split monomorphism

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## Definitions and terminology

A **split monomorphism** in a [[category]] $C$ is a [[morphism]] $m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ such that the [[composite]] $r \circ m$ equals the [[identity morphism]] $1_A$.  Then the morphism $r$, which satisfies the [[duality|dual]] condition, is a __[[split epimorphism]]__.

We say that:

* $r$ is a __[[retraction]]__ of $m$,

* $m$ is a __[[section]]__ of $r$,

* $A$ is a __[[retract]]__ of $B$,

* the pair $(r,m)$ is a __[[split idempotent|splitting]]__ of the [[idempotent]] $m \circ r\colon B \to B$.

A split monomorphism in $C$ can be equivalently defined as a morphism $m\colon A \to B$ such that for every [[object]] $X\colon C$, the [[function]] $C(m,X)$ is a [[surjection]] in $\mathbf{Set}$; a preimage of $1_A$ under $C(m,A)$ yields a retraction $r$.

Alternatively, it is also possible to define a split monomorphism as an __absolute monomorphism__: a morphism such that for every functor $F$ out of $C$, $F(m)$ is a [[monomorphism]].  From the definition as a morphism having a retraction, it is obvious that any split monomorphism is absolute; conversely, that the image of $m$ under the [[representable functor]] $C(1,A)$ is a monomorphism reduces to the characterization above.

## Properties

\begin{proposition}
Any split monomorphism is a [[monomorphism]], in fact a [[regular monomorphism]] (it is the [[equalizer]] of $m\circ r$ and $1_B$), and therefore also a [[strong monomorphism]], an [[extremal monomorphism]], and (of course) a [[monomorphism]].
\end{proposition}

Evident but important and in contrast to general [[monomorphisms]]:
\begin{proposition}
  All [[functors]] preserve split monomorphisms.
\end{proposition}

\begin{proposition}
  A morphism is an [[isomorphism]] if and only if it is an [[epimorphism]] and a split monomorphism.
\end{proposition}

For a proof, see [Yuan 2012](#Yuan12).

## Examples

### In vector spaces

In the category [[Vect]] of [[finite dimensional vector spaces]] (over any [[field]]) every monomorphism $V_1 \hookrightarrow V_2$ splits. The corresponding [[idempotent]] is the [[projection]] onto $V_1$ in $V_2$.

## In higher category theory

In [[higher category theory]], we may still consider the notion of "split monomorphism", i.e. a morphism $m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ with $r \circ m$ being [[equivalence|equivalent]] to the identity of $A$.  However, in a higher category, such a morphism $m$ will not necessarily be a "monomorphism", that is, it need not be $(-1)$-[[truncated object|truncated]].

In general, we can say that in an $(n,1)$-[[(n,1)-category|category]], a "split monomorphism" will be $(n-2)$-truncated.  Thus:

* in a [[(0,1)-category]] (a [[poset]]), a split mono is $(-2)$-truncated, i.e. an [[isomorphism]];

* in a [[1-category]], a split mono is $(-1)$-truncated, i.e. a [[monomorphism]];

* in a [[(2,1)-category]], a split mono is $0$-truncated, i.e. a [[discrete morphism]];

* in an [[(∞,1)-category]], a split mono is not necessarily truncated at any finite level.

## Related concepts

* [[monomorphism]]

* [[split epimorphism]]

* [[propositions as projections]]

## References

* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Yuan12} [[Qiaochu Yuan]], *[Split epimorphisms and split monomorphisms](https://qchu.wordpress.com/2012/10/01/split-epimorphisms-and-split-monomorphisms/)*, blog post (2012),

[[!redirects split monomorphism]]

[[!redirects split monomorphisms]]
[[!redirects split mono]]
[[!redirects split monos]]
[[!redirects split monic]]
[[!redirects split monics]]

[[!redirects absolute monomorphism]]
[[!redirects absolute monomorphisms]]
[[!redirects absolute mono]]
[[!redirects absolute monos]]
[[!redirects absolute monic]]
[[!redirects absolute monics]]


split octonions

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## Idea

Up to [[isomorphism]] there is only one [[split composition algebra]] of dimension 8; this is called the algebra of _split octonions_.

## Definition

The multiplication may be deduced from the fundamental result on doubling multiplication or may be expressed as follows. Denote scalars by letters like $r, s$ and 3-vectors by letters like $x, y$. Let $\langle x, y \rangle$ denote the standard inner product
$$x_1 y_1 + x_2 y_2 + x_3 y_3$$
and let $x \wedge y$ denote the standard cross-product, so that $\langle x \wedge y, z \rangle = det(x, y, z)$. Elements of $V$ are represented by $2 \times 2$ arrays
$$\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right)$$
and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:
$$\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right)
\cdot
\left(
\begin{aligned}
r' & x'\\
y' & s'
\end{aligned}
\right) =
\left(
\begin{aligned}
r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\
r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s'
\end{aligned}
\right)
$$
The norm is given by a kind of determinant formula
$$N\left(
\begin{aligned}
r & x\\
y & s
\end{aligned}
\right) = r s - \langle x, y \rangle
$$

## Related concepts

* [[octonions]]

* [[Albert algebra]]

* [[split-complex numbers]]

* [[split quaternions]]

## References

* [[John Baez]], [[John Huerta]], _$G_2$ and the Rolling Ball_, Trans. Amer. Math. Soc., Vol. 366 No. 10 (2014), 5257-5293 ([arXiv:1205.2447](https://arxiv.org/abs/1205.2447))

See also

* Wikipedia, _[Split-octonion](https://en.wikipedia.org/wiki/Split-octonion)_

[[!redirects split octonion]]

[[!redirects split-octonions]]
[[!redirects split-octonion]]


split property

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## Idea

A formal property of [[quantum field theories]] traditionally discussed in the context of [[algebraic quantum field theory]].

(...)

## References

Discussion in the generality of [[AQFT on curved spacetimes]]:

* [[Christopher J. Fewster]], *The split property for quantum field theories in flat and curved spacetimes* [[arXiv:1601.06936](https://arxiv.org/abs/1601.06936)]

Discussion in the context of [[quantum gravity]] and in relation to the apparent [[black hole information paradox]]:

* [[Suvrat Raju]], *Failure of the split property in gravity and the information paradox*,  Classical and Quantum Gravity, **39** 064002 [[doi:10.1088/1361-6382/ac482b](https://iopscience.iop.org/article/10.1088/1361-6382/ac482b), [arXiv:2110.05470](https://arxiv.org/abs/2110.05470)]


split quaternion

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## Definition

The **split-quaternions** are an [[algebra]] over the [[real numbers]]. Every split quaternion $q$ may be represented as
$$
q = a_0 + a_1 i + a_2 j + a_3k
$$
where the basis elements satisfy the following products:

| $\times$ | i | j | k |
|--|--|--|--|
| i | -1 | k | -j |
| j | -k | +1 | -i |
| k | j | i | +1 |

and conjugation $t^* = a_0 - a_1 i - a_2 j - a_3k$.

These are closely related to the [[quaternions]], as the generators satisfy similarly-looking relations. They are obtained from the [[split-complex numbers]] through the [generalization of the Cayley-Dickson construction](https://ncatlab.org/nlab/show/Cayley-Dickson+construction#cayleydicksonalbert_construction).

## Related concepts

* [[quaternion]]

* [[tessarines]]

* [[split-complex numbers]]

* [[split octonion]]

## References

See also

* Wikipedia, _[Split-quaternion](https://en.wikipedia.org/wiki/Split-quaternion)_

On [[projective spaces]] over split-quaternions:

* [[Konrad Voelkel]], _Motivic cell structures for projective spaces over split quaternions_, 2016 ([freidok:11448](https://freidok.uni-freiburg.de/data/11448), [pdf](https://freidok.uni-freiburg.de/fedora/objects/freidok:11448/datastreams/FILE1/content))

[[!redirects split quaternions]]

[[!redirects split-quaternion]]
[[!redirects split-quaternions]]


split supersymmetry

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## Idea

In  [[particle physics]] [[phenomenology]], the term _split supersymmetry_ refers to [[model (physics)|models]] of [[supersymmetry|supersymmetric]] extensions of the [[standard model of particle physics]] ([[MSSM]] or other) whose [[supersymmetry breaking]]-[[scale]] is very high (for instance at [[GUT]]-scale of about $10^{15}$ [[GeV]]), much higher than the [[electroweak symmetry breaking]]-[[scale]] (of about 246 [[GeV]]) hence violating the idea of "[[naturalness]]", but such that approximate [[chiral symmetry]] [[protection from quantum corrections|protects]] the [[masses]] of the [[fermion|fermionic]] [[superpartners]].

As a result, in such [[model (physics)|models]] there is a "split" of the [[mass]] [[scales]] of the [[superpartner]] [[particles]]:

1. the [[scalar field|scalar]] [[superpartners]] (the [[sfermions]], hence the [[squarks]] and [[sleptons]]) have high [[rest masses]], from tens of [[TeV]] (mass scale of the [[gravitino]]) up to the [[supersymmetry breaking]]-scale many order of magnitudes higher;

1. the fermionic [[superpartners]] ([[gauginos]] and [[higgsino]]) have masses much closer to the [[electroweak symmetry breaking]] scale, at about a few [[TeV]].

(see [Arkani-Hamed, Dimopoulos, Giudice, Romanino 05, p 2.](#ArkaniHamedDimopoulosGiudiceRomanino05))

Therefore one speaks of "split supersymmetry" ([Giudice-Romanino 04](#GiudiceRomanino04)).

Historically, the consideration of split supersymmetry ([Arkani-Hamed, Dimopoulos 04](#ArkaniHamedDimopoulos05)) was in contrast to the folklore of "[[naturalness]]" which governed [[supersymmetry]] [[model (physics)|model]]-building. Split supersymmetry was motivated by the observation that many other theoretical problems of low-scale susy models disappear if one does not insist on "[[naturalness]]". Indeed, after the [[LHC]] [[experiment]] produced first observational results, susy models exhibiting "[[naturalness]]" were ruled out, while only split supersymmetry models remained viable (this as of beginning of 2018).

## Examples

### $G_2$-MSSM

The class of [[model (physics)|physics]] called [[G2-MSSM]] generically exhibits "slightly" split supersymmetry.

From [Kane 17, p. 43-44 ((5-1)-(5-2))](#Kane17):

> It is important to understand that there are two measures relevant to understanding supersymmetry breaking, one the scale at which it is broken (about $10^{14}$ [[GeV]] as described above), and the other the resulting [[gravitino]] mass. In the [[KK mechanism|compactified]] [[M-theory]] case the [[gravitino]] mass is calculated, and comes out to be about 40 [[TeV]] (40 000 [[GeV]]). Sometimes even experts confuse these two scales if they are speculating about [[supersymmetry breaking]] without a real theory to
calculate.

> Thus 40 [[TeV]] is the natural scale for [[superpartner]] masses. That is not a surprising number in a theory starting with everything at the [[Planck scale]], but it is surprising if one expects the superpartner masses to be near the particle masses (all well below 1 [[TeV]]). The [[squarks]] and other masses are indeed predicted to be at the [[gravitino]] scale, tens of [[TeV|tera-electronvolts]].

> The theory has formulas (‘supergravity formulas’) for all the masses. When one calculates carefully the masses of the superpartners of the [[gauge bosons]] that mediate the [[standard model of particle physics|Standard Model]] forces they turn out to get no contribution from one large source, and the resulting value for the [[superpartners]] of the [[gauge bosons]] ([[gauginos]]) is about 1 [[TeV]], rather than about 40 [[TeV]]. They are the [[gluino]], photino, zino, and wino. The strong force gluino is heavier, about 1.5 [[TeV]] or somewhat more, and the [[electroweak field|electroweak]] ones (photino, zino, wino) are somewhat lighter, about half a [[TeV|tera-electronvolt]].

> ([Kane 17, p. 43-44 ((5-1)-(5-2))](#Kane17))

## References

The idea originates with

* [[James Wells]], _Implications of Supersymmetry Breaking with a Little Hierarchy between Gauginos and Scalars_,  Proceedings of [SUSY 2003](http://www.physics.arizona.edu/susy2003/) ([arXiv:hep-ph/0306127](https://arxiv.org/abs/hep-ph/0306127))

* {#ArkaniHamedDimopoulos04} [[Nima Arkani-Hamed]], [[Savas Dimopoulos]], _Supersymmetric Unification Without Low Energy Supersymmetry And Signatures for Fine-Tuning at the LHC_, JHEP 0506 (2005) 073 ([arXiv:hep-th/0405159](https://arxiv.org/abs/hep-th/0405159))

* {#GiudiceRomanino04} [[Gian Francesco Giudice]], A. Romanino, _Split Supersymmetry_, Nucl. Phys. B699:65-89,2004; Erratum-ibid.B706:65-89,2005 ([arXiv:hep-ph/0406088](https://arxiv.org/abs/hep-ph/0406088))

Further discussion includes

* {#ArkaniHamedDimopoulosGiudiceRomanino05} [[Nima Arkani-Hamed]], [[Savas Dimopoulos]], [[Gian Francesco Giudice]], A. Romanino, _Aspects of Split Supersymmetry_, Nucl. Phys. B709:3-46, 2005 ([arXiv:hep-ph/0409232](https://arxiv.org/abs/hep-ph/0409232))

* Fei Wang, Wenyu Wang, Jin Min Yang, _A split SUSY model from SUSY GUT_,  JHEP03(2015)050 ([arXiv:1501.02906](https://arxiv.org/abs/1501.02906))

Popular discussion of the "slightly split" [[G2-MSSM]] is in

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

Discussion of [[phenomenology]] in view of the observed near-criticality of the [[Higgs field]] [[vacuum stability]]:

* {#GiudiceStrumia11} [[Gian Giudice]], [[Alessandro Strumia]], _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_, Nuclear Physics B Volume 858, Issue 1, 1 May 2012, Pages 63-83 ([arXiv:1108.6077](https://arxiv.org/abs/1108.6077))

* {#IsidoriPattori17} Gino Isidori, Andrea Pattori, _On the tuning in the $(m_h, m_t)$ plane: Standard Model criticality vs. High-scale SUSY_, Physics Letters B Volume 782, 10 July 2018, Pages 551-558 ([arXiv:1710.11060](https://arxiv.org/abs/1710.11060))

See also

* Wikipedia, _[Split supersymmetry](https://en.wikipedia.org/wiki/Split_supersymmetry)_


split support

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# Split supports
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## Definition

Recall that the **[[support object]]** of an object $X$ of a [[regular category]] (or [[n-category]]) is its [[(-1)-truncation]], i.e. the [[image]] of the map $X\to 1$ to the [[terminal object]].  Let $\Vert X \Vert$ denote its support.

We say that $X$ has a **split support** if the canonical [[map]] $X\to {\Vert X \Vert}$ has a [[section]].  Since $\Vert X \Vert$ is a [[subterminal object]], it is equivalent to say that there exists any morphism ${\Vert X \Vert} \to X$.

If all objects has split supports, we say that **supports split** in the ambient category.

## Relation to axiom of choice

Supports splitting in a category is a weak form of the external [[axiom of choice]] (all regular epis split).  In fact, the splitting of supports is exactly the "difference" between the external axiom of choice and the *internal* axiom of choice, i.e.

$$ EAC \Leftrightarrow (IAC \;\text{ and }\; SS).$$

## In foundations

### In set theory

If we assume [[excluded middle]], then supports split in the category [[Set]].  This has little to do with the foundational [[axiom of choice]]; it is more like [[well-pointed category|well-pointedness]].  Note, however, that to say there is a *function* assigning to every $X$ a section of $X\to {\Vert X \Vert}$ is much stronger: such a function is a global [[choice operator]].

In [[constructive mathematics]], however, it may not be true that all supports split in Set; this can fail in the [[internal logic]] of some [[toposes]].  (Note that the "internal" version of "supports split in Set" in a topos may not be the same as the statement that supports split as an external statement about the topos itself.)  See [(Fourman-Scedrov)](#FourmanScedrov) and [(KECA)](#KECA).  Thus, splitting of supports can be regarded as a weaker form of [[excluded middle]].

### In (homotopy) type theory

In [[type theory]] under [[propositions as types]]), where assertions of existence are always witnessed, to say that "all supports split" would by default mean that there is a *function* as above called the [[choice operator]], assigning to every $X$ a section of $X\to {\Vert X \Vert}$, and imply the [[global axiom of choice]].  To recover the statement which depends only on excluded middle, we need an additional truncation:

$$ \prod_{(X:Type)} \Vert ( \Vert X \Vert \to X ) \Vert. $$

We might pronounce this version as "all supports [[mere proposition|merely]] split".

In [[homotopy type theory]], the pure constructive version of "all supports split" ($\prod_{(X:Type)} \Vert X \Vert \to X$) is in fact inconsistent: it contradicts the [[univalence axiom]].  As before, the truncated version is true under LEM but may fail otherwise.

#### Objects with split support

Since not all supports split in homotopy type theory, it is interesting to consider whether the support of some particular type is split.  For instance, for any $f:A\to B$, the type $qinv(f) \coloneqq \sum_{g:B\to A} (f g = id_B) \times (g f = id_A)$ has split support, since its support can be proven to be equal to the type of coherent equivalence data on $f$.

It is shown in [(KECA)](#KECA) that a type in homotopy type theory has split support if and only if it admits a [[steady function|steady]] endomap.  Thus, it has merely split support if and only if it merely admits a steady endomap.

Some general classes of types can be shown *not* to have split support, at least not uniformly, by arguments similar to the one which shows that not all types have split support.  For instance, the identity type $x=_A y$ has split support for all $x,y:A$ if and only if $A$ is an [[h-set]]; this is proven in [(KECA)](#KECA), and the "only if" direction is also a special case of Theorem 7.2.2 of [[the HoTT book]].

Similarly, we have:

+-- {: .un_theorem}
###### Theorem
If the type $steady(f)\coloneqq \prod_{(x,y:A+A)} (f x = f y)$ has split support for every [[endomap]] $f:A+A\to A+A$, then $A$ is an [[h-set]].
=--
+-- {: .proof}
###### Proof
Given $a,b:A$ with $\Vert a=b\Vert$, we show $a=b$; this suffices to show that $A$ is a set.  Define $g,h : A \to A+A$ to be constant at $inl(a)$ and $inl(b)$ respectively, and $f = [g,h] : A+A \to A+A$.  Then

$$steady(f) =  ( (A\times A) \to ((a=a) \times (a=b) \times (b=a) \times (b=b)) ) $$

Since $\Vert a=b\Vert$, we have $\Vert steady(f)\Vert$.  Thus, by assumption, $steady(f)$; hence $a=b$.
=--

In particular, not all types of the form $steady(f)$ have split support.  Thus, "steadiness" is not as well-behaved a notion as being quasi-invertible.

## References

* {#FourmanScedrov} M. P. Fourman and A. Scedrov. The "world's simplest axiom of choice" fails.
*Manuscripta Math.*, 38(3):325{332, 1982.

* {#KECA} [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]], [[Thorsten Altenkirch]], _Generalizations of Hedberg's theorem_, in M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. [PDF](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf).

  (In this paper, types with split support are called "h-stable")

[[!redirects split support]]
[[!redirects split supports]]
[[!redirects supports split]]

[[!redirects h-stable type]]
[[!redirects h-stable types]]


split-complex numbers

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## Definition

The algebra obtained from the [generalization of the Cayley-Dickson construction](https://ncatlab.org/nlab/show/Cayley-Dickson+construction#cayleydicksonalbert_construction) applied on the [[complex number|complex numbers]] applied to the [[real numbers]] with parameter $\gamma=-1$.

A split-complex number $z$ may be represented as
$$
z= x+ j y
$$
where $j^2=1$ (in contrast with the [[imaginary unit]] $i^2=-1$ in the complex numbers). Conjugation is similarly given by
$$
z^* = x- j y
$$

A consequence is that the product $z z^*$ is not non-negative anymore, since
$$
z z^* = x^2 - j^2 y^2 = x^2 - y^2
$$
meaning in particular that [[zero-divisors]] exist (for example $(1 - j)(1 + j)=1 - j^2=0$). Using the diagonal basis
$$
e = \frac{1 - j}{2}
$$
$$
e^* = \frac{1 + j}{2}
$$
of idempotent elements, hence for which $e^2=e$ and $(e^*)^2=e^*$, and fulfilling $e e^*=e^* e=0$ results in multiplication given by
$$
(a e+b e^*)(c e+d e^*)
=(a c) e+(b d) e^*
$$
which yields that the algebra $\mathbb{R}[j]=\mathbb{R}[e]$ of split-complex numbers is isomorphic to the algebra $\mathbb{R}\oplus\mathbb{R}$ with pointwise multiplication.

## Related concepts

* [[tessarines]]

* [[split quaternions]]

* [[split octonions]]

## References

* Robert Brown. *On generalized Cayley-Dickson algebras.* Pacific Journal of Mathematics 20, no. 3 (1967): 415-422. ([doi](http://dx.doi.org/10.2140/pjm.1967.20.415))


splitting field

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## Definition

Let $k$ be a [[field]], and $p \in k[x]$ a [[monic polynomial]] of degree $n$. A _splitting field_ for $p$ is a [[field extension]] $i: k \hookrightarrow E$ such that, regarding $p$ as a [[polynomial]] in $E[x]$ by applying the map

$$k[x] \cong k \otimes_k k[x] \stackrel{i \otimes 1}{\to} E \otimes_k k[x] \cong E[x],$$

the polynomial factors or "splits" as a product of linear factors (possibly repeated):

$$p(x) = (x - r_1)(x - r_2)\ldots (x - r_n)$$

with all the [[roots]] $r_i$ lying in $E$, and the smallest subfield of $E$ containing $k$ and these roots is $E$ itself (so that the roots generate $E$: $E = k(r_1, \ldots, r_n)$.

More generally, given a set $S \subseteq k[x]$ of monic polynomials, an extension field $E/k$ for which each $p \in S$ splits over $E$, and which as a field is generated over $k$ by the set of accompanying roots, is called a splitting field for $S$.

## Construction

The existence of a splitting field for a single polynomial $p \in k[x]$ can be proven by [[induction]] on $n = \deg(p)$. As $k[x]$ is a [[unique factorization domain]], $p$ is a product of irreducible polynomials $p_1 p_2 \ldots p_k$. The [[ideal]] $(p_1)$ is [[maximal ideal|maximal]] and so $E = k[x]/(p_1)$ is an [[finite-dimensional vector space|finite]] [[field extension]] of $k$ where the [[coset]] $r_1 \coloneqq x + (p_1) \in E$ is a [[root]] of $p_1(x) \in E[x]$, and $E = k(r_1)$. Then $q(x) = p(x)/(x - r_1) \in E[x]$ has degree $n-1$, and thus has a splitting field $F$ by induction, whence $q(x)$ splits as $(x - r_2)\ldots (x-r_n)$ in $F[x]$, and then $p(x) = (x - r_1)(x - r_2) \ldots (x - r_n)$ in $F[x]$. Finally we have the generation condition: $F = E(r_2, \ldots, r_n) = k(r_1)(r_2, \ldots, r_n) = k(r_1, r_2, \ldots, r_n).$

What is not clear from this construction is that splitting fields are unique (up to [[isomorphism]]). This is the case however:

+-- {: .num_theorem #unique}
###### Theorem
Splitting fields of a polynomial are unique up to (non-unique) isomorphism.
=--

+-- {: .proof}
###### Proof
(After [Conrad](#Con).) Suppose $E = k(r_1, \ldots, r_n)$ and $E' = k(r_1', \ldots, r_n')$ are two splitting fields of a polynomial $p \in k[x]$. The [[tensor product]] $E \otimes_k E'$ is a [[coproduct]] of $E$ and $E'$ in the category of [[commutative algebras]] over $k$; let $i: E \to E \otimes_k E'$ and $i': E' \to E \otimes_k E'$ be the coproduct [[coprojection|injections]]. Observe that as a $k$-algebra, $E \otimes_k E'$ is generated by the $i(r_j)$ together with the $i'(r_j')$.

Now let $m$ be any maximal ideal of $E \otimes_k E'$ (note that $E \otimes_k E'$ is finite-dimensional over $k$, so the existence of $m$ does not involve any choice principles), and consider the composite

$$\phi = (E \stackrel{i}{\to} E \otimes_k E' \stackrel{quot}{\to} E \otimes_k E'/m).$$

As is the case for any homomorphism between fields, this composite $\phi$ is an [[injective map]]. Moreover, letting $\rho_j$ denote the coset $i(r_j) + m$ and $\rho_j'$ the coset $i'(r_j') + m$, the polynomial $p(x)$ splits over the field $E \otimes_k E'/m$ as

$$\prod_j (x - \rho_j) = \prod_k (x - \rho_k')$$

and so by [[unique factorization domain|unique factorization]] of polynomials, for each $\rho_k'$ there is a $\rho_j = \phi(r_j)$ with $\rho_k' = \rho_j$. This means the $\phi(r_j)$ exhaust the generators $\rho_j, \rho_k'$ of $E \otimes_k E'/m$. In other words, $\phi$ is also surjective and provides an isomorphism $E \cong E \otimes_k E'/m$. By symmetry we also have an isomorphism $E' \cong E \otimes_k E'/m$, and thus $E \cong E'$.
=--

### Splitting fields of sets of polynomials

A splitting field for a finite set of polynomials $\{p_1, \ldots, p_k\}$ is just a splitting field of the product $p_1 p_2 \ldots p_k$.

Now let $S \subseteq k[x]$ be an arbitrary set of monic polynomials. For each $p \in S$, let $n_p$ denote the degree of $p$, and let us introduce a set of indeterminates $X_p = \{x_{p, 1}, \ldots, x_{p, n_p}\}$. Let $X$ be the coproduct $\sum_{p \in S} X_p$, and form the [[polynomial|polynomial algebra]] $R = k[X]$. For each $p \in S$ let us write

$$p(t) - \prod_{i = 1}^{n_p} (t - x_{p, i}) = \sum_{j = 0}^{n_p - 1} s_{p, j} t^j$$

for some elements $s_{p, j} \in R$. We claim that the set $\Sigma$ of all $s_{p, j}$ with $p$ ranging over $S$ generates a proper ideal in $R$. For given an $R$-linear combination $g = g_1 s_1 + \ldots + g_k s_k$ with $s_i \in \Sigma$, where $s_i$ is associated with a polynomial $p_i$, let $E$ be the splitting field of $p = p_1 \ldots p_k$, and define a homomorphism $k[X] \to E$ that sends distinct elements of the form $x = x_{p_i, j}$ to distinct roots $r_j$ of $p_i$, and otherwise sends $x = x_{f, j}$ with $f \notin \{p_1, \ldots, p_k\}$ to $0$. Then $g$ is sent to $0$ in $E$, hence $g$ cannot be $1$.

Let $\mathfrak{p}$ be a [[prime ideal]] containing the ideal generated by $\Sigma$; the existence of such $\mathfrak{p}$ is guaranteed under the [[ultrafilter principle]]. The ring $R/\mathfrak{p}$ is an [[integral domain]]; by construction every $p \in S$ splits over $R/\mathfrak{p}$ and the roots of such $p$ generate $R/\mathfrak{p}$ as a $k$-algebra. These roots thus generate its [[field of fractions]] $F$ as a field and $F$ is therefore a splitting field for $S$.

In particular, taking $S$ to be the set of all monic polynomials in $k[x]$, this gives an efficient construction of the [[algebraic closure]] of $k$ that invokes not the full strength of the [[axiom of choice]] (in the form of [[Zorn's lemma]]), but only of the weaker [[ultrafilter principle]].

+-- {: .num_theorem}
###### Theorem
Any two splitting fields of a given set $S$ of polynomials are isomorphic.
=--

Again we prove this only under the assumption of the [[ultrafilter principle]], and not under the assumption of full [[axiom of choice]].

+-- {: .proof}
###### Proof
(After [Caicedo](#AC).) Let $E, F$ be splitting fields for $S$; without loss of generality we may suppose $S$ is closed under multiplication of polynomials. For $p \in S$ let $E_p, F_p$ denote the corresponding subfields obtained by adjoining all roots of $p$ that occur in $E, F$ to the ground field $k$. Any isomorphism $E \to F$ restricts to an isomorphism $E_p \to F_p$. Notice that the $E_p$ form a directed system of fields, with an inclusion morphism $E_p \to E_q$ if $p$ divides $q$, and $E$ is the [[colimit]] of the $E_p$ over this directed system. We are required to show that the set of isomorphisms $Iso(E, F)$ is nonempty; we have restriction maps $Iso(E, F) \to Iso(E_p, F_p)$ which exhibit $Iso(E, F)$ as the [[inverse limit]] of $Iso(E_p, F_p)$ over the corresponding system, with transition maps $Res_{p q}: Iso(E_q, F_q) \to Iso(E_p, F_p)$ for $p|q$ also given by restriction. So we are required to show that this inverse limit is [[inhabited set|inhabited]].

By Theorem \ref{unique}, the set of isomorphisms $Iso(E_p, F_p)$ is inhabited, and it is also [[finite set|finite]] (being a [[torsor]] of the group $Aut(E_p)$ which has cardinality at most $n!$ where $n$ is the number of roots). It can then be proven from the [[ultrafilter theorem]] that $\Phi = \prod_{p \in S} Iso(E_p, F_p)$ is nonempty; see [here](/nlab/show/compactness+theorem#surj) for a proof. Similarly, the subset

$$\Phi_{p q} \coloneqq \{\phi = (\phi_p)_{p \in S} \in \Phi: Res_{p q}(\phi_q) = \phi_p\}$$

is also nonempty, as is any finite intersection of such subsets $\Phi_{p q}$ (which actually is another $\Phi_{p' q'}$, by considering the product of $p$'s and $q$'s).

Now $\Phi$ topologized as a product of finite discrete spaces is a [[compact Hausdorff space]]; this version of the [[Tychonoff theorem]] also follows from the ultrafilter theorem. We conclude from compactness that the full intersection $\bigcap_{p|q} \Phi_{p q}$ is inhabited. But this intersection is just the inverse limit which is $Iso(E, F)$, which concludes the proof.
=--

An alternative proof based on a similar pattern of reasoning, but more explicitly in the language of [[model theory]] and the [[compactness theorem]], was given by [Joel David Hamkins](#JDH), in an answer following Caicedo's at MathOverflow.

## References

* {#Ban}  Bernhard Banaschewski, _Algebraic closure without choice_, Mathematical Logic Quarterly, Vol. 38 Issue 1 (1992), 383–385. (doi: 10.1002/malq.19920380136, [paywall](http://onlinelibrary.wiley.com/doi/10.1002/malq.19920380136/pdf))

* {#Con} [[Keith Conrad]], *Isomorphisms of Splitting Fields*, Galois theory notes [[[Conrad-IsoSplittingFields.pdf:file]]]

* {#AC} [Andres Caicedo](http://mathoverflow.net/users/6085/andres-caicedo) _Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?_, URL (version: 2010-11-19): [http://mathoverflow.net/q/46568](http://mathoverflow.net/q/46568)

* {#JDH} [[Joel David Hamkins]], _Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?_, URL (version: 2010-11-20): [http://mathoverflow.net/q/46729](http://mathoverflow.net/q/46729)

[[!redirects splitting fields]]


splitting principle

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

#Contents#
* table of contents
{:toc}

## Idea
 {#SplittingPrinciple}

In [[algebraic topology]], a _splitting principle_ for a [[classifying space]] $B G$ for some [[topological group]] $G$ ([[delooping]] of some [[∞-group]] $G$) is a map from a [[torus]]-classifying space

$$
  B (T^n) \overset{f}{\longrightarrow} B G
$$

such that  the induced [[pullback in cohomology|pullback]] of [[cohomology rings]] is an [[injective function]]:

$$
  f^\ast \;\colon\; H^\bullet(B G) \hookrightarrow H^\bullet(B T^n)
  \,,
$$

hence allowing to view the cohomology of $G$-[[principal ∞-bundles]] in terms of that of plain [[torus]]-[[principal bundles]].

In particular for $G$ a classical [[compact Lie group]] such as the [[unitary group]], the splitting principle holds and allows to express [[Chern classes]] of [[complex vector bundles]] as algebraic expressions in just [[first Chern classes]] of [[complex line bundles]].

## Statement

Let $G$ be a [[connected topological space|connected]] [[compact Lie group]] and write $U(1)^n \simeq T \stackrel{i}{\hookrightarrow} G$ for a [[maximal torus]].

Let $X$ be a [[connected topological space]] and $P \to X$ a $G$-[[principal bundle]] over $X$ [[classifying space|classified]] by a map $g \colon X \to B G$.

Then consider the [[coset space]] $G/T$ and the $G/T$-[[fiber bundle]] $Y\to X$ [[associated bundle|associated]] to $P$, this is equivalently the [[homotopy pullback]] in the diagram

$$
  \array{
    Y
      &\stackrel{(g_1,\cdots, g_n)}{\longrightarrow}&
    B T
      & \simeq B U(1)^n
    \\
    \big\downarrow {}^{\mathrlap{p}}
      &&
    \big\downarrow {}^{\mathrlap{B i}}
    \\
    X &\stackrel{g}{\longrightarrow}& B G
  }
  \,.
$$

This diagram shows that the pullback of the $G$-principal bundle $P \to X$ along $p$ to $Y$ is equivalently a $T$-principal bundle splitting as [[circle group]]-principal bundles classified by $(g_1, \cdots, g_n)$.

That this is a useful splitting is the content of:

+-- {: .num_theorem #GeneralizedSplittingPrinciple}
###### Theorem
**(generalized splitting principle)**

Let $R$ be a [[commutative ring]] in which a [[prime number]] $p$ is a [[unit]] if $H_\bullet(B G,\mathbb{Z})$ has a $p$-[[torsion subgroup]].

Then

1. The $H^\bullet(B G,R)$-module $H^\bullet(B T, R)$ (via $B i^\ast$) is [[free module|free]] on the cohomology of $G/T$:

   $$
     H^\bullet(B T,R)\simeq H^\bullet(B G,R) \otimes H^\bullet(G/T, R)
     \,;
   $$

1. Analogously there is an isomorphism

   $$
     H^\bullet(Y,R) \simeq H^\bullet(X,R)\otimes H^\bullet(G/T,R)
   $$

   and hence $p^\ast$ is the canonical inclusion (and hence in particular is an [[injection]])

   $$
     p^\ast \;\colon\;  H^\bullet(X,R)\hookrightarrow H^\bullet(Y,R)
     \,.
   $$

=--

In this general form this is due to ([May](#May)).

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

Since the elements

$$
  c \in H^\bullet(B G,R)
$$

are the [[universal characteristic classes]] of $G$-principal bundles with [[coefficients]] in $R$ (hence by [[Chern-Weil theory]] the [[invariant polynomials]] of the [[Lie algebra]] $\mathfrak{g}$ if $R$ has [[characteristic]]-0), theorem \ref{GeneralizedSplittingPrinciple}
gives the following way to express the characteristic classes of $G$-principal bundles on $X$ by [[tuples]]
$$
  (c_1^1, \cdots, c_1^n) \coloneqq (B i)^\ast c
$$

of characteristic classes -- hence [[first Chern class|first Chern classes]] -- of just circle bundles ([[line bundles]]):

$$
  p^\ast (c(P)) \simeq (g_1^\ast c_1^1, \cdots, g_n^\ast c_1^n)
  \,.
$$

(Since $p^\ast$ is injective, this is a genuine characterization of $c(P)$).

=--

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

One way to see that $B i^*\colon H^*(B G) \to H^*(B T)$ is injective is by using [[Chern-Weil theory]] to recognise that this map is just $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$ for $G$ a compact [[Lie group]]. This tells us firstly that these cohomology rings are particularly nice.

One can define a [[Becker-Gottlieb transfer|transfer map]] $\tau\colon H^*(B T) \to H^*(B G)$ as in [this MO answer](http://mathoverflow.net/questions/61784/cohomology-of-bg-g-compact-lie-group/61943#61943), and then show, following ([Dupont 1978](#Dupont78), chapter 8), that $\tau\circ B i^*\colon H^*(B G)\to H^*(B G)$ is multiplication by the [[Euler class]] $\chi(G/T)$. Thus if $\chi(G/T) \gt 0 $ then $\tau\circ B i^*$ hence $B i^*$ is injective. One can calculate $\chi(G/T) = | N(T)/T |$, where $N(T)/T =: W_T$ is the [[Weyl group]] of the maximal torus $T$, using a [[Lefschetz trace formula|Lefshetz fixed point]]-argument, giving the result.

=--

## Examples
 {#Examples}

### Complex vectors bundles and their Chern roots
 {#ComplexVectorBundleAndTheirChernRoots}

For $G = U(n)$ the [[unitary group]], the [[universal characteristic classes]] are the [[Chern classes]] $c_k \in H^\bullet(B U(n), \mathbb{Z})$. By the discussion at _[Chern class -- Properties -- Splitting principle and Chern roots](Chern+class#SplittingPrinciple).
the universal splitting principle here says that

$$
  (B i)^\ast(\sum_k c_k) = (1 + x_1) \cdots (1+ x_n)
  \,,
$$

where the $x_i \in H^\bullet(B U(1)^n , \mathbb{Z})$ are the universal characteristic classes of the [[maximal torus]], hence are $n$ incarnations of the universal [[first Chern class]] (equivalently: the [weights](group+character#RelationToChernRootsAndSplittingPrinciple) of the [[group characters]] of $U(n)$).
It follows that every [[complex vector bundle]] $\xi$ of [[rank]] $n$ over a space $X$ when pulled back to its [[flag space]] bundle decomposes as a [[direct sum]] of [[complex line bundles]] $\zeta_i$ and has [[Chern classes]] $c_k$ expressed in terms of the [[first Chern classes]] of these line bundles as

$$
  c_k(p^\ast \xi ) = \sigma_k(c_1(\zeta_1), \cdots, c_n(\zeta_n))
  \,.
$$

This case is typically the default meaning of the "splitting principle".

For the [[special unitary group]] the situation is the same, only that here the splitting is into a sum of line bundles whose [[tensor product]] is constrained to be [[trivial vector bundle|trivializable]].

### Linear representations and Brauer induction

The [[Brauer induction theorem]] may be regarded as the splitting principle for [[linear representations]] ([Symonds 91](#Symonds91)), see also at _[[characteristic classes of linear representations]]_,

### Real vector bundles
 {#RealVectorBundles}

Under the _[Relation between Pontryagin classes and Chern classes](Pontryagin+class#FurtherRelationToChernClasses)_ the above translates into a splitting principle also for [[real vector bundles]].

### Genera and Hirzebruch characteristic series

The basic theorem of [[Hirzebruch series]] expresses [[genera]] via the splitting principle. The Hirzebruch characteristic series $K_\phi$ is a [[series]] in a single variable $x = c_1(L)$, to be thought of as the [[first Chern class]] of the universal [[complex line bundle]] over $B U(1)$.

The Hirzebruch formula for the value of the [[genus]] $\phi$ on an [[orientation|oriented]] [[manifold]] $X$

$$
  \phi(X) = \langle K_\phi(T X), [X]\rangle
$$

denotes the pairing of that class of the [[tangent bundle]] with the [[fundamental class]] which under the splitting principle pulls back on the [[flag space]] bundle to the class $\prod_k K_\phi(x_k)$ of the corresponding direct sum of line bundles.

## Related concepts

* [[Snaith's theorem]]

## References

* {#May} [[Peter May]], _A note on the splitting principle_, Topology and its Applications Volume 153, Issue 4, 1 November 2005, Pages 605-609 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Split.pdf), [doi:10.1016/j.topol.2005.02.007](https://doi.org/10.1016/j.topol.2005.02.007))

* Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 10.2 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 10 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

Discussion in the derivation of [[Chern classes]] and [[Stiefel-Whitney classes]] includes

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

See also

* Wikipedia, _[Splitting principle](https://en.wikipedia.org/wiki/Splitting_principle)_

* _Notes on the splitting principle_ ([pdf](http://www.math.sunysb.edu/~azinger/mat566/splitting.pdf))

* {#Dupont78} Johan L. Dupont, _Curvature and Characteristic Classes_, Lecture Notes in Mathematics, **640** (1978) doi:[10.1007/BFb0065364](http://dx.doi.org/10.1007/BFb0065364).

Discussion in the context of [[complex oriented cohomology theory]] and their [[generalized Chern classes]] includes

* [Kochmann 96, section 4.3](#Kochmann96)

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Section 3.3 and Theorem 3.22 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* [[Jacob Lurie]], lecture 4 of _[[Chromatic Homotopy Theory]]_, 2010 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf))

More expository discussion in the context of [[characteristic classes]] with applications in [[mathematical physics]] is in

* {#Zhang11} [[Yang Zhang]], _A brief introduction to characteristic classes from the differentiable viewpoint_, 2011 ([pdf](http://www.nbi.dk/~zhang/notes/A brief introduction to characteristic classes from the differentiable viewpoint.pdf))

The generalization to a splitting principle for [[twisted vector bundles]] ([[twisted cohomology]]) is discussed (in terms of [[bundle gerbe modules]]) in

* {#Tomoda07} [[Atsushi Tomoda]], _On the splitting principle of bundle gerbe modules_, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. ([Euclid](https://projecteuclid.org/euclid.ojm/1174324334), talk slides [pdf](http://ton.prosou.nu/official/ryousi2005.pdf))

For [[characteristic classes of linear representations]]:

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

[[!redirects Chern root]]


spontaneously broken symmetry

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

#Contents#
* table of contents
{:toc}

## Idea

For $Conf$ a [[configuration space]] and

$$
  S : Conf \to \mathbb{R}
$$

an [[action functional]] that is invariant under a [[group]] $G$ of symmetries [[action|acting]] on $Conf$, in that

$$
  \forall g \in G, \phi \in Conf : \,\,\, S(g(\phi)) = S(\phi)
$$

a solution $\phi_0$ to the [[Euler-Lagrange equations]] of motion is said to exhibit **spontaneously broken symmetry** if it is not a fixed-point of that group [[action]]: if there is $g \in G$ such that $g(\phi_0) \neq \phi_0$.

The "breaking" refers to the fact that the group no longer acts. It is called "spontaneous" because one imagines that by a physical process the theory "finds" one of its solutions. This comes from the class of examples where a statistical system is first considered at high temperature and then cooled down. At some point it will "spontaneously" freeze in one allowed configuration. A standard example is a [[ferromagnet]]: at high temperature its [[magnetization]] vanishes, while at very low temperature it spontaneously finds a direction of magnetization, thus "breaking" rotational symmetry.

One calls the subgroup $G_{\phi_0} \subset G$ that fixes the given configuration $\phi_0$ the _unbroken symmetry group_ .

In the context of the [[quantum field theory]] arising by [[quantization]] of this action functional one considers the given classical solution $\phi_0$ as a background about which to consider [[perturbation theory|perturbations]] of the remaining [[effective quantum field theory]].

The fields in this effective QFT are then small excitations $\delta \phi$ around the given $\phi_0$. Since the original symmetry group still acts on the full fields $\phi_0 + \delta \phi$, the remaining symmetry group of the effective field theory is $G_{\phi_0}$, whose elements $g$ send

$$
  g : (\phi_0 + \delta \phi) \mapsto g(\phi_0) + g(\delta \phi)
  = \phi_0 + g \delta \phi
 \,.
$$

Since in the effective theory around $\phi_0$ the [[vacuum]] state where all the $\delta \phi$ have no excitations (or rather: are in their ground state) corresponds to $\phi_0$ itself one says in this context that _a quantum theory exhibits spontaneouly broken symmetry if its vacuum state is not invariant under the pertinent symmetries_ .

## Formalization in cohesive homotopy-type theory
 {#FormalizaitonInCohesion}

We indicate the formalization of the concept in the axiomatics of [[cohesion]].

Let $\mathbb{G}$ be a cohesive [[abelian infinity-group]] (for instance $\mathbb{G}$ the [[circle group]] $U(1)$ in [[smooth infinity-groupoid|smooth cohesion]]).

Then a [[prequantum line bundle]] on a [[phase space]] $P$ is given by a [[modulating morphism]]

$$
  P \longrightarrow \mathbf{B}\mathbb{G}_{conn}
$$

to the [[moduli stack]] $\mathbf{B}\mathbb{G}_{conn}$ of $\mathbb{G}$-[[principal connections]].

A symmetry of the theory means that there is a cohesive [[infinity-group]] $G$ [[infinity-action|infinity-acting]] on $P$ such that the prequantum bundle descends to the [[homotopy quotient]]

$$
  \array{
     P  &\longrightarrow& \mathbf{B}\mathbb{G}_{conn}
     \\
     \downarrow & \nearrow
     \\
     P/G
   }
$$

Now given an [[infinity-action]] of $\mathbb{G}$ on some $V$ (take $\mathbb{G} = U(1)$ and $V = \mathbb{C}$ for traditional quantum mechanics) then a [[quantum state]] (a [[wavefunction]]) is a [[section]] of the [[associated infinity-bundle]], hence a diagram of the form

$$
   \array{
         P &\stackrel{\Psi}{\longrightarrow}& V/\mathbb{G}
         \\
         \downarrow &\swArrow_{\simeq}& \downarrow
         \\
         \mathbf{B}\mathbb{G}_{conn}
         &\longrightarrow&
         \mathbf{B}\mathbb{G}
    }
$$

So this is something defined on phase space $P$. If that also descends to the [[homotopy quotient]] $P/G$ (this is hard to draw here, but should be clear) then that makes the wavefunction also $G$-equivariant. If not, then the wavefunction $\Psi$ "breaks" the $G$-symmetry.

Now if on top of this we have that the given $\Psi$ is a "ground state", then if it does not descend to the homotopy quotient we say "the $G$-symmetry is spontaneously broken".

To axiomatize what "ground state" means: introduce another $\mathbb{R}$-action on $P$ which is a [[Hamiltonian action]], i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask $\Psi$ to  (be [[polarization|polarized]] and) be a minimal eigenstate of the respective Hamiltonian. That makes it a "ground state".

For more on the general translation between traditional [[geometric quantization]] and [[cohesion|cohesive]] homotopy theory see at _[[schreiber:Higher geometric prequantum theory]]_.

## Examples

### Scalars in mexican hat potential

A standard example which is both very simple but at the same time of central importance in one of the main applications in the [[standard model of particle physics]] -- the [[electroweak symmetry breaking]] via the [[Higgs mechanism]] -- is this:

Let $Conf = C^\infty(X = [0,1]^d, \mathbb{R}^N)$ be the configuration space of $N$ real scalar fields and take the action functional to be

$$
  S : \phi \mapsto \int_X \left( -\frac{1}{2}\vert \nabla \phi \vert^2
  -
  \frac{h}{2} \vert\phi\vert^2 - \frac{g}{4} \vert\phi\vert^4
  \right) d\mu_X
$$

for some $h, g \in \mathbb{R}$. This is manifestly invariant under the canonical action of the [[orthogonal group]] $G = O(N)$ on $Conf$.

This action functional has a class of critical points given by constant maps $\phi : X \to \mathbb{R}^n : \phi(x) = \Phi$. These extremize the [[action functional]] precisely if the $\Phi$ extremize the potential energy

$$
  \frac{h}{2} \vert\phi\vert^2 + \frac{g}{4} \vert\phi\vert^4
  \,.
$$

If both $g$ and $h$ are positive, then there is only one such critical point, given by $\Phi = 0$.  Therefore in this case the unique constant solution does _not_ break the symmetry, in that $g( \Phi = 0) = (\Phi = 0)$ for all $g \in O(N)$.

However, if $h$ is negative and $g$ positive, then the solutions are all those $\Phi$ with

$$
  \vert \Phi \vert^2 = - \frac{h}{g} \gt 0
  \,.
$$

The set of all these is closed under the action of $G = O(N)$ -- this group takes one of these solutions into another -- but none of these solutions is _fixed_ by this action.

One says in this case that any such solution $\phi : x \mapsto \Phi$ is a solution that _spontaneously breaks the symmetry_ of the theory.

### In gravity

The theory of [[gravity]] on a given [[topological manifold]] $X$ has as
[[configuration space|configurations]] [[pseudo-Riemannian metric]]s on $X$ and its [[action functional]] -- the [[Einstein-Hilbert action]] or one of its variants -- is invariant under the [[action]] of the [[diffeomorphism]] group on $X$.

The corresponding [[Euler-Lagrange equation]]s are _[[Einstein's equations]]_. A given solution $(X,g)$ _breaks_ the symmetry given by a [[diffeomorphism]] $f : X \to X$ unless $f$ is an [[isometry]]. This means that the unbroken symmetries connected to the identity correspond precisely to the [[Killing vector field]]s on $(X,g)$.

#### Kaluza-Klein reductions

Spontaneous symmetry breaking in [[gravity]] plays a central role for instance in the context of the [[Kaluza-Klein mechanism]]. For instance for $dim X = 5$ the [[effective field theory]] of gravity around a solution of the form $(X = X_4 \times S^1, g_4 \otimes g_1)$ is 4-dimensional gravity coupled to [[electromagnetism]] (and a [[dilaton]] field): the components of the field of gravity along the circle transmute into the electromagnetic field. The ansatz _breaks_ all the symmetries that would mix the remaining 4-dimensional gravitational excitations with these new electromagnetic excitations.

This is discussed in a bit of detail for instance in
([Strominger, lecture 1](#Strominger)).

#### Super Kaluza-Klein reductions

The above discussion has a direct analog in theories of higher
[[supergravity]]. By the same logic, one finds that the
[[effective quantum field theory]] around classical solutions that are
[[Kaluza-Klein mechanism|Kaluza-Klein reductions]] of the form
$(X^4 \times Y^d, (g_4 \otimes g_d))$ exhibits as global symmetries all
those [[diffeomorphism]]s that are not _spontaneously broken_ by this solution.

In this case, though, there are also [[supersymmetry]] analogs of the plain diffeomorphism action. Such a local supersymmetry remains unbroken in the given solution if it comes from a [[Killing spinor field]].

Therefore KK-reductions to 4-dimensional [[Minkowski space]] in supergravity that admit precisely four Killing spinors of the form $(\psi_4 = const \otimes \psi_d = covariantly const)$ give rise to [[effective field theories]] with exactly one remaining global [[supersymmetry]]. See also at _[[supersymmetry breaking]]_.

For more see [[supersymmetry and Calabi-Yau manifolds]].

This is discussed in a bit of detail for instance in
([Strominger, lecture 2](#Strominger)).

#### Scherk-Schwarz mechanism

Specifically, the _[[Scherk-Schwarz mechanism]]_ ([Scherk-Schwarz 79](#ScherkSchwarz79)) is the [[spontaneously broken symmetry|spontaneous]] [[supersymmetry breaking]] by [[KK-compactification]] on a [[circle]] whose [[spin structure]] imposes anti-periodic [[boundary conditions]] for [[fermion fields]].

## Related concepts

* [[gauge group]]

  * [[global gauge group]], [[local gauge group]]

* [[Goldstone boson]]

* [[QFT with defects|topological defects]] in the [[vacuum]]

  * [[domain wall]], [[cosmic string]], [[monopole]]

* [[enhanced gauge symmetry]]

## References

In

* [[Steven Weinberg]], _The quantum theory of fields_

sponaneously broken [[global gauge group]] symmetry is discussed in vol I, section 19, and spontaneously broken [[local gauge group]] symmetry in vol I, section 21.4.

Survey:

* Jose Bernabeu, _Symmetries and their breaking in the fundamental laws of physics_ ([arXiv:2006.13996](https://arxiv.org/abs/2006.13996))

Textbook discussion of broken symmetry in [[gravity]] and [[supergravity]] in the context of the [[Kaluza-Klein mechanism]] is in

* {#Strominger} [[Andrew Strominger]] (notes by [[John Morgan]]), _Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces_ , volume II, starting on page 1091 in

  [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,
[[David Kazhdan]],  [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.  , _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

Discussion of spontaneous [[supersymmetry breaking]] is in

* Yael Shadmi, _Supersymmetry breaking_ ([arXiv:hep-th/0601076](http://arxiv.org/abs/hep-th/0601076))

* [[Joseph Polchinski]], volume II, appendix B of _[[String theory]]_

The article

* [[Klaas Landsman]], Robin Reuvers, _A Flea on Schrödinger's Cat_,  Found. Phys. 43, 373-407 (2013) ([arXiv:1210.2353](http://arxiv.org/abs/1210.2353), [pdf](http://www.math.ru.nl/~landsman/Catresubmission.pdf))

points out that for symmetric systems with a symmetric ground state, already a tiny perturbation mixing the ground state with the first excited stated causes spontaneous symmetry breaking in the suitable limit, and suggests that this already resolves the [[measurement problem]] in [[quantum mechanics]].

[[!redirects spontaneously broken symmetries]]

[[!redirects spontaneous symmetry breaking]]

[[!redirects broken symmetry]]
[[!redirects broken symmetries]]

[[!redirects symmetry breaking]]


sporadic finite simple group

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## Idea

The sporadic finite simple groups are the [[exceptional structures]] among [[finite groups]]:

According to the [[classification of finite simple groups]], there are 18 [[countable set|countably infinite families]] and 26 _sporadic_ simple groups. The latter groups do not fit into any systematic classification, but there are a number of links between them. For example, the [[Monster group]] $M$, the largest of the sporadic groups, contains all but six (the '[[pariah groups]]', $J_1, J_3,Ru,ON,Ly,J_4$) of the other sporadic groups as [[subquotients]]. These 20 sporadic groups comprise what is termed the '[[Happy Family]]' by [[Robert Griess]], ([Griess 98](#Griess98)).

The _Happy Family_ is taken to be formed of three generations

* First: five subquotients of the [[Mathieu group]] $M_{24}$ = symmetries of the [[binary Golay code]]
* Second: seven subquotients of the [[Conway group]] $Co_1$ = symmetries of the [[Leech lattice]], $mod \{\pm 1\}$.
* Third: eight subquotients of the [[Monster group]] = conformal symmetries of the [[monster vertex operator algebra]]

## Examples

* [[Conway group]]

* [[monster group]]

* [[Mathieu group]]

## Related concepts

* [[Moonshine]]

* [[automorphisms of super vertex operator algebras]]

## References

* [[Robert A. Wilson]], _The Finite Simple Groups_. Springer, Graduate Mathematics series **251** (2009).

* {#Griess98} [[Robert Griess]], _Twelve Sporadic Groups_, Springer, 1998 ([doi:10.1007/978-3-662-03516-0](https://link.springer.com/book/10.1007/978-3-662-03516-0))

[[!redirects sporadic finite simple groups]]
[[!redirects sporadic simple group]]

[[!redirects exceptional finite group]]
[[!redirects exceptional finite groups]]

[[!redirects sporadic group]]
[[!redirects sporadic groups]]


Springer correspondence

The Springer theory is an important approach to the representation theory of [[Weyl group]]s or semisimple algebraic groups. In particular, it features so called Springer representations, which are associated to the unipotent conjugacy classes with an additional parameter, via so called Springer correspondence.

* wikipedia [Springer correspondence](http://en.wikipedia.org/wiki/Springer_correspondence)
* T. A. Springer, _A construction of representations of Weyl groups_, Invent. Math. __44__ (1978), no. 3, 279–293 [MR0491988](http://www.ams.org/mathscinet-getitem?mr=0491988) [doi](http://dx.doi.org/10.1007%2FBF01403165)
* N. Chriss, V. Ginzburg, _Representation theory and complex geometry_, Birkhäuser 1997. x+495 pp.

Kazhdan and Lusztig geometrically realized the Springer correspondence via the study of the [[Steinberg variety]] placing its study within the [[geometric representation theory]]:

* [[David Kazhdan]], [[George Lusztig]], _A topological approach to Springer's representations_, Adv. in Math. __38__ (1980), no. 2, 222–228 [MR82f:20076](http://www.ams.org/mathscinet-getitem?mr=597198) [doi](http://dx.doi.org/10.1016/0001-8708(80)90005-5)
* G. Lusztig, N. Spaltenstein, _On the generalized Springer correspondence for classical groups_, Advanced Studies in Pure Mathematics __6__ (1985) 289–316.
* N. Spaltenstein, _On the generalized Springer correspondence for exceptional groups_, Advanced Studies in Pure Mathematics __6__ (1985) 317–338.
* Ryoshi Hotta, _On Springer's representations_, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 863–876 (1982) [MR83h:20038](http://www.ams.org/mathscinet-getitem?mr=656061)
* D. Clausen, _Springer correspondence_, Harvard thesis, [pdf](http://www.math.harvard.edu/theses/senior/clausen/clausen.pdf)
* Mikhail Grinberg, _A generalization of Springer theory using nearby cycles_, Represent. Theory __2__ (1998), 410-431 [pdf](http://www.ams.org/journals/ert/1998-002-12/S1088-4165-98-00053-3/S1088-4165-98-00053-3.pdf)

* Julia Sauter, _A survey on Springer theory_, [arxiv/1307.0973](http://arxiv.org/abs/1307.0973)
* Gwyn Bellamy, [[Travis Schedler]], _Kostka polynomials from nilpotent cones and Springer fiber cohomology_, [arxiv/1509.02520](https://arxiv.org/abs/1509.02520)

Some higher categorical aspects are more explicitly present in the works of Nadler and coworkers:

* [[David Nadler]], _Springer theory via the [[Hitchin fibration]]_, [arxiv/0806.4566](http://arxiv.org/abs/0806.4566)

* [[David Ben-Zvi]], David Nadler, _Elliptic Springer theory_, [arxiv/1302.7053](http://arxiv.org/abs/1302.7053)

Elements of the analogue of Springer theory for [[global field]]s are in

* Zhiwei Yun, _Towards a Global Springer Theory I: The affine Weyl group action_, [arxiv/0810.2146](http://arxiv.org/abs/0810.2146)

[[!redirects Springer theory]]
[[!redirects Springer's theory]]
[[!redirects Springer's representation]]
[[!redirects Springer representation]]


Spyridon Michalakis

* [GoogleScholar page](https://scholar.google.se/citations?user=NqXEDpcAAAAJ&hl=sv)

## Selected writings

Providing rigorous theoretical foundations for the [[quantum Hall effect]]:

* [[Matthew Hastings]], [[Spyridon Michalakis]], _Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions_, Commun. Math. Phys., 334:433–471, (2015) ([arXiv:0911.4706](https://arxiv.org/abs/0911.4706), [doi:10.1007/s00220-014-2167-x](https://doi.org/10.1007/s00220-014-2167-x))

Review:

* [[Spyridon Michalakis]], _Why is the Hall conductance quantized?_, Nature Reviews Physics volume 2, pages 392–393 (2020) ([doi:10.1038/s42254-020-0212-6](https://doi.org/10.1038/s42254-020-0212-6))

category: people


square

> This entry is about squares in [[geometry]]. For squares in [[category theory]] see *[[commutative square]]*. For squares in [[ring theory]], see *[[square function]]*. For squares in [[type theory]], see *[[square type]]*.

***

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## Idea

As a [[polygon]] the square is the regular 4-gon.

As a [[topological space]] it is the [[product topological space]] of the [[bounded space|bounded]] [[closed interval]] with itself, $[0,1]^2$ .

## Quotients

The identification of opposite sides of the square yields the [[cylinder]] and the [[torus]].

And the identification with opposite [[orientation]] yields the [[Möbius strip]].

> graphics grabbed from [Lawson 03](#Lawson03)

## Related concept

* [[line]]

* [[plane]]

* [[circle]]

* [[triangle]]

* [[tetrahedron]]

* [[Peano curve]]

## References

* {#Lawson03} Terry Lawson, _Topology: A Geometric Approach_, Oxford University Press (2003) ([pdf](http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf))

[[!redirects squares]]


square integrable function

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## Idea

A [[function]] such that its [[square]] has a well-defined [[integration]] against a given [[measure]].

## Related concepts

* [[L-2 space]]

* [[continuous function]], [[differentiable function]], [[smooth function]]

* [[integrable function]], [[square-integrable function]]

* [[bounded function]]

* [[compactly supported function]]

* [[measurable function]]

* [[rapidly decreasing function]]

[[!redirects square integrable functions]]

[[!redirects square-integrable function]]
[[!redirects square-integrable functions]]


square matrix

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## Idea

A _square matrix_ is an $n \times m$-[[matrix]] for $n = m$.

Under [[matrix multiplication]] the $n \times n$ square matrices for fixed $n \in \mathbb{N}$ with values in some [[commutative ring]] $R$ form the _[[matrix algebra]]_ $Mat_n(R)$, inside which one finds [[matrix groups]] etc.

## Properties

### Gershgorin circle theorem

See at *[[Gershgorin circle theorem]]*.

## Related concepts

* [[diagonal matrix]]

* [[inverse matrix]]

* [[matrix algebra]]

* [[matrix group]]

[[!redirects square matrices]]


square root

# Square roots
* table of contents
{: toc}

## Definitions

If $K$ is a [[magma]], such as a [[monoid]], (which we write multiplicatively) and $x$ is an element of $K$, then the element $x^2$ is the __square__ of $x$.  Conversely, if $x^2 = y$, then $x$ is a __square root__ of $y$.

If $K$ is an [[integral domain]], then (in [[classical mathematics]]) $x$ and $-x$ are the only square roots of $x^2$.  If $y$ has a square root, then we often denote its square roots together as $\pm\sqrt{y}$, although there is no meaning of $\sqrt{y}$ itself.

If $K$ is a [[linearly ordered field]], then every element $y$ has a unique nonnegative square root if it has a square root at all; this is the __principal square root__ of $y$ and denoted $\sqrt{y}$.

## In constructive analysis

In [[constructive mathematics]] (here specifically: [[constructive analysis]]), it is not provable that $x$ and $-x$ are the only square roots of $x^2$.  In the [[ordered field]] of [[real numbers]], for example, the [[absolute value]] ${|x|}$ (like $-{|x|}$, for that matter) is also a square root of $x^2$, yet it is not constructively provable that ${|x|} = x$ or ${|x|} = -x$.  Without using the [[lesser limited principle of omniscience]], if $x$ is close to [[zero]] (and we do not yet know whether it is exactly zero), we cannot decide whether $x$ is nonnegative (so that ${|x|} = x$) or nonpositive (so that ${|x|} = -x$).

However, in any [[linearly ordered field]] with an [[absolute value]] (including any [[real-closed field]]), we still have a unique [[nonnegative number|nonnegative]] square root of $x^2$, which is in fact ${|x|}$.  Thus, we can still use the notation $\sqrt{y}$, but we cannot prove that every square root of $y$ is one of $\pm\sqrt{y}$.  However, we can prove, in any integral domain even, that if $x \neq \sqrt{y}$ and $x \neq -\sqrt{y}$, then $x^2 \neq y$.  (We are using the weak notions of field and integral domain so that $\mathbb{R}$ will be an example.)

If we accept results in the [[complex numbers]] (or in $K[\mathrm{i}]$ for $K$ any real-closed field), then every element of $K$ has a square root.  Even if $y$ is close to zero, we can still use
$$ \sqrt{y} \coloneqq \sqrt{\frac{{|y|} + y}2} + \sqrt{\frac{{|y|} - y}2}\mathrm{i} $$
to construct a square root of $y$.  However, we *cannot* prove that every complex number has a square root in the [[internal language]] of an arbitrary [[W-topos]], although [[weak countable choice]] is enough to prove this.  (See [Richman (1998)](#Richman).)  A counterexample is the topos of [[sheaves]] on the [[real line]], since there is no [[continuous map|continuous]] function $\sqrt{}\colon U \to \mathbb{C}$ for $U$ any [[neighbourhood]] of $0$ in $\mathbb{C}$.  This is important for interpreting the [[quadratic formula]].

+-- {: .query}
How much choice is needed to prove that every element of $K[\mathrm{i}]$ has a square root, where $K$ is *any* real-closed field?  Maybe [[COSHEP]] will do?
=--

Note that there is never any trouble finding a square root of $y$ if we assume that $y \neq 0$, nor (obviously) is there any trouble if we assume that $y = 0$.  Accordingly, the classical results hold for [[discrete fields]] and [[integral domain|discrete integral domains]], but this doesn\'t apply constructively to analysis.

## Related concepts

* [[root]]

* **square root**

* [[root of unity]]

* [[real square root function]]

* [[Euclidean field]]

## References

* [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*. Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]

More generally, square roots of [[positive operator|positive]] [[self-adjoint operators]]:

* Zoltán Sebestyén, Zsigmond Tarcsay, *On the square root of a positive selfadjoint operator*, Period Math Hung **75**  (2017) 268–272 [[doi:10.1007/s10998-017-0192-1](https://doi.org/10.1007/s10998-017-0192-1)]

[[!redirects square root]]
[[!redirects square roots]]

[[!redirects principal square root]]
[[!redirects principal square roots]]


square roots of line bundles - table

The following table lists classes of examples of [[square roots]] of [[line bundles]]

| line bundle |  square root | choice corresponds to |
|---|---|--|
| [[canonical bundle]] | [[Theta characteristic]] | over [[Riemann surface]] and [[Hermitian manifold]] (e.g.[[Kähler manifold]]): [[spin structure]] |
| [[density bundle]] | [[half-density bundle]] |  |
| [[canonical bundle]] of [[Lagrangian submanifold]] | [[metalinear structure]] | [[metaplectic correction]] |
| [[determinant line bundle]] | [[Pfaffian line bundle]] |  |
| quadratic [[secondary intersection pairing]] | [[partition function]] of [[self-dual higher gauge theory]] | [[integral Wu structure]] |


square type

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## Idea ##

In [[homotopy type theory]] a **square type** is the [[product type]] of the [[interval type]] with itself.

## See also ##

* [[square]]

* [[interval]], [[interval type]]

[[!redirects square types]]


square-free

* [[square-free integer]]

* [[square-free polynomial]]

* [[square-free element]]

category: disambiguation


square-free element

\tableofcontents

## Idea

The generalization of [[square-free integer]] and [[square-free polynomial]] to arbitrary [[unique factorization domains]].

## Definition

Let $R$ be a [[unique factorization domain]]. An element $r \in R$ is **square-free** if for all [[irreducible elements]] $x \in R$, $r$ is not in the [[ideal]] $x^2 R$.

## Examples

* A [[square-free integer]] is a square-free element in the integers $\mathbb{Z}$.

* Let $F$ be a [[discrete field]] and let $\overline{F}$ be the [[algebraic closure]] of $F$. A univariate [[square-free polynomial]] with coefficients in $\overline{F}$ is a square-free element in the polynomial ring $\overline{F}[x]$.

## Properties

Given a [[unique factorization domain]] $R$ such that for every [[irreducible element]] $p \in R$, the [[ideal]] $p R$ is a [[maximal ideal]], and a square-free element $x \in R$, the [[quotient ring]] $R/x R$ is a [[reduced ring]]. Since for every non-zero element $r \in R$, $R/x R$ is a [[prefield ring]], $R/x R$ is an [[integral domain]] and thus a [[field]] if and only if $x$ is an [[irreducible element]] in $R$.

## Square-free factorization

Given a non-zero element of a [[unique factorization domain]] $r \in R$, the **square-free factorization** or **square-free decomposition** of $r$ is a factorization of $r$ into powers of square-free elements $r_i$

$$r = \prod_{i = 0}^{n} r_i^i$$

where each of the elements $r_i$ are pairwise [[coprime]] and not in the [[group of units]].

## See also

* [[square-free integer]]

* [[square-free polynomial]]


square-free number

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## Idea

A square-free number is a [[natural number]] $n$ such that for all [[prime numbers]] $p$, $p^2$ does not divide $n$.

## Properties

Given a square-free number $n$ with prime factors
$$n = \prod p_i$$
the [[completion of a ring|completion]] of the [[sequence]] of finite cyclic rings $\mathbb{Z}/n^i\mathbb{Z}$ is the ring $\mathbb{Z}_n$, which is a subring of the [[profinite integers]] and the [[product]] of the [[p-adic integers]] of the prime factors of $n$:
$$\mathbb{Z}_n \coloneqq \prod \mathbb{Z}_{p_i}$$

Given a square-free integer $n$, the [[integers modulo n]] $\mathbb{Z}/n\mathbb{Z}$ is a [[reduced ring]]. Since for every integer $n$ the  $\mathbb{Z}/n\mathbb{Z}$ is a [[prefield ring]], $\mathbb{Z}/n\mathbb{Z}$ is an [[integral domain]] and thus a [[field]] if and only if $n$ is a [[prime number]].

Given any odd prime number $p$, the integer $2 p$ is square-free, and the ring $\mathbb{Z}/2 p\mathbb{Z}$ has [[idempotent elements]] of $0$, $1$, $p$, and $p + 1$. Every element of $\mathbb{Z}/2 p\mathbb{Z}$ could be written as a [[linear combination]] of $p$ and $p + 1$.

## See also

* [[prime number]]

* [[prime power]]

* [[square-free]]

* [[square-free polynomial]]

## External links

* Wikipedia, [Square-free integer](https://en.m.wikipedia.org/wiki/Square-free_integer)

[[!redirects square-free integer]]


square-free polynomial

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## Idea

Given a [[discrete field]] $F$, let $\overline{F}$ denote its [[algebraic closure]]. A *square-free polynomial* is a univariate [[polynomial]] $q \in \overline{F}[x]$ such that for all [[elements]] $a \in \overline{F}$, $(x - a)^2$ does not divide $q$.

## Properties

Given a [[discrete field]] $F$, let $\overline{F}$ denote its [[algebraic closure]]. Given a square-free polynomial $q \in \overline{F}[x]$, the [[quotient ring]] $\overline{F}[x]/q\overline{F}[x]$ is a [[reduced ring]]. Since for every polynomial $q \in \overline{F}[x]$, $\overline{F}[x]/q\overline{F}[x]$ is a [[prefield ring]], $\overline{F}[x]/q\overline{F}[x]$ is an [[integral domain]] and thus a [[field]] if and only if $q$ is a prime polynomial in $\overline{F}[x]$, a [[monic polynomial]] of degree one; the resulting quotient ring is equivalent to $\overline{F}$.

## See also

* [[square-free number]]

## External links

* Wikipedia, *[Square-free polynomial](https://en.m.wikipedia.org/wiki/Square-free_polynomial)*

[[!redirects square-free polynomials]]


squark

[[!redirects squarks]]

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## Idea

The (hypothetical) [[superpartners]] of [[quarks]] are called _squarks_; according to the general naming convention for [[sfermions]].

Experimental exclusion bounds of gluino [[rest masses]] due to the [[LHC]] [[experiment]] excludes gluinos/[[squarks]] of mass below about 3-4 [[TeV]] ([Particle Data Group Review 17, figure 113.2 and 113.9](#ParticleDataGroup17)):

[[LHCSuperpartnerMassExclusion2017.png:file]]

## Related concepts

* [[split supersymmetry]]

## References

* {#ParticleDataGroup17} [[Particle Data Group]] [Review 2017](http://pdg.lbl.gov/), 113. _Supersymmetry , part II (experiment)_ ([pdf](http://pdg.lbl.gov/2017/reviews/rpp2017-rev-susy-2-experiment.pdf))

See also:

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


squash type

\tableofcontents

## Idea

A version of [[propositional truncation]] which takes types and turns them into [[strict propositions]].

## Definition

### Direct definition

Provided that [[function types]] and [[dependent function types]] have [[judgmental equality|judgmental]] [[computation rules]], given a [[type of strict propositions]] $\mathrm{sProp}$, the squash type could be defined in the same manner as [[propositional truncations]]:

$$[A]_s \coloneqq \prod_{P:\mathrm{sProp}} (A \to P) \to P$$

### Inference rules

Without a [[type of strict propositions]] $\mathrm{sProp}$, or if either [[function types]] or [[dependent function types]] only have [[typal equality|typal]] [[computation rules]], the squash type would have to be defined  using [[inference rules]].

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash [A]_s \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:[A]_s, y:[A]_s \vdash x \equiv y:[A]_s} \qquad \frac{\Gamma \vdash x:A}{\Gamma \vdash \mathrm{squash}(x):[A]_s}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash P \; \mathrm{type} \quad \Gamma, p:P, q:P \vdash p \equiv q:P \quad \Gamma \vdash f:A \to P \quad \Gamma \vdash x:[A]_s}{\Gamma \vdash \mathrm{unsquash}_P(f, x):P}$$

Similarly to [[propositional truncations]], the recursion rule for squash types is sufficient to define the induction rule for squash types.

## Related concepts

* [[judgmental equality]]

* [[propositional truncation]]

* [[strict proposition]]

* [[type of strict propositions]]

## References

* [[Gaëtan Gilbert]], [[Jesper Cockx]], [[Matthieu Sozeau]], [[Nicolas Tabareau]], *Definitional Proof-Irrelevance without K*. Proceedings of the ACM on Programming Languages, Volume 3, Issue POPL, Article 3, pp 1–28, January 2019. ([doi:10.1145/3290316](https://doi.org/10.1145/3290316))

[[!redirects squash type]]
[[!redirects squash types]]


Sridhar Ramesh

I was a graduate student in the Logic program at Berkeley, broadly interested in categorical logic and foundations of mathematics, as well as in applications of category theory to the semantics of programming languages.

I left academia for industry in 2013, but I do still have many of the same interests. From 2021 to 2023, I worked in my free time to write my previously unwritten dissertation, and thus I finished the PhD I had originally left ABD.

## Selected writings

Unified account of [[Löb's theorem]]/[[Gödel's second incompleteness theorem]], [[Kripke semantics]] for [[provability logic]], and [[guarded recursion]], via [[essentially algebraic theories]] which "self-internalize":

* [[Sridhar Ramesh]], *Introspective Theories and Geminal Categories*, PhD thesis, Berkeley (2023) [[escholarship:3mn0c475](https://escholarship.org/uc/item/3mn0c475), [[Ramesh-IntrospectiveTheories.pdf:file]]]

category: people


Srikanth Iyengar

* [webpage](https://www.math.utah.edu/~iyengar/)

category: people


Srinivasa Ramanujan

[[!redirects Ramanujan]]

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

On Ranamujan's results as appearing in [[string theory]], [[black hole thermodynamics]] and [[moonshine]]:

* Jeffrey A. Harvey, _Ramanujan's influence on string theory, black holes and moonshine_ ([arXiv:1909.11477](https://arxiv.org/abs/1909.11477))

## Selected writings

On the [[asymptotic expansion]] of the [[partition function (number theory)|partition function]] in [[number theory]]:

* [[Godfrey Hardy]], [[Srinivasa Ramanujan]], *Asymptotic formulaæ in combinatory analysis*, Proceedings of the London Mathematical Society, 2(1):75–115, 1918 ([doi:10.1112/plms/s2-17.1.75](https://doi.org/10.1112/plms/s2-17.1.75))

category: people


SSet > history

< [[SSet]]
[[!redirects SSet -- history]]


SSet Cat

#Contents#
* table of contents
{:toc}

## Idea

**$SSet Cat$** or similar may denote the [[bicategory]] of [[sSet-enriched categories]].

## Related concepts

* [[category of enriched categories]]

category: category

[[!redirects sSet Cat]]


sSet-site

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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **$sSet$-site** is the incarnation of the notion of [[(∞,1)-site]] when [[(∞,1)-categories]] are incarnated as [[simplicially enriched categories]].

## Definition

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

An **$sSet$-site** is a [[simplicially enriched category]] $C$ together with the structure of a [[site]] on its [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(C)$.

=--

This appears as ([Toën & Vezzosi 005, def. 3.1.1](#ToënVezzosi05))

## Properties

### Relation to $(\infty,1)$-sites

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

Under the identification of [[simplicially enriched categories]] with models for [[(∞,1)-categories]], $sSet$-sites correspond to [[(∞,1)-site]]s.

=--

Because, as discussed at [[(∞,1)-site]], that is equivalently an [[(∞,1)-category]] equipped with the structure of a site on its [[homotopy category of an (∞,1)-category]].

### Relation to $(\infty,1)$-toposes

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

For $C$ an $sSet$-site, the local [[model structure on sSet-presheaves]] is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-topos]] $Sh_\infty(C)$ over the [[(∞,1)-site]] corresponding to $C$

$$
 ([C^{op}, sSet]_{loc})^\circ \simeq Sh_\infty(C)
  \,.
$$

=--

## Examples

* [[simplicial Stein site]]

## Related concepts

* [[site]]

* [[2-site]]

* [[(∞,1)-site]]

  * [[model site]], **simplicial site**

  * [[model topos]]

## References

* {#ToënVezzosi05} [[Bertrand Toën]], [[Gabriele Vezzosi]],  p. 4 and Def. 3.1.1 of:  _Homotopical Algebraic Geometry I: Topos theory_, Advances in Mathematics **193** 2 (2005) 257-372 [[arXiv:0207028](http://arxiv.org/abs/math/0207028)]

[[!redirects SSet-site]]
[[!redirects simplicial site]]
[[!redirects SSet-sites]]
[[!redirects simplicial sites]]

[[!redirects simplicially enriched site]]
[[!redirects simplicially enriched sites]]


Stab(infinity,1)Cat

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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
=--
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
=--
=--
=--

_$Stab(\infty,1)Cat$_ denotes the [[(∞,1)-category]] of [[stable (∞,1)-categories]].

category: category

[[!redirects Stab(∞,1)Cat]]


stability

Stability is a highly overloaded word in mathematics.

* property of a morphism can be [[pullback stability|stable under pullbacks]]

* the equations (for example differential equations) may depend on parameters like initial condition. If the parameters change a little bit, the solution may change more or less radically. Thus one can talk about stability of solutions of differential equations under perturbations. This is also related to the notion of stable equilibria in physics and engineering.See wikipedia [stability theory](http://en.wikipedia.org/wiki/Stability_theory)

* the notion of stability of particles and bound states in physics: stability under decay

* in [[model theory]] there is another [[stability in model theory|theory of stability]] including a more recent [[geometric stability theory]]

* in [[geometry]], specifically in the construction of [[moduli spaces]] there is a notion of stable and semistable objects (e.g. __[[GIT-stable point|Mumford stability]]__ for vector bundles over algebraic curves and used in [[geometric invariant theory]], Takemoto's stability over nonsingular algebraic surfaces, more generally, [[Gieseker stability]] conditions for  torsion-free coherent sheaves over algebraic manifolds); they can be taken with respect to some stability data; more recently those are important in physical applications and abstract versions appeared in the context of triangulated categories, most notably [[Bridgeland stability conditions]]. See also _[[stable vector bundle]]_


stability in model theory

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### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Stability theory, also referred to a classification theory, is a means to determine whether the isomorphism types of a given sort of [[structure in model theory|structure]] can be classified by means of intelligible invariants of the structure. It was largely created by Saharon Shelah.

The basic idea is that for quite general classes of algebraic objects, one can prove what Shelah calls a "structure/nonstructure theorem": either the isomorphism types are classifiable by a smallish number of invariants, or they are hopelessly wild in some sense, e.g., an arbitrary structure can be encoded set-theoretically in some isomorphism type of the class. An example of the "structure" case is the theory of [[algebraically closed fields]], whose isomorphism types can be classified according to characteristic and [[transcendence degree]]. An example of the "nonstructure" case is the family of linear orderings, where a proliferation of complicated linear orders can be constructed by various set-theoretic means.

In very rough outline, stability theory analyzes good (or "stable") notions of "free amalgams" $M_3 = M_1 \cup_{M_0} M_2$ where $M_0 \subset M_1$, $M_0 \subset M_2$ are substructures. In the "good" (structure) case, it is possible to analyze models by a series of free amalgams of small models, with the series indexed by a well-founded tree. Otherwise, if the class of algebraic objects does not admit a suitably good notion of free amalgam, we have a "bad" (nonstructure) case which permits arbitrarily wild models to be constructed.

## Definitions

(For the time being we are recording some definitions without any effort to motivate them. That should come later.)

For now we will be interested in [[complete theory|complete theories]] $\mathbf{T}$ over a countable [[signature (in logic)|signature]]. Let $\mathbf{M}$ be a [[theory|model]] of $\mathbf{T}$ with underlying set $M$. Let $A \subseteq M$ be a subset, and let $Def_A(M^n)$ be the [[Boolean algebra]] of subsets of $M^n$ that are definable by a formula in $\mathbf{T}$ with parameters in $A$.

Recall that an [[ultrafilter]] in a Boolean algebra $B$ is the set of elements that get mapped to the top element $1$ under some Boolean algebra homomorphism $\phi: B \to \mathbf{2}$; alternatively, we could define an ultrafilter as such a homomorphism.

+-- {: .num_defn}
###### Definition
The _space of complete $n$-types_ over $A$ is the [[Stone space]] $S_n^{\mathbf{M}}(A)$ of ultrafilters in the Boolean algebra $Def_A(M^n)$. Such an ultrafilter is called a complete $n$-type. More generally, an $n$-type is a [[filter]] in $Def_A(M^n)$.
=--

+-- {: .num_prop}
###### Proposition
Suppose $i: M \to N$ is an [[elementary embedding]] from $\mathbf{M}$ to $\mathbf{N}$. Then $i$ induces an isomorphism $S_n^{\mathbf{M}}(A) \cong S_n^{\mathbf{N}}(i(A))$.
=--

+-- {: .proof}
###### Proof
It is enough to show $i$ induces an isomorphism $Def_A(M^n) \to Def_A(N^n)$. As a Boolean algebra, $Def_A(M^n)$ is the Boolean quotient of formulas with $n$ free variables $\phi(\bar{x}, \bar{a})$ modulo the equivalence relation $E(\phi, \psi) \coloneqq \phi \Leftrightarrow \psi$ is satisfied in $\mathbf{M}$. By elementary equivalence, $\mathbf{M} \models E(\phi, \psi)(\bar{x}, \bar{a})$ iff $\mathbf{N} \models E(\phi, \psi)(\bar{x}, i(\bar{a}))$, as desired.
=--

+-- {: .num_defn}
###### Definition
A complete type is _realized_ by a point $\bar{a} \in M^n$ if it is of the form

$$Def_A(M^n) \hookrightarrow P(M^n) = \mathbf{2}^{M^n} \stackrel{eval_{\bar{a}}}{\to} \mathbf{2}$$

by restricting a principal ultrafilter generated by a point $\bar{a} \in M^n$. Such a type is denoted $tp(\bar{a}/A)$.
=--

+-- {: .num_defn}
###### Definition
For an infinite cardinal $\kappa$, a model $\mathbf{M}$ is called _$\kappa$-saturated_ if for every $A \subseteq M$ with ${|A|} \lt \kappa$, any complete $n$-type is realized in $\mathbf{M}$.
=--

+-- {: .num_defn}
###### Definition
For an infinite cardinal $\kappa$, the theory $\mathbf{T}$ is _$\kappa$-stable_ if for every model $\mathbf{M} \models \mathbf{T}$ and $A \subseteq M$ with ${|A|} = \kappa$, we have ${|S_n^{\mathbf{M}}(A)|} = \kappa$. A structure $\mathbf{M}$ of a countable language is called $\kappa$-stable if the complete theory $Th(\mathbf{M})$ is $\kappa$-stable.
=--

## Related concepts

* [[geometric stability theory]]

## References

* [[M. Makkai]], _A survey of basic stability theory, with particular emphasis on orthogonality and regular types_, Israel J. Math. __49__, n.1-3 (1984), 181-238, [doi](http://dx.doi.org/10.1007/BF02760649)

* John T. Baldwin, _Fundamentals of stability theory_, Perspectives in Math. Logic vol. 12 Springer Heidelberg 1988. ([toc](http://projecteuclid.org/euclid.pl/1235414194#toc))

* Steven Buechler, _Essential Stability Theory_ , Perspectives in Math. Logic vol. 4  Springer Heidelberg 1996. ([toc](http://projecteuclid.org/euclid.pl/1235420888#toc))

* Gregory L. Cherlin, Review of _Fundamentals of stability theory_, Bull. AMS, Vol. 20 No. 2 (April 1989), 185-190.


stability of matter

##Idea

The stability of matter was for many centuries a puzzle for physicists. [[Niels Bohr]], explaining the work he did on his model of the atom, said:

>My starting point was not at all the idea that an atom is a small-scale planetary system and as such governed by the laws of astronomy. I never took things as literally as that. My starting point was rather the stability of matter, a pure miracle when considered from the standpoint of classical physics.

An important step towards a rigorous argument was provided by [[Freeman Dyson]] and [[Andrew Lenard]] ([FD68](#FD68)). [[Elliott Lieb]] refined this argument.

The question of stability was divided by Lieb into the 'first kind', which concerns the fact that there is a lower bound to the energy of an atom, and the 'second kind', which relates to the energy of macroscopic systems.

##Stability of the first kind

For an individual atom, say a hydrogen atom, there must be a reason that the positively charged nucleus does not cause the negatively charged electron to approach it indefinitely closely. The uncertainty principle, best considered in the form of Sobolov's inequality, places a lower bound on the energy of such an atom. This places a bound on the proximity of the electron to the nucleus.

##Stability of the second kind

For macroscopic systems, we need to show that the dependence of the energy on the size of the system is at most linear. The energy of twice an amount of a substance should be roughly twice the energy of the amount itself. This implies the [[extensive quantity|extensivity]] of the quantity of matter. The argument relies on the [[Pauli exclusion principle]].

##References

* {#FD68} [[Freeman Dyson]] and [[Andrew Lenard]], _Stability of matter. I and II_, Jour. Math.
Phys. 8, 423–434, (1967); ibid. Jour. Math. Phys. 9, 698–711 (1968).

* [[Elliott Lieb]] and [[Robert Seiringer]], _Stability of Matter in Quantum Mechanics_, Cambridge University Press, Cambridge, 2010.

* Jan Philip Solovej, _Stability of Matter in Quantum Mechanics, by Elliott H. Lieb and Robert Seiringer_, ([arXiv:1111.0170](http://arxiv.org/abs/1111.0170))

* Michael Loss, _Stability of Matter_, ([pdf](http://people.math.gatech.edu/~loss/papertalks/Nov14review.pdf))


stability of persistence diagrams

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The central [[theorem]] in [[persistent homology]] states how the data retained in [[persistence diagrams]]/[[barcodes]] is stable under small deformations of the initial data.

Specifically, the original form of the stability theorem ([Cohen, Steiner, Edelsbrunner & Harer 2007](#CohenSteinerEdelsbrunnerHarer07)) applies to [[persistence modules]] $V(f)_\bullet$ given by the [[connected components]] of the [[level set|sub-level]] sets of a [[continuous function]] $X \xrightarrow{f} \mathbb{R}$ on some data set $X$

$$
  V(f)_l
    \;\coloneqq\;
  H_0\Big(  f^{-1}\big( (-\infty, l] \big) \Big)
$$

(equipped with the evident inclusions) and states that as the function $f$ is deformed to another continuous function $g$, the [[bottleneck distance]] $d_B$
([CSEH07, p. 3](#CohenSteinerEdelsbrunnerHarer07))

$$
  d_B
  \big(
    X
    ,\,
    Y
  \big)
  \;\coloneqq\;
  \underset{
    X \underoverset{\sim}{\gamma}{\to} Y
  }{inf}
  \;\;
  \underset{x \in X}{sup}
  \;
  \Vert
    x - \gamma(x)
  \Vert_\infty
$$

between the corresponding [[persistence diagrams]] $PDgr\big( V(f)_\bullet \big)$  is bounded by the [[supremum norm]] of the difference between the two functions:

$$
  d_B
  \Big(
    PDgr\big( V(f)_\bullet \big)
    ,\,
    PDgr\big( V(f)_\bullet \big)
  \Big)
  \;\leq\;
  \Vert
    f - g
  \Vert_\infty
  \,.
$$

Various generalizations of this stability result exist, notably the *algebraic stability theorem* ([CCGGO09](#CCGGO09)).

From [Botnan & Lesnick 18, p. 2](#BotnanLesnick18):

> The algebraic stability theorem is perhaps the central theorem in the theory of persistent homology; it provides the core mathematical justification for the use of persistent homology in the study of noisy data. The theorem is used, in one form or another, in nearly all available results on the approximation, inference, and estimation of persistent homology.

## Related concepts

* [[well group]]

## References

The stability theorem originates in:

* {#CohenSteinerEdelsbrunnerHarer07} [[David Cohen-Steiner]], [[Herbert Edelsbrunner]], [[John Harer]], *Stability of Persistence Diagrams*, Discrete & Computational Geometry **37** (2007) 103–120 $[$[doi:10.1007/s00454-006-1276-5](https://doi.org/10.1007/s00454-006-1276-5)$]$

The algebraic stability theorem:

* {#CCGGO09} [[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)$]$

* [[Ulrich Bauer]], [[Michael Lesnick]], *Induced Matchings and the Algebraic Stability of Persistence Barcodes*, Journal of Computational Geometry **6** 2 (2015) 162-191 $[$[arXiv:1311.3681](https://arxiv.org/abs/1311.3681), [doi:10.20382/jocg.v6i2a9](https://doi.org/10.20382/jocg.v6i2a9)$]$

Further developments:

* [[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)$]$

Generalization to [[zigzag persistence modules]]:

* {#BotnanLesnick18} [[Magnus Bakke Botnan]], [[Michael Lesnick]], *Algebraic Stability of Zigzag Persistence Modules*, Algebr. Geom. Topol. **18** (2018) 3133-3204 $[$[arXiv:1604.00655](https://arxiv.org/abs/1604.00655), [doi:10.2140/agt.2018.18.3133](https://doi.org/10.2140/agt.2018.18.3133)$]$

Refinement to [[persistent homotopy]]:

* {#BlumbergLesnick17} [[Andrew J. Blumberg]], [[Michael Lesnick]], *Universality of the Homotopy Interleaving Distance* $[$[arXiv:1705.01690](https://arxiv.org/abs/1705.01690)$]$

* {#Jardine19} [[J. F. Jardine]], *Data and homotopy types* $[$[arXiv:1908.06323](https://arxiv.org/abs/1908.06323)$]$

Version for [[persistent cohomotopy]]:

* {#FranekKrcal17} [[Peter Franek]], [[Marek Krčál]], Section 5 of: _Persistence of Zero Sets_, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))

[[!redirects stability in persistent homology]]

[[!redirects bottleneck distance]]
[[!redirects bottleneck distances]]


stability theory

As there are several notions of [[stability]] in mathematics there are several notions of stability including in dynamical systems and differential equations.

This entry is about the stability theory in the sense of [[model theory]].
See also [[geometric stability theory]] and [[categoricity]].

* [[Boris Zilber]], _Elements of geometric stability theory_, lecture notes, [pdf](http://people.maths.ox.ac.uk/zilber/est.pdf)
* John T. Baldwin, _Fundamentals of stability theory_


stability under pullback

[[!redirects pullback stability]]

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###Context###
#### Limits and colimits
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[[!include infinity-limits - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are several contexts in which it is of relevance that a certain property of a [[morphism]] $f : A \to B$ is preserved (or __stable__) under [[pullback]], i.e. also shared by the the morphism $\tilde{f}: X \times_B A \to X$ for any pullback diagram

$$
  \array{
     X \times_B A
       &\longrightarrow&
     A
     \\
     {}^{\mathllap{\tilde{f}}}
     \big\downarrow
       &&
     \big\downarrow
     {}^{\mathrlap{f}}
     \\
     X
       &\longrightarrow&
     B
  }
  \,.
$$

Geometers prefer to say "stable under [[base change]]".

## Examples

*  [[monomorphism|Monomorphisms]] are always stable under pullback; that is, if $f$ is a monomorphism, then so is $\tilde{f}$.

*  In many important kinds of categories; some or all [[colimits]] are stable under pullback; this is discussed at [[commutativity of limits and colimits]].

*  The right [[lifting property]]:  Generally, the property of a morphism of having a [[weak factorization system|right lifting property]] is stable under pullback. Therefore for instance [[fibrations]] and acyclic fibrations in a [[model category]] are stable under pullback. If also [[weak equivalences]] are stable under pullback along fibrations, then one speaks of a right proper [[model category]].

*  Similarly, the property of being right [[orthogonality|orthogonal]] to a class of morphisms is stable under pullback.  Thus, the right class in any [[orthogonal factorization system]] is stable under pullback.  If the left class is also pullback-stable, the OFS is called [[stable factorization system|stable]].

## Related concepts

* [[pullback-stable colimit]]

* [[stable factorization system]]

* [[extensive category]]

[[!redirects pullback-stability]]
[[!redirects pullback-stable]]