Stokes phenomenon

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### Context
#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
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## Overview

Matching [[asymptotic series]] expansions of certain functions like [[Airy functions]] and more generally of solutions to the [[wave equations]] (say in short wave approximation or solutions to non-Fuchsian meromorphic ODEs) comes with sharp changes (disconintuities) at certain phase angles (Stokes lines) described by certain factors, Stokes matrices.

G. G. Stokes has discovered these phenomena -- now also known more generally as [[wall crossing]] phenomena -- in the study of [[geometric optics]], more specifically the study of [[Airy function]] which is an example of a solution to a linear meromorphic [[ordinary differential equation]] which has an irregular singular point, hence the equation is not [[Fuchsian equation|Fuchsian]]. The Stokes phenomenon does not happen to Fuchsian equations. Their formal meromorphic solutions are automatically convergent. Exactly the presence of the irregular singularities makes the appearance of formal solutions with zero radius of convergence. Now look around the origin. One can try to prove that there are asymptotic expressions at best in some regions of argument. There are jumps at certain slopes. In fact there is Stokes sheaf and the first nonabelian cohomology of the Stokes sheaf measures the obstruction for a formal meromomorphic expansion to be build up of sectorial true meromorphic expansions.

Stokes factors (multipliers, matrices) appear also in the study of stability in the geometry of moduli spaces of sheaves. According to Toledano Laredo "Joyce's wall-crossing formulae for invariants counting semistable objects in an abelian category A may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of A.".

## Related entries

* [[Airy function]]

* [[semiclassical approximation]]

*  [[wall crossing]]

## Literature

### General

The historical work of G. Stokes is

* G. G. Stokes, _On the numerical calculation of a class of definite integrals and infinite series_, Trans. Camb. Phil. Soc., 9 (1847), 379-407

* [M. V. Berry](http://www.phy.bris.ac.uk/people/berry_mv/research.html), _Stokes' phenomenon; smoothing a Victorian discontinuity_,  Publ. Math. de l'IHÉS, 68 (1988), p. 211-221 (1988), [numdam](http://www.numdam.org/numdam-bin/item?id=PMIHES_1988__68__211_0).

Sibuya's has a textbook chapter on Stokes phenomena

* Yasutaka Sibuya, _Linear differential equations in the complex domain: Problems of analytic continuation_, Transl. of Math. Monographs __82__, Amer. Math. Soc. 1990 (Japanese original Kinokuniya, Tokyo, 1976, 1980)

* C. Sabbah, _Introduction to Stokes structures_, [http://arxiv.org/abs/0912.2762](http://arxiv.org/abs/0912.2762)

* Claude Sabbah, _Isomonodromic deformations and Frobenius manifolds_, Springer 2007, [doi](http://dx.doi.org/10.1007/978-1-84800-054-4)

* C. Sabbah, slides of a talk at Rennes, [pdf](http://www.math.polytechnique.fr/cmat/sabbah/sabbah_rennes091027-pro.pdf); _Wild geometry_, summary of Oberwolfach, [pdf](http://www.math.polytechnique.fr/cmat/sabbah/sabbah_oberwolfach090922.pdf)

* Y. Sibuya, _Stokes phenomena_, Bull. Amer. Math. Soc. __83__ (1977), 1075-1077 [MR0442337](http://www.ams.org/mathscinet-getitem?mr=0442337) [doi](http://dx.doi.org/10.1090/S0002-9904-1977-14391-7) [euclid](http://projecteuclid.org/euclid.bams/1183539486)

* T. Bridgeland, V. Toledano-Laredo, _Stability conditions and Stokes factors_, [arxiv/0801.3974](http://arxiv.org/abs/0801.3974); _Stokes factors and multilogarithms_ [arxiv/1006.4623](http://arxiv.org/abs/1006.4623)

* [[Valerio Toledano Laredo]], _Stability conditions and Stokes factors_, audio of the talk at UCSB, [link](http://doug-pc.itp.ucsb.edu/online/duality09/toledanolaredo1); slides [pdf](http://doug-pc.itp.ucsb.edu/online/duality09/toledanolaredo1/pdf/Toledanolaredo1_String_Duality_KITP.pdf)

* [[L. Katzarkov]], [[M. Kontsevich]], [[T. Pantev]], _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)

* M. Loday-Richaud, _Stokes cocycle and differential Galois groups_, J. Math. Sci. __124__, No. 5, 2004, [pdf](http://www.springerlink.com/content/h42h3vk6w59q0820/fulltext.pdf)

* T. Aoki, T. Kawai, Y. Takei, _On the exact steepest descent method: A new method for the description of Stokes curves_, J. Math. Phys. __42__ (2001), 3691-3713.

* R. Balian, G. Parisi, A. Voros, _Discrepancies from asymptotic series and their relation to complex classical trajectories_, Phys. Rev. Lett. __41__ (1978), 1141-1144.

* A. Voros, _The return of the quartic oscillator. The complex WKB method_, Annales de l'I.H.P. Physique théorique 39:3 (1983) 211-338 [eudml](http://eudml.org/doc/76217)

* E Delabaere, H Dillinger, F Pham, _Résurgence de Voros et périodes des courbes hyperelliptiques_, Annales de l'institut Fourier, 1993 [doi](https://doi.org/10.5802/aif.1326); _Exact semiclassical expansions for one-dimensional quantum oscillators_
J. Math. Physics __38__ (1997) 6126 [doi](https://doi.org/10.1063/1.532206)

* Eric Delabaere, Frédéric Pham, _Unfolding the quartic oscillator_ Annals of Physics
__261__:2 (1997) 180-218 [doi](https://doi.org/10.1006/aphy.1997.5737)

* R.B. Paris, A.D. Wood, _Stokes phenomenon demystified_, Bull. Inst. Math. Appl. __31__ (1995) 21-28.

* A R Its, A A Kapaev, _Quasi-linear Stokes phenomenon for the second
Painlevé transcendent_, Nonlinearity __16__ (2003) 363–386, [doi](http://dx.doi.org/10.1088/0951-7715/16/1/321)

* M. Hukuhara, _Sur les points singuliers des équations différentielles linéaires, II_, Jour.Fac. Sci. Hokkaido Univ. 5 (1937), 123-166; _Sur les points singuliers des équations différentielles linéaires, III_ , Jour. Fac. Sci. Kyushu Univ. 2 (1942), 125-137.

* D. G. Babbitt, V. S. Varadarajan, _Local moduli for meromorphic differential equations_,  Bull. Amer. Math. Soc. (N.S.) __12__, N. 1 (1985), 95-98.

* P. Boalch, _Symplectic manifolds and isomonodromic deformations_, [link](http://www.dma.ens.fr/~boalch/files/smid.pdf); _Geometry and braiding of Stokes data; fission and wild character varieties_, [arXiv:1111.6228](http://arxiv.org/abs/1111.6228); P. P. Boalch, _Stokes matrices, Poisson Lie groups and Frobenius manifolds_, Invent. Math. 146 (2001), no. 3, 479–506. [MR1869848](http://www.ams.org/mathscinet-getitem?mr=1869848) [doi](http://dx.doi.org/10.1007/s002220100170)

* A. A. Kapaev, _Quasi-linear Stokes phenomenon for the Painlevé first equation_, J. Phys. A: Math. Gen. __37__, 11149 (2004) [doi](http://dx.doi.org/10.1088/0305-4470/37/46/005)

* Kiran S. Kedlaya, _Good formal structures for flat meromorphic connections, I: Surfaces_, Duke Math. J. __154__, n. 2 (2010), 343-418, [MR2682186](http://www.ams.org/mathscinet-getitem?mr=2682186), [euclid](http://projecteuclid.org/euclid.dmj/1281963652)

* [[Marco Gualtieri]], Songhao Li, Brent Pym, _The Stokes groupoids_, [arxiv/1305.7288](http://arxiv.org/abs/1305.7288)

> We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.

Interconnection between exact WKB method, Stokes data and cluster algebras has been studied in

* Kohei Iwaki, Tomoki Nakanishi, _Exact WKB analysis and cluster algebras_, J. Phys. A 47 (2014) 474009 [arxiv/1401.7094](http://arxiv.org/abs/1401.7094); _Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras_, [arXiv:1409.4641](http://arxiv.org/abs/1409.4641)

* Xiaomeng Xu, _Stokes phenomenon and Yang-Baxter equations_, [arxiv/1808.07654](https://arxiv.org/abs/1808.07654), [doi](https://doi.org/10.1007/s00220-019-03565-7); _Stokes phenomenon, dynamical r-matrices and Poisson geometry_, [doi](https://doi.org/10.13097/archive-ouverte/unige:84496); _Frobenius manifolds and quantum groups_ [arxiv/1801.00123](https://arxiv.org/abs/1801.00123); _Stokes phenomenon, Gelfand-Zeitlin systems and relative Ginzburg-Weinstein linearization_ [arxiv/1701.08113](https://arxiv.org/abs/1701.08113)

* [[Kenzo Ishikawa]], Kenji Nishiwaki, Kin-ya Oda, _New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon_ ([arXiv:2102.12032](https://arxiv.org/abs/2102.12032))

> We find a new contribution in wave-packet scatterings, which has been overlooked in the standard formulation of [[S-matrix]]. As a concrete example, we consider a two-to-two scattering of light scalars ϕ by another intermediate heavy scalar Φ, in the Gaussian wave-packet formalism: ϕϕ→Φ→ϕϕ. This contribution can be interpreted as an "in-time-boundary effect" of Φ for the corresponding Φ→ϕϕ decay, proposed by Ishikawa et al., with a newly found modification that would cure the previously observed ultraviolet divergence. We show that such an effect can be understood as a Stokes phenomenon in an integral over complex energy plane: The number of relevant saddle points and Lefschetz thimbles (steepest descent paths) discretely changes depending on the configurations of initial and final states in the scattering.

Proceedings volumes:

* B L J Braaksma, G K Immink, M van der Put, eds, _The Stokes Phenomenon and Hilbert's 16th Problem_, World Sci. 1996

* B L J Braaksma, G K Immink, M van der Put, J Top, eds, _Differential equation and the Stokes phenomenon_, World Sci. 2002

### In string theory

On [[non-perturbative effects]] and [[Stokes phenomena]] in [[topological string]] [[string scattering amplitude|amplitudes]]:

* Murad Alim, Arpan Saha, [[Jörg Teschner]], Iván Tulli, *Mathematical structures of non-perturbative topological string theory: from GW to DT invariants* ([arXiv:2109.06878](https://arxiv.org/abs/2109.06878))

[[!redirects Stokes phenomena]]


Stokes theorem

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### Context
#### Cohomology
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#### Differential geometry
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# Contents
* table of contents
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## Idea

The __Stokes theorem__ (also _Stokes\' theorem_ or _Stokes\'s theorem_) asserts that the [[integration of differential forms|integral]] of an [[exterior differential form]] on the boundary of an [[orientation|oriented]] [[manifold with boundary]] (or [[submanifold]] or [[chain]] of such) equals the integral of the [[de Rham differential]] of the form on the manifold itself.  (The theorem also applies to exterior pseudoforms on a chain of pseudoriented submanifolds.)

## Statement

### Traditional statement

Let

$$
  \Delta_{Diff} : \Delta \to Diff
$$

be the [[cosimplicial object]] of standard $k$-[[simplices]] in [[SmoothMfd]]: in degree $k$ this is the standard $k$-[[simplex]] $\Delta^k_{Diff} \subset \mathbb{R}^k$ regarded as a [[smooth manifold]] [[manifold with corners|with boundary and corners]]. This may be parameterized as

$$
  \Delta^k = \{ t^1, \cdots, t^k \in \mathbb{R}_{\geq 0} | \sum_i t^i \leq 1\}
  \subset \mathbb{R}^k
  \,.
$$

In this parameterization the coface maps of $\Delta_{Diff}$ are

$$
  \partial_i : (t^1, \cdots, t^{k-1}) \mapsto
  \left\{
    \array{
       (t^1, \cdots, t^{i-1}, t^{i+1} , \cdots, t^{k-1}) & | i \gt 0
       \\
       (1- \sum_{i=1}^{k-1} t^i, t^1, \cdots, t^{k-1})
    }
  \right.
  \,.
$$

For $X$ any [[smooth manifold]] a **smooth $k$-simplex in $X$** is a [[smooth function]]

$$
  \sigma : \Delta^k \to X
  \,.
$$

The **boundary** of this simplex in $X$ is the [[chain]] (formal linear combination of smooth $(k-1)$-simplices)

$$
  \partial \sigma = \sum_{i = 0}^k (-1)^i \sigma \circ \partial_i
  \,.
$$

Let $\omega \in \Omega^{k-1}(X)$ be a degree $(k-1)$-[[differential form]] on $X$.

+-- {: .num_theorem}
###### Theorem
**(Stokes theorem)**

The [[integral]] of $\omega$ over the boundary of the simplex equals the integral of its [[de Rham differential]] over the simplex itself

$$
  \int_{\partial \sigma} \omega = \int_\sigma d \omega
  \,.
$$

=--

It follows that for $C$ any $k$-[[chain]] in $X$ and $\partial C$ its boundary $(k-1)$-chain, we have

$$
  \int_{\partial C} \omega = \int_{C} d \omega
  \,.
$$

More generally:

+-- {: .num_prop #StokesTheoremForFiberIntegration}
###### Proposition
**(Stokes theorem for [[fiber integration]])**

If $U$ is any [[smooth manifold]] and $\omega \in \Omega^\bullet(U \times \Sigma)$ is a differential form on the [[Cartesian product]], then with respect to [[fiber integration|fiber-wise]] [[integration of differential forms]]

$$
  \int_\Sigma \;\colon\;
  \Omega^{\bullet + dim(\Sigma)}(U \times \Sigma)
    \longrightarrow
  \Omega^\bullet(U)
$$

along $U \times \Sigma \overset{pr_1}{\to} U$ we have

$$
  \int_\Sigma d \omega
  \;=\;
  \int_{\partial_\Sigma} \omega
  +
  (-1)^{dim(\Sigma)} d \int_\Sigma \omega
  \,.
$$

=--

(e.g. [Gomi & Terashima 2000, remark 3.1](#GomiTerashima00))

### Abstract formulation in cohesive homotopy-type theory
 {#FormulationInCohesiveHomotopyTypeTheory}

We discuss here a general abstract formulation of differential forms, their integration and Stokes theorem in the axiomatics of [[cohesive homotopy type theory]] (following [Bunke-Nikolaus-Völkl 13, theorem 3.2](#BunkeNikolausVoelkl13)).

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] and write $T \mathbf{H}$ for its [[tangent cohesive (∞,1)-topos]].

Assume that there is an [[interval object]]

$$
  \ast \cup \ast \stackrel{(i_0, i_1)}{\longrightarrow}  \Delta^1
$$

"exhibiting the cohesion" (see at _[[continuum]]_) in that there is a (chosen) [[equivalence]] between the [[shape modality]] $\Pi$ and the [[localization of an (∞,1)-category|localization]] $L_{\Delta^1}$ at the the [[projection]] maps out of [[Cartesian products]] with this line $\Delta^1\times (-) \to (-)$

$$
  \Pi \simeq L_{\Delta^1}
  \,.
$$

This is the case for instance for the "standard [[continuum]]", the [[real line]] in $\mathbf{H} = $ [[Smooth∞Grpd]].

It follows in particular that there is a chosen [[equivalence of (∞,1)-categories]]

$$
  \flat(\mathbf{H})\simeq L_{\Delta^1}\mathbf{H}
$$

between the [[flat modality|flat]] [[modal types|modal homotopy-types]] and the $\Delta^1$-homotopy invariant homotopy-types.

Given a [[stable homotopy type]] $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ [[cohesion]] provides two objects

$$
  \Pi_{dR} \Omega \hat E \,,\;\;  \flat_{dR}\Sigma \hat E
  \;\;
  \in Stab(\mathbf{H})
$$

which may be interpreted as [[de Rham complexes]] with [[coefficients]] in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map

$$
  \array{
    \Pi_{dR}\Omega \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\Sigma \hat E
    \\
    & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}}
    \\
    &&
    \hat E
  }
$$

which interprets as the [[de Rham differential]] $\mathbf{d}$. See at _[[differential cohomology diagram]]_ for details.

Throughout in the following we leave the "inclusion" $\iota$ of "differential forms regarded as $\hat E$-connections on trivial $E$-bundles" implicit.

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

[[integration of differential forms|Integration of differential forms]] is the map

$$
  \int_{\Delta^1} \;\colon\;  [\Delta^1, \flat_{dR}\Sigma \hat E] \longrightarrow \Pi_{dR}\Omega \hat E
$$

which is induced via the [[homotopy cofiber]] property of $\flat_{dR}\Omega \hat E$ from the [[counit of a comonad|counit]] naturality square of the [[flat modality]] on $[(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -]$, using that this square exhibits a [[null homotopy]] due to the $\Delta^1$-homotopy invariance of $\flat \hat E$.

=--

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

Stokes' theorem holds:

$$
  \int_{\Delta^1} \circ \mathbf{d}
  \;\simeq\;
  i_1^\ast - i_0^\ast
  \,.
$$

=--

([Bunke-Nikolaus-Völkl 13, theorem 3.2](#BunkeNikolausVoelkl13))

## Classical forms

In early 20th-century [[vector analysis]] (and even today in undergraduate Calculus courses), the Stokes theorem took various classical forms about [[vector fields]] in the [[Cartesian space]] $\mathbb{R}^n$:

* if $n = 1$ and $k = 1$, then this is the second [[FTC|Fundamental Theorem of Calculus]]: $\int_{[a,b]} f' = f(b) - f(a)$, where $a \leq b$ are [[real numbers]] and $f$ is a [[continuously differentiable function]] on a [[neighbourhood]] of the [[interval]] $[a,b]$;

* if $k = 1$ more generally, then this is a generalized form of the FTC: $\int_C grad f \cdot \mathbf{T} = f(Q) - f(P)$, where $C$ is a continuously differentiable oriented [[curve]] in $\mathbb{R}^n$, $P$ and $Q$ are the beginning and ending points (respectively) of $C$, $\mathbf{T}$ is the [[unit vector]] field on $C$ tangent to $C$ in the direction given by its orientation, and $f$ is a continuously differentiable function on a neighbourhood of $C$;

* if $n = 2$ and $k = 2$, then this is [[Green's Theorem]] (see there for other forms; stated without proof by [[Cauchy]] in 1846, proved by [[Riemann]] in 1851): $\int\int_R (\partial{v}/\partial{x} - \partial{u}/\partial{y}) = \int_C (u \,\mathrm{d}x + v \,\mathrm{d}y)$, where $C$ is a continuously differentiable [[simple closed curve]] in $\mathbb{R}^2$ (oriented using the standard orientation on $\mathbb{R}^2$), $R$ is the region that it encloses (guaranteed by the [[Jordan curve theorem|Jordan Curve Theorem]]), and $u$ and $v$ are continuously differentiable functions of the coordinates $x$ and $y$ on a neighbourhood of $R$;

* if $n = 3$ and $k = 2$, then this is the __Kelvin--Stokes Theorem__ or __Curl Theorem__ (stated without proof by [[Stokes]] in 1854, proved by [[Hankel]] in 1861): $\int\int_R curl \mathbf{F} \cdot \mathbf{n} = \int_C \mathbf{F} \cdot \mathbf{T}$, where $R$ is a continuously differentiable [[pseudoorientation|pseudooriented]] [[surface]] in $\mathbb{R}^3$ with a continuously differentiable boundary $C$ (oriented to match the pseudoorientation of $R$ using the standard orientation on $\mathbb{R}^3$), $\mathbf{n}$ is the unit [[normal vector]] field on $R$ in the direction given by the pseudoorientation of $R$, $\mathbf{T}$ is the unit tangent vector field on $C$, and $\mathbf{F}$ is a continuously differentiable [[vector field]] on a neighbourhood of $R$;

* if $n = 3$ and $k = 3$, then this is the __Ostrogradsky--Gauss Theorem__ or __Divergence Theorem__ (special cases were proved by Gauss in 1813, the general case was proved by Ostrogradsky in 1826): $\int\int\int_D div \mathbf{F} = \int\int_R \mathbf{F} \cdot \mathbf{n}$, where $R$ is a continuously differentiable closed surface in $\mathbb{R}^3$, $D$ is the region that it encloses (guaranteed by the [[Jordan-Brouwer separation theorem|Jordan--Brouwer Separation Theorem]]), $\mathbf{n}$ is the outward-pointing unit normal vector field on $R$, and $\mathbf{F}$ is a continuously differentiable vector field on a neighbourhood of $D$;

* if $k = n$ more generally, then this is the generalized [[Divergence Theorem]] (proved by Ostrogradsky in 1836): $\int_D div \mathbf{F} = \int_R \mathbf{F} \cdot \mathbf{n}$, where $R$ is a continuously differentiable closed [[hypersurface]] in $\mathbb{R}^n$, $D$ is the region that it encloses, $\mathbf{n}$ is the outward-pointing unit normal vector field on $R$, and $\mathbf{F}$ is a continuously differentiable vector field on a neighbourhood of $D$.

## Related concepts

* [[Poincaré lemma]]

* **Stokes theorem**

  * [[nonabelian Stokes theorem]]

* [[de Rham theorem]]

* [[Hodge theorem]]

* a special case is [[Cauchy's integral theorem]]

## References
 {#References}

The basic statement:

* [[Raoul Bott]], [[Loring Tu]], Theorem 3.5 in: _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0))

In the generality of [[manifolds with corners]]:

* {#Lee12} [[John Lee]], Theorem 10.32 in: _Introduction to Smooth Manifolds_, Springer 2012 ([doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [pdf](https://lost-contact.mit.edu/afs/adrake.org/usr/rkh/Books/books/Introduction%20to%20Smooth%20Manifolds%20-%20J.%20Lee.pdf))

* [[Brian Conrad]], _Stokes' theorem with corners_ ([pdf](http://math.stanford.edu/~conrad/diffgeomPage/handouts/stokescorners.pdf), [[ConradStokesTheorem.pdf:file]])

Statement of the fiberwise Stokes theorem:

* {#GomiTerashima00} [[Kiyonori Gomi]], [[Yuji Terashima]], Remark 3.1 in: _A Fiber Integration Formula for the Smooth Deligne Cohomology_, International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf), [doi:10.1155/S1073792800000386](https://doi.org/10.1155/S1073792800000386))

* [[Liviu Nicolaescu]], Theorem 3.4.54 of: _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf), [[NicolaescuGeometryOfManifolds.pdf:file]])

and specifically for simplicial differential forms:

* Dupont, Ljungman, Theorem 5.10 in: _Integration of simllicial forms and Deligne cohomology_ ([arXiv:math/0402059](https://arxiv.org/abs/math/0402059))

Statement of the Stokes theorem in the full generality of [[fiber integration|fiberwise integration]] over fibers [[manifold with corners|with corners]]:

* [[Alberto Cattaneo]], Nima Moshayedi, Konstantin Wernli, equation (65) in: _Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary_ ([arXiv:1807.11782](https://arxiv.org/abs/1807.11782))

Discussion of [[chains]] of smooth [[singular simplices]]

* _Stokes' theorem on chains_ (pdf)

Discussion for manifolds with more general singularities on the boundary is in

* Friedrich Sauvigny, _Partial Differential Equations: Vol. 1 Foundations and Integral Representations_

Discussion in [[cohesive homotopy type theory]] is in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

[[!redirects Stokes Theorem]]
[[!redirects Stokes theorem]]
[[!redirects Stokes theorems]]

[[!redirects Stokes' theorem]]
[[!redirects Stokes' Theorem]]
[[!redirects Stokes’ theorem]]
[[!redirects Stokes’ Theorem]]
[[!redirects Stokes\' theorem]]
[[!redirects Stokes\' Theorem]]
[[!redirects Stokes's theorem]]
[[!redirects Stokes's Theorem]]
[[!redirects Stokes’s theorem]]
[[!redirects Stokes’s Theorem]]
[[!redirects Stokes\'s theorem]]
[[!redirects Stokes\'s Theorem]]

[[!redirects Kelvin-Stokes theorem]]
[[!redirects Kelvin-Stokes Theorem]]
[[!redirects Kelvin–Stokes theorem]]
[[!redirects Kelvin–Stokes Theorem]]
[[!redirects Kelvin--Stokes theorem]]
[[!redirects Kelvin--Stokes Theorem]]


Stolz conjecture

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###Context###
#### Index theory
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#### Riemannian geometry
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#### Cohomology
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[[!include cohomology - contents]]
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Stolz conjecture_ (also: Stolz-[[Gerald Höhn|Höhn]] conjecture) due to ([Stolz 96](#Stolz96)) asserts that if $X$ is a [[closed manifold]] with [[String structure]] which furthermore admits a [[Riemannian metric]] with [[positive number|positive]] [[Ricci curvature]], then its [[Witten genus]] vanishes.

See ([Dessai](#Dessai)) for a review.

## Motivation

One part of the reasoning that motivates the conjecture is the idea that [[string geometry]] should be a "[[looping and delooping|delooping]]" of [[spin geometry]], and that the [[Witten genus]] is roughly like the [[index of a Dirac operator]] on [[loop space]]. Now for [[spin geometry]] the [[Lichnerowicz formula]] implies that for [[positive number|positive]] [[scalar curvature]] there are no harmonic [[spinors]] on a [[Riemannian manifold]] $X$, and hence that the [[index]] of the [[Dirac operator]] vanishes. One might then expect that there is a sensible concept of scalar curvature of [[smooth loop space]] $\mathcal{L} X$ obtained by integrating the [[Ricci curvature]] on $X$ along loops ([[transgression]]). Therefore, in this reasoning, a positive [[Ricci curvature]] of $X$ would imply a positive [[scalar curvature]] of the [[smooth loop space]] $\mathcal{L}X$, thus a vanishing of the [[index]] of the "[[Dirac operator on smooth loop space]]", hence a vanishing of the Witten genus.

## Evidence

### From perturbation theory

The [[Witten genus]] is by construction the [[partition function]] of the [[heterotic string]], in the [[large volume limit]]. One may further approximate this in [[perturbative quantum field theory]]. Working equivalently with the [[2d (2,0)-superconformal QFT]] in [[perturbation theory]], one may say something. A decent review is in ([Yagi 10](#Yagi10)).

## Related concepts

* [[p-curvature]]

* [[elliptic cohomology]]

## References

The original article is

* {#Stolz96} [[Stephan Stolz]], _A conjecture concerning positive Ricci curvature and the Witten genus_, Mathematische Annalen Volume 304, Number 1 (1996) ([doi:10.1007/BF01446319](https://doi.org/10.1007/BF01446319))

Review:

* [[Anand Dessai]], _Some geometric properties of the Witten genus_, in: [[Christian Ausoni]], [[Kathryn Hess]], [[Jérôme Scherer]] (eds.) _Alpine Perspectives on Algebraic Topology_, Contemporary Mathematics 504 (2009) ([pdf](http://homeweb2.unifr.ch/dessaia/pub/papers/Arolla/DessaiArollaFinalRevised30June09.pdf), [[DessaiEllipticGenus.pdf:file]],
[doi:10.1090/conm/504](http://dx.doi.org/10.1090/conm/504))

Discussion in the context of [[perturbation theory]] for the [[2d (2,0)-superconformal QFT]] includes

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

[[!redirects Stolz-Höhn conjecture]]
[[!redirects Stolz-Hoehn conjecture]]
[[!redirects Stolz-Hoehn conjecture]]


Stone duality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

# Stone duality
* table of contents
{: toc}

## Idea

Stone duality is a subject comprising various [[dualities]] between [[space and quantity]] in the area of [[general topology]] and topological [[algebra]].

## Particular cases

### Locales and frames

Perhaps the most general duality falling under this heading is that between [[locales]] (on the [[space]] side) and [[frames]] (on the [[quantity]] side).  Of course, this duality is not very deep at all; the category [[Loc]] of locales is simply *defined* to be the [[opposite category|opposite]] of the category [[Frm]] of frames.  But there are several interesting dualities between [[subcategories]] of these.

### Topological spaces

Stone duality is often described for [[topological spaces]] rather than for [[locales]].  In this case, the most general duality is that between [[sober spaces]] and frames with enough points (which correspond to [[topological locale]]s).  In many cases, one requires the [[ultrafilter theorem]] (or other forms of the [[axiom of choice]]) in order for the duality to hold when applied to topological spaces, while the duality holds for locales even in [[constructive mathematics]].

### Coherent spaces and distributive lattices

Any [[distributive lattice]] generates a [[free object|free]] frame.  The locales which arise in this way can be characterized as the [[coherent locale]]s, and this gives a duality between distributive lattices and coherent locales.  Note that one must additionally restrict to "coherent maps" between coherent locales.  Also, at least assuming the [[axiom of choice]], every coherent locale is [[topological locale|topological]], so we may say "coherent space" instead.

### Stone spaces and Boolean algebras
 {#StoneSpacesAndBooleanAlgebras}

The duality which is due to [[Marshall Stone]], and which gives its name to the subject, is the duality between [[Stone spaces]] and [[Boolean algebras]].  Specifically, a [[distributive lattice]] is a Boolean algebra precisely when the free frame it generates is the topology of a Stone space, and any continuous map of Stone spaces is coherent.  Therefore, the category of Stone spaces is dual to the category of Boolean algebras.  The Boolean algebra corresponding to a Stone space consists of its [[clopen sets]].

This duality may be realized via a [[dualizing object]] as follows. The two-element [[Boolean algebra]] may be regarded as a Boolean algebra object $\mathbf{2}$ [[internalization|internal to]] the [[category]] of [[compact Hausdorff spaces]] $CH$. Thus, for each finitary Boolean algebra operation $\theta\colon \mathbf{2}^n \to \mathbf{2}$, there is a corresponding operation on the [[representable functor]] $CH(-, \mathbf{2}): CH^{op} \to Set$ given by

$$CH(-, \mathbf{2})^n \cong CH(-, \mathbf{2}^n) \stackrel{CH(-, \theta)}{\to} CH(-, \mathbf{2})$$

and therefore we obtain a lift

$$CH(-, \mathbf{2}): CH^{op} \to Bool$$

A _[[Stone space]]_ is by definition a [[totally disconnected topological space|totally disconnected]] [[compact topological space|compact]] [[Hausdorff topological space]]. Let $Stone \hookrightarrow CH$ denote the [[full subcategory]] of Stone spaces.

+-- {: .num_theorem}
###### Theorem (Stone representation)

The representable functor restricts to an [[equivalence of categories]] $Stone^{op} \to Bool$.
=--

This important theorem can be exploited to give a third description of the free Boolean algebra on a set $X$:

$$Bool(X) \cong CH(2^X, \mathbf{2})$$

where $2$ denotes the 2-element compact Hausdorff space, and $2^X$ the product space $\prod_X 2$. Indeed, the inverse equivalence

$$Bool^{op} \to Stone$$

takes a Boolean algebra $B$ to its [[spectrum]], i.e., the space of Boolean algebra maps $Bool(B, 2)$ (*this* $2$ is the two-element Boolean algebra $\mathbb{Z}_2$!) equipped with the [[Zariski topology]]. Applied to $B = Bool(X)$, we have

$$Bool(B, 2) \cong Set(X, 2) = 2^X$$

where the Zariski topology coincides with the [[product topology]] on $2^X$. By the equivalence, we therefore retrieve $Bool(X)$ as $CH(2^X, \mathbf{2})$. This in turn is identified with the Boolean algebra of [[clopen subset]]s of the generalised [[Cantor space]] $2^X$.

A second description of the inverse equivalence $Bool^{op} \to Stone$ comes about through the yoga of [[ambimorphic object|ambimorphic objects]]. Namely, the Boolean compact Hausdorff space $\mathbf{2}$ can equally well be seen as a [compact Hausdorff object](/nlab/show/compact+Hausdorff+object#comphaus) in the category of Boolean algebras. Thus, the representable functor $Bool(-, \mathbf{2}): Bool^{op} \to Set$ lifts canonically to a functor
$$Bool^{op} \to CH$$
and in fact part of the Stone representation theorem is that this factors through the inclusion $Stone \hookrightarrow CH$ as the inverse equivalence $Bool^{op} \to Stone$. In particular this lift determines the topology, providing an description alternative to the description in terms of the Zariski topology (although they are of course the same).

An extension of the classical Stone duality to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor) (see the references below).

### Stone spaces and profinite sets
 {#StoneSpacesAndProfiniteSets}

Note that a finite [[Stone space]] is necessarily [[discrete space|discrete]], and these correspond to the finite Boolean algebras, i.e. $FinSet \simeq FinStoneTop \simeq FinBool^{op}$.  However, since Boolean algebras form a [[locally finitely presentable category]], we have $Bool \simeq Ind(FinBool) \simeq Pro(FinSet)^{op}$ (see [[ind-object]] and [[pro-object]]).  In consequence, $StoneTop \simeq Pro(FinSet)$: i.e. [[Stone spaces]] are equivalent to *[[profinite sets]]*, in this context then often called _[[profinite spaces]]_.

One way of explaining this classical Stone duality is hence via the following sequence of [[equivalences of categories]]

$$
  Bool \simeq Ind(FinBool) \simeq Ind(FinSet^{op}) \simeq Pro(FinSet)^{op}
  \,,
$$

where "[[FinSet]]" is the [[category]] of [[finite sets]], "$Ind$" stands for [[ind-objects]], "$Pro$" for [[pro-objects]] and ${}^{op}$ for the [[opposite category]] and the [[equivalence of categories|equivalence]] $FinSet^{op} \simeq FinBool$ is that discussed at _[FinSet -- Opposite category](FinSet#OppositeCategory)_.

### Stonean spaces and complete Boolean algebras

The [[category]] of [[Stonean spaces]],
i.e., compact [[extremally disconnected]] Hausdorff topological spaces
equipped with open continuous maps as morphisms,
is contravariantly equivalent to the category
of [[complete Boolean algebras]] and continuous Boolean homomorphisms
as morphisms.

See [[complete Boolean algebra]] for more information.

### Profinite algebras

If $T$ is a [[Lawvere theory]] on $Set$, we can talk about *Stone $T$-algebras*, i.e. $T$-algebras with a compatible Stone topology, and compare the resulting category $T Alg(Stone)$ with the category $Pro(Fin T Alg)$ of pro-(finite $T$-algebras).  The previous duality says that these categories are equivalent when $T$ is the identity theory.  It is also true in many other cases, such as:

* the theory of [[groups]], resulting in the rich theory of [[profinite groups]]
* the theories of [[semigroups]] and [[monoids]]
* the theory of [[rings]] (with or without 1)
* the theories of [[distributive lattices]], [[Heyting algebras]], and [[Boolean algebras]]
* the theory of $M$-sets, where $M$ is a finite monoid
* the theories of $R$-modules and $R$-algebras, where $R$ is a finite ring

However it is false for some $T$, such as:

* the theory of $\mathbb{N}$-sets, i.e. sets equipped with an endomorphism
* the theory of [[Jónsson-Tarski algebras]]
* the theory of [[lattices]]

All of these can be found in chapter VI of Johnstone's book cited below.

The corresponding fact is also notably false for [[groupoids]], i.e. $Gpd(Stone)$ is not equivalent to $Pro(FinGpd)$, in contrast to the case for groups.  (Of course, groupoids are not described by a Lawvere theory.)

## Related concepts

* [[canonical extension]]

* [[syntax - semantics duality]]

* [[abstract Stone duality]]

* [[Gelfand duality]]

* [[Isbell duality]]

* [[Stone gamut]]

## References

* [[Peter Johnstone]], _[[Stone Spaces]]_

* [[Olivia Caramello]], _A topos-theoretic approach to Stone-type dualities_, [arxiv/1103.3493](http://arxiv.org/abs/1103.3493) 158 pp.

* G. D. Dimov, Some generalizations of the Stone Duality Theorem, Publ. Math. Debrecen 80/3-4 (2012), 255--293.

* G. Dimov, E. Ivanova-Dimova, [[W. Tholen]], _Categorical extension of dualities: From Stone to de Vries and beyond, I._ Appl. Categ. Struct. 30, 287–329 (2022) [doi](https://doi.org/10.1007/s10485-021-09658-6)

* H. P. Doctor, _The categories of Boolean lattices, Boolean rings and Boolean spaces_, Canad. Math. Bulletin 7 (1964), 245--252 [doi](https://doi.org/10.4153/CMB-1964-022-6)

* [[Jacob Lurie]], section A.1.1 of _[[Spectral Algebraic Geometry]]_.

There is a version in model theory, [[Makkai duality]],

* [[M. Makkai]], _Stone duality for first-order logic_, Adv. Math. __65__ (1987) no. 2, 97--170, doi, [MR89h:03067](http://www.ams.org/mathscinet-getitem?mr=900266); _Duality and definability in first order logic_, Mem. Amer. Math. Soc. __105__ (1993), no. 503

Other variants are in

* Henrik Forssell, _First-order logical duality_, Ph.D. thesis, Carnegie Mellon U. 2008, [pdf](http://www.andrew.cmu.edu/user/awodey/students/forssell.pdf)

* Spencer Breiner, _Scheme representation for first-order logic_, Ph.D. thesis, Carnegie Mellon U. 2014, [pdf](https://www.andrew.cmu.edu/user/awodey/students/breiner.pdf)

Discussion in [[E-∞ geometry]] is in

* [[Jacob Lurie]], section A of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

Discussion of an $(\infty, 1)$-version of Stone duality is in

* [[Jacob Lurie]], section 3.5.3 of [[Spectral Algebraic Geometry]]

[[!redirects Stone duality]]
[[!redirects Stone dualities]]

[[!redirects Stone representation theorem]]
[[!redirects Stone representation theorems]]
[[!redirects Stone's representation theorem]]
[[!redirects Stone's representation theorems]]
[[!redirects Stone's representation theorem]]
[[!redirects Stone's representation theorems]]

[[!redirects profinite algebra]]
[[!redirects profinite algebras]]

[[!redirects Stone algebra]]
[[!redirects Stone algebras]]

[[!redirects Stone topological algebra]]
[[!redirects Stone topological algebras]]


Stone gamut

# Stone gamut
* this block creates the table of contents, leave as is
{: toc}

## Idea

The Stone gamut is a unifying representation of several common categories in terms of [[Chu space|Chu spaces]]. It is named after [[Stone duality]]. The study of Chu duality and the Stone gamut is known as **chupology**. (Pratt 2006)

## Definition

Let **$Chu_2$** be the category of Chu spaces over the Booleans, with $r : Chu_2 \to \mathbb{N}$ and $c : Chu_2 \to \mathbb{N}$ counting the number of rows and columns respectively, and let **$FinChu_2$** be its [[full subcategory]] of finite Chu spaces. Let a **skeleton** of a Chu space be another Chu space with no repeated rows or repeated columns, and let the **dual** of a Chu space $A$, written $A^{\bot}$, be its transpose. Let a Chu space be **skeletal** if it is a skeleton. Let a *discreteness* be a function $\delta : FinChu_2 \to [-1, 1]$ such that:

* $\delta$ depends only on the number of rows and columns in its input's skeleton, and
* $\delta(A^{\bot}) = -\delta(A)$

For our purposes, we will consider Pratt discreteness, defined as:

$$
\delta(A) = \frac{P(A) - Q(A)}{P(A) + Q(A)}
$$

Where $P(A) = |2^{r(A)} - c(A)|$ and $Q(A) = |2^{c(A)} - r(A)|$. Intuitively, $P$ and $Q$ count the number of "missing" rows and columns in the skeleton.

A **property** of a skeletal Chu space is a superset of its columns. (Def. 5, Pratt 1999)

## Properties

Note that when $r(A) \gt c(A)$, $r(A) \leq 2^{c(A)}$; and dually, when $c(A) \gt r(A)$, $c(A) \leq 2^{r(A)}$, by the pigeonhole principle.

Pratt notes that their $\delta$ has the "odd property that the only discreteness possible in $[-\frac{1}{3}, \frac{1}{3}]$ is 0" (Pratt 1995).

## Examples

### Contravariant dualities

Category | $\delta$ |[[space and quantity|Spaces $\cong$ Quantities]] | $\delta$ | Category
---|---|---|---|---
[[CABA]] | -1 | [[complete Boolean algebra|CABAs]]/[[overlap algebra|overlap algebras]] $\cong$ sets | 1 | [[Set]]
[[SupLat]] | 0 | [[suplattice|suplattices]] $\cong$ inflattices | 0 | [[InfLat]]
[[Vect|$Vect_{\mathbb{F}_2}$]] | 0 | [[vector space|vector spaces]] over $\mathbb{F}_2$ $\cong$ [[dual vector space|dual vector spaces]] over $\mathbb{F}_2$ | 0 | $Vect_{\mathbb{F}_2}$

## References

* {#Pratt1995} [[Vaughan Pratt]], _The Stone gamut: a Coordinatization of Mathematics_, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science (1995) [[doi:10.1109/LICS.1995.523278](https://doi.org/10.1109/LICS.1995.523278), [pdf](http://boole.stanford.edu/pub/gamut.pdf)]
* {#Pratt1999} [[Vaughan Pratt]], _Chu Spaces_, School on Category Theory and Applications, University of Coimbra (1999) [[notes (pdf)](http://boole.stanford.edu/pub/coimbra.pdf)]
* {#Pratt2006} [[Vaughan Pratt]], _dualities_, *Categories list* mailing list (2006) [[list archive](http://www.mta.ca/~cat-dist/archive/2006/06-5)]


Stone locale

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

#Contents#
* table of contents
{:toc}

## Definition

A **Stone locale** is a [[compact space|compact]] [[zero-dimensional space|zero-dimensional]] [[locale]].

## Properties

In presence of the [[axiom of choice]],
every Stone locale is [[spatial]]
and the category of Stone locales
is equivalent to the category of [[Stone spaces]].

## Stone duality

The [[Stone duality]] theorem states
that the category of Stone locales is contravariantly equivalent
to the category of [[Boolean algebras]].

Unlike the corresponding statement for [[Stone spaces]],
this version is fully [[constructive]] and is valid
in any [[W-topos]].

In fact, the traditional [[Stone duality]] is an immediate consequence
of the localic Stone duality and the spatiality of [[Stone locales]].

## Related concepts

* [[Stone space]]
* [[Stonean locale]]

## References

[[!redirects Stone locales]]


Stone space

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

#Contents#
* table of contents
{:toc}

## Definition

A **Stone space** is a [[compact space|compact]], [[Hausdorff space|Hausdorff]] [[totally disconnected]] [[topological space]].

Stone spaces are sometimes called _[[profinite spaces]]_, since they are precisely the spaces which are small [[cofiltered limits]] of finite [[discrete spaces]], and moreover (as a consequence of [[Stone duality]]) the [[category]] of Stone spaces is [[equivalence of categories|equivalent]] to the category $pro(FinSet)$ of [[pro-objects]] in [[FinSet]] and finite sets sit $FinSet\hookrightarrow pro(FinSet)$ as finite discrete spaces.  This is especially common when talking about [[profinite groups]] and related topics.

## Related concepts

* [[Stone locale]]
* [[Stonean space]]
* [[profinite completion]]

## References

A standard textbook is

* [[Peter Johnstone]], _[[Stone Spaces]]_

See also

* MathOverflow: _[Why are profinite topologies important?](http://mathoverflow.net/questions/49483/why-are-profinite-topologies-important)_

See also

* [[S. B. Niefield]], [[K.I. Rosenthal]], _Sheaves of integral domains on stone spaces_, Journal of Pure and Applied Algebra Volume 47, Issue 2, August 1987, Pages 173–179

[[!redirects Stone spaces]]


Stone Spaces

This page compiles material related to:

* [[Peter Johnstone]]:

  **Stone Spaces**

  Cambridge Studies in Advanced Mathematics __3__,

  Cambridge University Press (1982, 1986)

  xxi+370 pages

  [ISBN:9780521337793](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/stone-spaces?format=PB&isbn=9780521337793)

  [MR85f:54002](http://www.ams.org/mathscinet-getitem?mr=698074), .

The monograph is ultimately about the [[Stone representation theorem]], but also a standard reference on using [[locales]] in place of [[topological spaces]].

(The same author wrote *[[Topos Theory]]* 1977 and the **[[Sketches of an Elephant]]*, both on [[topos theory]]).

Although it is a work of [[mathematics]] rather than [[foundations|metamathematics]], it shows clearly by example how (usually) results about locales do not require the [[axiom of choice]] even when analogous results about topological spaces do.  [[Paul Taylor]] has somewhat imprecisely written of this book

>the public theorems about topology \[...\] are marked with an asterisk, although the official meaning of that symbol is a dependence on the axiom of choice. ([ASD I](http://www.monad.me.uk/ASD/dedras/), page 3).

Unfortunately for [[constructive mathematics|constructive mathematicians]], [[excluded middle]] is *not* considered a form of choice by Johnstone.

A trailer for the book (according to its own words) is

* {#Johnstone83} [[Peter Johnstone]], _The point of pointless topology_ ,  Bull. Amer. Math. Soc. (N.S.) Volume 8, Number 1 (1983), 41-53. ([Bulletin AMS](http://www.ams.org/bull/1983-08-01/S0273-0979-1983-15080-2/home.html), [Euclid](http://projecteuclid.org/euclid.bams/1183550014))

## Contents with links to $n$Lab pages

Besides the usual prefaces, bibliography, and indexes, there is a historical introduction, and each chapter concludes with notes on historical and metamathematical aspects.  Otherwise, each of 7 chapters is divided into 4 sections, which in turn contain paragraphs that deal with essentially one idea each.  For the moment, we list (with minimal processing) the definitions from the index in each section.  There will also be some summaries of theorems; as in the book itself, an asterisk here indicates dependence on some form of choice beyond excluded middle (more precisely, a proof that cannot be internalised in an arbitrary [[boolean topos]]).

1.  Preliminaries

    1.  Lattices
        1.  [[poset]]
        2.  [[join]]
        3.  [[semilattice]]
        4.  [[meet]], [[bounded lattice]]
        5.  [[distributive lattice]]
        6.  [[complement]], [[Boolean algebra]]
        7.  (none)
        8.  [[symmetric difference]]
        9.  [[Boolean ring]]
        10. [[implication]], [[Heyting algebra]]
        11. [[pseudocomplement]]
        12. (none)
        13. [[regular element]] (in a Heyting algebra)
    2.  Ideals and filters
        1.  [[ideal]] (in a lattice or semilattice), [[lower set]], [[principal ideal]]
        2.  [[filter]] (in a lattice or semilattice), [[prime ideal]] (in a lattice), [[prime filter]] (in a lattice)
        3.  \* [[maximal ideal theorem]]
        4.  [[maximal ideal]]
        5.  \* discrete Stone representation theorem
        6.  (none)
    3.  Some categorical concepts
        1.  [[category]], [[functor]], [[natural transformation]]
        2.  [[concrete category]], [[locally small category]]
        3.  posets as categories
        4.  [[adjoint functors]], [[free functor]], [[reflective subcategory]], [[equivalence of categories]], [[opposite category]]
        5.  [[limit]], [[diagram]], [[small category]], [[colimit]], [[regular monomorphism]], [[complete category]], [[finitely complete category]]
        6.  [[monad]], [[algebra for a monad]], [[monadic adjunction]]
        7.  [[variety of algebras]]
        8.  [[algebraic category]], [[equationally presentable category]]
        9.  [[filtered category]], [[filtered colimit]], [[finitary functor]]
    4.  Free lattices
        1.  [[directed poset]], [[directed join]]
        2.  (none)
        3.  [[suplattice]], [[complete lattice]]
        4.  [[complete Boolean algebra]]; [[free semilattice]]
        5.  [[free suplattice]]
        6.  [[free lattice]]
        7.  [[free complete lattice]]
        8.  [[free distributive lattice]]
        9.  [[free boolean algebra]]
        10. [[free complete boolean algebra]]
        11. [[free Heyting algebra]]
2.  Introduction to locales

    1.  Frames and locales
        1.  [[frame]], [[locale]], [[subframe]]
        2.  [[free frame]]
        3.  [[point of a locale]], [[completely prime filter]], [[prime element]]
        4.  adjunction between $Loc$ and $Top$
        5.  [[spatial locale]]
        6.  [[irreducible closed subspace]], [[sober space]]
        7.  [[soberification]], $T_D$-[[T-D-space|space]]
        8.  [[specialization order]], [[Alexandroff topology]], [[upper set]], [[upper interval topology]]
        9.  [[Scott topology]]
        10. (none)
        11. enrichment of $Loc$ over $Pos$
    2.  Sublocales and sites
        1.  (none)
        2.  [[nucleus]]
        3.  [[sublocale]]
        4.  [[closed nucleus]], [[closed sublocale]], [[open nucleus]], [[open sublocale]], [[dense sublocale]], [[dense nucleus]], [[double-negation nucleus]]
        5.  (none)
        6.  (none)
        7.  (none)
        8.  (none)
        9.  (none)
        10. $Loc$ is not well-powered
        11. [[(0,1)-coverage|coverage]], [[(0,1)-site|site]], [[(0,1)-sheaf|sheaf]]
        12. [[localic product]]
        13. (none)
        14. (none)
    3.  Coherent locales
        1.  [[compact element]] (in a lattice)
        2.  [[coherent locale]]
        3.  [[coherent map]] (of locales); local Stone representation theorem for distributive lattices
        4.  [[coherent space]], [[prime spectrum]]; \* spatial Stone representation theorem for distributive lattices
        5.  [[maximal spectrum]]
        6.  [[normal distributive lattice]]
        7.  (none)
    4.  Stone spaces
        1.  [[totally disconnected space]], [[totally separated space]], [[zero-dimensional space]]
        2.  [[Stone space]]
        3.  (none)
        4.  \* Stone representation theorem for Boolean algebras, [[Stone duality]]
        5.  [[patch topology]]
        6.  (none)
        7.  [[totally order-separated space]], [[ordered Stone space]]
        8.  \* $Ord Sto Top \cong Coh Top$
        9.  (none)
3.  [[compactum|Compact Hausdorff spaces]]

    1.  Compact regular locales
        1.  [[compact locale]], [[regular locale]], [[well inside containment]], [[zero-dimensional locale]]
        2.  (none)
        3.  [[strongly Hausdorff locale]]
        4.  (none)
        5.  [[totally unordered locale]]
        6.  $Reg Loc$ is complete
        7.  $Comp Loc$ is complete ([[Tychonoff theorem]] for locales)
        8.  localic Stone--Čech compactification
        9.  (none)
        10. \* $Comp Reg Loc \cong Comp Haus Top$
        11. [[flat sublocale]]
    2.  [[Manes theorem|Manes' Theorem]]
        1.  [[ultrafilter]]
        2.  [[filter]] (on a set), [[neighbourhood filter]], [[convergence|limit]] (of a filter)
        3.  (none)
        4.  \* $Comp Haus Top$ is monadic
        5.  \* $Comp Haus Top A$ is monadic for $A$ a variety of algebras
    3.  [[Gleason's Theorem]]
        1.  [[projective object]]
        2.  (none)
        3.  (none)
        4.  (none)
        5.  [[extremally disconnected locale]], [[extremally disconnected space]]
        6.  (none)
        7.  [[projective compact Hausdorff space]]
        8.  [[proper map]]
        9.  (none)
        10. (none)
        11. [[MacNeille cut]], [[MacNeille completion]]
    4.  Vietoris locales
        1.  [[Vietoris space]]
        2.  [[Vietoris topology]], [[lower interval topology]]
        3.  [[Vietoris locale]]
        4.  (none)
        5.  (none)
        6.  (none)
        7.  (none)
        8.  (none)
4.  Continuous real-valued functions

    1.  Complete regularity and [[Urysohn lemma|Urysohn's Lemma]]
        1.  (none)
        2.  (none)
        3.  (none)
        4.  [[scale]], [[really inside containment]]
        5.  [[completely regular locale]]
        6.  [[normal locale]]
        7.  localic [[Tychonoff embedding theorem]]
    2.  The Stone--Čech compactification
        1.  [[Stone–?ech compactification]]
        2.  [[completely regular ideal]], [[regular ideal]]
        3.  [[completely regular filter]]
        4.  [[Wallman base]], [[Wallman compactification]]
        5.  [[cozero set]]
        6.  (none)
        7.  [[Alexandroff compactification]]
        8.  (none)
        9.  [[cozero element]], [[Alexandroff algebra]]
        10. (none)
        11. (none)
    3.  $C(X)$ and $C^*(X)$
        1.  (none)
        2.  [[lattice]]
        3.  [[Zariski topology]]
        4.  [[Gelfand-Kolmogorov theorem|Gelfand–Kolmogorov theorem]]
        5.  [[fixed maximal ideal]]
        6.  (none)
        7.  [[real point]] (of $\beta X$), [[realcompact space]], [[pseudocompact space]]
        8.  [[Hewitt realcompactification]]
        9.  (none)
        10. (none)
        11. (none)
        12. (none)
    4.  [[Gelfand duality]]
        1.  (none)
        2.  (none)
        3.  [[Stone-Weierstrass theorem|Stone–Weierstrass theorem]]
        4.  $C^*$-[[C*-algebra|algebra]]
        5.  (none)
        6.  (none)
        7.  (none)
        8.  (none)
        9.  (none)
        10. [[Stone-Gelfand-Naimark theorem|Stone–Gelfand–Naimark theorem]]
        11. [[Dedekind-complete poset]]
        12. $\mathit{MI}$-[[MI-space|space]]
5.  Representations of rings

    1.  A crash course in sheaf theory
        1.  (none)
        2.  [[bundle]]
        3.  [[trivial bundle]], [[sheaf]] (on a space), [[presheaf]] (on a space)
        4.  [[display space]]
        5.  [[local homeomorphism]]
        6.  (none)
        7.  (none)
        8.  [[direct image functor]], [[inverse image functor]]
        9.  (none)
        10. [[coherent logic]], [[field]]
        11. (none)
        12. [[cartesian logic]]
        13. [[regular logic]]
    2.  The Pierce spectrum
        1.  (none)
        2.  [[indecomposable ring]]
        3.  [[Pierce sheaf]], [[Pierce representation]], [[Pierce spectrum]]
        4.  (none)
        5.  (none)
        6.  [[von Neumann regular ring]]
        7.  [[local ring]], [[exchange ring]]
        8.  [[neat ideal]] (in a ring)
        9.  (none)
        10. (none)
    3.  The Zariski spectrum
        1.  [[Zariski spectrum]]
        2.  [[prime filter]] (in a ring), [[radical ideal]] (in a ring), [[semiprime ring]]
        3.  [[Zariski sheaf]]
        4.  (none)
        5.  [[Zariski representation]], [[local homomorphism]] (of rings)
        6.  [[nilradical]]
        7.  [[Gelfand ring]]
        8.  (none)
        9.  (none)
        10. (none)
        11. [[integral domain]], [[domain spectrum]], [[domain representable ring]]
        12. [[field spectrum]]
        13. (none)
    4.  Ordered rings and real rings
        1.  [[ordered ring]], [[positive cone]]
        2.  [[concave prime filter]] (in a ring), [[Brumfiel spectrum]]
        3.  [[shadow]], [[local ordered ring]]
        4.  $L$-[[L-ring|ring]]
        5.  (none)
        6.  $L$-[[L-ideal|ideal]] (in a ring), $F$-[[F-ring|ring]], [[Keimel spectrum]], [[irreducible L-ideal|irreducible]] $L$-ideal
        7.  (none)
        8.  (none)
        9.  $L$-[[L-local F-ring|local]] $F$-ring
        10. $L$-[[L-simple F-ring|simple]] $F$-ring
        11. [[formally real field]], [[real-closed field]], [[real point]] (of $spec A$), [[strictly positive filter]] (in a ring)
        12. [[real spectrum]]
        13. [[formally real local ring]], [[ordered local ring]]
6.  Profiniteness and duality

    1.  Ind-objects and pro-objects
        1.  (none)
        2.  [[ind-object]]
        3.  [[finitely continuous functor]]
        4.  (none)
        5.  [[final functor]]
        6.  (none)
        7.  (none)
        8.  [[cocompletion]], [[finitely-presentable object]]
        9.  [[pro-object]]
    2.  Profinite sets and algebras
        1.  (none)
        2.  (none)
        3.  [[profinite set]]
        4.  (none)
        5.  [[Jonsson-Tarski algebra|Jónsson–Tarski algebra]]
        6.  [[congruence]]
        7.  (none)
        8.  (none)
        9.  (none)
        10. (none)
    3.  [[Stone duality|Stone-type dualities]]
        1.  (none)
        2.  (none)
        3.  (none)
        4.  (none)
        5.  (none)
        6.  [[algebraic lattice]]
        7.  (none)
    4.  General [[concrete duality|concrete dualities]]
        1.  [[coseparator]], [[schizophrenic object]]
        2.  (none)
        3.  (none)
        4.  (none)
        5.  [[idempotent adjunction]]
        6.  [[Sierpinski topology|Sierpi?ski topology]], [[Sierpinski space|Sierpi?ski space]]
        7.  (none)
        8.  (none)
        9.  (none)
        10. (none)
        11. (none)
7.  Continuous lattices

    1.  [[compact lattice|Compact topological (semi)lattices]]
        1.  [[ordered space]], [[topological poset]], [[order-Hasudorff space]]
        2.  [[order-normal space]]
        3.  (none)
        4.  (none)
        5.  [[continuously distributive lattice]]
        6.  (none)
        7.  (none)
        8.  (none)
        9.  (none)
        10. [[completely distributive lattice]]
        11. [[interval topology]]
        12. (none)
        13. (none)
        14. (none)
        15. (none)
        16. (none)
        17. (none)
    2.  Continuous posets and lattices
        1.  [[ideal]] (in a poset)
        2.  [[way below containment]], [[continuous poset]], [[continuous lattice]]
        3.  [[algebraic poset]]
        4.  (none)
        5.  [[filter]] (in a poset), [[Scott-open filter]]
        6.  (none)
        7.  (none)
        8.  (none)
        9.  (none)
        10. [[Lawson map]]
        11. [[continuous semilattice]]
        12. [[stably continuous poset]]
    3.  Lawson semilattices
        1.  (none)
        2.  (none)
        3.  [[Lawson topology]], [[Lawson semilattice]]
        4.  (none)
        5.  (none)
        6.  (none)
        7.  (none)
        8.  (none)
    4.  Locally compact locales
        1.  (none)
        2.  [[locally compact locale]]
        3.  (none)
        4.  (none)
        5.  (none)
        6.  [[stably locally compact locale]]
        7.  [[injective sober space]]
        8.  (none)
        9.  [[injective locale]]
        10. [[exponentiable object]], [[exponentiable locale]]
        11. (none)
        12. [[exponentiable space]]

category: reference


Stone-von Neumann theorem

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

# Contents
* table of contents
{: toc}

## Idea

The _Stone--von Neumann theorem_ (due to [[Marshall Stone]] and [[John von Neumann]]) says that there is -- up to [[isomorphism]] -- a unique [[irreducible representation|irreducible]] [[unitary representation]] of the [[Heisenberg group]] on finitely many generators (equivalently: of the [[Weyl algebra]], [[Weyl relations]], [[canonical commutation relations]]).

The analogous statement does _not_ hold for infinitely many generators (as they appear in [[quantum field theory]]); this is _[[Haag's theorem]]_.

Explicitly, the [[canonical commutation relations]] on two generators ([[canonical coordinate]] $q$ and [[canonical momentum]] $p$) in the form

$$
  [q,p] = i \hbar
$$

may be represented as [[unbounded operators]] on the [[Hilbert space]] of [[square integrable functions]] $L^2(\mathbb{R})$ on the [[real line]] by defining them on the [[dense subspace]] of [[smooth functions]] $\psi \colon \mathbb{R} \to \mathbb{C}$ as

$$
  (q \psi)(x) \coloneqq x \psi(x)
  \phantom{AAAA}
  (p \psi)(x) \coloneqq -i \hbar \frac{\partial}{\partial x} \psi(x)
  \,,
$$

where on the right we have the [[derivative]] along the canonical [[coordinate function]] on $\mathbb{R}$.

This is often called the _Schrödinger representation_, after [[Erwin Schrödinger]], cf. eg. [Redei](#Redei) (to be distinguished from "[[Schrödinger picture]]" which is a related but different concept).

## Related concepts

* [[Weyl relation]], [[Weyl algebra]]

* [[canonical commutation relation]]

* For the Schrödinger representation obtained via [[geometric quantization]] see [there](geometric+quantization#ExamplesSchroedingerRepresentation).

## References

The original articles:

* [[John von Neumann]], *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104**  (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]

* [[John von Neumann]], _Über Einen Satz Von Herrn M. H. Stone_, Annals of Mathematics, Second Series **33** 3 (1932) 567-573 [[doi:10.2307/1968535](https://doi.org/10.2307/1968535), [jstor:1968535](https://www.jstor.org/stable/1968535)]

Review:

* {#Redei} [[Miklós Rédei]], _Von Neumann's proof of Uniqueness of Schrödinger representation of Heisenberg's commutation relation_ ([[RedeiCCRRepUniqueness.pdf:file]])

Further discussion:

* [[Marc Rieffel]], _On the uniqueness of the Heisenberg commutation relations_, Duke Math. J. **39**  4 (1972) 745-752 [[doi:10.1215/S0012-7094-72-03982-8](https://projecteuclid.org/journals/duke-mathematical-journal/volume-39/issue-4/On-the-uniqueness-of-the-Heisenberg-commutation-relations/10.1215/S0012-7094-72-03982-8.short)]

See also

* Wikipedia, _[Stone-von Neumann theorem](https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem)_

[[!redirects Stone von Neumann theorem]]
[[!redirects Stone-von Neumann theorem]]
[[!redirects Stone–von Neumann theorem]]
[[!redirects Stone--von Neumann theorem]]

[[!redirects Schrödinger representation]]
[[!redirects Schrödinger representations]]

[[!redirects Schroedinger representation]]
[[!redirects Schroedinger representations]]


Stone-Weierstrass theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _Stone--Weierstrass theorem_ says given a [[compact Hausdorff space]] $X$, one can uniformly approximate [[continuous functions]] $f: X \to \mathbb{R}$ by elements of any subalgebra that has enough elements to distinguish points. It is a far-reaching generalization of a classical theorem of Weierstrass, that real-valued continuous functions on a closed [[interval]] are uniformly approximable by [[polynomial]] functions.

## Precise statement

Let $X$ be a compact Hausdorff topological space; for a [[constructive mathematics|constructive]] version take $X$ to be a compact regular locale (see [[compactum]]). Recall that the algebra $C(X)$ of real-valued continuous functions $f: X \to \mathbb{R}$ is a commutative (real) [[Banach algebra]] with unit, under pointwise-defined addition and multiplication, and where the norm is the sup-norm

$$\|f\| := sup_{x \in X} |f(x)|$$

A **subalgebra** of $C(X)$ is a vector subspace $A \subseteq C(X)$ that is closed under the unit and algebra multiplication operations on $C(X)$. A **Banach subalgebra** is a subalgebra $A \subseteq C(X)$ which is closed as a subspace of the metric space $C(X)$ under the sup-norm metric. We say that $A \subseteq C(X)$ **separates points** if, given distinct points $x, y \in X$, there exists $f \in A$ such that $f(x) \neq f(y)$.

+-- {: .un_thm}
###### Theorem (Stone--Weierstrass)
A subalgebra inclusion $A \subseteq C(X)$ is dense if and only if it separates points. Equivalently, a Banach subalgebra inclusion $A \subseteq C(X)$ is the identity if and only if it separates points.
=--

## Outline of proof

* The first step is the classical Weierstrass approximation, that polynomial functions $p(x): [a, b] \to \mathbb{R}$ are dense in $C([a, b])$; without loss of generality, take $a = -\frac1{4}$ and $b = \frac1{4}$. We use the method of constructing polynomials which [[approximation of the identity|approximate the identity]] of the convolution product (the [[distribution|Dirac distribution]]). Explicitly, consider the normalizations $K_n = s_n/\|s_n\|_1$ where $s_n(x) = (1 - x^2)^n$ over $[-1, 1]$ and is compactly supported on $[-1, 1]$, so that $K_n$ approximates the Dirac distribution concentrated at 0. Given $f: [-\frac1{4}, \frac1{4}] \to \mathbb{R}$, extend to a function compactly supported on $[-\frac1{2}, \frac1{2}]$. The convolution product
$$(K_n * f)(x) = \int_{-1}^1 K_n(x-y) f(y) d y$$
is polynomial (since by differentiating under the integral enough times, we eventually kill the convolving polynomial factor $K_n$), and one may verify that $(K_n * f)$ converges to $f$ in sup norm (i.e., uniformly) as $n \to \infty$, and in particular when restricted over the interval $[-\frac1{4}, \frac1{4}]$.

Now suppose given a Banach subalgebra $A \subseteq C(X)$.

* Given $f \in A \subseteq C(X)$, the values of $f$ are contained in some interval $[a, b]$. If polynomials $p_n$ converge to the absolute value function $|\cdot |: [a, b] \to \mathbb{R}$ in sup norm, then $p_n(f) \in A$ converges to $|f| \in C(X)$ in sup norm. Since $A$ is closed in $C(X)$ with respect to the sup-norm, it follows that $|f| \in A$.

* Next, $A$ is partially ordered by $f \leq g$ if $f(x) \leq g(x)$ for all $x \in X$, and we claim the poset $A$ is closed under binary meets and binary joins. For,
$$f \vee g = \frac1{2}(|f - g| + (f + g))$$
and $f \wedge g = -((-f) \vee (-g))$, and we showed in the last step that $A$ is closed under the operation $f \mapsto |f|$.

Finally, suppose the Banach subalgebra $A$ separates points. Given $g \in C(X)$ and $\varepsilon \gt 0$, the last step is to show there exists $f \in A$ such that $\|f - g\| \leq \varepsilon$.

* +-- {: .un_lemma}
  ###### Lemma
  Given $x \in X$, there exists $f_x \in A$ such that $f_x(x) = g(x)$ and $g(y) \leq f_x(y) + \varepsilon$ for all $y \in X$.
  =--
  +-- {: .proof}
  ###### Proof
  For each $y \in X$, $y \neq x$ we can choose $s \in A$ such that $s(x) \neq s(y)$, since $A$ separates points. Thus, given any $x, y$, there exist $s \in A$ and scalars $a$ and $b$ such that
  $$a + b \cdot s(x) = g(x), \qquad a + b \cdot s(y) = g(y)$$
  Denote $a \cdot 1 + b s$ by $f_{x, y}$ (to indicate dependence on $x$ and $y$). For each $y$, choose a neighborhood $V_y$ so that $g(z) \leq f_{x, y}(z) + \varepsilon$ for all $z \in V_y$. Finitely many such neighborhoods $V_{y_1}, \ldots, V_{y_n}$ cover $X$; let $f_x$ be the join of $f_{x, y_1}, \ldots, f_{x, y_n}$. Then $g(y) \leq f_x(y) + \varepsilon$ for all $y \in X$.
  =--

* Given a choice of $f_x$ for each $x \in X$, as in the preceding lemma, we may choose a neighborhood $V_x$ such that $f_x(z) - \varepsilon \leq g(z)$ for all $z \in V_x$. Finitely many such neighborhoods $V_{x_1}, \ldots, V_{x_n}$ cover $X$; let $f$ be the meet of $f_{x_1}, \ldots, f_{x_n}$. Then
$$f(y) - \varepsilon \leq g(y) \leq f(y) + \varepsilon$$
for all $y \in X$, as was to be shown.

## Variations

There is a complex-valued version of Stone--Weierstrass. Let $C(X, \mathbb{C})$ denote the commutative $C^*$-[[C-star algebra|algebra]] of complex-valued functions $f: X \to \mathbb{C}$, where the star operation is pointwise-defined conjugation. A $C^*$-**subalgebra** is a subalgebra $A \subseteq C(X, \mathbb{C})$ which is closed under the star operation.

+-- {: .un_thm}
###### Theorem
A $C^*$-subalgebra $A \subseteq C(X, \mathbb{C})$ is dense if and only if it separates points.
=--

There is also a [[locally compact space|locally compact]] version. Let $X$ be a locally compact Hausdorff space and let $C_0(X)$ be the space of (say real-valued) functions $f$ which "vanish at infinity": for every $\varepsilon \gt 0$ there exists a compact set $K \subseteq X$ such that $|f(x)| \lt \varepsilon$ for all $x$ outside $K$. ($C_0(X)$ is no longer a Banach space, but it is [[locally convex topological vector space|locally convex]] and complete in its [[uniform space|uniformity]], and a Fréchet space if $X$ is second countable.) Under pointwise multiplication, $C_0(X)$ is a commutative algebra _without_ unit. As before, we have a notion of subalgebra $A \subseteq C_0(X)$.

+-- {: .un_thm}
###### Theorem
$A \subseteq C_0(X)$ is dense if and only if it separates points and for no $x \in X$ is it true that every $f \in A$ vanishes at $x$.
=--

## References

Named after [[Karl Weierstraß]] and [[Marshall Stone]].

See

* Wikipedia, _[Stone-Weierstrass theorem](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem)_

Discussion in [[constructive mathematics]]:

* [[B. Banaschewski]], [[C. J. Mulvey]], *A constructive proof of the Stone-Weierstrass theorem*, J. Pure Appl. Algebra __116__ (1997) 25-40 [doi:10.1016/S0022-4049(96)00160-0]

[[!redirects Stone–Weierstrass theorem]]
[[!redirects Stone--Weierstrass theorem]]


stone-weierstrass theorem


Stone-Čech compactification

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

#Contents#
* table of contents
{:toc}

## Definition

The [[forgetful functor]]/[[full and faithful functor|full and faithful]] [[subcategory]] embedding from [[compact topological spaces|compact]] [[Hausdorff topological spaces]]
into all [[topological spaces]]

$$
  U \;\colon\; Top_{CHaus} \hookrightarrow Top
$$

has a [[left adjoint]]

$$
  \beta \;\colon\; Top \longrightarrow Top_{CHaus}
$$

which sends a general [[topological space]] to a [[compact topological spaces|compact]] [[Hausdorff topological space]], called its _Stone-Čech compactification_. This hence exhibits $Top_{CHaus}$ as a [[reflective subcategory]] of all of $Top$.

The Stone-Čech compactification is in general "very large", even for "ordinary" non-compact spaces such as the [[real line]].

For more details see at _[compactum -- Stone-Čech compactification](compactum#StoneCechCompactification)_

## Properties

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

The [[unit of an adjunction|unit]]

$$
  X \longrightarrow \beta X
$$

of the compactification [[adjunction]] $(\beta \dashv U)$ is an [[embedding]] precisely for $X$ a [[Tychonoff space]].

=--

## Examples

+-- {: .num_example}
###### Example

The Stone-Čech compactification of a [[discrete topological space]]
is an [[extremally disconnected topological space]].
By a theorem by Gleason, these are precisely the [[projective objects]]
in the [[category]] of [[compact topological space|compact]]
[[Hausdorff topological spaces]].

Such spaces appear for instance as connected components of [[w-contractible rings]] as objects in the [[pro-étale site]]. See ([Bhatt-Scholze 13](#BhattScholze13)).

=--

## Related concepts

* [[compactification]]

* [[one-point compactification]]

* [[end compactification]]

* [[Bohr compactification]]

* [[Kaluza-Klein compactification]]

## References

Lecture notes include

* Tarun Chitra, _The Stone-Cech Compactification_ 2009 [pdf](http://www.math.cornell.edu/~riley/Teaching/Topology2009/essays/chitra.pdf)

* Brian Bockelman, _Functional Analysis Notes, [The Stone-Cech compactification](http://www.math.unl.edu/~s-bbockel1/929/node16.html)_

Discussion in the context of the [[pro-etale site]] is in

* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](https://arxiv.org/abs/1309.1198))
 {#BhattScholze13}

See also

* Wikipedia, _[Stone–Čech compactification](https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification)_

[[!redirects Stone-Cech compactification]]
[[!redirects Stone-Cech compactifications]]
[[!redirects Stone-Čech compactification]]
[[!redirects Stone-Čech compactifications]]
[[!redirects Stone–Cech compactification]]
[[!redirects Stone–Cech compactifications]]
[[!redirects Stone–Čech compactification]]
[[!redirects Stone–Čech compactifications]]
[[!redirects Stone--Cech compactification]]
[[!redirects Stone--Cech compactifications]]
[[!redirects Stone--Čech compactification]]
[[!redirects Stone--Čech compactifications]]


Stonean locale

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

#Contents#
* table of contents
{:toc}

## Definition

A **Stonean locale** is an [[extremally disconnected]] [[Stone locale]].
The category of Stonean locales has [[open maps]] of locales
as morphisms.

Thus, the category of Stonean locales is a (nonfull)
[[subcategory]] of the category of [[Stone locales]].

## Properties

In presence of the [[axiom of choice]],
every [[Stonean locale]] is spatial and the category of Stonean
locales is equivalent to the cateogry of [[Stonean spaces]].

As a variant of [[Stone duality]], the [[category]] of Stonean locales is contravariantly [[equivalence of categories|equivalent]] to the category
of [[complete Boolean algebras]] and continuous homomorphisms.
One can also reformulate this statement as an equivalence
between the categories of Stonean locales and [[Boolean locales]].

Unlike the corresponding statement for [[Stonean spaces]], this version is fully constructive and is valid in any [[W-topos]].

In fact, the traditional [[Stone duality|Stonean duality]] is an immediate consequence of the localic Stonean duality and the spatiality of Stonean locales.

## Related concepts

* [[Stone locale]]
* [[Stonean space]]

## References

The definition is due to [[Marshall Stone]], see

* Marshall H. Stone,
_Algebraic characterizations of special Boolean rings_.
Fundamenta Mathematicae 29:1 (1937), 223–303.

Further investigations of Stonean and [[hyperstonean]] spaces were done by Dixmier:

* Jacques Dixmier,
_Sur certains espaces considérés par M. H.Stone_.
Summa Brasiliensis Mathematicae 2 (1951), 151–182.
[PDF](https://dmitripavlov.org/scans/dixmier.pdf).

An expository account is available in Takesaki's book

* Masamichi Takesaki,
_Theory of operator algebras I_.
Encyclopaedia of Mathematical Sciences 124 (2002).

[[!redirects Stonean locales]]


Stonean space

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

#Contents#
* table of contents
{:toc}

## Definition

A **Stonean space** is a [[compact space|compact]], [[Hausdorff space|Hausdorff]] [[extremally disconnected]] [[topological space]].
Stonean spaces form a [[category]] if we take [[continuous function|continuous]] [[open maps]]
as [[morphisms]].

Thus, the category of Stonean spaces is a (nonfull)
[[subcategory]] of the category of [[Stone spaces]] and [[continuous maps]].

In presence of the [[axiom of choice]],
as a consequence of [[Stone duality]], the [[category]] of Stonean spaces is contravariantly [[equivalence of categories|equivalent]] to the category
of [[complete Boolean algebras]] and continuous homomorphisms.
This statement is known as the __Stonean duality__.

Without the [[axiom of choice]],
the [[category]] of [[Stonean locales]] is contravariantly [[equivalence of categories|equivalent]] to the category
of [[complete Boolean algebras]] and continuous homomorphisms.
See the article [[Stonean locale]] for more information.

## Related concepts

* [[Stone space]]

* [[Stonean locale]]

* [[hyperstonean space]]

## References

A standard textbook is

* [[Peter Johnstone]], _[[Stone Spaces]]_

[[!redirects Stonean spaces]]
[[!redirects Stonean]]
[[!redirects Stonean duality]]
[[!redirects Stonean dualities]]
[[!redirects compact extremally disconnected Hausdorff space]]
[[!redirects compact extremally disconnected Hausdorff spaces]]
[[!redirects compact extremally disconnected Hausdorff topological space]]
[[!redirects compact extremally disconnected Hausdorff topological spaces]]


store comonad

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

#Contents#
* table of contents
{:toc}

## Idea

The _store comonad_ (also _costate comonad_, as it is [[left adjoint]] to the *[[state monad]]*) is a [[comonad|co-]][[monad (in computer science)]] used to implement computational storage and retrieval of [[data]] ([[databases]]) in [[functional programming]].

Concretely, the [[coalgebra over a comonad|coalgebras]] over the store monad are equivalently the well-behaved *[[lenses (in computer science)|lens]]-data structures* ([O’Connor (2010)](#O’Connor10), [(2011)](#O’Connor11);
see [there](lens+in+computer+science#LensesAreCostateCoalgebras)) used to inspect and edit fields ("views") in [[databases]].

## Definition

### On a cartesian closed category
 {#OnACartesianClosedCategory}

By default, the costate comonad is understood to be defined on a [[cartesian closed category]] $(\mathcal{C}, \times \ast)$ whose [[internal hom]] we denote by $[\text{-}, \text{-}]$.

This means that given an [[object]] $S \in \mathcal{C}$ there is a pair of [[adjoint functors]]

\[
  \label{CartesianInternalHomAdjunction}
  \mathcal{C}
  \underoverset
    {\underset{[S,\text{-}]}{\longrightarrow}}
    {\overset{S \times (\text{-})}{\longleftarrow}}
    {\;\; \bot \;\;}
  \mathcal{C}
  \,.
\]

Notice that

* the [[unit of an adjunction|unit]] of this adjunction (being the [[adjunct]] of the [[identity morphism]] on ${S \times D}$) is:

  \[
    \label{InternalHomAdjunctionUnit}
    \array{
      D
      &
      \overset{
        ret^{ S State }_D
      }{\longrightarrow}
      &
      \big[ S ,\, S \times D \big]
      \\
      d &\mapsto& \big( s \mapsto (s,d)  \big)
      \mathrlap{\,,}
    }
  \]

* the [[counit of an adjunction|counit]] of this adjunction (being the [[adjunct]] of the [[identity morphism]] on ${[S,D]}$) is the [[evaluation map]]

  \[
    \label{InternalHomAdjunctionCoUnit}
    \array{
      S \times [S,D]
      &
      \overset{
        obt^{S Store}_D
      }{\longrightarrow}
      &
      D
      \\
      \big(
        s, f
      \big)
      &\mapsto&
      f(s)
      \mathrlap{\,.}
    }
  \]

While  the $S$-[[state monad]] is the [[monad]] [induced](monad#RelationBetweenAdjunctionsAndMonads) by this adjunction (eq:CartesianInternalHomAdjunction)

  $$
    S State
      \,\coloneqq\,
    \big[
      S
      ,\,
      S \times (\text{-})
    \big]
  $$

the *$S$-store comonad* is the induced comonad

$$
  S Store
  \,\coloneqq\,
  S \times \big[ S,\, (\text{-}) \big]
  \,.
$$

whose [[counit of a comonad|counit]] is (eq:InternalHomAdjunctionCoUnit) and whose duplicate operation (comonad coproduct) is given by the unit (eq:InternalHomAdjunctionUnit) as

$$
  \array{
    S \times [S, D]
    &
    \overset{
      S \times
      ret^{S State}_{[S,D]}
    }
    {\longrightarrow}
    &
    S \times \big[S, S \times [S,D] \big]
    \\
    (s, f)
    &\mapsto&
    \Big(
      s
      ,\,
      \big(
        s'
        \,\mapsto\,
        (s', f)
      \big)
    \Big)
  }
$$

Therefore the [[comonadic extend operation]] sends a [[coKleisli morphism]]

$$
  \mathcal{O}
  \,\colon\,
  S \times [S,D]
  \longrightarrow
  D'
$$

to

$$
  \array{
    \mathllap{
      extend^{ S Store }_{D,D'} \mathcal{O}
      \;\;\colon\;\;
    }
    S \times [S,D]
    &\overset{
      S \times ret^{S State}_{[S,D]}
    }{\longrightarrow}&
    S \times \big[ S \times [S,D]  \big]
    &\overset{
      S \times \big[ S, \mathcal{O} \big]
    }{\longrightarrow}&
    S \times [S, D']
    \\
    (s,f)
    &\mapsto&
    \Big(
      s
      ,\,
      \big(
        s'
        \,\mapsto\,
        (s', f)
      \big)
    \Big)
    &\mapsto&
    \Big(
      s
      ,\,
      \big(
        s'
        \,\mapsto\,
        \mathcal{O}(s',f)
      \big)
    \Big)
    \mathrlap{\,.}
  }
$$

In applications of [[comonads in computer science]] one thinks of

* $S$ as an address type,

* $[S,D]$ a data base of $S$-indexed $D$-data

hence of the $S Store$-comonad on $D$ as providing the computational context consisting of data base (of "stored $D$-values", whence the name *store comonad*) and an address.

Thus the *obtain*-operation (eq:InternalHomAdjunctionCoUnit) literally obtains (reads, extracts) the memory content at the given address.

### On closed monoidal categories

More generally, for $(\mathcal{C}, \otimes, \mathbb{1})$ a [[closed monoidal category]] (not necessarily [[cartesian closed category|cartesian]]), every object $D$ still still gives an [[internal-hom]] [[adjoint functor|adjunction]] of the form (eq:CartesianInternalHomAdjunction)

\[
  \label{MonoidalInternalHomAdjunction}
  \mathcal{C}
  \underoverset
    {\underset{[S,\text{-}]}{\longrightarrow}}
    {\overset{S \otimes (\text{-})}{\longleftarrow}}
    {\;\; \bot \;\;}
  \mathcal{C}
\]

and the [induced](monad#RelationBetweenAdjunctionsAndMonads) [[monad]] and [[comonad]] may be understood as [[substructural logic|substructural]] forms of the [[state monad]] and store comonad, respectively.

Particularly when $\mathcal{C} \,\equiv\, Mod_{\mathbb{C}}$ is the [[category of vector bundles|category of]] [[complex vector spaces]] ("$\mathbb{C}$-[[linear spaces]]") this gives a [[linear type theory|linear]] form of the costate comonad, discussed at

* *[[quantum costate comonad]]*.

## Properties

### Realization by base change

In a [[locally Cartesian closed category]]/[[dependent type theory]] $\mathbf{H}$, then to every type $W$ is associated its [[base change]] [[adjoint triple]]

$$
  \mathbf{H}_{/W}
   \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}}
  \mathbf{H}
  \,.
$$

In terms of this the store comonad is the composite

$$
  Store = \sum_W W^\ast \prod_W W^\ast
$$

of [[context extension]], followed by [[dependent product]] , followed by [[context extension]], followed by [[dependent sum]].

Here $\prod_W W^\ast = [W,-]$ is called the _[[function monad]]_ or _[[reader monad]]_ and $\sum_W W^\ast = W \times (-)$ is the _[[writer comonad]]_.

### Store-Comodules are Lenses
 {#LabelStoreComodulesAreLenses}

We spell out the data of [[comodule over a comonad|comodules]] over the classical Store comonad defined [above](#OnACartesianClosedCategory).

> (Beware that we are switching from using "$S$" for the state space to using it for the argument of the comonad -- this in order to be closer to the notation used at *[[lens in computer science]]*.)

Unwinding the definitions, a $V$-store [[comodule over a comonad|comodule]] [[structure]] on a type $S$ is a [[map]] of the form

\[
  \label{ComoduleMap}
  \array{
    S
    &
    \overset{
      \big(
        get
        ,\,
        \widetilde{put}
      \big)
    }{\longrightarrow}
    &
    V \times [V, S]
    \\
    s
    &\mapsto&
    \big(
      get(s)
      ,\,
      put(-,s)
    \big)
  }
\]

and hence consists, equivalently, of a [[pair]] of [[functions]] of the form

\[
  \label{LensStructureMaps}
  \begin{array}{l}
    get \;\colon\; S \to V \,,
    \\
    put \;\colon\; V \times S \to S \,.
  \end{array}
\]

On this data, the counitality condition of a comodule (eq:ComoduleMap) requires that the following composite is the [[identity morphism|identity]]

$$
  \array{
    S
    &
    \overset{
      \big(
        get
        ,\,
        \widetilde{put}
      \big)
    }{\longrightarrow}
    &
    V \times [V, S]
    &
    \overset{
      obt^{ V Store }_S
      \equiv
      ev
    }{\longrightarrow}
    &
    S
    \\
    s
    &\mapsto&
    \big(
      get(s)
      ,\,
      put(-,s)
    \big)
    &\mapsto&
    put\big(
      get(s)
      ,\,
      s
    \big)
    \mathrlap{\,,}
  }
$$

hence that

\[
  \label{GetPutLaw}
  s \in S
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  put\big(
    get(s)
    ,\,
    s
  \big)
  \;=\;
  s
  \mathrlap{\,,}
\]

while the coaction property requires that the following [[commuting diagram|diagram commutes]]

\begin{tikzcd}[
  column sep=4pt
]
  V \times [V,S]
  \ar[
    rrrrr,
    "{
      \mathrm{dupl}^{ V\mathrm{Store} }_S
    }"
  ]
  &[-5pt]
  &&&[-150pt]
  &[-30pt]
  V \times \Big[
    V
    ,\,
    \big[
      V
      \times
      [V,S]
    \big]
  \Big]
  \\
  &
  \big(
    \mathrm{get}(s)
    ,\,
    \mathrm{put}(-,s)
  \big)
  \ar[rr, phantom, "{ \mapsto }"]
  &&
  \Big(
    \mathrm{get}(s)
    ,\,
    v' \mapsto \big(v', \mathrm{put}(-,s)\big)
  \Big)
  \\
  &&&&
  \bigg(
    \mathrm{get}(s)
    ,\,
    v'
    \mapsto
    \Big(
      \mathrm{get}\big(\mathrm{put}(v',s)\big)
      ,\,
      \mathrm{put}\big(-, \mathrm{put}(v',s)\big)
    \Big)
  \bigg)
  \ar[ul, equals]
  \\
  &
  &&&
  \\
  &
  s
  \ar[rrr, phantom, "{ \mapsto }"]
  \ar[uuu, phantom, "{ \mapsto }"{sloped}]
  &&&
  \big(
    \mathrm{get}(s)
    ,\,
    v' \mapsto \mathrm{put}(v',s)
  \big)
  \ar[uu, phantom, "{ \mapsto }"{sloped}]
  \\
  S
  \ar[
    rrrrr,
    "{
      \big(
        \mathrm{get}
        ,\,
        \widetilde{\mathrm{put}}
      \big)
    }"{swap}
  ]
  \ar[
    uuuuu,
    "{
      \big(
        \mathrm{get}
        ,\,
        \widetilde{\mathrm{put}}
      \big)
    }"
  ]
  &&&&&
  V \times [V, S]
  \ar[
    uuuuu,
    "{
      V \times \Big[
        V
        ,\,
        \big(
          \mathrm{get}
          ,\,
          \widetilde{\mathrm{put}}
        \big)
      \Big]
    }"{swap}
  ]
\end{tikzcd}

which amounts to the two conditions

\[
  \label{PutGetLaw}
  v \in V
  ,\;
  s \in S
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  get\big( put(v,s) \big)
  \,=\,
  v
\]

and

\[
  \label{PutPutLaw}
  \label{}
  v, v' \in V
  ,\,
  s \in S
  \;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;
  put\big(
    v
    ,\,
    put(v', s)
  \big)
  \,=\,
  put(v,\, s)
\]

This data --- a pair of maps (eq:LensStructureMaps) satisfying the *get-put law* (eq:GetPutLaw), the *put-get law* (eq:PutGetLaw) and the *put-put law* (eq:PutPutLaw) -- is known as a *lawful [[lens in computer science]]* (see [there](#lens+in+computer+science#Lens)).

Hence, as first observed by [O’Connor 2010](#O’Connor10):

\begin{proposition}
Classical $V$-store [[comodule over a comonad|comodules]] are equivalently [[lawful lenses]] with "view" $V$.
\end{proposition}

## Related concepts

* [[quantum costate comonad]]

* [[maybe monad]],

* [[continuation monad]]

* [[state monad]]

* [[function monad]] (reader monad)

## References

### General

* {#Milewski19} [[Bartosz Milewski]] (compiled by Igal Tabachnik), §23.6 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

* {#Uustalu21} [[Tarmo Uustalu]], slide 14 of: *Monads and Interaction Lecture 3* , lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021) [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs3.pdf), [[Uustalu-Monads3.pdf:file]]]

### Relation to lenses

The observation that [[lenses (in computer science)]] are equivalently the [[coalgebra over a comonad|coalgebras]] of the [[costate comonad]] (cf.  *[[monads in computer science]]*) is due  to:

* {#O’Connor10} [[Russell O’Connor]], *Lenses Are Exactly the Coalgebras for the Store Comonad* (30th Nov 2010) [[r6research:23705](https://r6research.livejournal.com/23705.html)]

* {#O’Connor11} [[Russell O'Connor]], *Functor is to Lens as Applicative is to Biplate: Introducing Multiplate*, contribution to [ICFP '11: ACM SIGPLAN International Conference on Functional Programming](https://dl.acm.org/doi/proceedings/10.1145/2036918) (2011) [[arXiv:1103.2841](https://arxiv.org/abs/1103.2841)]

Early review:

* [[Jeremy Gibbons]], *[Lenses are the coalgebras for the costate comonad](https://patternsinfp.wordpress.com/2011/01/31/lenses-are-the-coalgebras-for-the-costate-comonad/)* (2011)

Further details:

* {#GibbonsJohnson12} [[Jeremy Gibbons]], [[Michael Johnson]], *Relating Algebraic and Coalgebraic Descriptions of Lenses*, Electronic Communications of the EASST **49** (2012) [[doi:10.14279/tuj.eceasst.49.726](http://dx.doi.org/10.14279/tuj.eceasst.49.726), [pdf](http://www.cs.ox.ac.uk/jeremy.gibbons/publications/colens.pdf)]

Further review:

* [[Bartosz Milewski]], *[Lenses, Stores, and Yoneda](https://bartoszmilewski.com/2013/10/08/lenses-stores-and-yoneda/)* (2013)

* [[Bartosz Milewski]], pp 240 in: *The Dao of Functional Programming* (2023) [[pdf](https://github.com/BartoszMilewski/Publications/blob/master/TheDaoOfFP/DaoFP.pdf), [github](https://github.com/BartoszMilewski/Publications/tree/master/TheDaoOfFP)]

[[!redirects store comonads]]

[[!redirects costate comonad]]
[[!redirects costate comonads]]


straightening functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

This is a subentry of [[(∞,1)-Grothendieck construction]].

#Contents#
* table of contents
{:toc}

## Idea

When [[(∞,1)-categories]] are modeled as [[quasi-categories]], the [[(∞,1)-Grothendieck construction]] takes the form of a [[Quillen equivalence]]
between
a [[model structure on sSet-enriched presheaves]]
and a [[slice model structure]] of [[simplicial sets]] which has been introduced under the name *straightening and unstraightening* in [Lurie 2009, Sec. 3.2](#Lurie09):

Concretely, in the notation recalled at *[[relation between quasi-categories and simplicial categories]]*:

For a [[simplicial set]] $S$, [[simplicially enriched category]] $C$, and [[sSet]]-[[enriched functor]] $\mathfrak{C}[S] \to C$, there is a [[pair]] of [[adjoint functors]]
$$
  (St_\phi\dashv Un_\phi)
  \;\colon\;
  sSet/S \xrightarrow{\;\; St \;\;} [C^{op}, sSet]
$$
which (under an assumption on the parameter $\phi$) can be shown to be a [[Quillen equivalence]] between

1. the [[slice category]] of [[sSet]] equipped with the [[model structure for right fibrations]] (also called contravariant model structure in [[Higher Topos Theory|HTT]])

1. the category of [[simplicial presheaves]] equipped with [[model structure on simplicial presheaves|global projective model structure]].

There is also a [[Quillen equivalence]]
$$
  (St_\phi \dashv Un_\phi)
  \;\colon\;
  sSet^+/S \stackrel{St}{\;\; \longrightarrow \;\;} [C^{op}, sSet^+]
$$
between

1. the [[model structure for Cartesian fibrations]]

1. the global projective [[model structure on functors]] with values in the [[model structure on marked simplicial sets]].

In both cases,

* $St_\phi$ is called the *straightening functor*

* $Un_\phi$ is called the *unstraightening functor*.

These names have been chosen due to the fact that objects in the left hand category are defined by existential assertions and choices where on the right side these properties become [[coherence laws]] being part of the structure.

## Related concepts

* [[cartesian fibration]]

* [[Grothendieck fibration]]

* [[(∞,1)-Grothendieck construction ]]

* [[cleavage]]

## References

The original result is due to:

* {#Lurie09} [[Jacob Lurie]], Sec. 3.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

Further discussion:

* {#Stevenson15} [[Danny Stevenson]], *Covariant Model Structures and Simplicial Localization*, North-West. Eur. J. Math. **3** (2017) 141-202 [[arXiv:1512.04815](http://arxiv.org/abs/1512.04815)]

* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II* [[arXiv:1602.01274](https://arxiv.org/abs/1602.01274)]

* [[Alexander Campbell]], *A modular proof of the straightening theorem*, talk at Macquarie University (2020) [[pdf](https://acmbl.github.io/straight_slides.pdf), [[Campbell-ModularProofofStraightening.pdf:file]]]

* [[Fabian Hebestreit]], [[Gijs Heuts]], [[Jaco Ruit]], _A short proof of the straightening theorem_ ([arXiv:2111.00069](https://arxiv.org/abs/2111.00069)).

Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

* [[Louis Martini]], *Cocartesian fibrations and straightening internal to an ∞-topos* [[arXiv:2204.00295](https://arxiv.org/abs/2204.00295)]

Discussion of straightening and unstraightening entirely within the context of [[quasi-categories]]:

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

which (along the lines of the discussion of the universal left fibration from [Cisinski 2019](universal+coCartesian+fibration#Cisinski19)) allows to understand the [[universal coCartesian fibration]] as [[categorical semantics]] for the [[univalence axiom|univalent]] [[type universe]] in [[directed homotopy type theory]]:

* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]

(see [video 3 at 1:16:58](https://youtu.be/P0Cfb4eUJo4?t=4618) and [slide 3.33](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33)).

Discussion of straightening and unstraightening for [[(∞,n)-categories]]:

* [[Lyne Moser]], [[Nima Rasekh]], [[Martina Rovelli]], _An (∞,n)-categorical straightening-unstraightening construction_ ([arXiv:2307.07259](https://arxiv.org/abs/2307.07259)).

[[!redirects straightening and unstraightening]]

[[!redirects straightening theorem]]


strain tensor

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

#Contents#
* table of contents
{:toc}

## In continuum mechanics / solid state physics

In [[elasticity theory]] (in the context of [[continuum mechanics]]/[[solid state physics]]) given _[[stress tensor]]_ expressing [[forces]] acting inside a [[solid body]], then the corresponding _strain tensor_ expresses the deformation of the body in reaction to these forces.

## Related concepts

* [[stress-energy tensor]]

## References

* {#LandauLifshitz59} [[Lev Landau]], [[Evgeny Lifshitz]], I.§1 of _Theory of Elasticity_, part VII of  _[[Course of Theoretical Physics]]_, 1959, 1970

[[!redirects strain tensors]]

[[!redirects strain]]


strange metal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] a *[[strange metal]]* is a [[metal|metallic]] [[phase of matter]] which is *not* described well by [[Landau's Fermi liquid theory]] of small [[perturbation theory|perturbations]] about the [[Fermi sea]].

Known strange metals exhibit [[topological order]] in that their [[ground state]] has long-range [[entanglement]] (a [[topological phase of matter]]). Instead of by Fermi liquid theory, strange metals are well described by [[AdS/CFT in condensed matter physics]].

## Related concepts

* [[Sachdev-Ye-Kitaev model]]

* [[superconductor]]

* [[AdS/CFT in condensed matter physics]]

* [[metal]], [[Landau's Fermi liquid theory]]

## References

Review:

* [Amoretti 17, Sec. I.2](#Amoretti17)

* Richard L. Greene, Pampa R. Mandal, Nicholas R. Poniatowski, and Tarapada Sarkar, *The Strange Metal State of the Electron-Doped Cuprates*, Annual Review of Condensed Matter Physics, Vol. 11:213-229 (2020) ([doi:10.1146/annurev-conmatphys-031119-050558](https://doi.org/10.1146/annurev-conmatphys-031119-050558))

See also:

* Wikipedia, _[Non-Fermi liquids](https://en.m.wikipedia.org/wiki/Fermi_liquid_theory#Non-Fermi_liquids)_

* [[Subir Sachdev]], Debanjan Chowdhury, *The novel metallic states of the cuprates: Fermi liquids with topological order, and strange metals*, Progress of Theoretical and Experimental Physics 2016, 12C102 ([arXiv:1605.03579](https://arxiv.org/abs/1605.03579))

* Eunseok Oh, Taewon Yuk, Sang-Jin Sin, *The emergence of Strange metal and Topological Liquid near Quantum Critical Point in a solvable model*, Annals of Physics Volume 417, June 2020, 168138 ([arXiv:2103.08166](https://arxiv.org/abs/2103.08166))

Models:

* Edwin W. Huang, Ryan Sheppard, Brian Moritz, Thomas P. Devereaux, *Strange metallicity in the doped Hubbard model*, Science  22 Nov 2019:
Vol. 366, Issue 6468, pp. 987-990 ([doi:10.1126/science.aau7063](https://science.sciencemag.org/content/366/6468/987))

More on relation of strange metals to high-temperature [[superconductivity]]:

* Konstantin Tikhonov, Mikhail Feigel'man, *Strange metal state near quantum superconductor-metal transition in thin films*, Annals of Physics
Volume 417, June 2020, 168138 ([arXiv:2002.08107](https://arxiv.org/abs/2002.08107), [doi:10.1016/j.aop.2020.168138](https://doi.org/10.1016/j.aop.2020.168138))

Relation of [[strange metals]]/[[high-temperature superconductors]] to [[confinement]]:

* J. P. Rodriguez, Pascal Lederer, _Confinement of Spin and Charge in High-Temperature Superconductors_, Phys. Rev. B 53, R11980(R) 1996 ([arXiv:cond-mat/9604109](https://arxiv.org/abs/cond-mat/9604109))

Discussion of [[strange metals]]/[[high-temperature superconductors]] via [[AdS/CFT in condensed matter physics]]:

* [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))

* {#Amoretti17} [[Andrea Amoretti]], _Condensed Matter Applications of AdS/CFT: Focusing on strange metals_, 2017 ([spire:1610363](http://inspirehep.net/record/1610363), [[AmorettiStrangeMetals.pdf:file]])

via the [[SYK model]]:

* [[Subir Sachdev]], *Statistical mechanics of strange metals and black holes* ([arXiv:2205.02285](https://arxiv.org/abs/2205.02285))

On [[strange metals]], high $T_c$-[[superconductors]] and [[AdS/CMT duality]]:

* [[Jan Zaanen]], _Planckian dissipation, minimal viscosity and the transport in cuprate strange metals_, SciPost Phys. 6, 061 (2019) ([arXiv:1807.10951](https://arxiv.org/abs/1807.10951))

* {#Zaanen21} [[Jan Zaanen]], *Lectures on quantum supreme matter* ([arXiv:2110.00961](https://arxiv.org/abs/2110.00961))

[[!redirects strange metals]]

[[!redirects high-temperature superconductor]]
[[!redirects high-temperature superconductors]]


strange quark

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]] the _strange quark_ is the down-type [[quark]] in the second [[generation of fermions]].

Regarded as a [[heavy quark]] in 2-flavor [[chiral perturbation theory]], and as a [[light quark]] in 3-flavor [[chiral perturbation theory]].

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

### General

* [[Howard Georgi]], §11  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

* Wikipedia, _[Strange quark](https://en.wikipedia.org/wiki/Strange_quark)_

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!redirects strange quarks]]


stratification

#Contents#
* table of contents
{:toc}

## Idea

A _stratification_ of a [[set]] or other object is some kind of decomposition into _strata_ that are often [[order|ordered]] by inclusion, or by a [[poset]] determining the relationships between the 'strata' as in a [[poset-stratified space]].

## Related concepts

* [[filtered object]]

* [[stratified space]]

* [[poset-stratified space]]

[[!redirects stratifications]]