $R \subseteq X \times Y$, and where the _composite_ $S \circ R$ of morphisms $R: X \to Y$ and $S: Y \to Z$ is defined as the usual relational composite:
$$\{(x, z) \in X \times Z: \exists_{y \in Y} (x, y) \in R \wedge (y, z) \in S\} \subseteq X \times Z .$$   The _identity_ $1_X: X \to X$ is then the [[equality relation]], in other words the usual diagonal embedding
$$\{(x, x): x \in X\} \subseteq X \times X.$$

Another important operation on relations is taking the _opposite_, also known as the _transpose_ or sometimes the _converse_.   Any relation $R: X \to Y$ induces a relation
$$R^\dagger = \{(y, x) \in Y \times X: (x, y) \in R\} \subseteq Y \times X$$
This makes $Rel$ into a [[dagger-category]]: in other words, we have $(S \circ R)^\dagger = R^\dagger \circ S^\dagger$, $1_X^\dagger = 1_X$, and $(R^{\dagger})^{\dagger} = R$.

We can enhance $Rel$ to a 2-category, which we can call $\mathbf{Rel}$ to distinguish it from the mere category, by taking [[2-morphisms]] or $2$-cells $R \Rightarrow S$ to be inclusions of relations.   Since there is at most one 2-morphism between parallel 1-morphisms, $\mathbf{Rel}$ is a [[2-poset]]: that is, an [[enriched category|category enriched]] in the category of [[posets]].

## Relations and spans

It is useful to be aware of the connections between the [[bicategory of relations]] and the [[bicategory]] of [[span|spans]]. Recall that a _span_ from $X$ to $Y$ is a diagram of the form
$$X \leftarrow S \to Y$$
and there is an obvious category whose objects are spans from $X$ to $Y$ and whose morphisms are morphisms between such diagrams. The terminal span from $X$ to $Y$ is
$$X \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y$$
and a relation from $X$ to $Y$ is just a [[subobject]] of the terminal span, in other words an isomorphism class of monos into the terminal span.

To each span $S$ from $X$ to $Y$, there is a corresponding relation from $X$ to $Y$, defined by taking the [[image]] of the unique morphism of spans $S \to X \times Y$ between $X$ and $Y$. It may be checked that this yields a lax morphism of bicategories
$$Span \to Rel$$

## Limits and colimits##
 {#LimitsAndColimit}

The category $Rel$ does have [[products]] and [[coproducts]]; they coincide (by self-duality) and are just [[disjoint unions]] of sets. However, otherwise $Rel$ has very few (co)limits; it doesn't even have splittings of all idempotents. All symmetric idempotents have splittings, but the order-relation $\leq \; \subseteq \{0,1\} \times \{0,1\}$ can't be split. It follows that it can't have (co)equalisers.

Since the category $Rel$ is the category of _free algebras_ ([[Kleisli category]]) for the [[powerset monad]], there is, indeed, very little chance of a limit of such algebras being free again. To get decent limits, one has to move to the [[Eilenberg-Moore category]] of the [[powerset]] [[monad]],
viz., the category of complete [[suplattices]].

### Weak equalizers and completion###

As the category $Rel$ has _[[weak limit|weak]]_ [[equalizers]], one can take its [[exact completion]]. This happens to be the category of complete
sup-lattices and sup-preserving maps. And the tensor product on $Rel$
extends to the [[exact completion]].

The Freyd completion of $Rel$, which is equivalent to the category of basic pairs which appear in formal topology, has all limits exactly because $Rel$ has products and weak equalizers. The Freyd completion adds freely a strong factorization system to a(ny) category $C$.  The Freyd completion has products if $C$ has products, and it has equalizers if $C$ has weak equalizers.

### In the double category of relations###

If you insert the category $Rel$ into the [[double category]]
$\mathrm{RRel}$ of sets, mappings and relations, one has a double
category with all [[double limits]] and colimits. For instance, the
obvious cartesian product $a \times b : X \times Y \to X' \times Y'$ (resp. sum
$a+b : X + Y \to X' + Y'$) of two relations is indeed a product
(resp. a sum) in the double category.

Similarly, many bicategories of spans, cospans, relations,
profunctors, etc. have poor (co)limits, but can be usefully embedded in
weak double categories (with the same objects, "strict morphisms",
"same morphisms", suitable double cells) that have all limits and
colimits.

## As an enriched category##

Writing $\mathcal{V}$ for the category of [[suplattices]], $Rel$ is a
$\mathcal{V}$-category (see [[enriched category]]). With that in mind, the parallel:

* in [[additive categories]] with [[zero object]], _finitary_ (co)[[products]]
  are [[biproducts]]
* $\mathcal{V}$-categories with [[zero object]], _arbitrary_
  (co)[[products]] are [[biproducts]]

is remarkable. This was probably first noticed by [[Dana May Latch]], in
the 20th century, for the category of $\mathcal{V}$ of complete
[[suplattices]].

The [[matrix algebra]] for maps to products, from coproducts, and most
especially, from coproducts to products, works just as it does in the
case of [[additive categories]], when it comes to these
$\mathcal{V}$-categories.

##In spans##
See [[van Kampen colimit]].

##Mono/epimorphisms##

It is not hard to see that a relation $R \subseteq A \times B$ is a
[[monomorphism]] $A \to B$ iff the map $\mathcal{P}A \to \mathcal{P}B$
sending a subset of A to the set of all R-relatives of its members is
injective; dually for [[epimorphisms]].

## 2-categorical aspects##

Recall that we are using $\mathbf{Rel}$ to stand for the 2-category of sets, relations and inclusions.   Various facts about relations can be recast in these terms.  For example, a relation $L : X \to Y$ is a function from $X$ to $Y$ iff it has a right adjoint $R : Y \to X$ in $\mathbf{Rel}$.  In this case, its right adjoint equals its transpose $L^\dagger : Y \to X$.

Similarly, a monad $T : X \to X$ in the 2-category $\mathbf{Rel}$ is exactly a preorder on $X$, since the existence of a unit $i: 1_X \Rightarrow T$ says that $T$ is reflexive and the existence of the multiplication $m: T^2 \Rightarrow T$ says that $T$ is transitive.    This monad has an [[Eilenberg-Moore category|Eilenberg-Moore object]] iff it is an equivalence relation, in which case the Eilenberg-Moore object is the set of equivalence classes of the relation.

## Relations in a category

More generally, given any [[regular category]] $C$, one can form a 2-category of relations $\mathbf{Rel}(C)$ in similar fashion. The objects of $\mathbf{Rel}(C)$ are objects of $C$, the morphisms $r: c \to d$ in $\mathbf{Rel}(C)$ are defined to be [[subobject|subobjects]] of the terminal span from $c$ to $d$, and 2-cells $r \to s$ are subobject inclusions. To form the composite of $r \subseteq c \times d$ and $s \subseteq d \times e$, one takes the [[image]] of the unique span morphism
$$r \times_c s \to c \times e$$
in the category of spans from $c$ to $e$, thus giving a mono into the terminal span from $c$ to $e$. The subobject class of this mono defines the relation
$$s \circ r \subseteq c \times e$$
and the axioms of a regular category ensure that $\mathbf{Rel}(C)$ is a 2-category with desirable properties. Similar to what was said above, there is again a lax morphism of bicategories
$$Span(C) \to \mathbf{Rel}(C)$$

There is also a functor
$$i: C \to \mathbf{Rel}(C)$$
that takes a morphism $f: c \to d$ to the functional relation defined by $f$, i.e., the relation defined by the subobject class of the mono
$$\langle 1, f\rangle: c \to c \times d$$

Such functional relations may also be characterized as precisely those 1-cells in $Rel(C)$ which are [[adjunction|left adjoints]]; the right adjoint of $\langle 1, f \rangle$ is the opposite relation $\langle f, 1\rangle$. The unit amounts to a condition
$$1_c \subseteq \langle f, 1 \rangle \circ \langle 1, f \rangle$$
which says that the functional relation is _total_, and the counit amounts to a condition
$$\langle 1, f \rangle \circ \langle f, 1 \rangle \subseteq 1_d$$
which says the functional relation is _well-defined_.

## Related categories

* [[Set]]

* [[FinRel]]

A [[category of correspondences]] is a generalization of a category of relations. The composition of relations is that of correspondences followed by [[(-1)-truncated|(-1)-truncation]].

For generalizations of $Rel$ see

* [[allegory]], [[cartesian bicategory]]

* [[bicategory of relations]]

* [[Prof]]

* [[2-poset of partial maps]]

* [[category of monic maps]]

## References ##

* Grandis, M., Pare, R. :[Limits in double categories](https://eudml.org/doc/91618). Cahiers Topologie Géom. Différentielle Catég. 40(3), 162–220 (1999)
* Bucalo, A., Rosolini, G., Completions, comonoids, and topological spaces
Annals of Pure and Applied Logic 137, 104-125 (2006)
* _Credits for the section on limits go to the contributors of the following threads on the categories list:_ [categories: Limits and colimits in Rel?](http://thread.gmane.org/gmane.science.mathematics.categories/8027), [categories: Limits in REL](http://thread.gmane.org/gmane.science.mathematics.categories/8186)

* [[Aurelio Carboni]], [[Stefano Kasangian]] and [[Ross Street]], _Bicategories of spans and relations_, Journal of Pure and Applied Algebra, Vol. 33, No. 3, 1984, pp. 259-267. [doi:10.1016/0022-4049(84)90061-6]

Relations may be considered in any category with a stable [[proper factorisation system]], but this is no more general than considering relations in a [[regular category]], as proven in

* [[Gregory Maxwell Kelly]]. *A note on relations relative to a factorization system*. Category Theory. Springer, Berlin, Heidelberg, 1991.

An axiomatization of the category of sets and relations is found in:

* [[Andre Kornell]], *Axioms for the category of sets and relations* ([arXiv:2302.14153](https://arxiv.org/abs/2302.14153))

category: category

[[!redirects Rel]]
[[!redirects REL]]
[[!redirects rel]]


relation

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

# Relations
* table of contents
{: toc}

## Idea

A _relation_ is the extension of a _[[predicate]]_.  That is, if you have a statement whose [[truth value]] may depend on some [[variables]], then you get a relation that consists of those instantiations of the variables that make the statement [[true]].  Equivalently, you can think of a relation as a [[function]] whose [[target]] is the set of truth values.

## Definitions

### General case
 {#DefinitionGeneralCase}

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

Given a [[family of sets|family]] $(A_i)_{i: I}$ of [[sets]], a **relation** on that family is a [[subset]] $R$ of the [[cartesian product]] $\prod_{i: I} A_i$.

Equivalently, this is a [[function]] from $\prod_{i: I} A_i$ to the set $\TV$ of [[truth values]] (because $\TV$ is the [[subobject classifier]] in [[Set]]).

Also equivalently, this is a [[monomorphism]] in/[[subobject]] of the [[cartesian product]]

$$
  R \hookrightarrow \prod_{i \colon I} A_i
  \,.
$$

=--

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

For $I$ the 2-element set (a _binary relation_) this is in particular a binary [[correspondence]]

$$
  \array{
     && R
     \\
     & \swarrow && \searrow
     \\
    A_1 && && A_1
  }
  \,.
$$

Hence relations are precisely the [[(-1)-truncated]] [[correspondences]].

This identification induces a natural notion of [[composition]] of relations from the composition in the [[category of correspondences]], see below at _[The 2-poset of binary relations](#The2PosetOfBinaryRelations)_.

Formulated this way, the notion of relation generalizes to [[categories]] other than [[Set]] and to [[higher category theory]]. For more on this see at _[Generalizations](#Generalization)_ below.

=--

### Special cases

A __nullary relation__ is a relation on the empty family of sets.  This is the same as a [[truth value]].

A __unary relation on $A$__ is a relation on the singleton family $(A)$.  This is the same as a [[subset]] of $A$.

A __binary relation on $A$ and $B$__ is a relation on the family $(A,B)$, that is a subset of $A \times B$.  This is also called a __relation from $A$ to $B$__, especially in the context of the $2$-category [[Rel]] described below, or sometimes called a __heterogenous relation__.

A __binary relation on $A$__ is a relation on $(A,A)$, that is a relation from $A$ to itself.  This is sometimes called a __homogenous relation__ on $A$, simply a __relation on $A$__, or just an __endorelation__ with its set implicit as a property if not explicitly mentioned.

An __$n$-ary relation__ on $A$ is a relation on a family of $n$ copies of $A$, that is a subset of $A^n$.

For a binary relation, one often uses a symbol such as $\sim$ and writes $a \sim b$ instead of $(a,b) \in \sim$.  Actually, even when a relation is given by a letter such as $R$, one often sees $a R b$ instead of $(a,b) \in R$, although now that does not look so good.

### Internal and external relations

In foundations of mathematics with a [[sort]] of [[propositions]] and a separate sort of [[sets]], such as most presentations of [[set theory]] in [[first-order logic]] with [[equality]], there are actually two notions of relation. The definition given above is the notion of **internal relation**, defined as a [[subset]] of the [[Cartesian product]] of a [[family of sets]]. There is a second definition of relation on a set: external relations, which are not defined purely inside of the set theory. Since each set has a type of elements inside of the set, an **external relation** is simply a [[proposition]] in the [[context]] of a family of [[variables]] $x_i$ inside of a family of sets $A_i$
$$\Gamma, x_0 \in A_0, x_1 \in A_1, x_2 \in A_2, \ldots, x_n \in A_n \vdash P \; \mathrm{prop}$$
In any [[unsorted set theory]], those variables would be expressed as
$$\Gamma, x_0, A_0, x_0 \in A_0 \; \mathrm{true}, x_1, A_1, x_1 \in A_1 \; \mathrm{true}, x_2, A_2, x_2 \in A_2 \; \mathrm{true}, \ldots, x_n, A_n, x_n \in A_n \; \mathrm{true} \vdash P \; \mathrm{prop}$$
It is in this second sense of external relation that [[equality]] in [[first-order logic with equality]] is an [[equivalence relation]], and [[inequality]] is a [[tight apartness relation]] in a first-order theory with equality presented in [[classical logic]].

In [[dependent type theory]], where propositions are considered to be the [[subsingletons]] or the [[sets]]/[[types]] themselves, there is only one notion of relation, the internal notion given above.

## Morphisms

If $A$ and $B$ are each sets equipped with a relation, then what makes a [[function]] $f: A \to B$ a _morphism_ of sets so equipped?

There are really two ways to do this, shown below.  (We will write these as if each set is equipped with a binary relation $\sim$, but any fixed arity would work.)

*  $f$ __preserves__ the relation if $x \sim y \;\Rightarrow\; f(x) \sim f(y)$ always;
*  *f* __reflects__ the relation if $x \sim y \;\Leftarrow\; f(x) \sim f(y)$ always.

Now, if $f$ is a [[bijection]], then it preserves the relation if and only if its inverse reflects it, so clearly an [[isomorphism]] of relation-equipped sets should do both.  What about a mere morphism?

In general, it\'s more natural to require only preservation; these are the morphisms you get if you consider a set with a relation as a models of a [[algebraic theory|finite-limit theory]] or a simply [[directed graph]].

But in some contexts, particularly when dealing only with [[irreflexive relation]]s, we instead require (only) that a morphism reflect the relation.  Sometimes an even stricter condition is imposed, as for [[well-order]]s.  But even in these cases, the definition of isomorphism comes out the same.

## Binary relations

Binary relations are especially widely used.

### Kinds of binary relations

Special kinds of relations from $A$ to $B$ include:

* [[functional relations]],
* [[entire relations]],
* [[one-to-one relation]]s,
* [[onto relation]]s.

Combinations of the above properties of binary relations produce:

* [[functions]],
* [[partial functions]],
* [[injections]],
* [[surjections]],
* [[bijections]].

Special kinds of binary relations on $A$ (so from $A$ to itself) additionally include:

* [[reflexive relation|reflexive]] and [[irreflexive relation|irreflexive]] relations;
* [[symmetric relation|symmetric]], [[antisymmetric relation|antisymmetric]], and [[asymmetric relation|asymmetric]] relations;
* [[transitive relations]] and [[comparisons]];
* left and right [[euclidean relations]];
* [[total relation|total]] and [[connected relation|connected]] relations;
* [[extensional relation|extensional]] and [[well-founded relation|well-founded]] relations.

Combinations of the above properties of binary relations produce:

* [[equivalence relations]],
* [[partial equivalence relations]],
* [[apartness relations]],
* the various kinds of [[orders]] (which see).

### Structure preserving relations

Given [[pointed sets]] $(A, a)$ and $(B, b)$, a binary relation $R$ from $A$ to $B$ is said to preserve the point if $R(a, b)$.

Given sets with endofunctions $(A, f_A)$ and $(B, f_B)$, a binary relation $R$ from $A$ to $B$ is said to preserve the endofunction if for every $a \in A$ and $b \in B$, if $R(a, b)$, then $R(f(a), f(b))$.

Given [[magmas]] $(A, \mu_A)$ and $(B, \mu_b)$, a binary relation $R$ from $A$ to $B$ is said to preserve the binary operation if for every $a \in A$, $c \in A$, $b \in B$ and $d \in B$, if $R(a, b)$ and $R(c, d)$, then $R(\mu_A(a, b), \mu_B(b, d))$.

And so forth.

### Properties

Every binary endorelation $R$ has an [[equivalence relation]] $\sim_R$ generated by $R$.

### Binary relations as a coalgebra

By [[currying]] the binary relation $R: \mathcal{P}(X \times X) \cong X \times X \rightarrow \Omega$, one gets a [[coalgebra for an endofunctor|coalgebra]] $\theta: X \rightarrow (X \rightarrow \Omega) \cong X \rightarrow \mathcal{P}(X)$ for the [[powerset]] [[endofunctor]] on [[Set]].

### The $2$-poset of binary relations
 {#The2PosetOfBinaryRelations}

Binary relations form a $2$-[[2-category|category]] (in fact a $2$-[[2-poset|poset]]) [[Rel]], which is the basic example of an [[allegory]].

The [[objects]] are [[sets]], the [[morphisms]] from $A$ to $B$ are the binary relations on $A$ and $B$, and there is a [[2-morphism]] from $R$ to $S$ (both from $A$ to $B$) if $R$ implies $S$ (that is, when $(a,b) \in R$, then $(a,b) \in S$).

The interesting definition is [[composition]]

+-- {: .num_defn #CompositionOfRelations}
###### Defintion

If $R$ is a relation from $A$ to $B$ and $S$ is a relation from $B$ to $C$, then their _composite relation_  -- written $S \circ R$ or $R;S$ -- from $A$ to $C$ is defined as follows:
$$(a,c) \in R;S \;\Leftrightarrow\; \exists b: B,\; (a,b) \in R \;\wedge\; (b,c) \in S.$$

The identity morphism is given by [[equality]].

=--

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

The [[composition]] operation of relation from def. \ref{CompositionOfRelations} is induced by the composition of the underlying [[correspondences]], followed by [[(-1)-truncated|(-1)-truncation]].

=--

The special properties of the kinds of binary relations listed earlier can all be described in terms internal to [[Rel]]; most of them make sense in any [[allegory]].  Irreflexive and asymmetric relations are most useful if the allegory\'s [[hom-object|hom-poset]]s have [[bottom]] elements, and linear relations require this.  Comparisons require the hom-posets to have finite [[union|unions]], and well-founded relations require some sort of higher-order structure.

As a [[function]] may be seen as a functional, entire relation, so the category [[Set]] of sets and functions is a [[subcategory]] of [[Rel]] (in fact a [[replete subcategory|replete]] and locally [[full subcategory|full]] sub-$2$-category).

### The quasitopos of endorelations {#endorel}

Endorelations on sets are the objects of the [[quasitopos]] __$EndoRel$__ or __$Bin$__. It is a  [[reflective subcategory]] of [[Quiv]] the [[category of presheaves|presheaf topos]] of quivers and its morphisms are quiver morphisms. Endorelations are the [[separated presheaf|separated presheaves]] for the [[double negation#in_topos_theory|double negation topology]] on $Quiv$. "Separated" here translates to a quiver having  at most one arc between pairs of verticies. The [[reflective subcategory|reflector]] $Quiv \to EndoRel$ collapses parallel arcs together. Such quivers might also be called __singular__ or __simple__  though sometimes "simple" also means "no loops".

#### Relation closures as reflexive subcategories of $EndoRel$

All of the sub-types  of endorelations with positive conditions ([[reflexive relation|reflexive]], [[symmetric relation|symmetric]], [[transitive relations|transitive]], and left and right [[euclidean relations|euclidean]]) and their combinations have an associated [[Moore closure|closure]] that can produce one from an arbitrary relation. Such a closure [[completion|completes]] a relation by adding the least number of arcs such that the conditions are satisfied.
Within $EndoRel$ these closures are reflectors that produce reflective subcategories.
For example the __symmetric closure__ $sym: EndoRel \to EndoSym$ will (possibly) enlarge  a quiver that contains any arc $v_a \to v_b$ to one that also contains  $v_b \to v_a$. The __transitive and reflexive closure__ $transRef: EndoRel \to EndoTransRef$ produces a category which is isomorphic to [[preorder|PreOrd]] though its objects are the underlying quivers of the preorders which are the objects of $PreOrd$.

In addition to being reflective, the categories from the _symmetric_, _reflexive_, and _symmetric and reflexive_  closures are also quasitoposes that can be [[quasitopos#exampsep|directly defined]] through  double negation separation. The _symmetric and reflexive_ closure ([[category of simple graphs|SimpGph]]) is also a [[quasitopos|Grothendieck quasitopos]]. On the other hand $PreOrd$ is not a quasitopos because it is not a [[regular category]].

## Ternary relations

Apart from binary relations, there are important ternary relations used in mathematics, which for the most part are [[families]] of binary relations indexed by a third set:

* [[uniformity]], where the index set is the set of [[entourages]].
* [[premetric]], where the index set is the set of [[real numbers]], or in some cases, the set of [[rational numbers]].

## Generalisation
 {#Generalization}

Most of the preceding makes sense in any [[category]] with enough [[products]]; giving rise to [[internal relation|internal relations]], for instance _[[congruences]]_ in the case of internal equivalence relations.

Probably the trickiest bit is the definition of [[composition]] of binary relations, so not every category with [[finite products]] has an [[allegory]] of relations.  In fact, in a certain precise sense, a category has an allegory of relations if and only if it is [[regular category|regular]].  It can then be recovered from this allegory by looking at the functional entire relations.

## Related concepts

* [[Rel]]

* [[graph]]

* [[logical relation]]

* [[quantum relation]]

[[!include flavors of modal logics and relations -- table]]

[[!redirects relation]]
[[!redirects relations]]
[[!redirects relation theory]]
[[!redirects theory of relations]]

[[!redirects nullary relation]]
[[!redirects nullary relations]]

[[!redirects unary relation]]
[[!redirects unary relations]]

[[!redirects binary relation]]
[[!redirects binary relations]]

[[!redirects binary endorelation]]
[[!redirects binary endorelations]]

[[!redirects internal relation]]
[[!redirects internal relations]]

[[!redirects external relation]]
[[!redirects external relations]]


relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section

### Relation between 5d Maxwell theory and self-dual 3-forms in 6d

Consider 5d- and 6d-dimensional [[Minkowski spacetime]] equipped with global [[orthonormal basis|orthonormal]] [[coordinate charts]]  $\{x^\kappa\}$, $\{x^\alpha\}$, respectively, adapated to an [[isometry|isometric]] [[embedding of smooth manifolds|embedding]]

$$
  \array{
   &
    \mathbb{R}^{4,1}
    &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}&
    \mathbb{R}^{5,1}
    \\
    \kappa = &   0, 1, 2, 3, 4\phantom{,}
    \\
    \alpha = & 0, 1, 2, 3, 4, && 5
  }
$$

With this notation, the [[pullback of differential forms]] along this embedding is notationally implicit.

Now any [[differential 3-form]] $H_3$ on $\mathbb{R}^{5,1}$ decomposes as

\[
  \label{5dDecompositionof3Form}
  H_3
  \;=\;
  \widehat{F} \wedge d x^{5}
  +
  \widehat{H}
\]

for unique differential forms of the form

$$
  \widehat F
    \;=\;
  \tfrac{1}{2}\hat F_{\kappa_1 \kappa_2}(x^\kappa, x^5)
  d x^{\kappa_1} \wedge d x^{\kappa_2}
$$

and

$$
  \widehat{H}
    \;=\;
  \tfrac{1}{3!} \widehat{H}_{\kappa_1 \kappa_2 \kappa_3}(x^\kappa, x^5)
  d x^{\kappa_1} \wedge d x^{\kappa_2} \wedge d x^{\kappa_3}
  \,.
$$

In the case that $H_3$ has vanishing [[Lie derivative]] along the $x^5$-direction,

\[
  \label{VanishingLieDerivateiveAlongx5}
  \mathcal{L}_5 H_3 \;=\; 0
\]

then also these components forms do not depend on $x^5$ are actualls [[pullback of differential forms|pullbacks]] of differential forms on $\mathbb{R}^{4,1}$.

In terms of this decomposition, the 6d [[Hodge dual]] of $H_3$ is equivalently given by the 5d [[Hodge duals]] of these components as (best seen by the relation to [[Hodge pairing]] according to [this Prop.](Hodge+star+operator#HodgePairing))

\[
  \label{6dHodgeDualityInTermsOf5d}
  \star_6 H_3
  \;=\;
  \big( \star_5 \widehat{H}\big) \wedge d x^{5}
  -
  \star_5 \widehat{F}
\]

Since the [[Hodge star operator]] squares to unity in the special case that it is applied to differential 3-forms on 6d Minkowski spacetime (by [this Prop.](Hodge+star+operator#HodgeStarFollowedByHodgeStar))

$$
  \star_6 \star_6 H_3 \;=\; + H_3
$$

we may ask for $H_3$ to he [[Hodge duality|Hodge self-dual]]. By (eq:6dHodgeDualityInTermsOf5d) this means equivalently that its 5d components are 5d Hodge duals of each other:

$$
  \big(
    H_3
    \;=\;
    \star_{6} H_3
  \big)
  \;\;\;
  \overset{
    H_3 = \widehat{F} \wedge d x^5 + \widehat{H}
  }{
    \Leftrightarrow
  }
  \;\;\;
  \big(
    \widehat{H} = \star_5 \widehat{F}
  \big)
  \,.
$$

It follows that if there is no $x^5$-dependence (eq:VanishingLieDerivateiveAlongx5) then the condition that $H_3$ be a [[closed differential form|closed]] and [[self-dual higher gauge theory|self-dual 3-form]] is equivalent to its 5d components $\widehat{F}$ ($\widehat{H}$) being the (dual) [[field strength]]/[[Faraday tensor]] satisfying the [[Maxwell equations]] of [[D=5 Maxwell theory]] (without [[source fields|source]] [[conserved current|current]]):

$$
  \underset{
    \color{blue}
    {
    {\phantom{A}}
    \atop
    {\text{D=6 self-dual 3-form theory}}
    }
  }{
  \left.
  \array{
    &
    d H_3 = 0
    \\
    &
    \star_6 H_3 = H_3
  }
  \right\}
  }
  \;\;\;
  \overset{
    {\mathcal{L}_{5} H_3 = 0}
    \atop
    {H_3 = \widehat{F}\wedge d x^5 + \cdots}
  }{
    \Leftrightarrow
  }
  \;\;\;
  \underset{
    \color{blue}
    {
    {\phantom{A}}
    \atop
    \text{D=5 Maxwell theory}
    }
  }{
  \left\{
  \array{
    &
    d \widehat{F} = 0
    \\
    &
    d \star_5 \widehat{F} = 0
  }
  \right.
  }
$$

This may be summarized as saying that the massless part of the [[Kaluza-Klein reduction]] of [[self-dual higher gauge theory|self-dual 3-form theory]] from 6d to 5d  is [[D=5 Maxwell theory]].

Essentially this relation underlies the formulation of the [[M5-brane]] via the [[Perry-Schwarz Lagrangian]].


relation between adjunctions and monads -- section

### Relation between adjunctions and monads
 {#RelationBetweenAdjunctionsAndMonads}

There is a close relation between [[adjunctions]] ([[adjoint functors]]) and [[monads]]:

* [Monad induced by an adjunction](#MonadInducedByAnAdjunction)

* [Category of adjunction-resolutions of a monad](#CategoryOfAdjunctionResolutionsOfAMonad)

* [Semantics-Structure adjunction](#SemanticsStructureAdjunction)

#### Monad induced by an adjunction
 {#MonadInducedByAnAdjunction}

Every [[adjunction]] $(L \dashv R)$ induces a [[monad]] $R \circ L$ and a [[comonad]] $L \circ R$.

([Huber 1961, §4](monad#Huber61); see eg. [MacLane 1971, §VI.1 (p 134)](monad#MacLane71); [Borceux 1994, vol. 2, prop. 4.2.1](monad#Borceux94)).

In detail:

\begin{proposition}

Let $(\mathcal{C},\mathcal{D},F,U,\eta,\epsilon)$ be a pair of [[adjoint functors]] ie $F \dashv U$ are adjoint functors where $F \colon \mathcal{C} \rightarrow \mathcal{D}$, $U \colon\mathcal{D} \rightarrow \mathcal{C}$, $\eta_{A}:A \rightarrow U(F(A))$ is the [[unit of an adjunction|unit]] and $\epsilon_{B} \colon F(U(B)) \rightarrow B$ is the [[counit of an adjunction|counit]]. Then:

* $U \circ F$ is a [[monad]] on $\mathcal{C}$, with [[unit of a monad|unit]] $\eta$ and multiplication $U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))$.

* $F \circ U$ is a [[comonad]] on $\mathcal{D}$, with [[counit of a comonad|counit]] $\epsilon$ and comultiplication $F(\eta_{U(B)}):F(U(B)) \rightarrow F(U(F(U(B))))$.

\end{proposition}
\begin{proof}

We verify that we obtain a monad, the argument for the comonad is [[formal duality|formally dual]].

**(1)**
We know that this diagram commutes:

\begin{tikzcd}
  F(A)
  \arrow[r, "F(\eta_{A})"]
  \arrow[rd, "="']
  &
  F(U(F(A))
  \arrow[d, "\epsilon_{F(A)}"]
  \\
  & F(A)
\end{tikzcd}
By applying $U$, we obtain the first part of the unitaly of the monad:
\begin{tikzcd}
U(F(A)) \arrow[r, "U(F(\eta_{A}))"] \arrow[rd, "="'] & U(F(U(F(A))) \arrow[d, "U(\epsilon_{F(A)})"] \\
                                                     & U(F(A))
\end{tikzcd}

**(2)**
We know that this diagram commutes:
\begin{tikzcd}
U(B) \arrow[r, "\eta_{U(B)}"] \arrow[rd, "="'] & U(F(U(B))) \arrow[d, "U(\epsilon_{B})"] \\
                                               & U(B)
\end{tikzcd}
By putting $B=F(A)$, we obtain the second part of the unitality of the monad:
\begin{tikzcd}
U(F(A)) \arrow[r, "\eta_{U(F(A))}"] \arrow[rd, "="'] & U(F(U(F(A)))) \arrow[d, "U(\epsilon_{F(A)})"] \\
                                                     & U(F(A))
\end{tikzcd}

**(3)**
The naturality of $\epsilon$ is written, for every $f:B \rightarrow B'$:

\begin{tikzcd}
F(U(B)) \arrow[d, "F(U(f))"'] \arrow[r, "\epsilon_{B}"] & B \arrow[d, "f"] \\
F(U(B')) \arrow[r, "\epsilon_{B'}"']                    & B'
\end{tikzcd}

We apply it to $f=\epsilon_{F(A)}$ and it gives:

\begin{tikzcd}
F(U(F(U(F(A))))) \arrow[d, "F(U(\epsilon_{F(A)}))"'] \arrow[r, "\epsilon_{F(U(F(A)))}"] & F(U(F(A))) \arrow[d, "\epsilon_{F(A)}"] \\
F(U(F(A))) \arrow[r, "\epsilon_{F(A)}"']                                                & F(A)
\end{tikzcd}

Finally apply $U$ to this diagram to obtain:

\begin{tikzcd}
U(F(U(F(U(F(A)))))) \arrow[d, "U(F(U(\epsilon_{F(A)})))"'] \arrow[r, "U(\epsilon_{F(U(F(A)))})"] & F(U(F(A))) \arrow[d, "U(\epsilon_{F(A)})"] \\
U(F(U(F(A)))) \arrow[r, "U(\epsilon_{F(A)})"']                                                   & U(F(A))
\end{tikzcd}

which is exactly the associativity of the multiplication of the monad.

**(4)**
The naturality of the multiplication $U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))$ is obtained by two [[whiskerings]] of the counit $\epsilon_{B}:U(F(B)) \rightarrow B$.
\end{proof}

#### Category of adjunction-resolutions of a monad
 {#CategoryOfAdjunctionResolutionsOfAMonad}

An adjunction inducing a monad $T$ (as [above](#MonadInducedByAnAdjunction)) is also called a *resolution of $T$*.

There is in general more than one such resolution, in fact there is a [[category]] of adjunctions for a given monad whose morphisms are "[[comparison functors]]" (eg. [MacLane 1971, §VI.3](#MacLane71)).

In this category:

* the [[initial object]] is the adjunction over the [[Kleisli category]] of the monad;

* the [[terminal object]] is that over the [[Eilenberg-Moore category]] of algebras, also called the _[[monadic adjunction]]_ (recognized by *[[monadicity theorems]]*).

(e.g. [Borceux 1994, vol. 2, prop. 4.2.2](monad#Borceux94))

#### Semantics-structure adjunction
 {#SemanticsStructureAdjunction}

The above passage from [[adjunctions]] to [[monads]] and back to their [[monadic adjunctions]] constitutes itself an [[adjunction]], sometimes called the _[[semantics-structure adjunction]]_.


relation between algebraic and functorial field theory -- references

### Relation between algebraic and functorial field theory
 {#ReferencesRelationBetweenAQFTAndFQFT}

On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):

* {#Schreiber09} [[Urs Schreiber]], *[[schreiber:AQFTfromFQFT.pdf:file]]*, Comm. Math. Phys. **291** 2 (2009) 357-401 [[arXiv:0806.1079](https://arxiv.org/abs/0806.1079), [doi:10.1007/s00220-009-0840-2](https://doi.org/10.1007/s00220-009-0840-2)]

* {#Johnson-Freyd21} [[Theo Johnson-Freyd]], *Heisenberg-picture quantum field theory*, in *Representation Theory, Mathematical Physics, and Integrable Systems*, Progress in Mathematics **340** (2021) [[arXiv:1508.05908](https://arxiv.org/abs/1508.05908), [doi:10.1007/978-3-030-78148-4_13](https://doi.org/10.1007/978-3-030-78148-4_13)]

* {#BunkMacManusSchenkel23} [[Severin Bunk]], [[James MacManus]], [[Alexander Schenkel]], *Lorentzian bordisms in algebraic quantum field theory* [[arXiv:2308.01026](https://arxiv.org/abs/2308.01026)]


relation between BV and BD

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Algbraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concepts of

1. _[[BV-algebras]]_

1. _[[BD-algebras]]_

have a superficial similarity and are both related to [[BV-BRST formalism]] in [[physics]]. Indeed the "[[BV-operad]]" was named such because its [[algebra over an operad|algebras]] resemble the structure seen on the [[BV-complexes]] in physics. However in the [[formal deformation quantization]] picture of BV-formalism used in [Costello-Gwilliam](#CostelloGwilliam), the quantum BV-complexes crucially are [[BD-algebras]] and _not_ [[BV-algebras]]. On the other hand, if one does not insist on a formal deformation parameter, one may also formalize quantum BV complexes using actual BV-algebras.

The following note means to indicate how all these structures are similar and yet different.

## Details

Consider all [[chain complexes]] in the following with [[differential]] $d$ of degree +1.

### BV

Write $E_2$ for the [[little disk operad]] and $E_2^{fr}$ for the [[framed little disk operad]]. The [[chain homology]] of these is the [[BV-operad]] $BV$ and the operad $P_2$ for [[Poisson 2-algebras]], respectively:

$$
  \array{
    BV &\simeq& H_\bullet(E_2^{fr})
    \\
    \uparrow && \uparrow
    \\
    P_2 &\simeq& H_\bullet(E_2)
  }
  \,.
$$

An [[algebra over an operad|algebra over]] these in chain complexes  -- a [[homotopy BV-algebra]] or [[Poisson 2-algebra]], respectively -- is a chain complex $(V^\bullet, d)$ equipped with the following operations of the degree shown:

$$
  \array{
     operation && d && (-)\cdot(-) & &  \{-,-\} && \Delta
     \\
     degree && +1 && 0 && -1 && -1
  }
  \,,
$$

where the [[BV-operator]] $\Delta$ exists on [[homotopy BV-algebra]]s but not on [[Poisson 2-algebras]], satisfying the relation

$$
  \Delta (a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a \vert} a \cdot \Delta b - \{a,b\}
  \,.
$$

If $A$ is a commutative smooth [[Poincaré duality algebra]] so that its [[Hochschild cohomology]] $HH^\bullet(A)$ is identified by the [[HKR theorem]] with the multi-[[derivations]], then $HH^\bullet(A)$ carries a [[BV-algebra]] structure and under this identification the above [[Gerstenhaber bracket]] $\{-,-\}$ is the [[Schouten bracket]] and the [[BV-operator]] $\Delta_{BV}$ is the image of the [[de Rham differential]] dualized. Hence this is then indeed the standard BV-complex as discussed there in the section _[Multivector fields dual to differential forms](BV-BRST+formalism#MultivectorFieldsDualToDifferentialForms)_.

If we disregard the [[BV-operator]] $\Delta$, then more generally there are [[Poisson n-algebras]] for all $n$, which are as above but with the bracket of degree $1-n$

$$
  \array{
    operation && \{-,-\}
    \\
    degree && 1-n
  }
  \,.
$$

Notice that this means that $P_2$ algebras are very similar to $P_0$-algebras: for the former the backet has degree -1, for the latter it has degree +1.

### BD

Now the [[BD-operad]], which is a chain complex of $\mathbb{R}[ [\hbar ] ]$-[[modules]] ([[formal power series]] ring in one [[variable]] $\hbar$) has as chain complex of binary operations

$$
  BD(2)
    \;\coloneqq\;
  \array{
    ( \cdots &\to& 0 &\to& \left\langle(-\cdot -)\right\rangle &\stackrel{d_{operad}}{\to}& \left\langle \hbar \{-,-\} \right\rangle &\to& 0 &\to& \cdots )
    \\
    && && 0 && 1
  }
  \,.
$$

Notice: The differential takes the single generator in degree 0 to the single generator in degree 1. As a differential in chain complexes of $\mathbb{R}$-modules this would be the null complex, but since the bracket carries a prefactor of $\hbar$, we get something non-null if we regard this suitably filtered over $\hbar$... In particular, setting "$\hbar = 0$" by forming the [[tensor product]] with $\mathbb{R} \simeq \mathbb{R}[ [\hbar] ]/(\hbar)$ turns it into the chain complex concetrated on a single generator in degree 0. This is the $E_0$-operad.
On the other hand, for finite $\hbar$, i.e. after tensoring with $\mathbb{R}[ [\hbar] ]/(\hbar-1)$ then it becomes the $\tilde E_0$-operad, which is equivalent to the $E_0$-operad (whose algebras are simply pointed chain complexes). See also [Gwilliam-Haugseng](#GwilliamHaugseng).

$$
  \array{
    && \tilde E_0 & \simeq E_0
    \\
    & {}^{\mathllap{\hbar = 1}}\nearrow
    \\
    BD
    \\
    & {}_{\mathllap{\hbar = 0}}\searrow
    \\
    && P_0
  }
  \,.
$$

In any case, an [[algebra over an operad]] in chain complexes over the [[BD-operad]] has operations

$$
  \array{
    operation && d && \hbar \{-,-\}
    \\
    degree && +1 && +1
  }
$$

subject to

$$
  d (a \cdot b) = (d a) \cdot b + (-1)^{\vert a \vert} a \cdot d b - \hbar\{a,b\}
  \,.
$$

This has the same structure as the equation above for [[BV-algebras]] if we syntactically replace $d$ by $\Delta$ and remove the factor of $\hbar$. But since $d$ and $\Delta$ are on a different footing, the similarity between these two equations is only superficial.

## References

* {#CostelloGwilliam} [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in quantum field theory Volume 2_ ([pdf](http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf))

* {#GwilliamHaugseng} [[Owen Gwilliam]], [[Rune Haugseng]], _Linear Batalin-Vilkovisky quantization as a functor of ∞-categories_ ([arXiv:1608.01290](https://arxiv.org/abs/1608.01290))


relation between compact Lie groups and reductive algebraic groups

#Contents#
* table of contents
{:toc}

## Idea

**Theorem** (Chevalley)

The [[functor]] that takes linear [[algebraic groups]] $G$ to their $\mathbb{R}$-points $G(\mathbb{R})$ constitutes an [[equivalence of categories]] between [[compact Lie groups]] and $\mathbb{R}$-aniosotropic [[reductive algebraic groups]] over $\mathbb{R}$ all whose connected components have $\mathbb{R}$-points.

For $G$ as in this equivalence, then the [[complex Lie group]] $G(\mathbb{C})$ is the [[complexification]] of $G(\mathbb{R})$.

(from ([Conrad 10](#Conrad10)))

## Related concepts

* [[form of an algebraic group]]

## References

* [[James Milne]], _Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups_ ([web](http://www.jmilne.org/math/CourseNotes/ala.html))

  _Lie algebras, algebraic groups and Lie groups_ ([pdf](http://www.jmilne.org/math/CourseNotes/LAG.pdf))

* A. L.Onishchik, E. B. Vinberg, _Lie groups and algebraic groups_, Springer 1990

* Richard Pink, _Compact subgroups of linear algebraic groups_ ([pdf](http://www.math.ethz.ch/~pink/ftp/LastVersion.pdf))

* {#Conrad10} [[Brian Conrad]], _[MO comment](https://mathoverflow.net/questions/6079/classification-of-compact-lie-groups/16269#16269)_ 2010

[[!redirects relation between reductive algebraic groups and compact Lie groups]]


relation between determinant and trace

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

#Contents#
* table of contents
{:toc}

## Statement

If $A$ is an $n \times n$ [[square matrix]], then the [[determinant]] of its [[exponential]] equals the [[exponential]] of its [[trace]]

$$
  \det\big(\exp(A)\big)
    \;=\;
  \exp\big(tr(A)\big)
  \,.
$$

More generally, the determinant of $A$ is a [[polynomial]] in the [[traces]] of the [[powers]] of $A$:

For $2 \times 2$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{2}\left( tr(A)^2 - tr(A^2) \right)
$$

For $3 \times 3$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{6}
  \left(
    (tr(A))^3
    -
    3 tr(A^2) tr(A)
    +
    2 tr(A^3)
  \right)
$$

For $4 \times 4$-matrices:

$$
  det(A)
  \;=\;
  \tfrac{1}{24}
  \left(
    (tr(A))^4
    -
    6 tr(A^2)(tr(A))^2
    +
    3 (tr(A^2))^2
    +
    8 tr(A^3) tr(A)
    -
    6 tr(A^4)
  \right)
$$

Generally for $n \times n$-matrices ([Kondratyuk-Krivoruchenko 92, appendix B](#KondratyukKrivoruchenko92)):

\[
  \label{DeterminantAsPolynomialInTracesOfPowers}
  det(A)
  \;=\;
  \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  \left(tr(A^l)\right)^{k_l}
\]

+-- {: .proof}
###### Proof of (eq:DeterminantAsPolynomialInTracesOfPowers)

It is enough to prove this for semisimple matrices $A$ (matrices that are [[diagonalizable matrix|diagonalizable]] upon passing to the [[algebraic closure]] of the ground field) because this [[subset]] of matrices is [[Zariski topology|Zariski]] [[dense subset|dense]] (using for example the nonvanishing of the [[discriminant]] of the [[characteristic polynomial]]) and the set of $A$ for which the equation holds is [[Zariski topology|Zariski]] [[closed subset|closed]].

Thus, without loss of generality we may suppose that $A$ is [[diagonal matrix|diagonal]] with $n$ [[eigenvalues]] $\lambda_1, \ldots, \lambda_n$ along the diagonal, where the statement can be rewritten as follows. Letting $p_k = tr(A^k) = \lambda_1^k + \ldots + \lambda_n^k$, the following identity holds:

\[
\prod_{i=1}^n \lambda_i = \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  p_l^{k_l}
\]

This of course is just a [[polynomial]] identity, one closely related to various of the [[Newton identities]] that concern symmetric polynomials in indeterminates $x_1, \ldots, x_n$. Thus we again let $p_k = x_1^k + \ldots + x_n^k$, and define the [[elementary symmetric polynomials]] $\sigma_k = \sigma_k(x_1, \ldots, x_n)$ via the generating function identity

\[
\sum_{k \geq 0} \sigma_k t^k = \prod_{i=1}^n (1 + x_i t).
\]

Then we compute

\[\array{
\sum_{k \geq 0} \sigma_k t^k & = & \prod_{i=1}^n (1 + x_i t) \\
 & = & \exp\left(\sum_{i=1}^n \log(1 + x_i t)\right) \\
 & = & \exp\left(\sum_{i=1}^n \sum_{k \geq 1} (-1)^{k+1} \frac{x_i^k}{k} t^k \right)\\
 & = & \exp\left( \sum_{k \geq 1} (-1)^{k+1} \frac{p_k}{k} t^k\right)
}\]

and simply match coefficients of $t^n$ in the initial and final series expansions, where we easily compute

\[
  x_1 x_2 \ldots x_n = \sum_{n = k_1 + 2k_2 + \ldots + n k_n} \prod_{l=1}^n \frac1{(k_l)!} \left(\frac{p_l}{l}\right)^{k_l} (-1)^{k_l+1}
  \,.
\]

This completes the proof.

=--

## Related concepts

* [[determinant]], [[trace]]

* [[characteristic form]]

## References

The proof of (eq:DeterminantAsPolynomialInTracesOfPowers) is spelled out in

* {#KondratyukKrivoruchenko92} L. A. Kondratyuk, I. Krivoruchenko, Appendix B of:  _Superconducting quark matter in $SU(2)$ colour group_, Zeitschrift für Physik A Hadrons and Nuclei March 1992, Volume 344, Issue 1, pp 99–115 ([doi:10.1007/BF01291027](https://doi.org/10.1007/BF01291027))

[[!redirects relation between trace and determinant]]


relation between L-infinity algebras and dg-Lie algebras

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[homotopy theory]] of [[L-infinity algebra|$L_\infty$-algebras]] happens to be equivalent to that of [[dg-Lie algebras]] (over any [[ground field]] of [[characteristic zero]]). In fact, [[dg-Lie algebras]], being [[chain complexes]] equipped with a binary bracket, manifestly form the [[full subcategory]] of the [[1-category]] of [[L-infinity algebra|$L_\infty$-algebras]] on those whose $k$-ary brackets vanish for all $k \geq 3$,

\[
  \label{InclusionOfdgLieAlgebrasIntoLInfinityAlgebras}
  dgLieAlgebras
  \xhookrightarrow{\;\;i\;\;}
  L_\infty Algebras
  \,,
  {\phantom{AAAA}}
  L_{\mathrm{W}}
  \big(
    dgLieAlgebras
  \big)
  \underoverset
    {\simeq}
    {\;\;L_{\mathrm{W}}(i)\;\;}
    {\longrightarrow}
  L_{\mathrm{W}}
  \big(
    L_\infty Algebras
  \big)
  \,
\]

and under [[simplicial localization]] $L_{\mathrm{W}}$ this inclusion
is an [[equivalence of (∞,1)-categories]].

Conversely, this means that [[dg-Lie algebras]] are *[[strict ∞-category|strict]]* [[L-infinity algebra|$L_\infty$-algebras]] -- namely those whose [[Jacobi identity]] is satisfied strictly, not just up to [[coherence|coherent]] [[higher homotopies]] -- and that every [[L-infinity algebra|$L_\infty$-algebra]] may be *[[rectification|rectified]]* to a strict one.

When formulated in terms of [[operads]] this means that the canonical [[functor]] from the category of [[algebras over an operad|algebras over]] the [[Lie operad]] in [[chain complexes]] (which are the [[dg-Lie algebras]]) to that of [[algebras over an operad|algebras over]] any of its [[cofibrant resolutions]] (which are the [[L-infinity algebra|$L_\infty$-algebras]]) is, at least, a surjection on [[quasi-isomorphism]] [[equivalence classes|classes]]. In this form the statement appears in [Kriz and May 1995, p. 28](#KrizMay95).

This refines to an [[equivalence of (∞,1)-categories|equivalence of homotopy theories]] as follows:

* [[L-infinity algebra|$L_\infty$-algebras]] are the [[fibrant objects]] in the [[model structure on dg cocommutative coalgebras]] ([Pridham 10](#Pridham10), [this Prop. ](model+structure+for+L-infinity+algebras#LInfinityAlgebraIsFibrantObjectIndgFormalSpace));

* [[dg-Lie algebras]] are [[fibrant objects]] in the [[model structure on dg-Lie algebras]] (trivially so)

and the inclusion (eq:InclusionOfdgLieAlgebrasIntoLInfinityAlgebras) of the latter [[category of fibrant objects]] into the former extends to a [[right Quillen functor]] with [[left adjoint]] being [[rectification]] ([this Prop.](model+structure+for+L-infinity+algebras#LeftAdjointFromDgCoAlgToDgAlg)) and which constitutes a [[Quillen equivalence]] ([Hinich 97](#Hinich97), [this Prop.](model+structure+for+L-infinity+algebras#QuillenEquivalenceBetweendgLieAnddgCoCAlg)).

## Related entries

* [[model structure for L-infinity algebras|model structure for $L_\infty$-algebras]]

* [[rectification]]

* [[rational homotopy theory]]

## References

* {#Quillen69} [[Dan Quillen]], _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

* {#KrizMay95} [[Igor Kriz]], [[Peter May]], p. 28 of: _Operads, algebras, modules and motives_, Astérisque __233__, Société Mathématique de France (1995) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf), [numdam:AST_1995__233__1_0](http://www.numdam.org/item/?id=AST_1995__233__1_0))

* {#Pridham10} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

* {#Hinich97} [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_, Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), Erratum: [arXiv:math/0309453](http://arxiv.org/abs/math/0309453))

[[!redirects relation between L-∞ algebras and dg-Lie algebras]]

[[!redirects relation between dg-Lie algebras and L-infinity algebras]]
[[!redirects relation between dg-Lie algebras and L-∞ algebras]]


relation between preorders and (0,1)-categories

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### Context
#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## General idea

The notions of *[[posets]]* and *[[prosets]]* (in [[order theory]]),  and of *[[thin categories]]*, *[[truth-value]]-[[enriched categories]]* and *[[(0,1)-categories]]* (in [[category theory]] and [[higher category theory]]) are all closely related and yet possibly subtly but crucially different (see also at *[[category theory vs order theory]]*):

In [[category theory]]-language:

* a *[[preordered set]]* (or *[[proset]]*, for short) is

  a [[strict category|strict]] and [[thin]] [[category]].

* a *[[partially ordered set]]* (or *[[poset]]*, for short) is

  a [[strict category|strict]] and [[thin]] **and [[skeletal category|skeletal]]** [[category]].

Now, while every [[poset]] is in particular a [[proset]],
a [[proset]] need not be [[isomorphism|isomorphic]] (namely as a [[strict category]]) to a [[poset]].

On the other hand, if we disregard [[strict category|strictness]] and assume the [[axiom of choice]], then every [[proset]] is *[[equivalence of categories|equivalent as a category]]* to a [[poset]]: This is the statement that every category has a [[skeletal category|skeleton]].

Finally, if [[prosets]] are regarded as actual [[categories]] this way (i.e. disregarding their [[strict category|strictness]]) then as such they are, equivalently:

1. [[thin categories]],

1. [[truth value]]-[[enriched categories]],

1. [[(0,1)-categories]].

Conversely, since the notion of [[skeletal category]] implies that of a [[strict category]], one may say that [[posets]] are the *[[skeletal category|skeletal]]* [[truth value]][[enriched category|-]]/[[thin category|thin-]]/[[(0,1)-categories|$(0,1)$-categories]].

In doing so, one should just keep in mind that in various contexts (such as in various [[foundations of mathematics]]) [[strict category|strictness]] may matter and/or the [[axiom of choice]] may fail, in which case [[prosets]] are *not* equivalently [[posets]] and neither may be really equivalent to [[thin categories]]/[[(0,1)-categories]].

Finally, beware that even these notions of categories may not always be equivalent: A context which uses the terminology  "[[(0,1)-category]]" is less likely to even consider the notion of a [[strict category]] than a context using the terminology "[[thin category]]" or "[[truth value]]-[[enriched category]]", which might even regard [[strict categories]] as the default notion.

## In Type theory

(...)