quotient space

> This entry is about the concept in topology. For _[[quotient vector spaces]]_ in linear algebras see there.

***

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#### Topology
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[[!include topology - contents]]
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#### Geometry
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[[!include higher geometry - contents]]
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# Quotient spaces
* table of contents
{: toc}

## Idea

A *quotient space* is a [[quotient object]] in some [[category]] of [[spaces]], such as [[Top]] (of [[topological spaces]]), or [[Loc]] (of [[locales]]), etc.

Often the construction is used for the quotient $X/A$ by a subspace $A \subset X$ (example \ref{QuotientBySubspace} below).

Beware that [[quotient objects]] in the [[category]] [[Vect]] of [[vector spaces]] also traditionally called 'quotient space', but they are really just a special case of [[quotient modules]], very different from the other kinds of quotient space.  However in _[[topological vector spaces]]_ both concepts come together.

## Definitions

### In $Top$

+-- {: .num_defn #QuotientTopologicalSpace}
###### Definition
**(quotient topological space)**

Let $(X,\tau_X)$ be a [[topological space]]  and let

$$
  R_\sim \subset X \times X
$$

be an [[equivalence relation]] on its underlying set. Then the _quotient topological space_  has

* as underlying set the [[quotient set]] $X/\sim$, hence the set of [[equivalence classes]],

and

* a subset $O \subset X/\sim$ is declared to be an [[open subset]] precisely if its [[preimage]] $\pi^{-1}(O)$ under the canonical [[projection map]]

  $$
    \pi \;\colon\; X \to X/\sim
  $$

  is open in $X$.

To see that this indeed does define a topology on $X/\sim$ it is sufficient to observe that taking pre-images commutes with taking unions and with taking intersections.

Often one considers this with input datum not the equivalence relation, but any [[surjection]]

$$
  \pi \;\colon\; X \longrightarrow Y
$$

of sets. Of course this identifies $Y = X/\sim$ with $(x_1 \sim x_2) \Leftrightarrow (\pi(x_1) = \pi(x_2)) $.
Hence the _quotient topology_ on the codomain set of a function out of any topological space has as open subsets
those whose pre-images are open.

Equivalently this is the _[[final topology]]_ or _[[strong topology]]_ induced on $Y$ by the function $X \to Y$, see at _[Top -- Universal constructions](Top#UniversalConstructions)_.

=--

For this construction the function $X \to Y$ need not even be surjective, and we could generalize to a [[sink]] instead of a single map; in such a case one generally says *[[final topology]]* or *[[strong topology]]*.  See also at _[[topological concrete category]]_.

### In $Loc$

A quotient space in $Loc$ is given by a [[regular subobject]] in [[Frm]].

(More details needed.)

## Examples

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

The [[trigonometric function]]

$$
  (cos(-),sin(-))
    \;\colon\;
  [0,2\pi]
    \longrightarrow
  S^1
   \subset
  \mathbb{R}^2
$$

from the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]] to the unit [[circle]] equipped with the [[subspace topology]] of the [[Euclidean space|Euclidean]] [[plane]]

descends to a [[homeomorphism]] on the quotient space $[0,2 \pi]/(0 \sim 2 \pi)$ by the equivalence relation which identifies the two endpoints of the open interval.

$$
  \array{
    [0,2\pi] &\overset{(cos(-),sin(-))}{\longrightarrow}& S^1
    \\
    \downarrow & \nearrow_{\mathrlap{\simeq}}
    \\
    [0,2\pi]/\simeq
  }
  \,.
$$

=--

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

By the [[universal property]] of the quotient it follows that $[0,2\pi]/(0 \sim 2\pi) \to S^1$ is a [[continuous function]]. Moreover, it is a [[bijection]] on the underlying sets by the $2\pi$-periodicity of sine and coside. Hence it is sufficient to see that it is an [[open map]] (by [this prop.](homeomorphism#HomeoContinuousOpenBijection)).

Since the open subsets of $[0,2\pi]$ are unions of

1. the [[open intervals]] $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,

1. the [[half-open intervals]]  $[0,b)$ and $(a,2\pi]$ with $0 \lt a,b  \lt 2\pi$

and since the projection map $\pi \colon [0,2\pi] \to [0,2\pi]/(0 \sim 2\pi)$ is injective on $(0, 2\pi)$, the open subsets of $[0,2\pi]/(0 \sim 2\pi)$ are unions of

1. the [[open intervals]] $(a,b)$ with $0 \lt a \lt b \lt 2\pi$,

1. the glued [[half-open intervals]]  $(b,2\pi]/(0\sim 2\pi) \cup [0,a)/(0 \sim 2\pi)$ for $0 \lt a,b \lt 2\pi$.

By the $2\pi$-periodicity of $(cos(-),sin(-))$, the image of the latter under $(cos(-),sin(-))$ is the same as the image of $(b, 2\pi + a)$. Since the function $(cos(-),sin(-)) \colon \mathbb{R} \to S^1$ is clearly an open map, it follows that the images of these open subsets in $S^1$ are open.

=--

+-- {: .num_example #QuotientBySubspace}
###### Example
**(quotient by a subspace)**

Let $X$ be a [[topological space]] and $A \subset X$ a [[inhabited set|non-empty]] [[subset]].
Consider the [[equivalence relation]] on $X$ which identifies all points in $A$ with each other. The resulting quotient space (def. \ref{QuotientTopologicalSpace}) is often simply denoted $X/A$.

Notice that $X/A$ is canonically a [[pointed topological space]], with base point the [[equivalence class]] $A/A \subset X/A$ of $A$.

If $A = \emptyset$ is the [[empty space]], then one defines

$$
  X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast
$$

to be the [[disjoint union space]] of $X$ with the [[point space]]. This is no longer a quotient space, but both constructions are unified by the _[[pushout]]_ $i \colon A \to X$ along the map $A \to \ast$, equivalently the [[cokernel]] of the inclusion:

$$
  \array{
    A &\overset{i}{\hookrightarrow}& X
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& X/A
  }
  \,.
$$

This kind of quotient space plays a central role in the discussion of [[long exact sequences in cohomology]], see at _[[generalized (Eilenberg-Steenrod) cohomology]]_.

=--

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

Consider the [[real numbers]] $\mathbb{R}$ equipped with their [[Euclidean space|Euclidean]] [[metric topology]]. Consider on $\mathbb{R}$ the [[equivalence relation]] which identifies all real numbers that differ by a [[rational number]]:

$$
  (x_1 \sim_{\mathbb{Q}} x_2)
    \Leftrightarrow
  \left(
     x_2 - x_1 \in \mathbb{Q} \subset \mathbb{Q}
  \right)
  \,.
$$

Then the quotient space $\mathbb{R}/\sim_{\mathbb{Q}}$ is a [[codiscrete topological space]].

=--

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

We need to check that the only open subsets of $X/\sim_{\mathbb{Q}}$ are the empty set and the entire set $X/\sim_{\mathbb{Q}}$.

So let $U \subset \mathbb{R}/\sim$ be a non-empty subset.

Write $\pi \colon \mathbb{R} \to \mathbb{R}/\sim_{\mathbb{Q}}$ for the quotient projection. By definition $U$ is open precisely if its pre-image $\pi^{-1}(U) \subset \mathbb{R}$ is open. By the Euclidean topology, this is the case precissely if $\pi^{-1}(U)$ is a union of [[open intervals]]. Since by assumption $\pi^{-1}(U)$ is non-empty, it contains at least one open interval $(a,b) \subset \mathbb{R}$, with $a \lt b$. By the density of the rational numbers, there exists a rational number $q \in \mathbb{Q} \subset \mathbb{R}$ with

$$
  0 \lt q \lt b - a
  \,.
$$

By definition of $\sim_{\mathbb{Q}}$ we have for all $n \in \mathbb{Z}$ that all elements in $(a + n q, b + n q) \subset \mathbb{R}$ are $\sim_{\mathbb{Q}}$-equivalent to elements in $(a,b)$, hence that also $(a+q,b+q) \subset \pi^{-1}(U)$. But the union of these open intervals is all of $\mathbb{R}$

$$
  \underset{n \in \mathbb{Z}}{\cup} (q + n q, b + n q)
  \;=\;
  \mathbb{R}
$$

and so $\pi^{-1}(U) = \mathbb{R}$.

=--

## Properties

1. Recall that a map $q \colon X \to Y$ is [[open map|open]] if $q(U)$ is open in $Y$ whenever $U$ is open in $X$. It is not the case that a quotient map $q \colon X \to Y$ is necessarily open. Indeed, the identification map $q \colon I \sqcup \{\ast\} \to S^1$, where the endpoints of $I$ are identified with $\ast$, takes the open point $\ast$ of the domain to a non-open point in $S^1$.

1. Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map $\pi_1 \colon \mathbb{R}^2 \to \mathbb{R}$, which projects the closed locus $x y = 1$ onto a non-closed subset of $\mathbb{R}$. (This is a quotient map, by the next remark.)

1. It is easy to prove that a continuous open surjection $p \colon X \to Y$ is a quotient map. For instance, projection maps $\pi \colon X \times Y \to Y$ are quotient maps, provided that $X$ is inhabited. Likewise, a continuous closed surjection $p: X \to Y$ is a quotient map: $p^{-1}(U)$ is open $\Rightarrow$ $p^{-1}(\neg U)$ is closed $\Rightarrow$ $p(p^{-1}(\neg U)) = \neg U$ is closed $\Rightarrow$ $U$ is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.

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

A [[continuous function]]

$$
  f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y)
$$

whose underlying function $f \colon X \longrightarrow Y$ is [[surjective function|surjective]]
exhibits $\tau_Y$ as the corresponding [[quotient topological space|quotient topology]]
precisely if $f$ sends open and $f$-[[saturated subsets]] in $X$ to open subsets of $Y$.
By [this lemma](saturated+subset#ComplementOfSaturatedSubsetIsSaturated)  this is the case precisely if it sends
closed and $f$-saturated subsets to closed subsets.

=--

## Related concepts

* [[Sullivan model of finite G-quotient]]

* [[image]]

* [[subspace]], [[subframe]]

* [[quotient]], [[quotient stack]], [[quotient type]]

[[!include universal constructions of topological spaces -- table]]

[[!redirects quotient space]]
[[!redirects quotient spaces]]
[[!redirects identification space]]
[[!redirects identification spaces]]

[[!redirects quotient topology]]
[[!redirects quotient topologies]]
[[!redirects identification topology]]
[[!redirects identification topologies]]

[[!redirects quotient locale]]
[[!redirects quotient locales]]

[[!redirects quotient topological space]]
[[!redirects quotient topological spaces]]

[[!redirects topological quotient space]]
[[!redirects topological quotient spaces]]

[[!redirects quotient space topology]]
[[!redirects quotient space topologies]]


quotient stack

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#Contents#
* table of contents
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## Idea

For $\mathcal{T}$ a [[sheaf topos]], $G \in Grp(\mathcal{T})$ a [[group object]] and $V \in \mathcal{T}$ any object, and for $\rho \colon V \times G \to V$ an [[action]] of $G$ on $V$  , the **quotient stack** $V// G$ is the [[quotient]] of this action but formed not in $\mathcal{T}$ but under the inclusion

$$
  \mathcal{T} \hookrightarrow \mathbf{H}
$$

into the [[(2,1)-topos]] over the given [[site]] of definition: it is the quotient after regarding the action as an [[infinity-action]] in $\mathbf{H}$.

This is the geometric version of the notion of _[[action groupoid]]_. A wider notion of the quotient stack may be defined using more general internal groupoids. Indeed,
the [[small fibration]] obtained by the [[externalization]] of an internal groupoid in a site with pullbacks will be a [[fibered category]] which is a candidate for a quotient stack in this context. For many interesting sites, sometimes under additional conditions on the internal groupoid, the resulting small fibration is indeed a stack.

If the [[stabilizer subgroups]] of the [[action]] are [[finite groups]], then the quotient stack is an [[orbifold]]/[[Deligne-Mumford stack]] --the "quotient orbifold".

## Motivation for definition of quotient stack.

Let $G$ be a Lie group action on a manifold $X$ (left action).

We define the quotient stack $[X/G]$ as
$$
[X/G](Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \,\text{is a G-bundle,}\, f \,\text{is}\, G\text{-equivariant}\}.
$$

Morphisms of objects are $G$-equivariant isomorphisms. This definition is taken from Heinloth's [Some notes on Differentiable stacks](https://www.uni-due.de/~hm0002/stacks.pdf).

Given a Lie group action of $G$ on $X$, if we want to associate a stack, we  start with simpler cases which allows us to guess how to define $[X/G]$ in general.

1. Suppose $X$ is trivial and $G$ acts trivially on $X=\{*\}$ then $[X/G]$ should only depend on $G$. We know what stack to associate for a Lie group $G$ i.e., $BG$. Thus, $[X/G]$ should just be $BG$.

2. Suppose $G$ is trivial and $G$ acts on $X$, $[X/G]$ should only depend on $X$. We know what stack to associate for a manifold $X$ i.e., $\underline{X}$. Thus, $[X/G]$ should just be $\underline{X}$.

3. Suppose $G$ is non trivial and $X$ is non trivial and that the action of $G$ on $X$ is free (and proper) so that $X/G$ is a manifold. We know what stack to associate for a manifold $X/G$ i.e., $\underline{X/G}$. Thus, $[X/G]$ should just be $\underline{X/G}$.

For general case of $G$ acting on $X$, we get a Lie groupoid, called the Translation groupoid (or action groupoid) usually denoted by $G\ltimes X$.

* Given a manifold $M$, we have a stack associated to it, namely $\underline{M}$. Given a Lie group $G$, we have a stack associated to it, namely $BG$. Given a Lie groupoid $\mathcal{G}$, we have a stack associated to it, namely $B\mathcal{G}$ i.e., the stack of principal groupoid $\mathcal{G}$ bundles.

For action groupoid $\mathcal{G}=G\ltimes X$, let $B\mathcal{G}$ be the corresponding stack of principal $\mathcal{G}$ bundles. It turns out that $B\mathcal{G}$ is same $[X/G]$ defined above. More details to be found in [this page](https://mathoverflow.net/questions/319038/motivation-for-definition-of-quotient-stack).

 * If action of the Lie group $G$ on the manifold $X$ is free and proper, what we get is   **a manifold** $X/G$. Stack associated to this manifold is $\underline{X/G}$ which we call to be the quotient stack, denote by $[X/G]$.

 * If the action of the Lie group $G$ on the manifold $X$ is not necessarily free and proper, what we get is   **a Lie groupoid** denoted (among other symbols) by $X//G$. Stack associated to this Lie groupoid $X//G$ is $B(X//G)$ which we call to be the quotient stack, denote by $[X/G]$.

## Universal property (??) for Quotient stack
 {#AsFibrationsOverBG}

> (references for what the following paragraphs are getting at are listed [below](#ReferencesAsFibrationsOverBG))

Let $G$ be a Lie group and $X$ be a manifold with a $G$-action.

Supposing that $G$ acts freely and properly on $X$, the quotient stack $[X/G]$ will be the stack $\underline{X/G}$.
This action yields a [[principal bundles|principal $G$-bundle]] of manifolds $X\rightarrow X/G$, which gives a morphism of stacks $\underline{X}\rightarrow \underline{X/G}$. We refer to this stack morphism $\underline{X}\rightarrow \underline{X/G}$ as a principal $G$-bundle of stacks.

More precisely, a stack morphism $\underline{M}\rightarrow \mathcal{D}$ is said to be **representable** if given a manifold $N$ and a stack morphism $\underline{N}\rightarrow \mathcal{D}$, the fiber product $\underline{M}\times_{\mathcal{D}}\underline{N}$ is a manifold.
A representable morphism of stacks is said to be a **principal $G$ bundle of stacks** if the map $\underline{M}\times_{\mathcal{D}}\underline{N}\rightarrow N$ is a principal $G$-bundle of manifolds.
The stack morphism $\underline{X}\rightarrow \underline{X/G}$ is a principal $G$-bundle of stacks, since the map
$X\rightarrow X/G$ is a principal $G$-bundle of manifolds.

The property "$\underline{X}\rightarrow \underline{X/G}$ is a principal $G$-bundle" is the main ingredient in the definition of the quotient stack $[X/G]$.  Irrespective of whether or not $G$ acts freely and properly on $X$, we still want to define a *quotient stack* as a stack $\mathcal{D}$ such that $\underline{X}\rightarrow \mathcal{D}$ is a principal $G$-bundle of stacks in a "minimal" way.

The **quotient stack of the action of $G$ on $X$** is a stack $\mathcal{D}$ equipped with a principal $G$-bundle of stacks $\underline{X}\rightarrow \mathcal{D}$ such that any other principal $G$-bundle of stacks $\underline{X}\rightarrow \mathcal{C}$ factors through $\underline{X}\rightarrow \mathcal{D}$.

If $G$ acts freely and properly, then an obvious choice for $\mathcal{D}$ is the stack $\underline{X/G}$. By the universal property, $\mathcal{D}$ must be precisely the stack appearing in the definition of quotient stacks, i.e.
$$
\mathcal{D}(Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \,\text{is a}\, G\text{-bundle,}\, f  \,\text{is}\, G\text{-equivariant}\}.
$$

Morphisms of quotient stacks are isomorphisms of principal $G$-bundles that commute with $G$-equivariant morphisms. Fixing notation, we write $[X/G]$ for $\mathcal{D}$ and refer to this as **the quotient stack**.

## Properties

### Relation to principal and associated bundles

For $V = *$ the [[terminal object]], one writes $\mathbf{B}G \coloneqq *// G$.  This is the [[moduli stack]] for $G$-[[principal bundles]]. It is also the trivial _$G$-[[gerbe]]_.

There is a canonical projection $\overline{\rho} \;\colon\; V// G \to \mathbf{B}G$. This is the [[universal associated infinity-bundle|universal rho-associated bundle]].

## Related concepts

* [[quotient]], [[quotient space]], [[quotient type]]

* [[action groupoid]], [[global quotient orbifold]]

* [[local quotient stack]]

* [[Borel construction]]

* [[(infinity,1)-colimit]], [[infinity-action]]

* [[mapping stack]]

* [[orbifold]], [[Deligne-Mumford stack]]

* [[geometric invariant theory]]

## References
 {#References}

### As stackification of action groupoids

The definition of quotient stacks as [[stackifications]] of ([[(2,1)-presheaves|presheaves of]]) [[action groupoids]] is considered for instance in:

* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], Prop. 4.3.1 in: _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]

* [[Stacks Project]], Definition 77.20.1 in: *Quotient Stacks* [[tag:044Q](https://stacks.math.columbia.edu/tag/044Q)]

### As presheaves of equivariant maps out of principal bundles

The construction of quotient stacks $X\sslash G$ as [[prestacks]] of $G$-[[principal bundles]] equipped with $G$-[[equivariant maps]] to $X$ is considered for instance in:

* [[Jochen Heinloth]], Exp. 1.5 in: *Notes on differentiable stacks* (2004) [[pdf](https://math.nyu.edu/~tschinke/WS04/pdf/heinloth.pdf), [[Heinloth-DifferentiableStacks.pdf:file]]]

* [[Frank Neumann]], p. 28 in: *Algebraic Stacks and Moduli of Vector Bundles*, impa (2011) [[pdf](https://impa.br/wp-content/uploads/2017/04/PM_36.pdf), [pdf](https://www.cimat.mx/~luis/seminarios/Pilas-algebraicas/neumann-Stacks.pdf)]

* [[Michael Groechenig]], Def. 5.23 in: *Algebraic Stacks*, Lecture notes (2014) [[web](http://individual.utoronto.ca/groechenig/stacks.html), [pdf](http://individual.utoronto.ca/groechenig/stacks.pdf), [[Groechenig-AlgebraicStacks.pdf:file]]]

* {#Halpern-Leistner20} [[Daniel Halpern-Leistner]], Section 7.1 of: *Moduli theory*, Lecture notes (2020) [[pdf](http://pi.math.cornell.edu/~danielhl/moduli_theory_notes.pdf), [[Halpern-Leistner_ModuliTheory.pdf:file]]]

Discussion of sufficient conditions for this construction to really yield a [[stack]] (instead of just a [[prestack]]):

* [[Elena Caviglia]], *Generalized principal bundles and quotient stacks*, Theory and Applications of Categories **39** 20 (2023) 567-597  [[arXiv:2302.07852](https://arxiv.org/abs/2302.07852), [tac:39-20](http://www.tac.mta.ca/tac/volumes/39/20/39-20abs.html)]

### As fibrations over $\mathbf{B}G$
 {#ReferencesAsFibrationsOverBG}

The characterization of quotient stacks $X \sslash G$ as fibrations over [[looping and delooping|delooping stacks]] $\ast \sslash G \simeq \mathbf{B}G$ with [[homotopy fiber]] $X$ (cf. also the discussion at *[[infinity-action|$\infty$-action]]*):

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Exp. 4.4 in: _[[schreiber:Principal ∞-bundles -- models and general theory|Principal $\infty$-bundles -- General theory]]_, J. Hom. Rel. Struc. **10** 4 (2015) 749-801 [[arXiv:1207.0248](https://arxiv.org/abs/1207.0248), [doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6)]

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], Prop. 2.79 in: _[[schreiber:Proper Orbifold Cohomology]]_ [[arXiv:2008.01101](https://arxiv.org/abs/2008.01101)]

* {#SatiSchreiber21} [[Hisham Sati]], [[Urs Schreiber]], Prop. 0.2.1 (ii) & Prop. 3.2.76 in: *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]* [[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)]

* [Halpern-Leistner (2020), Exc. 7.2](#Halpern-Leistner20)

### More

See also:

* [[Jack Morava]], _Theories of anything_ ([arXiv:1202.0684](http://arxiv.org/abs/1202.0684))

[[!redirects quotient stacks]]

[[!redirects orbifold quotient]]
[[!redirects orbifold quotients]]

[[!redirects quotient orbifold]]
[[!redirects quotient orbifolds]]

[[!redirects stacky quotient]]


quotient type

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### Context
#### Type theory
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[[!include type theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

In [[type theory]] the kind of [[type]] corresponding in [[categorical semantics]] to a [[quotient object]] / [[coequalizer]].

## Properties

### From higher inductive types

Quotient type may be constructed as [[higher inductive types]]. See [here](higher+inductive+type#QuotientsOfSets).

#### History and effectiveness

This notion of quotient type predates the general notion of higher inductive type, however; it appears already in [Martin Hofmann's thesis](#HofmannThesis).

Hofmann works in a theory that includes a [[type of propositions]] $Prop$ that may be assumed to satisfy [[proof irrelevance]] --- for instance, it might be the universe of [[h-propositions]] --- and considers the *effectiveness* of quotient types only for quotients of Prop-valued [[equivalence relations]].  This corresponds to the correct categorical notion of effectiveness (see [[Barr-exact category]]) and (as Hofmann noted) is provable in the presence of [[propositional extensionality]] (which in turn follows from the [[univalence axiom]]).

However, before the introduction of homotopy type theory and the realization of the importance of homotopy levels such as [[h-propositions]], some authors considered also quotients of arbitrary $Type$-valued "equivalence relations" (i.e. [[setoids]]).  [Maietti](#Maietti99) shows, by adapting [Diaconescu's argument](/nlab/show/excluded+middle#RelationToTheAxiomOfChoice), that "effectiveness" for quotients of this sort of "equivalence relation" implies the law of [[excluded middle]].

### From univalence
 {#FromUnivalence}

Quotient types are implied by adding [[univalence]] to [[Martin-Löf type theory]]. We indicate how this works:

A sufficient set of hypothesis that imply the existence of quotient types is this:

* ambient [[dependent product types]] [[dependent sum types]] and [[identity types]];
* availability of [[propositional truncation]];
* existence of a univalent family $U$;
* every [[h-prop]] is equivalent to some type in $U$.

In terms of [[categorical semantics]] in [[fibration categories]], this comes about as follows:

In [[fibration categories]] where every map is a fibration, "hprop" means
"[[subobject]]", propositional truncation becomes a stable [[image]]
factorisation, and from a univalent universe $U$ containing all hprops,
we can construct a [[subobject classifier]] as

$$
  \underset{x \colon U}{\sum} isProp(x)
  \,.
$$

(For the subobject classification axioms: existence comes from "every
hprop is equivalent to some type in $U$", and uniqueness comes from
univalence of $U$.)

So in this setting, the HoTT argument just becomes the usual construction (due to [Paré 74](#Pare), see for instance [MacLane-Moerdijk 92, IV.5](#MacLaneMoerdijk92)) of exact quotients from a [[subobject classifier]] in a [[regular category|regular]] [[locally cartesian closed category]]. For more on this see at _[quotient object -- in toposes](quotient+object#InToposes)_.

One thing to be careful about, in the [[fibration category]] setting, is
that "equivalences" as defined in type theory correspond not generally
to the original equivalences, but to the right homotopy equivalences,
which of course may be different.  This is why e.g. Mike Shulman's def
of "type-theoretic fibration category" doesn't include a class of
equivalences as separate data, but takes them to be the right homot
equivs.  So for the type-theoretic arguments to apply, "univalent
fibration" should be defined using right homot equivs.

## Related concepts

* [[setoid]]/[[Bishop set]]

* [[quotient]]

* [[quotient space]]

* [[quotient stack]]

## References

On the convoluted history of the notion of quotient types, from [[setoids]]/[[Bishop sets]] to [[higher inductive types]] and in [[univalent foundations of mathematics]]:

* Nuo Li, *Quotient Types in Type Theory*, Nottingham (2014) [[pdf](https://eprints.nottingham.ac.uk/28941/1/Nuo%20Li%27s_Thesis.pdf), [oai:eprints.nottingham.ac.uk:28941](https://core.ac.uk/display/33573841), [[Li-QuotientTypes.pdf:file]]]

* {#Murray22} [[Zachary Murray]], §4.3.2 in: *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

### As a higher inductive type

As [[higher inductive types]]:

* {#HofmannThesis} [[Martin Hofmann]], p. 111 in: _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

* {#Maietti99} [[Maria Emilia Maietti]], *About effective quotients in constructive type theory*, in "Types for proofs and programs" (1999)

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

### From univalence
 {#FromUnivalenceReferences}

The argument that adding [[univalence]] (plus universe resizing) to [[Martin-Löf type theory]] implies the existence of quotient types is attributed to Voevodsky in places like here: [pdf, slides 41 & 61](http://benediktahrens.net/talks/IRIT_slides.pdf).

Voevodsky's own account of this seems to be entirely in his Coq-code repository here:

* [[Vladimir Voevodsky]], _Equivalence classes with respect to a given relation_,
[here](https://github.com/vladimirias/Foundations/blob/df19211f602ab71ee44f0ae4de2a19170c5b9e4d/hlevel2/hSet.v#L466), _Set quotients of types_,
[here](https://github.com/vladimirias/Foundations/blob/df19211f602ab71ee44f0ae4de2a19170c5b9e4d/hlevel2/hSet.v#L555)

A more readable version of this is in section 6.10 of

* _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

Def. 6.10.5 there is the definition of the quotients.  This uses truncation, for which something like univalence and resizing is necessary and univalence and truncation is sufficient.
Then theorem 6.10.6 right below that definition checks that these quotient do indeed behave like quotients should, and this is where univalence proper comes in.

This discussion of quotients from univalence in the HoTT book
originally comes from

* [[Egbert Rijke]], [[Bas Spitters]], *Sets in homotopy type theory*, Mathematical Structures in Computer Science **25**  5 "From type theory and homotopy theory to Univalent Foundations of Mathematics" (2015) 1172-1202
[[doi:10.1017/S0960129514000553](https://doi.org/10.1017/S0960129514000553), [arXiv:1305.3835](http://arxiv.org/abs/1305.3835)]

The relevant section there is 2.4 "Voevodsky's impredicative
quotients", see def. 2.24 and the lines right below.

The [[categorical semantics]]-version of the argument for colimits from a [[subobject classifier]] in an [[elementary topos]] is due to

* {#Pare74} [[Robert Paré]], _Colimits in Topoi_, Bull. AMS 80 (1974) pp.556-661. ([pdf](http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13497-X/S0002-9904-1974-13497-X.pdf))

reviewed for instance in

* {#MacLaneMoerdijk92} [[Saunders MacLane]], [[Ieke Moerdijk]], section IV.5 of _[[Sheaves in Geometry and Logic]] – A first introduction to topos theory_, Springer Verlag, 1992

A more elementary discussion is in

* [[Todd Trimble]], _[[toddtrimble:An elementary approach to elementary topos theory]]_

[[!redirects quotient type]]
[[!redirects quotient types]]


QWIRE

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### Context
#### Computation
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[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

*QWIRE* [[Paykin, Rand & Zdancewic (2017)](#PaykinRandZdancewic17)] is a [[quantum programming language]] for [[quantum circuit]] [[certified programming|certification]] based on [[linear type theory]] combined with [[intuitionistic type theory]] via the [[exponential modality]]. More specifically, it is a [[domain specific programming language]] for [[quantum circuits]] meant to be [[domain specific embedded programming language|embedded]] into an [[intuitionistic type theory]]. As such it is similar to *[[Quipper]]*.

## Related

* [[Quipper]]

* [[CoqQ]]

* [[quantum programming languages]]

## References

The original article

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

Theoretical background:

* [[Robert Rand]], *Formally Verified Quantum Programming*, UPenn (2018) [[ediss:3175](https://repository.upenn.edu/edissertations/3175)]

  > emphasis on [[formal software verification]]:

  > [p. iv:] "We argue that [[quantum programs]] demand [[proof assistant|machine-checkable proofs]] of correctness. We justify this on the basis of the complexity of programs manipulating [[quantum states]], the expense of running [[quantum programs]], and the inapplicability of traditional debugging techniques to programs whose states cannot be examined."

  > [p. 3:] "Quantum programs are tremendously difficult to understand and implement, almost guaranteeing that they will have bugs. And traditional approaches to debugging will not help us: We cannot set breakpoints and look at our qubits without [[quantum state collapse|collapsing the quantum state]]. Even techniques like unit tests and random testing will be impossible to run on classical machines and too expensive to run on quantum computers – and failed tests are unlikely to be informative"

  > [p. 4:] "Thesis Statement: *[[quantum programming|Quantum programming]] is not only amenable to [[software verification|formal verification]]: it demands it.*"

  > "The overarching goal of this thesis is to write and verify quantum programs together. Towards that end, we introduce a quantum programming language called [[Qwire]] and embed it inside the [[Coq]] [[proof assistant]]. We give it a [[linear type system]] to ensure that it obeys the laws of [[quantum mechanics]] and a [[denotational semantics]] to prove that programs behave as desired."

* [[Jennifer Paykin]], *Linear/non-Linear Types For Embedded Domain-Specific Languages*, 2018 ([upenn:2752](https://repository.upenn.edu/edissertations/2752))

Application to [[verified programming]] after [[domain specific embedded programming language|embedding]] into [[Coq]]:

* {#RandPaykinZdancewic18} [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS **266** (2018) 119-132 [[arXiv:1803.00699](https://arxiv.org/abs/1803.00699)]

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

Using [[domain specific embedded programming language|embedding]] into [[homotopy type theory]]:

* {#PaykinZdancewic19}[[Jennifer Paykin]], [[Steve Zdancewic]], *A HoTT Quantum Equational Theory*, [talk at QPL2019](http://qpl2019.org/a-hott-quantum-equational-theory/) [[arXiv:1904.04371](https://arxiv.org/abs/1904.04371)]

For development *EWIRE*:

* [[Mathys Rennela]], [[Sam Staton]], *Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory*, Logical Methods in Computer Science **16** (2020) lmcs:6192 [[arXiv:1711.05159](https://arxiv.org/abs/1711.05159)doi:10.23638/LMCS-16(1:30)2020]

Fork development *SQIR*:

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

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

* [[Kesha Hietala]], *A verified software toolchain for quantum programming* (2022) [[pdf](https://khieta.github.io/files/drafts/khieta-dissertation.pdf), [blog](https://blog.sigplan.org/2021/06/02/verifying-a-quantum-compiler/)]

category: people

[[!redirects Qwire]]

[[!redirects SQIR]]
[[!redirects EWIRE]]


R-parity

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#### Fields and quanta
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#### Algebraic Quantum Field Theory
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#Contents#
* table of contents
{:toc}

## Idea

In the plain [[minimally supersymmetric standard model]] (MSSM), [[baryon]]/[[lepton]] [[number]] would not need to be preserved. Since however [[experimental observation]] indicates that these [[quantum observables]] are preserved to high accuracy, a common _ad hoc_ solution in [[bottom-up model building|bottom-up]] [[model (physics)|model]]-building is to impose a certain [[cyclic group of order 2|Z/2]]-[[action]] on the [[MSSM]] as a [[global symmetry]], called _R-parity, _ which does enforce baryon/lepton number conservation.

From a [[top-down model building]]-perspective there is the rough idea that R-parity may be thought of as a [[discrete group|discrete]] remnant of [[R-symmetry]] (e.g. [Chamseddine-Dreiner 96, p. 2](#ChamseddineDreiner96), [Frugiuele-Gregoire 11](#FrugiueleGregoire11))

There remain speculations that even if [[MSSM]]-like models are realistic, R-parity might in fact be violated in nature after all, such as as possibly indicated by the apparent [[flavour anomalies]] ([Wang-Yang-Yuan 19](#WangYangYuan19)).

## References

The original articles are

* [[Savas Dimopoulos]], [[Howard Georgi]], _Softly broken supersymmetry and $SU(5)$_, Nuclear Physics B Volume 193, Issue 1, 21 1981 (doi:10.1016/0550-3213(81)90522-8)

* N. Z. Sakai, _Naturalnes in supersymmetric GUTS_ Zeitschrift für Physik C Particles and Fields (1981) 11: 153 ([doi:10.1007/BF01573998](https://doi.org/10.1007/BF01573998))

* [[Hans-Peter Nilles]], [[Stuart Raby]], _Supersymmetry and the strong CP problem_, Nuclear Physics B Volume 198, Issue 1, 26 April 1982 (doi:10.1016/0550-3213(82)90547-8)

* N. Sakai, Tsutomu Yanagida, _Proton decay in a class of supersymmetric grand unified models_, Nuclear Physics B Volume 197, Issue 3, 12 April 1982, Pages 533-542 (doi:10.1016/0550-3213(82)90457-6)

* [[Savas Dimopoulos]], [[Stuart Raby]], [[Frank Wilczek]], _Proton decay in supersymmetric models_, Physics Letters B Volume 112, Issue 2, 6 May 1982, Pages 133-136 (doi:10.1016/0370-2693(82)90313-6)

See also

* Wikipedia _[R-symmetry](https://en.wikipedia.org/wiki/R-parity)_

Discussion of possible relation to [[R-symmetry]]:

* {#ChamseddineDreiner96} [[Ali Chamseddine]], Herbi Dreiner, _Anomaly-Free Gauged R-Symmetry in Local Supersymmetry_, Nucl.Phys. B458 (1996) 65-89 ([arXiv:hep-ph/9504337](https://arxiv.org/abs/hep-ph/9504337))

* {#FrugiueleGregoire11} Claudia Frugiuele, Thomas Gregoire, _Making the Sneutrino a Higgs with a $U(1)_R$ Lepton Number_, PhysRev D.85.015016 ([arXiv:1107.4634](https://arxiv.org/abs/1107.4634))

In [[SemiSpin(32)]]-[[heterotic string|heterotic]] [[string phenomenology]]:

* Saul Ramos-Sanchez, Section 5.4 of _Towards Low Energy Physics from the Heterotic String_,   Fortsch. Phys. 10:907-1036, 2009 ([arXiv:0812.3560](https://arxiv.org/abs/0812.3560))

Suggestion that the [[MSSM]] with R-parity violation could explain the [[flavour anomalies]]:

* {#WangYangYuan19} Dong-Yang Wang, Ya-Dong Yang, Xing-Bo Yuan, _$b \to c \tau \bar \nu$ decays in supersymmetry with R-parity violation_ ([arXiv:1905.08784](https://arxiv.org/abs/1905.08784))

* Quan-Yi Hu, Lin-Lin Huang,
_Explaining $b\to s \ell^+ \ell^-$ data by sneutrinos in the R-parity violating MSSM_ ([arXiv:1912.03676](https://arxiv.org/abs/1912.03676))

* Quan-Yi Hu, Ya-Dong Yang, Min-Di Zheng, _Revisiting the B-physics anomalies in R-parity violating MSSM_ ([arXiv:2002.09875](https://arxiv.org/abs/2002.09875))

* Wolfgang Altmannshofer, P. S. Bhupal Dev, Amarjit Soni, Yicong Sui, _Addressing $R_{D^\ast}$, $R_{K^\ast}$, muon $g-2$ and ANITA anomalies in a minimal R-parity violating supersymmetric framework_ ([arXiv:2002.12910](https://arxiv.org/abs/2002.12910))


r-reversible n-poset

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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=--
=--

# Contents
* table of contents
{: toc}

## Idea

Just as an [[(infinity,n)-category|$(\infinity,r)$-category]] could be $n$-connected, orthogonal to $n$-truncated $(\infinity,r)$-categories, which are just $(n,r)$-categories, an $(n,\infty)$-category, or [[n-poset|$(n+1)$-poset]], could be $r$-reversible, which are orthogonal to $r$-directed $(n+1)$-posets, which are just [[(n,r)-category|$(n,r)$-categories]].

## Definition

An $n$-poset is $r$-reversible if any $k$-morphism for $k \leq r$ is reversible. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $n$-poset.

## Examples

* A $0$-reversible $n$-poset is just a $n$-poset.

* A $n$-reversible $(n+1)$-poset is a $n$-groupoid.

## See also

* [[(n,r)-category]]

* [[k-simply connected n-category]]


R-symmetry

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#### Super-Geometry
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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The [[connected component]] of the [[subgroup]] of [[outer automorphisms]] of the [[super Poincaré group]] (in some given dimension for some real [[spin representation]] (= "number of supersymmetries")) which fixes the underlying [[Poincaré group]] is called its _R-symmetry_ group (e.g. [de Wit 02, p. 5-6](#deWit02)).

So these are transformations acting on just the spinors, leaving the vectors alone, such that in particular the super-bracket that pairs two spinors to a vector remains invariant.

For the real [[spin representations]] in dimension 3,4 and 6 which are induced by the [[normed division algebra]] $\mathbb{A} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, respectively (see at _[[division algebra and supersymmetry]]_) and for $N$ supersymmetries (i.e. the spin representation being the [[direct sum]] of $N$ copies of a real irrep) then the R-symmetry group at least includes the $N \times N$ $\mathbb{A}$-[[unitary matrices]] ([Varadarajan 04, section 6.7](#Varadarajan04)).

Geometrically, when realizing [[supergravity]] theories by [[KK-compactification]] of [[11-dimensional supergravity]] then this R-symmetry may be thought of as arising from residual [[isometries]] of the compactification space.

In [[gauged supergravity]], in turn, part of the [[global symmetry|global]] R-symmetry is promoted to an actual [[gauge group]].

For instance the $SO(8)$ R-symmetry and gauging of [[4d supergravity]] (...say which version...) may be understood geometrically by compactification of 11d supergravity on an $S^7$ fiber (e.g. [de Wit 02, section 5](#deWit02)).

## References

* [[Daniel Freed]], lectures 3 and 5 of _Five lectures on supersymmetry_, 1998

* {#deWit02} [[Bernard de Wit]], section 2.5 of _Supergravity_, 2002 ([pdf](http://www.staff.science.uu.nl/~wit00103/lhlect1.pdf))

* {#Varadarajan04} [[Veeravalli Varadarajan]], section 6.7 of _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

See also:

* Wikipedia, _[R-Symmetry](http://en.wikipedia.org/wiki/R-symmetry)_

Discussion in [[heterotic M-theory]]:

* Anthony Ashmore, Sebastian Dumitru, [[Burt Ovrut]], _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_ ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029))

[[!redirects R-symmetries]]

[[!redirects R-symmetry group]]
[[!redirects R-symmetry groups]]


R. J. Milgram > history

> See *[[R. James Milgram]]*


R. James Milgram

* [webpage](http://math.stanford.edu/~milgram/)

## Selected writings

On [[classifying spaces]] via (what came to be understood as) [[geometric realization of simplicial topological spaces]] (see [MacLane 1972](geometric+realization+of+simplicial+topological+spaces#MacLane)):

* {#Milgram67} [[R. James Milgram]], *The bar construction and abelian H-spaces*, Illinois J. Math. 11(2): 242-250 (1967) ([doi:10.1215/ijm/1256054662](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-11/issue-2/The-bar-construction-and-abelian-H-spaces/10.1215/ijm/1256054662.full))

On the [[homotopy type of spaces of rational maps]]:

* [[Fred Cohen]], [[Ralph Cohen]], [[Benjamin M. Mann]], [[R. James Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

*  [[Benjamin M. Mann]], [[R. James Milgram]],  *Some spaces of holomorphic maps to complex Grassmann manifolds*, J. Differential Geom. 33(2): 301-324 (1991) ([doi:10.4310/jdg/1214446318](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-2/Some-spaces-of-holomorphic-maps-to-complex-Grassmann/10.4310/jdg/1214446318.full))

* [[Charles P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994;173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))

On [[group cohomology]] of [[finite groups]]:

*  [[Alejandro Adem]], [[R. James Milgram]], _Cohomology of Finite Groups_, Springer 2004

## Related $n$Lab entries

* [[group cohomology]]

* [[Massey product]]

category: people

[[!redirects R. J. Milgram]]


R.Arthur

Graduate student in Mathematics, BSc in both Computer Science and Mathematics.
Specializing in Algebraic Topology.

category: people

[[!redirects R. Arthur]]


Rachid Marsli

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=270602)

## Selected writings

On the [[Gershgorin circle theorem]]:

* {#MarsliHall13} [[Rachid Marsli]], [[Frank J. Hall]], *Geometric Multiplicities and Geršgorin Discs*, The American Mathematical Monthly Vol. 120, No. 5 (May 2013), pp. 452-455 ([jstor:10.4169/amer.math.monthly.120.05.452](https://www.jstor.org/stable/10.4169/amer.math.monthly.120.05.452))

category: people


rack

# Racks
* table of contents
{: toc}

## Idea

A rack is a set equipped with a binary operation satisfying axioms analogous to the second and third [[Reidemeister moves]] in knot theory.  The most commonly appearing racks in nature are [[quandle|quandles]], which satisfy an additional axiom analogous to the first Reidemeister move.

While mainly used to obtain invariants of framed knots, racks are interesting algebraic structures in their own right: they capture the idea of an algebraic structure where every element acts as an automorphism of that structure.

## Definition

A **rack** may be defined as a set $R$ with a binary operation $\triangleright$ such that
for every $a, b, c \in R$ the **[[self-distributive law]]** holds:

$$a \triangleright (b \triangleright c) = (a  \triangleright b)\triangleright (a \triangleright  c)$$

and for every $a,b \in R$ there exists a unique
$c \in R$ such that

$$a \triangleright c = b$$

This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an [[existential quantifier]] which is not really necessary. To avoid this,  we may write the unique $c \in R$ such that $a \triangleright c = b$ as $b \triangleleft a$.  We then have

$$ a \triangleright c = b \iff  c = b \triangleleft a $$

and thus

$$ a \triangleright (b \triangleleft a) = b$$

and

$$(a \triangleright b) \triangleleft a = b$$

Using this idea, a rack may be equivalently defined as a set $R$ with two binary operations $\triangleright$ and $\triangleleft$ such that for all $a, b, c \in R$:

\[ a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)   \]

\[ (c \triangleleft b) \triangleleft a = (c \triangleleft a)\triangleleft (b \triangleleft a)  \]

\[ (a \triangleright b)\triangleleft a = b \]

\[ a \triangleright (b \triangleleft a) = a \]

It is convenient to say that the element $a \in R$ is **acting from the left** in the expression $a \triangleright b$, and **acting from the right** in the expression $b \triangleleft a$.  The third and fourth rack axioms then say that these left and right actions are inverses of each other.  Using this, we can eliminate either one of these actions from the definition of rack.  If we eliminate the right action and keep the left one, we obtain the terse definition given initially.   However, the longer definition makes it clear that racks are algebras of [[Lawvere theory]].

## Notations and Conventions

Many different conventions are used in the literature on racks and quandles.  For example, many authors prefer to work with just the _right_ action.  Furthermore, the use of the symbols $\triangleright$ and $\triangleleft$ is by no means universal: many authors use exponential notation

$$ a \triangleright b = {}^a b $$

and

$$ b \triangleleft a = b^{a} $$

while many others write

$$ b \triangleleft a = b \star a $$

We may equivalently define a rack in the following more conceptual manner: it is a set where each element acts on the left and right as [[automorphism|automorphisms]] of the rack, with the left action being the inverse of the right one.  In this definition, the fact that each element acts as automorphism encodes the left and right self-distributivity laws, and also these laws:

$$ a \triangleright (b \triangleleft c) = (a \triangleright b)\triangleleft (a \triangleright  c)$$

$$(c \triangleright b) \triangleleft a = (c \triangleleft a)\triangleright (b \triangleleft a)$$

which are consequences of the definition(s) given earlier.

## Shelves

Even simpler than a rack is a [[shelf]].  Briefly, a set with a binary operation $\triangleright$ obeying the left self-distributive law

$$ a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)   $$

is called a **left shelf**, and similarly a set with a binary operation $\triangleleft$ obeying the right self-distributive law is called a **right shelf**.

See [[shelf]] for more details.

## Related entries

* [[quandle]]

* [[shelf]]

## Links

* Gavin Wraith, [A personal story about knots](http://www.wra1th.plus.com/gcw/math/Rack.html).

* Wikipedia, [Racks and quandles](http://en.wikipedia.org/wiki/Racks_and_quandles).

* Sam Nelson, [Quandle theory](http://www.esotericka.org/cmc/quandles.html).

## References

* Nicolás Andruskiewitsch, Matiás Grañab, _From racks to pointed Hopf algebras_, Adv. Math. __178__(2):177--243 (2003, [arxiv:math/0202084](https://arxiv.org/abs/math/0202084) (doi)

* J. Scott Carter, _A survey of quandle ideas_, [arXiv](http://arxiv.org/abs/1002.4429).

The next reference makes it clear that racks are algebras of a [[Lawvere theory]], so that racks may be defined in any [[cartesian monoidal category]] (a category with finite [[products]]).

* Alissa Crans, _Shelves, racks, spindles and quandles_, [arXiv:math/0409602](https://arxiv.org/math/abs/pdf/0409602), in _Lie 2-Algebras_, p. 56.

* Roger Fenn, Colin Rourke, Brian Sanderson, _Trunks and classifying spaces_, Appl. Categ. Structures __3__ (1995), no. 4, 321--356 [MR1364012](https://www.ams.org/mathscinet-getitem?mr=1364012)[doi](https://doi.org/10.1007/BF00872903); _The rack space_, Trans. Amer. Math. Soc. __359__ (2007), no. 2, 701--740 [MR2255194](http://www.ams.org/mathscinet-getitem?mr=2255194)

* Colin Rourke, Roger Fenn, (1992). _Racks and links in codimension 2_. J. Knot Theory and Its Ramifications 1 (4): 343--406.

* Seiichi Kamada, _Knot invariants derived from quandles and racks_, [arXiv:math/0211096](https://arxiv.org/abs/math/0211096).

* Markus Szymik, _Permutations, power operations, and the center of the category of racks_, Comm. Algebra 46 (2018) 230--240 [arXiv:1609.08687](https://arXiv.org/abs/1609.08687)

A generalization of [[Lie integration]] to conjectural Leibniz groups has been conjectured by [[J-L. Loday]]. A local version via local Lie racks has been proposed in

* Simon Covez, _The local integration of Leibniz algebras_, [arXiv:1011.4112](https://arxiv.org/abs/1011.4112); _On the conjectural cohomology for groups_, [arXiv:1202.2269](https://arxiv.org/abs/1202.2269); _L'intégration locale des algèbres de Leibniz_, Thesis (2010), [pdf](http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf)

There are also crossed modules for racks:

* Alissa S. Crans, Friedrich Wagemann, _Crossed modules of racks_, [arXiv:1310.4705](https://arxiv.org/abs/1310.4705)

[[!redirects rack]]
[[!redirects racks]]


Radha Jagadeesan

* [webpage](http://reed.cs.depaul.edu/rjagadeesan/)

category: people


Radhakrishnan Chandrashekar

* [Institute page](https://shanghai.nyu.edu/academics/faculty/directory/chandrashekar-radhakrishnan-0)

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

## Selected writings

On [[quantum coherence]]:

* [[Radhakrishnan Chandrashekar|R. Chandrashekar]], P. Manikandan, J. Segar, [[Tim Byrnes]], *Distribution of quantum coherence in multipartite systems*, Phys. Rev. Lett. **116** 150504 (2016) [[arXiv:1602.00286](https://arxiv.org/abs/1602.00286), [doi:10.1103/PhysRevLett.116.150504](https://doi.org/10.1103/PhysRevLett.116.150504)]

* Md. Manirul Ali, Po-Wen Chen, [[Radhakrishnan Chandrashekar|R. Chandrashekar]], *Detecting quantum phase localization using Arnold tongue*, Physica A: Statistical Mechanics and its Applications
**633** (2024) 129436 [[doi:10.1016/j.physa.2023.129436](https://doi.org/10.1016/j.physa.2023.129436)]

category: people

[[!redirects Chandrashekar Radhakrishnan]]


Radhika Vathsan

* [institute page](https://universe.bits-pilani.ac.in/goa/radhikavathsan/profile)

* [GoogleScholar page](https://scholar.google.com/citations?user=XTuE_d0AAAAJ&hl=en)

## Selected writings

On [[anyon]]-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:

* {#MurthyShankar09} [[M.V.N. Murthy]], [[Ramamurti Shankar]], *Exclusion Statistics: From Pauli to Haldane* (1999, 2009) $[$[dspace:123456789/334](https://dspace.imsc.res.in/xmlui/handle/123456789/334) [pdf](https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1)$]$

* {#DMV03} G. Date, [[M.V.N. Murthy]], [[Radhika Vathsan]], *Classical and Quantum Mechanics of Anyons* $[$[arXiv:cond-mat/0302019](https://arxiv.org/abs/cond-mat/0302019)$]$

category: people


radial set

[[!redirects radial]]
In the theory of [[locally convex topological vector spaces]], certain types of subset turn up more frequently than others.  One such is _radial_ subsets.  In a radial subset, and line segment from the origin to a point in the subset is contained in the subset.  Thus the condition is similar to that of being convex, except that one of the end-points in the convexity condition is constrained to be the origin.
Thus a convex set containing the origin is automatically radial.

The precise definition is as follows.

+-- {: .num_defn #radial}
###### Definition
Let $E$ be a [[locally convex space|locally convex topological vector space]].  A subset $A$ of $E$ is **radial** if $r A \subseteq A$ for each $r \in [0,1]$.
=--

A radial set is clearly a [[star-shaped domain]].


radiation

## Related entries

* [[electromagnetic radiation]], [[electromagnetic waves]]

* [[gravitational waves]]

* [[radiative corrections]]

* [[ultra-high energy cosmic rays]]


radiative correction

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{:toc}

## Idea

In [[perturbative quantum field theory]] a _radiative correction_ is a contribution to the [[Feynman perturbation series]] corresponding to [[Feynman diagrams]] that describe exchange of [[virtual particle|virtual]] quanta of radiation, such as [[photons]] (for [[electromagnetic radiation]]).

## Example

* The main contribution to the _[[Lamb shift]]_ (often referred to as _the_ vertex correction) is due to the radiative correction induced by the [[electron]] emitting a [[virtual particle|virtual]] [[photon]], then interacting with the [[background field|background]] [[electromagnetic field]], and and re-absorbing the [[virtual particle|virtual]] [[photon]]; described by this [[Feynman diagram]]:

<img src="https://ncatlab.org/nlab/files/LambShiftMainVertexCorrection.png" width="100">

## Related concepts

* [[radiation]]

## References

* C. Was, _Radiative corrections_, 1994 ([pdf](https://cds.cern.ch/record/258038/files/P00020602.pdf))

Specifically for [[Lamb shift]]-like corrections:

* {#EidesGrotchShelyuto00} Michael Eides, Howard Grotch, Valery A. Shelyuto, _Theory of Light Hydrogenlike Atoms_, Phys. Rept. 342:63-261, 2001 ([arXiv:hep-ph/0002158](https://arxiv.org/abs/hep-ph/0002158))

On radiative corrections (ir)relevant for branching ratios in which [[flavour anomalies]] seem to be detected:

* {#BordoneIsidoriPattori16} [[Marzia Bordone]], [[Gino Isidori]], Andrea Pattori, *On the Standard Model predictions for $R_K$ and $R_{K^\ast}$*, Eur. Phys. J. C 76, 440 (2016). ([arXiv:1605.07633](https://arxiv.org/abs/1605.07633), [doi:10.1140/epjc/s10052-016-4274-7](https://doi.org/10.1140/epjc/s10052-016-4274-7))

[[!redirects radiative corrections]]


radical

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## Definition

For a _[[commutative ring]]_ one defines a __radical $\sqrt{I}$ of an ideal__ $I\subset R$ as an ideal
$$
\sqrt{I} = \{ r\in R \,|\, \exists n\in \mathbb{N}, r^n\in I \}
$$

An ideal is called a _radical ideal_ if it is equal to its own radical.

The [[nilradical]] of a commutative ring is the radical of the $0$ ideal.

For a [[noncommutative ring]] or an [[associative algebra]] there are many competing notions of a radical of a ring such as [[Jacobson radical]], Levitzky radical, and sometimes of radicals of ideals or, more often, of radicals of arbitrary modules of a ring.

## Radical functors

Each of the notions of radical mentioned above are functorial, and some of the abstract properties of such functors are used in noncommutative localization theory, when defining so called *radical functors*. Classically these were considered for module categories ${ }_R Mod$ (left modules over a ring $R$, but there are generalizations for arbitrary [[Grothendieck categories]], and there are also some notions of radical for nonadditive categories.  See [Shulgeifer 60](#Shulgeifer60).

We define here radical functors on ${ }_R Mod$, but warn that there are some terminological discrepancies across the literature.

However they are defined, all notions of radical involve [[additive functor|additive]] [[subfunctors]] $i: \sigma \hookrightarrow 1_{ _R Mod}$ of the identity on ${ }_R Mod$, the additive category of left $R$-modules. [[natural transformation|Naturality]] of $i$ implies the equation $i \circ \sigma i = i \circ i\sigma$, whence $\sigma i = i\sigma$ by [[monomorphism|monicity]] of $i$. Some authors refer to these as *preradical functors* (e.g., [Mirhosseinkhani 2010](#Mirhosseinkhani2010)).

Such a functor $\sigma: {}_R Mod\to {}_R Mod$ is __idempotent__ if $\sigma i = i\sigma: \sigma\sigma \to \sigma$ is an isomorphism, and is called a __radical functor__ if in addition $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. Note however that some authors call *this* a *preradical functor*, and define a radical functor to be such a preradical functor that is left exact.

Following [Goldman 1969](#Goldman1969), a left exact additive subfunctor of the identity is called an __idempotent kernel functor__. Observe that such is idempotent by the calculation

$$
\sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M\to M/\sigma M) = \sigma M
$$

where in the last step, we used that $\sigma$ is a subfunctor of the identity, hence the compositions $\sigma M\hookrightarrow M\to M/\sigma M$ and $\sigma M\to \sigma(M/\sigma M)\to M/\sigma M$ coincide.

However, beware that other authors call a left exact additive subfunctor $\sigma: {}_R Mod\to {}_R Mod$ of the identity functor a [[kernel functor]], and then call a kernel functor $\sigma$ an *idempotent kernel functor* if $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. In other words, their idempotent kernel functors coincide with what other authors call radical functors in the strong (left exact) sense above.

See [Bueso-Jara-Verschoren 95](#BuesoJaraVerschoren95)

## Examples

Example ([Bueso-Jara-Verschoren 95 2.3.4](#BuesoJaraVerschoren95)): Let $I$ be a two-sided ideal in a ring. Define a functor $\sigma : {}_R Mod\to {}_R Mod$ on objects by $\sigma M = \{ m\in M\,|\, \exists n, I^n M = 0\}$; it is left exact and idempotent. If $I$ is finitely generated as left $R$-ideal (i.e. as a left $R$-submodule of $R$) then $I$ is a left exact radical functor. It is clear that the formula for $\sigma M$ reminds the definition of the radical of an ideal of a commutative ring.

Nonexample: the subfunctor of identity which to any module $M$ assigns its [[socle]] is left exact but not a radical functor.

## References

* {#Shulgeifer60} E. G. Shulʹgeĭfer (Е. Г. Шульгейфер), _К общей теории радикалов в категориях_, Матем. сб., 51(93):4 (1960), 487–500 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=4826&what=fullt&option_lang=rus)

* {#BuesoJaraVerschoren95} J. L. Bueso, P. Jara, A. Verschoren, _Compatibility, stability, and sheaves_, Monographs and Textbooks in Pure and Applied Mathematics, 185. Marcel Dekker, Inc., New York, 1995. xiv+265 pp.

* {#Goldman1969} O. Goldman, _Rings and modules of quotients_, J. Algebra 13 (1969), 10-47.

See also

* Wikipedia, _[Radical of a ring](https://en.wikipedia.org/wiki/Radical_of_a_ring)_

[[!redirects radicals]]
[[!redirects radical functor]]

[[!redirects radical ideal]]
[[!redirects radical ideals]]


radical ring

A __radical ring__ is a ring $R$ whose [[Jacobson ideal]] equals the ring, $J(R) = R$.

Regarding that one characterization of the Jacobson ideal is that it is the intersection of all left annihilators of simple $R$-modules, that means that there are no simple modules.

For every ring $(R,+,\cdot,0,1)$ there is a Jacobson circle operation
$\circ$ defined by
$$
a\circ b = a\cdot b + a + b
$$
Now, $(R,\circ,0)$ is a semigroup. Jacobson has proven that the circle semigroup is a group iff $(R,+,\cdot,0,1)$ is a radical ring. In that case one calls this group the adjoint group $R^\circ$ of the radical ring $R$. Every radical ring $R$ is a right $R^\circ$-module via $r^a = r\cdot a + r$ for $r\in R$, $a\in R^\circ$. Indeed,
$$(r^a)^b = (r\cdot a + r)\cdot b + r\cdot a + r = r\cdot (a\cdot b + b + a)+r = r\cdot (a\circ b) + r$$

category: algebra


radius

#Contents#
* table of contents
{:toc}

## Idea

half the diameter of a [[sphere]]

## Related concepts

* [[circumference of a circle]]

* [[pi]]

* [[slope]]

* [[proton radius puzzle]]

## Literature

See also:

* Wikipedia, *[Radius](https://en.m.wikipedia.org/wiki/Radius)*

[[!redirects radii]]


radix notation

# Radix notation
* table of contents
{: toc}

## Idea

At least in [[classical mathematics]], every [[real number]] may be written in _base $n$_ for any [[natural number]] $n \geq 2$ (binary for $n = 2$, decimal for $n = 10$, etc), which can be generalized to fractional $n \gt 1$ (especially $n = \mathrm{e} \approx 2.718$) and beyond.

For [[integers]], this idea goes back to the Old Babylonians using base $60$ (around 2000 BCE), perfected in the Gupta Empire using base $10$ (around 400), and popularized in Europe by [[Fibonacci]] (in 1202).  For arbitrary real numbers, these were first used by [[Abu'l-Hasan al-Uqlidisi]] (around 952) and popularized by [[Simon Stevin]] (in 1585).  This notation (in base $10$) is now ubiquitous, and (despite the technical difficulties of doing so rigorously) serves as a de facto definition of the [[real numbers]] in elementary mathematics.

Radix notation in base $2$ is the basis of [[floating point arithmetic]], the fast but imprecise method of calculation with real numbers usually used in modern [[computing]].

## Definition

Given a [[natural number]] $n \geq 2$, let $[n]$ be the set of natural numbers (including [[zero]]) strictly less than $n$: $[n] = \{0, 1, 2, \ldots, n - 2, n - 1\}$; in this context, an element of $[n]$ is called a __digit__ (in base $n$).  Let $\mathbb{Z}$ be the set of [[integers]]: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$; in this context, an element of $\mathbb{Z}$ is called a __place value__.  Then a [[function]] from $\mathbb{Z}$ to $[n]$ is a doubly-infinite [[sequence]] of digits, one digit for each place value.  Such a sequence $a = (a_i)_{i\colon{\mathbb{Z}}}$ defines a doubly-infinite [[series]] of [[rational numbers]]:
$$ \sum_{-\infty \lt i \lt \infty} a_i n^{-i} .$$
This series converges (to a [[real number]]) if and only if there is a place value $N$ such that $a_i = 0$ for all $i \lt N$.  That is, we can write the series as
$$ \sum_{i = N}^\infty a_i n^{-i} .$$
In this way, a sequence of digits (finite on one end, infinite on the other) represents a nonnegative [[real number]].

As a representation of real numbers, this is almost unique.  Specifically, two distinct sequences $a$ and $b$ of digits represent the same real number (meaning that the sums of the series are equal) if and only if there is some place value $m$ such that

* $a_i = b_i$ for every place value $i \lt m$,
* $a_m = b_m + 1$ (or the reverse), and
* $a_i = n - 1$ for every $i \gt m$ while $b_i = 0$ for every $i \gt m$.

In this case, the sequence with $0$s is generally considered standard.  The real numbers arising in this way are precisely the positive $n$-[[adic rational]] numbers, that is the [[rational numbers]] that are positive integer multiples of $n^{-m}$ for some integer $m$ (the same $m$ as above).

In [[classical mathematics]], every nonnegative real number may be represented in this way.  Given such a number $x$ and a place value $i$, let $a_i$ be the [[remainder]] modulo $n$ of the [[floor]] of $x n^i$: that is, $a_i \coloneqq \lfloor{x n^i}\rfloor \mod n$; then $x = \sum_i a_i n^{-i}$.  (For $n$-adic rational $x$, this will produce the representation with $0$s; to automatically produce the other representation, use $a_i = (\lceil{x n^i} - 1\rceil) \mod n$.)  The number $0$ is represented only by a sequence of all $0$s.

Given $n$ unique symbols for the $n$ digits, the real number represented by a sequence of digits may be written by, beginning at the smallest place value whose digit is nonzero, writing the symbols in order, with a dot (or comma), called the __radix point__, between place value $0$ and place value $1$ (and padding zeroes after the radix point if the first nonzero digit has not yet been reached).  Of course, the sequence is still infinite and cannot be written down, but we may write any finite portion.  The $n$-adic rationals can be represented exactly by leaving out the infinitely repeating digit $0$; arbitrary [[rational numbers]] may be represented exactly by a bar over a list of digits that repeats infinitely.  (It is a theorem that every non-$n$-adic rational number can be uniquely represented in this way; in fact, only rational numbers can be so represented.)  A subscript may be used to indicate the base $n$ (with a default base $10$ in practice).  Finally, a negative real number is written by writing its [[absolute value]] after a minus sign.

## Generalizations

Given any number $n$ (not necessarily a natural number), still called the _base_, and any set $D$ of numbers, whose elements are called _digits_, we may consider finite or infinite sequences of digits, indexed by integers (still called _place values_).  Such a sequence should include a smallest place value $N$ as part of its data (since $0$ may not be one of the digits), so the place values are all (or perhaps only some) of the integers $i \geq N$.  Then the infinite series
$$ \sum_{i=N}^\infty a_i n^{-i} $$
still converges to a real number.

If $n \geq 2$ is a natural number and $D$ is $[n]$, then we recover the representation above of all nonnegative real numbers, unique except for the positive $n$-adic rationals.  More generally, if $n \gt 1$ is any real number and $D$ is $[\lceil{n}\rceil]$, then we get a representation of all nonnegative real numbers, which (for non-integer $n$) is never unique (although in practice one uses the last sequence in [[lexicographic ordering]]).

If $n \geq 2$ is a natural number and $D$ is $[n] + 1 = \{1, 2, 3, \ldots, n - 1, n\}$, then we get a representation of all positive real numbers (but not $0$ unless we allow the [[empty list]] of digits).  If $n = 1$ and $D$ is again $[n] + 1 = \{1\}$, then every infinite sequence gives a divergent series, but the finite sequences represent all positive integers, uniquely up to a [[shift]] of place value; this is called __tally notation__.  In the other direction, if $n \geq 3$ is a natural number and $D$ is a set of $n$ consecutive integers that includes both $-1$ and $1$, then we get a representation of all real numbers (positive, negative, and zero), which is unique except for a few rational numbers (in the case of $n = 3$ and so $D = \{-1, 0, 1\}$, those which are half of a power of $1/3$); this is called __balanced__ radix notation.

## Examples

Here are some representations of well-known real numbers in well-known radixes:

Number	Decimal ($n = 10$, $D = [10]$	Binary ($n = 2$, $D = \{0, 1\}$)	Positive binary ($n = 2$, $D = \{1, 2\}$)	Natural ($n = \mathrm{e}$, $D = \{0, 1, 2\}$)	Ternary ($n = 3$, $D = \{0, 1, 2\}$)	Balanced ternary ($n = 3$, $D = \{-1, 0, 1\}$, writing $-$ for $-1$)	Tally ($n = 1$, $D = \{1\}$)

$0$	$0$ (or $-0$)	$0$ (or $-0$)	$0$ (nonce symbol, or [[empty list]])	$0$ (or $-0$)	$0$ (or $-0$)	$0$	$0$ (nonce symbol, or [[empty list]])

$1$	$1$ (or $0.\overline{9}$)	$1$ (or $0.\overline{1}$)	$1$ (as a finite list, or $0.\overline{1}$ with padding $0$, or $0.0\overline{2}$)	$1$ (or $0.2121111212001\ldots$, etc)	$1$ (or $0.\overline{2}$)	$1$	$1$

$2$	$2$ (or $1.\overline{9}$)	$10$ (or $1.\overline{1}$)	$2$ (as a finite list, or $1.\overline{1}$, or $0.\overline{2}$ with padding $0$)	$2$ (or $1.2121111212001\ldots$, etc)	$2$ (or $1.\overline{2}$)	$2$	$11$

$\mathrm{e}$	$2.7182818284590\ldots$	$10.10110111111\ldots$	$1.21221222222\ldots$	$10$ (or $2.121111212001\ldots$, etc)	$2.2011011212211\ldots$	$10.{-}0111{-}{-}0{-}0{-}\ldots$	(not possible)

$3$	$3$ (or $2.\overline{9}$)	$11$ (or $10.\overline{1}$)	$11$ (as a finite list, or $2.\overline{1}$, or $1.\overline{2}$)	$10.020011200001\ldots$ (etc)	$10$ (or $2.\overline{2}$)	$10$	$111$

$\pi$	$3.14159265358979\ldots$	$11.001001000011\ldots$	$2.112112111122\ldots$	$10.101002020002$ (etc)	$10.010211012222\ldots$	$10.011{-}111{-}000{-}\ldots$	(not possible)

$1/2$	$0.5$ (or $0.4\overline{9}$)	$0.1$ (or $0.0\overline{1}$)	$0.1$ (as a finite list with padding $0$, or $0.0\overline{1}$ with padding $0$, or $0.00\overline{2}$)	$0.1021200201202\ldots$ (etc)	$0.\overline{1}$	$0.\overline{1}$ (or $1.\overline{-}$)	(not possible)

$1/3$	$0.\overline{3}$	$0.0\overline{10}$	$0.00\overline{21}$ (with padding $0$)	$0.021012102010201\ldots$ (etc)	$0.1$ (or $0.0\overline{2}$)	$0.1$	(not possible)

$-1$	$-1$ (or $-0.\overline{9}$)	$-1$ (or $-0.\overline{1}$)	$-1$ (as a finite list, or $-0.\overline{1}$ with padding $0$, or $-0.0\overline{2}$)	$-1$ (or $-0.2121111212001\ldots$, etc)	$-1$ (or $-0.\overline{2}$)	$-1$	$-1$

## Foundational issues

In [[constructive mathematics]], it is not generally true that every [[real number]] has a radix expansion. However, one does have the following results:

* In the presence of [[weak countable choice]], [[existential quantifier|there exists]] a [[radix notation|radix expansion]] in any base (e.g., a decimal expansion or binary expansion) for every [[Cauchy real number]] iff $\mathrm{LLPO}_\mathbb{N}$ holds; there exists a radix expansion for every [[Dedekind real number]] has iff the analytic $\mathrm{LLPO}$ holds. Without [[weak countable choice]], [[Lifschitz realizability]] gives a model in which $\mathrm{LLPO}_\mathbb{N}$ holds but it is not true that there exists a radix expansion in any base for every [[Cauchy real number]], which implies that there are models in which the analytic LLPO holds but it is not true that there exists a radix expansion in any base for every [[Dedekind real number]]. See [[Andrew Swan]]'s answer to [Birchfield (2024)](#Swan). In addition, the analytic $\mathrm{LLPO}$ holds for the [[Dedekind real numbers]] in [[condensed sets]], but every function from the Dedekind real numbers to the [[boolean domain]] $\mathbb{2}$ is a [[constant function]], which implies that it is not true that there exists a radix expansion in any base for every [[Dedekind real number]].

* In the presence of [[countable choice]], $\mathrm{LLPO}_\mathbb{N}$ is equivalent to the claim that the rings of radix expansions in any two bases are isomorphic. See Daniel Mehkeri\'s answer to [Feldman (2010)](#Mehkeri).

* That every Cauchy real number has a choice of radix expansion in any base implies that the [[weak limited principle of omniscience]] (WLPO) for the natural numbers holds; that every Dedekind real number has a choice of radix expansion implies that the [[analytic WLPO]] holds.

[[Fred Richman]] considered a number system (a noncancellable [[rig]]) of nonnegative decimal sequences in which $0.\overline{9} \lt 1$; the usual rig of nonnegative real numbers is a subrig; see [Richman 1999](#Richman1999).  Although Richman is a prominent constructivist, the development was not (and probably cannot be made) constructive.

Usenet legend [[Alexander Abian]], before branching into speculative physics, advocated polemically that the real numbers should be defined as finite or infinite sequences of decimal digits; see [Abian 1981](#Abian1981).  While not favoured by most mathematicians due to the inelegance of the definitions and proofs, this is the form in which the real numbers are often presented to elementary students.

## References

* [[Alexander Abian]]. 1981. Calculus must consist of the study of real numbers in their decimal representation and not of the study of an abstract complete ordered field or nonstandard real numbers. International Journal of Mathematical Education in Science and Technology 12(4). [Doi:10.1080/0020739810120417](https://doi.org/10.1080/0020739810120417).
  {#Abian1981}

* [[Fred Richman]], *Is 0.999… = 1?*. Mathematics Magazine, Volume 72, Issue 5, 1999, Pages 396-400, [[doi:10.1080/0025570X.1999.11996777](https://doi.org/10.1080/0025570X.1999.11996777)]
  {#Richman1999}

* {#Mehkeri} David Feldman (2010) on Math.Overflow, [Radix notation and toposes](http://mathoverflow.net/questions/49775/radix-notation-and-toposes/)

* {#Swan} Madeleine Birchfield (2024) on Category Theory Zulip, [Radix expansions in constructive mathematics](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Radix.20expansions.20in.20constructive.20mathematics)

[[!redirects radix notation]]
[[!redirects positional notation]]
[[!redirects place-value notation]]
[[!redirects place value notation]]

[[!redirects radix expansion]]
[[!redirects radix expansions]]
[[!redirects radix representation]]
[[!redirects radix representations]]
[[!redirects expansion in base]]
[[!redirects expansions in base]]
[[!redirects representation in base]]
[[!redirects representations in base]]

[[!redirects decimal notation]]
[[!redirects decimal expansion]]
[[!redirects decimal expansions]]
[[!redirects decimal representation]]
[[!redirects decimal representations]]


Radon measure

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#### Measure and probability theory
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* table of contents
{: toc}

## Definition

> NB: In the following the symbols "$\subset$" and "$\supset$" are used to denote nonstrict inclusions of [[subsets]], sometimes also denoted by "$\subseteq$" and "$\supseteq$".

Recall the following properties of a [[Borel measure]] $\mu$ on a [[Hausdorff topological space]]:

* $\mu$ is __outer regular__ if for every [[Borel subset]] $B$ we have
$$\mu(B)=\inf\{\mu(V)\mid V\supset B\; and\; V\; is\; open\}.$$

* $\mu$ is __locally finite__ if every point has a neighborhood with a finite $\mu$-measure.

* $\mu$ is __inner regular__ on some [[Borel subset]] $B$ if
$$\mu(B)=\sup\{\mu(K)\mid K\subset B\; and\; K\; is\; compact\}.$$

Also, if $m$ and $M$ are [[Borel measures]],
then $m$ is the __essential measure__ associated with $M$ if
$$m(A)=\sup\{m^*(C)\mid C\subset A,\; m^*(C)\; is\; finite\},$$
where
$$m^*(C)=\inf\{m(B)\mid B\supset C\; and\; B\; is\; Borel\}.$$
Equivalently, one can simply say that
$$
m(B)=\sup\{M(B')\mid B'\subset B,\; M(B')\; is\; finite,\; B'\; is\; Borel\}.
$$

We give three equivalent definitions of Radon measures.

\begin{definition}
If $X$ is a [[Hausdorff topological space]], then a __Radon measure__ on $X$
is a [[Borel measure]] $m$ on $X$ such that $m$ is locally finite and inner regular on all [[Borel subsets]].
\end{definition}

\begin{definition}
If $X$ is a [[Hausdorff topological space]], then a __Radon measure__ on $X$
is a [[Borel measure]] $M$ on $X$ such that $M$ is locally finite, outer regular, and inner regular on all open subsets.
\end{definition}

\begin{definition}
If $X$ is a [[Hausdorff topological space]], then a __Radon measure__ on $X$
is a pair of [[Borel measures]] $m$ and $M$ on $X$ such that $m$ is the
essential measure associated with $M$,
$M$ is outer regular (on all Borel subsets),
$M$ is locally finite,
$M$ is inner regular on all Borel subsets,
and $m(B)=M(B)$ whenever $B$ is open or $M(B)$ is finite.
\end{definition}

## Equivalence of definitions

In order to pass from $m$ to $M$, set
$$M(B)=\inf\{m(V)\mid V\supset B\; and\; V\; is\; open\}.$$

In order to pass from $M$ to $m$, set
$$m(B)=\sup\{M(B')\mid B'\subset B,\; M(B')\; is\; finite,\; B'\; is\; Borel\}.$$

If $m(X)$ or $M(X)$ is finite, then $m=M$.

If $B$ is a [[Borel subset]] such that $M(B)$ is finite or $B$ is open,
then $m(B)=M(B)$.

A Radon measure is __σ-finite__ if $m$ is σ-finite.

A Radon measure is __moderated__ if $M$ is σ-finite.

## Real and complex Radon measures

\begin{definition}
A real (respectively complex) Radon measure on a [[Hausdorff topological space]] $X$
is a real (respectively complex) valued function $\mu$
defined on relatively compact [[Borel subsets]] of $X$
that is (1) countably additive,
(2) every relative compact [[Borel subset]] $B\subset X$
can be presented as the union of countably many compact subsets
and a subset $N\subset B$ such that $\mu(N')=0$ for any [[Borel subset]] $N'\subset N$,
and (3) any point has a neighborhood $V$ such that $\sup |\mu(B)|$
is finite, where $B\subset V$ is a relatively compact subset of $X$.
\end{definition}

If $\mu\ge0$, then $\mu$ can be extended to a Radon measure $(m,M)$
in the previous sense.

## On locally compact Hausdorff topological spaces

Radon measures on locally compact Hausdorff topological spaces
admit yet another, Daniell-style definition, which is explored
in detail in Bourbaki's book.

\begin{definition}
Suppose $X$ is a locally compact Hausdorff topological space.
A __Radon measure__ on $X$ is a positive linear functional
$$\mu\colon C_c(X,\mathbf{R})\to\mathbf{R},$$
where $C_c$ refers to the vector space of continuous compactly
supported functions.
\end{definition}

Such $\mu$ induces a pair $(m,M)$ in the sense of above definitions
as follows: $M=\mu^*$ and $m=\mu^{\bar*}$,
where $\mu^*$ is the outer measure associated to $\mu$, i.e.,
$$\mu^*(A)=\inf\{\mu(\chi_B)\mid B\supset A,\; B\; is\; Borel\}$$
and $\mu^{\bar*}$ is the essential measure associated to $\mu^*$:
$$\mu^{\bar*}(A)=\sup\{\mu^*(C)\mid C\subset A,\; \mu^*(C)\; is\; finite\}.$$

(For σ-finite spaces we have $\mu^*=\mu^{\bar*}$.)

Vice versa, given $M$, we can reconstruct $\mu$ as the integral
with respect to $M$.

## Pushforwards of Radon measures

\begin{definition}
Suppose $X$ and $Y$ are [[Hausdorff topological spaces]]
and $\mu=(m,M)$ is a Radon measure on $X$.
A map of sets $H\colon X\to Y$ is __Lusin $\mu$-measurable__
if for any compact $K\subset X$ and $\epsilon\gt0$
there is a compact $K'\subset K$ such that $\mu(K\setminus K')\lt\epsilon$
and the restriction of $H$ to $K'$ is continuous.
We say that $H$ is __$\mu$-proper__ if, in addition,
every point of $Y$ has a neighborhood whose inverse image
is $\mu$-integrable.
\end{definition}

For example, continuous maps are Lusin $\mu$-measurable for any $\mu$.

Lusin $\mu$-measurable maps form a [[sheaf]] with respect to $X$.

Any Lusin $\mu$-measurable map is Borel $\mu$-measurable
(meaning preimages of Borel sets are $\mu$-measurable sets).

If $Y$ is [[metrizable]] and [[separable]], then any Borel $\mu$-measurable
map is Lusin $\mu$-measurable.

Step functions and lower semicontinuous maps are always Lusin $\mu$-measurable.

\begin{definition}
Suppose $H\colon X\to Y$ is a $\mu$-proper map.
The pushforward measure $H_*\mu$ is a Radon measure on $Y$ defined as follows.
Given a [[Borel subset]] $B\subset Y$, we set
$$H_*\mu(B)=\mu^\bullet(H^*(B)).$$
This yields a Radon measure $m$ on $Y$.
\end{definition}

\begin{theorem}
If $H\colon X\to Y$ is a [[continuous map]] of [[Hausdorff topological spaces]],
then a Radon measure $\nu$ on $Y$ is the pushforward along $H$
of a Radon measure on $X$ if and only if $\nu$
is concentrated in a countable union of images under $H$
of compact subsets of $X$.
\end{theorem}

## Properties

* If a Radon measure is $\sigma$-finite then it is regular (i.e. both inner and outer regular) on all Borel subsets.

* A Radon measure on a [[Hausdorff space]] is [[τ-additive]]. The converse is true on a [[compact Hausdorff space]].

* Radon [[probability measures]] on [[compact Hausdorff spaces]] form a [[monad]]: the [[Radon monad]]. Just as well, Radon probability measures of finite first moment on [[complete metric spaces]] give the [[Kantorovich monad]].

(See also [[monads of probability, measures and valuations]].)

## Examples

Most measures of interest in [[geometry]] are Radon. For example

* The [[Dirac measures]].

* The [[Lebesgue measure]] on the real line.

* The measure associated to a [[volume form]] on a Riemannian manifold.

* The left (right) [[Haar measure]] on a [[locally compact topological group]] is a nonzero Radon measure which is invariant under left (right) multiplications by elements in the group.

## See also

* [[Radon monad]], [[Kantorovich monad]], [[monads of probability, measures, and valuations]]

* [[Borel measure]], [[τ-additive measure]], [[continuous valuation]]

## References

The canonical references on Radon measures are

* [[Laurent Schwartz]], _Radon measures on arbitrary topological spaces and cylindrical measures_.

* [[Nicolas Bourbaki]], _Integration.  Chapter IX_.

More recent expositions include

* V. Bogachev, _Measure Theory_, vol. 2 (2007). doi:[10.1007/978-3-540-34514-5](https://doi.org/10.1007/978-3-540-34514-5)

* Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995

[[!redirects Radon measure]]
[[!redirects Radon measures]]


Radon monad

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#### Measure and probability theory
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#### $(0,1)$-Category theory
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#Contents#
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## Idea

The Radon monad is a [[measure monad]] on the [[category]] of [[compact Hausdorff spaces]].
Its functor assigns to each space $X$ the space of [[Radon measure|Radon]] [[probability measure|probability]] or subprobability [[measures]], equipped with the [[weak topology]].

There is also an [[order theory|ordered]] version of the Radon monad on the category of [[compact ordered spaces]] (see [below](#the_ordered_case)).

## Spaces of Radon measures

Let $X$ be a [[compact Hausdorff space]].
Denote by $R X$ the set of those [[Radon measures]] on $X$ which are _subnormalized_, i.e. those measures $\mu$ for which $\mu(X)\le 1$.
Denote by $P X$ its subset of Radon [[probability measures]].

Equip both sets with the [[weak topology|topology of weak convergence]] with respect to [[continuous functions]]. It is well known that the resulting spaces $R X$ and $P X$ are themselves [[compact Hausdorff spaces]].

### In terms of τ-additive measures

Since [[tau-additive measure#relationship_with_other_measuretheoretical_notions|on compact Hausdorff spaces Radon and τ-additive measures coincide]], we can equivalently define $R X$ and $P X$ as the sets of subnormalized and normalized [[τ-additive measures]].

For probability measures, the [[weak topology]] and the [[extended probabilistic powerdomain#the_measure_monad_on_top|A-topology]] coincide, so that $P$ an be extended to the [[extended probabilistic powerdomain#the_probability_monad_on_top|probability monad on Top]].

The same cannot be said for $R$, since the weak topology and the A-topology do not coincide for _subnormalized_ measures: given $\mu\in R X$, the measure $r \mu$ for $0 \le r \le 1$ is in the [[closure]] of $\mu$ for the $A$-topology, while the weak topology is [[Hausdorff]].

### In terms of valuations

Note that, since every [[continuous valuation]] on a [[compact Hausdorff space]] can be [[continuous valuation#extending_valuations_to_measures|extended to a τ-additive measure]], we could equivalently defined the Radon monad as a monad of [[continuous valuations]].

Just as above, for probability measures, the [[weak topology]] and the [[extended probabilistic powerdomain#spaces_of_valuations|pointwise topology of valuations]] coincide, so that $P$ an be extended to a submonad of the [[extended probabilistic powerdomain]] on [[Top]]. Again, the same cannot be said for $R$.

## Functoriality, unit and multiplication

Let $f:X\to Y$ be a [[continuous map]] between [[compact Hausdorff spaces]]. The [[pushforward of measures]] (or equivalently of [[continuous valuation#pushforward|valuations]]) gives a well-defined, continuous map $R X\to R Y$ which restricts to the map $P X \to P Y$.
This makes $R$ and $P$ [[endofunctors]] of [[Top]].

As it is usual for [[measure monads]], the unit is given by the [[Dirac measures]] and the multiplication is given by [[integration]].

More in detail, given a [[compact Hausdorff space]] $X$, we can assign to each $x\in X$ its [[Dirac measure]] (equivalently, [[continuous valuation#dirac_valuations|Dirac valuation]] $\delta_x$. The assignment $\delta:X\to P X$, or $X\to R X$ is continuous, and [[natural transformation|natural]] in $X$.

Just as well, given a measure $\mu\in P P X$, we can define the measure $E\mu\in P X$ as the one assigning to a measurable $A\subseteq X$ the number
$$
E\mu(A) \coloneqq \int_{P X} p(A) \,d\mu(p).
$$
(In terms of valuations, the same is given for [[open sets]] instead of [[measurable sets]].)
Again, the assignment $E: P P X \to P X$ is continuous and [[natural transformation|natural]] in $X$.
The multiplication for $R$ is defined analogously.

The unit and multiplication thus defined satisfy the usual [[axioms]] of a [[monad]].
The monads $R$ and $P$ are both known in the literature as the **Radon monad**.

The monad $P$ is the restriction to [[compact Hausdorff spaces]] of the [[extended probabilistic powerdomain#the_probability_monad_on_top|probability monad on Top]], which is itself a submonad of the [[extended probabilistic powerdomain]]. (See also [[monads of probability, measures, and valuations]].)

## Algebras

The [[algebra over a monad|algebras]] of the Radon monad $P$ are [[compact]] [[convex subsets]] of [[locally convex topological vector spaces]].

More in detail, a compact convex subset $C$ of a locally convex topological vector space is a [[compact Hausdorff space]], and it admits a canonical $P$-algebra structure $e:P C\to C$ via (vector-valued) [[integration]]:
$$
p \mapsto \int_{C} c \,dp(c) .
$$
Since the space is [[compact]], the integral above is well-defined, and it returns an element of $C$ which we can view as the "center of mass" of $p$.

Conversely, it can be proven that every $P$-algebra is of this form.

The [[Eilenberg-Moore category|morphisms of algebras]] are the continuous maps between algebras which commute with the operation of taking integrals.
It can be shown that these coincide with the _affine_ maps, i.e. those maps $f:A\to B$ which satisfy
$$
f\big( \lambda\, a + \mu\, b  \big) \;=\; \lambda\,f(a) + \mu\,f(b)
$$
for all $a,b\in A$ and $0\le\lambda,\mu\le 1$ (with $\lambda+\mu=1$ for the normalized case).

For more information, see [Swirszcz '74](#swirszcz) and the later [Keimel '08](#radonkeimel).

## The ordered case

The Radon monad can be lifted to the [[CompOrd#categories_of_compact_ordered_spaces|category of compact ordered spaces]] (and [[continuous map|continuous]] [[monotone maps]]).
This is done by means of the [[stochastic order]].

Here we sketch the construction. For more details, see [Keimel '08](#radonkeimel).

### Construction of the spaces

Given a [[compact ordered space]] $X$, we can form the spaces $R X$ and $P X$ as for the unordered case, and then equip them with the [[stochastic order]]. Over compact ordered spaces, the stochastic order is antisymmetric (i.e. it is a partial order), and it has [[closed graph]], so that $R X$ and $P X$ are again compact ordered spaces.

Given a monotone map $f:X\to Y$, the maps $R f$ and $P f$ are monotone for the stochastic order, and so are the monad structure maps. This way, $P$ and $R$ lift to a monad on [[CompOrd#categories_of_compact_ordered_spaces|CompOrd]].

For the canonical [[locally posetal 2-category]] structure of [[CompOrd#categories_of_compact_ordered_spaces|CompOrd]] given by the [[pointwise order]], $P$ and $R$ are even [[2-monads]], since whenever $f\le g:X\to Y$, we have $R f\le R g$ $P f\le P g$.

Both the resulting monads are known in the literature as **ordered Radon monad**.

### Algebras

The algebras of the ordered Radon monad $P$ are known to be [[compact]] [[convex space|convex]] subsets of [[locally convex topological vector space|locally convex]] [[ordered topological vector spaces]]. This can be seen as the ordered equivalent of the characterization [above](#algebras).

The algebras of $R$, similarly, can be seen as the ordered equivalent of [[Kegelspitze]].

See [Keimel '08](#radonkeimel) for more.

Just as for the unordered case, the [[Eilenberg-Moore category|algebra morphisms]] are the continuous affine maps, which here are also required to be monotone.

### Lax morphisms are concave maps

Differently from the unordered case, in the ordered setting we have a [[2-category]], and so it makes sense to talk about [[lax morphisms]] of algebras.
By definition, these amount to maps between algebras $f:A\to B$ with the [[stuff, structure, property|property]] that
$$
f \left( \int a \, dp(a) \right) \;\le\; \int f(a) \, dp(a)
$$
for all $p\in P A$ (resp. $R A$).

By the generalized [[Jensen's inequality]], these are precisely the (continuous, monotone) [[concave maps]], i.e. the maps that satisfy
$$
f\big( \lambda\, a + \mu\, b  \big) \;\le\; \lambda\,f(a) + \mu\,f(b) .
$$
(Compare with the strict case by replacing the order with equalities.)

Just as well, the oplax morphisms are the (continuous, monotone) [[convex maps]].

(For more information see the analogous discussion for the [[Kantorovich monad]] in [F-P '18](#orderedkantorovich).)

## See also

* [[monads of probability, measures, and valuations]]

* [[Giry monad]], [[extended probabilistic powerdomain]], [[Kantorovich monad]]

* [[locally convex topological vector space]], [[ordered vector space]], [[convex space]]

* [[compact Hausdorff space]], [[compact ordered space]], [[stably compact space]]

* [[Borel measure]], [[Radon measure]], [[τ-additive measure]], [[continuous valuation]]

## References

* {#swirszcz} T. Swirszcz, _Monadic functors and convexity_, Bulletin de l'Academie Polonais des Sciences 22, 1974 ([pdf](https://www.fuw.edu.pl/~kostecki/scans/swirszcz1974.pdf))

* {#radonkeimel} [[Klaus Keimel]], _The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras_, Topology and its Applications, 2008 ([doi:10.1016/j.topol.2008.07.002](https://doi.org/10.1016/j.topol.2008.07.002))

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

[[!redirects ordered Radon monad]]
[[!redirects radon monad]]
[[!redirects ordered radon monad]]


Radon transform

## Idea

The **Radon transform** converts between functions on $\mathbb{R}^n$ which are integrable on all hyperplanes and functions on the space of all hyperplanes of $\mathbb{R}^n$. It corresponds to pull-push through the incidence correspondence of points and hyperplanes.

It is a real version of the _[[Penrose transform]]_.

It is widely used in computerized tomography.

## References

* Sigurdur Helgason, _Radon Transform, second edition_, Birkhäuser, New York, 1999, ([book](http://www-math.mit.edu/~helgason/Radonbook.pdf)).

* Wikipedia, _[Radon transform](http://en.wikipedia.org/wiki/Radon_transform)_

[[!redirects Radon transforms]]

[[!redirects Radon transformation]]
[[!redirects Radon transformations]]