biquotient

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#### Lie theory
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#Contents#
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## Idea

In [[group theory]], but particularly in [[Lie group]]-[[Lie theory|theory]], the term "biquotient" tends to mean the [[quotient space]] of a [[topological group]] or [[Lie group]] $G$ by the [[action]] of _two_ [[subgroups]] $H_1, H_2 \subset G$, hence by the action of their [[direct product group]] $H_1 \times H_2$, one factor regarded as [[action|acting]] by group multiplication from the left, the other (more precisely: its [opposite](opposite+category#OppositeGroup)) acting by multiplication from the right.

This is typically and suggestively denoted as

$$
  H_1 \backslash G / H_2
  \;\coloneqq\;
  G/( H_1 \times H_2^{op} )
  \;\coloneqq\;
  G/( g \sim h_1 \cdot g \cdot h_2 \vert h_i \in H_i )
  \,.
$$

Another way to think of a biquotient is as a [[double coset space]], see there for more.

Typically extra conditions are imposed on $H_1, H_2 \subset G$, such as that $H_i \subset G$ are [[closed subspaces|closed]] [[subgroups]] and notably that the induced action of $H_1$ on the single [[quotient space]]/[[coset space]] $G/H_2$ of the other, is still [[free action|free]].

More generally, one can consider biquotients of $G$ by [[subgroups]] of the [[direct product group]] (e.g. [Kapovitch](#Kapovitch)).

## Examples

### Gromoll-Meyer sphere

The [[Gromoll-Meyer sphere]] -- an [[exotic 7-sphere]] -- arises as the [[biquotient]] of [[Sp(2)]] by two copies of [[Sp(1)]].

## References

For instance:

* Burt Totaro, _Cheeger manifolds and the classification of biquotients_ ([arXiv:math/0210247](https://arxiv.org/abs/math/0210247))

In [[rational homotopy theory]]:

* {#Kapovitch} Vitali Kapovitch, _A note on rational homotopy of biquotients_ ([pdf](http://www.math.utoronto.ca/vtk/biquotient.pdf))

[[!redirects biquotients]]

[[!redirects biquotient space]]
[[!redirects biquotient spaces]]


birational geometry

Let $X$ be an [[algebraic variety]] (in the modern approach an irreducible reduced [[scheme]] of finite type over a [[field]], but we will work here with the usual [[maximal spectrum|maximal spectra]] to simplify the exposition). To a variety one can associate a [[function field]] $k(X)$ whose elements are [[rational function]]s on $X$ (they are regular functions on the big-Zariski open subvarieties of $X$). A [[partial function|partially defined map]] from a variety to another variety is [[rational map|rational]] if it is defined and regular on a Zariski open set. One can check easily that in fact rational maps compose. Rational maps which have an inverse on a Zariski open subset are called [[birational maps]] or birational isomorphisms (see there for a more precise definition).

**Birational geometry** considers properties of varieties which depend only on the birational class, i.e., they are equivalent when they have isomorphic function fields. In fact, one can define the appropriate [[category]] by starting with the category of varieties over a fixed field and then [[localization|localizing]] at all birational equivalences.

## Literature

Related articles: [[Mori program]], [[function field]]

* [[János Kollár]], [[Shigefumi Mori]], _Birational geometry of algebraic varieties_, Cambridge Tracts in Mathematics __134__, Cambridge University Press 2010 ([doi:10.1017/CBO9780511662560](https://doi.org/10.1017/CBO9780511662560))

* wikipedia: [birational geometry](http://en.wikipedia.org/wiki/Birational_geometry), [rational variety](http://en.wikipedia.org/wiki/Rational_variety)

* Caucher Birkhar, _Birationali geometry_, online notes short version 35 pages, [pdf](https://www.dpmms.cam.ac.uk/~cb496/birgeom.pdf); _Lectures on birational geometry_, 85 pages, [arxiv/1210.2670](http://arxiv.org/abs/1210.2670)

* Bruno Kahn, R. Sujatha, _Birational geometry and localisation of categories_, [arxiv/0805.3753v1](http://arxiv.org/abs/0805.3753v1)

A recent new approach is in

* S. Cantat, S. Lamy, _Normal subgroups of the Cremona group_, Acta Mathematica 210 (2013), 31–94

Probing birational geometry via the derived category of coherent sheaves is explained in

* Alexander Kuznetsov, _Derived categories view on rationality problems_, [arxiv/1509.09115](http://arxiv.org/abs/1509.09115)

category: algebraic geometry
[[!redirects birational algebraic geometry]]


birational map

A [[rational map]] $f: X \dashrightarrow Y$ of varieties is __birational__ if there is a rational map $g: Y \dashrightarrow X$ such that both compositions $g\circ f$ and $f\circ g$ are defined as rational maps and equal the identity. Two varieties are birational (synonyms: birationally isomorphic, birationally equivalent) if there is a birational map between them. See [[birational geometry]].

* [[eom]]: [birational morphism](http://eom.springer.de/B/b016510.htm)

[[!redirects birational isomorphism]]
[[!redirects birational morphism]]
[[!redirects birational maps]]
[[!redirects birational isomorphisms]]
[[!redirects birationally isomorphic]]
[[!redirects birational equivalence]]
[[!redirects birationally equivalent]]


Birch and Swinnerton-Dyer conjecture

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## Idea

The _Birch and Swinnerton-Dyer conjecture_ is a [[conjecture]] about the form of the first non-vanishing [[derivative]] of the [[Hasse-Weil L-function]] of an [[elliptic curve]] at $s= 1$, expressed in terms of a [[higher regulator]], analogous to the [[class number formula]] for a [[Dedekind zeta function]].

This is hence a conjecture about [[special values of L-functions]]. It influenced the more far-reaching [[Beilinson conjectures]].

## Statement

Let $E$ be an elliptic curve over $\mathbb{Q}$. The Mordell-Weil theorem states that the Mordell-Weil group $E(\mathbb{Q})$ (defined to be the group of rational points of $E$) is a finitely generated abelian group, i.e. it is isomorphic to the product of $r$ copies of $\mathbb{Z}$ with some torsion subgroup. This $r$ is called the algebraic rank.

The rank part of the Birch and Swinnerton-Dyer conjecture states that the algebraic rank $r$ is given by the order of vanishing of the [[Hasse-Weil L-function]] $L(E,s)$ of $E$ at $s=1$. For instance, if $L(E,s)$ does not vanish at $s=1$, then the conjecture states that $E$ has no rational points of infinite order.

Note that more generally it is not known if the Hasse-Weil L-function of an elliptic curve (over a more general field) is even defined at $s=1$. However the [[modularity theorem]] states that every elliptic curve over $\mathbb{Q}$ is modular, so this Hasse-Weil L-function is equal to the L-function of some [[modular form]], which is known to have analytic continuation to all of $\mathbb{C}$, by the work of [[Erich Hecke]].

There is an even stronger statement of the Birch and Swinnerton-Dyer conjecture that gives the first non-vanishing Taylor coefficient of the Hasse-Weil L-function of $E$:

$$\frac{L^{(r)}(E,1)}{r!}=\frac{\#Sha(E) \Omega_{E} R_{E}\prod_{p\vert N}c_{p}}{(#E(\mathbb{Q})_{Tor})^{2}}.$$

Here

* $\#Sha(E)$ is the order of the [[Tate-Shafarevich group]] of $E$

* $\Omega_{E}$ is the real period of $E$

* $R_{E}$ is the regulator of $E$

* $c_{p}$ is the [[Tamagawa number]] of $E$ at a prime $p$ dividing the conductor $N$ of $E$

* $\#E(\mathbb{Q})_{Tor}$ is the order of the torsion subgroup of the Mordell-Weil group $E(\mathbb{Q})$ of $E$.

## Progress

It is known that if the L-function does not vanish at $s=1$ then the elliptic curve has no rational points of infinite order. It is also known that if the L-function has a zero of order $1$ at $s=1$ then the rank of the elliptic curve is $1$.

These results follow from the work of [[Benedict Gross]] and [[Don Zagier]] on the Gross-Zagier formula and Heegner points, and [[Victor Kolyvagin]]'s work on [[Euler systems]] (which bound the [[Selmer group]], which in turn control the rational points).

## Related concepts

* [[Picard group]], [[Tate-Shafarevich group]]

* [[Tamagawa number]]

## References

* Wikipedia, _[Birch and Swinnerton-Dyer conjecture](http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture)_

* Frank Gounelas, _The BSD cconjecture, regulators and special values of L-functions_ ([[GounelasBSD.pdf:file]])

* {#Bloch80} [[Spencer Bloch]], _A note on height pairings, Tamagwawa numbers, and the Birch and Swinnerton-Dyer conjecture_, Inventiones math. 58, 65-76 (1980) ([[BlochTamagawa.pdf:file]])

* Conrad, Venkatesh, et al., _BSD Seminar_ _[Introduction to the BSD conjecture](http://math.stanford.edu/~conrad/BSDseminar/Notes/L1.pdf)_, _[All lecture notes](http://math.stanford.edu/~conrad/BSDseminar/Notes/)_
[[!redirects BSD conjecture]]
[[!redirects BSD-conjecture]]


bireflective subcategory

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## Idea

By a _bireflective subcategory_ one might mean:

1. A [[subcategory]] $\iota \,\colon\, \mathcal{C} \hookrightarrow \mathcal{B}$ which is both [[reflective subcategory|reflective]] and [[coreflective subcategory|coreflective]] --- i.e. a [[fully faithful functor]] possessing both [[left adjoint|left]] and [[right adjoints]]: $L \dashv \iota \dashv R$ (*[[reflective localization]]* and *coreflective localization*).

   Such subcategories are called **essential localizations** by [Kelly & Lawvere (1989)](#KellyLawvere89), following the terminology of *[[essential geometric morphisms]]* from [[topos theory]], see also at *[[level of a topos]]*.

   The [[adjoint pair]] $\iota \circ L \dashv \iota \circ R$ is then an [[adjoint modality]]; see there for more.

1. More specifically, one may in addition ask that $L \simeq R$, hence that the subcategory inclusion $\iota$ has an [[ambidextrous adjoint]] $\beta \dashv \iota \dashv \beta$.

   Such subcategories are called **quintessential localizations** by [Johnstone (1996)](#JS96).

   The corresponding ([[comonad|co]]-)[[monad]] $\natural \coloneqq \iota \circ \beta$ is then a *[[Frobenius monad]]* (see at *[[classical modality]]*).

1. Finally, one may moreover ask that the [[points-to-pieces transform]] $\natural \overset{\epsilon}{\longrightarrow} id \overset{\eta}{\longrightarrow} \natural$ of a quintessential localization be the [[identity morphisms]].

   This strong form of the notion is the case actually called **bireflective subcategories** by [Freyd et al. (1999)](#FOPTST99).

## Definition
 {#AmbidextrouslyBireflectiveSubcategories}

\begin{definition}
  \label{BireflectiveSubcategory}
\linebreak
  A [[full subcategory]]-inclusion
  $$
    \iota
    \;\colon\;
    \mathcal{C} \hookrightarrow \mathcal{B}
  $$
  is *bireflective* in the sense of [FOPTST99, Def. 8](#FOPTST99)
  iff

(1.) the inclusion has an [[ambidextrous adjoint]]

  \begin{tikzcd}[column sep=20pt]
    \mathcal{C}
    \ar[
      from=rr,
      shift right=18pt,
      "{ \beta }"{description}
    ]
    \ar[
      rr,
      hook,
      "{ \iota }"{description}
    ]
    \ar[
      from=rr,
      shift left=18pt,
      "{ \beta }"{description}
    ]
    \ar[
      rr,
      shift left=8pt,
      phantom,
      "{
        \scalebox{.7}{$\bot$}
      }"
    ]
    \ar[
      rr,
      shift right=8pt,
      phantom,
      "{
        \scalebox{.7}{$\bot$}
      }"
    ]
    &&
    \mathcal{B}
    \ar[out=+50, in=-50,
      looseness=4.5,
      shorten=-3pt,
      shift left=0pt,
      "\scalebox{1.1}{${
          \hspace{-18pt}
          \mathclap{
            {
              \natural
            }
          }
          \hspace{4pt}
        }$}"{
          pos=.5,
          xshift=14pt
        }
    ]
  \end{tikzcd}

(2.) such that for all $E \in \mathcal{B}$ the [[counit of an adjunction|counit]] of the right adjunction followed by the [[unit of an adjunction|unit]] of the left adjunction (the [[points-to-pieces transform]]) equals the [[identity morphisms|identity]]:

  \begin{tikzcd}
    \natural E
    \ar[
      r,
      "{
        \epsilon^{\iota \,\dashv\, \beta}_{A}
      }"
    ]
    \ar[
      dr,
      equals
    ]
    &
    E
    \ar[
      d,
      "{
        \eta^{\beta \,\dashv\, \iota}_{A}
      }"
    ]
    \\
    &
    \natural E
  \end{tikzcd}

\end{definition}

\begin{remark}
  The first clause in Def. \ref{BireflectiveSubcategory} does not in general by itself enforce the second clause. A [[counter-example]] is given in [FOPTST99, Rem. 12](#FOPTST99).

However, the second clause does follow automatically when $\mathcal{B}$ is a [[Cauchy-complete category]] ([Johnstone (1996)](#JS96)).
\end{remark}

\begin{remark}
  In the other direction, the second clause in Def. \ref{BireflectiveSubcategory} implies that

  \begin{tikzcd}[column sep=20pt]
    \mathrm{id}_{\mathcal{B}}
    \ar[rr, "{ \eta^{\beta \,\dashv\, \iota} }"]
    &&
    \iota \beta
    \ar[rr, "{ \epsilon^{\iota \,\dashv\, \beta} }"]
    &&
    \mathrm{id}_{\mathcal{B}}
  \end{tikzcd}

is an [[idempotent]] [[natural transformation|natural]] [[endomorphism]], exhibited as a [[split idempotent]].
\end{remark}

\begin{prop}\label{BireflectionsInCCorrespondToSplitIdempotentsOnIdC}
  Given a [[category]] $\mathcal{B}$,
its bireflective subcategories (Def. \ref{BireflectiveSubcategory}) are in correspondence with the [[split idempotent]] [[natural transformation|natural]] [[endomorphisms]] on $id_\mathcal{B}$ with specified splitting, i.e.

  \begin{tikzcd}[column sep=20pt]
    \mathrm{id}_{\mathcal{B}}
    \ar[rr, "{ \eta^\natural }"]
    &&
    \natural
    \ar[rr, "{ \epsilon^\natural }"]
    &&
    \mathrm{id}_{\mathcal{B}}
  \end{tikzcd}

with

\[
  \label{EtaCircEpisolonIsIdentity}
  \eta^\natural \,\circ\, \epsilon^\natural
  \;=\;
  \mathrm{id}
  \,.
\]

\end{prop}
([FOPTST99, Thm. 13](#FOPTST99))

Concretely, from a split idempotent on $id_{\mathcal{B}}$ we recover the [[Frobenius monad]] $\natural$ as follows:

\begin{lemma}
  Given a split idempotent transformation as in Prop. \ref{BireflectionsInCCorrespondToSplitIdempotentsOnIdC} we have for all $E \,\in\, \mathcal{B}$
$$
  \natural
  (\eta^{\natural}_{E})
  \;=\;
  \eta^{\natural}_{\natural E}
  \;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;
  \natural
  (\epsilon^{\natural}_{E})
  \;=\;
  \epsilon^{\natural}_{\natural E}
  \,.
$$
\end{lemma}
([RFL21, Lem. 7.7](#RFL21))
\begin{proof}
  By [[natural transformation|naturality]] of the transformation $\eta$

\begin{tikzcd}[sep=30pt]
  E
  \ar[r, "{ \eta^\natural_E }"{description}]
  \ar[d, "{ \eta^\natural_E }"{description}]
  &
  \natural E
  \ar[d, "{ \natural \eta^\natural_E }"{description} ]
  \\
  \natural E
  \ar[r, "{ \eta^{\natural}_{\natural E} }"{description}]
  &
  \natural \natural E
\end{tikzcd}

and using (eq:EtaCircEpisolonIsIdentity) we get the first statement:

$$
  \natural
  (\eta^{\natural}_{E})
  \;=\;
  \natural
  (\eta^{\natural}_{E})
  \circ
  \overset
    { \mathrm{id}_E }
    {
      \overbrace{
        \eta^\natural_E
        \circ \epsilon^\natural_E
      }
    }
  \;=\;
  \eta^\natural_{\natural E}
  \circ
  \eta^{\natural}_{E}
  \circ
  \epsilon^\natural_E
  \;=\;
  \eta^{\natural}_{\natural E}
$$

The second follows analogously.
\end{proof}

\begin{prop}
  Given a split idempotent transformation as in Prop. \ref{BireflectionsInCCorrespondToSplitIdempotentsOnIdC}, $\natural \colon \mathcal{B} \to \mathcal{B}$ becomes

1. an [[idempotent monad]] $(\natural, \eta, \mu)$ by setting

   $$
     \mu \,\coloneqq\, \natural(\epsilon^\natural_{(-)})
   $$

1. an [[idempotent comonad]] $(\natural, \epsilon, \delta)$ by setting

   $$
     \delta \,\coloneqq\, \natural(\eta^\natural_{(-)})
   $$

\end{prop}
([RFL21, Prop. 7.8](#RFL21))
\begin{proof}
For the [[associativity]] of the monoid product we need to show the [[commuting square|commutativity]] of

  \begin{tikzcd}[sep=30pt]
    \natural
    \natural
    \natural
    E
    \ar[
      d,
      "{
        \natural(
          \natural \epsilon^\natural_E
        )
      }"{description}
    ]
    \ar[r, "{ \natural \epsilon^\natural_{\natural E} }"{description}]
    &
    \natural \natural E
    \ar[
      d,
      "{
        \natural
        \epsilon^\natural_E
      }"{description}
    ]
    \\
    \natural \natural E
    \ar[
      r,
      "{ \natural \epsilon^\natural_E }"{description}
    ]
    &
    \natural E
  \end{tikzcd}

By [[functor|functoriality]] of $\natural$ this follows from the commutativity of

  \begin{tikzcd}[sep=30pt]
    \natural
    \natural
    E
    \ar[
      d,
      "{
          \natural \epsilon^\natural_E
      }"{description}
    ]
    \ar[r, "{  \epsilon^\natural_{\natural E} }"{description}]
    &
    \natural E
    \ar[
      d,
      "{
        \epsilon^\natural_E
      }"{description}
    ]
    \\
    \natural E
    \ar[
      r,
      "{ \epsilon^\natural_E }"{description}
    ]
    &
    E
  \end{tikzcd}

which holds by [[natural transformation|naturality]] of $\epsilon$.

The other claims follow similarly.
\end{proof}

## Modal type theory
 {#ModalTypeTheory}

A [[modal type theory]] for bireflection [[modalities]] has been proposed in [Riley, Finster & Licata (2021)](#RFL21), [Riley (2022)](#Riley22).

We show the [[inference rules]] which these authors propose to add to plain [[intuitionistic type theory]] for encoding the presence of a bireflection [[classical modality]] $\natural$ on the type system, and we show corresponding [[categorical semantics]] via [[commuting diagrams]] in an interpreting [[locally cartesian closed category]]  (a more syntactic semantics is indicated in [RFL21, Sec. 7, esp. Fig. 6](#RFL21)).

\begin{remark}
  Given a bireflection $\natural \colon \mathcal{B} \to \mathcal{B}$ as above, then for $\Gamma \in \mathcal{B}$ we obtain a [[monad]] on the [[slice category]], to be denoted

\[
  \label{TheMonadOnSlices}
  \natural_\Gamma
  \;\colon\;
  \mathcal{B}_{/\Gamma} \to \mathcal{B}_{/\Gamma}
\]

by setting

$$
  \natural_\Gamma
  \left[
  \begin{array}{c}
    A
    \\
    \big\downarrow \mathrlap{\!\! {}^{p_A} }
    \\
    \Gamma
  \end{array}
  \;\;\;\;
  \right]
  \;\;\coloneqq\;\;
  \big(\eta^\natural_\Gamma\big)^\ast
  \,\natural\,
  \left[
  \begin{array}{c}
    \natural A
    \\
    \big\downarrow \mathrlap{\!\! {}^{\natural p_A} }
    \\
    \natural \Gamma
  \end{array}
  \;\;\;\;
  \right]
$$

\end{remark}

\begin{remark}\label{OdditiesOfTheRFLSyntax}
Beware of some non-obvious properties of the following type-theoretic [[syntax]]:

1. There is an asymmetry in the meaning of $\underline{(-)}$:

   An underline of a context denotes application of $\natural$, but an underline of a type in context only means its pullback along the [[counit of a comonad|counit]] $\epsilon^{\natural}$. The idea is that underlines indicate *dependency* on $\natural$-modal variables, to be distinguished of a type itself being $\natural$-[[modal type|modal]] or not.

1. The syntactic symbol "$\natural$" is not exactly [[categorical semantics|interpreted]] as the bireflective monad of that name. Rather:

   * An unadorned "$\natural$" in the syntax [[categorical semantics|interprets]], in [[context]] $\Gamma$, as the [[relative monad]]

     $$
       \classical^{rel}_\Gamma
       \;\;
       \colon
       \;\;
       \array{
         \mathcal{B}_{/\natural \Gamma}
         &
           \xrightarrow{
             (\eta^\natural_\Gamma)^\ast
           }
         &
         \mathcal{B}_{/\Gamma}
         &
           \xrightarrow{
             \natural_\Gamma
           }
         &
         \mathcal{B}_{/\Gamma}
       }
     $$

   * The combination $\natural \underline{(-)}$ is what actually [[categorical semantics|interprets]] as $\natural_\Gamma$ (eq:TheMonadOnSlices).

     This transpires syntactically in [RFL21, Cor. 2.14](#RFL21); [Riley22, Cor. 1.1.16](#Riley22), and semantically in (eq:InternalUnitConstruction) below.

\end{remark}

\linebreak

The list of [[inference rules]] and their [[categorical semantics]] (first the [[axioms]] then derived rules):

Using these ingredients, one finds that the function $a \,\mapsto\, \underline{a}^\natural$ is interpreted simply as postcomposition with the $\eta^{\natural}$-[[naturality square]]:

\linebreak
\linebreak

\[
  \label{InternalUnitConstruction}
  \;
\]

With this, one finds the following rule for $\natural$ respecting [[dependent pair types]]:

\linebreak
\linebreak

\[
  \label{NaturalRespectsDependentPairs}
  \;
\]

Using this one finds that every type $A$ is the [[dependent sum type]] of its fiber [[linear types]] $A_{\underline{x}}$ (NB: we show 1-categorical semantics, interpreting [[identification types]] as [[diagonal morphisms]]):

\linebreak
\linebreak

\[
  \label{LinearTpes}
  \;
\]

\linebreak

## Examples

\begin{example}\label{ReractiveSpaces}
**(spaces among retractive spaces)**
\linebreak
  Let $\mathcal{B}$ be the category of [[retractive space|retractive]] [[topological spaces]] (or just of [[Sets]]), i.e. the category of [[diagrams]] in [[Top]] ([[Set]]) of the form $B \xrightarrow{i} E \xrightarrow{p} B$ such that $p \circ i \,=\, id_B$. Then the functor

$$
  \begin{array}{ccc}
    Top &\hookrightarrow& Top_{\mathcal{R}}
    \\
    B &\mapsto& \big[ B \xrightarrow{id} B \xrightarrow{id} B \big]
  \end{array}
$$

is a bireflective subcategory inlusion (Def. \ref{BireflectiveSubcategory}) with [[ambidextrous adjoint]] given by

$$
  \begin{array}{ccc}
    Top_{\mathcal{R}} &\longrightarrow& Top
    \\
    \big[
      B \xrightarrow{i} E \xrightarrow{p} B
    \big]
    &\mapsto&
    B
  \end{array}
$$

\end{example}

\begin{example}\label{ZeroVectorBundlesAmongVectorBundles}
**([[zero-vector bundles]] among all [[vector bundles]])**
\linebreak
  For any [[ground field]],
  let $\mathcal{B} \coloneqq Vect_{Set} \coloneqq \int_{S \in Set} Vect S$ be the category [[VectBund]] of [[vector bundles]] over [[Sets]], hence of [[indexed set|indexed]] [[vector spaces]]. Then the functor
$$
  Set \xhookrightarrow{\;\; 0 \;\;} Vect_{Set}
$$
which sends any set $S$ to the [[zero vector bundle]] $S \times \{0\}$ over $S$ is a bireflective subcategory in the sense of Def. \ref{BireflectiveSubcategory}.
\end{example}

\begin{example}\label{ZeroSpectrumBundlesAmongSpectrumBundles}
**([[zero-spectrum bundles]] among all [[parameterized spectra]])**
\linebreak
Let $T Grpd_\infty$ denote the [[(infinity,1)-category|$\infty$-category]] of [[parameterized spectra]] over varying [[infinity-groupoids|$\infty$-groupoids]] (the [[tangent (infinity,1)-category|tangent $\infty$-category]] of [[∞Grpd|$Grpd_\infty$]]). Then the inclusion
$$
  Grpd_\infty \hookrightarrow T Grpd_\infty
$$
given by sending $X \in Grpd_\infty$ to the [[zero-spectrum bundle]] over $X$ ought to count as a "bireflective [[sub-(infinity,1)-category|sub-$\infty$-category]]" (cf. [here](https://ncatlab.org/nlab/show/infinitesimal+cohesion#GoodwillieTangentCohesion) at *[[infinitesimal cohesion]]*).

A [[model category]]-presentation of this situation by a (left and right) [[Quillen functor]] to a [[model structure for parameterized spectra]] [[underlying]] which is a bireflective sub-1-category inclusion (Def. \ref{BireflectiveSubcategory}) may be recognized in [Hebestreit, Sagave & Schlichtkrull (2020), Lem. 3.19](model+structure+for+parameterized+spectra#HebestreitSagaveSchlichtkrull20), there incarnated as the canonical inclusion
$$
  \mathcal{S}^{\mathcal{I}}
  \hookrightarrow
  Sp^{\Sigma}_{\mathcal{R}}
$$
of [[I-spaces|$\mathcal{I}$-spaces]] into [[symmetric spectra]] [[internalization|internal to]] [[retractive spaces]] (Exp. \ref{ReractiveSpaces}).

An analogous statement is in [Braunack-Mayer (2019), Cor. 2.45](model+structure+for+parameterized+spectra#BraunackMayer19).
\end{example}

## Related concepts

* [[infinitesimal cohesion]]

* [[classical modality]]

* [[VectBund]]

* [[parameterized spectra]]

## References

On subcategory inclusion which have both a left and a right adjoint ([[essential localizations]]):

* {#KellyLawvere89} [[G. M. Kelly]], [[F. W. Lawvere]], *On the complete lattice of essential localizations*, Bull. Soc. Math. Belg. Sér. A **41** (1989) 289–319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

On subcategory inclusions with ambidextrous adjoints ([[quintessential localizations]]):

* {#JS96} [[Peter Johnstone]], _Remarks on Quintessential and Persistent Localizations_, TAC **2** 8 (1996) 90-99. [[pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf), [2-08](http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html)]

On subcategory inclusions with ambidextrous adjoints whose [[points-to-pieces transform]] is the [[identity]] (actual *bireflective subcategories*):

* {#FOPTST99} [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

Discussion of a [[modal type theory]] for bireflective [[modalities]], in a context of [[linear homotopy type theory]]:

* {#RFL21} [[Mitchell Riley]], [[Eric Finster]], [[Daniel R. Licata]], Section 7.1 in: *Synthetic Spectra via a Monadic and Comonadic Modality* [[arXiv:2102.04099](https://arxiv.org/abs/2102.04099)]

* {#Riley22} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269), [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

[[!redirects bireflective subcategories]]

[[!redirects essential localization]]
[[!redirects essential localizations]]


Birgit Richter

* [website](http://www.math.uni-hamburg.de/home/richter/)

## Selected writings

On [[ring spectra]]:

* {#BakerRichter04} [[Andrew Baker]], [[Birgit Richter]] _Structured ring spectra_, London Mathematical Society Lecture Notes Series 315, Springer 2004 ([ISBN:9780521603058](https://www.cambridge.org/de/academic/subjects/mathematics/algebra/structured-ring-spectra?format=PB))

* [[Birgit Richter]], _Commutative ring spectra_ ([arXiv:1710.02328](https://arxiv.org/abs/1710.02328), [pdf](https://www.math.uni-hamburg.de/home/richter/rev-commringspectra.pdf)), in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press.

Interpretation of [[algebraic K-theory]] of [[topological K-theory]] ([[iterated algebraic K-theory]]) as the [[K-theory]] of [[BDR 2-vector bundles]]:

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Stable bundles over rig categories_, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 ([doi:10.1112/jtopol/jtr016](https://doi.org/10.1112/jtopol/jtr016))

On [[category theory]] and [[homotopy theory]]:

* {#Richter19} [[Birgit Richter]], _From categories to homotopy theory_, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 ([doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf))

category: people


biring

#Contents#
* table of contents
{:toc}

## The idea

Just as a [[bimonoid]] is both a [[monoid]] and a [[comonoid]] in a compatible way, a 'biring' is both a commutative [[ring]] and a commutative [[coring]] in a compatible way.

##Definition##

A **biring** is a [[commutative ring]] $R$ equipped with ring [[homomorphisms]] called **coaddition**:

$$ R \to R \otimes R $$

**cozero**:

$$ R \to \mathbb{Z} $$

**co-additive inverse**:

$$ R \to R $$

**comultiplication**:

$$ R \to R \otimes R $$

and the **multiplicative counit**:

$$ R \to \mathbb{Z} $$

satisfying the usual axioms of a commutative ring, but 'turned around'.

More tersely, and also more precisely, a biring is a [[internalization|commutative ring object]] in the [[opposite category|opposite]] of the category of commutative rings (also known as the category of [[affine schemes]]).

Equivalently, a biring is a commutative ring $R$ equipped with a lift of the functor

$$ hom(R, -) : CommRing \to Set $$

to a functor

$$ hom(R, -) : CommRing \to CommRing $$

Birings form a [[monoidal category]] thanks to the fact that functors of this form are closed under composition.  A [[monoid object]] in this monoidal category is called a [[plethory]].  A plethory is an example of a [[Tall–Wraith monoid]].

The most important example of a biring is $\Lambda$, the ring of [[symmetric polynomials]].  This is actually a [[plethory]].

## Categorified Birings ##

The biring $\Lambda$ is the [[Grothendieck group]] of the category of [[Schur functors]], which is equivalent to the [[functor category]]

$$[\mathbb{P}, FinDimVect] $$

where $\mathbb{P}$ is the [[permutation groupoid]] and [[FinDimVect]] is the category of [[finite-dimensional vector spaces]] over a [[field]] $k$ of [[characteristic zero]].  $\Lambda$ is also the [[Grothendieck group]] of

$$[\mathbb{P}, Vect ]$$

where we drop the finite-dimensionality restriction on our vector spaces and work with all of [[Vect]].

This suggests that the biring structure of $\Lambda$ may emerge naturally from a 'categorified biring' structure on $[\mathbb{P}, Vect ]$.  In this section we sketch how such a categorified biring might be constructed, based on the assumption that there is a tensor product of [[cocomplete category|cocomplete]] [[linear categories]] with good universal properties.

Namely, we assume that given cocomplete linear categories $X$ and $Y$, there is a cocomplete linear category $X \otimes Y$ such that:

* There is a linear functor $i: X \times Y \to X \otimes Y$ which is [[cocontinuous functor|cocontinuous]] in each argument.

* For any cocomplete linear category $Z$, the category of linear functors $X \otimes Y \to Z$ is equivalent to the category of linear functors $X \times Y \to Z$ that are [[cocontinuous functor|cocontinuous]] in each argument, with the equivalence being given by precomposition with $i$.

With any luck these two assumptions will let us show that for any categories $A$ and $B$,
\[ [A \times Y, Vect] \simeq [A,Vect] \otimes [B, Vect] \label{tensor} \]
where we use $[-,-]$ to denote the [[functor category]].

Assuming all this, we obtain the following operations on the category $[\mathbb{P}, Vect]$:

1. **Addition**: form the composite functor
$$ [\mathbb{P}, Vect] \times  [\mathbb{P}, Vect]  \to [\mathbb{P}, Vect \times Vect] \to  [\mathbb{P}, Vect] $$
where the last arrow comes from postcomposition with
$$ \oplus : Vect \times Vect \to Vect $$
This composite is our **addition**:
$$ \oplus :  [\mathbb{P}, Vect] \times  [\mathbb{P}, Vect] \to  [\mathbb{P}, Vect] $$
It's really just the coproduct in $[\mathbb{P}, Vect]$.

1. **Multiplication**: first form the composite functor
$$ [\mathbb{P}, Vect] \times  [\mathbb{P}, Vect]  \to [\mathbb{P}, Vect \times Vect] \to  [\mathbb{P}, Vect] $$
where the last arrow comes from postcomposition with
$$ \otimes : Vect \times Vect \to Vect $$
This composite is our **multiplication**:
$$ \otimes :  [\mathbb{P}, Vect] \times  [\mathbb{P}, Vect] \to  [\mathbb{P}, Vect] $$
Since this product preserves colimits in each argument, if we use the hoped-for universal property of the tensor product of cocomplete linear categories, we can reinterpret this as a cocontinuous functor
$$  \otimes:  [\mathbb{P}, Vect] \otimes  [\mathbb{P}, Vect]  \to  [\mathbb{P}, Vect] $$

1. **Coaddition**: Form the composite functor
$$  [\mathbb{P}, Vect] \to [\mathbb{P} \times \mathbb{P} , Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect] $$
where the first arrow comes from precomposition with the addition operation on $\mathbb{P}$ (a restriction of [[coproduct]] in [[FinSet]]), and the second comes from our hoped-for relation \eqref{tensor}.  This is our **coaddition**:
$$  coadd: [\mathbb{P}, Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect] $$

1.  **Comultiplication**: Form the composite functor
$$  [\mathbb{P}, Vect] \to [\mathbb{P} \times \mathbb{P} , Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect] $$
where the first arrow comes from precomposition with the multiplication operation on $\mathbb{P}$ (a restriction of [[product]] in [[FinSet]]), and the second comes from our hoped-for relation \eqref{tensor}.  This is our **comultiplication**:
$$  comult: [\mathbb{P}, Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect] $$

The additive and multiplicative unit and counit may be similarly defined.  Note that we are using rather little about $\mathbb{P}$ and $Vect$ here.  For example, the category of ordinary non-linear [[species]], $[\mathbb{P}, Set]$, should also become a categorified biring if there is a tensor product of cocomplete categories with properties analogous to those assumed for cocomplete $k$-linear categories above.  But we could also replace $\mathbb{P}$ by any [[rig category]].  So, 'biring categories', or more precisely 'birig categories', should be fairly common.

## References

* D. Tall and [[Gavin Wraith|G. Wraith]], Representable functors and operations on rings, _Proc. London Math. Soc._ **3** (1970), 619--643.

[[!redirects birings]]
[[!redirects bi-ring]]
[[!redirects bi-rings]]


Birkhoff decomposition

__Riemann-Birkhoff factorization/decomposition__ is a realization of a holomorphic matrix function of a circle as a product of a matrix holomorphic on a neighborhood of closed disk and a function of a matrix holomorphic on a neighborhood of an exterior of the disk including infinity and the circle itself. This decomposition is in the essence of
Riemann-Hilbert problem. The interpretation in terms of loop groups is related to Bruhat decomposition.

There is an algebraic Birkhoff decomposition discovered in the study of
Connes-Kreimer Hopf algebraic approach to renormalization in QFT.

## References

Related $n$Lab entries: [[loop group]], [[Riemann-Hilbert problem]], [[Wiener-Hopf decomposition]]

* wikipedia [Birkhoff factorization](http://en.wikipedia.org/wiki/Birkhoff_factorization), [Birkhoff–Grothendieck theorem](http://en.wikipedia.org/wiki/Birkhoff%E2%80%93Grothendieck_theorem)
* [[eom]]: [Birkhoff factorization](http://www.encyclopediaofmath.org/index.php/Birkhoff_factorization)
* G. D. Birkhoff, _The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations_, Proc. Amer. Acad. Arts and Sci. __49__ (1913), 531-568.
* G. Segal, A. Pressley, _Loop groups_, Oxford University Press
* I. Z. Gohberg, M. G. Krein, _Systems of integral equations on a half-line with kernels depending on the difference of the arguments_, Transl. Amer. Math. Soc. __14__ (1960) pp. 217–284
* K. F. Clancey, I. Z. Gohberg, _Factorization of matrix functions and singular integral operators_, Birkhäuser (1981)
[[!redirects Birkhoff factorization]]
[[!redirects Riemann-Birkhoff decomposition]]
[[!redirects Riemann-Birkhoff factorization]]


Birkhoff's HSP theorem

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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_example #BirkhoffHSP}
###### Theorem
**(Birkhoff's HSP theorem)**

Given a [[language]] $L$ generated by a set of (single-sorted) finitary operations, and a [[class]] $C$ of [[structures]] for $L$. Then $C$ is the class of models for a set of universally quantified equations between terms of $L$ (a [[Lawvere theory]]) if and only if

1. (H) The class is closed under homomorphic [[images]] (see [below](#HomomorphicImages)),

1. (S) The class is closed under subalgebras,

1. (P) The class is closed under taking products.

=--

See also at _[Lawvere theory -- Characterization of examples](Lawvere+theory#CharacterizationOfExamples)_

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

Here "closed under homomorphic images" means that if $A$ and $B$ are structures in the class, and $\phi \colon A \to B$ is a [[homomorphism]] between them, then also its [[image]] $im(\phi) \hookrightarrow B$ is an element of the class.

=--

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

The [[first order logic|first-order]] analogue of HSP (theorem \ref{BirkhoffHSP}) is the characterization (see e.g. Chang and Keisler's original text ([Chang-Keisler 66](#ChangKeisler66)) on continuous model theory) of [[elementary classes of structures]] of structures: they're precisely those closed under [[elementary substructures]], [[elementary embeddings]], [[ultraproducts]], and [[ultraroots]] (if an ultrapower of something is in your class, that something was in your class.)

=--

## References

* {#ChangKeisler66} Chang, Keisler _Continuous Model Theory_, Princeton University Press (1966) [ISBN: 9780691079295]

* [[Michael Barr]], *HSP type theorems in the category of posets*, in: *Proc. 7th International Conf. Mathematical Foundation of Programming Language Semantics*, Lecture Notes in Computer Science **598** (1992) 221–234 [[doi:10.1007/3-540-55511-0_11](https://doi.org/10.1007/3-540-55511-0_11), [[Barr-HSPinPosets.pdf:file]]]

* [[Michael Barr]], *Functorial semantics and HSP type theorems*, Algebra Universalis **31** (1994) 223–251 [[doi:10.1007/BF01236519](https://doi.org/10.1007/BF01236519), [pdf](https://www.math.mcgill.ca/barr/papers/hsp.pdf), [[Barr-FuncSemanticsHSP.pdf:file]]]

* [[Michael Barr]], *HSP subcategories of Eilenberg-Moore algebras*, Theory Appl. Categories **10** 18 (2002), 461–468 [[tac:10-18](http://www.tac.mta.ca/tac/volumes/10/18/10-18abs.html)]

See also:

* Wikipedia, Birkhoff's theorem

[[!redirects Birkhoff HSP theorem]]

[[!redirects HSP theorem]]


Birkhoff's theorem

[[Zoran Škoda]] writes:

>There are several things called "Birkhoff's theorem" in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff's. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff's factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal and Pressley).

For now we have:

* the [[Birkhoff–von Neumann theorem]]
* [[distributive lattice|Birkhoff Duality of distributive lattices]]
* [[Birkhoff's HSP theorem]]

## References

For the theorem as in [[universal algebra]]:

* [[Garrett Birkhoff]], _Subdirect unions in universal algebra_, Bull. Amer. Math. Soc. 50 (1944), 764-768.

* [Wolfram Mathworld](http://mathworld.wolfram.com/BirkhoffsTheorem.html)

[[William Lawvere]] considers this Birkhoff's theorem through the eyes of [[Isbell duality]] and [[cohesive toposes]] in

* {#Lawvere16} [[William Lawvere]], _Birkhoff's Theorem from a geometric perspective: A simple example_, to appear in Categories and General Algebraic Structures with Applications (2016), [journal page](http://www.cgasa.ir/article_12425_0.html).

[[!redirects Birkhoff's theorem]]
[[!redirects Birkhoff's theorem]]
[[!redirects Birkhoff theorem]]
[[!redirects Birkhoff's theorems]]
[[!redirects Birkhoff's theorems]]
[[!redirects Birkhoff theorems]]


Birkhoff-von Neumann theorem

#Contents#
* table of contents goes here
{:toc}

## The Birkhoff-von Neumann theorem

Given a [[permutation]] $\sigma \in S_n$, the [[permutation matrix]] that is associated with $\sigma$ is the $n \times n$ matrix $P_{\sigma}(j,k)$, where $1 \le j,k \le n$, whose entries are given by

$P_{\sigma}(j,k)=
\left \{
\begin{aligned}
1 & if \sigma (j)=k \\
0 & otherwise.
\end{aligned}
\right.$

An $n \times n$ doubly stochastic matrix $D$ is a square matrix whose elements are real and whose rows and columns sum to unity.  Given such a matrix, there exist nonnegative weights $\{w_{\sigma}:\sigma \in S_n\}$ such that

$\sum_{\sigma \in S_n} w_{\sigma}=1$ and $\sum_{\sigma \in S_n} w_{\sigma} P_{\sigma}=D$.

The set of such matrices of order $n$ is said to form the convex hull of permutation matrices of the same order where the latter are the vertices (extreme points) of the former.

## Quantum extensions

In the quantum context, doubly stochastic matrices become doubly stochastic channels, i.e. completely positive maps preserving both the trace and the identity.  Quantum mechanically we understand the permutations to be the unitarily implemented channels.  That is, we expect doubly stochastic quantum channels to be convex combinations of unitary channels (that is channels that can be represented by some combination of [[unitary]] transformations).  Unfortunately it is well-known that some quantum channels $cannot$ be written that way.

However, large tensor powers of a channel may be easier to represent in this way, because one need not use only product unitaries in the decomposition.  Thus one proposed solution to the problem is to denote, for any doubly stochastic channel $T$, by $d_{B}(T)$ the **Birkhoff defect**, defined as the cb-norm distance from $T$ to the convex hull of the unitarily implemented channels.  Then the key is to determine whether $d_{B}(T^{\otimes n})$ goes to zero as $n\to\infty$.

## References

* Liviu Paunescu, Florin Radulescu, _A generalisation to Birkhoff - von Neumann theorem_ ([arXiv:1506.01685](https://arxiv.org/abs/1506.01685))

[[!redirects Birkhoff-von Neumann theorem]]
[[!redirects Birkhoff–von Neumann theorem]]
[[!redirects Birkhoff--von Neumann theorem]]


bisection of a Lie groupoid

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

#Bisections of Lie groupoids#
* table of contents
{:toc}

## Definition

### In components

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

Let $X=(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0)$ be a [[Lie groupoid]].

A **bisection** of $X$ is a [[smooth function]] $\sigma : X_0 \to X_1$ such that

1. $\sigma$ is a [[section]] of $d_1$;

1. $d_0 \circ \sigma : X_0 \to X_0$ is a [[diffeomorphism]].

Bisections naturally form a [[group]] under pointwise composition in $X$, the **group of bisections** of the Lie groupoid.
=--

One can prove that the bisection group is a [[infinite-dimensional Lie group]] in the sense of Milnor (see Neeb's survey) (under some mild assumptions on the underlying Lie groupoid).
The infinite-dimensional Lie group of bisections is closely connected to the underlying Lie groupoid (see references below), e.g.

1. From the knowledge of the smooth structure of the bisection group and the manifold of units, one can even reconstruct the underlying Lie groupoid (again under some assumptions).

1. The construction is functorial in a suitable sense and extending this one can even relate (smooth) representations of Lie groupoids to smooth representations of its bisection group

### Abstractly

Let $\mathbf{H} = $ [[Smooth∞Grpd]]. Let $X \in \mathbf{H}$ be equipped with an _atlas_, hence with an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] $U \to X$ out of a [[0-truncated object]].

We may regard this atlas as an object in the [[slice (∞,1)-topos]]
$\mathbf{X} \in \mathbf{H}_{/X}$

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

The **[[smooth ∞-group]] of bisections** of $\mathbf{X}$ is its [[automorphism ∞-group]]

$$
  \mathbf{BiSect}(X,U) \coloneqq
  \mathbf{Aut}_{/X}(\mathbf{X}, \mathbf{X})
  \,.
$$

=--

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

For $X$ a 1-groupoid as above and $U = X_0$, a bisection is precisely a smooth [[natural transformation]] of the form

$$
  \array{
    U &&\stackrel{\simeq}{\to}&& U
    \\
    & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow
    \\
    && X
  }
  \,.
$$

Here the top morphism is a [[diffeomorphism]] $\phi : X \to X$
and since the diagonal morphisms are identities onto the object manifold the component map of $\eta$ is

$$
  x \mapsto  (x \stackrel{\eta(x)}{\to} \phi(x))  \,.
$$

This is precisely the bisection in the traditional sense of
def. \ref{TraditionalDefinition}.

=--

## Properties

### Relation to Lie-Rinehart algebras

For $U \to X$ a Lie groupoid with atlas as above, write
$\mathfrak{g} = Lie(\mathbf{BiSect}(X,U))$
for the [[Lie algebra]] of the group of bisections.
Then $(C^\infty(X), \mathfrak{g})$ is the [[Lie-Rinehart algebra]] corresponding to the [[Lie algebroid]] of the Lie groupoid.

### Relation to Atiyah groupoids

> for the moment see at _[[Atiyah groupoid]]_ and _[[higher Atiyah groupoid]]_.

## References

* [[Ieke Moerdijk]], [[Janez Mrčun]], p. 114 of _[[Introduction to foliations and Lie groupoids]]_, Cambridge Studies in Advanced Mathematics __91__, 2003. x+173 pp. ISBN: 0-521-83197-0

* [[Alexander Schmeding]], [[Christoph Wockel]], _The Lie group of bisections of a Lie groupoid_ ([arXiv:1409.1428](http://arxiv.org/abs/1409.1428))

* [[Alexander Schmeding]], [[Christoph Wockel]], _(Re)constructing Lie groupoids from their bisections and applications to prequantisation_ ([arXiv:1506.05415](https://arxiv.org/pdf/1506.05415.pdf))

* [[Alexander Schmeding]], [[Christoph Wockel]], _Functorial aspects of the reconstruction of Lie groupoids from their bisections_ ([arXiv:1506.05587](https://arxiv.org/abs/1506.05587))

* Habib Amiri, [[Alexander Schmeding]], _Linking Lie groupoid representations and representations of infinite-dimensional Lie groups_ ([arXiv:1805.03935
](https://arxiv.org/pdf/1805.03935.pdf))

[[!redirects bisections of a Lie groupoid]]

[[!redirects group of bisections]]

[[!redirects bisection]]
[[!redirects bisections]]

[[!redirects ∞-bisection]]
[[!redirects ∞-bisections]]

[[!redirects ∞-group of bisections]]
[[!redirects ∞-groups of bisections]]


Bishop set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Bishop sets
* table of contents
{: toc}

## Idea

A _Bishop set_ is a notion of _[[set]]_ in [[constructive mathematics]], commonly used in [[Bishop's constructive mathematics]] (cf. [[constructive analysis]]).

In ([Bishop](#Bishop)) the notion of _[[set]]_ is specified by stating that a set has to be given by a description of how to build [[elements]] of this set and by giving a binary [[relation]] of [[equality]], which has to be an [[equivalence relation]].

A [[function]] from a set $A$ to a set $B$ is then given by an operation, which is compatible with the equality (i.e. two elements which are equal in A are mapped to two elements which are equal in B), and is described as "a finite routine $f$ which assigns an element $f(a)$ of $B$ to each given element $a$ of $A$". This notion of routine is left informal but must "afford an explicit, finite, mechanical reduction of the procedure for constructing $f(a)$ to the procedure for constructing $a$."

These ideas have been formalized in [[type theory]], where they serve to set up [[set theory]] in an ambient type theoretic [[logical framework]]. See _[Formalization in type theory](#FormalizationInTypeTheory)_ below.

## Formalization in type theory
 {#FormalizationInTypeTheory}

Formally, the definition of Bishop set presumes some prior notion of "collection" (the elements) and "relation" (the equality) that can be put together.  These collections are sometimes called [[presets]], and can be formalized in various ways.  In particular, presets generally lack operations such as [[quotients]] and even [[subsets]].

One of the most common approaches is to use [[type theory]], where the collections are simply types.  Note that this is a different approach to the use of type theory in foundations of mathematics than that of, for instance, [[homotopy type theory]] --- in the latter, type theory is enriched with operations on types such as quotients, enabling them to be the basic set-like objects out of which mathematics is built; while in the Bishop-set approach one defines "sets" as types equipped with extra structure, so that the sets have quotient-type operations even though the types do not.  There is an analogy to the construction of the [[exact completion]] of a category, which has quotients even if the original category does not.

Frequently, the type theories used for this purpose do not even have a separate notion of "relation" or "proposition", and thus the untruncated [[propositions as types]] interpretation is used for the equivalence relations in a Bishop set.  That is, one has a type $X:Type$ of elements and a type family $R:X\to X \to Type$ with witnesses that it is "reflexive, symmetric, and transitive", but nothing prevents $R(x,y)$ from having more than one element.  (If the type theory has [[identity types]], one could demand that $R$ consist of [[h-propositions]], but at least historically this is not often done.)  This means that categorically, the Bishop sets are most like the [[ex/lex completion]] of the category of types --- although the morphisms in the ex/lex completion are defined as a *quotient* in the ambient set theory, whereas the morphisms between Bishop sets simply form another Bishop set.

For example, see ([Palmgren05](#Palmgren05)) for an exposition and ([Coquand-Spiwack, section 3](#CoquandSpiwack)) for a brief list of the axioms.

In the context of [[type theory]] Bishop sets are also called _[[setoids]]_.

## Properties

### The predicative topos of Bishop sets

In much of [[set theory]] the [[category]] [[Set]] of all sets is a [[Topos]]. For Bishop sets formalized in [[type theory]] this is not quite the case. Instead:

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

The category of Bishop sets in [[Martin-Löf dependent type theory]] is a [[strong predicative topos]].

=--

This is ([van den Berg, theorem 6.2](#vdBerg)), based on ([Moerdijk-Palmgren, section 7](#MoerdijkPalmgren)).

## Related concepts

* [[setoid]], [[quotient type]]

* [[constructive mathematics]]

* [[ETCS]]

[[!include oidification - table]]

## References

The original texts, with focus on [[constructive analysis]]:

* {#Bishop} [[Errett Bishop]]: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]]: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

Discussion of Bishop sets --- under the name *[[setoids]]* ---  as [[categorical semantics]] for [[dependent type theory]]:

* {#Hofmann95} [[Martin Hofmann]], _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)]

Review and relation to [[ETCS]]:

* {#Palmgren05} [[Erik Palmgren]],  _Bishop's set theory_ (2005) [[pdf](http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf), [[Palmgren-BishopSetTheory.pdf:file]]]

* [[Erik Palmgren]], _Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets_, Annals of Pure and Applied Logic **163** 10 (2012) 1384-1399 (2009) [[doi:10.1016/j.apal.2012.01.011](https://doi.org/10.1016/j.apal.2012.01.011), [arXiv:1201.6272](https://arxiv.org/abs/1201.6272)]

* [[Erik Palmgren]], *Bishop-style constructive mathematics in type theory -- a tutorial* (2013) [[pdf](https://staff.math.su.se/palmgren/Palmgren_Nis.pdf), [[Palmgren-BishopConstructiveMathematics.pdf:file]]]

The term "Bishop set" and their formalization in [[type theory]] (essentially as for [[setoids]], but not mentioning this term) appears (in a context of formalization of [[homological algebra]]) in:

* {#CoquandSpiwack} [[Thierry Coquand]], [[Arnaud Spiwack]], section 3 of: *Towards constructive homological algebra in type theory*, in: *Towards Mechanized Mathematical Assistants. MKM Calculemus 2007*,  Lecture Notes in Computer Science **4573** Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), [pdf](https://hal.inria.fr/inria-00432525/document)]

Some of the text above is taken from this section.

The [[predicative topos]] formed by Bishop sets in type theory is discussed in

* {#MoerdijkPalmgren} [[Ieke Moerdijk]], [[Erik Palmgren]], _Wellfounded trees in categories. Ann. Pure Appl. Logic, 104(1-3):189–218, (2000)

* [[Benno van den Berg]], _Predicative toposes_  ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959))
 {#vdBerg}

A formalization of constructive set theory in terms of setoids in [[intensional type theory|intensional]] [[type theory]] coded in [[Coq]] is discussed in

* {#PalmgrenWilander} [[Erik Palmgren]], Olov Wilander, _Constructing categories of setoids in type theory_, ([pdf](http://www2.math.su.se/~palmgren/coq/czf_and_setoids/czf&setoids.pdf), [Coq](http://www2.math.su.se/~palmgren/coq/czf_and_setoids))

See also

* [[UF-IAS-2012]], _On the setoid model of type theory_ ([video](http://video.ias.edu/univalent/palmgren))

* [[UF-IAS-2012]], _[Setoid model of type theory formalized](http://uf-ias-2012.wikispaces.com/Setoid+model+of+type+theory+formalized)_

* [Discussion](http://golem.ph.utexas.edu/category/2009/10/syntax_semantics_and_structura.html#c029271) on categorical aspects of the setoid construction.

[[!redirects Bishop set]]
[[!redirects Bishop sets]]


Bishop's constructive mathematics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Bishop's constructive mathematics

* table of contents
{: toc}

## Overview

In 1967, [[Errett Bishop]] revived interest and practice of [[constructive mathematics]] with his book [[Foundations of Constructive Analysis]], in which he showed that contrary to previous perceptions, large swaths of mathematics could be developed constructively with only minor changes from the classical theory.

While [[Brouwer]] had insisted that mathematics was primary to [[logic]], the [[intuitionism|intuitionists]] who followed him went more into intuitionistic [[foundations]] and [[philosophy]] than into mathematics as such.  The [[Russian constructivism|Russian school]] of constructivism focussed on [[recursion theory]] and was far removed from modern [[Bourbaki]]-style mathematics.  There was a pervasive feeling that constructive mathematics, particularly constructive [[analysis]], was simply unworkable; Bishop attacked this idea, not by arguing against it, but by simply demonstrating how constructive analysis could be done.

Bishop\'s work was primarily in [[analysis]], because constructive mathematics is more difficult at an elementary level there than in some other fields such as [[combinatorics]].  Bishop\'s style was extended to constructive [[algebra]] by his disciple [[Fred Richman]].  On the other hand, other fields such as point-set [[topology]] are so much *more* difficult constructively that Bishop did not even consider them; later authors have attacked the problem with tools such as [[apartness spaces]] and [[locales]]/[[formal topologies]].  In fact, even Bishop's treatment of analysis has actual flaws regarding the notion of continuous function (see [below](#Continuity)) that seem to be best fixed by using locales.

Bishop's work is important not only for its historical role in reviving interest in constructive mathematics and belief in its feasibility, but because the framework for constructive mathematics that he laid out has played a central role in a great deal of later work.  Known as **Bishop's constructive mathematics** and abbreviated **BISH** or **BCM**, this framework is often viewed as a sort of "common core" of all kinds of constructive mathematics.  However, this is not really a true statement about the mathematics that Bishop himself did, which in particular is not compatible with the [[internal logic]] of [[elementary toposes]] (see [below](#WhatItIs)) --- but for added confusion, some mathematicians who say they are working in "Bishop's constructive mathematics" are in fact working in a theory about which this statement *is* true.  Thus, unless one really does want to work in the mathematics that Bishop himself did, it is better to work in "pure neutral" constructive mathematics and say so explicitly, while clearly mentioning any additional principles one may be assuming.

This page will discuss Bishop's constructive mathematics *as Bishop did it*; see [[constructive mathematics]] for "pure neutral" constructive mathematics that is valid in any topos.  Unfortunately, the confusion surrounding BISH means that any individual paper claiming to use BISH must be read carefully to find out exactly what framework it *is* using.

## What is it?
 {#WhatItIs}

Bishop's constructive mathematics is often described as "obtained from classical mathematics by removing the law of [[excluded middle]] and the [[axiom of choice]]".  It is also claimed that "every other variety of constructive mathematics, including classical mathematics, can be obtained from Bishop's constructive mathematics by adding further assumptions" (e.g. [BridgesRichman](#BridgesRichman)).  However, neither of these statements is really true.

First of all, Bishop accepted the principle of [[countable choice]].  Since countable choice fails in the [[internal logic]] of many [[toposes]], this means that if we want to do constructive mathematics because it is valid in toposes, we cannot use Bishop's framework.  (Regarding countable choice see also [Richman](#Richman00).)  Similarly, although it is true (as often claimed) that "every constructive proof in BISH is also a classical proof" if by "classical" one includes both excluded middle and choice, it is not true if "classical" includes foundational theories like [[ZF]] (without the axiom of choice).

However, focusing attention on countable choice is misleading, because it may lead one to think that we could simply "remove that assumption" from BISH, and that proofs in BISH that don't happen to invoke countable choice would be topos-valid.  In fact it is not really possible to "remove countable choice" from BISH, because it is not an axiom but a *theorem*, and BISH also proves other non-topos-valid principles like the [[presentation axiom]].

The central point is the following: although the carriers of mathematical structure in BISH are, like in other mathematical frameworks, called [[sets]], BISH is not an *axiomatic [[set theory]]* but a *definitional set theory*.  That is, the notion of "set" in BISH is not an undefined one given meaning by axioms and rules the way it is in [[ZFC]] or [[ETCS]] (and also some ways of using [[type theory]] as a foundation, such as [[homotopy type theory]]).  Instead, the notion of "set" is *defined* in terms of more basic notions such as "construction" and "method".  (See [[Bishop set]].)  This definition then allows us to *prove* that sets satisfy various properties, including countable choice.  But Bishop directly uses "constructions" and "methods" throughout in addition to the "sets" that are defined from them, so his mathematics cannot be simply interpreted into an axiomatic set theory with certain assumptions added such as countable choice.

Bishop did not formalize his foundations, and in particular did not specify rules for how "constructions" and "methods" should behave.  However, many people believe that a fairly faithful reading is obtained by viewing "methods" as the [[terms]] in a [[dependent type theory]], whose [[types]] are classifiers for families of "constructions".  (Indeed, making this work was one of [[Per Martin-Lof]]'s original motivations for defining dependent type theory.)  Bishop's *logic* is then the *untruncated* [[propositions as types]] logic of this type theory, with for instance "or" interpreted as the [[sum type]] and "there exists" interpreted as the [[dependent sum type]]. Bishop: "choice is implied by the very meaning of existence".  (Other possible formalizations of Bishop's mathematics use Feferman's [[explicit mathematics]] or [[NuPrl]]. "Nuprl can be seen as expressing the philosophical ideas of Brouwer and Bishop" (expressed [here](http://www.nuprl.org/book/Overview.html) ).  Alternatively, one can try to isolate the principles implied by Bishop's definitional set theory to formulate an axiomatic set theory in which Bishop's work can be formalized --- one common proposal is [[CZF]] with the [[presentation axiom]] --- but this is getting further from what Bishop actually did.)

Thus, for instance, when Bishop says that a set is defined by specifying how to construct an element of it and how to say when two such elements are equal, we take this as meaning that a set is specified by a type $A$ and an equality relation $E : A \to A \to Type$ satisfying the laws of an [[equivalence relation]] (a.k.a. a [[setoid]] in the type theory).  And whenever Bishop says "for all $x$ there exists a $y$", he means $\prod_{x:A} \sum_{y:B}$, which by the commutation of $\Pi$ and $\Sigma$ entails the existence of a function $A\to B$.  This "function" is in the sense of the underlying type theory, i.e. an element of a function type; inside BISH it would be called a "method" or "operation", with the word "function" reserved for operations that respect the specified setoid equalities.  Since not every operation is a function, the full axiom of choice does not hold in BISH; but it does hold whenever the equality on $A$ is "minimal" or "identity" or "Leibniz" (that is, the [[identity type]] of the type theory).  In particular, since this is true for $\mathbb{N}$, countable choice holds.

Now we can see the more basic problem with saying that BISH specializes to other theories such as topos-logic or ZF.  It's not just that BISH proves statements that "don't hold" in these other theories; rather, the latter *have* no "constructions/methods" layer underneath, so there isn't even any natural way to *try* to interpret BISH into them.  (One might even argue that the same is true for the other flavors of constructivism that are generally believed to be specializations of BISH, such as Brouwer's [[intuitionistic mathematics]] and [[Russian constructivism]], but I'm not familiar enough with the philosophy of those theories to make that claim.)  The only halfhearted possibility seems to be to interpret the objects of the topos (or sets of ZF) as being the *types* in the underlying type theory; but in this case BISH is not talking about the same things as the other theory, and theorems of BISH do not specialize to the corresponding theorems of the other theory.  (For classical mathematics *with* choice, though, it happens that this specialization is correct: every setoid is isomorphic to a set, namely the actual quotient of its equality equivalence relation.)

## Relation to triposes
 {#Triposes}

One of the most distinctive aspects of Bishop's framework for mathematics is the notion that a "set" is something *equipped with an equality relation*, so that for instance we can construct a "quotient" of a set by simply equipping the same underlying "collection" with a new equality relation.  There is a way to preserve this idea in a theory that is general enough to include topos logic, but the result looks rather different from Bishop's.

First we have to allow a more general "logic" than untruncated propositions-as-types, such as the topos logic of [[subobjects]].  Semantically, this can be described as working in a [[first-order hyperdoctrine]].  Second, such a more general logic will no longer satisfy the [[unique choice|function comprehension]] (unique choice) principle *with respect to setoids* (though it may with respect to the "natural" equality of objects of the topos).  In fact it is nearly impossible for it to do so: if it does, then the morphisms in the base category of the hyperdoctrine satisfy the full axiom of choice!  For if $R$ is a total relation from $A$ to $B$ in such a hyperdoctrine, define a setoid equality on $A\times B$ by setting $(x_1,y_1) = (x_2,y_2)$ iff $x_1=x_2$ (in the minimal Leibniz sense) and $R(x_1,y_1) \leftrightarrow R(x_2,y_2)$.  Then $R$ induces a total functional relation from $A$ to $A\times B$ by $R'(x,(x',y)) = (x=x') \wedge R(x,y)$, and if there is an operation $f:A\to A\times B$ such that $R'(x,f(x))$ then $\pi_1 f : A \to B$ satisfies $R(x,\pi_1 f(x))$ and so is a choice function.

Thus, to obtain function comprehension, which is a *sine qua non* for much mathematics, we have to essentially generalize the notion of "function" to be not just an operation (a term in the underlying type theory) that preserves equality, but a total functional relation.  The "sets" in a hyperdoctrine, with the "functions" of this sort between them, are most studied in the case of [[triposes]] and are called the *tripos-to-topos construction*.

Thus, working with "sets" in tripos logic (defined there as [[partial equivalence relations]], since a tripos may lack "comprehensions" to allow us to assert conditions on a construction to define an element of a set) can be viewed as a generalization of BISH to include topos logic and ZF.  But a proof in BISH is not always valid in tripos logic, even if it doesn't explicitly invoke principles like countable choice, since it can use facts such as that every function has an underlying "operation" and that every existence statement comes with an "method" selecting its witnesses.

## Bishop's notion of continuity
 {#Continuity}

Bishop rejected the usual notion of pointwise [[continuous function]] (and more generally [[topological spaces]]) as inadequate for analysis and instead defined a "continuous function" (from $\mathbb{R}$ to $\mathbb{R}$) to be "locally uniformly continuous".  However, there is some confusion about this definition diagnosed by [Waaldijk](#Waaldijk).  In particular, the two books _FCA_ and _CA_ use different definitions, which can be reconciled only using the [[fan theorem]] (which Bishop rejected).  There is a similar problem with the notion of [[locally compact space]].  In fact, Waaldijk showed more strongly that if there is *any* notion of "kontinuous" function between subsets of $\mathbb{R}$ which is (1) closed under composition, (2) specializes on closed intervals to uniformly continuous functions, and (3) includes $f(x)= \frac{1}{x}$, then the fan theorem holds.  This seems quite damaging to the prospect of a treatment of analysis as Bishop envisioned it.

A more modern perspective is to use [[locale]] theory (or the predicative variant [[formal topologies]]) instead.  [Palmgren](#Palmgren03) shows that the continuous endomaps of the [[locale of real numbers]] $\mathbb{R}$ coincide with Bishop's locally uniformly continuous functions, whereas for spaces other than $\mathbb{R}$ itself locale maps are better-behaved than Bishops': in particular, they are of course closed under composition, and they contain inversion (suitably interpreted).  Similarly, locally compact spaces and measure theory are arguably better done with locales than spaces.  In particular, see [[localic completion]] and the references below.

Alternatively, one can consider sheaves over the localic unit interval to model the [[fan theorem]]. This approach is pushed even further in [[continuous truth]]. This may be seen as a modern approach to Brouwer's intuitions, where instead of postulating [[choice sequences]] to justify the [[fan theorem]], one works directly with representations of topological spaces and continuous functions between them. These are not unlike Bishop's basic "operations".

## References

Books about or using Bishop's constructive mathematics include:

* [[Errett Bishop]], _[[Foundations of Constructive Analysis]]_ (1967), the work which made him famous in the foundations of mathematics circles;

* [[Errett Bishop]] and [[Henry Cheng]], _Constructive Measure Theory_ (1972), featuring a better [[measure theory]] than in _FCA_;

* [[Errett Bishop]] and [[Douglas Bridges]], _Constructive Analysis_ (1985), a revised version of _FCA_, incorporating the measure theory of _CMT_.

* [[Ray Mines]], [[Fred Richman]], and [[Wim Ruitenberg]], _A course in constructive algebra_ (1988)

* [[Douglas Bridges]] and [[Luminiţa Vîţă]], _Apartness and Uniformity: A Constructive Development_ (2011)

* {#BridgesRichman}[[Douglas Bridges]] and [[Fred Richman]], _Varieties of constructive mathematics_ (1987)

Other papers mentioned above:

* {#Waaldijk} [[Frank Waaldijk]], _On the foundations of constructive mathematics - especially in relation to the theory of continuous functions_  (2001).  [web site](http://www.fwaaldijk.nl/mathematics.html)

* [[Fred Richman]], *Constructive Mathematics without Choice*,
in: *Reuniting the Antipodes -- Constructive and Nonstandard Views of the Continuum*, Synthese Library **306**, Springer (2001) 199-206 [[doi:10.1007/978-94-015-9757-9_17](https://doi.org/10.1007/978-94-015-9757-9_17)]
 {#Richman00}

For more about the localic approach, see:

* [[Erik Palmgren]], _Continuity on the real line and in formal spaces_. (2003). Revised version of U.U.D.M. Report 2003:32. [link](http://www2.math.uu.se/~palmgren/crealform-rvd2.pdf)
 {#Palmgren03}

* [[Erik Palmgren]], _From intuitionistic to point-free topology_ (2005) U.U.D.M. Report 2005:47. [link](http://www2.math.uu.se/~palmgren/homotopy.pdf)
 {#Palmgren05}

* [[Erik Palmgren]], _A constructive and functorial embedding of locally compact metric spaces into locales_. Topology and its Applications, 154
(2007), 1854--1880.

* [[Erik Palmgren]], _Resolution of the uniform lower bound problem in constructive analysis_, [link](http://onlinelibrary.wiley.com/doi/10.1002/malq.200710034/full)

* [[Bas Spitters]], _Locatedness and overt sublocales_, doi:10.1016/j.apal.2010.07.002 APAL

* [[Thierry Coquand]], [[Erik Palmgren]], [[Bas Spitters]], _Metric complements of overt closed sets_, Mathematical Logic Quarterly, 57(4), 373-378, 2011.  [link](http://dx.doi.org/10.1002/malq.201010011)

* T Kawai, _A point-free characterisation of Bishop locally compact metric spaces_, [link](http://www.logicandanalysis.com/index.php/jla/article/viewFile/239/120)

* T Kawai, _Localic completion of uniform spaces_, [arxiv](https://arxiv.org/pdf/1703.02255)

[[!redirects BCM]]
[[!redirects BISH]]


Bishop-compact space

# Bishop-compact spaces

* table of contents
{: toc}

## Idea

The classical notion of [[compact space]] (defined e.g. using the finite subcover property) is not very useful in point-based [[constructive mathematics|constructive]] topology (as opposed to [[locale]] theory or [[formal topology]]).  In particular, it is impossible to prove constructively that familiar spaces like the [[unit interval]] $[0,1]\subseteq \mathbb{R}$ are compact in this sense.  Since classically, a [[metric space]] (or more generally a [[uniform space]]) is compact if and only if it is [[complete space|complete]] and [[totally bounded space|totally bounded]], [[Errett Bishop]] decided to simply take the conjunction of the latter two properties as a *definition* of compactness in [[Bishop's constructive mathematics|his]] constructive mathematics.

## Definition

A [[uniform space]] is **Bishop-compact** if it is both [[complete space|complete]] and [[totally bounded space|totally bounded]].

## Properties

### Equivalence to open-cover compactness

In [[classical mathematics]] (more precisely, assuming the [[ultrafilter principle]]), a uniform space is Bishop-compact if and only if its underlying [[topological space]] is [[compact space|compact]] in the usual sense.  In fact, more can be said: a compact [[Hausdorff space]] admits a *unique* compatible uniformity, which is complete and totally bounded.  So Bishop-compactness is really equivalent to ordinary compactness in a strong sense.  One can also use Bishop-compactness to develop various parts of the theory of compactness directly; for instance, [Shulman](#ShulmanSOI) constructs the [[Stone-Cech compactification]] using Bishop-compactness for [[gauge spaces]].

### Topological invariance

Completeness and total boundedness are not individually topological invariants.  In other words, if two uniform spaces are topologically isomorphic (i.e. their underlying [[topological spaces]] are homemorphic), it doesn't follow that if one is complete or totally bounded then so is the other.  For example, $\mathbb{R} \cong (0,1)$ topologically, but the former is complete and not totally bounded, while the latter is totally bounded and not complete.

However, the fact that Bishop-compactness is classically equivalent to open-cover compactness means that it *is* topologically invariant: if two uniform spaces are topologically isomorphic, and one is both complete *and* totally bounded, then so is the other.

In constructive mathematics, though, even this much cannot be proven: the statement that Bishop-compactness is a topological invariant implies the [[fan theorem]], which is (classically true but) unprovable constructively.  This is essentially the content of section 3.2 of [Diener's thesis](#Diener08), though it is phrased there in contrapositive form (assuming a counterexample to the fan theorem, we can construct a counterexample to the topological invariance of Bishop-compactness).

This constructive failure of topological invariance for Bishop-compactness has been one of the impetuses for the constructive study of [[proximity spaces]], in the hope of finding a structure at least somewhat weaker than uniformity under which a point-based notion of compactness is invariant.

## References

* {#Diener08} Hannes Diener, *Compactness under constructive scrutiny* (Ph.D. thesis), 2008, [pdf](http://www.math.canterbury.ac.nz/~h.diener/publications/thesis.pdf)

* {#ShulmanSOI} [[Mike Shulman]], *The shape of infinity*, [arxiv](https://arxiv.org/abs/1209.2735)

[[!redirects Bishop-compact space]]
[[!redirects Bishop-compact spaces]]
[[!redirects Bishop-compact]]
[[!redirects Bishop-compactness]]


bisimplicial group

#Contents#
* table of contents
{:toc}

## Idea

[[bisimplicial object]]s in [[Group]].

## Bisimplicial abelian groups

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

Let $A,B : \Delta^{op} \times \Delta^{op} \to Ab$ be bisimplicial abelian groups.
A morphism $f : A \to B$ which is degreewise in one argument a weak equivalence
$f_{n,\bullet} : A(n,\bullet) \to B(n,\bullet)$ induces a weak equivalence
$d(f) : d(A) \to d(B)$ of the associated diagonal complexes.

=--

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

This is Lemma 2.7 in [chapter 4](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) of ([GoerssJardine](#GoerssJardine))

=--

## References

* [[Paul Goerss]] [[Rick Jardine]], _Simplicial Homotopy Theory_ ([dvi](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))
{#GoerssJardine}

* [[Rick Jardine]], Lecture 008 (2010) ([pdf](http://www.math.uwo.ca/~jardine/papers/HomTh/lecture008.pdf))

* [[Samuel Issacson]], _Excercises in homotopy colimits_ ([pdf](http://www-math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf))
{#Isaacson}


bisimplicial object

#Contents#
* table of contents
{:toc}

## Definition

A _bisimplicial object_ in a [[category]] $C$ is a [[functor]]

$$
  F : \Delta^{op} \times \Delta^{op} \to C
$$

where $\Delta$ is the [[simplex category]].

This is the same as a [[simplicial object]] in the [[category]] of [[simplicial object]]s in $C$.

## Examples

* [[bisimplicial set]]

* [[bisimplicial group]]

## Related concepts

* [[simplicial object]]

[[!redirects bisimplicial objects]]


bisimplicial set

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

# Contents
* table of contents
{: toc}

## Definition

A bisimplicial set is a [[bisimplicial object]] in [[Set]].

## Properties

### Diagonal
 {#Diagonal}

+-- {: .num_defn }
###### Definition
**([[diagonal of a bisimplicial set|diagonal]])**

For $X_{\bullet,\bullet}$ a bisimplicial set, its **[[diagonal of a bisimplicial set|diagonal]]** is the simplicial set that is the precomposition with $(Id, Id) \colon \Delta^{op} \to \Delta^{op} \times \Delta^{op}$ with the [[diagonal morphism|diagonal]] [[functor]] on the [[opposite category|opposite]] of the [[simplex category]], i.e. the simplicial set with components

$$
  d(X)_n \;\coloneqq\; X_{n,n}
  \,.
$$

=--

+-- {: .num_defn }
###### Definition
**(realization)**

The **realization** $|X|$ of a bisimplicial set $X_{\bullet,\bullet}$ is the [[simplicial set]] that is given by the [[coend]]

$$
  |X| =  \int^{[n] \in \Delta} X_{n, *} \times \Delta[n]
$$

in [[sSet]].

=--

+-- {: .num_prop }
###### Proposition
**(diagonal is realization)**

For $X$ a bisimplicial set, its diagonal $d(X)$ is (isomorphic to) its realization $|X|$:

$$
  |X| \simeq d(X)
  \,.
$$

=--

This is for instance exercise 1.6 in  in [chapter 4](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) [Goerss-Jardine](#GoerssJardine). For a derivation see the examples at _[[homotopy colimit]]_.

\begin{proposition}
\label{DiagonalAsSimplicialHomotopyColimit}
**(diagonal is homotopy colimit)**

The diagonal of a bisimplicial set $X_{\bullet,\bullet}$ is also (up to weak equivalence) the [[homotopy colimit]] of $X$ regarded as a simplicial diagram in the [[model structure on simplicial sets]]

$$
  diag X \simeq hocolim (X : \Delta^{op} \to sSet_{Quillen})
  \,.
$$

\end{proposition}

This appears for instance as theorem 3.6 in ([Isaacson](#Isaacson)).

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

This follows with the above equivalence to the [[coend]] $diag X \simeq \int^{[k] \in \Delta} \Delta[k] \cdot X_k$ and general expression of  [[homotopy colimit]]s by coends (as discussed there) in terms of the [[Quillen bifunctor]]

$$
  \int^\Delta (-) \cdot (-)
    :
   [\Delta, sSet_{Quillen}]_{Reedy} \times
   [\Delta, sSet_{Quillen}]_{Reedy}
  \to
    sSet_{Quillen}
$$

in [[Reedy model structure]]s (as discussed there) by using that $\Delta[-] : \Delta \to sSet_{Quillen}$ is a Reedy cofibrant resultion of the point in $[\Delta, sSet_{Quillen}]$ and that every object in $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$ is cofibrant.
=--

+-- {: .num_prop }
###### Proposition
**(degreewise weak equivalences)**

Let $X,Y : \Delta^{op} \times \Delta^{op} \to Set$ be bisimplicial sets.
A morphism $f : X \to Y$ which is degreewise in one argument a weak equivalence
$f_{n,\bullet} : X(n,\bullet) \to Y(n,\bullet)$ induces a weak equivalence
$d(f) : d(X) \to d(Y)$ of the associated diagonal simplicial sets
(with respect to the standard [[model structure on simplicial sets]]).

=--

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

This is prop 1.9 in [chapter 4](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) of

* Goerss-Jardine, _Simplicial Homotopy Theory_ ([dvi](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))

=--

### Total décalage and total simplicial sets
 {#TotalSimplicialSets}

There is a [[functor]] called _[[ordinal sum]]_ (see also at [[simplex category]])

$$
  + : \Delta^\op \times \Delta^{op} \to \Delta^{op}
  \,.
$$

$$
  + : [k], [l] \mapsto [k+l+1]
  \,.
$$

This induces an [[adjoint triple]]

$$
  ssSet
   \stackrel{\overset{+_!}{\longrightarrow}}{\stackrel{\overset{+^*}{\longleftarrow}}{\underset{+_*}{\longrightarrow}}}
  sSet
  \,.
$$

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

Here

* $T \coloneqq +_*$ is called the **total simplicial set** functor or **Artin-Mazur codiagonal** (we will use the first of these as codiagonal also has another accepted meaning, see [[codiagonal]]);

* $Dec \coloneqq +^*$ is called the [[total décalage]] functor (inside which is plain [[décalage]]);

=--

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

$T$ preserves degreewise [[weak equivalences]] of [[simplicial sets]].

=--

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

For $X$ any bisimplicial set

* the canonical morphism

  $$
    diag X \to T X
  $$

  from the [[diagonal]] to the total simplicial set is a [[weak equivalence]] in the [[model structure on simplicial sets]];

* the [[unit of an adjunction|adjunction unit]]

  $$
    X \to T Dec X
  $$

  is a weak equivalence.

=--

These statements are for instance in ([CegarraRemedios](#CegarraRemedios)) and ([Stevenson](#Stevenson)). They may be considered as a non-additive versions of the [[Eilenberg-Zilber theorem]].

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

By prop. \ref{TotalAndDiagonal} and the  usual _[[Eilenberg-Zilber theorem]]_ it follows
that under forming chain complexes for [[simplicial homology]], total simplicial sets correspond to [[total complexes]] of [[double complexes]].

=--

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

After [[geometric realization]] these spaces are even related by a [[homeomorphism]].

=--

(This seems to be due to Berger and Hübschmann, but related results were known to Zisman as they are so mentioned by Cordier in his work on homotopy limits.)

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

The [[simplicial classifying space]]-[[functor]] for [[simplicial groups]]

$$
  \overline{W}
    \;\colon\;
  sGrp \longrightarrow sSet_*
$$

is the [[composition|composite]]

$$
  \bar W
   \colon
  sGrp
   \stackrel{\mathbf{B}}{\longrightarrow}
  sGrpd
   \stackrel{N}{\longrightarrow}
  ssSet
   \stackrel{T}{\longrightarrow}
  sSet
  \,.
$$

=--

We have the following explicit formula for $T X$, attributed to [[John Duskin]]:

+-- {: .num_lemma }
###### Lemma

For $X$ a [[bisimplicial set]] the total simplicial set
$T X$ is in degree $n$ the [[equalizer]]

$$
  (T X)_n
  \to
  \prod_{i = 0}^n X_{i, n-i}
  \stackrel{\longrightarrow}{\longrightarrow}
  \prod_{i = 0}^{n-1}
  X_{i, n-i-1}
$$

where the components of the two morphisms on the right are

$$
  \prod_{i = 0}^n X_{i,n-i}
   \stackrel{p_i}{\to}
  X_{i, n-i}
 \stackrel{d_0^v}{\to}
  X_{i, n-i-1}
$$

and

$$
 \prod_{i = 0}^n X_{i,n-i}
  \stackrel{p_{i+1}}{\to}
  X_{i+1,n-i-1}
  \stackrel{d_{i+1}^h}{\to}
  X_{i,n-i-1}
 \,.
$$

The face maps $d_i : (T X)_n \to (T X)_{n-1}$
are given by

$$
  d_i
    =
  (d_i^v p_0,
   d_{i-1}^v p_1,
   \cdots,
   d_1^v p_{i-1},
   d_i^h p_{i+1},
   d_i^h p_{i+2},
   \cdots,
   d_i^h p_n
  )
$$

and the degeneracy maps are given by

$$
  s_i
  =
  (s_i^v p_0, s_{i-1}^v p_1, \cdots,
   s_0^v p_i, s_i^h p_{i+1}, \cdots,
   s_i^h p_n)
  \,.
$$

The $(Dec \dashv T)$-[[unit of an adjunction|adjunction unit]] $\eta : X \to T Dec X$ is given in degree $n$ by

$$
  \eta : x \mapsto
   (s_0(x), s_1(x), \cdots, s_n(x))
  \,.
$$

=--

### Geometric realization

See [[geometric realization of simplicial topological spaces]].

## Model structures

There are various useful [[model category]] structures on the category of bisimplicial sets.

### Induced from the diagonal

There is an [[adjunction]]

$$
  (L \dashv diag)   :
  ssSet
  \stackrel{\overset{L}{\longleftarrow}}{\underset{diag}{\longrightarrow}}
  sSet
  \,.
$$

The [[transferred model structure]] along this adjunction of the standard [[model structure on simplicial sets]] exists and with respect to it the above [[Quillen adjunction]] is a [[Quillen equivalence]].

This is due to ([Moerdijk 89](#Moerdijk))

### Induced from codiagonal $T$.

The [[transferred model structure]] on $ssSet$ along the total simplicial set functor $T$ exists. And for it

$$
  (Dec \dashv T) : ssSet
    \stackrel{\overset{Dec}{\longleftarrow}}{\underset{T}{\longrightarrow}}
   sSet
$$

is a [[Quillen equivalence]].

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

Every diag-fibration is also a $T$-fibration.

=--

This is ([CegarraRemedios, theorem 9](#CegarraRemedios)).

#### Remark on notation

There are two uses of $\bar W$ in this area, one is as used in ([CegarraRemedios](#CegarraRemedios)) where it is used for the codiagonal (sometimes denoted "$\nabla$" or $T$ as above), the other is for the [[classifying space]] functor for a [[simplicial group]]. This latter is not only the older of the two uses, but also comes with a related $W$ construction.  The relationship between the two is that given a simplicial group or simplicially enriched groupoid, $G$, applying the [[nerve]] functor in each dimension gives a bisimplicial set and $\bar{W}G = T Ner G$. Because of this, some care is needed when using these sources.

## Bisimplicial abelian groups

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

Let $A,B : \Delta^{op} \times \Delta^{op} \to Ab$ be bisimplicial abelian groups.
A morphism $f : A \to B$ which is degreewise in one argument a weak equivalence
$f_{n,\bullet} : A(n,\bullet) \to B(n,\bullet)$ induces a weak equivalence
$d(f) : d(A) \to d(B)$ of the associated diagonal complexes.

=--

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

This is Lemma 2.7 in [chapter 4](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) of ([GoerssJardine](#GoerssJardine))

=--

## Related concepts

* [[π-Kan condition]]

* [[decalage]]

* [[bicubical set]]

## References

Some standard material is for instance in

* {#GoerssJardine} [[Paul Goerss]] and [[Rick Jardine]], _Simplicial Homotopy Theory_

* [[Rick Jardine]], Lecture 008 (2010) ([pdf](http://jardine.math.uwo.ca/HomTh/lecture008a.pdf))

* {#Isaacson} [[Samuel Isaacson]], _Excercises in homotopy colimits_ ([pdf](http://www-math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf))

The total simplicial set functor goes back to

* [[M. Artin]], [[B. Mazur]], _On the Van Kampen theorem_ , Topology 5 (1966) 179–189.

The [[diagonal of a bisimplicial set|diagonal]], total décalage and [[total simplicial set]] constructions are discussed in

* [[Antonio Cegarra]], [[Josué Remedios]], _The relationship between the diagonal and the bar constructions on a bisimplicial set_, Topology and its applications, volume 153 (1)  (2005) ([pdf](http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf), [doi:10.1016/j.topol.2004.12.003](https://doi.org/10.1016/j.topol.2004.12.003))

* [[Antonio Cegarra]], [[Josué Remedios]], _The behaviour of the $\bar W$-construction on the homotopy theory of bisimplicial sets_, Manuscripta Mathematica, volume 124 (4) Springer (2007)
 {#CegarraRemedios}

* [[Danny Stevenson]], _Décalage and Kan's simplicial loop group functor_ ([arXiv:1112.0474](http://arxiv.org/abs/1112.0474)){#Stevenson}

* [[Viktoriya Ozornova]], [[Martina Rovelli]], _The unit of the total décalage adjunction_, Journal of Homotopy and Related Structures (2020) 15:333–349, [doi.org/10.1007/s40062-020-00257-1](https://doi.org/10.1007/s40062-020-00257-1)

The diagonal-induced model structure on $ssSet$ is discussed in

* {#Moerdijk} [[Ieke Moerdijk]], _Bisimplicial sets and the group completion theorem_  in _Algebraic K-Theory: Connections with Geometry and Topology_, pp 225–240. Kluwer, Dordrecht (1989)

On the behaviour of fibrations under [[geometric realization]] of bisimplicial sets:

* {#Anderson78} [[Donald Werner Anderson]], _Fibrations and geometric realization_,  Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. ([euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139))

Discussion of the [[simplicial classifying space]]-construction $\overline{W}$ respecting [[fibrant objects]] is in Fact 2.8 of:

* [[Antonio Martínez Cegarra]], Benjamın A. Heredia, [[Josué Remedios]],  _Double groupoids and homotopy 2-types_, Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also [arXiv:1003.3820](http://arxiv.org/abs/1003.3820).

[[!redirects bisimplicial set]]
[[!redirects bisimplicial sets]]

[[!redirects simplicial simplicial set]]
[[!redirects simplicial simplicial sets]]

[[!redirects total simplicial set]]
[[!redirects total simplicial sets]]

[[!redirects total décalage]]


bisimulation

# Contents
* table of contents
{:toc}

## Definition

### In set theory
 {#InSetTheory}

Let $S$ be a set equipped with a [[binary relation]] $\prec$. Then a **bisimulation** is a binary relation $\sim$ on $S$ such that for all $x, y \in S$ with $x \sim y$, the following conditions hold:

* for all $a \in S$ such that $a \prec x$, there exists a $b \in S$ such that $b \prec y$ and $a \sim b$

* for all $b \in S$ such that $b \prec y$, there exists a $a \in S$ such that $a \prec x$ and $a \sim b$

For [[labelled transition systems]], [Walker 1994](#Walker94) gives a definition equivalent to the following:

\begin{definition}
On a system with points $P,Q,\ldots$ and transition relations $\{\overset{\alpha}{\rightarrow} \colon \alpha\in L\}$, a *bisimulation* is a [[symmetric relation]] $\sim$ such that whenever $P\sim Q$, $\alpha\in L$ and $P\overset{\alpha}{\rightarrow}P'$, then $Q\overset{\alpha}{\rightarrow}Q'$ with $P'\sim Q'$.
\end{definition}

\begin{definition}
\label{Bisimilarity}
 The [[union]] of all bisimulation relations, as [[subsets]] of $S \times S$, is a relation called **bisimilarity**.
\end{definition}

Let $\mathrm{traces}(s)$ be the set of all possible (finite or infinite) transition sequences starting as $s$.
Trace equivalence, defined as $s\sim t$ if and only if $\mathrm{traces}(s)=\mathrm{traces}(t)$, is a bisimulation.

### In category theory

In Joyal-Nielsen-Winskel (p.13) is given the following definition. For what a "path category in a category of models" is, see there.

\begin{definition}
Let $P$ be a path category in a category of models $M$. Two objects $X_1,X_2$ are called to be *$P$-bisimilar* if there is a span of $P$-open maps $X_1\leftarrow X\to X_2$.
\end{definition}

The relation of $P$-open maps and [[open map]]s is given by Proposition 11, p.32:

\begin{proposition}
If $P$ is a dense full subcategory of $M$, then $f$-is $P$-open iff $M(-,f)$ is an [[open map]].
\end{proposition}

## See also

* [[simulation]]

* [[extensional relation]]

* [[transition system]]

* [[cotopos]]

* [[coinduction]]

## References

* Wikipedia (English), _[Bisimulation](http://en.wikipedia.org/wiki/Bisimulation)_

*  [[Peter Aczel]], *[Non-Well-Founded Sets](https://web.archive.org/web/20161020002632/https://standish.stanford.edu/pdf/00000056.pdf)*, especially Chapter 4.

* [[Sam Staton]], _[Relating coalgebraic notions of bisimulation](http://arxiv.org/abs/1101.4223)_
* Davide Sangiorgi, _[On the Origins of Bisimulation and Coinduction](http://www.cs.unibo.it/~sangio/DOC_public/history_bis_coind.pdf)_

* [[André Joyal]], Mogens Nielsen, [[Glynn Winskel]], Bisimulation from [[open map|open maps]], [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)

* Bard Bloom, Sorin Istrail, Albert Meyer, Bisimulation can't be traced, [pdf](http://www.cse.psu.edu/~catuscia/DEA/General/BloomIstrailMeyer.pdf)

* Yde Venema, Algebras and Coalgebras, §6 (p.332-426).11(p.398-403) in Blackburn, van Benthem, Wolter, Handbook of modal logic, Elsevier, 2007.

* [[Pedro Resende]], _Quantales, finite observations and strong bisimulation_, Theor. Comp. Sci. __254__:1–2 (2001) 95–149, doi

* David Walker, _On Bisimulation in the [[pi-calculus|$\pi$-calculus]]_, CONCUR '94: Concurrency Theory (1994).

[[!redirects bisimulations]]
[[!redirects bisimilarity]]
[[!redirects bisimilarities]]


Bismut-Chern character

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

#Contents#
* table of contents
{:toc}

## Idea

A version of the [[Chern character]] on [[free loop space]].

## Related concepts

* [[superconnection]]

* [[free loop space]]

## References

The original article:

* [[Jean-Michel Bismut]], *Index Theorem and Equivariant Cohomology on the Loop Space*, Comm. in Math. Phys. **98** (1985) 213-237 [[doi:10.1007/BF01220509](https://doi.org/10.1007/BF01220509), [euclid](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-98/issue-2/Index-theorem-and-equivariant-cohomology-on-the-loop-space/cmp/1103942357.full)]

Further development:

* [[Fei Han]], *Supersymmetric QFT, Super Loop Spaces and Bismut-Chern Character* [[arXiv:0711.3862](https://arxiv.org/abs/0711.3862)

* [[Fei Han]], [[Varghese Mathai]], *Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality*,  Commun. Math. Phys. **337** (2015) 127-150 [[arXiv:1405.1320](https://arxiv.org/abs/1405.1320),  [doi:10.1007/s00220-014-2270-z](https://doi.org/10.1007/s00220-014-2270-z)]

reviewed in:

* [[Varghese Mathai]], *T-duality for loop spaces, or equivalently for the 1D sigma model*,  talk  at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#Mathai2023)*, NYU Abu Dhabi (Jan  2023) [[web](/nlab/show/M-Theory+and+Mathematics#Mathai2023)]

[[!redirects Bismu-Chern characters]]


Bisognano-Wichmann theorem

<div class="rightHandSide toc">
[[!include physicscontents]]
***
[[!include AQFT and operator algebra contents]]
</div>

#Contents#
* the following line creates the automatic table of contents
{:toc}

## Idea ##
In the [[Haag-Kastler approach]] to [[quantum field theory]] the central object is a [[local net]] of [[operator algebras]]. The [[modular theory]] says that the local algebras have an associated modular group and a modular conjugation (see [[modular theory]]). The result of Bisognano-Wichmann that this page is about, describes the relation of these to the [[Poincare group]].

## Abstract ##
...

## Definition ##
...

## The Theorem ##
Assume a Wightman field theory of a scalar neutral field is such that the smeared field operators generate the local algebras, see [[Wightman axioms]].

Then:

1. The modular group of the algebra associated with a wedge and the
vacuum vector coincides with the unitary representation of the group
of Lorentz boosts which maps the wedge onto itself.

2. The modular conjugation of the wedge W is given by the formula
$$
J_W = \Theta U(R_W(\pi))
$$
Here $\Theta$ denotes the PCT-operator of the Wightman field theory and
$U(R_W(\pi))$ is the unitary representation of the rotation which leaves the
characteristic two-plane of the wedge invariant. The angle of rotation
is $\pi$.

3. The theory fulfills wedge duality, that is the commutant of the algebra associated to a wedge is the algebra associated to the causal complement of the wedge.

## References ##

The original work is:

* Bisognano, J. and Wichmann, E.H.: _On the duality condition for a
Hermitian scalar field_  , J. Math. Phys. 16 (1975), 985-1007.

There are a lot of secondary references, one is for example:

* Daniele Guido: _Modular Theory for the von Neumann Algebras of Local Quantum Physics_ ([arXiv] (http://xxx.uni-augsburg.de/abs/0812.1511))

[[!redirects Bisognano-Wichmann relations]]