stereographic projection

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#### Topology
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#### Manifolds and cobordisms
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#Contents#
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    "caption": "From [Apostol 1973](#Apostol73)"
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## Idea

_Stereographic projection_ is the name for a specific [[homeomorphism]] (for any $n \in \mathbb{N}$) from the [[n-sphere]] $S^n$ with one point $p \in S^n$ removed to the [[Euclidean space]] $\mathbb{R}^n$

$$
  S^n \backslash \{p\} \overset{\simeq}{\longrightarrow} \mathbb{R}^n
  \,.
$$

One thinks of both the $n$-sphere as well as the Euclidean space $\mathbb{R}^n$ as [[topological subspaces]] of $\mathbb{R}^{n+1}$ in the standard way, such that they [[intersection|intersect]] in the [[equator]] of the $n$-sphere.

\begin{imagefromfile}
    "file_name": "StereographicProjection.png",
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For $p \in S^n$ one of the corresponding poles, the _stereographic projection_ is the  map which sends a point $x \in S^{n}\backslash \{p\}$ along the line connecting it with $p$ to the equatorial plane.

If one applies stereographic projection to _both_ possible poles $p_+, p_- \in S^n$ of the sphere given a fixed equatorial plane, then one obtains two different [[homeomorphisms]]

$$
  \left\{
     \mathbb{R}^n
       \underoverset{\simeq}{\phi_i}{\longrightarrow}
     S^n\backslash \{p_i\}
  \right\}_{i \in \{+,-\}}
  \,.
$$

The set of these two projections constitutes an [[atlas]] that exhibits the [[n-sphere]] as a [[topological manifold]], in fact a [[differentiable manifold]] and in fact as a [[smooth manifold]].

For $n = 2$ and with $\mathbb{R}^2 \simeq \mathbb{C}$ regarded as the [[complex plane]], then this atlas realizes the [[2-sphere]] as a [[complex manifold]]: the _[[Riemann sphere]]_.

## Definition

We consider the ambient canonical [[coordinates]] of $\mathbb{R}^{n+1}$, in terms of which the $n$-sphere is the [[topological subspace]] whose underlying [[subset]] is presented as follows:

$$
  S^n
   \;\simeq\;
  \left\{
     x = (x_1, \cdots, x_{n+1})  \in \mathbb{R}^{n+1}
     \,\vert\,
     \underoverset{i = 1}{n+1}{\sum} (x_i)^2 = 1
  \right\}
  \;\subset\;
  \mathbb{R}^{n+1}
  \,.
$$

Without restriction, we may identify the given pole point in the coordinates with

$$
  p = (1,0,\ldots,0)
$$

hence the corresponding equatorial hyperplane with

$$
  \mathbb{R}^n
    \;\simeq\;
  \left\{
    x = (x_1, x_2, \cdots, x_{n+1}) \in \mathbb{R}^{n+1}
    \,\vert\,
    x_1 = 0
  \right\}
   \;\subset\;
  \mathbb{R}^{n+1}
  \,.
$$

+-- {: .num_prop #StandardStereographicProjection}
###### Proposition
**(standard stereographic projection)**

The function which sends a point $x \in S^{n} \backslash p \subset \mathbb{R}^{n+1}$ to the  intersection of the line through $p$ and $x$ with the equatorial hyperplane is a [[homeomorphism]] which is given in terms of ambient coordinates by

$$
  \array{
    \mathbb{R}^{n+1} \supset \;\;\;
    &
    S^n \backslash (1,0, \cdots, 0)
      &\overset{\phantom{AA} \phantom{AA}}{\longrightarrow}&
    \mathbb{R}^{n}
    &
    \;\;\;
    \subset \mathbb{R}^{n+1}
    \\
    &
    (x_1, x_2, \cdots, x_{n+1})
    &\overset{\phantom{AAAA}}{\mapsto}&
    \frac{1}{1 - x_1}
    \left(
      0 , x_2, \cdots, x_{n+1}
    \right)
  }
  \,.
$$

=--

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

First consider more generally the stereographic projection

$$
  \sigma
    \;\colon\;
  \mathbb{R}^{n+1} \backslash (1,0,\cdots, 0)
    \longrightarrow
  \mathbb{R}^n
   =
  \{x \in \mathbb{R}^{n.1} \,\vert\, x_1 = 0 \}
$$

of the entire ambient space minus the point $p$ onto the equatorial plane, still given by mapping a point $x$ to the unique point $y$ on the equatorial hyperplane such that the points $p$, $x$ any $y$ sit on the same straight line.

This condition means that there exists $d \in \mathbb{R}$ such that

$$
  p +  d(x-p) = y
  \,.
$$

Since the only condition on $y$ is that $y_1 = 0$ this implies that

$$
  p_1 + d(x_1-p_1) = 0
  \,.
$$

This equation has a unique solution for $d$ given by

$$
  d = \frac{1}{1 - x_1}
$$

and hence it follow that

$$
  \sigma(x_1, x_2, \cdots, x_{n+1})
    =
  \frac{1}{1-x_1}(0,x_2, \cdots, x_n)
  \,
$$

Since [[rational functions are continuous]], this function $\sigma$ is continuous and since the topology on $S^n\backslash p$ is the [[subspace topology]] under the canonical embedding $S^n \backslash p \subset \mathbb{R}^{n+1} \backslash p$ it follows that the restriction

$$
  \sigma\vert_{S^n \backslash p}
    \;\colon\;
  S^n\backslash p
    \longrightarrow
  \mathbb{R}^n
$$

is itself a [[continuous function]] (because its pre-images are the restrictions of the pre-images of $\sigma$ to $S^n\backslash p$).

To see that $\sigma \vert_{S^n \backslash p}$ is a [[bijection]] of the underlying sets we need to show that for every

$$
  (0, y_2, \cdots, y_{n+1})
$$

there is a unique $(x_1, \cdots , x_{n+1})$ satisfying

1. $(x_1, \cdots, x_{n+1}) \in S^{n} \backslash \{p\}$, hence

   1. $x_1 \lt 1$;

   1. $\underoverset{i = 1}{n+1}{\sum} (x_i)^2 = 1$;

1. $\underset{i \in \{2, \cdots, n+1\}}{\forall} \left(y_i = \frac{x_i}{1-x_1} \right)$.

The last condition uniquely fixes the $x_{i \geq 2}$ in terms of the given $y_{i \geq 2}$ and the remaining $x_1$, as

$$
  x_{i \geq 2} = y_i \cdot (1-x_1)
  \,.
$$

With this, the second condition says that

$$
  (x_1)^2 + (1-x_1)^2 \underset{r^2}{\underbrace{\underoverset{i = 2}{n+1}{\sum}(y_i)^2}} = 1
$$

hence equivalently that

$$
  (r^2 + 1) (x_1)^2 - (2 r^2) x_1 + (r^2 - 1) = 0
  \,.
$$

By the [[quadratic formula]] the solutions of this equation are

$$
  \begin{aligned}
    x_1
      & =
    \frac
      { 2 r^2 \pm \sqrt{ 4 r^4 - 4 (r^4 - 1)  } }
      { 2 (r^2 + 1) }
    \\
    & =
    \frac
      { 2 r^2 \pm 2  }
      { 2 r^2 + 2 }
  \end{aligned}
  \,.
$$

The solution $\frac{ 2 r^2 + 2  }{ 2 r^2 + 2 } = 1$ violates the first condition above, while the solution $\frac{ 2 r^2 - 2  }{ 2 r^2 + 2 } \lt 1$ satisfies it.

Therefore we have a unique solution, given by

$$
  \left(
    \sigma\vert_{S^n \backslash \{p\}}
  \right)^{-1}(0,y_2, \cdots, y_{n+1})
  \;=\;
  \left(
    \frac{2 r^2 - 2}{2 r^2 +2}
    ,
    \left( 1- \frac{2 r^2 - 2}{2 r^2 +2} \right) y_2
    ,
    \cdots
    ,
    \left( 1- \frac{2 r^2 - 2}{2 r^2 +2} \right) y_{n+1}
  \right)
$$

In particular therefore also an [[inverse function]] to the stereographic projection exists and is a [[rational function]], hence continuous. So we have exhibited a homeomorphism as required.

=--

## Generalizations

Of course the construction in prop. \ref{StandardStereographicProjection} does not really depend on the specific [[coordinates]] chosen there.

More generally, for any point $p\in S^n$, considered as a [[vector]] in $\mathbb{R}^{n+1}$, let $W\subset \mathbb{R}^{n+1}$ be the [[linear subspace]] that is the [[orthogonal complement]] to the [[linear span]] $span\{v\}$. Then there is a [[homeomorphism]]

$$
  \sigma \;\colon\; S^n\setminus \{v\} \stackrel{\simeq}{\longrightarrow} W
$$

given by sending a point $p\in S^n\setminus \{v\}$ to the point in $W$ that is the intersection of the line through $v$ and $p$ with $W$.

More generally, one may take for $W$ any _[[affine space|affine]]_ subspace of $\mathbb{R}^{n+1}$ normal to $span\{v\}$ and not containing $v$. For instance one option is to take $W$ to be the tangent space to $S^n$ at $-v$, embedded as a subspace of $\mathbb{R}^{n+1}$.

(see e.g. [Cook-Crabb 93, section 1](#CookCrabb93))

## Properties

### One-point compactification

The inverse map $\sigma^{-1}$ exhibits $S^n$ as the [[one-point compactification]] of $W$.

### Extra geometric structure

In most cases of interest one is doing [[geometry]] using stereographic projection, so the sphere and the subspace $W$ are equipped with extra structure. Taking the explicit case above ($v=(1,0,\ldots,0)$), as the formula is so simple, some structures are automatically preserved, for instance:

* [[conformal structure]]

* [[orthogonal group|orthogonal]] [[group action]] fixing $v,-v$ pointwise (note that these two points are sent to "$\infty$" and $0$, respectively, under $\sigma$)

## Over other fields

Stereographic projection is a general [[algebraic geometry|geometric]] technique which can be applied to other [[commutative rings]] $k$ besides $k = \mathbb{R}$. We give a brief taste of this for the case of [[fields]]. For simplicity, we focus on [[conic sections]], i.e., solutions sets in the [[projective plane]] $\mathbb{P}^2(k)$ to [[polynomial|homogeneous polynomials]] of [[degree]] $2$.

Via stereographic projection, all *pointed*[^1] nonsingular conic sections $C \subset \mathbb{P}^2(k)$ are isomorphic and can be identified explicitly with a projective line $\mathbb{P}^1(k)$ by means of a [[stereographic projection]].

Geometrically, if $p$ is the chosen basepoint of $C$ and $L \subset \mathbb{P}^2(k)$ is a line not incident to $p$, then for any other point $q$ of $C$ the unique line $L(p, q)$ incident to $p$ and $q$ intersects $L$ in exactly one point, denoted $\phi(q)$. (Here $\phi(p)$ is taken to be the intersection of the _tangent_ to $p$ at $C$ with $L$; this can be considered the basepoint of $L$.) In the opposite direction, to each point $x$ of $L$, the line $L(p, x)$ intersects $C$ in $p$ and (since a quadratic with one root will also have another root) another point $q$ (which might be the same as $p$; this happens precisely when $L(p, x)$ is the tangent to $C$ at $p$); this gives the inverse $\phi^{-1}(x) = q$. In this way we obtain an isomorphism $\phi: C \to L$ of subvarieties.

+-- {: .num_example #PythagoreanTriples}
###### Example
For the case $k = \mathbb{Q}$ and the rational conic $C = \{(x, y) \in \mathbb{Q} \times \mathbb{Q}: x^2 + y^2 = 1\}$, stereographic projection from the point $(-1, 0)$ on $C$ to the rational line $x = 0$ defines an isomorphism of [[algebraic varieties]] $C \to \mathbb{P}^1(\mathbb{Q})$. As the calculations above show, its inverse takes $(0, t)$ (for $t \in \mathbb{Q}$) to $(\frac{1 - t^2}{1 + t^2}, \frac{2 t}{1 + t^2})$.

A Pythagorean triple is a triple of integers $(a, b, c)$ such that $a^2 + b^2 = c^2$, so that $(\frac{a}{c}, \frac{b}{c})$ lies on $C$. The isomorphism indicates that for each such point on $C$ (written as a pair of fractions in reduced form), there is a rational number $t = \frac{p}{q}$ (again in reduced form) so that

$$\frac{a}{c} = \frac{1 - t^2}{1 + t^2} = \frac{q^2 - p^2}{q^2 + p^2}, \qquad \frac{b}{c} = \frac{2 t}{1 + t^2} = \frac{2 p q}{q^2 + p^2}.$$

If we assume $a, b, c$ are coprime, and further (re)arrange the triple so that $a$ is odd and $b$ is even, one may quickly conclude from the first of this pair of equations that $p, q$ must have opposite parity. A little further argumentation then shows $q^2 - p^2, 2 p q, q^2 + p^2$ must be mutually coprime, so that in fact $a = q^2 - p^2, b = 2 p q$, and $c = q^2 + p^2$. Thus we arrive at the classical parametrization of Pythagorean triples, essentially on the basis of the geometry of stereographic projection!
=--

## Related entries

* [[Riemann sphere]]

* [[representation sphere]]

## References

Textbook accounts:

* {#Apostol73} [[Tom Apostol]], Fig. 1.3 in: *Mathematical Analysis* 1973 ([pdf](http://www.ru.ac.bd/wp-content/uploads/sites/25/2019/03/205_04_Apostol-Mathematical-Analysis-1973.pdf))

See also:

* Wikipedia, _[Stereographic projection](https://en.wikipedia.org/wiki/Stereographic_projection)_

* {#CookCrabb93} A. L. Cook, M.C. Crabb, _Fiberwise Hopf structures on sphere bundles_, J. London Math. Soc. (2) 48 (1993) 365-384 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf))

[[!redirects stereographic projections]]


stereotype space

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#### Functional analysis
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#### Topology
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#Contents#
* table of contents
{:toc}

## Idea

In [[functional analysis]] and related areas of mathematics _stereotype spaces_ are [[topological vector spaces]] defined by a special variant of reflexivity condition. They form a class of spaces with a series of remarkable properties, in particular, this class is very wide (for instance, it contains all [[Fréchet spaces]] and thus, all [[Banach spaces]]), it consists of spaces satisfying a natural condition of completeness, and it forms a [[closed monoidal category]] with the standard analytical tools for constructing new spaces, like taking closed subspace, quotient space, projective and injective limits, the space of operators, tensor products, etc.

* From Wikipedia: [Stereotype space](https://en.wikipedia.org/wiki/Stereotype_space)

## Definition

Let $X$ be a [[topological vector space]] over the [[complex numbers]] $\mathbb{C}$. Let $X^\ast$ be the vector space of continuous linear functionals $X \to \mathbb{C}$ endowed with the [[topology]] of uniform convergence on [[totally bounded space|totally bounded]] subsets of $X$[^fine]. This $X^\ast$ is a topological vector space, and may be regarded as a form of dual space of $X$. This construction defines a [[contravariant functor]] $(-)^\ast$ on the category of topological vector spaces.

There is a canonical [[natural transformation]] $i_X: X \to X^{\ast\ast}$, sending $x \in X$ to the functional defined by $i(x)(f) \coloneqq f(x)$.

+-- {: .num_defn}
###### Definition
A TVS $X$ is a **stereotype space** if $i_X$ is an [[isomorphism]].
=--

There is another way of characterizing stereotype spaces $X$ directly in terms of the TVS structure on $X$. We say that a TVS $X$ is *pseudo-complete* if every totally bounded [[Cauchy net]] in $X$ converges. Next, call a subset $C$ of $X$ *capacious* if, for every totally bounded set $A \subset X$, there exists a finite set of elements $x_1, \ldots, x_n$ such that every $y \in \bar{A}$ is contained in some translate $x_i + C$. Then, say that $X$ is *pseudo-saturated* if every closed convex balanced capacious subset of $0$ is a neighborhood of $0$ (this can be regarded as a weakening of the notion of barreledness).

+-- {: .num_theorem}
###### Theorem
A TVS $X$ is a stereotype space if and only if it is locally convex, pseudo-complete, and pseudo-saturated.
=--

## Basic results

+-- {: .num_prop}
###### Proposition
The inclusion of pseudo-saturated locally convex TVS, as a [[full subcategory]] of locally convex TVS, is [[coreflection|coreflective]]. The right adjoint to the inclusion takes $X$ to a space denoted $X^\triangle$, called the *pseudo-saturation* of $X$. If $X$ is pseudo-complete, so is $X^\triangle$.
=--

Let $X$, $Y$ be stereotype spaces, and let $L(X, Y)$ be the space of continuous linear maps $X \to Y$. Then $L(X, Y)$ is locally convex, and as $Y$ is pseudo-complete, it follows that $L(X, Y)$ is also pseudo-complete. However, $L(X, Y)$ might not be pseudo-saturated. Let $Hom(X, Y)$ be the pseudo-saturation of $L(X, Y)$, so that $Hom(X, Y)$ is a stereotype space. This construction endows the category of stereotype spaces with a [[closed category]] structure, with unit $\mathbb{C}$. We have $X^\ast \cong Hom(X, \mathbb{C})$.

+-- {: .num_theorem}
###### Theorem
The category of stereotype spaces, as a closed category with duality $(-)^\ast$, acquires a structure of complete and cocomplete $\ast$-[[star-autonomous category|autonomous]] category.
=--

In slightly more detail, a tensor product on stereotype spaces may be defined by the formula $X \otimes Y \coloneqq Hom(X, Y^\ast)^\ast$. Then for stereotype spaces $X, Y, Z$ there is a natural isomorphism

$$\hom(X \otimes Y, Z) \cong \hom(X, Hom(Y, Z))$$

and there are associativity, symmetry, and unit constraints on $\otimes$ so that in this way, stereotype spaces form a [[closed monoidal category|symmetric monoidal closed category]]. By taking $D = I = \mathbb{C}$ as dualizing object, it is moreover $\ast$-autonomous. It is not, however, autonomous in the sense of [[compact closed categories]], i.e., we do not have an isomorphism between $(X \otimes Y)^\ast$ and $X^\ast \otimes Y^\ast$. Rather, we get a second tensor product via the formula

$$X \odot Y \coloneqq (X^\ast \otimes Y^\ast)^\ast$$

and this tensor product plays the role of multiplicative disjunction in [[linear logic]] (where we regard $\ast$-autonomous categories as models of multiplicative linear logic).

## References

* S.S. Akbarov, _Pontryagin duality in the theory of topological vector spaces and in topological algebra_, Journal of Mathematical Sciences 113 (2) (2203), 179–349.

[^fine]: In other words, the smallest topology on $X^\ast$ that includes all sets $L(B, U) \coloneqq \{f: X \to \mathbb{C}: f(B) \subseteq U\}$ where $B$ is totally bounded in $X$ and $U$ is open in $\mathbb{C}$.


Sterling Ebel

## Selected writings

On the [[model structure on compactly generated topological spaces]] and [[model structure on Delta-generated topological spaces|on Delta-generated topological spaces]], and on a [[model category]] of [[locales]] which makes the [[reflective subcategory|reflection]] of [[sober topological spaces]] a [[Quillen adjunction]] to the sober-restriction of the [[classical model structure on topological spaces]]:

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

category: people


Stern-Gerlach experiment

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

## Idea

The _Stern-Gerlach experiment_ showed that [[electrons]] are [[spinors]]. It was the first [[experimental observation]] of the elementary [[spin]] $\frac{1}{2}$ of [[fermions]].

## Implications

Via the [[Pauli exclusion principle]]/[[spin-statistics theorem]], this implies that [[wavefunctions]] depending on fermionic variable commute with each other only up to a sign, forming a  [[supercommutative superalgebra]]. This way the Stern-Gerlach experiment is ultimately the observation that physical reality is described by [[supergeometry]].

(Notice that there is [[supergeometry]] independently of [[supersymmetry]], which may or may not be present in addition).

## Related entries

* [[Bell's inequality]]

## References

* [[Roland Omnès]], §2 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

See also:

* Wikipedia, _[Stern-Gerlach experiment](http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment)_ .

[[!redirects Stern-Gerlach device]]
[[!redirects Stern-Gerlach devices]]


Steve Awodey

Steve Awodey is a professor at Carnegie Mellon University interested in category theory, logic,
philosophy of mathematics, history of logic and analytic philosophy. For the 'n-community', one should point out his recent work on homotopical ideas in type theory.

* [website](http://www.andrew.cmu.edu/user/awodey)

## Selected writings

On [[propositional truncation]] in [[dependent type theory]]:

* [[Steve Awodey]], [[Andrej Bauer]], *Propositions as $[$Types$]$*, Journal of Logic and Computation, **14**  (2004) 447-471 [[doi:10.1093/logcom/14.4.447](https://doi.org/10.1093/logcom/14.4.447), [pdf](http://andrej.com/papers/brackets_letter.pdf)]

On [[algebraic set theory]]:

* [[Steve Awodey]]. *Notes on algebraic set theory*, Notes for lectures given at the Summer School on Topos Theory, Haute-Bodeux, Belgium. May 29 to June 5, 2005. Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005. ([pdf](https://www.phil.cmu.edu/projects/ast/Papers/bnotes.pdf))

On [[category theory]]:

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

Early note on what came to be known as *[[homotopy type theory]]*:

* {#Awodey09} [[Steve Awodey]], _Homotopy and Type Theory_, grant proposal project description ([pdf](https://ncatlab.org/homotopytypetheory/files/proposal2009.pdf))

Proof that every [[simplicial model category]] in which the [[cofibrations]] are [[monomorphisms]] provides a sound [[model]] for [[intensional type theory|intensional]] [[Martin-Löf type theory]] (i.e. including the [[identity types]] used in [[homotopy type theory]]):

* {#AwodeyWarren07} [[Steve Awodey]], [[Michael Warren]], *Homotopy theoretic models of identity type*,  Mathematical Proceedings of the Cambridge Philosophical Society **146** 1 (2009) [[arXiv:0709.0248](http://arxiv.org/abs/0709.0248), [doi:10.1017/S0305004108001783](https://doi.org/10.1017/S0305004108001783)]

Stating [[Awodey's conjecture]]:

* {#Awodey10} [[Steve Awodey]], _Type theory and homotopy_, Epistemology versus Ontology. Springer Netherlands, 2012. 183-201  ([arXiv:1010.1810](http://arxiv.org/abs/1010.1810))

On [[inductive types]] in [[homotopy type theory]] via homotopy-[[initial algebras over an endofunctor]]:

* {#AwodeyGambinoSojakova12} [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Inductive types in homotopy type theory*, LICS'12: (2012) 95–104 [[arXiv:1201.3898](http://arxiv.org/abs/1201.3898), [doi:10.1109/LICS.2012.21](https://doi.org/10.1109/LICS.2012.21), [Coq code](https://github.com/HoTT/HoTT/tree/master/Coq/IT)]

  \linebreak

  Exposition:

  [[Steve Awodey]], *Inductive types in HoTT* (Jan 2012) [[blog post](http://homotopytypetheory.org/2012/01/19/inductive-types-in-hott/)]

and analogously on [[higher inductive types]]:

* [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Homotopy-initial algebras in type theory*, Journal of the ACM **63** 6 (2017) 1–45 [[arXiv:1504.05531](http://arxiv.org/abs/1504.05531), [doi:10.1145/3006383](https://doi.org/10.1145/3006383)]

On [[first-order set theory]] and [[categorical logic]]:

* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], *Relating first-order set theories, toposes and categories of classes*, Annals of Pure and Applied Logic **165** 2 (2014) 428-502 [[doi:10.1016/j.apal.2013.06.004](https://doi.org/10.1016/j.apal.2013.06.004)]

On [[natural models of homotopy type theory]]:

* {#Awodey2016} [[Steve Awodey]], *Natural models of homotopy type theory*, Mathematical Structures in Computer Science, **28** 2 (2016) 241-286 [[arXiv:1406.3219](https://arxiv.org/abs/1406.3219), [doi:10.1017/S0960129516000268](https://doi.org/10.1017/S0960129516000268)]

On [[Kripke-Joyal semantics]] and [[forcing]] in [[homotopy type theory]]:

* {#AGH21} [[Steve Awodey]], [[Nicola Gambino]], [[Sina Hazratpour]], *Kripke-Joyal forcing for type theory and uniform fibrations* (2021) [[arXiv:2110.14576](https://arxiv.org/abs/2110.14576)]

On the [[model category]] structure on the category of cartesian [[cubical sets]]:

* [[Steve Awodey]], _Cartesian cubical model categories_ ([arXiv:2305.00893](https://arxiv.org/abs/2305.00893))

## Related entries

* [[algebraic set theory]]

* [[type theory]], [[homotopy type theory]]

* [[Awodey's conjecture]]

category: people

[[!redirects Steven Awodey]]
[[!redirects SteveAwodey]]


Steve Carlip

* [webpage](http://www.physics.ucdavis.edu/Text/Carlip.html)

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

## Selected writings

On [[3d gravity]] as a [[Chern-Simons theory]] and its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):

* [[Steven Carlip]], _Inducing Liouville theory from topologically massive gravity_, Nuclear Physics B Volume 362, Issues 1–2, 16 September 1991, Pages 111-124 (doi:10.1016/0550-3213(91)90558-F)

* [[Steven Carlip]], _Lectures in (2+1)-Dimensional Gravity_, J. Korean Phys. Soc. 28: S447-S467, 1995 ([arXiv:gr-qc/9503024](https://arxiv.org/abs/gr-qc/9503024))

* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions_, Cambridge Monographs on Mathematical Physics (2003) ([publisher](http://www.cambridge.org/de/academic/subjects/physics/cosmology-relativity-and-gravitation/quantum-gravity-21-dimensions))

* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe_, Living Rev. Rel. 8:1, 2005 ([arXiv:gr-qc/0409039](https://arxiv.org/abs/gr-qc/0409039))

* [[Steven Carlip]], _[My Research -- (2+1)-Dimensional quantum gravity](http://www.physics.ucdavis.edu/Text/Carlip.html#2+1)_

* [[Steve Carlip]], *Quantum Gravity in 2+1 Dimensions*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12596](https://arxiv.org/abs/2312.12596)]

## Related entries

* [[3d quantum gravity]]

* [[black hole entropy]]

category: people

[[!redirects Steven Carlip]]


Steve Halperin > history

see at _[[Stephen Halperin]]_


Steve Lack > history

see _[[Stephen Lack]]_


Steve Vickers

Steve Vickers is a theoretical computer scientist based in Birmingham (U.K.).

* [home page](http://www.cs.bham.ac.uk/~sjv/)

* [Wikipedia page](http://en.wikipedia.org/wiki/Steve_Vickers_%28computer_scientist%29)

## Writings

On [[topology]] via [[formal logic]] ([[locales]], [[sober topological spaces]], ...):

* _Topology via Logic_, Cambridge University Press (1989) ([toc pdf](http://www.gbv.de/dms/ilmenau/toc/21309293X.PDF))

On [[point-free topology]]:

* {#Vickers04} [[Steve Vickers]], _The double powerlocale and exponentiation: a case study in geometric logic_, Theory and Applications of Categories **12** (2004) pp. 372-422 ([pdf](http://www.tac.mta.ca/tac/volumes/12/13/12-13.pdf))

* {#Vickers07} [[Steve Vickers]], _Locales and toposes as spaces_, Chapter 8 in _Handbook of Spatial Logics_ (ed. Aiello, Pratt-Hartman, van Bentham), Springer, 2007, pp. 429-496.

* {#Vickers14} [[Steve Vickers]], _Continuity and geometric logic_, Journal of Applied Logic **12 (1)** (2014), pages 14-27 ([pdf](http://www.cs.bham.ac.uk/~sjv/GeoAspects.pdf))

* [[Ming Ng]], [[Steve Vickers]], *Point-free Construction of Real Exponentiation*, Logical Methods in Computer Science, Volume 18, Issue 3 (August 2, 2022), ([doi:10.46298/lmcs-18(3:15)2022](https://doi.org/10.46298/lmcs-18%283%3A15%292022), [arXiv:2104.00162](https://arxiv.org/abs/2104.00162))

* [[Steven Vickers]], *Generalized point-free spaces, pointwise* [[arXiv:2206.01113](https://arxiv.org/abs/2206.01113)]

* [[Ming Ng]], [[Steven Vickers]], *A Point-Free Look at Ostrowski's Theorem and Absolute Values* ([arXiv:2308.14758](https://arxiv.org/abs/2308.14758))

* [[Steve Vickers]], *The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms*, ([arXiv:2312.05228](https://arxiv.org/abs/2312.05228))

On [[arithmetic pretoposes]]:

* {#MaettiVickers12} [[Maria E. Maietti]], [[Steve Vickers]], _An induction principle for consequence in arithmetic universes_, JPAA **216** (2012) pp.2049-2067. [[doi:10.1016/j.jpaa.2012.02.040](https://doi.org/10.1016/j.jpaa.2012.02.040), [pdf](http://www.math.unipd.it/~maietti/papers/aumv12.pdf)]

* {#Vickers16} [[Steve Vickers]], _Sketches for arithmetic universes_, arXiv:1608.0159 (2016) &lbrack[arXiv:1608.01559](http://arxiv.org/abs/1608.01559)]

* {#Vickers17} [[Steve Vickers]], _Arithmetic universes and classifying toposes_ [[arXiv:1701.04611](https://arxiv.org/abs/1701.04611)]

On [[geometric logic]] [[internal logic|internal to]] [[Grothendieck toposes]]:

* [[Steven Vickers]], *Continuity and geometric logic*, Journal of Applied Logic Volume 12, Issue 1, March 2014, Pages 14-27 ([doi:10.1016/j.jal.2013.07.004](https://doi.org/10.1016/j.jal.2013.07.004))

category:people

[[!redirects Steven Vickers]]


Steve Y. Oudot

* [personal page](https://geometrica.saclay.inria.fr/team/Steve.Oudot/)

## Selected writings

On [[persistence modules]] and [[stability of persistence diagrams]]:

* [[Frédéric Chazal]], [[David Cohen-Steiner]], [[Marc Glisse]], [[Leonidas J. Guibas]], [[Steve Y. Oudot]], *Proximity of persistence modules and their diagrams*, SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 237–246 $[$[doi:10.1145/1542362.1542407](https://doi.org/10.1145/1542362.1542407)$]$

* [[Frédéric Chazal]], [[Vin de Silva]], [[Marc Glisse]], [[Steve Oudot]],  *The structure and stability of persistence modules*, SpringerBriefs in Mathematics, Springer (2016) $[$[arXiv:1207.3674](https://arxiv.org/abs/1207.3674), [doi:10.1007/978-3-319-42545-0](https://doi.org/10.1007/978-3-319-42545-0)$]$

On [[persistent homology]] and [[topological data analysis]], with focus on the use of [[quiver representation]]-theory (such as [[Gabriel's theorem]]):

* {#Oudot15} [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/), [pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf)$]$

On [[level set]] [[persistent homology]] and [[sheaf theory]]:

* {#BerkoukGinotOudot19} [[Nicolas Berkouk]], [[Grégory Ginot]], [[Steve Oudot]], *Level-sets persistence and sheaf theory* $[$[arXiv:1907.09759](https://arxiv.org/abs/1907.09759), [talk](Center+for+Quantum+and+Topological+Systems#GinotAtHomotopicalPerspectivesOnTDAJune2022)$]$

category: people

[[!redirects Steve Oudot]]


Steve Zdancewic

* [personal page](https://www.cis.upenn.edu/~stevez/)

## Selected writings

On [[linear algebra]] ([[finite-dimensional vector spaces]] over [[finite fields]]) as [[categorical semantics]] for [[linear logic]]/[[linear type theory]]:

* [[Benoît Valiron]], [[Steve Zdancewic]], *Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi*, in: *Proc. of ICTAC'14*, Lecture Notes in Computer Science **8687**, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), [doi:10.1007/978-3-319-10882-7_26](https://doi.org/10.1007/978-3-319-10882-7_26), slides:[pdf](https://www.cs.bham.ac.uk/~drg/bll/steve.pdf), [[Zdancewic-LinearLogicAlgebra.pdf:file]]]

Introducing *[[QWIRE]]*, a [[quantum programming language]] based on [[linear type theory]] connected with [[intuitionistic type theory]] via the [[exponential modality]]:

* {#PaykinRandZdancewic17} [[Jennifer Paykin]], [[Robert Rand]], [[Steve Zdancewic]], *QWIRE: a core language for quantum circuits*, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesJanuary 2017 Pages 846–858 ([doi:10.1145/3009837.3009894](https://doi.org/10.1145/3009837.3009894))

Application to [[verified programming]] after implementation in [[Coq]]:

* [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS 266, 2018, pp. 119-132 ([arXiv:1803.00699](https://arxiv.org/abs/1803.00699))

* [[Robert Rand]], [[Jennifer Paykin]], [[Dong-Ho Lee]], [[Steve Zdancewic]], *ReQWIRE: Reasoning about Reversible Quantum Circuits*, EPTCS **287** (2019) 299-312 [[arXiv:1901.10118](https://arxiv.org/abs/1901.10118), [doi:10.4204/EPTCS.287.17](https://doi.org/10.4204/EPTCS.287.17)]

Using ambient [[homotopy type theory]]:

* [[Jennifer Paykin]], [[Steve Zdancewic]], *A HoTT Quantum Equational Theory*, [talk at QPL2019](http://qpl2019.org/a-hott-quantum-equational-theory/) [[arXiv:1904.04371](https://arxiv.org/abs/1904.04371), [pdf](https://jpaykin.github.io/talks/HoTT-3-8-2019.pdf), [[PaykinZdancewic-HoTTQuantum.pdf:file]]]

category: people


Steven Costenoble

* [webpage](http://people.hofstra.edu/steven_r_costenoble/)

## Selected writings

Generalization of the [[Conner-Floyd isomorphism]] to [[equivariant cohomology theory]], relating [[equivariant cobordism cohomology]] to [[equivariant K-theory]]:

* [[Steven Costenoble]], *Equivariant Conner-Floyd isomorphism*, Trans. Amer. Math. Soc. **304** 2 (1987) 801--818, [[jstor:2000742](http://www.jstor.org/stable/2000742)]

On [[equivariant complex oriented cohomology theory]]:

* [[Steven Costenoble]], [[Peter May]], [[Stefan Waner]], _Equivariant orientation theory_, Homology Homotopy Appl.
Volume 3, Number 2 (2001), 265-339 ([euclid:hha/1139840256](https://projecteuclid.org/euclid.hha/1139840256))

On [[equivariant ordinary cohomology]]:

* [[Steven Costenoble]], [[Stefan Waner]], _Equivariant ordinary homology and cohomology_, Springer, 2016 ([arXiv:math/0310237](https://arxiv.org/abs/math/0310237))

category: people


Steven Duplij

* [personal page](https://ivv5hpp.uni-muenster.de/u/douplii)

* [InSpire page](https://inspirehep.net/authors/1011195)

* [GoogleScholar page](https://scholar.google.com/citations?user=BeLVm2QAAAAJ&hl=en)

## Selected writings

On the early history of [[supergravity]] ([[D=4 supergravity]])

* [[Steven Duplij]], _Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn't it?_, East Eur. J. Phys. p. 81-82 (2019) ([arXiv:1910.03259](https://arxiv.org/abs/1910.03259))

On unconventional paradigms for [[quantum computation]]:

* [[Steven Duplij]], [[Raimund Vogl]], *Innovative Quantum Computing*, IOP (2023) [[ISBN:978-0-7503-5281-9](https://iopscience.iop.org/book/mono/978-0-7503-5281-9)]

category: people


Steven G. Krantz

* [Institute page](https://math.wustl.edu/people/steven-g-krantz)

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

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=8339)

## Selected writings

On [[geometric integration theory]]:

* [[Steven G. Krantz]], [[Harold Parks]], *Geometric integration theory*, Springer (2008) [[pdf](http://www.math.wustl.edu/~sk/books/root.pdf)]

## Related entries

* [[current (distribution theory)]]

category: people

[[!redirects Steven Krantz]]
[[!redirects Steven George Krantz]]


Steven Giddings

* [webpage](http://web.physics.ucsb.edu/~giddings/)

## related $n$Lab entries

* [[super Riemann surface]]

* [[Randall-Sundrum model]]

category:people

[[!redirects Steve Giddings]]


Steven Gubser

* [webpage](http://wwwphy.princeton.edu/~ssgubser/)

## Selected writings

Introducing the dictionary of the [[AdS-CFT correspondence]]:

* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) [[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), doi:10.1016/S0370-2693(98)00377-3]

> Relations between gauge fields and strings present an old, fascinating and unanswered question. The full answer to this question is of great importance for theoretical physics. It will provide us with a theory of quark [[confinement]] by explaining the dynamics of color-electric fluxes.

comprehensive review:

* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. 323:183-386, 2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

On [[single trace operators]] in the [[AdS/CFT correspondence]]:

* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *A semi-classical limit of the gauge/string correspondence*, Nucl. Phys. B **636** (2002) 99-114 [[arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051), doi:10.1016/S0550-3213(02)00373-5]

On [[double trace operators]] in [[AdS/CFT]]:

* [[Steven Gubser]], Indrajit Mitra, _Double-trace operators and one-loop vacuum energy in AdS/CFT_, Phys. Rev. D67 (2003) 064018 ([arXiv:hep-th/0210093](https://arxiv.org/abs/hep-th/0210093))

* [[Steven Gubser]], [[Igor Klebanov]], _A universal result on central charges in the presence of double-trace deformations_, Nucl. Phys. B656 (2003) 23-36 ([arXiv:hep-th/0212138](https://arxiv.org/abs/hep-th/0212138))

On the [[p-adic AdS/CFT correspondence]]:

* [[Steven Gubser]], Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, _$p$-adic AdS/CFT_ ([arXiv:1605.01061](https://arxiv.org/abs/1605.01061))

  (the [[Bruhat-Tits tree]] as AdS/CFT spacetime]])

* [[Steven Gubser]], Sarthak Parikh, _Geodesic bulk diagrams on the Bruhat-Tits tree_ ([arXiv:1704.01149](https://arxiv.org/abs/1704.01149))

* [[Steven Gubser]], Christian Jepsen, Brian Trundy, _Spin in $p$-adic AdS/CFT_, Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 14 ([arXiv:1811.02538](https://arxiv.org/abs/1811.02538))

  (including [[spinors]])

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))

## Related $n$Lab entries

* [[p-adic AdS-CFT]]

category: people


Steven H. Simon

* [webpage](https://www-thphys.physics.ox.ac.uk/people/SteveSimon/)

* [GoogleScholar](https://scholar.google.com/citations?user=Snr-5TcAAAAJ&hl=en)

## Selected writings

On [[topological quantum computation]] with [[anyons]]:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083 (2008) ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889))

* {#Simon21} [[Steven H. Simon]], *Topological Quantum*, 2021 $[$[pdf](http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/TopoBook-Sep1-2021.pdf), [webpage](http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/topocourse2021.html)$]$

On [[topological quantum compilation]] and approximating (cf. the [[Solovay-Kitaev theorem]]) given [[quantum gates]] by (i.e. compiling them to) [[quantum circuit|cicuits]] of [[anyon]] [[braid representation|braid]] [[quantum gates|gates]] (generally considered for [[su(2)-anyons]] and here mostly for universal [[Fibonacci anyons]], to some extent also for non-universal [[Majorana anyons]]):

* [[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)$]$

* [[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)$]$

* [[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)$]$

* M. Baraban, [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Resources required for topological quantum factoring*, Phys. Rev. A **81** 062317 (2010) $[$[doi:10.1103/PhysRevA.81.062317](https://doi.org/10.1103/PhysRevA.81.062317), [arXiv:1002.0537](https://arxiv.org/abs/1002.0537)$]$

  > (focus on compiling [[Shor's algorithm]])

and particularly by just the "weaves" among all [[braids]]:

* [[Steven H. Simon]], [[Nick E. Bonesteel]], [[Michael H. Freedman]], N. Petrovic, [[Layla Hormozi]], *Topological Quantum Computing with Only One Mobile Quasiparticle*, Phys. Rev. Lett. 96 (2006) 070503 ([arXiv:quant-ph/0509175](https://arxiv.org/abs/quant-ph/0509175), [doi:10.1103/PhysRevLett.96.070503](https://doi.org/10.1103/PhysRevLett.96.070503))

On [[topological phases of matter]] via [[higher gauge theory|higher]] lattice gauge theory:

* [[Joe Huxford]], [[Steven H. Simon]], *Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview* [[arXiv:2202.08294](https://arxiv.org/abs/2202.08294)]

category: people

[[!redirects Steven Simon]]


Steven Kleiman

**Steven Kleiman** is an American [[algebraic geometry|algebraic geometer]].

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

* I. Altman, S. Kleiman, _Introduction to Grothendieck duality theory_, Springer 1970.

* Steven L. Kleiman, _The Picard scheme_, [[FGA explained]], 235--321,  MR2223410 (draft [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s6t3&ifd=15022&down=1&type=lecture_notes))

category:people


Steven L. Kleiman

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

* [institute page](https://math.mit.edu/directory/profile.php?pid=134)

## Selected writings

On basics of [[algebraic geometry]] following [[Grothendieck]]'s [[FGA]]:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:

  **[[FGA explained]]** (ICTP, Trieste 2003--2005)

  Mathematical Surveys and Monographs __123__

  Amer. Math. Soc. 2005. x+339 pp.

  * [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)

    [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

    [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)

category: people

[[!redirects Steven Kleiman]]


Steven M. Girvin

* [Institute page](https://physics.yale.edu/people/steven-girvin)

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

## Selected writings

On the [[fractional quantum Hall effect]] and the [[Laughlin wavefunction]]:

* [[Steven M. Girvin]], Section 2.1 of: *Introduction to the Fractional Quantum Hall Effect*, Séminaire Poincaré **2** (2004) 53–74, reprinted in *The Quantum Hall Effect*, Progress in Mathematical Physics **45**, Birkhäuser (2005)  $[$[pdf](http://www.bourbaphy.fr/girvin.pdf), [doi:10.1007/3-7643-7393-8_4](https://doi.org/10.1007/3-7643-7393-8_4)$]$

category: people

[[!redirects Steven Girvin]]


Steven Rayan

- [web page](http://www.math.usask.ca/~rayan/)


Steven Rosenberg

* [webpage](http://math.bu.edu/people/sr/)

category: people


Steven Shnider

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

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=69034)

## selected writings

* [[Mikhail Katz]], [[Steven Shnider]], _Cayley 4-form, comass, and triality isomorphisms_ ([arXiv:0801.0283](http://arxiv.org/abs/0801.0283))

  (on [[Cayley 4-forms]])

## related $n$Lab entries

* [[operad]], [[associahedron]]

* [[supersymmetry]]

* [[Cayley 4-form]]

category: people

[[!redirects Steve Shnider]]


Steven V Sam

* [personal page](https://mathweb.ucsd.edu/~ssam/)

## Selected writings

On [[representation stability]]:

* [[Steven V Sam]], *Representation stability*, lecture notes (2017) [[pdf](https://mathweb.ucsd.edu/~ssam/old/17F-847/notes.pdf)]

category: people

[[!redirects Steven V. Sam]]
[[!redirects Steven Sam]]


Steven Weinberg

* [website](http://www.ph.utexas.edu/~weintech/weinberg.html)

* [Wikipedia entry](http://de.wikipedia.org/wiki/Steven_Weinberg)

* Rohini M. Godbole, Urjit Yajnik: *Steven Weinberg (1933 - 2021)*, Current Science [**121** 5](https://www.currentscience.ac.in/show.issue.php?volume=121&issue=5) (10 Sep 2021) [[arXiv:2302.02704](https://arxiv.org/abs/2302.02704)]

## Selected writings

(Re-)Introducing [[light-cone gauge quantization]]:

* [[Steven Weinberg]], *Dynamics at Infinite Momentum*, Phys. Rev. **150** (1966) 1313,  Erratum Phys. Rev. 158, 1638 (1967) [[doi:10.1103/PhysRev.150.1313](https://doi.org/10.1103/PhysRev.150.1313)]

Introducing the [[electroweak field|electroweak]] [[symmetry breaking]]:

* {#Weinberg67} [[Steven  Weinberg]], *A model of Leptons*, Phys. Rev. Lett. **19** (1967) 1264 [[doi:10.1103/PhysRevLett.19.1264](https://doi.org/10.1103/PhysRevLett.19.1264)]

and personal recollection:

* [[Steven Weinberg]], *The Red Camaro*, in *George* (1997), reprinted in: [[Steven Weinberg]]: *Facing Up: Science and its Cultural Adversaries*, Harvard University Press (2001) [[pdf](https://www.ictp.it/media/1867057/100reasons-Weinberg.pdf), [[Weinberg-RedCamaro.pdf:file]], [ISBN:9780674011205](https://www.hup.harvard.edu/catalog.php?isbn=9780674011205)]

Introducing [[quark-hadron duality]]:

* E. C. Poggio, H. R. Quinn, and [[Steven Weinberg]], _Smearing method in the quark model_, Phys. Rev. D 13, 1958 (1976) ([doi:10.1103/PhysRevD.13.1958](https://doi.org/10.1103/PhysRevD.13.1958))

On [[effective field theory]] and [[chiral perturbation theory]]:

* {#Weinberg79} [[Steven Weinberg]], _Phenomenological Lagrangians_, Physica A: Statistical Mechanics and its Applications Volume 96, Issues 1–2, April 1979, Pages 327-340 (doi:10.1016/0378-4371(79)90223-1)

* [[Steven Weinberg]], _On the Development of Effective Field Theory_ ([arXiv:2101.04241](https://arxiv.org/abs/2101.04241))

Deriving the [[soft graviton theorem]]:

* {#Weinberg65} [[Steven Weinberg]], _Infrared photons and gravitons_, Phys. Rev. 140, B516 (1965) ([doi:10.1103/PhysRev.140.B516](https://doi.org/10.1103/PhysRev.140.B516))

On classification of possible long-range [[forces]], hence of [[scattering]] processes of [[mass|massless]] [[field (physics)|fields]], by classification of suitably factorizing and decaying [[Poincaré group|Poincaré]]-[[invariant]] [[S-matrices]] depending on particle spin, leading to uniqueness statements about [[Maxwell theory|Maxwell]]/[[photon]]-, [[Yang-Mills theory|Yang-Mills]]/[[gluon]]-, [[gravity]]/[[graviton]]- and [[supergravity]]/[[gravitino]]-[[interactions]]:

* [[Steven Weinberg]], _Feynman Rules for Any Spin. 2. Massless Particles_, Phys. Rev. 134 (1964) B882 ([doi:10.1103/PhysRev.134.B882](https://doi.org/10.1103/PhysRev.134.B882))

* [[Steven Weinberg]], _Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass_, Phys. Rev. 135 (1964) B1049 ([doi:10.1103/PhysRev.135.B1049](https://doi.org/10.1103/PhysRev.135.B1049))

* [[Steven Weinberg]], _Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 ([doi:10.1103/PhysRev.138.B988](https://doi.org/10.1103/PhysRev.138.B988))

On [[quantum field theory]]:

* {#Weinberg95} [[Steven Weinberg]] *The Quantum Theory of Fields Vol 1: Foundations*, Cambridge University Press (1995) [[doi:10.1017/CBO9781139644167](https://doi.org/10.1017/CBO9781139644167)]

* {#Weinberg96} [[Steven Weinberg]] *The Quantum Theory of Fields Vol 2: Applications*, Cambridge University Press (1996) [[doi:10.1017/CBO9781139644174](https://doi.org/10.1017/CBO9781139644174)]

On [[supersymmetry]]:

* [[Steven Weinberg]], *The quantum theory of fields. Vol. 3: Supersymmetry*, Cambridge University Press (2000) [[ISBN:9781139632638](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-theory-fields-volume-3?format=AR&isbn=9780521670555),  [spire:527189](https://inspirehep.net/literature/527189), [pdf](http://www.stat.ucla.edu/~ywu/research/documents/weinberg3.pdf)]

On [[quantum mechanics]]:

* [[Steven Weinberg]], *Lectures on Quantum Mechanics*, Cambridge University Press (2015) [[doi:10.1017/CBO9781316276105](https://doi.org/10.1017/CBO9781316276105)]

* [[Steven Weinberg]]: _Scientist: Four golden lessons_, Nature **426**, 389 (27 November 2003) [[doi:10.1038/426389a](http://www.nature.com/nature/journal/v426/n6965/full/426389a.html)]

* [[Steven Weinberg]], *Against philosophy*, in: _Dreams of a final theory_ [[pdf](http://emilkirkegaard.dk/en/wp-content/uploads/Steven-Weinberg-%E2%80%9CAgainst-Philosophy%E2%80%9D.pdf)]

## Related entries

* [[perturbative quantum field theory]]

* [[S-matrix]]

category: people


Stewart Priddy

Stewart Beauregard Priddy is a mathematician at Northwester University.

He got his PhD in 1968 at MIT, advised by [[Daniel M. Kan]].

* [webpage](http://www.math.northwestern.edu/~priddy/)

## Selected writings

On the [[group completion theorem]]:

* [[Michael Barratt]], [[Stewart Priddy]], _On the homology of non-connected monoids and their associated groups_, Commentarii Mathematici Helvetici (1972) 47: 1 ([doi:10.1007/BF02566785](https://doi.org/10.1007/BF02566785))

## Related $n$Lab entries

* [[simplicial Lie algebra]]

* [[model structure on simplicial Lie algebras]]

category: people

[[!redirects Stewart B. Priddy]]


Stewart S. Cairns

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=5658)

## Selected writings

The [[triangulation theorem]] for [[smooth manifolds]]:

* {#Cairns35} [[Stewart S. Cairns]], *Triangulation of the manifold of class one*, Bull. Amer. Math. Soc. 41(8): 549-552 ([euclid:1183498332](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-41/issue-8/Triangulation-of-the-manifold-of-class-one/bams/1183498332.full))

with a simpler argument in

* {#Cairns61} [[Stewart S. Cairns]], *A simple triangulation method for smooth manifolds*, Bull. Amer. Math. Soc. 67 (1961), 389-390 ([doi:10.1090/S0002-9904-1961-10631-9](https://doi.org/10.1090/S0002-9904-1961-10631-9))

    > (but see [MO:a/177199](https://mathoverflow.net/a/177199/381), where this "simpler argument" is claimed to be wrong)

category: people

[[!redirects Stewart Cairns]]


Stiefel manifold

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$, write $O(n)$ for the [[orthogonal group]] acting on $\mathbb{R}^n$. For the following we regard these groups as [[topological groups]] in the canonical way.

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

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th **real Stiefel manifold** of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

$$
  V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n)
  \,,
$$

where the [[action]] of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.

=--

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

The group $O(k)$ [[transitive action|acts transitively]] on the set of $n$-dimensional [[linear subspaces]] equipped with an [[orthonormal basis]], and given any such, then its [[stabilizer subgroup]] in $O(k)$ is isomorphic to $O(k-n)$. In this way the underlying set of $V_n(\mathbb{R}^k)$ is in [[natural bijection]] to the set of $n$-dimensional linear subspaces in $\mathbb{R}^k$ equipped with [[orthonormal basis]]. The realization as a [[coset]] as above serves to equip this set naturally with a [[topology|topological space]].

=--

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

By def. \ref{StiefelManifold} there are canonical inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ that are compatible with the $O(n)$-[[action]]. The [[colimit]] (in [[Top]], see there) over these inclusions is denoted

$$
  E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)
  \,.
$$

=--

This is a model for the total space of the $O(n)$-[[universal principal bundle]].

## Properties

### Homotopy groups

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

The Stiefel manifold $V_n(k)$ is [[n-connected topological space|(n-1)-connected]].

=--

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

Consider the [[coset]] [[quotient]] [[projection]]

$$
  O(n)
    \longrightarrow
  O(k)
    \longrightarrow
  O(k)/O(n)
    =
  V_n(\mathbb{R}^k)
  \,.
$$

By [this prop.](orthogonal+group#OrthogonalGroupIsCompact) and by [this corollary](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal) the projection $O(k)\to O(k)/O(n)$ is a [[Serre fibration]]. Therefore there is the [[long exact sequence of homotopy groups]] of this [[fiber sequence]] and by [this prop.](orthogonal+group#InclusionOfOnIntoOkIsnMinus1Equivalence) it has the following structure in degrees bounded by $n$:

$$
  \cdots
    \to
  \pi_{\bullet \leq n-1}(O(k))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq n-1}(O(n))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq n-1}(V_n(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt n-1}(O(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt n-1}(O(n))
    \to
  \cdots
  \,.
$$

This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

=--

+-- {: .num_cor}
###### Corollary

The colimiting space $E O(n) = \underset{\longleftarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. \ref{EOn} is [[weakly contractible topological space|weakly contractible]].

=--

+-- {: .proof}
###### Proof
This follows from the proposition, together with the fact that the sequence of maps $V_n(\mathbb{R}^k) \to V_n(\mathbb{R}^{k+1})$ are closed $T_1$-inclusions. See the Proposition [[compact object#CompactObjectsInTop|here]] together with the accompanying commentary.
=--

### CW-complex structure

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

The Stiefel manifold $V_n(\mathbb{R}^k)$ admits the structure of a [[CW-complex]].

=--

e.g. ([James 59, p. 3](#James59), [James 76, p. 5 with p. 21](#James76), [Hatcher, p. 302](#Hatcher)[Blaszczyk 07](#Blaszczyk07))

{#CellularInclusionOfStiefelManifolds} And it should be true that with that cell structure the inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions:

According to ([Yokota 56](#Yokota56)) the inclusions $SU(n)\hookrightarrow SU(k)$ are cellular and this is compatible with the group action (reviewed [here](http://web.stanford.edu/~kupers/liegroups.pdf) in 3.3 and 3.3.1). This implies that also the projection $SU(k) \to SU(k)/SU(k-n)$ is cellular (e.g. [Hatcher, p. 302](#Hatcher)).

### Relation to Grassmannians and universal bundles

Similarly, the _[[Grassmannian manifold]]_ is the [[coset]]

$$
  Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n)\times O(k-n))
  \,.
$$

The [[quotient]] [[projection]]

$$
  V_{n}(\mathbb{R}^k)\longrightarrow Gr_n(\mathbb{R}^k)
$$

is an $O(n)$-[[principal bundle]], with [[associated bundle]] $V_n(\mathbb{R}^k)\times_{O(n)} \mathbb{R}^n$ a [[vector bundle]] of [[rank]] $n$. In the limit ([[colimit]]) that $k \to \infty$ is this gives a presentation of the $O(n)$-[[universal principal bundle]] and of the [[universal vector bundle]] of rank $n$, respectively.. The base space $Gr_n(\infty)\simeq_{whe} B O(n)$ is the [[classifying space]] for $O(n)$-[[principal bundles]] and rank $n$ vector bundles.

## Related concepts

* [[Grassmannian manifold]]

* [[classifying space]]

## References

* {#Stiefel35} [[Eduard Stiefel]], _Richtungsfelder und Fernparallelismus in
$n$-dimensionalen Mannigfaltigkeiten_, Comment. Math. Helv. , 8(1935/6),
3-51.

* {#Yokota56} I. Yokota, _On the cells of symplectic groups_, Proc. Japan Acad. 32 (1956), 399-400.

* {#James59} [[Ioan Mackenzie James]], _Spaces associated with Stiefel manifolds_, Proc. Lond. Math. Soc. (3) 9 (1959)

* [[Ioan Mackenzie James]], _On the homotopy type of Stiefel manifolds_, Proceedings of the AMS, vol. 29, Number 1, June 1971

* {#James76} [[Ioan Mackenzie James]], _The topology of Stiefel manifolds_, Cambridge University Press, 1976

* {#Kochmann96} [[Stanley Kochmann]], section 1.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Blaszczyk07} Zbigniew Błaszczyk, _On cell decompositions of $SO(n)$_, 2007 ([pdf](http://knm.mat.umk.pl/pliki/konferencje/zb_lsait3_2007.pdf))

* {#Hatcher} Hatcher, _Algebraic topology_

* Wikipedia, _[Stiefel manifold](https://en.wikipedia.org/wiki/Stiefel_manifold)_

* {#Saito55} Yoshihiro Saito, _On the homotopy groups of Stiefel manifolds_, J. Inst. Polytech. Osaka City Univ. Ser. A Volume 6, Number 1 (1955), 39-45. [Project Euclid](http://projecteuclid.org/euclid.ojm/1353054734)

[[!redirects Stiefel manifolds]]

[[!redirects Stiefel space]]
[[!redirects Stiefl spaces]]

[[!redirects Stiefel variety]]
[[!redirects Stiefel varieties]]


Stiefel-Whitney class

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

#Contents#
* table of contents
{:toc}

## Idea

A family of [[characteristic class]]es that obstruct [[orientation]], [[spin structure]], [[spin^c structure]], [[w4-orientation of EO(2)-theory|orientation of EO(2)-theory]] etc.

Stiefel-Whitney classes have coefficients in $\mathbb{Z}_2$, but via the [[Bockstein homomorphism]] they are lifted to [[integral Stiefel-Whitney class]]es.

## Definition
 {#Definition}

+-- {: .num_defn #AxiomaticDefinition}
###### Definition
**(axiomatic definition)**

The _Stiefel-Whitney classes_ are [[characteristic classes]] $w_i \in H^{i}(B O(n), \mathbb{Z}_2)$ on the [[classifying space]] of the [[orthogonal group]] in [[dimension]] $n$, defined by

1. $w_0 = 1$ and if $i \gt n$ then $w_i = 0$;

1. for $n = 1$,  $w_1 \neq 0$;

1. for the inclusion $\iota : B O(n) \hookrightarrow B O(n+1)$ we have $\iota^* w_i^{(n+1)} = w_i^{(n)}$;

1. **sum rule**: for all $k,l \in \mathbb{N}$ with the canonical inclusion

   $$
     \iota : B O(k) \times B O(l) \to B O(k+l)
   $$

   we have for all $i \in \mathbb{N}$ that

   $$
    \iota^* w_i = \sum_{j = 0}^i w_j \cup w_{i-j}
   $$

   (on the right the [[cup product]]).

=--

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

For $E \to X$ a real [[vector bundle]]/[[orthogonal group]]-[[principal bundle]], the **total universal Stiefel-Whitney class** $w(E)$ is

$$
  w(E) \coloneqq \sum_i w_i(E) \in H^\bullet(X, \mathbb{Z}_2)
$$

as an element in the [[cohomology ring]].

=--

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

For the total SW class of def. \ref{TotalSWClass}, the sum rule of def. \ref{AxiomaticDefinition} says equivalently  that for $E_1, E_2$ two real [[vector bundles]], then the total SW class of their [[direct sum of vector bundles]] is the [[cup product]] of the separate classes:

$$
  w(E_1 \oplus E_2) = w(E_1) \cup w(E_2)
  \,.
$$

=--

## Constructions

(...)

## Properties

### Spanning the cohomology ring of $B SO$

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

Every class in the [[ordinary cohomology]] $H^\bullet(B O(n), \mathbb{Z}_2)$ and $H^\bullet(B S O(n), \mathbb{Z}_2)$ of the [[classifying spaces]] of the ([[special orthogonal group|special]]) [[orthogonal group]] with [[coefficients]] in [[cyclic group of order 2|$\mathbb{Z}_2$ is  uniquely a [[polynomial]] in the Stiefel-Whitney classes. In fact the [[cohomology rings]] are [[polynomial]] algebras over $\mathbb{Z}_2$ in the SW classes:

$$
  H^\bullet(B O(n), \mathbb{Z}_2)
  \simeq
  \mathbb{Z}_2[w_1, \cdots, w_n]
  \,,
$$
$$
  H^\bullet(B S O(n), \mathbb{Z}_2)
  \simeq
  \mathbb{Z}_2[w_2, \cdots, w_n]
  \,.
$$

=--

([Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.](#MilnorStasheff74))

### Whitney duality formula

For $X \hookrightarrow \mathbb{R}^q$ an [[embedding]] of a compact manifold, write $\tau := T X$ for the [[tangent bundle]] and $\nu$ for the corresponding [[normal bundle]]. Then since

$$
  \tau \oplus \nu \simeq T \mathbb{R}^q
$$

and the class of the vector bundle on the right is trivial, the sum rule for the SW classes says gives the [[cup product]] duality

$$
  w(\tau) \cup w(\nu) = 1
  \,.
$$

### Relation to Chern-classes
 {#RelationToChernClasses}

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

If $E_{\mathbb{C}}$ is a complex [[vector bundle]]/ $U(n)$-[[principal bundle]] and $E_{\mathbb{R}}$ is the underlying real vector bundle / $O(2n)$-[[principal bundle]] then the second Stiefel-Whitney class is given by the [[first Chern class]] mod 2:

$$
  w_2(E_{\mathbb{R}}) = c_1(E_{\mathbb{C}}) \; mod 2
  \,.
$$

=--

+-- {: .num_cor}
###### Corollary

An [[almost complex structure]] on the  [[tangent bundle]] of a [[manifold]] induces a [[spin^c structure]].

=--

This is discussed at _[Spin^c-structure -- From almost complex structures](spin^c+structure#FromAlmostComplexStructures)_.

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

More generally, the SW classes are then given by the [[Chern character]]. See for instance [Milnor-Stasheff, p. 171](#MilnorStasheff).

=--

## Related concepts

* [[orientation]], [[Spin structure]], [[w4-structure]]

* [[Chern class]]

* [[Pontryagin class]], [[Wu class]], [[one-loop anomaly polynomial I8]]

* [[orthogonal calculus]]

## References

Named after [[Eduard Stiefel]] and [[Hassler Whitney]].

Textbook accounts include

* {#MilnorStasheff} [[John Milnor]], [[Jim Stasheff|James D. Stasheff]], _Characteristic Classes_, Annals of Mathematics Studies 76, Princeton University Press (1974).

* {#Kochmann96} [[Stanley Kochmann]], section 2.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Peter May]], chapter 23, section 3 of _A concise course in algebraic topology_ ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

Discussion with an eye towards [[mathematical physics]]

* {#RudolhSchmidt} Gerd Rudolph, Matthias Schmidt, _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

Stiefel-Whitney classes generating the [[cohomology ring]] of $BO(n)$

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press (1974) ([ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf))

See also

* Wikipedia, _[Stiefel-Whitney class](http://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class)_

[[!redirects Stiefel-Whitney classes]]

[[!redirects first Stiefel-Whitney class]]
[[!redirects second Stiefel-Whitney class]]


Stieltjes integral

##Stieltjes Integral##

The Stieltjes integral can refer to two main integrals: the Riemann-Stieltjes integral and the Lebesgue-Stieltjes integral.


Stigler's law of eponymy

\tableofcontents

## Introduction

__Stigler's law of eponymy__ is the principle that fundamental discoveries are named after the person who made them famous, not the person who made them first, and these are often not the same person.

It is also called the __Matthew effect__, after Matthew 25:29. It is a variant of Arnold's law, which also applies to itself.

In certain parts of the mathematics community this is also called **Baez\'s law**, which states that

> "Any effect, constant, theorem or equation named after Professor X was first discovered by Professor Y, for some value of Y not equal to X."

## Examples

Stigler himself named the sociologist Robert K. Merton as the discoverer of [[Stigler's law of eponymy]], so that it is itself an example of [[Stigler's law of eponymy]].

### In mathematics

* [[currying|Currying]] was first identified by [[Moses Schonfinkel|Moses Schönfinkel]], rather than by [[Haskell Curry]].
* [[Kripke frame]]s go back before Kripke to 1957; see [nForum discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1991).
* [[Conduché functors]] were introduced by [[Jean Giraud]] 8 years before [[F. Conduché]].
* [[Russell's paradox]] was first proved by [[Ernst Zermelo]] rather than [[Bertrand Russell]]

There are several more already on the nLab; they just don't link here yet.

### In physics

* [[Hubble's law]] was first formulated by [[Georges Lemaître]] rather than [[Edwin Hubble]]

## References

* Wikipedia, *[Stigler's law of eponymy](https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy)*

* Wikipedia, *[List of examples of Stigler's law](https://en.wikipedia.org/wiki/List_of_examples_of_Stigler%27s_law)*

*  This Week's Finds in Mathematical Physics (Week 112), [reply on `sci.math`](https://web.archive.org/web/20150326195705/http://mathforum.org/kb/message.jspa?messageID=104026)

[[!redirects Stigler's law]]
[[!redirects Baez's law]]


Stijn Vermeeren

* [webpage](http://stijnvermeeren.be/english)

## related $n$Lab entries

* [[net]]

* [[realizability]], [[realizability topos]]

* [[effective topos]]

category: people


Stinespring factorization theorem

**Under Construction**

This page is mostly a stub right now.

See also Wikipedia's [Stinespring factorization theorem](http://en.wikipedia.org/wiki/Stinespring_factorization_theorem)

+-- {: .un_theorem}
###### Stinespring's dilation theorem

Let $T: S_{1}(\mathcal{H}) \to S_{2}(\mathcal{H})$ be a completely positive trace-preserving (CPTP) map between states on a finite-dimensional [[Hilbert space]], $\mathcal{H}$.  Then there exists a Hilbert space $\mathcal{A}$ and a unitary operation $U$ on $\mathcal{H}\otimes\mathcal{A}$ such that

$$
   T(\rho) = Tr_{\mathcal{A}}U(\rho \otimes \rho_{\mathcal{A}})U^{\dagger}
$$

for all $\rho \in S_{n}(\mathcal{H})$, where $Tr_{\mathcal{A}}$ is the [[partial trace]] on the $\mathcal{A}$-system.  We refer to $\mathcal{A}$ as an _ancilla_ system and it is chosen such that dim$\mathcal{A}\le$dim$^{2}\mathcal{H}$.  This representation is unique up to unitary equivalence.

=--

+-- {: .query}
TO DO: Write down the proof.
=--

This dilation construction of channels and states is commonly termed the "Church of the Larger Hilbert Space" (coined by John Smolin) since it incorporates the Hilbert space of both the state under consideration as well the environment, even though the operation is on the system only in most cases (for an exception, see the discussion of open quantum systems below).  Note that Stinespring's theorem can also be used to show that every [[mixed state]] can be thought of as arising from some [[mixed state|pure state]] on a larger Hilbert space.


Stirling's approximation

\tableofcontents

## Idea

Stirling's approximation, or Stirling's formula, gives an asymptotic approximation to the [[factorial|factorial function]] in terms of [[elementary function|elementary functions]].

## Statement

Define $n!$ for $n \in \mathbb{N}$ as usual by [[recursion]]: $0! = 1$ and $(n+1)! = (n+1) \cdot n!$. Stirling's approximation says

$$n! \sim \frac{n^n \sqrt{2\pi n}}{e^n}$$

or in other words that

$$\underset{n \to \infty}{\lim}\; \frac{n^n \sqrt{2\pi n}}{e^n \cdot n!} = 1.$$

## Proof

Many proofs are known. The following is meant to bring out the essential kinship between this formula and the famous [[Gaussian integral]] identity

$$\int_{-\infty}^\infty e^{-x^2}\; d x = \sqrt{\pi},$$

or equivalently

$$\int_{-\infty}^\infty e^{-x^2/2}\; d x = \sqrt{2\pi}.$$

First we establish an elementary fact, accessible to a student of elementary calculus.

\begin{proposition} For some [[constant function|constant]] $C$,

$$n! \sim C\frac{n^n \sqrt{n}}{e^n}.$$
\end{proposition}

\begin{proof}
Consider a [[trapezoidal sum estimate]] for the integral

$$I_n = \int_1^n \log x\; dx = \left. x\log x - x\right|_1^n = n \log n - n + 1 = 1 + \log \frac{n^n}{e^n}.$$
Partitioning the interval $[1, n]$ into subintervals of unit length, the corresponding trapezoidal sum is

$$T_n = \frac{\log 1 + \log 2}{2} + \frac{\log 2 + \log 3}{2} + \cdots + \frac{\log (n-1) + \log n}{2} = \log (n!) - \frac1{2} \log n = \log \frac{n!}{\sqrt{n}}.$$
$T_n$ underestimates $I_n$ because the graph of $\log$ is "concave down" (meaning $-\log x$ is a [[convex function]]). In other words, the error terms

$$\int_k^{k+1} \log x\; dx - \frac{\log k + \log (k+1)}{2}  $$
are positive, so that the sum of these error terms from $k=1$ to $n-1$, which is $I_n - T_n$, increases with $n$. However, using the error term estimate for the trapezoidal rule, the term (1) is on the order of $1/k^2$ (the maximum value of $|(\log)''(x)| = 1/x^2$ over $[k, k+1]$). Since $\sum 1/k^2$ converges, we see that the increasing sequence $I_n - T_n$ has a uniform upper bound and therefore a limit. Hence the limit

$$\underset{n \to \infty}{\lim} (I_n - 1) - T_n =
\underset{n \to \infty}{\lim} \log \frac{n^n}{e^n} - \log \frac{n!}{\sqrt{n}} =
\underset{n \to \infty}{\lim} \log \frac{n^n \sqrt{n}}{e^n \cdot n!}$$
exists, and therefore $n! \sim C\frac{n^n \sqrt{n}}{e^n}$ for some constant $C$.
\end{proof}

The following lemma makes reference to the [[Gamma function]].

\begin{lemma}
$\frac{\sqrt{2\pi}}{C} = \underset{n \to \infty}{\lim} \frac{\Gamma(n + (1/2))}{\Gamma(n) \cdot \sqrt{n}}$; in particular, this limit exists.
\end{lemma}

\begin{proof}
From $n! \sim C\frac{n^n \sqrt{n}}{e^n}$, we easily derive

$$\frac{(2n)!}{n! \cdot n!} \sim \frac{2^{2n} \sqrt{2n}}{C n}$$
so that

$$\frac1{C} \sim \frac{(2n)!}{2^{2n} n! \cdot n!} \cdot \frac{n}{\sqrt{2n}} = \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac1{\sqrt{2n}}.$$
Then, using $\Gamma\left(\frac1{2}\right) = 2\int_0^\infty e^{-x^2} dx = \sqrt{\pi}$, we have

$$\frac{\sqrt{2\pi}}{C} \sim \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac{\sqrt{\pi}}{\sqrt{n}} = \frac{\left(n-\frac1{2}\right)\left(n-\frac{3}{2}\right) \ldots \left(\frac1{2}\right) \Gamma\left(\frac1{2}\right)}{(n-1)! \cdot \sqrt{n}} = \frac{\Gamma\left(n + \frac1{2}\right)}{\Gamma(n) \cdot \sqrt{n}}.$$
\end{proof}

\begin{proposition}
$\underset{n \to \infty}{\lim} \frac{\Gamma(n + (1/2))}{\Gamma(n) \cdot \sqrt{n}}  = 1$.
\end{proposition}

\begin{proof}
From log-convexity of $\Gamma$, we may derive

$$\frac{\Gamma(x-1/2)}{\Gamma(x - 1)} \leq \frac{\Gamma(x)}{\Gamma(x-1/2)} \leq \frac{\Gamma(x+1/2)}{\Gamma(x)}$$
which implies that the values of $\frac{\Gamma(x+1/2)}{\Gamma(x) \cdot \sqrt{x}}$, with $x$ ranging over half-integers, tend to the same limit $L$ as with $x$ ranging over whole integers. Therefore

$$L^2 = \underset{n \to \infty}{\lim} \frac{\Gamma(n+1)}{\sqrt{n+(1/2)}\cdot \Gamma(n + (1/2))} \cdot \frac{\Gamma(n + (1/2))}{\sqrt{n}\Gamma(n)}
= \underset{n \to \infty}{\lim} \frac{n\Gamma(n)}{\sqrt{(n+1/2)n}\cdot \Gamma(n)} = \underset{n \to \infty}{\lim} \frac{n}{\sqrt{(n+1/2)n}},$$
whence $L^2 = 1$, and therefore $L = 1$.
\end{proof}

## References

* [n-Category Café post](https://golem.ph.utexas.edu/category/2021/10/stirlings_formula.html)

* Dan Romik, *Stirling's Approximation for n!: the Ultimate Short Proof?*,
The American Mathematical Monthly
Volume 107 Issue 6 (2000), 556-557. doi.org/10.1080/00029890.2000.12005235


Stoch

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## Idea

The [[category]] whose [[objects]] are [[measurable spaces]] and whose [[morphisms]] are [[stochastic kernels]] (or [[Markov kernels]]) is often denotes _Stoch_, or similar.

It is the motivating example of a [[Markov category]], and together with its subcategory [[Stoch#borelstoch|BorelStoch]], one of the most important categories of [[categorical probability]].

## Definition

_Stoch_ is the category whose

* [[Objects]] are [[measurable spaces]], i.e. pairs $(X,\mathcal{A})$ where $X$ is a set and $\mathcal{A}$ is a [[sigma-algebra]] on $X$;
* [[Morphisms]] $(X,\mathcal{A})\to(Y,\mathcal{B})$ are [[Markov kernels]] of entries $k(B|x)$, for $x\in X$ and $B\in\mathcal{B}$;
* The [[identities]] $(X,\mathcal{A})\to(X,\mathcal{A})$ are given by the [[Dirac delta]] kernels
$$
\delta(A|x) = 1_A(x) = \begin{cases}
1 & x\in A ; \\
0 & x\notin A ;
\end{cases}
$$
* The [[composition]] of kernels $k:(X,\mathcal{A})\to(Y,\mathcal{B})$ and $h:(Y,\mathcal{B})\to(Z,\mathcal{C})$ is given by the [[Lebesgue integral]]
$$
(h\circ k) (C|x) = \int_Y h(C|y)\,k(dy|x)
$$
for all $x\in X$ and $C\in\mathcal{C}$.
This is sometimes called _Chapman-Kolmogorov formula_. (Compare with the composition formula for [[stochastic matrices]].)

## As a Kleisli category

_Stoch_ can be equivalently described as the [[Kleisli category]] of the [[Giry monad]] on [[Meas]].

(...)

## Notable subcategories

### BorelStoch

_BorelStoch_ is the [[full subcategory]] of _Stoch_ whose objects are [[standard Borel space|standard Borel]] measurable spaces.
This is particularly important as a [[Markov category]] because it has [[Markov category#conditionals|conditionals]] and countable [[Markov category#kolmogorov_products|Kolmogorov products]].
It is the Kleisli category of the [[Giry monad]] restricted to [[standard Borel spaces]].

As proven [here](#markov_support), in _BorelStoch_ all [[idempotent splitting|idempotents split]], making it a [[Cauchy-complete category]].

### FinStoch

[[FinStoch]] is the [[full subcategory]] of _Stoch_ whose objects are discrete [[finite sets]]. Equivalently, it is the category of finite [[stochastic matrices]].
It is closely related to the Kleisli category of the [[distribution monad]].

## Related concepts

* [[Markov category]]
* [[Markov kernel]]
* [[category of couplings]]
* [[probability monad]]
* [[Giry monad]]
* [[Kleisli category]]
* [[categorical probability]]

## References

* {#Lawvere62} [[Bill Lawvere]], _The category of probabilistic mappings_, ms. 12 pages, 1962
([[lawvereprobability1962.pdf:file]])

* {#Chentsov65} N. N. Chentsov, _The categories of mathematical statistics_, Dokl. Akad. SSSR 164, 1965.

* {#panangaden_cat_markov} [[Prakash Panangaden]], _The category of Markov kernels_, ENTCS, 1999. ([full text](https://www.sciencedirect.com/science/article/pii/S1571066105806024))

* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

* {#markov_support} Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone, Dario Stein, _Absolute continuity, supports and idempotent splitting in categorical probability_, ([arXiv:2308.00651](https://arxiv.org/abs/2308.00651))

category: category, probability

[[!redirects BorelStoch]]
[[!redirects category of kernels]]
[[!redirects category of Markov kernels]]


stochastic dependence and independence

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## Idea

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## Intuition

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## In measure-theoretic probability

Let $(\Omega,\mathcal{F},p)$ be a [[probability space]], and let $A,B\in\mathcal{F}$ be [[events]], i.e. measurable subsets of $\Omega$.
We say that $A$ and $B$ are **independent** if and only if
$$
p(A\cap B) = p(A)\,p(B) ,
$$
i.e. if *the [[joint probability]] is the product of the probabilities*.

More generally, if $f:(\Omega,\mathcal{F})\to(X,\mathcal{A})$ and $g:(\Omega,\mathcal{F})\to(Y,\mathcal{B})$ are [[random variables]] or [[random elements]], one says that $f$ and $g$ are **independent** if and only if all the events they induce are independent, i.e. for every $A\in\mathcal{A}$ and $B\in\mathcal{B}$,
$$
p\big(f^{-1}(A)\cap g^{-1}(B)\big) = p\big(f^{-1}(A)\big)\,p\big(g^{-1}(B)\big) .
$$

Equivalently, one can form the [[joint distribution|joint random variable]] $(f,g):(\Omega,\mathcal{F})\to(X\times Y,\mathcal{A}\otimes\mathcal{B})$ and form the [[joint distribution]] $q=(f,g)_*p$ on $X\times Y$.
We have that $f$ and $g$ are independent as random variables if and only if for every $A\in\mathcal{A}$ and $B\in\mathcal{B}$,
$$
q\big(\pi_1^{-1}(A)\cap \pi_2^{-1}(B)\big) = q\big(\pi_1^{-1}(A)\big)\,q\big(\pi_2^{-1}(B)\big) .
$$
If we denote the [[marginal distributions]] of $q$ by $q_X$ and $q_Y$, the independence condition reads
$$
q(A \times B) = q_X(A)\,q_Y(B) ,
$$
meaning that the joint distribution $q$ is the product of its marginals.

(...)

### In terms of the Giry monad

(...)

### In the category of Markov kernels

(...)

## In Markov categories

(For now see [[Markov category#stochastic_independence|Markov category - Stochastic independence]])

## See also

* [[probability theory]], [[categorical probability]]
* [[joint and marginal probability]]
* [[Kolmogorov extension theorem]]
* [[probability monad]], [[Giry monad]]
* [[Markov category]], [[Markov kernel]]

## References

* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

category: probability

[[!redirects stochastic dependence]]
[[!redirects stochastic independence]]
[[!redirects statistical dependence]]
[[!redirects statistical independence]]
[[!redirects conditional dependence and independence]]
[[!redirects conditional dependence]]
[[!redirects conditional independence]]
[[!redirects stochastic interaction]]
[[!redirects stochastic interactions]]
[[!redirects statistical interaction]]
[[!redirects statistical interactions]]
[[!redirects independent events]]


stochastic Loewner equation

* wikipedia: [Schramm-Loewner evolution](https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution)
* Oded Schramm, _Conformally invariant scaling limits: an overview and a collection of problems_, International Congress of Mathematicians Zürich 2006, Vol. I, 513–543 Eur. Math. Soc. 2007 [pdf 1.7Mb](http://www.mathunion.org/ICM/ICM2006.1/Main/icm2006.1.0513.0544.ocr.pdf) [djvu 415kb](http://www.mathunion.org/ICM/ICM2006.1/Main/icm2006.1.0513.0544.ocr.djvu) [math.PR/0602151](https://arxiv.org/abs/math/0602151)

category: probability

[[!redirects Schramm-Loewner evolution]]


stochastic map

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## Idea

In [[probability theory]], a *stochastic map* between sets $X$ and $Y$ is a way of assigning numbers to elements $(x,y)$ of $X\times Y$ to model a probabilistic transition from $X$ to $Y$. The entries of a stochastic map form a matrix often called a *stochastic matrix* or *transition matrix*, which has nonnegative entries whose columns (or rows, depending on the convention) sum to one.

A generalization to arbitrary measurable spaces is called a [[stochastic kernel]] (or *Markov kernel*).
(Sometimes the term *stochastic map* is used to denote a stochastic kernel.)

If a stochastic map also preserves the uniform distribution, then it is called a *doubly stochastic map*.
This means that the stochastic matrix has not just columns but also rows which sum to unity, then called a *doubly stochastic matrix*.

## Definitions

Let $X$ and $Y$ be sets. A **stochastic map** or **stochastic matrix** from $X$ to $Y$ is a function $f:X\times Y\to[0,1]$, whose entries we denote by $f(y|x)$, such that

* For all $x\in X$ and $y\in Y$, the number $f(y|x)$ is non-negative;
* For all $x\in X$, the numbers $f(y|x)$ are nonzero for only finitely many $y\in Y$;
* For all $x\in X$, $\sum_Y f(y|x) = 1$.

The quantity $f(y|x)$ can be thought of as a transition [[probability]] or [[conditional probability]] of moving to $y$ if the current state or current knowledge is $x$.

### Composition

The product of stochastic matrices is again a stochastic matrix (and so they form a category, see below).
Given stochastic matrices $f$ from $X$ to $Y$ and $g$ from $Y$ to $Z$, the entries of the product are
$$
(g\circ f) (z|x) = \sum_X g(z|y)\,f(y|x) .
$$
Its probabilistic interpretation is that the random transition from $x$ to $y$ and from $y$ to $z$ are independent, as in a Markov process (hence their product is taken), and all intermediate states $y$ are mutually exclusive, so that the probabilities are summed over all $y$. This formula is sometimes called the *Chapman-Kolmogorov formula*. (See also the generalization to [[Markov kernels]].)

As identity matrices are stochastic, stochastic matrices form a [[category]], see below.

### Measure-preserving stochastic maps

Let $(X,p)$ and $(Y,q)$ be discrete [[probability spaces]], i.e. sets equipped with a functions $p:X\to[0,1]$ and $q:Y\to[0,1]$ which sum to one.
A stochastic map $f$ from $X$ to $Y$ is called **measure-preserving** if for every $y\in Y$,
$$
\sum_{x\in f^{-1}(y)} p(x) = q(y) .
$$
(Compare this with the [[Markov kernel#measurepreserving_kernels|analogous notion for stochastic kernels]].)

### Stochastic maps from deterministic functions

The [[identity]] function on a set $X$ defines a stochastic map as follows,
$$
\delta(x'|x) = \delta_{x,x'} \begin{cases}
1 & x'=x ; \\
0 & x'\neq x
\end{cases}
$$
for every $x,x'\in X$. This gives the identity morphisms in the categories of stochastic maps [[stochastic map#categories_of_stochastic_maps|below]].

More generally, let $f:X\to Y$ be a function. We can define the stochastic map $\delta_f:X\to Y$ as follows,
$$
\delta_f(y|x) = \delta_{f(x),y} = \begin{cases}
1 & f(x) = y ; \\
0 & f(x) \ne y
\end{cases}
$$
for every $x\in X$ and $y\in Y$.
Intuitively, this kernel represents the deterministic transition, from $x$ to $f(x)$ with probability one.
This construction induces a [[functor]] from the category [[Set]] to the categories of stochstic maps [[stochastic map#categories_of_stochastic_maps|below]].
(Compare with the [[Markov kernel#kernels_from_deterministic_functions|analogous construction for stochastic kernels]].)

## As Kleisli morphisms

A stochastic map can be equivalently written as a function
\begin{tikzcd}
X \ar{r}{f^\sharp} & DY
\end{tikzcd}
where $D$ denotes the [[distribution monad]].
Therefore, stochastic maps can be seen as the [[Kleisli morphisms]] of the distribution monad on [[Set]].

(Compare with the [[Markov kernel#as_kleisli_morphisms|analogous statement for stochastic kernels]].)

## Categories of stochastic maps

* The category of [[finite sets]] and stochastic matrices between them is often called [[Stoch#finstoch|FinStoch]], and is an example of a [[Markov category]].

* The category of finite [[probability spaces]] and measure-preserving stochastic matrices between them form a category as well.

## Related entries

* [[Markov kernel]]
* [[Markov category]]
* [[Stoch#finstoch|FinStoch]]
* [[Stoch]]
* [[quantum channel]]

## References

See also:

* Wikipedia, *[Stochastic matrix](https://en.wikipedia.org/wiki/Stochastic_matrix)*

* James Fullwood, [[Arthur Parzygnat]], Section 4 of: *The information loss of a stochastic map*, Entropy **2021** 23(8) 1021 [[arXiv:2107.01975](https://arxiv.org/abs/2107.01975), [doi:10.3390/e23081021](https://doi.org/10.3390/e23081021)]

category: probability

[[!redirects stochastic maps]]

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[[!redirects doubly stochastic maps]]

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stochastic order

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## Idea

The stochastic order is [[partial order]] or [[preorder]] between [[probability measures]] or [[continuous valuations|valuations]] over an ordered space, which compares them in terms of "how higher up in the order the mass is".

## Definition

Let $X$ be a [[topological space]] equipped with a [[preorder]] relation, and let $p$ and $q$ be [[Borel measures]] or [[continuous valuations|valuations]] on $X$. We say that $p \le q$ in the **stochastic order** if and only if for every [[upper set|upper]] [[open set]] $U\subseteq X$, we have that $p(U)\le q(U)$.

Note that, if one does not require the measures to be normalized, one has for example $p\le 2p$ whenever the measure $p$ is non-signed.
However, usually one is interested in comparing [[probability measures]] (or normalized [[valuation (measure theory)|valuations]]). In this case, assigning a higher mass to upper sets can be indeed interpreted as "having the mass placed at higher places".

Often one does not start with both an order and a topology on $X$, but obtains one structure from the other. If one starts with a [[partial order]] or a [[preorder]], one can take either the [[Alexandrov topology]] or the [[Scott topology]]:

 * In the former case, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every [[upper set]] $U$.

 * In the latter case, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every [[Scott-open]] set $U$, i.e. [[upper set|upper]] and inaccessible by [[directed suprema]]. This is how one construct the [[probabilistic powerdomain]].

If one starts with a [[topological space]], one can take as preorder the [[specialization preorder]] relative to the given topology. This way, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every [[open set]] $U$.
Equivalently, this is the [[pointwise order]] of [[valuations]] as functions on the frame of open sets.
This is how one constructs the [[extended probabilistic powerdomain]].

### For random variables

The stochastic order can be extended from probability measures to [[random variables]] by comparing their [[image measures|laws]].

Let $X$ be an ordered topological space, and let $f,g:\Omega\to X$ be [[random elements]] with values in $X$ on the [[probability space]] $(\Omega,\mu)$.
We say that $f\le g$ in the **stochastic order** if and only if $f_*\mu\le g_*\mu$ in the stochastic order of measures defined above.

In decision theory, economics and related fields, this order is also known as **first-order stochastic dominance**, especially on the real line.

## As a 2-functor

Categories of [[preorders]] (possibly with extra [[structure]]) and monotone maps (possibly preserving the extra structure) can be canonically equipped with the structure of a [[locally posetal 2-category]].
Explicitly this is given by the [[pointwise order]]: given preorders $X$ and $Y$ and [[monotone maps]] $f,g:X\to Y$, we say that $f\le g$, or we draw a 2-cell $f\Rightarrow g$, if and only if for every $x\in X$, $f(x)\le g(x)$.

The stochastic order can be seen as a way to turn a [[probability monad]] on a category of preorders into a [[2-monad]] by equipping it with an action on the 2-cells.
In particular, let $P$ be a [[probability monad]] on a category of preorders. In order to make it a [[2-functor]], and so a [[2-monad]], we want to say that if $f,g:X\to Y$ are such that $f\le g$, then also $P f\le P g$ as maps $P X\to P Y$. The latter condition means that for all $p\in P X$, $(P f)(p) \le (P g)(p)$ in $P Y$, so we need a preorder relation on $P Y$ that is suitably compatible with the order on $Y$, in such a way that $P$ preserves these 2-cells (inequalities). The stochastic order is the canonical choice of such a preorder.

One can actually _define_ the stochastic order directly in terms of 2-cells as follows: given $f,g:X\to Y$ as above, $P f\le P g$ _if and only if_ $f\le g$.

Similarly, the stochastic order can be seen as a way of making the functor $P$ [[enriched functor|enriched]] in [[truth values]].

## See also

* [[monads of probability, measures, and valuations]]
* [[probabilistic powerdomain]], [[extended probabilistic powerdomain]]
* [[valuation (measure theory)]]

## References

* {#orderedkantorovich} [[Tobias Fritz]] and [[Paolo Perrone]], _Stochastic order on metric spaces and the ordered Kantorovich monad_, Advances in Mathematics 366, 2020. ([arXiv:1808.09898](https://arxiv.org/abs/1808.09898))

(...)

[[!redirects stochastic ordering]]
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stochastic process

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## Idea

A __stochastic process__ is a function from a [[linearly ordered set]] $T$ ("time", typically the integers or positive real axis) into a set of [[random variable]]s (on fixed [[probability space]]), that is a collection $(X_t)_{t\in T}$ of random variables.

By a __random process__ physicists usually mean the same, but mathematicians sometimes allow random processes indexed by more general sets, not usually with meaning of time or equipped with a linear order; maybe "random collection" would be a better term in that case.

The most studied examples include [[Brownian motion]], [[Ornstein-Uhlenbeck process]] and [[Lévy process]]es.

## References

* Wikipedia [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process)

With an eye towards [[quantum noise]];

* [[Stéphane Attal]], *Stochastic Processes*, Lecture 4 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Stochastic_Processes.pdf), [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

[[!redirects stochastic processes]]

[[!redirects random process]]
[[!redirects random processes]]


stochastic quantization

## Idea

A kind of [[quantization]]

## References

The standard review article is

* P. Damgaard, H. Huffel, _Stochastic quantization_, Physics Reports 152 (5 and 6): 227–398; doi [pdf](https://homepage.univie.ac.at/helmuth.hueffel/PhysRep.pdf)

Relation to [[supersymmetry]], to the generalization of the correspondence between [[Chern-Simons theory]] and [[WZNW model]] and conjectural relation to M-theory is explained in

* [[Robbert Dijkgraaf]], Domenico Orlando, Susanne Reffert, _Relating field theories via stochastic quantization_, Nucl.Phys. __B824__, 365--386 (2010) [arxiv/0903.0732](http://arxiv.org/abs/0903.0732) [doi](http://dx.doi.org/10.1016/j.nuclphysb.2009.07.018)

Related items: [[Langevin equation]]