Planck's constant

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

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

What is called _Planck's constant_ in [[physics]] and specifically in [[quantum physics]] (after [[Max Planck]]) is a [[physical unit]] of
"[[action functional|action]]" which sets the [[scale]] at which effects of [[quantum physics]] are genuinely important and physics is no longer well approximated by [[classical mechanics]]/[[classical field theory]].

This is discussed in the section:

* _[As a physical constant](#AsAPhysicalConstant)_

In the [[mathematics|mathematical]] formulation of the [[theory (physics)|theory]], Planck's constant $h$ is the choice of [[unit]] $h \in \mathbb{R}^\times$ in the [[short exact sequence]]  $\mathbb{Z}\stackrel{h\cdot(-)}{\longrightarrow} \mathbb{R} \to U(1)$
which governs the [[prequantization]] lift from real ([[differential cohomology|differential]]) [[cohomology]] to ([[differential cohomology|differential]]) [[integral cohomology]]. The [[integer|integrality]] of $\mathbb{Z}$ here is the very "quantum"-ness of quantum theory, and this is what Planck's constant parameterizes. This is discussed in the section:

* _[In geometric quantization](#InGeometricQuantization)_.

Finally, when infinitesimally approximating this [[quantization]] step in [[perturbation theory]] in $\hbar$ (see at [[formal deformation quantization]]), then Planck's constant is the very [[formal geometry|formal expansion parameter]] of the [[deformation theory|deformation]]. This is discussed in the section:

* _[In perturbative quantization](#InFormalDeformationQuantization)_.

Applied to the key example of [[perturbative quantum field theory]] it turns out that the powers of $\hbar$ in contributions to the [[Feynman perturbation series]]  essentially correspond to the [[loop order]] of the given [[Feynman diagram]]. This is discussed in the section

* _[In the Feynman perturbation series](#InFeynmanPerturbationSeries)_

## As a physical constant
 {#AsAPhysicalConstant}

The original __Planck constant__ $h$ is a quantum of [[action functional|action]]. It may be illustrated in the case of the [[electromagnetic field]] by the fact that each of its [[quanta]] -- a [[photon]] -- carries an [[energy]] $E$ that is fixed by its [[frequency]] (cycles per second) $\nu$ according to the relation $E = h\nu$. Thus, the energy emitted by a [[laser]] beam of fixed frequency $\nu$ is an integer multiple $n h \nu$ of a packet of energy $h\nu$, where $n$ is the number of photons emitted.

As a fundamental physical constant, $h$ has dimension $(mass)(length)^2(time)^{-1}$. In meter-kilogram-second (MKS) [[physical units|units]], its value is exactly

$$h = 6.62607015 \cdot 10^{-34} m^2 kg / s$$

by the SI definition of the kilogram.

The *reduced Planck constant* or *Dirac constant* [[Dirac 1933](#Dirac33) [p. 208](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/1933.pdf#page=235)]

$$
  \hbar \,\coloneqq\, h/2\pi
$$

is the proportionality constant that relates energy (of a photon) to angular frequency $\omega$ (radians per second as opposed to cycles per second), so that $E = \hbar \omega$.

## In geometric quantization
 {#InGeometricQuantization}

### Basic definition
 {#BasicDefinition}

The step of [[prequantization]] is about refining data in ([[differential cohomology|differential]]) real [[cohomology]] to ([[differential cohomology|differential]]) [[integral cohomology]]. Often this is understood in terms of the canonical inclusion

$$
  \mathbb{Z} \hookrightarrow \mathbb{R}
$$

of the [[integers]] as an addiditve [[subgroup]] of the [[real numbers]]. But since strictly speaking what appears in [[physics]] is the [[real line]] on which a [[unit]] is chosen as part of the identification of mathematical formalism with physical reality, one should really consider _all_ possible additive group homomorphisms $\mathbb{Z}\to \mathbb{R}$. These are parameterized by

$$
  h \in (\mathbb{R}- \{0\}) \hookrightarrow \mathbb{R}
$$

$$
  (-)\cdot h \;\colon\; \mathbb{Z} \longrightarrow \mathbb{R}
$$

and this "physical [[unit]]" $h$ is what is called _Planck's constant_.

In particular the induced [[circle group]] is identified as the [[quotient]] of $\mathbb{R}$ by $h \mathbb{Z}$, in this sense

$$
  U(1) \simeq \mathbb{R}/h \mathbb{Z}
$$

and under this identification its [[quotient]] map is expressed in terms of the [[exponential function]] $\exp \colon z \mapsto  \sum_{k = 0}^\infty \frac{z^k}{k!} \in \mathbb{C}$ as

$$
  \exp\big(
    2 \pi \tfrac{\mathrm{i}}{h}(-)
  \big)
  \;=\;
  \exp\big(
    \tfrac{\mathrm{i}}{\hbar} (-)
  \big)
  \;\colon\; \mathbb{R} \longrightarrow U(1)
  \,,
$$

where

\[
  \label{ReducedPlanckViaPrequantization}
  \hbar \,\coloneqq\, h/2\pi
\]

is the "reduced Planck constant", or *Dirac constant* [[Dirac 1933](#Dirac33) [p. 208](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/1933.pdf#page=235)].

The resulting [[short exact sequence]] is the real [[exponential exact sequence]]

$$
  0
   \to
  \mathbb{Z}
    \longrightarrow
  \mathbb{R}
    \stackrel{\exp\left(\tfrac{i}{\hbar}(-)\right)}{\longrightarrow}
  U(1)
    \to
  0
  \,.
$$

This is the source of the ubiquity of the expression  $\exp(\tfrac{i}{\hbar} (-))$ in [[quantum physics]], say in the [[path integral]], where the exponentiated [[action functional]] appears as $\exp(\tfrac{i}{\hbar} S)$.

### In relation to the symplectic form

In the context of [[geometric quantization]] Planck's constant appears as the inverse scale of the [[symplectic form]].

For instance in the simple case that [[phase space]] is $T^* \mathbb{R} \simeq \mathbb{R}^2$ with standard coordinates $\{p,q\}$, then the normalization of the symplectic form $\sim d p \wedge dq$ actually needed in physics is

$$
  \omega = \frac{1}{\hbar} d p \wedge d q
  \,.
$$

This is because after [[geometric quantization]] of this form the [[observables]] will obey

$$
  [\hat q, \hat p] = i (\omega_{p,q})^{-1}
$$

and this is supposed to be

$$
  \cdots = i \hbar
  \,.
$$

Accordingly, it follows that if $(E, \nabla)$ is a [[prequantum line bundle]] for $\omega$, then its $k$-fold [[tensor product]] with itself, for $k \in \mathbb{N}$, is a line bundle $(E^{\otimes k}, \nabla_k)$ with [[curvature]] $k \omega$. By the above this corresponds to rescaling

$$
  \hbar \to \hbar / k
  \,.
$$

This implies in particular

1. a global rescaling of the [[periods]] of the symplectic form may be absorbed in a rescaling of Planck's constant, see at _[[geometric quantization of non-integral forms]]_;

1. for $(E, \nabla)$ a given [[prequantum line bundle]] the limit of the tensor powers $(E^{\otimes k}, \nabla_k)$ as $k$ tends to [[infinity]] roughly corresponds to taking a [[classical limit]]. See also ([Donaldson 00](#Donaldson00)).

### Examples

In [[Chern-Simons theory]] Planck's constant corresponds to the inverse _level_ of the theory, hence the inverse of the [[characteristic class]] that defines the theory, regarded as an element in $\mathbb{Z}$.

Similarly for [[schreiber:infinity-Chern-Simons theory]].
For instance ordinary [[spin group]] Chern-Simons theory may be taken to have as the fundamental value $\hbar = 2$, because the [[first Pontryagin class]] that defines the theory is divisible by 2, the [[prequantum circle n-bundle|prequantum 3-bundle]] that defines the theory of the [[moduli stack]] of $Spin$-[[principal connections]] is

$$
  \tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}
  \,.
$$

Similarly for 7-dimensional [[String 2-group]] [[schreiber:infinity-Chern-Simons theory]] the fundamental value is $\hbar = 6$, with the extended Lagrangian being

$$
  \tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}
  \,.
$$

See at _[[higher geometric quantization]]_ for more on this.

## In perturbative quantization
 {#InFormalDeformationQuantization}

In [[perturbative quantum field theory]] Planck's constant (together with the [[coupling constant]], which indicates the strength of [[interactions]]) is regarded as tiny, in fact as [[infinitesimal]], in that all [[observables]] are expressed as (generally non-converging, [[asymptotic series|asymptotic]]) [[formal power series]] in the coupling constant.

This is explicitly realized by [[formal deformation quantization]], which regards [[quantization]] as as deformation of the classical [[algebra of observables]] to a non-commutative algebra on [[formal power series]] with [[coefficients]] the original observables.

## In perturbative quantum field theory

In [[perturbative quantum field theory]] the [[scattering amplitudes]] in the [[S-matrix]] are expressed as [[formal power series]] in (the [[coupling constant]] and) in [[Planck's constant]] $\hbar$. This formal power series may be expressed as a formal sum of contributions labeled by [[Feynman diagrams]]. The _loop order_ refers to something like the "number of loops" of [[edges]] in the [[Feynman diagram]] that contibutes to a given [[scattering amplitude]]. It turns out that the loop order corresponds to the order in $\hbar$ that is contributed by this diagram (see [below](#RelationToPowersInPlancksConstant)).

Therefore contributions of graphs at zero without loops (these are [[trees]], and hence these contributions are referred to as being at "tree level") correspond to the limit of [[classical field theory]] with $\hbar \to 0$. Indeed tree level Feynman diagrams yield [[perturbation theory|perturbative]] solutions of the [[classical field theory|classical]] [[equations of motion]] (see [Helling](#Helling)).

Most predictions of the [[standard model of particle physics]] have very good agreement with [[experiment]] already to very low loop order, first or second; inclusion of third loop order is used (at least in [[QCD]]) to constrain theoretical uncertainties of the result (see [Cacciari 05, slide 5](#Cacciari05), e.g. in [[Higgs field]] computation, see [ADDHM 15](#ADDHM15)).
In rare cases higher loop orders are used (for instance in the computation of the [[anomalous magnetic moments]] [AHKN 12](#AHKN12), but this is not a scattering experiment).

This usefulness of low loop order is forturnate because

1. the [[S-matrix]] [[formal power series]] for all [[theory (physics)|theories]] of interest has _vanishing_ [[radius of convergence]] ([Dyson 52](perturbation+theory#Dyson52)), hence is at best an [[asymptotic series]] for which the [[sum]] of more than some low order terms is meaningless;

1. the computational effort increases immensely with loop order.

## In the Feynman perturbation series
 {#InFeynmanPerturbationSeries}

In the computation of [[scattering amplitudes]] for [[field (physics)|fields]]/[[particles]] via [[perturbative quantum field theory]] the [[scattering matrix]] ([[Feynman perturbation series]]) is a [[formal power series]] in (the [[coupling constant]] and) [[Planck's constant]] $\hbar$ whose contributions may be labeled by [[Feynman diagrams]]. Each Feynman diagram $\Gamma$ is a finite labeled [[graph]], and the order in $\hbar$ to which this graph contributes is

$$
  \hbar^{ E(\Gamma) - V(\Gamma) }
$$

where

1. $V(\Gamma) \in \mathbb{N}$ is the number of [[vertices]] of the graph

1. $E(\Gamma) \in \mathbb{N}$ is the number of [[edges]] in the graph.

This comes about (see at _[S-matrix -- Feynman diagrams and Renormalization](S-matrix#ExistenceAndRenormalization)_ for details) because

1. the explicit $\hbar$-dependence of the [[S-matrix]] is

   $$
     S\left(\tfrac{g}{\hbar} L_{int} \right)
     =
     \underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )
   $$

1. the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is

   $$
     T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)}  \right) ( L_{int} \otimes L_{int} )
     \,,
   $$

where $\omega_F$ denotes the [[Feynman propagator]] and $\phi(x)$ the field observable at point $x$ (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as [[scalar fields]], because this is all that affects the counting of the $\hbar$ powers).

The resulting terms of the S-matrix series are thus labeled by

1. the number of factors of the [[interaction]] $L_{int}$, these are the [[vertices]] of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$

1. the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$.

Now the formula for the [[Euler characteristic of planar graphs]] says that the number of regions in a plane that are encircled by edges, the _faces_ here thought of as the number of "loops", is

$$
  L(\Gamma) =  1 + E(\Gamma) - V(\Gamma)
  \,.
$$

Hence a planar Feynman diagram $\Gamma$ contributes with the "[[loop order]]" $L(\Gamma)$ as

$$
  \hbar^{L(\Gamma)-1}
  \,.
$$

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

$$
  \langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right)
  \vert \psi_{in} \rangle
  \,,
$$

where

$$
  \vert \psi_{in}\rangle
    \propto
  \tfrac{1}{\sqrt{\hbar^{n_{in}}}}
  \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle
$$

is a state of $n_{in}$ free field quanta and similarly

$$
  \vert \psi_{out}\rangle
    \propto
  \tfrac{1}{\sqrt{\hbar^{n_{out}}}}
  \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle
$$

is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$.

This means that an actual [[scattering amplitude]] given by a [[Feynman diagram]] $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as

$$
  \hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 }
  \,.
$$

(For the analogous discussion of the dependence on the actual [[quantum observables]] on $\hbar$ given by [[Bogoliubov's formula]], see [there](Bogoliubov's+formula#PowersInPlancksConstant).)

## History
 {#History}

[[Max Planck]] introduced the constant named after him in the discussion of [[black body radiation]]. In [[classical field theory]] black body radiation comes out completely wrong ("[[ultraviolet catastrophe]]"). Planck fixed this in an ad-hoc but succesful manner by postulating that [[energy]]/[[frequency]] of the [[harmonic oscillators]] in the black body ([[atoms]], [[molecules]]) is quantized in units measured by some quantum of action. Eventually this ad hoc postulate led to a change of the foundations of physics from [[classical physics]] to [[quantum physics]], which now predicts the quantization of energy/frequency from more conceptual, fundamental principles.

## Related concepts

* [[theory (physics)]]

* [[quantization]]

* [[coupling constant]]

[[!include fundamental scales -- contents]]

## References
 {#References}

Planck's constant originates in:

* [[Max Planck]] (transl. M. Martius), [p. 164](https://www.gutenberg.org/files/40030/40030-pdf.pdf#page=178) in: *The Theory of Heat Radiation* (1914) [[pdf](https://www.gutenberg.org/files/40030/40030-pdf.pdf)]

\begin{imagefromfile}
    "file_name": "DiracPrinciples-DiracConstant.jpg",
    "float": "right",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [Dirac 1930, p. 87](#Dirac30))"
\end{imagefromfile}

The notation $\hbar$ for the reduced Planck constant (eq:ReducedPlanckViaPrequantization) is  due to:

* {#Dirac30} [[Paul Dirac]], *[[The Principles of Quantum Mechanics]]*, International series of monographs on physics, Oxford University Press (1930) [[ISBN:9780198520115](https://global.oup.com/academic/product/the-principles-of-quantum-mechanics-9780198520115)]

\begin{imagefromfile}
    "file_name": "DiracIntroducingHBar.jpg",
    "float": "right",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [Dirac 1933, p. 208](#Dirac33))"
\end{imagefromfile}

but Dirac did not stick to this notation in his publications, until:

* {#Dirac33} [[Paul A. M. Dirac]], [p. 208](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/1933.pdf#page=235) in: *Théorie du Positron*, Proceedings of *Structure et propriétés des noyaux atomiques*, [Solvay](http://www.solvayinstitutes.be/) Conference (1933) 203-212 [full:[pdf](http://www.solvayinstitutes.be/pdf/Proceedings_Physics/1933.pdf)]

Further discussion may be found in any monograph on [[quantum mechanics]] (see references [there](quantum+mechanics#References)).

See also:

* {#Donaldson00} [[Simon Donaldson]], _Planck's constant in complex and almost-complex geometry_, XIIIth International Congress on Mathematical Physics (London,
2000), 63–72, Int. Press, Boston, MA, 2001

* Wikipedia, _[Planck's constant](https://en.wikipedia.org/wiki/Planck_constant)_

[[!redirects Planck constant]]
[[!redirects Planck's constant]]
[[!redirects Planck’s constant]]
[[!redirects Planck\'s constant]]

[[!redirects reduced Planck constant]]
[[!redirects reduced Planck's constant]]
[[!redirects reduced Planck’s constant]]
[[!redirects reduced Planck\'s constant]]

[[!redirects Dirac constant]]
[[!redirects Dirac's constant]]
[[!redirects Dirac’s constant]]
[[!redirects Dirac\'s constant]]


plane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

The **plane** is the [[Cartesian space]] $\mathbb{R}^2$. This is naturally a [[topological space]], a [[manifold]], and a [[smooth manifold]].  If we take one of the axes (traditionally the second) to be imaginary, then this __real plane__ may be identified with the [[complex plane]] $\mathbb{C}^1$.  As a stage for [[Euclidean geometry]], it may be called the __Cartesian plane__, __Euclidean plane__, or __coordinate plane__.

## Related concepts

* [[point]]

* [[real line]]

* **plane**, [[surface]], [[hyperplane]]

* [[slope of a line]]

* [[area]]

## References

For discussion of the plane via [[axioms]] for the points and lines in it ([[synthetic geometry]]) see the references at _[[Euclidean geometry]]_.

[[!redirects plane]]
[[!redirects planes]]

[[!redirects real plane]]
[[!redirects real planes]]

[[!redirects coordinate plane]]
[[!redirects coordinate planes]]

[[!redirects Cartesian plane]]
[[!redirects Cartesian planes]]

[[!redirects Euclidean plane]]
[[!redirects Euclidean planes]]


plane geometry

The [[Euclidean geometry]] of the [[plane]].


plane graph

\tableofcontents

## Idea

The term "plane graph" is used (with a slight bit of ambiguity) to refer to a crossing-free drawing of a [[graph]], or more abstractly to an [[embedding]] of a graph into a [[surface]] of [[genus of a surface|genus]] 0 (also called a _planar map_), or even more abstractly to a graph equipped with a cyclic ordering of the half-edges around each vertex which determines such an embedding up to isomorphism (see: [[combinatorial map]]).
In any case, when the terminology is employed, it is usually meant to be distinguished from a "[[planar graph]]" as being a [[structure]] rather than a [[property]] (in the sense of [[stuff, structures, and properties]]), although sometimes the word "planar graph" is also used in the structure sense.

## Related pages

* [[planar graph]]
* [[four-colour theorem]]
* [[link diagram]]
* [[string diagram]]
* [[topological map]]

[[!redirects plane map]]
[[!redirects planar map]]

## References

* Sergei K. Lando and Alexander K. Zvonkin, _Graphs on Surfaces and Their Applications_, Springer, 2004.

* [[Phillipe Flajolet]], [[Robert Sedgewick]]: _Analytic Combinatorics_. First Edition. Cambridge University Press. 2009. (See Chapter VII. 8.2)
{#FlajoletSedgewick}


plane wave

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ for the [[Cartesian space]] of [[dimension]] $n$. Then a _plane wave_ on $\mathbb{R}^n$ is a [[function]]

$$
  \mathbb{R}^n \longrightarrow \mathbb{C}
$$

given by the [[exponential function]] with [[complex number|complex]] argument of the form

$$
  \vec x \;\mapsto\; A e^{2 \pi i \vec k \cdot \vec x}
  \,.
$$

Here $\vec k \in \mathbb{R}^n$ is the _[[wave vector]]_ of this plane wave (and $A \in \mathbb{C}$ is its [[amplitude]]).

In [[Fourier analysis]] over [[Cartesian space]], the [[Fourier transform]] expresses every [[function with rapidly decreasing partial derivatives]] as a [[superposition]] of plane waves.

If here $\mathbb{R}^n \simeq \mathbb{R}^{p,1}$ is identified with [[Minkowski spacetime]] with canonical [[coordinates]] labeled $(x^0, x^1, \cdots x^p)$, then  the 0-component of the [[wave vector]]

$$
  \nu \coloneqq k_0
$$

is called the _[[frequency]]_ of the wave (in this chosen [[coordinate system]]). If in this situation the [[wave vector]]  satisfies $k_\mu k^\mu = 0$, then the plane wave is a solution to the [[wave equation]] on Minkowski spacetime. If more generally it satisfies $k_\mu k^\mu + m^2 = 0$ for some $m^2 \in \mathbb{R}$, the it is a solution to the [[Klein-Gordon equation]] on Minkowski spacetime.

[[!include plane waves -- table]]

## Related concepts

* [[wave]]

* [[wave polarization]]

* [[Fourier transform]]

* [[wave function]]

## References

See also

* Wikipedia, _[Plane wave](https://en.wikipedia.org/wiki/Plane_wave)_

[[!redirects plane waves]]

[[!redirects planewave]]
[[!redirects planewabes]]


plane waves -- table

**[[plane waves]] on [[Minkowski spacetime]]**

$$
  \array{
    \mathbb{R}^{p,1}
    &\overset{\psi_k}{\longrightarrow}&
    \mathbb{C}
    \\
    x  &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right)
    \\
    (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right)
    \\
    (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right)
  }
$$

| symbol | name |
|--------|------|
| $c$ | [[speed of light]] |
| $\hbar$ | [[Planck's constant]] |
| $\,$ | $\,$ |
| $m$ | [[mass]] |
| $\frac{\hbar}{m c}$ | [[Compton wavelength]] |
| $\,$ |  $\,$ |
| $k$, $\vec k$ | [[wave vector]] |
| $\lambda = 2\pi/{\vert \vec k \vert}$  | [[wave length]] |
| ${\vert \vec k \vert} = 2\pi/\lambda$| [[wave number]] |
| $\omega \coloneqq  k^0 c = -k_0 c = 2\pi \nu$  | [[angular frequency]] |
| $\nu = \omega / 2 \pi$ | [[frequency]] |
| $p = \hbar k$, $\vec p = \hbar \vec k$ | [[momentum]] |
| $E = \hbar \omega$ | [[energy]] |
| $\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }$ | [[Klein-Gordon equation|Klein-Gordon]] [[dispersion relation]] |
| $E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }$ | [[energy-momentum relation]] |


Planet Math

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

_[Planet Math](http://planetmath.org/)_ was a virtual community that aimed to help make mathematical knowledge more accessible. PlanetMath's content was created collaboratively: the main feature is the
[mathematics encyclopedia](http://planetmath.org/encyclopedia), a Wiki on [[mathematics]] with entries written and reviewed by members.

The project appears to have gone inactive not later than 2013,
the year in which any nontrivial editing activity seems to have stopped.

[[!redirects PlanetMath]]

category: reference


plasma

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

#Contents#
* table of contents
{:toc}

## Idea

A _plasma_ is a [[phase of matter|phase]] of [[baryon|baryonic]] [[matter]] at [[temperature]] high enough that [[electrons]] and [[nuclei]] do not form [[bound states]] ([[atoms]]) but each propagate freely.

## Related concepts

* [[quark-gluon plasma]]

## References

* Wikipedia, _Plasma_(physics)_

[[!redirects plasmas]]


plasmon

## References

See also:

* Wikipedia, _[Plasmon](https://en.m.wikipedia.org/wiki/Plasmon)_

[[!redirects plasmons]]


Plato

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

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

## Related entries

* [[Parmenides dialogue]]

* [[Timaeus dialogue]]

* [[Republic dialogue]]

* [[Gorgias dialogue]]

* [[doctrine of ideas]]

* [[Socrates]]

## References

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Plato](Lectures%20on%20the%20History%20of%20Philosophy#Plato)_

[[!redirects Plato's]]

category: people


Platonic 2-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Platonic 2-group_ ([Epa 10](#Epa10)) is a [[2-group]] [[infinity-group extension|higher extension]] of a "Platonic group" in the [[ADE classification]], i.e. of a [[finite subgroup of SO(3)]]; or else of its [[double cover]], hence of [[finite subgroup of SU(2)]].

[[!include ADE -- table]]

## Properties

### Relation to the string 2-group
 {#RelationToTheString2Group}

+-- {: .num_prop #UniversalPlatonic2GroupIsRestrictionOfString2}
###### Proposition

The universal Platonic 2-group extensions $\mathcal{G}_{uni}[i]$ of  $G_{ADE}$ [[finite subgroups of SU(2)]] by $U(1)$ ([Epa-Ganter 16, def.2.11](#EpaGanter16)) are equivalently the restrictions of the [[string 2-group]] extension $String(SU(2)) \to SU(2)$ of $SU(2)$, hence the classifying [[cocycle]] is the restriction of the smooth [[second Chern class]]:

$$
  \array{
    \mathcal{G}_{uni}[i]
      &\longrightarrow&
    \mathbf{B}G_{ADE}
      &\longrightarrow&
    \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert}
    \\
    \big\downarrow
      &&
    \big\downarrow
      &&
    \big\downarrow^{\mathrlap{\mathbf{B}^3 i}}
    \\
    String(SU(2))
      &\longrightarrow&
    \mathbf{B} SU(2)
      &\underset{\mathbf{c}_2}{\longrightarrow}&
    \mathbf{B}^3 U(1)
  }
  \,,
$$

where $i$ is the canonical inclusion of [[roots of unity]], $i: \mathbb{Z}/{|G|} \hookrightarrow U(1)$.

=--

([Epa-Ganter 16, prop. 4.1](#EpaGanter16))

See also at _[[finite subgroup of SU(2)]] -- [Discrete torsion](finite+rotation+group#DiscreteTorsion)_

## References

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([[Epa_Platonic2Groups.pdf:file]])

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))

[[!redirects Platonic 2-groups]]

[[!redirects platonic 2-group]]
[[!redirects platonic 2-groups]]


Platonic solid

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

#Contents#
* table of contents
{:toc}

## Idea

The five [[regular polyhedron|regular]] [[convex set|convex]] [[polyhedron]] in 3-[[dimension|dimensional]] [[Cartesian space]]:

[[tetrahedron]], [[cube]], [[octahedron]], [[dodecahedron]], [[icosahedron]]

## Platonic symmetry groups
 {#PlatonicSymmetryGroups}

Regarding a Platonic solid, determined uniquely up to [[isometry]] on $\mathbb{R}^3$ as a regular convex [[polyhedron]], as a [[metric space|metric]] subspace $S$ of $\mathbb{R}^3$. Then is [[symmetry]] [[group]] may be defined as the group of [[isometries]] of $S$.

The groups arising this way are called the groups of ADE-type:

[[!include ADE -- table]]

## Related concepts

* [[Johannes Kepler]], _[[Music of the Spheres]]_

* Plato's [[Timaeus]]

* [[120-cell]]

## References

The Platonic solids are named after their discussion in

* [[Plato]], _[[Timaeus dialogue]]_

Their construction and the [[proof]] that there is exactly five of them appears in

* [[Euclid]], [Book XIII](http://aleph0.clarku.edu/~djoyce/elements/bookXIII/bookXIII.html) of the _[[Elements]]_, $\sim$300 BC

Modern textbook accounts include

* {#Lamotke86} [[Klaus Lamotke]], section 1 of _Regular Solids and Isolated Singularities_, Vieweg 1986

* {#Rees05} [[Elmer Rees]], from p. 23 (32 of 124) on in _Notes on Geometry_, Springer 2005

See also

* Wikipedia, _[Platonic solid](http://en.wikipedia.org/wiki/Platonic_solid)_

[[!redirects Platonic solids]]

[[!redirects Platonic solids]]

[[!redirects platonic solid]]
[[!redirects platonic solids]]


Platonism

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

#Contents#
* table of contents
{:toc}

## Idea

**Platonism** is a position in [[philosophy]] committed to the existence of abstract entities that derives from the theory of ideas of the Greek philosopher [[Plato]].

Since this attitude provides a convenient ontological underpinning for mathematical concepts, platonism is one of the major schools of thought in the [[philosophy of mathematics]] as well as in spontaneous affinity with the practice of 'every-day mathematics' with its feel of objectivity and free availability of non-constructive classical modes of reasoning.

Famous mathematicians like [[Georg Cantor]], [[Gottlob Frege]], [[Kurt Gödel]] or more recently [[Alain Connes]] and [[Roger Penrose]] have advocated views akin to platonism.

## Related entries

* [[intuitionism]]

* [[formalism]]

* [[continuum hypothesis]]

* [[mathematical object]]

## Link

* Wikipedia, _[Philosophy of mathematics -- Platonism](http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism)_

## References

A classical reference is

* [[Paul Bernays]] , _Sur le platonisme dans les mathématiques_ , L'enseignement mathématique **34** (1935) pp.52-69. ([pdf](http://www.e-periodica.ch/cntmng?pid=ens-001:1935:34::14))

Other texts include

* Mark Balaguer, _Platonism & Anti-Platonism in Mathematics_ , Oxford University Press, 1998.

* [[Barry Mazur]], _Mathematical Platonism and its Opposites_, Newsletter of the European Mathematical Society 68 (2008), 19–21. [PDF](https://ems.press/content/serial-issue-files/13674), [PDF](https://dash.harvard.edu/bitstream/handle/1/2623649/Mazur_Platonism.pdf), ([pdf](http://www.math.harvard.edu/~mazur/papers/plato4.pdf))

[[!redirects platonism]]
[[!redirects platonic]]
[[!redirects Platonic]]


Plebanski formulation of gravity

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

#Contents#
* table of contents
{:toc}

## Idea

A certain formulation of the theory of [[gravity]].

## References

The original article is

* Plebanski, _On the separation of Einsteinian substructures_, J. Math. Phys.  18, 2511 (1977) ([doi](http://dx.doi.org/10.1063/1.523215))

An account of the general formalism is in

* R. Capovilla, T. Jacobson, J. Dell, L. Mason, _Selfdual 2-forms and gravity_ Classical and Quantum Gravity 8, 41 (1991) ([doi](http://dx.doi.org/10.1088/0264-9381/8/1/009)).

A discussion of practical computations in this formalism is in

* Kirill Krasnov, _Plebanski Formulation of General Relativity: A Practical Introduction_ ([arXiv](http://arxiv.org/abs/0904.0423))


plethory

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

#Contents#
* table of contents
{:toc}

## Idea

The concept **plethory** derives from that of [[plethysm]] ([Borger-Wieland 05](#BW05), p. 2), a certain form of composition in the theory of [[symmetric functions]].

In what follows, all [[rings]] are assumed to be [[commutative ring|commutative]] and [[unital ring|unital]].

Since the ring $\mathbb{Z}[x]$ of [[integer]]-[[coefficient]] [[polynomials]] in one [[variable]] is the [[free ring]] on $1$, we have an [[isomorphism]] $R \stackrel \sim \to \mathsf{Ring}(\mathbb{Z}[x], R)$ which sends an element $r \in R$ to [[evaluation]] at $r$, $p(x) \mapsto p(r)$. If we let $R$ also be $\mathbb{Z}[x]$, this gives exactly "substitution" or "composition" of polynomials. A **plethory** is a ring which carries a substitution structure. The ring $\Lambda$ of [[symmetric functions]] is another example.

The data of a plethory is also what is needed to represent a "natural operation" on the category of rings. For example $\mathsf{Ring}(\mathbb{Z}[x],-)$ is the identity functor on $\mathsf{Ring}$, and $\mathsf{Ring}(\Lambda,-)$ sends every ring to its [[ring of Witt vectors]].

## Definitions

A [[biring]] is a (commutative) ring object in $\mathsf{Ring}^{op}$. The category of birings is $\mathsf{Ring}(\mathsf{Ring}^{op})^{op}$. The extra op ensures that a biring homomorphism is a ring homomorphism, not a reversed ring homomorphism.

The category of birings is equivalent to $\mathsf{LAdj}(\mathsf{Ring},\mathsf{Ring})$, the category of left adjoints $\mathsf{Ring} \to \mathsf{Ring}$, and also to $\mathsf{RAdj}(\mathsf{Ring},\mathsf{Ring})^{op}$, opposite of the category of right adjoints $\mathsf{Ring} \to \mathsf{Ring}$. Under this equivalence, the category of birings inherits a monoidal structure induced from endofunctor composition. The monoidal product is called the *substitution product*, denoted by $\odot$. The unit object is the ring of polynomials $\mathbb{Z}[x]$. A generators-and-relations description of $\odot$ can be found in ([TW70](#TW70)).

A **plethory** is a monoid in $(\mathsf{Biring}, \odot, \mathbb{Z}[x])$. Equivalently, a plethory is a right adjoint comonad $\mathsf{Ring} \to \mathsf{Ring}$.  Equivalently, a plethory is a left adjoint monad $\mathsf{Ring} \to \mathsf{Ring}$.

### Plethory over a ring

For any (commutative) ring $k$, a $k$-**plethory** is a [[monoid object]] in the [[monoidal category]] of $k$-$k$-[[biring|birings]], with respect to the composition product. That is, it is a biring $P$ equipped with an associative map of birings $\circ: P \odot_k P \to P$ and unit $k \langle e \rangle \to P$.

In other words,

>a $k$-plethory is a commutative k-algebra together with a comonad structure on the covariant functor it represents, much as a k-algebra is the same as a $k$-module that represents a comonad. So, just as a $k$-algebra is exactly the structure that knows how to act on a $k$-module, a $k$-plethory is the structure that knows how to act on a commutative $k$-algebra. ([BW05](#BW05))

## Related concepts

* [[biring]]

* [[Tall-Wraith monoid]]

* [[2-plethory]]

##References

The idea was introduced here, where it was called a "biring triple":

* {#TW70} D. Tall and [[Gavin Wraith|G. Wraith]], Representable functors and operations on rings, _Proc. London Math. Soc._ **3** (1970), 619--643.

The term "plethory" was introduced here:

* {#BW05} [[James Borger]], [[Ben Wieland]],  Plethystic algebra, _Advances in Mathematics_ **194** (2005), 246--283.  ([web](http://wwwmaths.anu.edu.au/~borger/papers/03/paper03.html))

* {#BMT21} [[John Baez]], [[Joe Moeller]], [[Todd Trimble]], Schur functors and categorified plethysm, [arXiv:2106.00190](https://arxiv.org/abs/2106.00190)

[[!redirects plethories]]
[[!redirects biring triple]]


plethysm

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

\tableofcontents

## Overview

The composite of [[Schur functors]] is again a Schur functor.  The process of composing Schur functors is known as **plethysm**, especially in situations where we want an explicit "formula" for how the Schur functors compose.

For example, the $n^{th}$ [[exterior power]] functor $\Lambda^n$ is a Schur functor, as is the $n^{th}$ [[symmetric power]] functor $S^n$.  So, we might want to describe the composite

$$  V \mapsto \Lambda^2 S^2(V) $$

as a direct sum of Schur functors coming from [[Young diagrams]].    This is the sort of problem people study when they talk about "plethysm".

## Plethysm product and symmetric operads

Let $S$ be the skeletal category of finite sets and bijections and $C$ a symmetric additive monoidal category with monoidal product $\otimes$ and unit object $\mathbf{1}$. Objects in the category of contravariant functors $C^{S^{op}}$ can be descibed as collections $M = \{M(n), n\geq 0\}$ of objects $M(n)$ in $C$ with action of a symmetric group $\Sigma_n$ on $n$ letters. The category $C^{S^{op}}$ acts on $C$ by the polynomial functors
$$
M : V \mapsto \oplus_{n\geq 0} M(n)\otimes_{\Sigma_n} V^{\otimes n}
$$
The composition of such functors defines a monoidal product on $C^{S^{op}}$ called the __plethysm product__. This way we obtain a monoidal category. The monoids in that category are the (symmetric) $C$-[[operads]].

## Related concepts

* [[plethory]]

## History and references

In Richard P. Stanley's book _Enumerative Combinatorics_, he discusses the origin of the term 'plethysm' in Volume 2, Appendix 2.  He says that the term was introduced in

* [[D. E.  Littlewood]], _Invariant theory, tensors and group characters_, Philos. Trans. Roy. Soc. London. Ser. A. **239**, (1944), 305--365.

The term 'plethysm' was suggested to Littlewood by M. L. Clark after the Greek word _plethysmos_, or πληθυσμός, which means 'multiplication' in modern Greek (though apparently the meaning goes back to ancient Greek).  The related term _plethys_ in Greek means 'a big number' or 'a throng', and this in turn comes from the Greek verb _plethein_, which means 'to be full', 'to increase', 'to fill', etc.

Some aspects of plethysm appear (partly through exercises) in the textbook W. Fulton, J. Harris, _Representation theory_.

* A. M. Garsia, G. Tesler, _Plethystic formulas for Macdonald $q, t$-Kostka coefficients_, Advances in Math. __123__ (1996) 144–222, [MR1420484](http://www.ams.org/mathscinet-getitem?mr=1420484); A. M. Garsia, J. Remmel, _Plethystic formulas and positivity for $q,t$-Kostka coefficients_, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 245–262, Progr. Math. __161__, Birkhäuser  1998, [MR99j:05189d](http://www.ams.org/mathscinet-getitem?mr=1627327)

For an application in the study of [[characteristic classes]] see

* Dragutin Svrtan, _New plethysm operation, Chern characters of exterior and symmetric powers with applications to Stiefel-Whitney classes of Grassmannians_, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).  Theoret. Comput. Sci.  __117__ (1993),  no. 1-2, 289--301, <http://dx.doi.org/10.1016/0304-3975(93)90320-S>

An application in a counting problem:

*  Thomas Kahle, Mateusz Michalek, _Plethysm and lattice point counting_ [arxiv/1408.5708](http://arxiv.org/abs/1408.5708)


plot

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[generalized space]] $X$ conceived as a [[concrete sheaf]] $X \in Sh(Charts)_{\sharp_1} \hookrightarrow Sh(Charts)$ on a [[site]] $Charts$ of local model spaces, a _plot_ of $X$ over a $U \in Charts$ is an [[element]] of $X(U) \in Set$, hence (under the [[Yoneda lemma]]) is a [[morphism]] $U \to X$ from that local model space to $X$.

## References

For [[diffeological spaces]] the term was coined in

* {#PIZ} [[Patrick Iglesias-Zemmour]], p. 26 of: _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [ ISBN:978-0-8218-9131-5](https://bookstore.ams.org/surv-185))

[[!redirects plots]]


plump ordinal

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

# Plump ordinals

* table of contents
{: toc}

## Idea

The [[ordinal numbers]] admit two [[inequalities]]: a strict one $\lt$ and a non-strict one $\le$.  When the ordinals are defined in [[material set theory]] in the von Neumann style as "the set of all smaller ordinals", these relations are identified with $\in$ (set [[membership]]) and $\subseteq$ ([[subset|containment]]).  And when ordinals are defined in a structural theory as certain well-ordered sets, the relation $\le$ refers to inclusion as an [[initial subset]] (a [[simulation]]) while $A\lt B$ refers to such an inclusion as the strict slice $\{ y | y \lt x \}$ of some element $x$ of $B$.

In [[classical mathematics]], $\lt$ and $\le$ are definable in terms of each other's negation: $A\lt B$ if and only if $\neg(B\le A)$, and similarly $A\le B$ if and only if $\neg(B\lt A)$.  However, in [[constructive mathematics]] this is no longer true (although $\leq$ can still be defined from $\lt$ in a more round-about way).  This phenomenon occurs for other ordered structures as well; for instance, for the [[real numbers]].  (For the most commonly used notions of real number, $\le$ is definable as the negation of $\gt$, but not conversely.)   In general, there are certain laws we may expect such a pair of inequalities to satisfy, such as:

* $x\le x$.
* Not $x \lt x$.
* If $x\le y\le z$, then $x\le z$.
* If $x\lt y$, then $x\le y$.
* If $x\lt y\lt z$, then $x\lt z$.
* If $x\lt y\le z$, then $x\lt z$.
* If $x\le y\lt z$, then $x\lt z$.

For the constructive ordinal numbers, these are all satisfied *except for the last one*.  In fact, it's easy to see that if $x\le y\lt z$ implies $x\lt z$ for all $x,y,z$, then [[excluded middle]] holds: consider $z = 2 = \{0,1\}$ and $y = 1 = \{0\}$ and $x = \{0 | P\}$ for some proposition $P$.

However, Paul Taylor [(Taylor)](#Taylor) realized that this last property can be recovered by essentially "restricting to the subclass of ordinals $z$ for which it holds hereditarily".  These are the *plump ordinals*.

## Definition

An [[ordinal]] $A$ is **plump** if

1. every ordinal $B\lt A$ is plump, and
2. whenever $C\le B\lt A$ and $C$ is a plump ordinal, then $C\lt A$.

It is not obvious that this definition is sound (i.e. non-circular), but it can be shown to be so based on a well-founded relation (other than $A$ itself).  It does, however, require the existence of [[power sets]], so it is not obviously sensible in [[predicative mathematics]] which denies the existence of those.

## Examples

$0 = \varnothing$ is trivially plump.

Since $0$ is plump, $1 = \mathcal{P}0 = \{0\}$ is also plump.

More generally, any subset of $1$ (which may be thought of as a [[truth value]]) is a plump ordinal.

But $2 = \{0,1\}$ (the set of classical truth values) with $0 \lt 1$ is plump if and only if excluded middle holds.

In contrast, $\Omega = \mathcal{P}1$ (the set of all truth values) with $0 \lt 1$ (as the only instance of ${\lt}$) is plump.  So $\Omega$ is the plump version of $2$, equal to $2$ iff excluded middle holds.

The plump version of $3$ may be quite large; it consists of all [[lower sets]] (downwards-closed collections) of truth values.  So it includes $0$, $1$, and $\Omega$, the other truth values between $0$ and $1$, the [[downset]] ${\downarrow}p$ (the lower set generated by $p$) of any given truth value $p$ (not just $1 = {\downarrow}0$ and $\Omega = {\downarrow}1$), the lower set generated by two not-necessarily-comparable truth values $p$ and $q$, etc.  But if excluded middle holds, then it\'s just $\{0,1,2\}$.

Generalizing most of these examples, if $\alpha$ is a plump ordinal, then the successor $\alpha^+$, consisting of all lower sets of $\alpha$, is also a plump ordinal.  This is bigger than $\alpha + 1 = \alpha \cup \{\alpha\}$, in that if $0 \lt \alpha$ and $\alpha^+ = \alpha + 1$, then excluded middle holds.

Any downwards-closed subset of a plump ordinal is also a plump ordinal.

## References

* [[Paul Taylor]], *Intuitionistic sets and ordinals*, Journal of Symbolic Logic, 61:705-744, 1996, available [here](http://www.paultaylor.eu/ordinals/#intso)

* See also Section 6.7 of [[Paul Taylor]], *Practical Foundations of Mathematics*, [here](http://www.paultaylor.eu/prafm/html/s67.html).

 {#Taylor}


pluralism

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

\tableofcontents

## Idea

Pluralism is a philosophical position which states that there are multiple equally valid approaches to a particular field. This contrasts with [[monism]], which states that there is only one valid approach to a particular field.

## Pluralism in the foundations of mathematics

In the context of the [[foundations of mathematics]], pluralism states that rather than singling out a single approach to the foundations of mathematics, such as [[ZFC]] or [[ETCS]], as the universal approach to the foundations of mathematics, each different approach are all regarded as valid approaches to the foundations of mathematics.

Approaches to the foundations of mathematics could be distinguished by its characterization of [[equality]] (whether as a proposition, a judgment, or as a type), [[equivalence]] (whether it is more fundamental than equality for collections), and [[propositions]]/[[logic]] (whether it is primitive or derived), which distinguishes [[material set theory]], [[structural set theory]], [[categorical set theory]], [[allegorical set theory]], [[higher order logic]], [[dependent type theory]], [[FOLDS]], [[class theory]], [[formal category theory]], [[preset theory]], and so forth, from each other.

There is another dimension in which approaches to the foundations of mathematics are distinguished, which is the strength of the theory; these concerns are orthogonal to the above concern regarding the characterization of equality, equivalence and propositions/logic: these distinguish [[predicative mathematics]], [[strongly predicative mathematics]], [[weakly predicative mathematics]], [[constructive mathematics]], [[classical mathematics]], [[set-level foundations]], [[higher foundations]], and so forth, from each other.

To show that these two dimensions are independent of each other, we give some examples. On one hand, one could have a [[material set theory]] in which all [[sets]] have a [[choice operator]]; on the other hand, one could merely have a material set theory which is strongly predicative. In addition, we could have a [[categorical set theory]] in which all sets have a [[choice operator]], and a categorical set theory which is strongly predicative. Formal category theory is usually considered to be [[higher foundations]], but set-level formal category theory could be considered by adding an [[axiom]] or rule that every category is a [[discrete category]].

There is also mathematics based in [[linear logic]]/[[linear dependent type theory]], as well as [[paraconsistent mathematics]], which from the pluralistic view are also true in their own ways.

## See also

* [[philosophy]]

* [[reductionism]]

* [[epistemology of mathematics]]

## References

* Justin Clarke-Doane (2022). *Mathematics and Metaphilosophy (Elements in the Philosophy of Mathematics)*. Cambridge: Cambridge University Press. ([doi:10.1017/9781108993937](https://doi.org/10.1017/9781108993937), [philsci-archive:20728](https://philsci-archive.pitt.edu/20728/))

* [[Mirna Džamonja]], *A New Foundational Crisis in Mathematics, Is It Really Happening?*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_11](https://doi.org/10.1007/978-3-030-15655-8_11), [arXiv:1802.06221](https://arxiv.org/abs/1802.06221)]

* [[Michèle Friend]], *Varieties of Pluralism and Objectivity in Mathematics*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_15](https://doi.org/10.1007/978-3-030-15655-8_15)]

* [[Michèle Friend]], *Pluralism in Mathematics: A New Position in Philosophy of Mathematics*, Logic, Epistemology and the Unity of Science, Springer, 2014.  [ISBN 978-94-007-7057-7, [doi:10.1007/978-94-007-7058-4](https://doi.org/10.1007/978-94-007-7058-4), [excerpt](https://content.e-bookshelf.de/media/reading/L-2680820-cee701bdbb.pdf)]

On pluralism in [[set theory]]:

* Neil Barton (2017). *Independence and Ignorance: How agnotology informs set-theoretic pluralism.* ([philsci-archive:14636](https://philsci-archive.pitt.edu/14636/)

## External links

* Wikipedia, *[Epistemological pluralism](https://en.wikipedia.org/wiki/Epistemological_pluralism)*


plus construction

The terminology 'plus construction' is used for three very different types of construction.

 *  The version which sends a presheaf to a separated presheaf to a sheaf is described at _[[plus construction on presheaves]]_ and _[[sheaf]]_.

*  The process of killing a perfect normal subgroup of the fundamental group of a space is described under [[Quillen plus construction]].

* The assignment, due to [[John Baez]] and [[James Dolan]], $O \mapsto O^+$ sending a [[operad]] (with colours $S$) to a new operad whose algebras are $S$-coloured operads equipped with an operad morphism to $O$ (i.e. [[slice category|slices]] over $O$).

## References

* {#BaezDolan97} [[John Baez]], [[James Dolan]], Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, _Adv. Math._ **135** (1998), 145--206. ([arXiv:q-alg/9702014](http://arxiv.org/abs/q-alg/9702014))

[[!redirects Grothendieck plus construction]]


plus construction on presheaves

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

This is a subentry of _[[sheaf]]_ about the _plus-construction on presheaves_. For other constructions called _[[plus construction]]_, see there.

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The _plus construction_ $(-)^+ : PSh(C) \to PSh(C)$ on presheaves over a [[site]] $C$ is an operation that makes a presheaf "closer" to being a sheaf.

For ordinary sheaves of sets, the plus construction replaces a [[presheaf]] first by a [[separated presheaf]] and then by a [[sheaf]].

$$
  PSh(C) \stackrel{(-)^+}{\to} SepPSh(C)
  \stackrel{(-)^+}{\to} Sh(C)
  \,.
$$

(The first of these steps, however, is not a [[reflective subcategory|reflection]] into $SepPsh(C)$.)  Notice that in terms of [[n-truncated]] morphisms, a presheaf is

* separated precisely if every comparison map $A(U) \to Desc(A,U)$ to the set of [[descent data]] is [[(-1)-truncated]], namely a [[monomorphism]];

* a sheaf precisely if every such comparison map is [[(-2)-truncated]], namely an [[equivalence]].

More generally, in [[(n,1)-topos]] theory, the plus-construction is applied $(n+1)$-times in a row to an [[(n,1)-presheaf]] (a presheaf of $(n-1)$-groupoids), at each step decreasing the truncatedness of the comparison map until at the last step it becomes an actual [[(n,1)-sheaf]], a.k.a. stack; see [Anel and Leena Subramaniam, 2020](#ALS).  When $n=\infty$, even a countable sequence of applications does not suffice, but a transfinite one does; see [Lurie, section 6.5.3](#Lurie).

## Definition

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

Let $C$ be a small site equipped with a [[Grothendieck topology]] $J$, let $A:C^{op}\to Set$ be a functor. Then the __plus construction__ is a functor $(-)^+ : PSh(C) \to PSh(C)$ defined by the following equivalent descriptions:

1. $A^+(U) = colim_{(R\hookrightarrow  y U)\in J(U)} Hom_{Psh(C)}(R,A)$ where $J(U)$ can be defined in any of the following equivalent ways:
   * The poset of covering [[sieves]] on $U$ for the topology.  (It is important here that we use covering *sieves*, or at least families closed under composition and identities.)
   * The class of [[dense monomorphisms]] with [[representable functor|representable]] codomain $y U$.
   * The class of monomorphisms with codomain $y U$ that are inverted by the [[sheafification]] functor.  (Of course, this definition is not available unless we have shown that sheafification exists in some other way.)

1. For $U\in C^{op}$ we define $A^+(U)$ to be the set of [[equivalence classes]] of pairs $(R,s)$ where $R\in J(U)$ and $s=(s_f\in A(dom f)|f\in R)$ is a [[matching family|compatible family]] of elements of $A$ relative to $R$, and $(R,s)\sim (R', s')$ iff there is a $J$-covering sieve $R'' \subseteq R \cap R'$ on which the restrictions of $s$ and $s'$ agree.

1. In the [[internal language]] of the presheaf topos $PSh(C)$, we can define $ A^+ = \{ K \subseteq A \,|\, j(K\,\text{ is a singleton}) \}/{\sim}$, where $\sim$ is the equivalence relation given by $K \sim L$ if and only if $j(K = L)$ and $j$ is the [[Lawvere-Tierney topology]].

1. $A^+ = Lan_{r^{op}} Ran_{s^{op}} A$, where $r : J \to C$ is the [[Grothendieck construction]] of the functor $Cov : C^{op} \to Pos$ sending each $U$ to its set of covering [[sieves]], and $s:C\to J$ sends each $U\in C$ to its maximal sieve $M_U$ (the set of all morphisms with codomain $U$).
=--

The first and last of these definitions also work for higher sheaves on higher sites, i.e. when $C$ is an [[(infinity,1)-site]] and $A:C^{op} \to \infty Gpd$ is an [[(infinity,1)-presheaf]], as long as limits, colimits, and Kan extensions are interpreted in the $(\infty,1)$-categorical sense.

## Construction of sheafification

The basic property of the plus-construction is that we can construct [[sheafification]] (a.k.a. stack completion, in the higher-categorical case) by iterating it.  For ordinary presheaves of sets, two iterations suffice: for any presheaf $A$, $A^+$ is separated, while if $A$ is separated then $A^+$ is a sheaf; thus $A^{++}$ is the sheafification of any presheaf $A$.

+-- {: .num_remark}
###### Remark
Note, though, that $(-)^+ : PSh(C) \to SepPSh(C)$ is not left adjoint to the inclusion $\iota : SepPSh(C) \hookrightarrow PSh(C)$ of the full subcategory of separated presheaves. If it were, it would be a [[reflector]] and therefore satisfy $(-)^+ \circ \iota \cong Id$. But this is false, since the plus construction applied to separated presheaves yields their sheafification. See [this MathOverflow question](http://mathoverflow.net/questions/49486/the-single-plus-construction-is-not-the-left-adjoint-of-the-inclusion-of-separat) for details.
=--

More generally, when working with presheaves of $n$-groupoids for some finite $n$, it takes $n+2$ iterations for the plus-construction to converge to a sheaf/stack.  In the case $n=\infty$, no finite number of iterations suffices, and not even a countable number, but a transfinite number does.

We now sketch a proof of these facts, using the [[implicit infinity-category convention]] so that our results apply to higher sheaves/stacks as well as ordinary ones.

We use the fourth definition above $A^+ = Lan_{r^{op}} Ran_{s^{op}} A$, which we will write more shortly as $A^+ = r_! s_* A$.  There is a canonical map $A\to A^+$, which we can more easily construct and study with the following observation:

\begin{lemma}\label{AnAdjointQuadruple}
  $s:C\to J$ is right adjoint to $r:J\to C$.  Therefore, we have an adjunction $r^* : Psh(C) \rightleftarrows Psh(J) : s^*$, and hence an [[adjoint quadruple]]:
  $$ r_! \dashv r^* \dashv s^* \dashv s_*. $$
  Moreover, $r^*$ and $s_*$ are [[fully faithful]] and $r_!$ preserves finite limits, so $Psh(J)$ is [[cohesive topos|cohesive]] and even [[totally connected topos|totally connected]] over $Psh(C)$.
\end{lemma}
\begin{proof}
  Since the maximal sieve $M_U$ is the terminal object of the fiber of $r$, the functor $s$ picks out a fiberwise terminal object in the fibration $r$; thus it is right adjoint to $r$.  This implies the existence of the adjoint quadruple as discussed [[adjoint quadruple#KanExtensionOfAdjointPairIsAdjointQuadruple|here]].  Since $r s = 1_C$, we have that $s$ is fully faithful, and hence so are $s_*$ and $s_! \cong r^*$.  Finally, the fibers of $r^{op}$ are filtered since covering sieves are closed under intersections, so $r_!$ preserves finite limits since finite limits commute with filtered colimits.
\end{proof}

Thus, we can identify $Psh(C)$ with either the [[discrete objects]] or [[codiscrete objects]] in $Psh(J)$, via the fully faithful functors $r^*$ or $s_*$.  If we regard $A\in Psh(C)$ as the discrete object $r^*A$, then the plus-construction is similarly identified with $r^* r_! s_* s^* r^* A$.  Thus $r^*(A^+)$ can be written as $ʃ \sharp r^* A$, where $ʃ = r^* r_!$ is the reflector into the discrete objects and $\sharp = s_* s^*$ is the reflector into the codiscrete objects.

The rest of the proof can actually be carried out "synthetically" in the [[internal logic|internal]] [[homotopy type theory]] of the cohesive topos $Psh(J)$.  This argument is formalized [in the HoTT/Coq library](#HoTTplus).  But here we formulate the argument externally and categorically.

\begin{lemma}\label{PlusWellPointed}
The plus-construction is a [[well-pointed endofunctor]].  That is, we have a natural transformation $\eta : A \to A^+$ and a homotopy (an equality in the case $n=0$) $\theta : \eta_{A^+} \sim (\eta_A)^+$.  Moreover, the whiskered homotopy $\theta \eta_A$ is homotopic to the naturality square $\eta_{A^+} \circ \eta_A \sim (\eta_A)^+ \circ \eta_A$.
\end{lemma}
\begin{proof}
This follows fairly formally from its identification with $ʃ \sharp$, the composite of two reflectors.
\end{proof}

\begin{lemma}
The plus-construction preserves finite limits.
\end{lemma}
\begin{proof}
Both functors $r_!$ and $s_*$ preserve finite limits.
\end{proof}

\begin{lemma}
For $A\in Psh(C)$, the following are equivalent:

1. $A$ is a sheaf.
1. The canonical map $r^* A \to s_* A$ is an equivalence.
1. $r^* A$ is codiscrete.
1. $\eta_A : A \to A^+$ is an equivalence.
1. $\eta_A : A \to A^+$ has a retraction.

\end{lemma}
\begin{proof}
To see that the first two are equivalent, note that for a covering sieve $R$ of $U$ we have $(r^* A)(R) = A(U)$ while $(s_* A)(R)$ is the space of descent data for $R$.  Now we have $s_* A \cong s_* s^* r^* A = \sharp r^* A$, so the second and third statements are equivalent.

For the fourth, if $r^*A $ is codiscrete, then it is equivalent to $\sharp r^* A$; hence the latter is also discrete, and thus equivalent to $ʃ \sharp r^* A$.  Of course if $\eta_A$ is an equivalence, it has a retraction.  Finally, if it has a retraction, then the composite $\sharp r^* A \to ʃ \sharp r^* A = r^* A^+$ is a retraction for $r^* A \to \sharp r^* A$, so that $r^*A$ is codiscrete.
\end{proof}

\begin{theorem}
If some (perhaps transfinite) iteration of the plus-construction on $A\in Psh(C)$ stabilizes, it is a sheaf and the sheafification of $A$.
\end{theorem}
\begin{proof}
This follows formally from the fact that the plus-construction is a well-pointed endofunctor whose fixed points are the sheaves.
\end{proof}

Now since the plus-construction is an accessible endofunctor of a locally presentable category, its iteration always converges at some transfinite stage.  It follows that sheafification exists.  (Roughly this argument appears in [Lurie, section 6.5.3](#Lurie).)

We now want to prove that if $A$ is a presheaf of $n$-groupoids for some finite $n$, the iteration in fact converges already after $n+2$ steps.  Though long expected, it seems that this was first actually proven by [Anel and Leena Subramaniam, 2020](#ALS); what follows is inspired by their proof.  We will actually prove something slightly more general.

\begin{definition}
A presheaf $A\in Psh(C)$ is **$n$-separated** if the map $\eta_A : A\to A^+$ is [[n-truncated]].
\end{definition}

Thus $A$ is a sheaf if and only if it is $(-2)$-separated.  Moreover, since the plus-construction preserves finite limits, it preserves truncated objects; thus if $A$ is $n$-truncated (i.e. a presheaf of $n$-groupoids) then so is $A^+$ and hence so is $\eta_A$.  Thus, any $n$-truncated presheaf is $n$-separated.

We will prove that if $A$ is $n$-separated, then its plus-construction converges after $n+2$ steps.  The central auxiliary definition is:

\begin{definition}
A map $f:A\to B$ in $Psh(C)$ is **plus-connected** if there is a lift in its $\eta$-naturality square.  That is, there is a map $h:B \to A^+$ and homotopies $f^+ h \sim \eta_B$ and $h f \sim \eta_A$ that paste to the naturality square $f^+ \eta_A \sim \eta_B f$.
\end{definition}

\begin{lemma}
For any $A\in Psh(C)$, the map $\eta_A : A\to A^+$ is plus-connected.
\end{lemma}
\begin{proof}
Lemma \ref{PlusWellPointed} says precisely that $1_{A^+}$ is a lift in the $\eta$-naturality square for $\eta_A$.
\end{proof}

\begin{lemma}
If $f:A\to B$ is plus-connected, so is its diagonal $\triangle f : A \to A\times_B A$.
\end{lemma}
\begin{proof}
Since the plus-construction preserves finite limits, it suffices to observe that if a given commutative square has a lift, so does the induced square between its diagonals.
\end{proof}

So far everything has been very formal.  The concrete input from the fact that we are talking about a Grothendieck topology comes here:

\begin{lemma}\label{Composing}
If $f:A\to B$ is a [[monomorphism in an (infinity,1)-category|monomorphism]] that is plus-connected, then $f^+ : A^+ \to B^+$ is an equivalence.
\end{lemma}
\begin{proof}
Since the plus-construction preserves finite limits, it preserves monomorphisms; thus it suffices to show that $f^+$ is surjective.  An object $x\in B^+(U)$ is determined by a matching family over a covering sieve $R\hookrightarrow y U$, consisting of $x_V \in B(V)$ for all $V\to U$ in $R$.  Since $f$ is plus-connected, we have a map $h:B\to A^+$ assigning to each $x_V$ an object $h(x_V) \in A^+(V)$, determined by a matching family over a covering sieve $S_V \hookrightarrow y V$.  Now we can compose the sieves $S_V$ and $R$ to obtain a covering sieve $T$ on $U$, such that the restriction of $x$ to $T$ (which is equal to $x$) is in the image of $f^+$.
\end{proof}

\begin{lemma}
For $n\ge -1$, if $f:A\to B$ is $n$-truncated and plus-connected, then $f^+ : A^+ \to B^+$ is $(n-1)$-truncated.
\end{lemma}
\begin{proof}
By induction.  The base case $n=-1$ is the previous lemma.  For the inductive step, $f$ is $(n+1)$-truncated and plus-connected, then its diagonal $\triangle f$ is $n$-truncated and plus-connected.  Thus, by induction $(\triangle f)^+$ is $(n-1)$-truncated.  But since the plus-construction preserves finite limits, $(\triangle f)^+$ is equivalent to $\triangle (f^+)$, so $f^+$ is $n$-truncated.
\end{proof}

\begin{theorem}
If $A$ is $n$-separated, then its plus-construction converges after $n+2$ steps.
\end{theorem}
\begin{proof}
If $A$ is $n$-separated, then by definition $\eta_A : A\to A^+$ is $n$-truncated, and we have shown it is always plus-connected.  Thus, $(\eta_A)^+$ is $(n-1)$-truncated.  Hence so is $\eta_{A^+}$, in other words $A^+$ is $(n-1)$-separated.

Thus, by induction, if $A$ is $n$-separated, the $(n+2)$-th iteration of the plus-construction is $(n-(n+2))$-separated, i.e. $(-2)$-separated, i.e. a sheaf.
\end{proof}

\begin{remark}
Note that we do not need to know that the category $J$ is obtained as the category of covering sieves for a topology on $C$.  We only need that there is a projection $r:J\to C$ that is a fibration with cofiltered fibers and a fully faithful right adjoint, plus the result of Lemma \ref{Composing}.  In particular, $J$ could be a "mono-saturated lex modulator" in the sense of [ALS](#ALS), generating a non-topological localization of $Psh(C)$.
\end{remark}

\begin{remark}
We can also construct left exact localizations of non-presheaf toposes in this way.  If $E$ is a topos that is itself a left exact localization of some presheaf topos $Psh(C)$, then a lex modulator on $E$ induces a $J$ as above, and by composing the reflectors $ʃ$ and $\sharp$ with the reflection into $E$ we can make the same argument work to construct a left exact localization of $E$.
\end{remark}

## Related concepts

* [[sheafification]]

* [[(∞,1)-sheafification]] / [[∞-stackification]]

## References

Related entries: [[sheafification]]

A standard textbook reference in the context of 1-[[topos theory]] is:

* [[Peter Johnstone]], _[[Sketches of an elephant]] - a topos theory compendium_, Section C.2.2, proof of proposition 2.2.6, p.551

Remarks on the plus-construction in [[(infinity,1)-topos theory]] is in section 6.5.3 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

The plus-construction for presheaves in values in abelian categories is also called the Heller-Rowe construction:

* [[Alex Heller]], K. A. Rowe, _On the category of sheaves_
Amer. J. Math. 84 1962 205–216, [MR144341](http://www.ams.org/mathscinet-getitem?mr=144341), [doi](http://dx.doi.org/10.2307/2372759)

The plus-construction is studied for higher sheaves, and shown to converge after $n+2$ steps for $n$-presheaves, in

* {#ALS} [[Mathieu Anel]], Chaitanya Leena Subramaniam.  _Small object arguments, plus-construction, and left-exact localizations_.  [arxiv:2004.00731](https://arxiv.org/abs/2004.00731), 2020.

The above argument is formalized in the internal logic of $Psh(J)$ here:

* {#HoTTplus} Canonical binary meets of subuniverses, <https://github.com/HoTT/HoTT/blob/master/theories/Modalities/Meet.v>

[[!redirects plus-construction on presheaves]]
[[!redirects plus constructions on presheaves]]
[[!redirects plus-constructions on presheaves]]


Po Hu

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=142142)

## Selected writings

On the [[Adams spectral sequence]] for [[Real cobordism cohomology]]:

* [[Po Hu]], *The cobordism of Real manifolds and calculations with the Real Adams-Novikov spectral sequence*, (1998) [[hdl:2027.42/130996](http://hdl.handle.net/2027.42/130996), [pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/130996/9825253.pdf?sequence=2&isAllowed=n)]

* {#Hu99} [[Po Hu]], *The cobordism of Real manifolds*, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 [[dml:212395](https://eudml.org/doc/212395), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm161/fm16115.pdf)]

* {#Kriz01} [[Po Hu]], [[Igor Kriz]], *Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence*, Topology **40** (2001) 317-399 [doi:10.1016/S0040-9383(99)00065-8, [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf)]

* [[Po Hu]], [[Igor Kriz]], *Some Remarks on Real and Algebraic Cobordism*, K-Theory **22** (2001) 335–366 [[pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43150/10977_2004_Article_321295.pdf?sequence=1), [doi:10.1023/A:1011196901303](http://dx.doi.org/10.1023/A:1011196901303)]

see also:

* [[Po Hu]], [[Igor Kriz]], *Topological Hermitian Cobordism*, Journal of Homotopy and Related Structures, **11** (2016) 173–197 [[arXiv:1110.5608](https://arxiv.org/abs/1110.5608), [doi:10.1007/s40062-014-0100-9](https://doi.org/10.1007/s40062-014-0100-9)]

On [[six operations]] and [[Wirthmüller contexts]]:

* [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

category: people


Po-Hsiang Chu

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=20797)

## Selected writings

Introducing the [[Chu construction]]:

* {#Chu79} [[Po-Hsiang Chu]], *Constructing $\ast$-autonomous categories*, appendix to: [[Michael Barr]], _$\ast$-Autonomous Categories_, Lecture Notes in Mathematics **752**, Springer (1979) 103-138 [[doi:10.1007/BFb0064579](https://doi.org/10.1007/BFb0064579)]

category: people

[[!redirects Peter Chu]]


Po-Shen Hsin

* [institute page](https://pma.caltech.edu/people/po-shen-hsin)

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

## Selected writings

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from a [[higher gauge theory]]-perspective

* [[Francesco Benini]], [[Clay Cordova]], [[Po-Shen Hsin]], *On 2-Group Global Symmetries and Their Anomalies*,  J. High Energ. Phys. **2019** 118 (2019) ([arXiv:1803.09336](https://arxiv.org/abs/1803.09336), doi:10.1007/JHEP03(2019)118)

category: people


Pochhammer loop

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Integration theory
+-- {: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *Pochhammer loop* or *Pochhammer contour* ([Pochhammer 1890](#Pochhammer1890)) is a [[loop (topology)|loop]] in the [[complement]] of a [[pair]] of points in side the [[plane]], hence a [[map]]

$$
  \gamma_P
  \;\colon\;
  S^1
  \longrightarrow
  \mathbb{C} \setminus \{0,1\}
  \,,
$$

which represents:

1. a non-trivial element in the [[fundamental group]], $0 \neq [\gamma_P] \,\in\,\pi_1\big( \mathbb{C}\setminus \{0,1\}\big)$,

   but such that the winding number around each point separately vanishes;

1. a trivial [[cycle]] in [[ordinary homology|homology]] $0 = [\gamma_P] \,\in\, \pi_1\big( \mathbb{C}\setminus \{0,1\}\big)$.

Due to this property, Pochhammer loops may underlie non-trivial [[cycles]] in [[twisted homology]] (e.g. [Varchenko 1995, Fig. 1.1](#Varchenko95), [Etingof, Frenkel & Kirillov 1998, Fig. 4.1](#EtingofFrenkelKirillov98)).

## References

Original articles:

* {#Pochhammer1890} [[Leo Pochhammer]], *Zur Theorie der Euler'schen Integrale*, Mathematische Annalen **35** (1890) 495–526 $[$[doi:10.1007/BF02122658](https://doi.org/10.1007/BF02122658)$]$

See also

* Wikipedia, *[Pochhammer contour](https://en.wikipedia.org/wiki/Pochhammer_contour)*

Discussion in the context of the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* {#Varchenko95} [[Alexander Varchenko]], *Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups*, Advanced Series in Mathematical Physics **21**, World Scientific (1995) ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* {#EtingofFrenkelKirillov98} [[Pavel Etingof]], [[Igor Frenkel]], [[Alexander Kirillov]], Lecture 7 in: *Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations*, Mathematical surveys and monographs **58**, American Mathematical Society (1998) $[$[ISBN:978-1-4704-1285-2](https://bookstore.ams.org/surv-58), [review pdf](http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf)$]$

[[!redirects Pochhammer loops]]

[[!redirects Pochhammer contour]]
[[!redirects Pochhammer contours]]


Podleś sphere

#Contents#
* table of contents
{:toc}

## Idea

__Podleś spheres__ are a family of examples of [[quantum group]]-analogues of [[homogeneous spaces]] which are $q$-deformations of the usual [[2-sphere]].

## References

* Piotr Podleś, _Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups_, Commun.Math.Phys. 170 (1995) 1-20 [doi](https://doi.org/10.1007/BF02099436) [hep-th/9402069](https://arxiv.org/abs/hep-th/9402069)
* F. Bonechi, N. Ciccoli, N. Staffolani, M. Tarlini, _The quantization of the [[symplectic groupoid]] of the standard Podleś sphere_, J. Geom. Phys. __62__:8 (2012) 1851-1865 [doi](https://doi.org/10.1016/j.geomphys.2012.04.00) [arXiv:1004.3163](http://arxiv.org/abs/1004.3163)

[[!redirects Podles sphere]]
[[!redirects Podles spheres]]
[[!redirects Podleś spheres]]


Poincar%C3%A9 duality > history

< [[Poincar%C3%A9 duality]]


Poincaré complex

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

#Contents#
* table of contents
{:toc}

## Idea

A special kind of [[Poincaré duality space]]: a [[CW-complex]] which satisfies [[Poincaré duality]] and in this respect thus behaves like a [[closed manifold]]. Also called a _homology manifold_, therefore.

## References

* [[C. T. C. Wall]], _Poincaré complexes I_, Annals of Mathematics, Second Series, Volume 86 (1967) [pdf](http://www.maths.ed.ac.uk/~aar/papers/poincarecx.pdf
)

* [[Andrew Ranicki]], _45 slides on chain duality_, September 1996 ([pdf](http://www.math.uiuc.edu/K-theory/0154/45slides.pdf))
 {#Ranicki96}

* [pdf](http://www.dm.ufscar.br/~ebt2012/main_files/mc-Hillman.pdf)

With an eye towards generalization in [[spectral geometry]] ([[spectral triples]]):

* {#Connes95} [[Alain Connes]], page 10 of _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](http://www.alainconnes.org/docs/reality.pdf))

[[!redirects Poincaré complexes]]

[[!redirects Poincare complex]]
[[!redirects Poincare complexes ]]

[[!redirects Poincaré duality complex]]
[[!redirects Poincaré duality complexes]]

[[!redirects Poincare duality complex]]
[[!redirects Poincare duality complexes]]


Poincaré conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_theorem}
###### Theorem
**(Poincaré conjecture)**

Every [[simply connected]] [[compact space|compact]] [[topological manifold|topological]] [[3-manifold]] without boundary is [[homeomorphism|homeomorphic]] to the 3-sphere.

=--

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

A proof strategy was given by [[Richard Hamilton]]: imagine the manifold is equipped with a [[metric]]. Follow the [[Ricci flow]] of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.

The hard technical part of this program is to show that the passage through the  singularities can be controlled. This was finally shown in ([Perelman 02](Perelman02)).

=--

See at _[[Ricci flow]]_ for more.

## Related entries

* [[geometrization conjecture]]

## References

### In dimension 3

* {#Perelman02} [[Grigori Perelman]], _The entropy formula for the Ricci flow and its geometric applications_ ([arXiv:math/0211159](http://arxiv.org/abs/math/0211159))

* {#BessiersEtAl} Laurent Bessieres, Gerard Besson, Michel Boileau, Sylvain Maillot, Joan Porti, _Geometrisation of 3-manifolds_ ([pdf](http://www-fourier.ujf-grenoble.fr/~besson/book.pdf))

Notes from a survey talk:

* [Huisken on Uniformization I](http://golem.ph.utexas.edu/category/2007/02/huisken_on_uniformization.html)

* [Huisken on Uniformization II](http://golem.ph.utexas.edu/category/2007/02/huisken_on_uniformization_ii.html)

See also

* {#Martelli16} Bruno Martelli, _An Introduction to Geometric Topology_ ([arXiv:1610.02592](https://arxiv.org/abs/1610.02592))

### In higher dimesnions

The analog of the Poincaré conjecture in higher dimensions:

* {#Newman66} M. H. A. Newman, Theorem 7 in: *The Engulfing Theorem for Topological Manifolds*, Annals of Mathematics Second Series, *84** 3 (1966) 555-571  ([jstor:1970460](https://www.jstor.org/stable/1970460))

[[!redirects Poincare conjecture]]


Poincaré conjecture - diagrammatic formulation

#Contents#
* table of contents
{:toc}

## Introduction

The Poincaŕe conjecture can be re-formulated as a conjecture concerning link diagrams. After recalling some preliminaries, we present this diagrammatic formulation.

## Kirby equivalence ##

The _framed Reidemeister moves_ on a link diagram are depicted [[framed_reidemeister_moves.pdf|here:file]].

The _Kirby moves_ on a link diagram are depicted [[kirby_moves.pdf|here:file]].

A pair of link diagrams are _Kirby equivalent_ if there is a finite sequence of framed Reidemeister moves and Kirby moves taking one to the other.

## Fundamental theorems on 3-manifolds

We shall rely on the following fundamental theorems, which allow for a diagrammatic approach to the Poincaré conjecture.

+-- {: .un_thm}
###### Theorem (Lickorish-Wallace)

Let $M$ be a closed, connected, orientable 3-manifold. There is a link diagram $L$ such that $M$ is isomorphic to the 3-manifold obtained by the integral Dehn surgery on $L$ in $S^{3}$ with respect to the blackboard framing of $L$.

=--

+-- {: .un_thm}
###### Theorem (Kirby)

Let $M_{0}$ and $M_{1}$ be closed, connected, orientable 3-manifolds. Let $L_{0}$ (respectively $L_{1}$) be a link diagram such that the 3-manifold obtained by the integral Dehn surgery on $L_{0}$ (respectively $L_{1}$) in $S^{3}$ with respect to the blackboard framing of $L_{0}$ (respectively $L_{1}$). Then $M_{0}$ is isomorphic to $M_{1}$ if and only if $L_{0}$ and $L_{1}$ are Kirby equivalent.

=--

## Fundamental group of a link diagram

Let $L$ be a link diagram, with some choice of orientation. We denote the free group on the arcs of $L$ by $F(L)$.

We define $\pi_{1}(L)$, the fundamental group of $L$, to be the quotient of $F(L)$ by the normal subgroup generated by words of the form $a_3^{-1} a_2^{-1} a_1^{-1} a_2$, for any crossing of $L$ as depicted [[labelled_crossing.pdf|here:file]], irrespective of the orientation of the horizontal arcs.

## Longitude of the component of a link diagram

Let $L$ be a link diagram, with some choice of orientation. The longitude of a component of $L$ is defined to be the word $w$ which we obtain after carrying out the following procedure.

1. Pick any arc of $L$, say $a$. Let $w$ be the empty word.
2. Walk around $L$, following the orientation. When we walk under an arc $b$, whether or not $b$ belongs to same component of $L$ or a different one, we add $b$ to the end of $w$ if the configuration of orientations at the crossing is as depicted in the first figure [[longitude.pdf|here:file]], and add $b^{-1}$ to the end of $w$ if the configuration of orientations at the crossing is as depicted in the second figure [[longitude.pdf|here:file]].
3. Stop when we return to the arc we started with, namely $a$.

The following is a consequence of the van Kampen theorem.

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

Let $M$ be a closed, connected, orientable 3-manifold. Let $L$ be a link diagram such that the 3-manifold obtained by the integral Dehn surgery on $L$ in $S^{3}$ with respect to the blackboard framing of $L$ is isomorphic to $M$. Then the group $\pi_{1}(M)$ is isomorphic to the group $\pi_{1}(L) / \langle l_1, \ldots, l_n \rangle$, where $l_1$, $\ldots$, $l_{n}$ are the longitudes of the components of $L$, and $\langle l_1, \ldots, l_n \rangle$ is the normal subgroup generated by these.

=--

## Diagrammatic formulation of the Poincaré conjecture

Let $M$ be a closed, connected $3$-manifold. The _Poincaré conjecture_ is that if $\pi_{1}(M)$ is trivial (that is to say, isomorphic to a group with one element), then $M$ is isomorphic to $S^{3}$.

If $\pi_{1}(M)$ is trivial, then $M$ is orientable. It thus follows from the Lickorish-Wallace theorem, the Kirby theorem, the preceding proposition, and the fact that integral Dehn surgery on the empty link diagram gives $S^{3}$, that the Poincaré conjecture is equivalent to the following: if a link diagram $L$ has the property that the group $\pi_{1}(L) / \langle l_1, \ldots, l_n \rangle$ is trivial, then $L$ is Kirby equivalent to the empty link diagram.


Poincaré duality

[[!redirects Poincare duality]]

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Poincaré duality over some [[space]] is an [[equivalence]]  (if it exists) relating [[cohomology]] with [[homology]] on that space.

The canonical example is the Poincaré duality in [[ordinary cohomology]] $H^\bullet(X)$/[[ordinary homology]] $H_\bullet(X)$ which exists over an [[orientation|orientable]] [[closed manifold]] $X$ of [[dimension]] $n$: any choice of [[volume form]] $\omega$ induces by the [[cap product]] an [[isomorphism]]

$$
  (-)\cap \omega \;\colon\; H_\bullet(X) \stackrel{\simeq}{\to} H^{n-\bullet}(X)
  \,.
$$

More generally Poincaré duality is about [[dual objects]] in a [[generalized cohomology theory]]. For instance if $X$ above is furthermore [[K-orientation|K-oriented]], hence [[orientation in generalized cohomology |oriented]] in the [[K-theory]] [[cohomology theory]] (hence if it has [[spin^c structure]]) then there is an isomorphism between its [[K-homology]] and [[K-theory]]. For more on this see at _[[Poincaré duality algebra]]_.

Poincaré duality is the mechanism behind [[Umkehr maps]]/[[push-forward in generalized cohomology]]: given a map of [[spaces]] $f \colon X \to Y$ which enjoy Poincaré duality with respect to some [[generalized cohomology theory]] $R$, one can pass from the canonically given pullback morphism

$$
  f^\ast \colon R^\bullet(Y) \to R^\bullet(X)
$$

to the [[dual morphism]]

$$
  f_! \colon R^\bullet(X) \simeq R_\bullet(X) \stackrel{f_\ast}{\to} R_\bullet(Y) \simeq R^\bullet(Y)
  \,.
$$

Here the duality is typically exhibited in two steps:

1. an [[Atiyah duality]] identifies the dual of $R^\bullet(X)$ with the $R$-cohomology of a [[Thom space]] of $X$;

1. a [[Thom isomorphism]] identified the $R$-cohomology of the Thom space back with that of $X$.

More generally, spaces $X$ are not [[self-dual]] in this way, but may at least be dual to themselves but equipped with a [[twisted cohomology|twist]]. This yields the _[[twisted Umkehr maps]]_ in [[twisted cohomology]] (see for instance at [[Freed-Witten-Kapustin anomaly cancellation]]).

For more on this see at _[[twisted Umkehr map]]_.

## Definition and statement
 {#DefinitionAndStatement}

We first state the

* _[Traditional formulation](#TraditionalFormulation)_

and then its

* _[Refinement in homotopy theory](#RefinementInHomotopyTheory)_.

### Traditional formulation
 {#TraditionalFormulation}

Around 1895 [[Henri Poincaré]] made an observation about [[Betti numbers]] of [[closed manifolds]], which in the 1930s was then formulated by [[Eduard Čech]] and [[Hassler Whitney]] in the following modern form:

+-- {: .num_theorem #TraditionalPoincareDuality}
###### Theorem

Let $X$ be a [[closed manifold]] of [[dimension]] $n$ that is [[orientation|orientable]]. Then the [[cap product]] with any choice of [[orientation]] -- in the form of a [[fundamental class]] $[X] \in H_n(X)$ -- induces [[isomorphisms]] of the form

$$
  (-)\cap [X] \;\colon\; H^k(X) \stackrel{\simeq}{\to} H_{n-k}(X)
$$

between the [[ordinary cohomology]] and the [[ordinary homology]] groups of $X$.

=--

Later this was turned around and more general [[topological spaces]] satisfying this condition were considered:

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

A [[topological space]] for which there is $d \in \mathbb{N}$ and a  class $[X] \in H_d(X)$ such that the [[cap product]] induces [[isomorphisms]]

$$
  (-) \cap [X] \;\colon\; H^\bullet(X) \stackrel{\simeq}{\to} H_{d-\bullet}(X)
$$

is called a **[[Poincaré duality space]]**.

=--

### Refinement to homotopy theory
 {#RefinementInHomotopyTheory}

Traditionally Poincaré duality is stated as a [[dual object|duality]] of [[chain homology|chain]] [[homology groups]]. The passage from [[chain complexes]] to their [[homology groups]], hence the passage from full [[homotopy theory]] to just some invariants, however forgets a lot of information. But it turns out that this can always be lifted:

+-- {: .num_theorem}
###### Theorem

Let $X$ be a [[Poincaré duality space]] of [[dimension]] $n$, def. \ref{PoincareDualitySpace}. Then there is a [[quasi-isomorphism]]

$$
  C^\bullet(X) \stackrel{\simeq}{\to} \Sigma^{-d} C_\bullet(X)
$$

between the [[chain complex]] of [[ordinary cohomology]] of $X$ with that $d$-fold [[suspension|de-suspension]] of the [[chain complex]] of [[ordinary homology]] of $X$.

This is such that its image in [[chain homology]] is (up to sign) the traditional Poincaré duality isomorphism

$$
  H^\bullet(X) \stackrel{\simeq}{\to} H_{d -\bullet}(X)
$$

of theorem \ref{TraditionalPoincareDuality}.  When $\bullet=r$, the sign is $-1^{r(r+1)/2}$.

=--

This is ([EM 11, theorem, 2.5.2](#EM11)).

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

Since $C^\bullet(X) \simeq [C_\bullet(X,), \mathbb{I}] \simeq (C_\bullet(X))^\vee$ is the [[dual object]] to $C_\bullet(X)$ in the [[(∞,1)-category of unbounded chain complexes]], this says that for a [[Poincaré duality space]] $X$ the [[chain complex]] $C_\bullet(X)$ is almost a self-[[dual objects]], except for a degree-shift, hence except for a [[twisted cohomology|twist]] of (itself) degree 0.

=--

## Properties

### Recognition of manifolds

+-- {: .query}

From [ALGTOP-L], Oct 5, 2010.

* [[Jim Stasheff]]:

  Any one have a reference for obstructions which detect whether a space whose cohomology has Poincare duality is actually a manifold? thanks

* John Klein:

  Ranicki's total surgery obstruction does it in the top case, in
surgery dimensions.

  * The total surgery obstruction. Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pp. 275--316, Lecture Notes in Math., 763, Springer, Berlin, 1979. ([pdf](http://www.maths.ed.ac.uk/~aar/papers/total.pdf))

* Nathanien Rounds:

  I believe this question is answered (simply connected case, over Q) in
Dennis Sullivan's paper Infinitesmal Computations in Topology (Theorem 13.2).  The answer, as I understand it, is that outside dimension 4k any graded commutative algebra over Q wtih first betti number 0 satisfying Poincare Duality can be realized as the cohomoloyg ring of a manifold.  In dimension 4k there is an obstruction related to the signature which is given in that paper.  There won't in general be a unique manifold corresponding to the ring; for example one can choose different rational Pontragin classes and change the homemorphism type but not the cohomology.

  The general answer (not simply connected, over Z) is given by Ranicki's total surgery obstruction, as John Klein has already pointed out. One way to interpret this obstruction geometrically is that the Poincare duality map is always "local" but it need not have a "local" inverse, and the lack of a local inverse is an obstruction to having a anifold structure.  See for example McCrory's paper "A Characterization of Homology Manifolds" and also my thesis, which if you're interested is here:

  www.math.purdue.edu/~nrounds

=--

See also ([Ranicki 96](#Ranicki96)) and see at _[[Poincaré complex]]_.

### Relation to Thom isomorphism

The Poincaré dual of a [[submanifold]] can be identified with the [[Thom class]] on its [[normal bundle]]

(...)

### Relation to push-forward in cohomology

Given Poincare duality and hence (twisted-)[[self-dual objects]], the [induced dagger-structure](self-dual+object#RelationToDaggerCompactStructure) allows torevert morphisms in [[cohomology]]. These "[[Umkehr maps]]" describe *[[fiber integration]]* in cohomology.

### Generalizations

The [[six operations]] of [[Grothendieck]] and [[Grothendieck duality]] are designed to ensure a generalized and relative versions of Poincaré duality and related phenomena in the setups like sheaf and [[topos theory]], algebraic geometry. See _[[Grothendieck duality]]_ for references.

Another generalization, for singular spaces, is with help of stratifications and via [[intersection cohomology]].

## Variants and generalizations

* in [[topology]]: [[Poincaré duality complex]]

* in [[noncommutative topology]]: [[Poincaré duality algebra]]

* in [[derived category|derived]] [[abelian sheaf|abelian]] [[sheaf and topos theory]] (for [[abelian sheaf cohomology]]): [[Verdier duality]]

* in [[nonabelian cohomology]]: [[nonabelian Poincaré duality]]

## Related concepts

* [[Poincaré complex]]

* [[Poincaré duality algebra]]

* [[KO-dimension]]

* [[Serre duality]]

## References

### General

This historical work of [[Henri Poincaré]] is reviewed around page 28. pf

* [[Jean Dieudonné]], _A History of Algebraic and Differential Topology, 1900 - 1960_

Summaries of the traditional modern statement of Poincaré duality are for instance in

* [[eom]], _[Poincaré duality](http://www.encyclopediaofmath.org/index.php/Poincar%C3%A9_duality)_

* Manifold Atlas, _[Poincaré Duality Spaces](http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9_Duality_Spaces)_

Discussion of generalizations to "chain duality" (see also at _[[Poincare complex]]_) is at

* [[Andrew Ranicki]], _45 slides on chain duality_, September 1996 ([pdf](http://www.math.uiuc.edu/K-theory/0154/45slides.pdf))
 {#Ranicki96}

Discussion in [[etale cohomology]] is in

* [[James Milne]], sections 14 and 24 of _[[Lectures on Étale Cohomology]]_

With an eye towards generalization in [[spectral geometry]] ([[spectral triples]]):

* {#Connes95} [[Alain Connes]], page 10 of _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](http://www.alainconnes.org/docs/reality.pdf))

### For equivariant cohomology

In [[equivariant cohomology]]:

For [[finite groups]]

* [[Steven Costenoble]], [[Stefan Waner]], _Equivariant Poincaré duality_, Michigan Math. J. 39 (1992), no. 2, 325–351

For [[compact Lie groups]]

* [[Michel Brion]], _Poincaré duality and equivariant (co)homology_, Michigan mathematical journal, Volume 48, Issue 1 (2000), 1-624 ([EUCLID](http://projecteuclid.org/euclid.mmj/1030132709))

* [[Steven Costenoble]], [[Stefan Waner]], _Equivariant ordinary homology and cohomology_ ([arXiv:math/0310237](http://arxiv.org/abs/math/0310237))

* Christopher Allday, Matthias Franz, [[Volker Puppe]], _Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property_ ([arXiv:1303.1146](http://arxiv.org/abs/1303.1146))

In [[equivariant K-theory]]:

* {#Tu06} [[Jean-Louis Tu]], _Twisted K-theory and Poincare duality_ ([arXiv:0609556](http://arxiv.org/abs/math/0609556))

* [[Heath Emerson]], [[Ralf Meyer]], _Dualities in equivariant Kasparov theory_, New York J. Math. 16 (2010), 245-313 ([arXiv:0711.0025](http://arxiv.org/abs/0711.0025))

* [[Heath Emerson]], _Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey_ ([arXiv:0904.4744](http://arxiv-web3.library.cornell.edu/abs/0904.4744))

### For Hochschild cohomology

[[nLab:Poincaré duality]] on [[nLab:Hochschild cohomology|Hochschild (co)homology]]

* M. Van den Bergh, _A relation between Hochschild homology and cohomology for Gorenstein rings_ . Proc. Amer. Math. Soc. 126 (1998), 1345–1348; ([JSTOR](http://www.jstor.org/stable/118786))

  Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.
{#VanDenBergh}

with more on that in

* U. Krähmer, _Poincaré duality in Hochschild cohomology_
  ([pdf](http://www.maths.gla.ac.uk/~ukraehmer/brussels.pdf))

That this Poincare duality takes the [[Connes coboundary operator]] to the [[BV operator]] is shown in

* [[nLab:Victor Ginzburg]], _Calabi-Yau Algebras_, [arXiv/math.AG/0612139](http://arxiv.org/abs/math.AG/0612139)
{#Ginzburg}

### For spaces in higher geometry

Poincaré duality on [[Deligne-Mumford stacks]]/[[orbifolds]] is discussed in

* Dan Abramovich, Tom Graber, [[Angelo Vistoli]], _Gromov-Witten theory of Deligne-Mumford stacks_ ([arXiv:math/0603151](http://arxiv.org/abs/math/0603151))

* Weimin Chen, [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](http://arxiv.org/abs/math/0004129))

For disucssion in [[noncommutative topology]]/[[KK-theory]] see at _[[Poincaré duality algebra]]_.

### On the level of chains

Poincaré duality is traditional considered on the level of [[cohomology groups]]. One may asks if it lifts to a duality on the underlying [[chain complexes]].

Such "derived" Poincaré duality on the level of chain complexes is claimed in theorem 2.5.2 of

* Eric J. Malm, _String topology and the based loop space_ ([arXiv:1103.6198](http://arxiv.org/abs/1103.6198))
 {#EM11}

based on -- or at least inspired by -- the discussion in (section 10) of

* [[William Dwyer]], [[John Greenlees]], S. Iyengar, _Duality in algebra and topology_ ([arXiv:math/0510247](http://arxiv.org/abs/math/0510247))

The article

* [[Thomas Tradler]], [[Mahmoud Zeinalian]], [[Dennis Sullivan]], _Infinity Structure of Poincare Duality Spaces_ ([arXiv:math/0309455](http://arxiv.org/abs/math/0309455))

claims a chain duality (even of $A_\infty$-comodule chains but then possibly only exhibited by a [[bimodule]]) for the [[singular chain complex]] of any Poincaré duality topological space.

A review is in

* [[Thomas Tradler]], _Infinity inner products on A-infinity algebras_ ([arXiv:0806.0065](http://arxiv.org/abs/0806.0065))

[[!redirects Poincare duality]]
[[!redirects Poincaré duality]]
[[!redirects Poincar%C3%A9 duality]]