solid module

# Contents
* table of contents
{: toc}

## Definition

\begin{definition}
([Mann (2022)](#Mann22), Definition 1.6.3)
\linebreak
Let $A$ be a [[ring]] and let $\mathcal{D}(A)$ be the ("[[derived category|derived]]") [[(infinity,1)-category|$\infty$-category]] [[(infinity,1)-category of chain complexes|of chain complexes of]] [[condensed module|condensed $A$-modules]]. For any [[profinite set]] $S=\underset{\underset{i}{\leftarrow}}\lim S_{i}$ consider the [[limit in an (infinity,1)-category|$\infty$-(co)limit]]
$$
  A_{\square}[S]
  \;\coloneqq\;
  \underset
    {\underset{ A'\subseteq A }{\rightarrow}}
    {\lim}
  \;
  \underset
    {\underset{i}{\leftarrow}}
    {\lim}
  \;
  A'\big[S_{i}\big]
  \,,
$$
where $A'$ ranges over all [[subrings]] of $A$ which are of [[finite type]] over [[integers|$\mathbb{Z}$]].

An [[object]] $M \in \mathcal{D}(A)$ is called _solid_ if for any [[profinite set]] $S$ the canonical [[map]] of [[enriched (infinity,1)-category|mapping objects]]
$$
  \underline{Hom}
  \big(
    A_{\square}[S]
    ,\,
    M
  \big)
  \longrightarrow
  \underline{Hom}
  \big(
    A[S]
    ,\,
    M
  \big)
$$
is an [[equivalence in an (infinity,1)-category|equivalence]] in $\mathcal{D}(A)$.

The [[full subcategory]] of solid modules in $\mathcal{D}(A)$ is denoted $\mathcal{D}_{\square}(A)$.

\end{definition}

## Globalization

Using the notion of [[adic spaces]], we can glue together solid modules and consider the derived category $\mathcal{D}((\mathcal{O}_{X},\mathcal{O}_{X}^{+})_{\square})$, for $X$ an adic space. By passing to the homotopy category we get the triangulated category $D((\mathcal{O}_{X},\mathcal{O}_{X}^{+})_{\square})$ ([ScholzeLCM](#ScholzeLCM), Lectures IX and X).

## Coherent duality

There exists a notion of coherent duality (analogous to [[Grothendieck duality]]) for solid modules ([ScholzeLCM](#ScholzeLCM), Theorem 11.1).

For brevity, given a scheme $X$ with associated adic space $X^{\ad}$, let us define $D(\mathcal{O}_{X,\square}):=D((\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad}}^{+})_{\square})$.

\begin{theorem}
Let $F:X\to \mathrm{Spec}(R)$ be a separated and smooth map of finite type, of dimension $d$. Let $\omega_{X/R}=\bigwedge^{d}\Omega_{X/R}^{1}$. There is a canonical functor

$$f_{!}:D(\mathcal{O}_{X,\square})\to D(R_{\square})$$

that agrees with $R\Gamma(X,-)$ in the case that $f$ is proper. It preserves compact objects. There is a natural trace map

$$f_{!}\omega_{X/R}[d]\to R$$

such that for all $C\in D(\mathcal{O}_{X,\square})$, the natural map

$$R\Hom_{\mathcal{O}_{X}}(C,\omega_{X/R})[d]\to R\Hom_{R}(f_{!}C,R)$$

is an isomorphism.
\end{theorem}

## Six-functor formalism

The category $D(\mathcal{O}_{X,\square})$ admits the [[six operations]] ([ScholzeLCM](#ScholzeLCM), Lecture XI). The first four functors $-\otimes -$, $\Hom(-,-)$, $f_{*}$, $f^{*}$ are classical and do not require condensed mathematics. The functor $f_{!}$ has to be constructed, and $f^{!}$ will be defined to be its [[right adjoint]].

Let $f:X\to Y$ be a separated map of finite type. Then the corresponding map of [[adic spaces]] $f_{\ad}:X^{\ad}\to Y^{\ad}$ factors as $f_{\ad}:X^{\ad}\xrightarrow{j} X^{\ad/ Y} \xrightarrow{f^{\ad /Y}} Y^{\ad}$, where the map $j$ is an open immersion and an isomorphism if $f$ is proper.

\begin{proposition}([ScholzeLCM](#ScholzeLCM) Proposition 11.2)

The functor

$$j^{*}:D(\mathcal{O}_{X^{\ad /Y}},\mathcal{O}_{X^{\ad /Y}}^{+})\to D(\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad}}^{+})$$

admits a left adjoint

$$j_{!}:D((\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad }}^{+})_{\square})\to D(\mathcal{O}_{X^{\ad} /Y},\mathcal{O}_{X^{\ad /Y}}^{+})_{\square}).$$

\end{proposition}

\begin{definition}([ScholzeLCM](#ScholzeLCM) Definition 11.3)
The functor $f_{!}:D(\mathcal{O}_{X},\square)\to D(\mathcal{O}_{Y,\square})$ is defined to be the composition

$$f_{!}=f_{*}^{\ad /Y}\circ j_{!}.$$
\end{definition}

It turns out that the functor $f_{!}$ commutes with all direct sums and therefore admits a right adjoint, which will be the sixth functor we call $f^{!}$.

## Solid $\mathcal{O}_{X}^{+}/\pi$ almost-modules

In [Mann22](#Mann22), Mann combines the theory of solid modules with the theory of [[almost modules]] to construct the derived $\infty$-category (in the sense of 1.3.2 of [LurieHA](#LurieHA)) $\mathcal{D}_{\square}^{\a}(\mathcal{O}_{X}^{+}/ \pi)$ of solid $\mathcal{O}_{X}^{+}/ \pi$ [[almost modules]] and with it the [[six operations]] on [[rigid analytic spaces]], in order to prove the following "mod p" version of [[Poincare duality]]:

\begin{proposition} ([Mann22](#Mann22), Theorem 1.1.1)

Let $K$ be an algebraically closed field of characteristic $0$ whose residue field is of characteristic $p$. Let $X$ be a proper smooth rigid-analytic variety of pure dimension $d$ over $K$. Then for all $i\in\mathbb{Z}$ there is a natural perfect pairing

$$H_{et}^{i}(X,\mathbb{F}_{p})\otimes_{\mathbb{F}}H_{et}^{2d-i}(X,\mathbb{F}_{p})\to\mathbb{F}_{p}(-d).$$
\end{proposition}

"mod p" Poincare duality for rigid analytic spaces had also previously been proven by Zavyalov in [Zavyalov21](#Zavyalov21), using different methods.

The theory of [[almost modules]] is necessary in order to make the structure sheaf $\mathcal{O}_{X}^{+}/\pi$ [[acyclic]] on [[affinoid]] [[perfectoid spaces]] (compare the analogous classical situation for [[abelian sheaf cohomology]] or [[Cech cohomology]] for [[schemes]]).

In the course of proving Poincare duality for rigid analytic spaces, Mann also proves a version of a p-torsion [[Riemann-Hilbert correspondence]] for small v-stacks ([Mann22](#Mann22), Theorem 3.9.23).

## Related concepts

* [[condensed mathematics]]

* [[analytic ring]]

* [[six operations]]

## References

* {#Mann22} [[Lucas Mann]], *A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry* [[arXiv:2206.02022](https://arxiv.org/abs/2206.02022)]

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)

Zavyalov's proof of Poincare duality for rigid analytic spaces can be found in

* {#Zavyalov21} Bogdan Zavyalov, _Mod-p Poincaré Duality in p-adic Analytic Geometry_, [arXiv:2111.01830](https://arxiv.org/abs/2111.01830)

* {#LurieHA} [[Jacob Lurie]], _Higher Algebra_, ([pdf](https://people.math.harvard.edu/~lurie/papers/HA.pdf))

[[!redirects solid modules]]


solid state physics

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

#Contents#
* table of contents
{:toc}

## Idea

The [[physics]] of [[solid]] condensed [[matter]] (made of [[fermions]], due to the [[Pauli exclusion principle]]), also _[[continuum mechanics]]_.

## Properties

### Topological phases of matter

See at _[[K-theory classification of topological phases of matter]]_.

## Related concepts

* [[quantum many-body physics]]

* [[mathematical physics]]

* [[high energy physics]]

* [[Bloch's theorem]]

* [[phonon]]

* [[degeneracy pressure]]

* [[Bose-Einstein condensate]]

* [[Levin-Wen model]]

* [[quantum material]]

  * [[superconductivity]]

  * [[topological phases of matter]]

  * [[topological order]]

  * [[symmetry protected trivial order]]

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

* [[phase of matter]]

* [[Josephson effect]]

* [[mesoscopic physics]]

* [[high energy physics]], [[nuclear physics]]

## References

### General

Textbook accounts:

* {#Kittel53} [[Charles Kittel]], *Introduction to Solid State Physics*, Wiley (1953-) $[$[ISBN:978-0-471-41526-8](https://www.wiley.com/en-us/Introduction+to+Solid+State+Physics%2C+8th+Edition-p-9780471415268), [pdf](http://metal.elte.hu/~groma/Anyagtudomany/kittel.pdf),  [Wikipedia entry](https://en.wikipedia.org/wiki/Introduction_to_Solid_State_Physics)$]$

* {#JS75} [[John Slater]], *Solid-State and Molecular Theory: A Scientific Biography*, Wiley (1975) [[archive](https://archive.org/details/solidstatemolecu00slat/)]

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

* [[John W. Negele]], [[Henri Orland]], *Quantum Many-Particle Systems*, Westview Press (1988, 1998) [[doi:10.1201/9780429497926](https://doi.org/10.1201/9780429497926)]

* [[Naoto Nagaosa]], *Quantum Field Theory in Condensed Matter Physics*, Texts and Monographs in Physics, Springer (1999) [[doi:10.1007/978-3-662-03774-4_2](https://doi.org/10.1007/978-3-662-03774-4_2)]

* [[Alexander L. Fetter]], [[John Dirk Walecka]], *Quantum theory of many-particle systems*, Mcgraw-Hill (1991); Dover (2003) [[archive.org](https://archive.org/details/quantum-theory-of-many-particle-systems-by-alexander-l.-fetter-john-dirk-walecka-physics-z-lib.org/mode/thumb)]

* {#Roessler04} [[Ulrich Rößler]], _Solid State Theory: An Introduction_, Springer (2004, 2009) $[$[doi:10.1007/978-3-540-92762-4](https://link.springer.com/book/10.1007/978-3-540-92762-4)$]$

In terms of [[quantum field theory]]:

* [[Eduardo Fradkin]], *Field Theories of Condensed Matter Physics*, Cambridge University Press (2013) [ISBN: 9781139015509, [doi:10.1017/CBO9781139015509](https://doi.org/10.1017/CBO9781139015509)]

* Eduardo C. Marino, *Quantum Field Theory Approach to Condensed Matter Physics*,  Cambridge University Press (2017) [[doi:10.1017/9781139696548](https://doi.org/10.1017/9781139696548)]

Lecture notes:

* {#Tong2017} [[David Tong]], *Lectures on solid state physics* (2017) $[$[pdf](http://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf), [webpage](http://www.damtp.cam.ac.uk/user/tong/solidstate.html)$]$

Specifically on [[Bloch-Floquet theory]]:

* [[Michael Reed]], [[Barry Simon]], Sec. XIII.16 *Schrödinger operators with periodic potentials*, of:  *[[Methods of Modern Mathematical Physics]] -- IV: Analysis of Operators*, Academic Press (1978)
([ISBN:9780080570457](https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7))

With focus on [[semiconductor]]-theory:

* [[Karlheinz Seeger]], *Semiconductor Physics*, Advanced texts in physics, Springer (2004) $[$[doi:10.1007/978-3-662-09855-4](https://doi.org/10.1007/978-3-662-09855-4)$]$

* {#Li06} [[Sheng San Li]], *Semiconductor Physical Electronics*, Springer (2006)  61-104 $[$[doi:10.1007/0-387-37766-2](https://link.springer.com/book/10.1007/0-387-37766-2)$]$

See also:

* Wikipedia, _[Solid state physics](http://en.wikipedia.org/wiki/Solid_state_physics)_

and maybe also

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

### Examples and applications

Discussion of possible realization of the [[SYK-model]] in [[condensed matter physics]]:

* D. I. Pikulin, M. Franz, _Black hole on a chip: proposal for a physical realization of the SYK model in a solid-state system_, Phys. Rev. X 7, 031006 (2017) ([arXiv:1702.04426](https://arxiv.org/abs/1702.04426))

### AdS/CMT correspondence

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

* {#HartnollLucasSachdev16} [[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))

Proposed realization of aspects of [[p-adic AdS/CFT correspondence]] in [[solid state physics]]:

* Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, [[Steven Gubser]], Monika Schleier-Smith, _Treelike interactions and fast scrambling with cold atoms_, Phys. Rev. Lett. 123, 130601 (2019) ([arXiv:1905.11430](https://arxiv.org/abs/1905.11430))

### K-Theory classification of gapped topological phases of matter

Classification of [[condensed matter physics|condensed matter]] with [[gapped Hamiltonians]] ([[topological insulators]], [[topological phases of matter]]) by [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[topological K-theory]]:

* [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, talk at: L.D.Landau Memorial Conference "Advances in Theoretical Physics", June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) ([arXiv:0901.2686](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/10.1063/1.3149495))

* [[Daniel Freed]], [[Gregory Moore]], _Twisted equivariant matter_, Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

* Guo Chuan Thiang, _On the K-theoretic classification of topological phases of matter_, Annales Henri Poincare 17(4) 757-794 (2016) ([arXiv:1406.7366](https://arxiv.org/abs/1406.7366))

* Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann, _Topological insulators and K-theory_ ([arXiv:1510.08001](https://arxiv.org/abs/1510.08001), [spire:1401095/](http://inspirehep.net/record/1401095/))

* [[Daniel Freed]], [[Michael Hopkins]], _Reflection positivity and invertible topological phases_ ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527))

* [[Daniel Freed]], _Lectures on field theory and topology_ ([cds:2699265](https://cds.cern.ch/record/2699265))

* Charles Zhaoxi Xiong, _Classification and Construction of Topological Phases of Quantum Matter_ ([arXiv:1906.02892](https://arxiv.org/abs/1906.02892))

### Tensor networks in solid state physics

Discussion of exotic [[phases of matter]] via [[tensor network states]]:

General:

* [[Roman Orus]], _A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States_, 	Annals of Physics 349 (2014) 117-158  ([arXiv:1306.2164](https://arxiv.org/abs/1306.2164))

* [[Jens Eisert]], _Entanglement and tensor network states_, Modeling and Simulation 3, 520 (2013) ([arXiv:1308.3318](https://arxiv.org/abs/1308.3318))

* [[Thorsten B. Wahl]], _Tensor network states for the description of quantum many-body systems_ ([arXiv:1509.05984](https://arxiv.org/abs/1509.05984))

Specifically [[tree tensor networks]]:

* Valentin Murg, Örs Legeza, Reinhard M. Noack, Frank Verstraete, _Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks_, Phys. Rev. B 82, 205105 (2010) ([arXiv:1006.3095](https://arxiv.org/abs/1006.3095))

Concrete materials:

* A. Kshetrimayum, C. Balz, B. Lake, [[Jens Eisert]], _Tensor network investigation of the double layer Kagome compound $Ca_{10} Cr_{7} O_{28}$_ ([arXiv:1904.00028](https://arxiv.org/abs/1904.00028))

[[!redirects condensed matter physics]]
[[!redirects condensed matter theory]]

[[!redirects condensed matter]]

[[!redirects solid body]]
[[!redirects solid bodies]]

[[!redirects solid-state physics]]


solid state physics -- contents

* basics

  * [[Slater determinant]]

  * [[degeneracy pressure]]

* [[crystal]]

  * [[crystallographic group]]

  * [[Bravais lattice]]

* [[Bloch theory]]

* [[electronic band structure]]

  * [[valence bundle]], [[conduction bundle]]

  * [[valence band]], [[conduction band]]

* [[Berry connection]]

  * [[Berry phase]], [[Zak phase]]

* [[phase of matter]]

  * [[insulator]], [[topological insulator]]

  * [[metal]]/[[conductor]], [[semi-conductor]], [[semi-metal]],

  * [[superconductor]]

  * [[strange metal]]

* [[topological phase of matter]], [[topological state of matter]]

  * [[symmetry protected topological phase]]

  * [[K-theory classification of topological phases of matter]]

* [[topological order]]

  [[anyons]]

* Examples

  * [[graphene]], [[Haldane model]],

  * [[quantum Hall effect]])

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

  * [[tensor network state]]

* [[Levin-Wen model]]

* [[Josephson effect]]


solid topos

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _[[solid topos]]_ is meant to be a characterization of _[[gros toposes]]_ which further refines that of [[elastic toposes]] and [[cohesive toposes]]. The idea is that we may consistently regard the [[objects]] of such toposes as [[generalized spaces]] for some flavor of [[geometry]], and that the axioms on the topos determine aspects of the geometric nature of these [[generalized spaces]], as follows:

| $\phantom{A}$[[gros topos]]$\phantom{A}$ | $\phantom{A}$[[generalized spaces]] obey...$\phantom{A}$ |
|-----|-----|
| $\phantom{A}$[[cohesion]] | $\phantom{A}$principles of [[differential topology]]$\phantom{A}$ |
| $\phantom{A}$[[elastic topos|elasticity]] | $\phantom{A}$principles of [[differential geometry]]$\phantom{A}$ |
| $\phantom{A}$[[solid topos|solidity]]$\phantom{A}$ | $\phantom{A}$principles of [[supergeometry]]$\phantom{A}$ |
{: style='margin:auto'}

## Definition

+-- {: .num_defn #SuperDifferentialCohesion}
###### Definition
**([[solid topos]])**

Let $\mathbf{H}_{bos}$ be an [[elastic topos]] ([this Def.](geometry+of+physics+--+categories+and+toposes#DifferentialCohesion)) over a [[cohesive topos]] $\mathbf{H}_{red}$ ([this Def.](geometry+of+physics+--+categories+and+toposes#CohesiveTopos)). Then a _solid topos_ or _super-differentially cohesive topos_ over $\mathbf{H}_{bos}$ is a [[sheaf topos]] $\mathbf{H}$, which is

1. a [[cohesive topos]] over [[Set]] ([this Def.](geometry+of+physics+--+categories+and+toposes#CohesiveTopos)),

1. an [[elastic topos]] over $\mathbf{H}_{red}$ ([this Def.](geometry+of+physics+--+categories+and+toposes#DifferentialCohesion)).

1. equipped with a [[adjoint quadruple|quadruple]] of [[adjoint functors]] ([this Def.](geometry+of+physics+--+categories+and+toposes#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets)) to $\mathbf{H}_{bos}$ of the form

   $$
     \mathbf{H}_{bos}
       \array{
         \overset{\phantom{A} even \phantom{A} }{\longleftarrow}
         \\
         \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow}
         \\
         \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow}
         \\
         \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow}
       }
     \mathbf{H}
   $$

   hence with $\iota_{sup}$ and $Disc_{sup}$ being [[fully faithful functors]].

=--

+-- {: .num_lemma #ProgressionOfSubcategoriesOfSolidTopos}
###### Lemma
**(progression of ([[coreflective subcategory|co-]])[[reflective subcategories]] of [[solid topos]])**

Let $\mathbf{H}$ be a [[solid topos]] (Def. \ref{SuperDifferentialCohesion}) over an [[elastic topos]] $\mathbf{H}_{red}$:

$$
  Set
    \array{
      \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow}
      \\
      \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow}
      \\
      \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow}
      \\
      \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow}
    }
  \mathbf{H}_{red}
    \array{
      \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow}
      \\
      \phantom{A}
      \\
      \phantom{A \atop A}
    }
  \mathbf{H}_{bos}
    \array{
      \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow}
      \\
      \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow}
      \\
      \phantom{A}
      \\
      \phantom{A \atop A}
      \\
      \phantom{A \atop A}
    }
  \mathbf{H}
$$

Then these adjoint functors arrange and decompose as shown in the following [[diagram]]:

Here the composite [[adjoint quadruple]]

$$
  Set
    \array{
      \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow}
      \\
      \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow}
      \\
      \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow}
    }
  \mathbf{H}
$$

exhibits the [[cohesion]] of $\mathbf{H}$ over [[Set]], and the composite adjoint quadruple

$$
  \mathbf{H}_{red}
    \array{
      \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow}
      \\
      \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow}
      \\
      \overset{Disc_{inf} Disc_{red}}{\hookrightarrow}
      \\
      \overset{ \Gamma_{sup} }{\longleftarrow}
    }
  \mathbf{H}
$$

exhibits the [[elastic topos|elasticity]] of $\mathbf{H}$ over $\mathbf{H}_{red}$.

=--

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

As in the proof of [this Prop.](geometry+of+physics+--+categories+and+toposes#ProgressionOfSubcategoriesOfElasticTopos),  this is immediate by the essential uniqueness of adjoints ([this Prop.](geometry+of+physics+--+categories+and+toposes#UniquenessOfAdjoints)) and of the [[global section]]-[[geometric morphism]] ([this Example](geometry+of+physics+--+categories+and+toposes#GlobalSectionsGeometricMorphism)).

=--

+-- {: .num_defn #SuperDiffCohesiveModalities}
###### Definition
**([[adjoint modalities]] on [[solid topos]])**

Given a [[solid topos]] $\mathbf{H}$ over $\mathbf{H}_{bos}$ (Def. \ref{SuperDifferentialCohesion}), [[composition]] of the functors in Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos} yields, via [this Prop.](geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories), the following [[adjoint modalities]] ([this Def.](geometry+of+physics+--+categories+and+toposes#AdjointModality))

$$
  \rightrightarrows
  \;\dashv\;
  \rightsquigarrow
  \;\dashv\;
  Rh
  \;\;\colon\;\;
  \mathbf{H}
    \array{
      \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even  }{\longleftarrow}
      \\
      \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}  }{\longrightarrow}
      \\
      \overset{ Rh \;\coloneqq\; Disc_{sup}  \circ \Pi_{sup} }{\longleftarrow}
    }
  \mathbf{H}
  \,.
$$

Since $\iota_{sup}$ and $Disc_{sup}$ are [[fully faithful functors]] by assumption, these are ([[comodal operator|co-]])[[modal operators]] ([this Def.](geometry+of+physics+--+categories+and+toposes#ModalOperator)) on the [[cohesive topos]], by [this Prop.](geometry+of+physics+--+categories+and+toposes#ModalOperatorsEquivalentToReflectiveSubcategories).

We pronounce these as follows:

| $\phantom{A}$ [[fermionic modality]] $\phantom{A}$ | $\phantom{A}$ [[bosonic modality]] $\phantom{A}$ | $\phantom{A}$ [[rheonomy modality]] $\phantom{A}$ |
|--------------------|-------------------|--------------------|
|   $\phantom{A}$  $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$  $\phantom{A}$ | $\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$  | $\phantom{A}$ $ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} $ $\phantom{A}$  |
{: style='margin:auto'}

and we refer to the corresponding [[modal objects]] ([this Def.](geometry+of+physics+--+categories+and+toposes#ModalObjects)) as follows:

* a $\rightsquigarrow$-[[comodal object]]

  $$
    \overset{\rightsquigarrow}{X}
      \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow}
    X
  $$

  is called a _[[bosonic object]]_;

* a $Rh$-[[modal object]]

  $$
    X
      \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow}
    Rh X
  $$

  is called a _rheonomic object_;

=--

+-- {: .num_prop #ProgressionOfModalitiesOnElasticTopos}
###### Proposition
**(progression of [[adjoint modalities]] on [[solid topos]])**

Let $\mathbf{H}$ be a [[solid topos]] (Def. \ref{SuperDifferentialCohesion}) and consider the [[adjoint modalities]] which it inherits

1. for being a [[cohesive topos]], from [this Def.](geometry+of+physics+--+categories+and+toposes#CohesiveModalities),

1. for being an [[elastic topos]], from [this Def.](geometry+of+physics+--+categories+and+toposes#DiffCohesiveModalities),

1. for being a [[solid topos]], from Def. \ref{SuperDiffCohesiveModalities}:

| $\phantom{A}$ [[shape modality]] $\phantom{A}$ | $\phantom{A}$ [[flat modality]] $\phantom{A}$ | $\phantom{A}$ [[sharp modality]] $\phantom{A}$ |
|--------------------|-------------------|--------------------|
|   $\phantom{A}$  $ʃ \;\coloneqq\; Disc \Pi$  $\phantom{A}$ | $\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$  | $\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma $ $\phantom{A}$  |
|   |   |   |
| $\phantom{A}$ **[[reduction modality]]** $\phantom{A}$ | $\phantom{A}$ **[[infinitesimal shape modality]]** $\phantom{A}$ | $\phantom{A}$ **[[infinitesimal flat modality]]** $\phantom{A}$ |
|   $\phantom{A}$  $\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup}$  $\phantom{A}$ | $\phantom{A}$ $\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup}$ $\phantom{A}$  | $\phantom{A}$ $ \& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup} $ $\phantom{A}$  |
|   |    |    |
| $\phantom{A}$ **[[fermionic modality]]** $\phantom{A}$ | $\phantom{A}$ **[[bosonic modality]]** $\phantom{A}$ | $\phantom{A}$ **[[rheonomy modality]]** $\phantom{A}$ |
|   $\phantom{A}$  $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$  $\phantom{A}$ | $\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$  | $\phantom{A}$ $ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} $ $\phantom{A}$  |
{: style='margin:auto'}

Then these arrange into the following progression, via the [[preorder]] on modalities from [this Def.](geometry+of+physics+--+categories+and+toposes#PreorderOnModalities):

$$
  \array{
    id &\dashv& id
    \\
    \vee && \vee
    \\
    \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh
    \\
    && \vee && \vee
    \\
    && \Re &\dashv& \Im &\dashv& \&
    \\
    && && \vee && \vee
    \\
    && && ʃ &\dashv& \flat &\dashv& \sharp
    \\
    && && && \vee && \vee
    \\
    && && && \emptyset &\dashv& \ast
  }
$$

where we are displaying, for completeness, also the [[adjoint modalities]]  at the [[bottom]] $\emptyset \dashv \ast$ and the [[top]] $id \dashv id$  ([this Example](geometry+of+physics+--+categories+and+toposes#InitialAndFinalAdjointModality)).

=--

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

By [this Prop.](geometry+of+physics+--+categories+and+toposes#ProgressionOfModalitiesOnElasticTopos), it just remains to show that for all [[objects]] $X \in \mathbf{H}$

1. $\Im X$ is an $Rh$-[[modal object]], hence $Rh \Im X \simeq X$,

1. $\Re X$ is a [[bosonic object]], hence $\overset{\rightsquigarrow}{\Re X} \simeq \Re X$.

The proof is directly analogous to the proof of [that Prop.](geometry+of+physics+--+categories+and+toposes#ProgressionOfModalitiesOnElasticTopos), now using the decompositions from Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos}:

$$
  \begin{aligned}
    Rh \Im
    & =
    Disc_{sup}
    \underset{
      \simeq id
    }{
    \underbrace{
      \Pi_{sup} \;\; Disc_{sup}
    }
    }
    Disc_{inf} \Pi_{inf} \Pi_{sup}
    \\
    & \simeq
    Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup}
    \\
    & = \Im
  \end{aligned}
$$

and

$$
  \begin{aligned}
    \rightsquigarrow \Re
    & =
    \iota_{sup}
    \underset{\simeq id}{\underbrace{
    \Pi_{sup} \;\; \iota_{sup}}}
    \iota_{inf} \Pi_{inf}\Pi_{sub}
    \\
    & \simeq
    \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub}
    \\
    & \simeq \Re
  \end{aligned}
$$

=--

## Examples

### Super formal smooth sets

The [[sheaf topos]] of [[super formal smooth sets]] is solid over that of [[formal smooth sets]], which is [[elastic topos|elastic]] over that of [[smooth sets]], which is [[cohesive topos|cohesive]] over [[Set]].

See [this Prop](geometry+of+physics+--+supergeometry#SuperSmoothSetsSystemOfAdjunctions) at _[[geometry of physics -- supergeometry]]_.

[[!redirects super-differentially cohesive topos]]

[[!redirects solid model topos]]
[[!redirects solid model toposes]]

[[!redirects ∞-solid site]]
[[!redirects ∞-solid sites]]


solid torus

#Contents#
* table of contents
{:toc}

## Idea

A _solid torus_ is a [[topological space]] (a [[manifold with boundary]]) [[homeomorphism|homeomorphic]] to the [[product topological space]] $S^1 \times D^2$ of [[generalized the|the]] [[circle]] with [[generalized the|the]] [[2-disk]], whose [[boundary]] is hence [[generalized the|the]] actual [[torus]].

## Related concepts

* [[Dehn surgery]]

[[!redirects solid tori]]


solidification functor

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

## Idea

[[Dustin Clausen]] says in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057781):

> Namely, I claim that if you’re willing to pass to suspension spectra, then you do get a fully well-defined functor which takes a CW-complex (viewed as a condensed set) to the suspension spectrum of its associated anima. But now this is not a functor to ordinary spectra, but a functor to condensed spectra which happens to take “discrete” values (discrete in the condensed direction) on CW-complexes.

> Actually, what you really have is a colimit-preserving endofunctor of condensed spectra, which when applied to the suspension spectrum of a CW complex (viewed as a condensed set) gives the suspension spectrum of its associated anima. This is the “solidification” functor, which works for spectra much in the same way it works for abelian groups.

## See also

* [[condensed spectrum]]
* [[solid spectrum]]
* [[solid abelian group]]


solidity

#Contents#
* table of contents
{:toc}

## In solid state physics

In [[physics]], _solidity_ of [[matter]] (the subject of [[solid state physics]]) is fundamentally due to the [[Pauli exclusion principle]] whose induced [[degeneracy pressure]] prevents collections of [[fermions]] (such as [[electrons]] and [[quarks]] which consitute [[atoms]]) from collapsing to a [[condensate]].

## Related concepts

* [[elasticity]]

* [[cohesion]]

## References

Exposition includes

* [[Jim Pivarski]], _[Why stuff is hard](https://cornellmath.wordpress.com/2007/08/08/why-stuff-is-hard/)_

Textbook accounts include

* Ulrich Rössler, _Solid State Theory: An Introduction_, 2009

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


soliton

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

#Contents#
* table of contents
{:toc}

## Idea

The term _soliton_ comes from an abbreviation of "solitary wave".

A **soliton solution** of a nonlinear [[wave equation]] is a solution whose large amplitude part is **localized in space** and is **asymptotically stable in time**. This asymptotic stability (more precisely non-damping and asymptotic preservation of shape, up to translation) is typically a feature of an infinite number of [[conservation law]]s, and many models of equations allowing soliton solutions are in fact [[integrable system]]s (with infinitely many degrees of freedom). Soliton solution often combine to multisoliton solutions in a nonlinear way, with a period of interaction when they "meet", but after a passage of some time, the waves gradually uncouple and regain their original shape when outgoing to infinity.  A typical example of a nonlinear wave equation exhibiting soliton solutions is the exactly solvable "nonlinear Schroedinger equation" appearing in [[optics]].

Solitons appear in description of many natural phenomena. For example, Davydov soliton ([wikipedia](https://en.wikipedia.org/wiki/Davydov_soliton)) has a role in stabilizing dynamics of
proteins.

## Examples

* solitons in [[photonic crystals]]: see [there](photonic+crystal#ReferencesSolitons)

## Related concepts

* [[vortex]], [[instanton]], [[caloron]], [[skyrmion]]

* [[integrable model]], [[Korteweg de Vries equation]]

* [[solitonic brane]]

  [[black hole]], [[black brane]]

## Refereneces

> (See also the general references at *[[non-perturbative quantum field theory]]*.)

* D J Wallace, *Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory*,  Phys. Bull. **34** 29 (1983) [[doi:10.1088/0031-9112/34/1/043](https://iopscience.iop.org/article/10.1088/0031-9112/34/1/043)]

* [[Michio Jimbo]], [[Tetsuji Miwa]], [[Etsuro Date]], (transl. from Japanese by [[Miles Reid|M. Reid]]), *Solitons: differential equations, symmetries and infinite dimensional algebras*,  Cambridge Tracts in Mathematics **135** (2012) [[ISBN:9781107404199](https://www.cambridge.org/us/universitypress/subjects/mathematics/geometry-and-topology/solitons-differential-equations-symmetries-and-infinite-dimensional-algebras)]

* A. Newell, _Solitons in mathematics and physics_

* [[Ludwig D. Faddeev]], [[Leon Takhtajan]], *Hamiltonian methods in the theory of solitons*, Springer (2007) [[doi:10.1007/978-3-540-69969-9](https://doi.org/10.1007/978-3-540-69969-9)]

  (transl. from Russian Гамильтонов подход в теории солитонов. — М.: Наука. 1986.)

* [[Ludwig D. Faddeev]], V. E. Korepin, _Quantum theory of solitons_ (1987)

* [[David Tong]], _TASI Lectures on Solitons_ ([web](http://www.damtp.cam.ac.uk/user/tong/tasi.html)),

* Anastasia Doikou, Iain Findlay, _Solitons: conservation laws & dressing methods_ ([arXiv:1812.11914](https://arxiv.org/abs/1812.11914))

* Fritz Gesztesy, Helge Holden, _Soliton equations and their algebro-geometric solutions_ (vol. I), Cambridge Univ. Press 2003; vol. II, with Johanna Michor, Gerald Teschl, 2008.

* [[Emma Previato]], _Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation_, Duke Math. J. 52(2): 329--377 (1985) [doi](https://doi.org/10.1215/S0012-7094-85-05218-4)

[[!redirects solitons]]


solitonic brane

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] the term *solitonic brane* refers to [[brane]]-like solutions of [[equations of motion]] in ([[supergravity|super]]-)[[gravity]] whose presence is imprinted in topologically stable configurations of [[background fields]] of the ambient [[spacetime]], notably in the [[gravitational field]] and the [[higher gauge field]]-[[flux]] for which the brane is a source. This is in contrast to *[[fundamental branes]]* or *[[probe branes]]* described by [[sigma models]] with [[target space]] the given [[spacetime]], whose trajectories are effected by such [[background fields]] but which do not themselves [[backreaction|backreact]] on them.

More specifically, when the term "solitonic brane" was introduced in [Duff, Khuri & Lu 1992](#DuffKhuriLu92), [1994](#DuffKhuriLu94) and [Duff & Lu 1993](#DuffLu93), [1994](#DuffLu94) it referred concretely to "topologically stable" but *non-singular* [[supergravity]]-solutions, hence to those for which the [[background field]] [[field strength|strength]]/[[curvature]] is finite on all of [[spacetime]] as one may expect to be the case for *[[solitons]]* --- this in contrast to *singular-* or *[[black branes]]* whose would-be locus in spacetime is a "[[singularity]]" where these fields diverge, in higher dimensional generalization of the situation for [[black holes]].

However, while the term "solitonic brane" has become common place, authors often use it with somewhat different meaning.
(For better or worse, already the reference term *[[soliton]]* in [[field theory]] is used in a variety of neighbouring but different meanings.)

Another convention [[Stelle (1998)](#Stelle98), [Bergshoeff & Riccioni (2012)](#BergshoeffRiccioni12)] is to say "solitonic" for the singular magnetically charged brane solutions (eg. the [[black brane|black]] [[NS5-brane]]) which are the [[electromagnetic duals]] to the "elementary" (namely just as singular but) electrically charged brane solutions (eg. the [[black string]]). In this convention the adjective "solitonic" is not distinguishing between singular/non-singular fields, but identifies the secondary magnetic sources for given primary degrees of freedom. For instance in this terminology and regarding the [[electron]] as the [[fundamental particle]] that it is, the [[magnetic monopole]] would be its "solitonic" dual version.

But nowadays many authors (eg. [Smith (2002), p. 5](#Smith02)) use "solitonic brane" to subsume all singular and non-singular brane-like supergravity solutions, hence use the term generally as the antonym to fundamental/elementary branes (a terminology that is not used fully consistently across authors, either), ie. as antonym to branes described by [[sigma-models]].

## Related concepts

* [[brane]]

  [[black brane]]

  [[probe brane]], [[flavor brane]]

* [[soliton]]

## References

The original use of "solitonic branes" as "topological sugra solutions with everywhere finite flux densities":

* {#DuffKhuriLu92} [[Michael Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String and Fivebrane Solitons: Singular or Non-singular?*, Nucl.Phys. B **377** (1992) 281-294 [[arXiv:hep-th/9112023](https://arxiv.org/abs/hep-th/9112023), doi:10.1016/0550-3213(92)90025-7]

* {#DuffLu93} [[Michael Duff]], [[Jian Xin Lu]], *Type II $p$-branes: the brane-scan revisited*, Nuclear Physics B **390** 2 (1993) 276-290 [doi:10.1016/0550-3213(93)90457-Z]

* {#DuffKhuriLu94} [[Mike Duff]], [[Ramzi R. Khuri]], [[Jian Xin Lu]], *String Solitons*, Phys. Rept. **259** (1995) 213-326 [[arXiv:hep-th/9412184](https://arxiv.org/abs/hep-th/9412184), doi:10.1016/0370-1573(95)00002-X]

* {#DuffLu94} [[Michael Duff]], [[Jian Xin Lu]], *Black and super $p$-branes in diverse dimensions*, Nucl. Phys. B **416** (1994) 301-334 [[arXiv:hep-th/9306052](http://arxiv.org/abs/hep-th/9306052), doi:10.1016/0550-3213(94)90586-X]

Understanding non-singular [[solitonic brane|solitonic]] [[D-branes]] (though not using this term) and their [[T-duality]] in terms of the [[D-brane charge quantization in K-theory]]:

* {#BergmanGimonHořava99} [[Oren Bergman]], [[Eric G. Gimon]], [[Petr Hořava]], *Brane Transfer Operations and T-Duality of Non-BPS States*, JHEP 9904 (1999) 010 \[doi:10.1088/1126-6708/1999/04/010, [arXiv:hep-th/9902160](https://arxiv.org/abs/hep-th/9902160)\]

Other uses of the word "solitonic brane":

* {#Stelle98} [[Kellogg Stelle]], *BPS Branes in Supergravity*, in: *Quantum Field Theory: Perspective and Prospective*, NATO Science Series **530** (1999) 257-351 [[arXiv:hep-th/9803116](http://arxiv.org/abs/hep-th/9803116), [doi:10.1007/978-94-011-4542-8_12](https://doi.org/10.1007/978-94-011-4542-8_12)]

* {#Smith02} [[Douglas J. Smith]], *Intersecting brane solutions in string and M-theory*, Class. Quant. Grav. **20** R233 (2003) [[arXiv:hep-th/0210157](https://arxiv.org/abs/hep-th/0210157), [doi:10.1088/0264-9381/20/9/203](https://doi.org/10.1088/0264-9381/20/9/203)]

* {#BergshoeffRiccioni12} [[Eric A. Bergshoeff]], [[Fabio Riccioni]], *Solitonic branes and wrapping rules*, Phys. Part. Nuclei **43** (2012) 557–561 [[doi:10.1134/S106377961205005X](https://doi.org/10.1134/S106377961205005X)]

  > (see also at *[[exotic branes]]*)

[[!redirects solitonic branes]]


Solomon Feferman

**Solomon Feferman** (1928-2016) was an American logician and set theorist with important contributions to recursion theory.

A former student of [[Alfred Tarski]] he has also coauthored a biography of Tarski.

He has accompanied the development of categorical foundations of mathematics with a series of critical papers over a period of 30 years.

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

* [homepage](http://math.stanford.edu/~feferman/)

* [list of publications](http://math.stanford.edu/~feferman/papers.html)

## Related entries

* [[continuum hypothesis]]

category:people

[[!redirects S. Feferman]]
[[!redirects Feferman]]
[[!redirects Solomon Fefermann]]


Solomon Lefschetz

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

## Selected writings

Introducing the [[Lefschetz fixed point theorem]]:

* [[Solomon Lefschetz]], _On the fixed point formula_, Ann. of Math. (2), **38**  (1937) 819–822 ([jstor:1968838](https://www.jstor.org/stable/1968838))

## Related entries

* [[Lefschetz trace formula]]
* [[Lefschetz theorem on (1,1)-classes]]

category: people

[[!redirects Lefschetz]]


Solovay-Kitaev theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[finitely generated group|finitely generated]] and [[dense subspace|dense]] [[subgroup]] $\langle g_1, g_1^{-1} \cdots, g_n, g_n^{-1} \rangle$ of the [[topological group]] [[SU(2)]], the *Solovay-Kitaev theorem* [[Kitaev (1997)](#Kitaev97), [Nielsen & Chuang (2000, §A3)](#NielsenChuang00)] gives a strong bound on the length of sequences of [[elements]] from $\{g_1, \cdots, g_n\}$ which are needed in order to approximate any given element $g \,\in\, SU(2)$ to given accuracy.

In [[quantum computation]]-theory one interprets:

1. the set $\{g_0, \cdots, g_n\}$ as a given set of (operations of) [[quantum gates]] on [[qbits]]

   (that the subgroup these generate is [[dense subspace|dense]] in [[SU(2)]] is said to mean that this is a *universal set* of quantum gates);

1. the target element $g \,\in\, SU(2)$ as an intended operation on qbits (an intended quantum program);

1. the sequence of elements $g_i$ used to approximate $g$ as a [[quantum circuit]] realizing the quantum program ("[[quantum compilation]]"),

and then the Solovay-Kitaev theorem guarantees that the maximum *depth* of the quantum circuit grows only slowly with its required accuracy.

In fact, the theorem holds in the generality of [[SU(n)|$SU(d)$]], $d \in \mathbb{N}_{\geq 1}$, which in terms of [[quantum computation]] corresponds to working with *[[qdits]]* instead of just [[qbits]].

Regarding the practical relevance, it has been argued [[MO:a/22197](https://quantumcomputing.stackexchange.com/a/22197/19363)] that [[quantum computing]]-hardware such as based on [[superconductors]] will already allow to approximate any operation on a "physical qbit" by a single [[quantum gate]], but that for such hardware the Solovay-Kitaev theorem is still relevant on the level of "[[logical qbits]]", i.e. taking [[quantum error correction]] into account.

Similarly, in [[topological quantum computation]] the available set of basic [[anyon]] [[braiding|braid gates]] is typically far from the usual [[QBit]]-basis requiring substantial effort in "topological quantum compilation" (e.g. [Bonesteel, Hormozi, Zikos & Simon (2005)](#BonesteelHormoziZikosSimon05), [Hormozi, Zikos, Bonesteel & Simon (2007)](#HormoziZikosBonesteelSimon07), [Hormozi, Bonesteel & Simon (2009)](#HormoziBonesteelSimon09)).

{#BraidImplementingCNotGate} For example, the following lengthy braid has been proposed [[Hormozi, Zikos, Bonesteel & Simon (2007)](#HormoziZikosBonesteelSimon07)] as a possible topological quantum compilation of a *single* [[CNOT gate]]:

## References

### General

According to [Dawson & Nielsen (2006, p. 13)](#DawsonNielsen06) the first announcement of the theorem was

* "by [[Robert Solovay|Solovay]] in 1995 on an email discussion list".

A proof was then outlined in:

* {#Kitaev97} [[Alexei Kitaev]], Thm. 4.8 of: *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), pdf]

followed by talk announcement:

* {#Solovay00} [[Robert Solovay]], *Lie Groups and Quantum Circuits*, talk at MSRI (2000) [[webpage](https://www.msri.org/workshops/189/schedules/12826), [mp4](https://s3-us-west-1.amazonaws.com/msri-videos/paperclip/videos/data/000/049/315/original/189-12826-Lie_Groups_and_Quantum_Circuits-Robert_Solovay.mp4)]

Following [Kitaev (1997)](#Kitaev97), a full proof was spelled out in:

* {#NielsenChuang00} [[Michael A. Nielsen]], [[Isaac L. Chuang]], *Solovay-Kitaev theorem*,  Appendix 3 of: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

Review:

* {#KitaevShenVyalyi02} [[Alexei Kitaev]], A. H. Shen, M. N. Vyalyi, §8.3 in: *Classical and Quantum Computation*, Graduate Studies in Mathematics **47**, Amer. Math. Soc. (2002) [[doi:10.1090/gsm/047](http://dx.doi.org/10.1090/gsm/047)]

* {#DawsonNielsen06} [[Christopher M. Dawson]], [[Michael A. Nielsen]], *The Solovay-Kitaev algorithm*, Quantum Information & Computation **6** 1 (2006) 81–95 [[arXiv:quant-ph/0505030](https://arxiv.org/abs/quant-ph/0505030), [doi:10.5555/2011679.2011685](https://dl.acm.org/doi/10.5555/2011679.2011685)]

See also:

* Wikipedia, *[Solovay-Kitaev theorem](https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem)*

* Maris Ozols, *The Solovay-Kitaev theorem* [[pdf](http://home.lu.lv/~sd20008/papers/essays/Solovay-Kitaev.pdf)]

Further refinement of the algorithm:

* Attila B. Nagy, *On an implementation of the Solovay-Kitaev algorithm*, 10th Rhine Workshop on Computer Algebra [[arXiv:quant-ph/0606077](https://arxiv.org/abs/quant-ph/0606077)]

### Quantum circuit compilation
 {#QuantumCircuitCompilationReferences}

More on quantum circuit compilation:

* [[Neil J. Ross]], Chapters 6-7 in: *Algebraic and Logical Methods in Quantum Computation*, Ph.D. thesis, Dalhousie University (2015) [[arXiv:1510.02198](https://arxiv.org/abs/1510.02198)]

* Adi Botea, Akihiro Kishimoto, Radu Marinescu, *On the Complexity of Quantum Circuit Compilation*, Proceedings of the International Symposium on Combinatorial Search, **9** 1 (2018) [[doi:10.1609/socs.v9i1.18463](https://doi.org/10.1609/socs.v9i1.18463)]

* Robert Wille; Stefan Hillmich; Lukas Burgholzer, *Efficient and Correct Compilation of Quantum Circuits*, in *2020 IEEE International Symposium on Circuits and Systems (ISCAS)* IEEE Explore (2020) [[doi:10.1109/ISCAS45731.2020.9180791](https://doi.org/10.1109/ISCAS45731.2020.9180791)]

* Nathanial Stemen, *Quantum Circuit Compilation from the Ground Up* (2022) [[pdf](https://uwaterloo.ca/applied-mathematics/sites/ca.applied-mathematics/files/uploads/files/quantum_circuit_compilation_from_the_ground_up.pdf)]

* Giulia Meuli, Fereshte Mozafari, Bruno Schmitt, Heinz Riener, Mathias Soeken, Giovanni De Micheli: *[Quantum Compilation](https://si2.epfl.ch/~demichel/research/quantum.html)*

A textbook account:

* Marco Maronese, Lorenzo Moro, Lorenzo Rocutto, Enrico Prati, *Quantum Compiling*, in *Quantum Computing Environments*, Springer (2021) 39-74 [[arXiv:2112.00187](https://arxiv.org/abs/2112.00187), [doi:10.1007/978-3-030-89746-8_2](https://doi.org/10.1007/978-3-030-89746-8_2)]

Quantum compilation with a [[proof assistant]] based on [[QWIRE]]:

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Xiaodi Wu, [[Michael Hicks]], *A verified optimizer for Quantum circuits*, Proceedings of the ACM on Programming Languages **5** Issue POPL 37 (2021) 1–29 [[doi:10.1145/3434318](https://doi.org/10.1145/3434318)]

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Liyi Li, [[Michael Hicks]], *Proving Quantum Programs Correct*, in *12th International Conference on Interactive Theorem Proving (ITP 2021)*, Leibniz International Proceedings in Informatics (LIPIcs) **193** (2021) [[arXiv:2010.01240](https://arxiv.org/abs/2010.01240)]

* [[Kesha Hietala]], *A verified software toolchain for quantum programming* (2022) [[pdf](https://khieta.github.io/files/drafts/khieta-dissertation.pdf)]

### Topological quantum compilation
 {#ReferencesInTopologicalQuantumComputation}

Discussion of the Solovay-Kitaev theorem for quantum compilation in the context of [[topological quantum computation]]:

* {#BonesteelHormoziZikosSimon05} [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) [[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)]

* {#HormoziZikosBonesteelSimon07} [[Layla Hormozi]], [[Georgios Zikos]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Compiling*, Phys. Rev. B **75** 165310 (2007) [[arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111), [doi:10.1103/PhysRevB.75.165310](https://doi.org/10.1103/PhysRevB.75.165310)]

* {#HormoziBonesteelSimon09} [[Layla Hormozi]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Computing with Read-Rezayi States*, Phys. Rev. Lett. **103** 160501 (2009) [[doi:10.1103/PhysRevLett.103.160501](https://doi.org/10.1103/PhysRevLett.103.160501), [arXiv:0903.2239](https://arxiv.org/abs/0903.2239)]

* [[Joren W. Brunekreef]], *Topological Quantum Computation and Quantum Compilation*, Utrecht (2014) [[studenttheses.uu.nl:20.500.12932/17738](https://studenttheses.uu.nl/handle/20.500.12932/17738)]

* Keisuke Fujii, *Quantum Computation with Topological Codes: from qubit to topological fault-tolerance*,  SpringerBriefs in Mathematical Physics **8** (2015) [[arXiv:1504.01444](https://arxiv.org/abs/1504.01444), [doi:10.1007/978-981-287-996-7](https://doi.org/10.1007/978-981-287-996-7)]

* Vadym Kliuchnikov, Alex Bocharov, Krysta M. Svore, *Asymptotically Optimal Topological Quantum Compiling*, Phys. Rev. Lett. **112** (2014) 140504 [[arXiv:1310.4150](https://arxiv.org/abs/1310.4150), [doi:10.1103/PhysRevLett.112.140504](https://doi.org/10.1103/PhysRevLett.112.140504)]

* Caitlin Carnahan, Daniel Zeuch, [[Nicholas E. Bonesteel]], *Systematically generated two-qubit anyon braids*, Phys. Rev. A **93** 052328 (2016) [[arXiv:1511.00719](https://arxiv.org/abs/1511.00719), [doi:10.1103/PhysRevA.93.052328](https://doi.org/10.1103/PhysRevA.93.052328)]

*  [[Emil Génetay-Johansen]], [[Tapio Simula]], Section IV of: *Fibonacci anyons versus Majorana fermions -- A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models*, PRX Quantum **2** 010334 (2021) [[arXiv:2008.10790](https://arxiv.org/abs/2008.10790), [doi:10.1103/PRXQuantum.2.010334](https://doi.org/10.1103/PRXQuantum.2.010334)]

[[!redirects quantum compilation]]
[[!redirects quantum circuit compilation]]

[[!redirects topological quantum compilation]]
[[!redirects topological quantum circuit compilation]]


solution

A _solution_ to an [[equation]] $f(x) = g(x)$ is a choice of value $v$ for the [[free variable]] $x$ such that the statement $f(v) = g(v)$ becomes [[true]].

If we think of the equation as embodied by a pair of [[parallel morphisms]]

$$
  X_1 \underoverset{\underset{f_2}{\longrightarrow}}{\overset{f_2}{\longrightarrow}}{\phantom{AAAA}} X_2
$$

then a solution is a [[cone]] over this [[diagram]]. The solution space itself is the [[limit]] [[cone]], hence the [[equalizer]] of the diagram.

[[!redirects solutions]]


solution set condition

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# Contents
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## Idea

The _solution set condition_ appears as part of the hypothesis in Freyd's [[adjoint functor theorem|General Adjoint Functor Theorem]].

## Statement

A [[functor]] $F : C \to D$ satisfies the **solution set condition** if for every [[object]] $Y$ of $D$ there exists a [[small set]] $I$, an $I$-indexed family $(X_i)_{i \in I}$ of objects of $C$ and an $I$-indexed family $(f_i\colon Y \to F(X_i))_{i \in I}$ of morphisms in $D$ such that each morphism $h\colon Y \to F(X)$ in $D$ can be factored, for some index $i$ and some morphism $t\colon X_i \to X$ in $C$, as

$$
  F(t) \circ f_i
  \colon
  Y \stackrel{f_i}{\to}
  F(X_i)
  \stackrel{F(t)}{\to}
  F(X).
$$

This is a smallness condition in that the family is required to be indexed by a [[small set]].

A restatement of this condition is that the comma categories $(Y \downarrow F)$ all admit weakly initial families of objects.

Here is the connection with the adjoint functor theorem: when small products exist in those comma categories, this is equivalent to saying that they all admit weakly initial objects. When small equalizers exist in those comma categories also, this is equivalent to saying that they all admit initial objects, and this is equivalent to $F$ being a right adjoint.

## Examples

* To define the [[projective tensor product]] of [[Banach spaces]] using the [[adjoint functor theorem]], let $C$ and $D$ each be [[Ban]] (the category of Banach spaces and [[short linear maps]]), fix a Banach space $B$, and let $F(X)$ be $BdLin(B,X)$, the Banach space of [[bounded linear maps]] from $B$ to $X$.

  If $f\colon X_1 \to X_2$ is a short linear map and $\phi\colon B \to X_1$ is a bounded linear map, then $BdLin(B,f)(\phi) \coloneqq f \circ \phi\colon B \to X_2$ is a bounded linear map with norm ${\|BdLin(B,f)(\phi)\|} \leq {\|f\|} {\|\phi\|}$, which proves that ${\|BdLin(B,f)\|} \leq {\|f\|} \leq 1$, so that $BdLin(B,f)$ is also a short map, as it must be for $F$ to be a functor.  (That $F$ is linear and preserves identities and composition is trivial.)

  The solution set condition now states:  For every Banach space $Y$ there exists a small set $I$, an $I$-indexed family $(X_i)_{i \in I}$ of Banach spaces and an $I$-indexed family $(f_i\colon Y \to BdLin(B,X_i))_{i \in I}$ of short linear maps such that each short linear map $h\colon Y \to BdLin(B,X)$ can be factored, for some index $i$ and some short linear map $t\colon X_i \to X$, as $h = BdLin(B,t) \circ f_i$.

  (to be completed)

## Related pages

The solution set condition weakens the notion of adjoint functor by allowing a family of objects rather than a single one, and also by removing the uniqueness of the factorization.  If we weaken the definition in only one of these ways, we obtain a [[multi-adjoint]] or a [[weak adjoint]] respectively.

[[!redirects solution set condition]]
[[!redirects solution set conditions]]
[[!redirects small solution set condition]]
[[!redirects small solution set conditions]]


solvable group

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* table of contents
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## Definition

\begin{definition}
**(solvable group)**\linebreak
A [[group]] is __solvable__ if it is a [[finite number|finite]] iterated [[group extension]] of an [[abelian group]] by [[abelian groups]].

In other words, a group $G$ is solvable if and only if there exists a finite sequence

$$
  \{ 1\}
    \;\subset\;
  G_1
    \;\subset\;
  G_2
    \;\subset\;
  \ldots
    \;\subset\;
  G_k
   \;=\;
  G
  \,,
$$

in which

1. $G_{j-1}$ is [[normal subgroup|normal]] in $G_j$

1. the [[quotient groups]] $G_j/G_{j-1}$ are [[abelian group|abelian]].

\end{definition}

Solvability can equivalently be expressed in terms of breaking a group down rather than building it up.
\begin{definition}
**(solvable group)**\linebreak
Given elements $g,h$ of a group $G$, the [[group commutator|commutator]] is $[g,h] := g^{-1} h^{-1} g h$. The [[commutator subgroup]] of $G$ is the subgroup of $G$ generated by the commutators $[g,h]$ for all $g, h \in G$. A group is solvable if the series of groups produced by repeatedly taking the commutator subgroup, the [[derived series]], terminates with the [[trivial group]] after finitely many steps.
\end{definition}

Similar to nilpotent groups, solvable groups have an inductive definition via group extensions.

The class of **solvable groups** is defined inductively by the following clauses:

1. All abelian groups are solvable.

2. If $1\to G' \to G \to G''\to 1$ is a group extension and both $G'$ and $G''$ are solvable, then $G$ is solvable.

## Examples

\begin{example}\label{GaloisGroups}
**([[Galois groups]])**\linebreak
The terminology "solvable groups" comes from elementary [[Galois theory]]: every [[polynomial]] [[equation]] $\phi$ over an [[integral domain]] $K$ has a corresponding [[Galois group]] $Gal(\phi/K)$, and $Gal(\phi/K)$ is a solvable group if and only if $\phi$ is a [[solvable polynomial equation|solvable equation]] (meaning that all its solutions in an [[algebraic closure]] of $K$ are expressible using the elements of $K$, the field operations, and extraction of roots).
\end{example}

Informally, this can be thought of as each step in the derived series of the Galois group requiring an additional level of nesting of radicals. If the derived series does not terminate, then the solutions of the polynomial equation cannot be expressed by radicals, no matter how deeply nested.

\begin{example}\label{NilpotentGroups}
**([[nilpotent groups]])**
A [[nilpotent group]] is a solvable group given by *[[central extensions|central]]* [[group extensions]].
\end{example}

## Related concepts

* [[commutator subgroup]]
* [[nilpotent group]]

## References

Textbooks:

* [[Cornelia Druţu]], [[Michael Kapovich]], Chapter 13 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))
* Ian Stewart, *Galois Theory*, fourth edition. CRC Press, 2015. ISBN:978-1-4822-4582-0. Chapter 14.

Articles:

* L. Goldmakher, "Arnold's elementary proof of the insolubility of the quintic". [pdf](https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf)

See also:

* [[eom]]: [solvable group](http://eom.springer.de/s/s086130.htm), [Lie group, solvable](http://eom.springer.de/l/l058690.htm)

* Wikipedia, *[Solvable group](http://en.wikipedia.org/wiki/Solvable_group)*

[[!redirects solvable group]]
[[!redirects solvable groups]]
[[!redirects solvable subgroup]]
[[!redirects solvable subgroups]]

[[!redirects soluble group]]
[[!redirects soluble groups]]
[[!redirects soluble subgroup]]
[[!redirects soluble subgroups]]


solvable polynomial

A [[polynomial]] $\phi$ over an [[integral domain]] $K$ (or equivalently a polynomial equation, which is the generality that one should consider if $K$ is only an integral cancellable [[rig]]) is __solvable__ if all of its roots in an [[algebraic closure]] (or any other sufficiently large [[splitting field]]) of $K$ are expressible using elements of $K$, the field operations, and extraction of roots.

The basic result of elementary [[Galois theory]] is that $\phi$ is solvable over $K$ if and only if its [[Galois group]] over $K$ is a [[solvable group]] (whence the name of the latter).

[[!redirects solvable polynomial]]
[[!redirects solvable polynomials]]
[[!redirects solveable polynomial]]
[[!redirects solveable polynomials]]
[[!redirects soluble polynomial]]
[[!redirects soluble polynomials]]

[[!redirects solvable polynomial equation]]
[[!redirects solvable polynomial equations]]
[[!redirects solveable polynomial equation]]
[[!redirects solveable polynomial equations]]
[[!redirects soluble polynomial equation]]
[[!redirects soluble polynomial equations]]


Some geometric perspectives in concurrency theory

This entry is about the text:

[[Eric Goubault]], Some geometric perspectives in [[concurrency theory]], appears in : Homology Homotopy Appl. Volume 5, Number 2 (2003), 95-136.][pdf](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.hha/1088453323)

#Abstract#

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of "geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the "direction" of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: "directed" or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, $\omega$-categorical and topological techniques.

category: reference


Some remarks on axiomatized set theory

* [[Thoralf Skolem]],
  _Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre_, 1922

is a talk on occasion of the Fith Congress of Scandinavian Mathematicians in which some major ideas of [[set theory]] appear and the idea of set theoretic [[foundation of mathematics|foundation]] is discussed.

## Content

Axiomatic set theory at the stage of this talk is not a formal theory, i.e. not conceived as being a manipulation of strings.
Instead the axioms are considered to describe properties of a certain universe or range of things $B$ (German "Bereich").
This $B$ would be called underlying set of a model in modern terminology.

### The peculiar fact that, in order to treat "sets", we must begin with "domains" that are constituted in a certain way

First, Skolem points out the problem of how to think of $B$ itself: if we conceive $B$ as a set itself, we can not do so as a thing of the theory we are developing.
Second, it is pointed out that $B$ is not uniquely determined by the set of axioms.

### A definition, much to be desired, that makes Zermelo's notion "definite proposition" precise

In modern language a (well-formed) formula.

### The fact that in every thoroughgoing axiomatization set-theoretic notions are unavoidably relative

Downward version of [[Löwenheim-Skolem theorem]] is proved, i.e. existence of countable models.
[[Skolem's paradox]] is explained.

### The fact that Zermelo's system is not sufficient to provide a foundation for ordinary set theory

[[axiom of replacement|Axioms of replacement]] is suggested.

### The difficulties caused by nonpredicative stipulations when one wants to prove the consistency of the axioms

Skolem argues that the consistency of set theory can not be proved (note that at that time Gödel's [[incompleteness theorem]] was still unknown).
Such a proof is in Skolem's opinion not possible as in set theory sets are formed in a "non-predicative" way, that is to say they are formed from all sets of $B$ and not just a finite number of sets, e.g. the intersection of all sets with some property.

Skolem remarks that a the "non-predicative requirement of reproduction of sets" also occurs in [[Principia Mathematica|Russell's type theory]] in form of the [[axiom of reducibility]].

### The nonuniqueness [Mehrdeutigkeit] of the domain $B$

In modern language: Based on a given model of set theory, a new model is constructed by extending the underlying set by a new element.
In a footnote Skolem speculates that the [[continuum hypothesis]] can not decided in the set theory of his time.

### The fact that mathematical induction is necessary for the logical investigation of abstractly given systems of axioms

A skeptic view on [[Hilbert's program]], that was ongoing at that point, is explained.
In modern language the argument amounts to saying that in the metalanguage of mathematical logic the induction axiom is indispensable.

### A remark on the principle of choice

Skolem criticizes opponents of the [[axiom of choice]] by rooting their opposition to a "non-axiomatic" understanding of sets:
sets are for them not things subject to some axioms but just a collection on _explicitely_ given things.

### Concluding remark

> ... I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it.
  But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics ...

> ... glaubte ich, dass es so klar sei, dass diese Mengenaxiomatik keine befriedigende letzte Grundlage der Mathematik wäre,
  dass die Mathematiker grösstenteils sich nicht so sehr darum kümmern würden.
  In der letzten Zeit habe ich aber zu meinem Erstauenen gesehen,
  dass sehr viele Mathematiker diese Axiome der Mengenlehre als ideale Begründung der Mathematik betrachten; ...

## References

Originally published in German

* [[Thoralf Skolem]], _Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre_, Mathematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse: 217–232

English translation

* _Some remarks on axiomatized set theory_ in _From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931_, Cambridge, Massachusetts: Harvard University Press

[[!redirects Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre]]

category: reference


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This entry is about the article

* [[William Lawvere]],

  _Some Thoughts on the Future of Category Theory_

  ([doi:10.1007/BFb0084208](https://link.springer.com/chapter/10.1007/BFb0084208))

  In: A. Carboni, M. Pedicchio, G. Rosolini (ed.s)

  _Category Theory_,

  [[Como|Proceedings of the International Conference held in Como]],

  Lecture Notes in Mathematics 1488, Springer (1991)

about the formalization of the "[[objective logic]]" of [[Hegel|Hegelian]] [[metaphysics]] ([[Science of Logic]]) and in particular (implicitly) the notion of [[cohesive toposes]]. (See there for further references and other background material).

A closely related text is _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_.

#Contents#
* table of contents
{:toc}

## Exegesis

The article is written in a style typical for Lawvere, where precise general abstract [[category theory|category theoretic]] and [[topos theory|topos theoretic]] situations are discussed more in prose than in the usual style of mathematical writing. The thoughts revolve around a topic that Lawvere takes up in various later articles, which are all listed in the References-section at _[[cohesive topos]]_. In the following we try to illuminate what the article here is saying. Of course such an _exegesis_ may or may not accurately reflect some of the original author's actual intentions.

Pages 6 to 8 in section II of the text is to a large extent a proposal that there is a useful formalization of the [[unity of opposites]] and their "[[Aufhebung]]" that governs [[Georg Hegel]]'s [[metaphysics]] as laid out in his _[[Science of Logic]]_. The proposal is that a "determination of [[being]] and [[becoming]]" is expressed by an [[adjoint modality]]. The tautological example $(\emptyset \dashv \ast)$ is "pure being" and refinements thereof such as $(\flat \dashv \sharp)$ ([[flat modality]] $\dashv$ [[sharp modality]]) characterize more determinate ways of entities to be.

Specifically, the notion of a **[[category of being]]** that Lawvere discusses (following terminology in [[Hegel|Hegel's]] _[[Science of Logic]]_) in _Some Thoughts on the Future of Category Theory_ is the notion that more recently he has been calling a _category of cohesion_ . The following tries to illuminate a bit what's going on .

We restrict attention to the case that the [[category]] "of [[being|Being]]" is a [[topos]] and say _[[cohesive topos]]_ for short. This is a topos that satisfies a small collection of simple but powerful axioms that are supposed to ensure that its [[object]]s may consistently be thought of as _[[geometry|geometric]] [[space]]s_ built out of [[point]]s that are equipped with "cohesive" structure (for instance [[topology|topological structure]], or [[smooth structure]], etc.). So the idea is to axiomatize [[big topos]]es in which [[geometry]] may take place.

We walk through the main bits of the article:

One [[axiom]] on a [[cohesive topos]] $\mathcal{E}$ is that the [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E} \to \mathcal{S}$  to the given [[base topos]] $\mathcal{S}$ has a further [[left adjoint]] $\Pi_0 := \Gamma_! : \mathcal{E} \to \mathcal{S}$ to its [[inverse image]] $\Gamma^{\ast}$, which we shall write $\mathrm{Disc} := \Gamma^{\ast}$, for reasons discussed below. This extra left adjoint has the interpretation that it sends any object $X$ to the set $\Pi_0(X)$ "of [[connected]] components". What Lawvere calls a [[locally connected topos|connected object]] in the article (p. 4) is hence one that is sent by $\Pi_0$ to the [[terminal object]].

Another [[axiom]] is that $\Pi_0$ preserves finite [[product]]s. This implies by the above that the collection of connected objects is closed under finite products. This appears on page 6. What he mentions there with reference to Hurewicz is that given a topos with such $\Pi_0$, it becomes canonically [[enriched category|enriched]] over the [[base topos]] in a second way, a geometric way.

The meaning of this, like that of various other aspects of cohesive toposes, may be clearer as we make the evident step to
[[cohesive (∞,1)-topos]]es.  (But notice that this, while inspired by Lawvere, is not due to him.)

In this more encompassing context the extra [[left adjoint]] $\Pi_0$ becomes a left [[adjoint (∞,1)-functor]] $\Pi_\infty$ which we just write $\Pi$: it sends, one can show, any object to its [[fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos
|geometric fundamental ∞-groupoid]], for a notion of geometric paths intrinsic to the $\infty$-topos. The fact that this preserves finite [[product]]s then says that there is a notion of _[[concordance]]_ of [[principal ∞-bundle]]s in the $(\infty,1)$-topos.

The next [[axiom]] on a cohesive topos says that there is also a further [[right adjoint]] $\mathrm{coDisc} := \Gamma^! : \mathcal{S} \to \mathcal{E}$ to a total [[adjoint quadruple]]

$$
  (\Pi_0 \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{coDisc}) :=
  (\Gamma_! \dashv \Gamma^* \dashv \Gamma_* \dashv \Gamma^!) : \mathcal{E} \to \mathcal{S}
$$

such that both $\mathrm{Disc}$ as well as $\mathrm{coDisc}$ are [[full and faithful functor|full and faithful]].

This is what Lawvere is talking about from the bottom of p. 6 on. The _downward functor_ that he mentions is $\Gamma : \mathcal{E} \to \mathcal{S}$. This has the interpretation of sending a cohesive space to its underlying set of points, as seen by the [[base topos]] $\mathcal{S}$. The left and right adjoint inclusions to this are $\mathrm{Disc}$ and $\mathrm{coDisc}$. These have the interpretation of sending a set of points to the corresponding space equipped with either _[[discrete space|discrete cohesion]]_ or _[[codiscrete space|codiscrete (indiscrete) cohesion]]_ . For instance in the case that cohesive structure is [[topology|topological structure]], this will be the [[discrete topology]] and the [[indiscrete space|indiscrete topology]], respectively, on a given set. Being full and faithful, $\mathrm{Disc}$ and $\mathrm{coDisc}$ hence make $\mathcal{S}$ a [[full subcategory]] of $\mathcal{E}$ in two ways (p. 7), though only the image of $\mathrm{coDisc}$ will also be a [[subtopos]], as he mentions on page 7.

(This has, by the way, an important implication that Lawvere does not seem to mention: it implies that we are entitled to the corresponding [[quasi-topos]] induced by the sub-topos. That, one can show, may be identified with the collection of _[[concrete sheaf|concrete]]_ cohesive spaces. In the case of the cohesive topos for differential geometry, the concrete objects in this sense are precisely the _[[diffeological space]]s_ .  )

He calls the subtopos given by the image of $\mathrm{coDisc} : \mathcal{S} \to \mathcal{E}$ that of "pure [[becoming|Becoming]]" further down on p. 7, whereas the subcategory of discrete objects he calls that of "non [[becoming|Becoming]]". One way one might understand this terminology is as follows:

whereas any old [[(∞,1)-topos]] is a collection of _[[space]]s_ , a [[cohesive (∞,1)-topos]] comes with the extra adjoint $\Pi$ which, as mentioned above, has the interpretation of sending any space to its [[fundamental ∞-groupoid]]. Therefore there is an intrinsic notion of _geometric paths_ in any cohesive $\infty$-topos. This allows notably to define [[parallel transport]] along paths and [[higher parallel transport]] along higher [[dimension]]al paths, hence a kind of _dynamics_ . In fact there is [[differential cohomology]] in every cohesive $(\infty,1)$-topos.

Now, in a [[discrete space|discrete object]] there are no non-trivial paths (formally because   by the fact that $Disc$  is [[full and faithful functor|full and faithful]] and [[left adjoint]] to $\Pi$ we have $\Pi Disc \simeq Id$), so there is "no dynamics" in a discrete object hence "no [[becoming]]", if one wishes. Conversely in a [[codiscrete space|codiscrete object]] every sequence of points whatsoever counts as a path, hence the distinction between the space and its "dynamics" disappears and so we have "pure [[becoming]]", if one wishes.

Notice next that every [[adjoint triple]] induces an [[adjoint monad]]. In the present situation we get

$$
  (\mathrm{Disc} \;\Gamma \dashv \mathrm{coDisc}\; \Gamma) : \mathcal{E} \to \mathcal{E}
$$

This is what Lawvere calls the _skeleton_ and the _coskeleton_ on p. 7. In the [[(∞,1)-topos]] context the left adjoint $\mathbf{\flat} := \mathrm{Disc} \; \Gamma$ has the interpretation of sending any object $A$ to the coefficient for cohomology of [[local system]]s with coefficients in $A$.

The paragraph wrapping from page 7 to 8 comments on the possibility that the [[base topos]] $\mathcal{S}$ is not just that of sets. $\mathcal{S} =$ [[Set]], but something richer. An example of this is that of _super cohesion_ (in the sense of [[superalgebra]] and [[supergeometry]]): the topos of [[smooth super ∞-groupoid|smooth super-geometry]] is cohesive over the base topos of [[super ∞-groupoid|bare super-sets]].

What follows on page 9 are thoughts which it seems Lawvere has not formalized further later on. But then on the bottom of p. 9 he gets to the axiomatic identification of [[infinitesimal space|infinitesimal]] or _formal_ spaces in the cohesive topos. In the more recent article _Axiomatic Cohesion_ what he says here on p. 9 is formalized as follows: he says an object $X \in \mathcal{E}$ is infinitesimal if the canonical morphism $\Gamma X \to \Pi_0 X$ is an [[isomorphism]]. To see what this means, suppose that $\Pi_0 X = *$, hence that $X$ is connected. Then the isomorphism condition means that $X$ has exactly one global point. But $X$ may be bigger: it may be a formal neighbourhood of that point, for instance it may be [[infinitesimally thickened point]]  $\mathrm{Spec} \;k[x]/(x^2)$ that is formally dual to the [[ring of dual numbers]]. A general $X$ for which $\Gamma X \to \Pi_0 X$ is an iso is hence a disjoint union of formal neighbourhoods of points.

Again, the meaning of this becomes more pronounced in the context of [[cohesive (∞,1)-topos]]es: there objects $X$ for which $\Gamma X \simeq * \simeq \Pi X$ have the interpretation of being _[[∞-Lie algebroid|formal ∞-groupoid]]s_ , for instance [[Lie integration|formally exponentiated]] [[L-∞ algebra]]s. And so there is [[∞-Lie theory]] canonically in every cohesive $\infty$-topos.

More discussion of all this is at [[schreiber:differential cohomology in a cohesive topos]].

## Related entries

18 years later at the same place, Lawvere gives a lecture series on this topic: _[[Cohesive Toposes -- Combinatorial and Infinitesimal Cases]]_.

## References

* A sequence of videos covering the talk in Como is available [on YouTube](http://www.youtube.com/playlist?list=PLRIsuk3ZOitzDqeVKOZhe0MrzEUtRNAh5)

category: reference


This entry provides links related to

* [[Pavol Ševera]],

  _Some title containing the words "homotopy" and "symplectic", e.g. this one_, based on a talk at [Poisson 2000](http://www.lpthe.jussieu.fr/~dito/poissongeometry/Poisson2000/index.html), [CIRM Marseille Luminy](https://www.cirm-math.fr/), June 2000 ([arXiv:0105080](http://arxiv.org/abs/math/0105080))

on this table:

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

category: reference


Sonia Haddad

* [GoogleScholar page](https://scholar.google.com/citations?user=Jtm9xZQAAAAJ&hl=fr)

## Selected writings

Models for [[Chern insulators]] akin to the [[Haldane model]]:

* [[Marwa Mannaï]], [[Jean-Noël Fuchs]], [[Frédéric Piéchon]], [[Sonia Haddad]], *Stacking-induced Chern insulator* [[arXiv:2208.02491](https://arxiv.org/abs/2208.02491)]

On the behaviour of [[topological insulators]]/[[semi-metals]] ([[Haldane model]]) under external [[strain]]:

* [[Marwa Mannaï]], [[Sonia Haddad]], *Strain tuned topology in the Haldane and the modified Haldane models*., J of Physics: Condens. Matter **32** 225501 (2020) $[$[arXiv:1907.11213](https://arxiv.org/abs/1907.11213), [doi:10.1088/1361-648X/ab73a1](https://doi.org/10.1088/1361-648X/ab73a1)$]$

* [[Marwa Mannaï]], [[Sonia Haddad]], *Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides*, Phys. Rev. B **103** 201112 (2021) $[$[arXiv:2011.08818](https://arxiv.org/abs/2011.08818), [doi:10.1103/PhysRevB.103.L121112](https://doi.org/10.1103/PhysRevB.103.L121112)$]$

category: people


Sonia Natale

__Sonia Natale__ is a mathematician at the National University of Cordoba ([web](https://www.famaf.unc.edu.ar/~natale/)). She works mainly in categorical algebra, with [[fusion categories]] and [[Hopf algebras]].

## Selected writings

On [[fusion categories]]:

* Sonia Natale, _On the classification of fusion categories_, Proc. Int. Congr. Mathematicians, Rio de Janeiro (2018): 173-200.

* Sebastian Burciu, Sonia Natale, _Fusion rules of equivariantizations of fusion categories_, Journal of Mathematical Physics 54.1 (2013)

On [[Hopf algebras]]:

* Sonia Natale, _On semisimple Hopf algebras of dimension $p q^2$_, J. Algebra 221.1 (1999): 242-278 [doi](https://doi.org/10.1006/jabr.1999.7968); _On semisimple Hopf algebras of dimension $p q^2$, II_, Algebras and Representation Theory 4, 277--291 (2001) [doi](https://doi.org/10.1023/A:1011434104084)

category: people


Sophia E. Economou

* [institute page](https://www.phys.vt.edu/About/people/Faculty/sophia-economou.html)

* [research group](https://www1.phys.vt.edu/~economou/)

* [GoogleScholar page](https://scholar.google.com/citations?user=60Wn3gYAAAAJ&hl=en)

## Selected writings

On [[quantum repeaters]] for [[quantum networks]]:

* [[Koji Azuma]], [[Sophia E. Economou]], [[David Elkouss]], Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, Ilan Tzitrin, *Quantum repeaters: From quantum networks to the quantum internet* [[arXiv:2212.10820](https://arxiv.org/abs/2212.10820)]

category: people

[[!redirects Sophia Economou]]


Sophia Prime


Sophie Bernard

* [website](https://univ-cotedazur.fr/annuaire/sophie-bernard)

## Publications

* [[Sophie Bernard]], [[Cyril Cohen]], [[Assia Mahboubi]], [[Pierre-Yves Strub]], *Unsolvability of the Quintic Formalized in Dependent Type Theory*, ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France ([hal:hal-03136002](https://hal.inria.fr/hal-03136002))

category: people


Sophie Morel

* [webpage at ENS Lyon](http://perso.ens-lyon.fr/sophie.morel/)

*  [Wikipedia (English)](http://en.wikipedia.org/wiki/Sophie_Morel)

* [Wikipedia (francais)](http://fr.wikipedia.org/wiki/Sophie_Morel)

category: people


Sophocles J. Orfanidis

* [institute page](https://www.ece.rutgers.edu/sophocles-orfanidis)

* [personal page](http://eceweb1.rutgers.edu/~orfanidi/)

## Selected writings

On [[electromagnetic waves]]

* [[Sophocles J. Orfanidis]], *Electromagnetic Waves and Antennas* (1999-2016) [[web](http://eceweb1.rutgers.edu/~orfanidi/ewa), [pdf](http://eceweb1.rutgers.edu/~orfanidi/ewa/ewa-1up.pdf)]

category: people

[[!redirects Sophocles Orfanidis]]


Sophus Lie

**Marius Sophus Lie** (1842--1899) was a pioneer of study of (and almost solely devoted to) the groups of continuous and smooth transformations, mainly motivated by symmetries of differential equations. He is considered a founder of the theory of [[Lie groups]] and [[Lie algebras]], whose global theory however has been completed only with works of [[Élie Cartan]]. See also [[Lie groupoid]], [[Lie's three theorems]], [[Lie theory]].

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

category: people


Sorgenfrey line

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The *Sorgenfrey line* (after [[Robert Sorgenfrey]], a mathematical descendant of [[R.L. Moore]]) is the [[real line]] $\mathbb{R}$, but [[topological space|topologized]] by taking as [[base|topological basis]] the collection of [[half-open intervals]] $[a, b)$. This is sometimes called the *[[lower limit topology]]*, and the Sorgenfrey line is often denoted by $\mathbb{R}_l$.

The Sorgenfrey line and spaces derived from it, notably the *Sorgenfrey plane* $\mathbb{R}_l \times \mathbb{R}_l$ (with the [[product topology]]), is a rich source for counterexamples in [[general topology]]. For example, the Sorgenfrey line is a ($T_1$) [[normal space]] and even a [[paracompact space]], but the Sorgenfrey plane fails to be normal (and thus is not paracompact either). The Sorgenfrey plane is also an example of a [[separable space]], but admitting an [[uncountable set|uncountable]] [[discrete space|discrete]] [[subspace]] (which is thus not separable), showing that separability is not a hereditary property.

[[!redirects Sorgenfrey lines]]
[[!redirects Sorgenfrey plane]]


Sorin Dragomir

* [webpage](https://unibas.etrasparenza.it/index.php?id_oggetto=3&id_cat=0&id_doc=54101)

## Selected writings

On [[CR-manifolds]]:

* {#DragomiTomassini06} [[Sorin Dragomir]], Giuseppe Tomassini, _Differential Geometry and Analysis on CR Manifolds_, Birkhäuser, 2006 ([doi:10.1007/0-8176-4483-0](https://link.springer.com/book/10.1007/0-8176-4483-0))

On [[CR-orbifolds]]:

* [[Sorin Dragomir]], Jun Masamune, _Cauchy-Riemann orbifolds_, Tsukuba J. Math. Volume 26, Number 2 (2002), 351-386 ([euclid:tkbjm/1496164430](https://projecteuclid.org/euclid.tkbjm/1496164430))

category: people


soul

## In superalgebra

In the context of [[superalgebra]], the the ideal of a superalgebra generated by its odd part is called its _soul_.

The [[quotient]] by that ideal is called the _[[body]]_.

See at _[superalgebra -- Adjoints to inclusion of plain algebras](super+algebra#AdjointsToInclusionOfPlainAlgebra)_.

This terminology is (for the special case of [[Grassmann algebras]] at least) due to [DeWitt 92, p. 1](#DeWitt92)

## References

* {#DeWitt92} [[Bryce DeWitt]], _Supermanifolds_, 1992

[[!redirects soul]]
[[!redirects souls]]


sound doctrine of limits

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# Sound doctrines

* table of contents
{: toc}

## Idea

A [[class]] $\Phi$ of [[diagram]] shapes for [[limits]] --- or more generally a class of [[weighted limits|weights]] for limits --- is called *sound* as a *[[doctrine]] of limits* [[Adámek, Borceux, Lack & Rosicky (2002)](#AdámekBorceuxLAckRosicky2002)] if it behaves nicely when paired with the class $\Phi^+$ of all [[colimit]] shapes (or weights) that [[commutativity of limits and colimits|commute]] with $\Phi$-limits in [[Set]] (or more generally in the [[enriched category|base of enrichment]]).

## Definition

...

## References

* {#AdámekBorceuxLackRosicky2002} [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiri Rosicky]], *A classification of accessible categories*, Journal of Pure and Applied Algebra **175** 1–3 (2002)  7-30 [doi:10.1016/S0022-4049(02)00126-3]

* [n-Cafe discussion about the above paper](https://golem.ph.utexas.edu/category/2014/05/classifying_by_generalizing_th.html)

* [[Stephen Lack]], [[Jiri Rosicky]], *Notions of Lawvere theory*, [arxiv](https://arxiv.org/abs/0810.2578)

* Matěj Dostál, Jiří Velebil, *An elementary characterisation of sifted weights*, [arxiv](https://arxiv.org/abs/1405.3090)

* [[G. M. Kelly]], V. Schmitt, *Notes on enriched categories with colimits of some class*, [arXiv](https://arxiv.org/abs/math/0509102)

[[!redirects sound doctrines of limits]]

[[!redirects doctrine of limits]]
[[!redirects doctrines of limits]]

[[!redirects sound doctrine]]
[[!redirects sound doctrines]]


Sourav Chatterjee

* [personal page](http://statweb.stanford.edu/~souravc/)

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

## Selected writings

On formulating and proving [[confinement]] and the [[mass gap problem]] in [[lattice QCD]] with [[probability theory|propabilistic methods]]:

* [[Sourav Chatterjee]], _Yang-Mills for probabilists_, in: _Probability and Analysis in Interacting Physical Systems_, PROMS **283** (2019) Springer ([arXiv:1803.01950](https://arxiv.org/abs/1803.01950), [doi:10.1007/978-3-030-15338-0](https://link.springer.com/book/10.1007/978-3-030-15338-0))

* {#Chatterjee20} [[Sourav Chatterjee]], _A probabilistic mechanism for quark confinement_, Comm. Math. Phys. 2020 ([arXiv:2006.16229](https://arxiv.org/abs/2006.16229))

> The confinement of quarks is one of the enduring mysteries ofmodern physics. $[\ldots]$ In spite of many decades of research, physically relevant quantum gauge theories have not yet been con-structed in a rigorous mathematical sense. $[$ [[non-perturbative quantum field theory|non-perturbatively]], that is $]$ $[\ldots]$ Perhaps the most important example is four-dimensional [[SU(3)]]-[[lattice gauge theory]]. If one can show that this theory has a [[mass gap]] at all values of the [[coupling constant|coupling strength]], that would explain why particles known as [[glue-balls]] in the [[QCD|theory of strong interactions]] have mass. All such questions remain open.

> The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.

category: people


source

> This entry is about the basic notion of "source" of a morphism in [[category theory]]. For another use in category theory see at _[[sink]]_ and for the use in [[physics]] see at _[[source field]]_. There is also [[source forms]] in [[variational calculus]].

> This entry is about the basic notion of "domain" of a morphism in [[category theory]]. For other uses, see at _[[domain]]_.

***

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#Contents#
* table of contents
{:toc}

## Definition

The **source object**, or simply **source**, of a [[morphism]] $f: x \to y$ in some [[category]] $C$ is the [[object]] $x$.  The source of $f$ is also called its _[[domain]]_, although that can be confusing in categories of [[partial functions]].

Given a [[small category]] $C$ with set of objects $C_0$ and set of morphisms $C_1$, the **source function** of $C$ is the function $s: C_1 \to C_0$ that maps each morphism in $C_1$ to its source object in $C_0$.

Generalising this, given an [[internal category]] $C$ with object of objects $C_0$ and object of morphisms $C_1$, the **source morphism** of $C$ is the morphism $s: C_1 \to C_0$ that is part of the definition of internal category.

Warning: there is another meaning of 'source' in category theory; see [[sink]].

## Related concepts

* [[target]], [[codomain]]

* [[simplicial identities]]

[[!redirects source object]]
[[!redirects sources]]


source field

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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

In the [[path integral quantization]] formulation of [[quantum field theory]] the [[correlation functions]] ([[expectation values]]) are schematically [[path integrals]] of the form

$$
  \langle \phi(x_1) \cdots \phi(x_n)\rangle
  \coloneqq
  \frac{
    \int [D\phi] \,\phi(x_1) \cdots \phi(x_n) \, \exp\left(i S\left(\phi\right)\right)
  }{
    \int [D\phi]\,\exp\left(i S\left(\phi\right)\right)
  }
  \,.
$$

Therefore, as for ordinary [[moments]] (and explicitly so under [[Wick rotation]], if possible), there is [[generating functional]] for the correlators of the schematic form

$$
  \Psi(J)
  =
  \int [D \phi] \, \exp\left(i S\left(\phi\right) + i \int_X J \phi d\mu\right)
  \,.
$$

Here in the exponent one may regard

$$
  S'(\phi,J) = S(\phi) + \int_X J \phi d\mu
$$

as a new [[action functional]] defined on a larger space of [[field (physics)|fields]] that also contains the parameters $J$ as fields. In this context one calls $J$ a **source field**.

This is in the corresponding [[equations of motion]] of $S'$ $J$ will act like a source term. The [[Euler-Lagrange equations]] for the modified action are:

$$
  EL(S') = EL(S) + J = 0
  \,.
$$

Notably if $EL(S)$ is a homogeneous [[wave equation]] (as for a [[free field theory]]) then $J$ is the inhomogeneous term in such a wave equation which describes indeed a "source" of wave excitations.

## Related concepts

[[!include holographic principle -- table]]

[[!redirects source fields]]


source form

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#Contents#
* table of contents
{:toc}

## Definition

Given a [[smooth function|smooth]] [[bundle]] $E \to \Sigma$, then a [[differential form]] on its [[jet bundle]] $J^\infty_\Sigma(E)$ is called a _source form_ if it is of vertical degree 1 (with respect to the [[variational bicomplex]]) and its evaluation on a [[vector field]] depends only on the projection of that vector field to a vector field on $E$ itself (e.g. [Zuckerman 87, p. 6](#Zuckerman87)):

$$
  \Omega^{k,1}_{\Sigma,source}(E)
  \;\coloneqq\;
  \Omega^{k,0}_\Sigma(E) \wedge \delta C^\infty(E)
  \,.
$$

## In variational calculus

Given a [[local Lagrangian]] $\mathbf{L}$, i.e. a horizontal form on $Jet(E)$ of maximal horizontal degree, then there is a unique source form $\mathbf{E}$ such that

$$
  d \mathbf{L} = \mathbf{E} - d_H \theta
$$

for some form $\theta$. This $\mathbf{E}$ is the [[Euler-Lagrange form]] of $\mathbf{L}$.

The sum

$$
  \rho \coloneqq \mathbf{L} + \theta
$$

is the corresponding [[Lepage form]].

## Related concepts

* [[variational bicomplex]]

* [[evolutionary vector field]], [[evolutionary derivative]]

## References

* {#Zuckerman87} [[Gregg J. Zuckerman]], *Action principles and global geometry*, in: [[Shing-Tung Yau]] (ed.) *Mathematical Aspects of String Theory*, World Scientific (1987) 259-284 [[[ZuckermanVariation.pdf:file]], [doi:10.1142/0383](https://doi.org/10.1142/0383)]

[[!redirects source forms]]


sources in computational topology

# Sources in computational topology
* table of contents
{: toc}

## Books

* {#EdelsbrunnerHarer10} [[Herbert Edelsbrunner]], [[John L. Harer]], *[[Computational Topology, An Introduction]]*, American Mathematical Society, 2010 ([ISBN:978-0-8218-4925-5](http://www.ams.org/bookstore-getitem/item=MBK-69), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf))

* A. Zomorodian, _Topology for Computing_, Cambridge University Press, 2005, [publishers' webpage](http://www.cambridge.org/gb/knowledge/isbn/item1171674/?site_locale=en_GB)

* Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek, _Computational Homology_, Appl. Math. Sci. Vol. 157, Springer Verlag, NY 2004, [webpage](http://www.springer.com/mathematics/applications/book/978-0-387-40853-8)

* Sergei Matveev, _Algorithmic Topology and Classification of 3-Manifolds_, Second Edition, Algorithms and Computations in mathematics, Volume 9, Springer Verlag, NY 2007, ISBN 978-3-540-45898-2

## Websites

* [Chomp](http://chomp.rutgers.edu)

* [Comptop](http://comptop.stanford.edu/) -- applied and computational topology group at Stanford University.

* [HAP](http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/aboutContents.html) --  although primarily concerned with Homological Algebra Programming, (whence HAP), this library of GAP based programs includes some of relevance to topology. (See also [CHA](http://hamilton.nuigalway.ie/CHA/).

*  [Kenzo](http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/) -- effective algebraic topology as viewed by Francis Sergeraert (Grenoble).

## Software

* [javaplex](http://code.google.com/p/javaplex) - Java library for computing persistent topological invariants
* [Perseus](http://www.sas.upenn.edu/~vnanda/perseus) -- C++ persistent homology software based on Discrete Morse theory.
* [jPlex](http://comptop.stanford.edu/u/programs/jplex/index.html)
* [Dionysus](http://www.mrzv.org/software/dionysus) -- C++ library for computing persistent homology
* Plex for Matlab: older, [C++ based version of Plex](http://comptop.stanford.edu/u/programs/plex.html)

category: topology, applications, reference
[[!redirects sources in computational topology]]