valuation

> This page is about valuations on [[rings]]/[[fields]]. For valuation in [[measure theory]] see [[valuation (measure theory)]].

***

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## Idea

[Wikipedia](https://en.wikipedia.org/wiki/Valuation_%28algebra%29) says very succinctly

> A **valuation** is a [[function]] on a [[field]] that provides a measure of size or multiplicity of elements of the field. They generalize to commutative [[algebra]] the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in [[complex analysis]], the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry.

Sometimes one also discusses *exponential* (or *multiplicative*) *valuations* (also called valuation functions, and viewed as generalized [[absolute values]]) which look more like [[norms]], and their [[equivalence classes]], [[places]]. See at _[[absolute value]]_ for more on this common sense.

See also [[discrete valuation]] and [[valuation ring]].

## Definition

Given a [[total order|totally ordered]] [[abelian group]] $G$, a $G$-valued __valuation__ $v$ on a (commutative) [[field]] $K$ is a (typically required to be surjective) [[function]] $v:K\to G\cup \infty$ such that $v(K^\times)\subset G$ and

* $v$ defines a [[homomorphism]] of [[groups]] $v|_K : K^\times\to G$ where $K^\times$ is the [[multiplicative group]] of $K$

* $v(0) = \infty$

* $v(x+y) \geq min\{ v(x),v(y)\}$

with usual conventions for $\infty$.  A field equipped with a valuation is a __valued field__.

If the abelian group is the group of [[integers]] $\mathbb{Z}$, then we talk about [[discrete valuation | discrete valuations]].

## Properties

### Valuative criteria

In [[algebraic geometry]] there are very important theorems, due to Chevalley, called the [[valuative criterion of properness]] and [[valuative criterion of separatedness]].

### Valuation rings and ideals

The **valuation ring** of a valued field is the subring of elements of valuation $\ge 0$.  The **valuation ideal** is the ideal in the valuation field consisting of elements of valuation $\gt 0$.

### Valuation uniformity

Any valued field admits a **valuation [[uniformity]]**, where each $g\in G$ generates an [[entourage]] $\{ (x,y) \mid v(x-y)\ge g \}$.  For a discrete valuation, the restriction of this uniformity to the valuation ring coincides with the [[adic topology]] generated by the valuation ideal; but in general this need not be the case.

A **complete valued field** is a valued field whose valuation unifomity is complete.  A non-complete valued field can be completed.

## Examples

### Rational functions

For any field $k$, the field $k(t)$ of formal [[rational functions]] in a variable $t$ has a discrete valuation given by the difference in the *lowest* nonzero degree of the numerator and denominator.  Thus for instance

$$\frac{3t + 2t^3- 7t^4}{2t^2 - 8t^9}$$

would have valuation $1-2 = -1$.  The valuation ring contains the [[polynomial ring]] $k[t]$, but is strictly larger, for instance it contains $\frac{x^2}{x+1}$.  The restriction of the valuation ideal to $k[t]$ does coincide with $(t)$ however.  The valued field $k(t)$ is not complete; its completion is the field $k[[t]]$ of [[formal power series]].

### Generalized rational functions

For any field $k$, a **[[generalized polynomial]]** with exponents in $G$ is a finite formal sum of monomials $a q^\ell$, where $a\in k$ and $\ell\in G$; this defines a ring $k[q^G]$.  The field $k(q^G)$ of **[[generalized rational function|generalized rational functions]]** is the [[field of fractions]] of $k[q^G]$.  It is a valued field with value group $G$, with valuation defined analogously to the case of ordinary rational functions.  As there, its valuation ring contains $k[q^G]$ but is larger.

It is not generally complete.  At least in good cases (e.g. if $G$ is Archimedean in the sense that positive integers are cofinal therein, such as $\mathbb{R}$), its completion should be the subfield of the field $k((q^G))$ of [[Hahn series]] consisting of those Hahn series of order type $\omega$ that converge to themselves in the topology of $k((q^G))$.

### $p$-adic valuations

The field $\mathbb{Q}$ admits a $p$-adic valuation, whose completion is the field $\mathbb{Q}_p$ of [[p-adic numbers]].

## Related concepts

[[!include analytic geometry ingredients -- table]]

## Literature

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

* Wikipedia, *[valuation (algebra)](http://en.wikipedia.org/wiki/Valuation_%28algebra%29)*

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* {#Cassels86} [[J. W. S. Cassels]], Chapter 2 in: *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

* [[Serge Lang]], _Algebraic number theory_, GTM __110__, Springer 1970, 2000

* [[Ehud Hrushovski]], [[David Kazhdan]], _The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL 2$_, [math.LO/0609115](http://arxiv.org/abs/math/0609115); _Integration in valued fields_, in Algebraic geometry and number theory, 261–405, Progress. Math. __253__, Birkhäuser Boston, [pdf](http://math.huji.ac.il/~ehud/papers/intv-060428.pdf)
* A H Lightstone, Abraham Robinson, _Nonarchimedean fields and asymptotic expansions_, North-Holland Publ. 1976

check out ([Scholze 11, def. 22, remark 2.3](perfectoid%20space#Scholze11))

[[!redirects valuations]]
[[!redirects valued field]]
[[!redirects valued fields]]


valuation (measure theory)

> This page is about valuation in [[measure theory]]. For valuation in [[algebra]] (on [[rings]]/[[fields]]) see at _[[valuation]]_.

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## Idea

A **valuation** is a construction analogous to that of a [[measure]], which is however more compatible with [[constructive mathematics]], and readily generalizable to contexts such as [[point-free topology]].

On most spaces of interest for measure theory and probability (such as [[metric spaces]]) the notions of suitably continuous valuations and measures coincide (see [below](#extending_valuations_to_measures)).

## Definition

### Valuations on lattices

Let $L$ be a [[distributive lattice]] with a [[bottom]] element $\bottom$. A **valuation** or **evaluation** on $L$ is a [[map]] $\nu$ from $L$ into the space of non-negative [[one-sided real number|lower reals]], with the following properties:

* **Monotonicity**: for all $x,y$ in $L$, $x\le y$ implies $\nu(x)\le\nu(y)$;

* **Strictness** (or *[[unitality]]*): $\nu(\bottom)=0$;

* **Modularity**: for all $x,y$ in $L$,

$$
\nu(x) + \nu(y) = \nu(x \vee y) + \nu(x \wedge y) .
$$

Moreover, we call a valuation *continuous* if the following property holds, which is an instance of [[Scott topology|Scott continuity]], as well as of [[tau-additive measure|$\tau$-additivity]]:

* **Continuity**: for every [[directed set|directed]] [[net]] $\{x_\lambda\}_{\lambda\in\Lambda}$ in $L$ admitting a supremum,

$$
\nu \big( \sup_{\lambda} x_\lambda \big) = \sup_\lambda \nu(x_\lambda) .
$$

Note that, differently from measures, there is no explicit mention of [[complements]]. Moreover, the continuity condition can be interpreted, measure-theoretically, both as an analogue of $\sigma$-additivity, and as an analogue of a regularity condition (more precisely it corresponds to [[τ-additivity]]).
See [[correspondence between measure and valuation theory]] for more on this.

### Valuations on locales and topological spaces

Let $L$ be a [[locale]]. Then a valuation on $L$ is by definition a valuation on its [[frame]] $\mathcal{O}(L)$. Similarly, a valuation on a [[topological space]] is a valuation on the [[lattice of open subsets|lattice of its open sets]].

Valuations on locales are used in the [[topos approach to quantum mechanics]] and the [[Bohr topos]].

## Examples

### Dirac valuation

Let $X$ be a topological space, and let $x\in X$ be a point. The **Dirac valuation** at $x$, which we denote by $\delta_x$, maps an open set $U\subseteq X$ to
$$
\delta_x(U) \;\coloneqq\;
\begin{cases}
1 & x\in U \\
0 & x\notin U.
\end{cases}
$$

These are the valuation analogue of [[Dirac measures]], and give the unit map of [[valuation monads]].
Note the analogy with [[neighborhood filters]]: Dirac measures can be considered a quantitative analogue thereof.

### Simple valuations

On a topological space, a **simple valuation** is a finite convex (or linear) combination of Dirac valuations, i.e. a valuation in the form
$$
\nu = \sum_{i=1}^n \alpha_i\,\delta_{x_i} ,
$$
for $x_i\in X$, and for positive (lower) real numbers $\alpha_i$, possibly summing to one.

### Borel measures

For more on this, see [[tau-additive measure|$\tau$-additive measure]].

Let $X$ be a topological space, and let $\mu$ be a measure defined on the [[Borel subset|Borel $\sigma$-algebra]] of $X$. Then the restriction of $\mu$ to the open subsets of $X$ is a valuation. The valuation is continuous if and only if $\mu$ is [[tau-additive measure|$\tau$-additive]].

The converse problem of whether a valuation is the restriction of a Borel measure is more difficult, see [below](#extending_valuations_to_measures).

## Measure-theoretic structures and properties

Valuations share a number of constructions which are similar to those for [[measures]]. The constructions below are given for the case of continuous valuations on [[topological spaces]].

### Pushforward

Given spaces $X$ and $Y$ and a continuous map $f:X\to Y$, we can map a valuation $\nu$ on $X$ to a valuation $f_*\nu$ on $Y$ in the following way. For every open set $U\subseteq Y$, we set
$$
f_*\nu (U) \coloneqq \nu(f^{-1}(U)) .
$$
The assignment gives a well-defined valuation, which is continuous if $\nu$ is continuous. We call $f_*\nu$ the **pushforward valuation** of $\nu$ along $f$.

Compare with the [[pushforward measure|analogous construction for measures]].

This construction allows to makes the assignment [[functor]], which is even a [[monad]] (see [[monads of probability, measures and valuations#functor_ unit_and_multiplication|measure monads]] for more on this).

### Joints, marginals and products

Given a valuation $\nu$ on a product space $X\times Y$, one calls its pushforward along the projection map $X\times Y\to X$ the **marginal** of $\nu$ on $X$. The same can be done for $Y$.
The assignment from $\nu$ to the ordered pair of its marginals can be seen as an [[monoidal functor|oplax monoidal structure]] of the [[monads of probability, measures and valuations|valuation monad]].

Given valuations $\nu$ on $X$ and $\rho$ on $Y$, there may be many possible valuations on $X\times Y$ which have $\nu$ and $\rho$ as their marginals. Any such valuation on $X\times Y$ is called a **joint valuation** or **coupling** of $\nu$ and $\rho$.

Between the many joints of $\nu$ and $\rho$ there is always a canonical choice, namely the **product valuation** $\nu\otimes\rho$.
From the point of view of [[probability theory]], this corresponds to a distribution exhibiting [[independence]] between $\nu$ and $\rho$.
On the [[basis]] of the [[product topology]] of $X\times Y$ given by sets in the form $U\times V$ for $U\subseteq X$ and $V\subseteq Y$ open, the product valuation is given by
$$
\nu\otimes\rho (U\times V) \coloneqq \nu(U)\cdot\rho(V).
$$
Note that this is not enough to define a valuation a priori. For continuous valuations, however, this turns out to be the well-defined, as proven for example in [Heckmann '96](#Heckmann96).

The assignment to $\nu$ and $\rho$ of their product valuation can be seen as a [[monoidal functor|lax monoidal structure]] of the [[monads of probability, measures and valuations|valuation monad]].
See also [[monads of probability, measures and valuations#monoidal_structure]].

### Null sets and support

Valuations admit a notion of [[support]] similar to that of measures. In particular, continuous valuations, just as [[tau-additive measure|$\tau$-additive measures]], have a well-defined and well-behaved support.

Let $\nu$ be a valuation on a [[locale]] or [[topological space]] $X$, and $U$ an open set of $X$ (i.e. an element of the corresponding [[frame]]). We say that $U$ is a **null** or **measure zero set** for $\nu$ if $\nu(U)=0$. The [[complement]] of $U$, which is a [[closed subspace#locales|closed subspace]] of $X$, is said to have **full measure**.

Since a finite union of null sets is null, null sets form a directed net in the frame. Therefore, if $\nu$ is a continuous valuation, it admits a unique maximal null open set. The [[complement]] of this set, which is the largest closed subspace of full measure, is called the **[[support]]** of $\nu$.

The support induces a [[monad#the_bicategory_of_monads|morphism of monads]] (see [Fritz-Perrone-Rezagholi](#support), section 5).

### Integration

There is an integration theory for valuation analogous to that of measures, where the [[open sets]] play the role of the [[measurable sets]], and lower [[semicontinuous functions]] play the role of [[measurable functions]]
(see also [[correspondence between measure and valuation theory]]).
Sometimes integration of valuation is known as **lower integration**, since approximations are done from below.

The way to define integration, mutatis mutandis, parallels usual [[Lebesgue integral]] construction.
We sketch the construction for the case of topological spaces.

Let $\nu$ be a valuation on a space $X$.

* Given an open set $U\subseteq X$, and denoting by $1_U$ its [[indicator function]], we define
$$
\int 1_U \, d\nu \;\coloneqq\; \nu(U) .
$$

* A [[simple function|simple]] lower semicontinuous function is a lower semicontinuous function assuming only finitely many values. Such functions can be expressed (nonuniquely) as finite positive linear combinations of indicator functions:
$$
f \;=\; \sum_i r_i \, 1_{U_i} .
$$
We define the integral of a simple $f$ as
$$
\int f \, d\nu \;\coloneqq\; \sum_{i}r_i \nu(U_i) .
$$
This is well-defined, i.e.~it depends only on $f$ and not on the particular way of expressing $f$ as a linear combination of indicators.

* Every lower semicontinuous function can be written as pointwise directed supremum of simple lower semicontinuous functions. So suppose $g:X\to[0,\infty]$ is lower semicontinuous. Take an increasing net $(g_\alpha)_{\alpha\in A}$ of nonnegative simple lower semicontinuous functions. Then we define
$$
\int g \, d\nu \;\coloneqq\; \sup_\alpha \int g_\alpha\,d\nu ,
$$
where the integral on the right is the one defined above for simple functions, and the supremum on the right is either the one of real numbers (or lower real numbers), or $+\infty$.

This integral satisfies analogous properties to the Lebesgue integral, such as [[linear map|linearity]] and [[Scott continuity]] (cfr. the [[monotone convergence theorem|sequential monotone continuity]] of the Lebesgue integral).

## Monads of valuations

Just as there are several [[monads of measures]] (such as the [[Giry monad]]), there are a number of analogous [[monads]] of valuations.
The most famous are

* The [[extended probabilistic powerdomain]] on the [[Top|category of topological spaces]], which was introduced by [Heckmann 96](#Heckmann96).

* The [[valuation monad on locales]], defined by [Steve Vickers](#vmonad).

* The [[probabilistic powerdomain]] on the category of [[dcpo|dcpos]], defined by Jones and Plotkin, of wide use in theoretical computer science.

See also the list at [[monads of probability, measures, and valuations]].

## Extending valuations to measures

As we have seen [above](#borel_measures), a [[Borel measure]] always restricts to a valuation. It is natural to ask the converse question of whether a valuation can always be extended to a Borel measure. In general, the answer is negative. In the case of *continuous* valuations, however, one would expect that in many cases the valuation can be extended to a [[tau-additive measure|$\tau$-additive Borel measure]].

The question is known, for example, to be true on all regular Hausdorff ($T_3$) spaces:

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

A locally finite continuous valuation on a [[regular topological space]]
extends uniquely to a regular τ-smooth [[Borel measure]].
A locally finite continuous valuation on a [[locally compact]] [[sober space]]
extends uniquely to a τ-smooth [[Borel measure]].

=--

([Manilla 02, Theorems 4.4 and 4.12](#Manilla02))

This includes in particular every [[metric space]], and every compact Hausdorff space. So, in many spaces of interest for analysis and probability theory, working with measures and working with valuations is only a difference in the language.

The more general question of whether one can extend a finite continuous valuation to a Borel measure on any sober space, at the present time, is still open.

### Particular cases

* Every Dirac valuation can be extended to the corresponding [[Dirac measure]].

* The pushforward of an extendable valuation along a [[continuous map]] is again extendable, and the resulting [[measure]] is the [[pushforward measure]] of the extension. (Equivalently, the restriction of measures to valuations is a [[natural transformation]]).

* The above specializes to the fact that marginals of extendable valuations are extendable, and the resulting measure is the [[marginal measure]].

* Somewhat conversely, the product of extendable valuations is extendable, and the resulting measure is the [[product measure]] of the extensions.

* The integration, given by the multiplication map $E: P P X \to P X$ of the [[extended probabilistic powerdomain]], maps extendable valuations to extendable valuations, and the resulting measure is the integral of the extension. Therefore there is a [[monad#the_bicategory_of_monads|morphism of monads]] exhibiting [[τ-additive measures]] as a submonad of valuations. (See also [[extended probabilistic powerdomain#the_measure_monad_on_top|the measure monad on Top]].)

## See also

* [[monads of probability, measures, and valuations]]
* [[correspondence between measure and valuation theory]]
* [[monad]], [[Giry monad]], [[monads in computer science]]
* [[extended probabilistic powerdomain]], [[probabilistic powerdomain]], [[valuation monad on locales]]
* [[probability valuation]]
* [[sigma-continuous valuation]], [[sigma-continuous probability valuation]]
* [[content (measure theory)]], [[probability content]]
* [[measure]], [[τ-additive measure]], [[probability measure]]

## References

* {#Heckmann96} [[Reinhold Heckmann]], *Spaces of valuations*, Annals of the New York Academy of Sciences, **806** 1 *Papers on General Topology and Applications*, (1996) $[$[doi:10.1111/j.1749-6632.1996.tb49168.x](https://doi.org/10.1111/j.1749-6632.1996.tb49168.x), [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf)$]$

* Mauricio Alvarez-Manilla, Achin Jung, [[Klaus Keimel]], _The probabilistic powerdomain for stably compact spaces_, Theoretical Computer Science 328, 2004 ([doi:10.1016/j.tcs.2004.06.021](https://doi.org/10.1016/j.tcs.2004.06.021))

* {#algebras} [[Jean Goubault-Larrecq]] and Xiaodong Jia, _Algebras of the extended probabilistic powerdomain monad_, ENTCS 345, 2019
([doi:10.1016/j.entcs.2019.07.015](https://doi.org/10.1016/j.entcs.2019.07.015))

* {#support} [[Tobias Fritz]], [[Paolo Perrone]] and Sharwin Rezagholi, _Probability, valuations, hyperspace: Three monads on Top and the support as a morphism_, 2019 ([arXiv:1910.03752](https://arxiv.org/abs/1910.03752))

For valuations on locales, see

* {#vmonad} [[Steve Vickers]], _A monad of valuation locales_, 2011 ([pdf](https://www.cs.bham.ac.uk/~sjv/Riesz.pdf))

For the theory of [integration over valuations](#integration), see

* {#kirch} Olaf Kirch, _Bereiche und Bewertungen_ (in German), Master Thesis, Technische Hochschule Darmstadt, 1993 ([ps.gz](http://fb04286.mathematik.tu-darmstadt.de/fbereiche/logik/research/Domains/papers/kirch/diplom.ps.gz))

* {#jung} Achim Jung, _Stably compact spaces and the probabilistic powerspace construction_, ENTCS 87, 2004 ([doi:10.1016/j.entcs.2004.10.001](https://doi.org/10.1016/j.entcs.2004.10.001)).

* [[Thierry Coquand]] and [[Bas Spitters]], _Integrals and Valuations_, 2009,  	Logic and Analysis (2009) 1(3) p.1-22 ([arXiv:0808.1522](https://arxiv.org/abs/0808.1522))

Integration on locales can be found in

* {#vintegral} [[Steve Vickers]], _A localic theory of lower and upper integrals_, 2008 ([doi:10.1002/malq.200710028](https://doi.org/10.1002/malq.200710028))

For the problem of [extending valuations to measures](#extending_valuations_to_measures), see

* Mauricio Alvarez-Manilla, Abbas Edalat, and Nasser Saheb-Djahromi, _An extension result for continuous valuations_, 1998 (doi:10.1016/S1571-0661(05)80210-5)

* {#Manilla00} Mauricio Alvarez-Manilla, _Measure theoretic results for continuous valuations on partially ordered spaces_, Dissertation, 2000 ([ps.gz](http://www.cs.tufts.edu/~nr/cs257/archive/mauricio-alvarez-manilla/public.ps.gz))

*  {#Manilla02} Mauricio Alvarez-Manilla, _Extension of valuations on locally compact sober spaces_, Topology and its Applications Volume 124, Issue 3, 20 October 2002, Pages 397-433 (doi:10.1016/S0166-8641(01)00249-8)

* Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).

* [[Klaus Keimel]] and [[Jimmie D. Lawson]], _Measure extension theorems for_
$T_0$ _spaces_, 2004 ([doi:10.1016/j.topol.2004.02.019](https://doi.org/10.1016/j.topol.2004.02.019))

[[!redirects continuous valuation]]
[[!redirects continuous valuations]]


valuation ring

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## Definition and Properties

A **valuation ring** is an [[integral domain]] $O$ which satisfies any of the following equivalent conditions:

1. for every nonzero element $x$ in its [[field of fractions]] $K$, either $x \in O$ or $x^{-1} \in O$;

1. the ideals of $O$ are totally ordered by inclusion;

1. the principal ideals of $O$ are totally ordered by inclusion.

These equivalences are not difficult to establish. For example, if 1. holds and $I, J$ are distinct ideals, then some element $x$ belongs to one but not the other, say $x \in J$, $x \notin I$. Then for each $y \in I$, the condition $x/y \in O$ leads to $x = y(x/y) \in I$ which is false; therefore $y/x \in O$, whence $y = x(y/x) \in J$, and we conclude $I \subset J$. That 2. implies 3. is trivial, and if 3. holds and $x \in K$, write $x = a/b$ where $a, b \in O$, and conclude either $(a) \subseteq (b)$ or $(b) \subseteq (a)$, where $x \in O$ in the former case and $x^{-1} \in O$ in the latter.

+-- {: .num_prop}
###### Proposition
A valuation ring $O$ is a [[local ring]]. Its maximal ideal is said to be the [[valuation ideal]].
=--

+-- {: .proof}
###### Proof
Suppose $x$, $y$ are nonzero non-invertible elements of $O$. Either $x/y$ or $y/x$ belongs to $O$, say $x/y$. Then $(x+y)/y$ belongs to $O$ as well. If $x+y$ were a unit of $O$, it would follow that $1/y$ belongs to $O$, i.e., both $y$ and $1/y$ belong to $O$, so that $y$ is a unit of $O$, contradiction. It follows that non-unit elements of $O$ are closed under addition. It is also clear that if $x$ is a non-unit element of $O$ and $y$ is any element of $O$, then $x y$ is a non-unit element of $O$. Therefore the non-units form an ideal of $O$, clearly the unique maximal ideal of $O$.
=--

+-- {: .num_prop}
###### Proposition
A valuation ring $O$ is integrally closed in its field of fractions $F$.
=--

+-- {: .proof}
###### Proof
Suppose $x \in F$ satisfies an equation $x^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0$ where the $a_i$ belong to $O$. Either $x$ or $1/x$ belongs to $O$, and if $1/x$ belongs to $O$, then so does $x$ because
$$-x = a_{n-1} + a_{n-1}x^{-1} + \ldots + a_0 x^{-n+1}$$
and this completes the proof.
=--

+-- {: .num_theorem}
###### Theorem
A valuation ring is a [[Prüfer domain]].
=--

+-- {: .proof}
###### Proof
There are many characterizations of Prüfer domains, but one is that the [[lattice]] of [[ideals]] is [[distributive lattice|distributive]], which is obviously the case if the lattice is [[linear order|linearly ordered]].
=--

### Example: germs of definable functions

Local rings often arise as stalks of sheaves of real or complex-valued functions, and similarly, some of the more interesting examples of valuation rings arise by considering germs of functions at an "ideal" point, for example at an ultrafilter or infinite point where the functions are not formally defined. Examples such as these are often rich sources of rings and fields with _infinitesimal elements_. The following example should give the flavor of this phenomenon.

Consider the class of all functions $\mathbb{R} \to \mathbb{R}$ which can be defined by a first-order formula, starting with the basic operations
$+, -, \cdot, \exp$, any constant $a \in \mathbb{R}$, and the relations $\lt$ and $=$. This is an enormous class of functions: it includes the [[logarithm]] function and any function which can be built from [[polynomials]], [[exponential map|exponentials]], and logarithms using the four basic arithmetic operations and composition, and also implicitly defined functions such as:

$$f(x) = max \{y: y^5 - (e^{e^x - (\log x)^2 + 1/x})y^2 + \log(1 + x^{\sqrt{2}}))y - 17 = 0\}$$

and many, many more. Of interest are _rates of growth_ (cf. [[O notation]]) of these functions, or, at an even more refined level, the precise ordering of these functions $f(x) \leq g(x)$ when $x$ is large. A quite remarkable and deep fact is the following:

+-- {: .un_thm}
######Theorem
Any such definable function $f(x)$ is either positive for all sufficiently large $x$, $0$ for all sufficiently large $x$, or negative for all sufficiently large $x$.
=--

This theorem implies that the germs at infinity of such functions ($\sim$-equivalence classes of functions where $f \sim g$ if $f(x) = g(x)$ for all sufficiently large $x$) form a totally [[ordered field]], in fact a [[real closed field]]. The real numbers are embedded in this field as germs of constant functions, but lying between ordinary real numbers are other "numbers" infinitesimally close to reals, such as $2 - [1/x]$, $1 + [1/\log \log(x)]$, as well as infinite numbers such as $[e^x]$.

Sitting inside this field $Germ(\mathbb{R}_{exp})$ is the valuation ring of germs of bounded definable functions, in other words the ring of finite "numbers", which contains infinitesimals of incredibly rich variety.

Such examples are close in spirit to [[hyperreal number]]s, which form a considerably larger real closed field. In this case, the procedure is similar, except that one takes germs of _all_ functions $\mathbb{N} \to \mathbb{R}$ in the neighborhood of a non-principal ultrafilter on $\mathbb{N}$, which can be considered an ideal point at "infinity". This is called an [[ultrapower]] of the standard real numbers. Again the finite hyperreals form a valuation ring sitting inside.

## The valuation function

(also called multiplicative or exponential valuation)

The nonzero elements of $K$ may be partially ordered as follows: write $x \leq y$ if $x/y$ belongs to $O$. For any two nonzero elements $x$, $y$ of $K$, exactly one of the following conditions holds:

* $x/y$ is a non-unit of $O$;

* $x/y$ is a unit of $O$: $x/y$ and $y/x$ belong to $O$;

* $y/x$ is a non-unit of $O$.

We call this the _trichotomy law_. Thus, if we write $O^*$ for the units of $O$, it follows from trichotomy that the partial order on $K^*$ descends to a [[total order]] on the quotient group $G = K^*/O^*$. This totally ordered group is called the **value group** of the valuation ring $O$. When the value group is isomorphic to $\mathbb{Z}$, the ring $O$ is called a **discrete valuation ring**. Many local rings which arise in practice, for example localizations of rings of [[algebraic integer]]s $O$ with respect to a prime ideal $\mathcal{p}$, or their completions as rings of $\mathcal{p}$-adic integers $O_{(\mathcal{p})}$, are discrete valuation rings.

We may then define a **valuation** function

$$v: K \to K^*/O^* \cup \{0\}$$

where $v(x)$ is the coset $x O^*$ if $x \in K^*$, and $v(0) = 0$. The codomain becomes a totally ordered monoid if $0$ is regarded as the bottom element and is _absorbent_ ($0 x = 0$ for all $x$). This function satisfies the following conditions:

1. $v(x) = 0$ if and only if $x = 0$;

1. $v(x y) = v(x)v(y)$;

1. $v(x + y) \leq \max \{v(x), v(y)\}$

1. $v$ is surjective.

### Example: rates of growth

Let us return to our earlier example of a valuation ring $O$, consisting of germs of bounded functions which are first-order definable in the theory of the reals as ordered field with exponentiation. Two "numbers" $[f]$, $[g]$ have the same coset, $[f]O^* = [g]O^*$, if both $[f/g]$ and $[g/f]$ are bounded. But this is precisely to say $f$ is $O(g)$ and $g$ is $O(f)$.

Thus, the elements of the value group in this case can be described as the various "rates of growth" of definable functions. The order relation is that $[f]O^* \leq [g]O^*$ if $f$ is $O(g)$. Thus the classical analysis notion of 'O notation' fits within the theory of valuation rings.

Rates of growth, as elements of the value group $K^*/O^*$, can also be regarded as "numbers" containing infinitesimal and infinite quantities. Thus, ordinary real numbers $a$ would correspond to rates of growth of power functions $x^a$, whereas the rate of growth $\log x$ is infinitesimal and the rate of growth of $e^x$ is infinite. The fact that rates of growth of definable functions are totally ordered is essentially due to G.H. Hardy, and in his honor, fields of germs of definable functions are frequently called "Hardy fields".

* Hardy actually studied the class of functions definable from polynomials, exponential, and logarithmic functions using the four arithmetic operations and composition. This of course is a small subclass of the definable functions considered above, but contains all functions likely to be of interest in analytic number theory (for example).

## Valuation rings from valuation functions

Quite generally, we may define a **valuation** on a field $K$ to be a function

$$v: K \to G \cup \{0\}$$

(where $G$ is a totally ordered group, extended to a totally ordered monoid $G \cup \{0\}$ as above), satisfying conditions 1 - 4 listed above. Two valuations $v$, $v'$ are **equivalent** if there is an isomorphism

$$\phi: G \cup \{0\} \to G' \cup \{0\}$$

of totally ordered monoids such that $v' = \phi \circ v$. In fact, valuations $v$ may be [[preorder|preordered]]: if we regard $v$ as a special sort of group homomorphism $v: K^* \to G$, then define $v \leq v'$ if there is a surjective homomorphism of ordered groups $\phi: G \to G'$ such that $v' = \phi \circ v$.

From the data of a valuation on $K$, we may construct a valuation ring $O_v$ inside $K$:

$$O_v = \{x \in K: v(x) \leq 1\}$$

where $1$ is the identity element of $G$. (The fact that $O_v$ is a ring follows straightforwardly from the axioms 1 - 3, and that $O_v$ is a valuation ring follows from the fact that $G$ is totally ordered.)

In summary, we have the following result whose proof is straightforward:

+-- {: .un_prop}
######Proposition
There is a natural bijective correspondence between equivalence classes of valuations on $K$ and valuation rings in $K$.
=--

To express the naturality more precisely: there are two functors

$$V: Field^{op} \to Pos, \qquad V': Field^{op} \to Pos$$

from fields to posets. Here $V$ assigns to a field $K$ the poset of equivalence classes of valuations on $K$, where the partial order is inherited from the preorder described above; $V'$ assigns to a field $K$ the poset of valuation rings inside $K$, ordered by inclusion. If $i: K \to K'$ is a field homomorphism, then $V'(i): V'(K') \to V'(K)$ takes a valuation ring $O' \subseteq K$ to the valuation ring $i^{-1}(O') \subseteq K$, and $V(i)$ is defined similarly. The proposition asserts that the functors $V$, $V'$ are naturally isomorphic. We freely conflate them, denoting either functor as $Val: Field^{op} \to Pos$.

### Examples

1. The algebraic notion of Riemann surface from 19th century is constructed in a way in which valuations are used to construct points of Riemann surfaces over various fields. See [[Riemann surface via valuations]].

2. There is a very general construction which takes as input an arbitrary field $k$ and a totally ordered abelian group $G$, and produces as output a valuation ring whose value group is naturally identified with $G$. This is the ring of [[Hahn series]], see there.

## Equivalent characterisations

\begin{proposition}(Homological characterisation)
For a commutative [[local ring]] $V$, the following are equivalent :

1. $V$ is a valuation ring ;
1. all [[torsion-free]] $V$-modules are flat ;
1. the [[weak global dimension]] of $V$ satisfies $wgldim(V) \leq 1$ ;
1. $V$ is [[semi-hereditary]].

\end{proposition}

\begin{proof}
$1 \Rightarrow 2$. Let $M$ be a torsion-free $V$-module and let $\sum_{0 \leq i\leq n} v_i m^i = 0$ be a linear relation in $M$. One can assume that $v_0 \in V$ is the element with the minimal valuation. It follows that for every $i \gt 0$ , $v_i$ is divisible by $v_0$ in V. Now since $M$ is torsion-free, it means that we can divide the whole linear equation by $v_0$ and reduce to the case where $v_0 = 1$, that is $m^0 + \sum_{1 \leq i \leq n} v_i m^i = 0$.
Let $b_i \coloneqq m_i$ for $i \gt 0$ and define $a^i_j$ for $0 \leq i \leq n$ and $0 \lt j \leq n$ with $a^{j+1}_j \coloneqq 1$, $a^0_j \coloneqq - v_j$ for every $0 \lt j$ and $a^i_j  \coloneqq 0$ otherwise.
Then $m^i = \sum_j a^i_j b^j$ for every $0 \leq i$ and $\sum_i v_i a^i_j = 0$ for every $0 \lt j$.

$2 \Rightarrow 3$. We must show that $Tor_2^V(M, N) = 0$ for every $V$-modules $M, N$. Since $Tor_2$ commutes with [[filtered colimits]], it is enough to show this for every finitely generated $M$. Using the [[long exact sequence]] for Tor, one can further reduce to the case where $M$ is a cyclic module $V/I$.
By assumption, since all ideals $I \subset V$ are torsion-free, they are flat, that is $Tor_1^V(I, N) = 0$ for every $V$-module $N$. Tensoring the short exact sequence $0 \to I \to V \to V/I \to 0$ with $N$, one gets $Tor_2^V(V/I, N) = Tor_1^V(I, N) = 0$ for every $N$.

$3 \Rightarrow 4$. Let $I \subset V$ be a finitely generated ideal. By assumption $Tor_2^V(V/I, M) = 0$ for every $V$-module $M$. So $Tor_1^V(I, M) = 0$ for every $M$ ; $I$ is flat. Finitely generated flat modules over a local ring are free.

$4 \Rightarrow 1$. Let $I \subset V$ be a finitely generated ideal. Since $V$ is semi-hereditary, $I$ is projective. Because $V$ is local this means that $I$ is free ; this is possible only if $I$ is principal and generated by a regular element of $V$ or the ideal $(0)$. It follows that $V$ must be an [[integral domain]] and that every finitely generated ideal is principal.
Let $x, y \in V$ be distinct, since $(x, y)$ is finitely generated, there is a regular $z \in V$ such that $(x) + (y) = (z)$. Then there exists $\alpha, \beta, \lambda, \mu$ such that $x = \alpha z$, $y = \beta z$ and $z = \lambda x + \mu y$. Then $z (1 - \alpha \lambda - \beta \mu) = 0$ and since $z$ is regular $\alpha \lambda + \beta \mu = 1$. Now because $V$ is a local ring, one of these two terms must be invertible. If $\alpha \lambda$ is invertible, then again since $V$ is local, $\alpha$ must be invertible and $(y) \subset (x)$. In the other case one has $(x) \subset (y)$.
We have thus proved that principal ideals in $V$ are totally ordered by inclusion and thus $V$ is a valuation ring.

\end{proof}

## Related concepts

* [[valuation]]

* [[discrete valuation]]

* [[class field theory]]

## Literature

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* [[Serge Lang]], _Algebraic number theory_. GTM 110, Springer 1970, 2000

* O. Zariski, Samuel, _Commutative algebra_

[[!redirects valuation rings]]


valuative criterion of properness

The valuative criterion of [[proper morphism|properness]] [[EGA II]], 7.3.8 ([numdam](http://www.numdam.org/item?id=PMIHES_1961__8__5_0)). It has been found by Chevalley.

>Théorème (7.3.8). — Soient Y un préschéma (resp. un préschéma localement noethérien),
>$X\to Y$ un morphisme quasi-compact séparé (resp. de type fini). Les conditions suivantes sont équivalentes :

>a) f est [[universally closed morphism|universellement fermé]] (resp. *propre*).

>b) Pour tout $Y$-schéma de la forme $Y'=Spec(A)$, où $A$ est un anneau de valuation (resp. anneau de valuation discrète) de corps des fractions $K$, l'application canonique
>$$
>Hom_Y(Y', X) \to Hom_Y(Spec(K), X)
>$$
>correspondant à l'injection canonique $A\to K$, est surjective (resp. bijective).

In other words, the class of proper morphisms satisfies the *unique right lifting property* with respect to the class of morphisms $Spec K\to Spec R$ formally dual to the injections of valuation rings into their fields of fractions.

There is a pattern-terminological reinterpretation in Kontsevich-Rosenberg, _Noncommutative spaces_, 6.8.1:

Let $M_v'$ be the family of canonical injections of valuation rings to their fields of fractions, and let $M_v$ be its image in the category of functors $CRing\to Set$. Consider schemes as covariant presheaves on $CRing$. Then a morphism $f:X\to Y$ of schemes is [[separated morphism of schemes|separated]] iff it is formally $M_v$-unramified, [[universally closed morphism|universally closed]] iff it is formally $M_v$-smooth and proper iff it is formally $M_v$-étale (in the sense of KR 6.3).


valuative criterion of separatedness

The __valuative criterion of separatedness__ is [[EGA II]], 7.2.3. ([numdam](http://www.numdam.org/item?id=PMIHES_1961__8__5_0)) which translated into English says

**Proposition.** Let $X$ be a [[scheme]] (resp. a [[locally noetherian scheme]]),
 $f: X\to Y$ a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent

a) $f$ is [[separated morphism of schemes|separated]].

b) The diagonal morphism $X\to X\times_Y X$ is [[quasicompact morphism|quasicompact]], and for every affine scheme $Y' = Spec A$ in which $A$ is a [[valuation ring]] (resp. a discrete valuation ring), any two morphisms from $Y'\to X$ which coincide at the generic point of $Y'$ are equal.

c) The diagonal morphism $X\to X\times_Y X$ is quasicompact, and for every affine scheme of the form  $Y' = Spec A$ in which $A$ is a valuation ring (resp. a discrete valuation ring), any two sections of $X' = X(Y')$ which coincide at the generic point of $Y'$ are equal.

Compare the [[valuative criterion of properness]], [[EGA II]], 7.3.8.

* MathOverflow [can-the-valuative-criteria-for-separatedness-properness-be-checked-formally](http://mathoverflow.net/questions/79/can-the-valuative-criteria-for-separatedness-properness-be-checked-formally)

category: algebraic geometry


value

#Contents#
* table of contents
{:toc}

## Idea

### Value of a variable

(...)

### Value of a function

In [[set theory]], the value of a *[[function]]* $f \,\colon\, X \longrightarrow A$ at an element $x \in X$ is the element $f(x) \in A$.

More generally, in [[type theory]] the value of a function on a [[term]] $x$ of [[type]] $X$ is the term $f(x)$ of type $A$.

(...)

## Properties

### Relation to images

The [[image]] of a function $f \,\colon\, X \to A$ on a [[subset]] $S \subset X$ is the *value* at $S$ of the function $f_\ast \;\colon\; P(X) \to P(A)$, which in turn is the value at $f$ of the [[covariant power set functor]] $P \,\colon\, Set \to Set$ on the [[category]] [[Set]] of [[sets]].

## Related concepts

* [[image]]

* [[absolute value]]

## References

See also:

* Wikipedia, *Value_(mathematics)*

[[!redirects values]]


Van Eric Mayes

* [webpage](https://www.uhcl.edu/science-engineering/faculty/mayes-van)

## Selected writings

On [[model (in theoretical physics)|model building]] with realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys.Rev.D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))

* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, Van E. Mayes, _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))

category: people


van Est isomorphism

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

#Contents#
* table of contents
{:toc}

## Idea

The content of a [[van Est isomorphism]] is that the canonical comparison map from [[Lie group cohomology]] to [[Lie algebra cohomology]] (by [[differentiation]]) is an [[isomorphism]] whenever the Lie group is sufficiently connected.

In particular when the underlying [[topological space]] of a Lie group is a [[contractible topological space]], then [[Lie group cohomology]] (both the naive and the correct stacky version) coincides with Lie algebra cohomology [vanEst 53, theorem 14.1](#vanEst53) (see e.g. the cocycle which gives the [[Heisenberg group]] extension)

## References

The original article is

* {#vanEst53} [[Willem van Est]], _Group cohomology and Lie algebra cohomology in Lie groups. I, II._  Nederl. Akad. Wetensch. Proc. Ser. A. 56, Indagationes Math. 15, (1953). 484–492, 493–504. ([MR0059285](http://www.ams.org/mathscinet-getitem?mr=59285))

A comprehensive statement is around theorem 3.7 in

* {#Wockel14} [[Christoph Wockel]], _Topological group cohomology of Lie groups and Chern-Weil theory for compact symmetric spaces_, ZMP-HH/14-1, Hamburger Beiträge zur Mathematik Nr. 498 ([arXiv:1401.1037](http://arxiv.org/abs/1401.1037))

The generalizations to [[Lie groupoids]] and [[Lie algebroids]] is considered in

* [[Alan Weinstein]], [[Ping Xu]], _Extensions of symplectic groupoids and quantization_, Journal für die reine und angewandte Mathematik 417 (1991): 159-190 ([EuDML](https://eudml.org/doc/153329))

* [[Marius Crainic]], _Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes_ ([arXiv:0008064](http://arxiv.org/abs/math/0008064))

[[!redirects van Est isomorphisms]]

[[!redirects van Est map]]
[[!redirects van Est maps]]

[[!redirects van Est theorem]]
[[!redirects van Est's theorem]]


van Kampen colimit

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

# Van Kampen colimits

* table of contents
{: toc}

## Idea

A [[colimit]] in a [[category]] or [[higher category]] is called *van Kampen* [[Sobocinski & Heindel 2011](#SobocinskiHeindel11)] if it is both [[universal colimit|universal]] (i.e. [[pullback-stability|stable]] under [[pullback]]) and satisfies [[descent]].  Many [[exactness properties]] can be phrased in terms of certain colimits being van Kampen.

## Definition

Let $\mathcal{C}$ be a [[category]] with [[pullbacks]].
Then there is a (pseudo) [[2-functor]]
\[
  \label{SelfIndexingFunctor}
  S
  \,\colon\,
  \mathcal{C}^{op} \to Cat
\]
defined by $S(x) \,\coloneqq\, \mathcal{C}/x$ (the [[slice category]]), called the *[[self-indexing]]* of $\mathcal{C}$.  Its [[Grothendieck construction]] is the [[codomain fibration]].

\begin{definition}\label{VanKampenColimit}
A [[colimit]] in $\mathcal{C}$ is **van Kampen** if it is [[preserved limit|preserved]] by the functor $S$ (eq:SelfIndexingFunctor), i.e. it is taken to a (weak) [[2-limit]] in $Cat$.
\end{definition}

### Universality and descent

We need the following notation:

* For $\mathcal{D}$ any [[category]], write $\mathcal{D}^{\triangledown}$ for the result of adjoining to it a [[terminal object]]. This comes with a canonical [[full subcategory]] inclusion $\mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown}$

* For $\mathcal{D}$ a [[small category]], and $G \,\colon\, \mathcal{D} \to \mathcal{C}$ a $\mathcal{D}$-shaped [[diagram]] with [[colimit]] $\underset{\longrightarrow}{\lim} G \,\in\, \mathcal{C}$, write $G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}$ the extension of $G$ to $\mathcal{D}^{\triangledown}$ by assigning $\underset{\longrightarrow}{\lim} G$ to the adjoined terminal object.

\begin{theorem}\label{UniversalityAndDescent}
**(equifibrancy characterization of van Kampen colimits)**
\linebreak
If $\mathcal{C}$ has all [[colimits]] of $\mathcal{D}$-shaped diagrams, then the colimit $\underset{\longrightarrow}{\lim}G$ of $G \colon \mathcal{D}\to \mathcal{C}$ is van Kampen (Def. \ref{VanKampenColimit}) if and only if the following condition holds:

* For any diagram $F^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \to \mathcal{C}$ and [[natural transformation]] $\alpha^{\triangledown} \,\colon\, F^{\triangledown} \Rightarrow G^{\triangledown}$ whose [[restriction]] $\alpha \colon F \Rightarrow G$ along $\mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown}$ is [[equifibered natural transformation|equifibered]], the following are equivalent:

  1. $\alpha^{\triangledown}$ is [[equifibered natural transformation|equifibered]].

  1. $F^{\triangledown}$ is a [[colimit|colimiting]] [[cocone]].

\end{theorem}
\begin{proof}
First note that the [[2-limit]] of $S\circ G$ is [[equivalence of categories|equivalent]] to the [[full subcategory]] of the [[slice category]] of the [[functor category]]
$$
  \big([\mathcal{D},\mathcal{C}] \Downarrow G\big)
  \hookrightarrow
  [\mathcal{D},\mathcal{C}]/G
$$
consisting of the [[equifibered natural transformation|equifibered transformations]].  Moreover, under this equivalence, the comparison map $S(x) \to \lim (S\circ G)$ is identified with the pullback functor
$$
  \mathcal{C}/x \longrightarrow ([\mathcal{D},\mathcal{C}] \Downarrow G)
  \,.
$$
Now, this functor has a [[left adjoint]] given by taking [[colimits]]. [Therefore](adjoint+functor#FullyFaithfulAndInvertibleAdjoints) it is an [[equivalence of categories|equivalence]] if and only if the [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]] of the adjunction are isomorphisms.

The unit is the map from an equifibered transformation over $G$ into the pullback of its colimit.  The latter underlies an equifibered $\alpha'$ by construction, so the unit is an isomorphism just when $\text{(2)}\Rightarrow \text{(1)}$.  Similarly, the counit is the map into an object over $x$ from the colimit of its pullback.  Thus, it is an isomorphism just when $\text{(1)}\Rightarrow\text{(2)}$.
\end{proof}

\begin{remark}
The condition $\text{(1)}\Rightarrow\text{(2)}$ is precisely the statement that the colimit of $G$ is [[universal colimit|universal]], i.e. preserved by pullback.  The condition $\text{(2)}\Rightarrow\text{(1)}$ is a form of [[descent]].
\end{remark}

### Colimits in Span

+-- {: .num_theorem #ColimitsInSpan}
###### Theorem
A colimit in $C$ is van Kampen if and only if it is preserved by the inclusion $C\to Span(C)$ into the [[bicategory]] of [[spans]] in $C$.
=--
([Sobocinski & Heindel 2011](#SobocinskiHeindel11)).

## Examples

\begin{example}

* A category with [[pullbacks]] is [[lextensive category|lextensive]] just when [[coproducts]] are van Kampen.

* A category with [[pullbacks]] is [[adhesive category|adhesive]] just when [[pushouts]] of [[monomorphisms]] are van Kampen.

* A category with [[pullbacks]] is [[exhaustive category|exhaustive]] just when transfinite [[unions]] of monomorphisms are van Kampen.

\end{example}

\begin{example}\label{RegularCatWithVanKampColimitsIsExact}
A [[regular category]] with van Kampen [[quotients]] of [[congruences]] is [[exact category|exact]].
\end{example}
\begin{proof}
Let $p_1, p_2 : R \rightrightarrows A$ be a congruence. We need to show that $R$ is the [[kernel pair]] of the [[quotient map]] $A \to A/R$.

Let the transitivity of the congruence be witnessed by $t \,\colon\, R \times_A R \to R$. Then, in the following [[diagram]]

$$
\array{
  R \times_A R
    &
  \underoverset{\pi_2}{t}{\rightrightarrows}
    &
  R & \stackrel{p_2}{\to} & A
  \\
  {{}^\mathllap{\pi_1}}\big\downarrow && \big\downarrow{{}^\mathrlap{p_1}} && \big\downarrow
  \\
  R & \underoverset{p_2}{p_1}{\rightrightarrows} & A & \to & A/R
}
$$

the left squares are [[pullbacks]] and the top diagram is a [[split coequalizer]] (with the splitting maps given by reflexivity), [hence](split+coequalizer#AsCoequalizers) a [[coequalizer]]. Now since the bottom coequalizer is assumed to be van Kampen, Thm. \ref{UniversalityAndDescent} implies that also the right square is a pullback. But this is the desired statement that $R$ is the [[kernel pair]] of $A \to A/R$.
\end{proof}

\begin{example}
In [[Set|$Set$]], the [[pushout]] square
$$
  \array{
    2 & \to & 1
    \\
    \big\downarrow && \big\downarrow
    \\
    1 & \to & 1
  }
$$
is not van Kampen, neither is the [[coequalizer]] diagram $\mathbf 1 \rightrightarrows \mathbf 1 \to \mathbf 1$.

On the other hand, in an [[(∞,1)-topos]] (say [[∞Grpd]]), the descent property of the [[circle]] seen [[circle type#types_equivalent_to_the_circle_type|as a coequalizer]] gives a concise proof that the [[loop space]] of the circle is equivalent to the [[integers]]: in the diagram below, the two squares on the left are pullbacks because both $\mathrm{id}$ and $s$ are isomorphisms, and the top row is a [[homotopy coequalizer]] diagram because the [[line type]] is [[contractible]]. Furthermore, there is a higher coherence relating the two homotopies obtained by composing the top cofork with the back and right squares and by composing the front and right squares with the bottom cofork, ensuring that this diagram is [[homotopy coherent]].

$$
\array{
  \mathbb{Z}
    &
  \underoverset{\mathrm{id}}{s}{\rightrightarrows}
    &
  \mathbb{Z} & \to & \mathbf{1}
  \\
  \big\downarrow && \big\downarrow && \big\downarrow
  \\
  \mathbf{1} & \rightrightarrows & \mathbf{1} & \to & S^1
  \mathrlap{\,.}
}
$$

Then, descent implies that the square on the right is a pullback, which says exactly that the [[path space]] between any two points of the circle is [[homotopy equivalence|homotopy equivalent]] to the [[integers]] $\mathbb{Z}$.
\end{example}

## In higher categories
 {#InHigherCategories}

There is an evident generalization of the definition to [[higher category theory|higher categories]], and in particular to [[(∞,1)-categories]].  In the latter case, van Kampen colimits exactly characterize descent.  In particular,we have:

* A [[locally presentable (∞,1)-category]] is an [[(∞,1)-topos]] just when *all* colimits are van Kampen.

If we take Theorem \ref{UniversalityAndDescent} as the definition of "van Kampen colimit", this follows from Theorem 6.1.3.9 of [[HTT]], see also around ([[(infinity,2)-Categories and the Goodwillie Calculus|Lurie 2Cats+Goodwillie, example 1.2.3]]).  The $(\infty,1)$-categorical version of Theorem \ref{UniversalityAndDescent} may not exactly be in the literature, however.  The cited theorem of HTT essentially gives Theorem \ref{UniversalityAndDescent} under the additional global hypothesis that all colimits in $C$ are universal.

In this case, being van Kampen is also equivalent to being preserved by the composite of $S$ with the [[core]] functor $Core : (\infty,1)Cat \to \infty Gpd$.  Thus, the [[adjoint functor theorem]] (specifically, the [[representable functor theorem]]) implies that (assuming all colimits to be universal), all colimits are van Kampen just when all small versions of $Core \circ S$ are [[representable functor|representable]], i.e. when [[object classifiers]] exist.

## Related pages

* [[exactness property]]

* [[object classifier in an (∞,1)-topos]]

* [[descent]] / [[universal colimit]]

* [[strict initial object]]

## References

* {#SobocinskiHeindel11} [[Pawel Sobocinski]], [[Tobias Heindel]], *Being Van Kampen is a universal property*, Logical Methods in Computer Science, **7** 1 (2011) [[arXiv:1101.4594](https://arxiv.org/abs/1101.4594), doi:10.2168/LMCS-7(1:14)2011]

[[!redirects van Kampen colimits]]


van Kampen theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The *Seifert-van Kampen theorem* is a classical theorem in [[algebraic topology]] which computes the [[fundamental group]] of a pointed [[topological space]]   in terms of a decomposition into [[open subsets]].

It is most naturally expressed by saying that the [[fundamental groupoid]] [[functor]] preserves certain [[colimits]].  Here there is a bifurcation in possible generalizations, however.  The colimits of spaces we consider (covers by open subsets) are both strict colimits and [[homotopy colimits]].  But we can ask the resulting colimit of groupoids to be either a [[homotopy colimit]] (i.e. a [[2-colimit]]) or a [[strict 2-colimit|strict colimit]].  The first is more natural for formal work; the second is more useful for computation.  Accordingly, we will speak of *homotopy van Kampen theorems* and *strict van Kampen theorems*.

## Statement

### For fundamental groups

The most classical version of the **Seifert-van Kampen theorem** for the [[fundamental group]] is the following.

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

Let $X$ be a [[topological space]] [[covering|covered]] by [[open subsets]] $U,V \subset X$ such that $U \cap V$ is [[path-connected space|path connected]]. Then for every choice of basepoint $x \in U \cap V$, the [[diagram]] of [[homotopy group]]s

$$
  \array{
    \pi_1(U \cap V, x) &\to& \pi_1(U,x)
    \\
    \downarrow && \downarrow
    \\
    \pi_1(V,x) &\to& \pi_1(X,x)
  }
$$

is a [[pushout]] square in [[Grp]].

=--

### From groups to groupoids

We can regard the classical theorem as being a statement about fundamental groupoids as follows.  Recall that any group $G$ can be considered as a 1-object groupoid $\mathbf{B}G$ by [[delooping]].  It is easy to see that the functor $\mathbf{B}$ takes colimits of groups to strict colimits of groupoids.  Moreover, a strict pushout of one-object groupoids is also a 2-pushout, because the maps we are pushing out are [[bo functor|bijective on objects]], hence [[cofibrations]] in the [[canonical model structure]] on $Gpd$.  Thus, under the hypotheses of the classical van Kampen theorem, we have a square

$$
  \array{
    \mathbf{B}\pi_1(U \cap V, x) &\to& \mathbf{B}\pi_1(U,x)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}\pi_1(V,x) &\to& \mathbf{B}\pi_1(X,x)
  }
$$

which is both a strict pushout and a [[2-colimit|2-pushout]] in $Gpd$.

Now let $X'$ be the [[path component]] of $X$ which contains $U\cap V$ (there is exactly one such, because $U\cap V$ is path-connected), and similarly $U'$ and $V'$.  Then the [[fundamental groupoid]] $\Pi_1(X')$ is equivalent to $\mathbf{B}\pi_1(X,x)$, and similarly in the other cases.  Since 2-pushouts are invariant under equivalence of groupoids, we also have a 2-pushout square

$$
  \array{
    \Pi_1(U \cap V) &\to& \Pi_1(U')
    \\
    \downarrow && \downarrow
    \\
    \Pi_1(V') &\to& \Pi_1(X')
  }
$$

This is an instance of what we referred to above as a *homotopy van Kampen theorem*.  But we can easily make it a *strict van Kampen theorem* as well.

Note that the functors $\Pi_1(U\cap V) \to \Pi_1(U')$ and $\Pi_1(U\cap V) \to \Pi_1(U')$ are injective on objects, hence cofibrations in the canonical model structure, and so their strict pushout is also a 2-pushout.  This strict pushout is therefore equivalent to $\Pi_1(X')$.  (It need not be isomorphic to it, but this can be remedied by defining the *fundamental groupoid on a set of specified basepoints* as below.)

We now consider more general versions of the van Kampen theorem which do not require $U\cap V$ to be path-connected, and which compute all of $\Pi_1(X)$ rather than just $\Pi_1(X')$.

### Strict van Kampen theorem for groupoids

Let us define $\Pi_1(X,X_0)$ of a space $X$ and a subset $X_0 \subseteq X$ to be the full subgroupoid of $\Pi_1(X)$ on the set $X_0$.  More generally, if $U$ is given as a subspace of $X$ and $X_0 \subseteq X$, we write $\Pi_1(U,X_0)$ for $\Pi_1(U,X_0\cap U)$.

If $X_* =(X,X_0)$ is a pair consisting of a space $X$ and a set $X_0$ of base points, we say $X_*$ is _connected_ if $X_0$ meets each path component of $X$.  This is clearly a necessary and sufficient condition for $\Pi_1(X,X_0)$ to be equivalent to $\Pi_1(X)$.

Now let $\{U^i \to X\}$ for $i \in I$ be an [[open cover]] of $X$.  If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$, and similarly for $n$-fold intersections.  For any $n$-tuple $e\in I^n$, let $U^e_*$ be the pair $(U^e,X_0 \cap U^e)$.  We then have a [[coequaliser]] diagram of pairs of spaces, where $a,b,c$ are determined by inclusions:

$$
  \bigsqcup_{d \in I^2} U^d_*
    \stackrel{\overset{a}{\to}}{\underset{b}{\to}}
  \bigsqcup _{i \in I} U^i_* \xrightarrow{c} X_*
  \,.
$$

+--{.un_theorem}
###### Theorem

If the pairs of spaces $U^f_*$ are connected for all 1-, 2-, or 3-fold  intersections $U^f_*$ of the pairs $U^i_*$, then

1. (Conn) The pair $X_*$ is connected; and

1. (Iso) The [[fundamental groupoid]] functor $\Pi_1$ takes the above coequaliser diagram of pairs of spaces to a (strict) coequaliser diagram of groupoids.

=--

The limit "3" on the connectivity hypothesis may be analyzed as "$1+2$", where $1$ is the categorical dimension of the fundamental groupoids we are considering and $2$ is a universal constant.  This becomes evident when generalizing to [[higher homotopy van Kampen theorems]].

In one proof of the theorem (due to Brown and Razak, see below), the number 3 arises from the Lebesgue covering dimension of $\mathbb{R}^2$.  This proof verifies the universal property for a coequaliser, using basic techniques: subdivide a path;  deform a subdivision so that it is product of paths joining points of $X_0$; subdivide a homotopy rel end points, and deform this subdivision so that _all_ subpaths join points of $X_0$; any composition of [[commutative square]]s in a groupoid is commutative.

In another, more categorical proof explained in the next section, the number $3$ arises from the fact that 2-colimits can be computed using coproducts and [[descent objects]] of 2-truncated [[simplicial objects]].

Note that the theorem about groupoids does _not_ reduce to a theorem about groups. For example, $X$ may be a connected space which is the union of two open sets each with many components and whose  intersection has many path components. Such examples arise commonly from applications to combinatorial group theory; consider for example a covering space of a wedge of spaces.    Or the connected $X$ may be the union of 23 open sets whose three fold intersections have 123 path components. In each case the fundamental group one might want to calculate  is in the middle of this complicated combinatorial situation, but at least the theorem has turned a topological problem into a group theory and combinatorial problem, and the remarkable fact is that the fundamental groups are _completely_ determined by the theorem. This is an "anomaly" for traditional algebraic topology, where invariants relating adjacent dimensions may be determined by exact sequences which do not give complete information. The reason for the success seems to be that groupoids have structure in dimensions 0 and 1, and so can model the gluing of spaces.

This suggests that [[higher homotopy van Kampen theorem]]s could give new kinds of homotopical information, i.e. colimit theorems for higher homotopy invariants, which has proved to be so.

### Homotopy van Kampen theorem for groupoids

From a 2-categorical point of view, the strict van Kampen theorem above can be decomposed into the following parts:

1. When $X$ is the union of a family of open sets, then it is the homotopy colimit of a diagram involving these open sets.

1. The fundamental groupoid functor preserves homotopy colimits (i.e. takes them to homotopy colimits, or equivalently 2-colimits, of groupoids).

1. Certain 2-colimits of groupoids can be computed over "2-truncated" subdiagrams.

1. In certain situations, strict colimits of groupoids are also 2-colimits.

We can then prove the theorem by proving each of these three parts separately.  This is essentially accomplished in [Farjoun](#Farjoun).

+-- {: .num_theorem #Cechcover}
###### Theorem
If a space $X$ is the union of a family of open sets $(U^i)_{i\in I}$, then we have a [[weak equivalence]]
$$ hocolim C_\bullet(U) \xrightarrow{\sim} X, $$
where $C_\bullet(U)$ is the [[Cech cover]] of $X$, a [[simplicial space]] defined by
$$C_n(U) = \coprod_{J\subseteq I,\, |J|=n} \bigcap_{i\in J} U^i. $$
=--

See [Dugger-Isaksen 01](#DuggerIsaksen01).

+-- {: .num_theorem #Pi1Hocolim}
###### Theorem
Suppose $I$ is a small category and $(X^i)_{i\in I}$ is a [[diagram]] of spaces.  Then we have an equivalence of groupoids
$$ \Pi_1(\hocolim_i X^i) \simeq \hocolim_i \Pi_1(X^i). $$
=--
+-- {: .proof}
###### Proof
The functor $\Pi_1$ is, up to equivalence, the composite of itself with the 1st [[Postnikov approximation]] functor $P_1 : Top \to Top$.  We can regard $P_1$ as a [[Bousfield localization]] functor for $Top$ (for localizing at the map $S^2 \to \ast$), so it preserves homotopy colimits.  And $\Pi_1$ is part of a [[Quillen equivalence]] between $Gpd$ and the 1-localized model structure on $Top$, so it also preserves homotopy colimits.
=--

+-- {: .num_theorem #Codescent}
###### Theorem
If $X$ is the union of $(U^i)_{i\in I}$, and $X_0\subseteq X$ is a set of basepoints which intersects each 1-, 2-, and 3-fold intersection of opens $U^i$, then $\Pi_1(X,X_0)$ is equivalent to the [[2-colimit]] (i.e. [[codescent object]]) of the following diagram
$$ \coprod_{i,j,k} \Pi_1(U^{i j k},X_0) \;\underoverset{\to}{\to}{\to}\;
\coprod_{i,j} \Pi_1(U^{i j},X_0) \;\rightrightarrows\;
\coprod_i \Pi_1(U^i,X_0).
$$
=--
+-- {: .proof}
###### Proof
If we neglect the $X_0$'s, then what we have above is the 2-truncated part of the homotopy colimit of groupoids obtained by applying Theorem \ref{Pi1Hocolim} to Theorem \ref{Cechcover}.  Since homotopy colimits of simplicial objects in $Gpd$ are equivalent to codescent objects of their 2-truncations (essentially because groupoids are 1-groupoids), this has an equivalent homotopy colimit.  Finally, the assumption on $X_0$ ensures that restricting to it as a set of basepoints does not change the equivalence type of any of the groupoids in the 2-truncated diagram.
=--

+-- {: .num_theorem #CodescentCoeq}
###### Theorem
Under the hypotheses of Theorem \ref{Codescent}, we have that $\Pi_1(X,X_0)$ is isomorphic to the strict coequalizer of
$$
\coprod_{i,j} \Pi_1(U^{i j},X_0) \;\rightrightarrows\;
\coprod_i \Pi_1(U^i,X_0).
$$
=--
+-- {: .proof}
###### Proof
Let $A = P(I)$ be the power set of the indexing set $I$, regarded as a poset, and consider the functor $F:A\to Gpd$ defined by $F(J) = \Pi_1(\bigcap_{i\in J} U^i, X_0)$.  We can verify that the 2-colimit of $F$ is equivalent to the codescent object in Theorem \ref{Codescent}, for essentially the same reasons cited there.  (Formally, one can say that the functor $P_{\le 3}(I) \to A$ is "2-final".)

However, $F$ is cofibrant in the [[projective model structure]] on $Gpd^A$, which is to say that the functor $A\xrightarrow{F} Gpd \xrightarrow{ob} Set$ is free (relative to the forgetful functor $Set^A \to Set^{ob(A)}$): its generators are the points $x\in F(J)$ such that $J = \{ i\in I \,|\, x\in U^i \}$.  Thus, the 2-colimit of $F$ is equivalent to its strict colimit, which in turn is easily seen to be the coequalizer shown above.

This shows that $\Pi_1(X,X_0)$ is equivalent to the strict coequalizer shown above.  But this equivalence is clearly also bijective on objects, so it is an isomorphism of groupoids.
=--

### Examples

Let $X=S^1$, $U= S^1 \setminus \{N\}$, and $V= S^1 \setminus \{S\}$, where $N$ and $S$ are antipodal points.  Then $U\cong V\cong (0,1)$ are contractible, while $U\cap V \cong (0,1) \sqcup (0,1)$.  Thus the homotopy van Kampen theorem gives a 2-pushout of groupoids that is equivalent to
$$ \array{ \ast\sqcup \ast & \to & \ast \\
  \downarrow && \downarrow \\
  \ast & \to & \Pi_1(S^1) } $$
It's not hard to compute that this 2-pushout must be $\mathbf{B}\mathbb{Z}$.  But we can also obtain this directly from the strict van Kampen theorem with a set $X_0 = \{u,v\}$ of two basepoints with $u\in U$ and $v\in V$, which yields a strict pushout of groupoids
$$ \array{ \ast\sqcup \ast & \to & \mathcal{I} \\
  \downarrow && \downarrow \\
  \mathcal{I} & \to & \Pi_1(S^1,X_0) } $$
where $\mathcal{I}$ is the invertible [[interval category]], which is (literally) $\Pi_1(U,X_0)$ and $\Pi_1(V,X_0)$.

Note that the version of the theorem for fundamental groups does not apply, since $U\cap V$ is not path-connected.

### van Kampen theorem in homotopy type theory

The van Kampen theorem can be expressed in [[homotopy type theory]], a type theory under which one could do [[synthetic homotopy theory]], as follows:

Let $A, B, C$ be [[types]] and $f : A \to B$ and $g : A \to C$ [[functions]]. Let $P\equiv B \sqcup_A C$ be their [[pushout]].

By defining a special *code* function using the [[encode-decode method]] the van Kampen theorem says that:

$$\Pi_1 P(u, v) \simeq code(u,v)$$

There are (at least) two code functions that can be given, which are listed in section 8.7 of the [[HoTT Book]]. (it would be good if somebody could transcribe the code functions over here).  The first is a "naive" van Kampen theorem. The second version is the van Kampen at a set of basepoints.

## Generalizations

* [[van Kampen theorem for toposes]]

* [[higher homotopy van Kampen theorem]]

## References

See also this [mathoverflow discussion](http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one)

The use of a set of base points for a pushout theorem and so determining the fundamental group of the circle was published in

* [[Ronnie Brown]], _Groupoids and van Kampen's theorem_, Proc. London Math. Soc. (3) 17 (1967) 385-401.

The more general theorem for the fundamental groupoid on a set of base points:

* [[Ronnie Brown]] and A. Razak, _A van Kampen theorem for unions of non-connected  spaces_, Archiv. Math. 42 (1984) 85-88. [pdf](http://pages.bangor.ac.uk/pdffiles/brown-razak.pdf)

The pushout theorem is among  applications of groupoids to $1$-dimensional homotopy theory (e.g. gluing theorem, covering spaces, orbit spaces) and to the Jordan Curve Theorem in:

* [[Ronnie Brown]], _[Topology and Groupoids](http://groupoids.org.uk/topgpds.html)_, Booksurge (2006). pdf available;  book available from amazon.

The proof for  pushouts  is discussed in some detail in

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_

as a background to the following chapters  on versions for crossed modules and crossed complexes,  i.e. for higher dimensional versions.

The proof via homotopy colimits is roughly taken from

* {#Farjoun} [[Emmanuel Dror Farjoun]], _Fundamental group of homotopy colimits_, Adv. in Math. 182 (2004), 1-27.  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/farjoun.pdf))

Other references cited above:

* {#DuggerIsaksen01} [[Dan Dugger]], [[Daniel Isaksen]], _Hypercovers in topology_ ([arxiv:math/0111287](https://arxiv.org/abs/math/0111287))

Discussion of [[persistent homotopy]] with focus on the [[van Kampen theorem]], [[excision]] and the [[Hurewicz theorem]]:

* Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, *Persistent Homotopy* $[$[arXiv:1909.08865](https://arxiv.org/abs/1909.08865)$]$

For the van Kampen theorem in [[homotopy type theory]]:

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

Formalization of the van Kampen theorem in [[proof assistants]]:

* [In Agda](https://github.com/HoTT/HoTT-Agda/tree/e48a16bcd719e2fcb409d79e3f7df6c6b81223bb/theorems/homotopy/vankampen)
* [In Lean](https://github.com/leanprover/lean2/blob/master/hott/homotopy/vankampen.hlean)

[[!redirects Seifert-van Kampen theorem]]


van Kampen theorem for toposes

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[van Kampen theorem]] from [[Top]] to [[Topos]].

## References

* [[Marta Bunge]], [[Steve Lack]], _van Kampen theorem for toposes_ ([ps](http://maths.mq.edu.au/~slack/papers/vkt.ps.gz))


Van Quine

Willard Van Orman Quine (1908–2000) was an analytic philosopher and logician / set theorist.

## References

* Stanford Encyclopedia of Philosophy
    - _[Willard Van Orman Quine](https://plato.stanford.edu/entries/quine/)_
    - _[Quine's New Foundations](https://plato.stanford.edu/entries/quine-nf/)_

* ⋮

## Selected writings

* Roger F. Gibson (ed.) *Quintessence -- Basic readings from the philosophy of W. V. Quine*, The Belknap Press of Harvard University Press (2004) [[ISBN:9780674027558](https://www.hup.harvard.edu/catalog.php?isbn=9780674027558)]

## Related entries

* [[salva veritate]]

category: reference

[[!redirects Willard Van Orman Quine]]
[[!redirects W. V. O. Quine]]


Vandermonde determinant

The __Vandermonde determinant__ or Vandermonde polynomial $\Delta(x_1,\ldots,x_n)$ is the determinant of the Vandermonde matrix

$$
V(x_1,\ldots,x_n) = \left( \array{1 & x_1 & \cdots & x_1^{n-1}\\ 1 & x_2 &\cdots & x_2^{n-1}\\ \cdot &\cdot &\cdot &\cdots\\ 1 & x_n &\cdots &x_n^{n-1}}\right)
$$

+-- {: .num_prop}
###### Proposition
$\Delta(x_1,\ldots,x_n)= \prod_{1\leq i\lt j\leq n} (x_j - x_i)$.
=--

+-- {: .proof}
###### Proof
We argue by induction. WLOG, assume $x_1, \ldots, x_{n-1}$ are distinct (else the value is zero as claimed), and treat these as constants. Write

$$\det \left( \array{1 & x_1 & \cdots & x_1^{n-1}\\ 1 & x_2 &\cdots & x_2^{n-1}\\ \cdot &\cdot &\cdot &\cdots\\ 1 & x &\cdots &x^{n-1}}\right) = f(x) = \sum_{i=1}^{n-1} a_i x^i.$$

Then the leading coefficient $a_{n-1}$ is $\prod_{1\leq i\lt j\leq n-1} (x_j - x_i)$ by inductive hypothesis, and $f(x)$ has simple roots at $x = x_1, \ldots, x_{n-1}$ since each of those values makes two rows equal (hence with zero determinant). It follows that

$$f(x) = a_{n-1}\prod_{i=1}^{n-1} (x - x_i) = (\prod_{1\leq i\lt j\leq n-1} (x_j - x_i))\prod_{i=1}^{n-1} (x - x_i)$$

and the inductive step follows by setting $x = x_n$.
=--

The Vandermonde determinant appears in many important situations, as a square root of a [[discriminant]], sometimes as a [[Wronskian]], and also in Lagrange interpolation (see wikipedia [Lagrange polynomial](http://en.wikipedia.org/wiki/Lagrange_polynomial)), [[Selberg integral]] etc.

* wikipedia [Vandermonde matrix](http://en.wikipedia.org/wiki/Vandermonde_matrix)
* A. Varchenko, Multidimensional Vandermonde Determinant, Uspekhi Mat. Nauk 43:4 (1988), p. 190.
* A. Varchenko, _Beta-function of Euler, Vandermonde determinant, Legendre equation and critical values of linear functions of configuration of hyperplanes_, I. Izv. Akademii Nauk USSR, Seriya Mat., 53:6 (1989), 1206-1235.
[[!redirects Vandermonde matrix]]


vanishing at infinity

# Contents
* table of contents
{: toc}

## Idea

A [[map]] $f$ between [[spaces]] (say, a [[continuous map]] between [[topological spaces]]) _vanishes at infinity_ if $f(x)$ gets arbitrarily close to [[zero]] as $x$ gets sufficiently close to infinity.

For a map $f\colon X \to Y$, we need a notion of being close to $0$ in $Y$, so take $Y$ to be a [[pointed space]]; then getting arbitrarily close to $0$ means entering any [[neighbourhood]] of the basepoint.  We also need a notion of being close to infinity in $X$, so take $X$ to be a [[locally compact Hausdorff space]]; then getting sufficiently close to infinity means entering the [[exterior]] of some [[compact subspace]].  (To interpret 'getting', of course, we may use [[nets]].)  It is likely, however, that further generalisations are possible.

## Definitions

Let $X$ and $Y$ be [[topological spaces]], and let $f$ be a [[continuous map]] (or potentially any [[function]]) from $X$ to $Y$.  Let $Y$ be [[pointed space|pointed]], and let $X$ be [[locally compact Hausdorff space|locally compact Hausdorff]].

+-- {: .num_defn}
###### Definition
The map $f\colon X \to Y$ __vanishes at infinity__ if for every [[neighbourhood]] $N$ of the [[basepoint]] in $Y$, there is [[compact subspace]] $K$ of $X$ such that $f(x)$ belongs to $N$ whenever $x$ lies in the [[exterior]] of $K$ in $X$.
=--

In case $Y$ is a pointed [[metric space]] (such as a [[Banach space]], with basepoint $0$; or in particular the [[real line]], with basepoint $0$), then we may equivalently say:

* For every [[positive number]] $\epsilon$, there is [[compact subspace]] $K$ of $X$ such that ${\|f(x)\|} \lt \epsilon$ whenever $x$ lies in the [[exterior]] of $K$ in $X$.

(Here, ${\|{-}\|}$ is the [[norm]] in a Banach space, or more generally the distance from the basepoint in any pointed metric space.)

## Properties

### Relation to compactifications

One way of considering this definition is that one can adjoin to $X$ a point "at infinity", denoted $\infty$, by declaring that the [[open neighborhoods]] of $\infty$ are sets of the form $\{\infty\} \cup Ext(K)$ for $K \subset X$ compact. This is called the [[one-point compactification]], denoted $X^{cpt}$. Then a [[continuous function]] $f \colon X \to A$ vanishes at infinity equivalently if it [[extension|extends]] $f$ to a map $X^{cpt} \to A$, continuous at $\infty$ (at least) that sends $\infty$ to $0$ -- thus _literally- "vanishing at $\infty$".

For more see

* at _[[one-point compactification]] -- [Relevance for Monopoles and Instantons](one-point+compactification#RelevanceForMonopolesAndInstantons)_

* at _[Yang-Mills instanton -- SU(2)-instantons from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_.

## References

* Wikipedia, _[Locally compact space](http://en.wikipedia.org/wiki/Locally_compact_space)_

[[!redirects vanishing at infinity]]
[[!redirects vanish at infinity]]
[[!redirects vanishes at infinity]]


vanishing cycle

There is a special classical case due to Lefschetz. In SGA 7.II, [[Deligne]] proposed the framework of vanishing and nearby cycle functors. Cf. also [[Milnor fiber]].

* Pierre Deligne, _Le formalisme des cycles évanescents_, Exposés XIII, XIV in:Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II). Dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics __340__,  Springer-Verlag 1973. x+438 pp, [MR354657](http://www.ams.org/mathscinet-getitem?mr=354657)

* wikipedia [vanishing cycle](http://en.wikipedia.org/wiki/Vanishing_cycle); eom: [vanishing cycle](http://eom.springer.de/v/v096070.htm), [monodromy transformation](http://eom.springer.de/M/m064700.htm)

* S. Lefschetz, _L'analysis situs et la géométrie algébrique_, Gauthier-Villars, Paris, 1924; Jbuch 50, 663 (reprinted in his _Selected papers_, Chelsea Publ. 1971, [MR299447](http://www.ams.org/mathscinet-getitem?mr=299447))

* [[V. Ginzburg|V. Ginsburg]], _Characteristic varieties and [[vanishing cycle]]s_, Inv. Math. __84__, 327–402 (1986)

* [[Luc Illusie]], _Vanishing cycles over general bases after P. Deligne, O. Gabber, G. Laumon and F. Orgogozo_, [pdf](http://www.math.u-psud.fr/~illusie/vanishing1b.pdf)

+ MathOverflow: [is-there-a-good-way-to-think-of-vanishing-cycles-and-nearby-cycles](http://mathoverflow.net/questions/1590/is-there-a-good-way-to-think-of-vanishing-cycles-and-nearby-cycles), [higher-vanishing-cycles](http://mathoverflow.net/questions/871/higher-vanishing-cycles)

* Chris Peters, _Tata lectures on motivic aspects of Hodge theory_, [pdf](http://www-fourier.ujf-grenoble.fr/~peters/motivic.f/Tatalects.pdf)

* [[Masaki Kashiwara]], _Vanishing cycle sheaves and holonomic systems of differential equations_, [pdf](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/vanishing.pdf); in: Algebraic geometry (Tokyo/Kyoto, 1982), 134–142, Lecture Notes in Math. __1016__, Springer 1983.

* Mihai Tibăr, _Polynomials and vanishing cycles_, Cambridge Tracts in Mathematics __170__, 2007. xii+253 pp. [MR2008m:32048](http://www.ams.org/mathscinet-getitem?mr=2360234)

* [[A.A. Beilinson]], _How to glue perverse sheaves_, [pdf transcription](
http://www.math.harvard.edu/~gaitsgde/grad_2009/Beilinson%20--%20How%20to%20glue%20perverse%20sheaves.pdf).

* [[Ryan Reich]], _Notes on Beilinson's "How to glue perverse sheaves"_, [arxiv/1002.1686](http://arxiv.org/abs/1002.1686)

* [[Sam Lichtenstein]], _Vanishing cycles for algebraic D-modules_, Harvard senior thesis (adv. [[D. Gaitsgory]]) March 2009, pdf

* [[Alexander Beilinson]], [[Dennis Gaitsgory]], _A corollary of the b-function lemma_, [arXiv:0810.1504](http://xxx.lanl.gov/abs/0810.1504).

* [[A. Beilinson]], [[J. Bernstein]], _A proof of Jantzen conjectures_, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf).

* Ofer Gabber, _Gersten's conjecture for some complexes of vanishing cycles_, [MR1305746](http://www.ams.org/mathscinet-getitem?mr=1305746), [doi](http://dx.doi.org/10.1007/BF02568202)
* There is a number of related preprints of D. Massey, cf. [here](http://www.massey.math.neu.edu/Massey/Massey_docs/preprints.html) including lecture notes on [[perverse sheaves]] and v.c. [pdf](http://www.massey.math.neu.edu/Massey/Massey_preprints/perverse_notes.pdf)
* C. Sabbah, _Proximité évanescente_, Compos. Math. __62__, 283-328, 1987.
* Dirk Siersma, _Vanishing cycles and special fibres_, in: Singularity theory and its applications I_ (Coventry, 1988/1989), 292–301, Springer LNM __1462__, 1991, [MR92j:32129](http://www.ams.org/mathscinet-getitem?mr=1129039), [doi](http://dx.doi.org/10.1007/BFb0086389)
* P. Seidel, _Fukaya categories and Picard-Lefschetz theory_, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. [MR2009f:53143](http://www.ams.org/mathscinet-getitem?mr=2441780), [doi](http://dx.doi.org/10.4171/063)
* A. Dimca, _Sheaves in topology_, Springer 2004, chap. 4.
[[!redirects vanishing cycles]]


vanishing ideal

## Idea

Let $X$ be some kind of [[space]] with $\mathcal{O}(X)$ some kind of [[ring]] of [[functions]] on $X$. Then for $S \subset X$ a subspace, of kind, its _vanishing ideal_ is the [[ideal]] $I(S) \subset \mathcal{O}(X)$ of all those functions that vanish when restricted to $S$.

## Related concepts

* [[Zariski topology]]

[[!redirects vanishing ideals]]


Vardan Oganesyan

* [personal page](https://www.vardanoganesyan.com/)

## Selected writings

A proposal for [[homotopy theory]] of [[symplectic manifolds]]:

* [[Vardan Oganesyan]], *The first step towards symplectic homotopy theory* [[arXiv:2304.10529] (https://arxiv.org/abs/2304.10529)]

category: people


variable

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

# Contents
* table of contents
{: toc}

## Idea

In [[logic]] and [[type theory]], a variable is a symbol that stands for an arbitrary instantiation of some given [[type]] (although in single-typed theories the type is trivial).  Thus, every variable of type $T$ is a [[term]] of type $T$ (although typically there are other terms).

Logicians have developed several ways to precisely specify what variables are and how to keep track of them, and the only thing more annoyingly pedantic than learning one of these is translating between different ones.

## Free and bound variables

If we keep track of [[context]], every [[context extension|introduction of a variable]] changes the context.  Thus, whatever [[terms]] of [[type]] $T$ there may (or may not) be in a given context $\Gamma$, in the context $(\Gamma, x\colon T)$ ---which is $\Gamma$ extended by a __free variable__ $x$ of type $T$--- there is (at least) one more term of type $T$, and that term is $x$ itself.

Conversely, we may be able to take a term $a$ in the context $(\Gamma, x\colon T)$ and apply some operation $f$ to it to create a term $f(a)$ (possibly of a different type) in the context $\Gamma$; then any appearances of the variable $x$ in the term $a$ have become __bound variables__ in the term $f(a)$.  That is, the appearances of $x$ are bound by the operator $f$, and $x$ has no meaning outside of its scope.

So for example, if (say in something simple like [[Peano arithmetic]], supplemented with the abbreviations used in practice) I write $2x$, this is a term (for a natural number) in a context with a free variable $x$, whereas if I write $3 \sum_{x=0}^4 2x + 5$, this is a term (also for a natural number) in a context with no free variables; here, $x$ is bound.  The variable $x$ has a meaning after the operator $\sum_{x=0}^4$ (whose notation also specifies the variable that will have a meaning inside it) and before the plus sign (which, following the standard order of operations, marks the end of the scope of $\sum_{x=0}^4$) but has no meaning outside of that (where the $3$ and $5$ are, or even where the $0$ and $4$ are).  And indeed, the ultimate meaning of $2x$ depends on what $x$ is, but the ultimate meaning of $3 \sum_{x=0}^4 2x + 5$ is simply $65$.

## In categorical semantics

In the [[internal language]] of a [[category]] $\mathcal{C}$ a [[morphism]]

$$
  f\colon B \to A
$$

is a _[[term]]_ $f(x)$ of _[[type]]_ $A$ where $x$ is
a **free variable** of _type_ $B$, which in symbols is given by

$$
  x\colon B \vdash f(x)\colon A
  \,.
$$

We may think of the _free variables_ here as being placeholders for all the [[generalized elements]]
$U \stackrel{x}{\to} B$ of $B$.

Then the assertion

$$
  x\colon B \vdash f(x)\colon A
$$

indicates that with $B \stackrel{f}{\to} A$ given we may send $U \stackrel{x}{\to} B$ to the [[composition]]

$$
  (U \stackrel{x}{\to} B \stackrel{f}{\to}A ) = (U \stackrel{f(x)}{\to} A)
  \,.
$$

A free variable becomes a **bound variable** after application of a [[quantifier]]: for instance the image of $f\colon P \to X$ under [[base change]]

$$
  \mathcal{C}_{/X} \stackrel{\to}{\stackrel{\leftarrow}{\to}}
  \mathcal{C}
$$

represents, for the [[left adjoint]] the [[type]]

$$
  \exists x\colon X , P(x)
$$

([[existential quantifier]]), and the the [[right adjoint]] the type

$$
  \forall x\colon X , P(x)
$$

([[universal quantifier]]) in which now $x$ is a _bound variable_.

## Related concepts

* [[quantification]]

* [[variable rule]]

* [[context extension]]

* [[proposition]]

## References
 {#References}

A textbook account with an eye towards applications in [[computer science]] is in section 1.2 of

* {#Harper} [[Robert Harper]], _Practical Foundations for Programming Languages_ ([web](http://www.cs.cmu.edu/~rwh/plbook/book.pdf))

An exposition on the relation between free variables and [[universal quantification]] is in

* [[Andrej Bauer]], _Free variables are not "implicitly universally quantified"!_ ([web](http://math.andrej.com/2012/12/25/free-variables-are-not-implicitly-universally-quantified/))

[[!redirects variable]]
[[!redirects variables]]

[[!redirects free variable]]
[[!redirects free variables]]

[[!redirects bound variable]]
[[!redirects bound variables]]

[[!redirects dependent variable]]
[[!redirects dependent variables]]

[[!redirects metavariable]]
[[!redirects metavariables]]


variable rule

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

#Contents#
* table of contents
{:toc}

## Idea

In [[dependent type theory]], what is known as the *variable rule* (or *projection rule*) is the [[structural rule|structural]] [[inference rule]] which, informally speaking, asserts for any [[dependent type]] $A$ the apparent [[tautology]] that "given a variable $a \colon A$" (as a [[hypothesis]]/[[antecedent]]) "then we have that variable $a \colon A$" (as a [[succedent]]).

$$
  VAR
  \frac{
    \Gamma
    \;\;\;\;
    \vdash
    \;\;\;\;
    A \colon Type
  }{
    \Gamma
    ,\;
    a \colon A
    \;\;\;\;
    \vdash
    \;\;\;\;
    a \colon A
  }
$$

As usual in [[dependent type theory]], the meaning of this rule is a little less trivial than it may superficially seem by the generic [[dependent type|type dependency]] involved: In the [[antecedent]] the variable may be picked out of a [[context]] [[type telescope|telescope]], and in the [[succedent]] it is implicitly [[weakening rule|weakened]] to depend on that full context, -- which includes $a \colon A$ itself!

Accordingly, the [[categorical semantics]] [[categorical semantics of dependent types|as dependent types]] of the variable rule is given by the [[diagonal]] morphism on the [[object]] interpreting $A$, regarded as a [[section]] of the [[pullback bundle|pullback]] of the [[bundle]]/[[display map]] $A \to \Gamma$ to its own total space:

## Related concepts

* [[variable]]

* [[structural rule]]

  [[weakening rule]]

  [[substitution rule]]

  [[exchange rule]]

## References

* {#Jacobs98} [[Bart Jacobs]], p. 121 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

* {#UFP13} [[Univalent Foundations Project]], p. 422 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

  > (in the context of [[homotopy type theory]])

* {#Rijke18} [[Egbert Rijke]], p. 4 of: *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

  > (in the context of [[homotopy type theory]])

[[!redirects identity rule]]