Beck-Chevalley condition

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### 2-category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
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## Idea

The **[[Jon Beck|Beck]]--[[Claude Chevalley|Chevalley]] condition**, also sometimes called just the *Beck condition* or the *Chevalley condition*, is a "commutation of [[adjoint functor|adjoint]]s" property that holds in many "[[base change|change of base]]" situations.

## Motivation

The Beck-Chevalley condition may be understood as a natural compatibility  condition for

**[(1)](#MotivationFromIntegralTransforms)** [[integral transforms]] in [[geometry]]

**[(2)](#MotivationFromLogicAndTypeTheory)** [[quantifiers]] in [[formal logic]]/[[type theory]]

### From integral transforms
 {#MotivationFromIntegralTransforms}

From the point of view of [[geometry]], in contexts such as of [[integral transforms]] one considers [[correspondences]] between [[spaces]] $A$, $B$ given by [[spans]] of [[maps]] between them:

\begin{tikzcd}[sep=20]
  &
  X
  \ar[dl, "f"{swap}]
  \ar[dr, "g"]
  \\
  A && B
\end{tikzcd}

Via [[composition]] of such correspondences by [[fiber product]] of adjacent legs, they form the [[1-morphisms]] in a [[2-category]] *[[Span]]*.

Assuming some [[base change]] [[adjoint pair]] $f^\ast \vdash f_\ast$, thought of as

* *push-forward* $f_\ast \,\colon\, \mathcal{D}(X) \longrightarrow \mathcal{D}(A)$

and

* *pull-back* $f^\ast \,\colon\, \mathcal{D}(A) \longrightarrow \mathcal{D}(X)$

is [[functor|functorially]] associated with [[maps]] $f$ (i.e. such that there are [[natural isomorphisms]] $(f_2 f_1)_\ast \,\simeq\, (f_2)_\ast (f_1)_\ast$ and dually), the [[integral transform]] encoded by any span as above is the "pull-push" operation given by $g_* f^*$.

Now the Beck-Chevalley condition (on such assignment of [[base change]] adjoints to maps) essentially says that this pull-push construction is ([[2-functor|2-]])[[functor|functorial]] under [[composition]] of [[spans]]:

Concretely, given a pair of composable spans:

\begin{tikzcd}[sep=20]
  &
  X
  \ar[dl, "f"{swap}]
  \ar[dr, "g"]
  &&
  Y
  \ar[dl, "h"{swap}]
  \ar[dr, "k"]
  \\
  A && B && C
\end{tikzcd}

and their composite in [[Span]] formed by the [[fiber product]] of the adjacent legs

\begin{tikzcd}[sep=20]
  &&
  X \times_B Y
  \ar[dl, "q"{swap}]
  \ar[dr, "p"]
  \ar[dd, phantom, "{\lrcorner}"{rotate=-45, pos=.3}]
  \\
  &
  X
  \ar[dl, "f"{swap}]
  \ar[dr, "g"]
  &&
  Y
  \ar[dl, "h"{swap}]
  \ar[dr, "k"]
  \\
  A && B && C
\end{tikzcd}

then functoriality of pull-push means that the
 *two* different ways to pull-push through the diagram coincide:

Pull-pushing through the spans separately results in

$$
  (k_\ast h^\ast)
  \circ
  (g_\ast f^\ast)
  \;=\;
  k_\ast
  \,
  \underbrace{h^\ast \, g_\ast}
  \,
  f^\ast
  \,,
$$

while pull-pushing through the composite span results in

$$
  (k p)_* (f q)^*
  =
  k_*
  \,
  \underbrace{p_* \, q^*}
  \,
  f^*
  \,.
$$

For these two operations to coincide, up to [[isomorphism]], hence for pull-push to respect composition of spans, we need an isomorphism between the operations shown over braces, hence of the form

\[
  \label{BCConditionInMotivationFromIntegralTransform}
  h^\ast \, g_\ast
  \;\simeq\;
  p_* \, q^*
\]

between the two ways of  push-pulling along the sides of any [[pullback square]]

\begin{tikzcd}[sep=20]
  &
  X \times_B Y
  \ar[dl, "q"{swap}]
  \ar[dr, "p"]
  \ar[dd, phantom, "{\lrcorner}"{rotate=-45, pos=.3}]
  \\
  X
  \ar[dr, "g"]
  &&
  Y
  \ar[dl, "h"{swap}]
  \\
  & B &
\end{tikzcd}

This (eq:BCConditionInMotivationFromIntegralTransform) is the Beck-Chevalley condition:  A natural necessary condition to ensure that [[integral transforms]] by pull-push [[base change]] through [[correspondences]] is functorial under [[composition]] of [[correspondences]] by [[fiber product]] of adjacent legs.

More motivation along these lines also be found at *[[dependent linear type theory]]*. As a formulation of propagation in cohomological [[quantum field theory]] this is [Sc14](dependent+linear+type+theory#Schreiber14) there.

### From logic and type theory
 {#MotivationFromLogicAndTypeTheory}

From the point of view of [[formal logic]] and [[dependent type theory]], the three items $f_! \dashv f^\ast \vdash f_\ast$ in a [[base change]] [[adjoint triple]] constitute the [[categorical semantics]] of [[quantification]] and [[context extension]] ([[existential quantification]]/[[dependent pair type]] $\dashv$ [[context extension]] $\dashv$ [[universal quantification]]/[[dependent product type]]), and so the Beck-Chevalley condition says that these are compatible with each other: concretely that [[substitution]] of [[free variables]] commutes with [[quantification]] --- a condition which [[syntax|syntactically]] is "self-evident", at least it is evidently desirable.

Original articles with this logical perspective on the BC condition: [Lawvere (1970), p. 8](#Lawvere70); [Seely (1983), p. 511](#Seely83); [Pavlović (1991), §1](#Pavlović91); [Pavlović (1996), p. 164](#Pavlović96).

For more on this logical aspect see [below](#InTypeTheory).

## Definition

Suppose given a [[commutative square]] (up to [[isomorphism]]) of [[functors]]:
$$\array{ & \overset{f^*}{\to} & \\
  ^{g^*}\downarrow && \downarrow^{k^*}\\
  & \underset{h^*}{\to} & }$$
in which $f^*$ and $h^*$ have [[left adjoint]]s $f_!$ and $h_!$, respectively. (The classical example is a [[Wirthmüller context]].)  Then the [[natural isomorphism]] that makes the square commute

$$
  k^* f^* \to h^* g^*
$$

has a [[mate]]

$$
  h_! k^* \to g^* f_!
$$

defined as the composite

$$
  h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_!
  \,.
$$

We say the original square satisfies the **Beck--Chevalley condition** if this mate is an [[isomorphism]].

More generally, it is clear that for this to make sense, we only need a transformation $k^* f^* \to h^* g^*$; it doesn't need to be an isomorphism.  We also use the term *Beck--Chevalley condition* in this case,

### Left and right Beck--Chevalley condition

Of course, if $g^*$ and $k^*$ also have [[left adjoints]], there is also a Beck--Chevalley condition stating that the corresponding mate $k_! h^* \to f^* g_!$ is an isomorphism, and this is not equivalent in general.  Context is usually sufficient to disambiguate, although some people speak of the "left" and "right" Beck--Chevalley conditions.

Note that if $k^* f^* \to h^* g^*$ is not an isomorphism, then there is only one possible Beck-Chevalley condition.

### Dual Beck--Chevalley condition

If $f^*$ and $h^*$ have *right* adjoints $f_*$ and $h_*$, there is also a dual Beck--Chevalley condition saying that the mate $g^* f_* \to h_* k^*$ is an isomorphism.  By adjointness, if $f^*$ and $h^*$ have right adjoints and $g^*$ and $k^*$ have left adjoints, then $g^* f_* \to h_* k^*$ is an isomorphism if and only if $k_! h^* \to f^* g_!$ is.

### For bifibrations

Originally, the Beck-Chevalley condition was introduced in ([Bénabou-Roubaud, 1970](#BenabouRoubaud70)) for [[bifibrations]] over a base category with pullbacks. In *loc.cit.* they call this condition **Chevalley condition** because he introduced it in his 1964 seminar.

A [[bifibration]] $\mathbf{X} \to \mathbf{B}$ where $\mathbf{B}$ has pullbacks satisfies the **Chevalley condition** iff for every commuting square
$$\array{ & \overset{\psi^\prime}{\rightarrow} & \\
  \downarrow^{\varphi^\prime} && \downarrow^{\varphi}\\
  & \underset{\psi}{\rightarrow} & }$$
in $\mathbf{X}$ over a pullback square in the base $\mathbf{B}$ where $\varphi$ is [[cartesian morphism|cartesian]] and $\psi$ is cocartesian it holds that $\varphi^\prime$ is cartesian
iff $\psi^\prime$ is cocartesian. Actually, it suffices
to postulate one direction because the other one follows.
The nice thing about this formulation is that there is no
mention of "canonical" morphisms and no mention of [[cleavages]].

A fibration $P$ has products satisfying the Chevalley condition iff the opposite fibration $P^{op}$ is
a bifibration satisfying the Chevalley condition in the above sense.

According to the [[Benabou–Roubaud theorem]], the Chevalley condition  is crucial for establishing the connection between the descent in the sense of fibered categories and the [[monadic descent]].

### "Local" Beck--Chevalley condition {#Local}

Suppose that $f^*$ and $h^*$ do not have entire left adjoints, but that for a particular object $x$ the left adjoint $f_!(x)$ exists.  This means that we have an object "$f_! x$" and a morphism $\eta_x\colon x \to f^* f_! x$ which is initial in the [[comma category]] $(x / f^*)$.  Then we have $k^*(\eta) \colon k^* x \to k^* f^* f_! x \to h^* g^* f_! x$, and we say that the square satisfies the *local Beck-Chevalley condition at $x$* if $k^*(\eta)$ is initial in the comma category $(k^* x / h^*)$, and hence exhibits $g^* f_! x$ as "$h_! k^* x$" (although we have not assumed that the entire functor $h_!$ exists).

If the functors $f_!$ and $h_!$ do exist, then the square satisfies the (global) Beck-Chevalley condition as defined above if and only if it satisfies the local one defined here at every object.

## In logic / type theory
 {#InTypeTheory}

If the functors in the formulation of the Beck-Chevalley condition are [[base change]] functors in the [[categorical semantics]] of some [[dependent type theory]] (or just of a [[hyperdoctrine]]) then the BC condition is equivalently stated in terms of logic as follows.

A [[commuting diagram]]

$$
  \array{
    D &\stackrel{h}{\to}& C
    \\
    {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    A &\stackrel{f}{\to}& B
  }
$$

is interpreted as a morphism of [[contexts]]. The [[pullback]] (of [[slice categories]] or of fibers in a [[hyperdoctrine]]) $h^*$ and $f^*$ is interpreted as the [[substitution]] of [[variables]] in these contexts. And the [[left adjoint]] $\sum_k \coloneqq k_! $ and $\sum_q \coloneqq g_!$, the [[dependent sum]] is interpreted (up to [[truncated|(-1)-truncation]], possibly) as [[existential quantifier|existential quantification]].

In terms of this the Beck-Chevalley condition says that if the above diagram is a [[pullback]], then **substitution commutes with existential quantification**

$$
  \sum_k h^* \phi \stackrel{\simeq}{\to} f^* \sum_g \phi
  \,.
$$

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

Consider the diagram of [[contexts]]

$$
  \array{
    [\Gamma, x : X] &\stackrel{}{\to}& [\Gamma, x : X, y : Y]
    \\
    \downarrow && \downarrow
    \\
    \Gamma &\to& [\Gamma, y : Y]
  }
  \;\;\;
  \simeq
  \;\;\;
  \array{
    \Gamma \times X &\stackrel{(p_1,p_2,t)}{\to}& \Gamma \times X \times Y
    \\
    {}^{\mathllap{p_1}}\downarrow && \downarrow^{\mathrlap{(p_1,p_3)}}
    \\
    \Gamma &\stackrel{(id,t)}{\to} & \Gamma \times Y
  }
  \,,
$$

with the horizontal morphism coming from a [[term]] $t : \Gamma \to Y$ in [[context]] $\Gamma$ and the vertical morphisms being the evident [[projection]], then the condition says that we may in a [[proposition]] $\phi$ substitute $t$ for $y$ before or after quantifying over $x$:

$$
 \sum_{x : X} \phi(x,t) \simeq (\sum_{x : X} \phi(x,y))[t/y]
 \,.
$$

=--

## Examples

### Various

* The [[codomain fibration]] of any [[category]] with [[pullbacks]] is a bifibration, and satisfies the Beck--Chevalley condition at every pullback square. In particular, [[locally cartesian closed categories]] always satisfy the Beck--Chevalley condition.

* If $C$ is a [[regular category]] (such as a [[topos]]), the bifibration $Sub(C) \to C$ of [[subobjects]] satisfies the Beck--Chevalley condition at every pullback square.

* The [[family fibration]] $Fam(C)\to Set$ of any category $C$ with small sums satisfies the Beck--Chevalley condition at every pullback square in $Set$.

* [Mackey's restriction formula](double+coset#MackeyFormula) for group representations.

### Images and pre-images

\begin{example}
\label{BeckChevalleyForPreImageBetweenPowerSets}
**(Beck-Chevalley for images and pre-images of sets)**
\linebreak
  Given a [[set]] $S \,\in\, Set$, write $\mathcal{P}(S)$ for its [[power set]] regarded as a [[poset]], meaning that

1. $\mathcal{P}(S)$ is the set of [[subsets]] $A \subset S$,

1. regarded as a [[category]] by declaring that there is a [[morphisms]] $A \to B$ precisely if $A \subset B$ as subsets of $S$.

Then for $f \,\colon\, S \longrightarrow T$ a [[function]] between sets, we obtain the followin three [[functors]] between their [[power sets]]:

1. $f_! \,\coloneqq\, f(-) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$

   forms [[images]] under $f$

1. $f^\ast \,\coloneqq\, f^{-1} \,\colon\, \mathcal{P}(T) \longrightarrow \mathcal{P}(S)$

   forms [[preimages]] under $f$,

1. $f_ast \,\coloneqq\, T \setminus f\big(S \setminus (-)\big) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$

   forms the [[complement]] of [[images]] of [[complements]].

These three functors constitute an [[adjoint triple]]

$$
  f(-) \;\dashv\; f^{-1} \;\dashv\; f_\ast
$$

as is readily verified and also discussed at some length in the entry *[[interactions of images and pre-images with unions and intersections]]*.

In particular, the familiar/evident fact that forming [[images]] and [[preimages]] [[preserved colimit|preserves]] [[unions]] may be understood as a special case of the fact that [[left adjoints preserve colimits]].

Now given a [[pullback diagram]] in [[Set]]

   \[
     \label{PullbackOfSets}
     \array{
       D'
         &\overset{\beta}{\longrightarrow}&
       D
       \\
       \mathllap{{}^{\psi}}\big\downarrow
       &{}^{{}_{(pb)}}&
       \big\downarrow\mathrlap{{}^{\phi}}
       \\
       C'
         &\underset{\alpha}{\longrightarrow}&
       C
     }
   \]

then these operations satisfy the left Beck-Chevalley condition in the sense that the following [[diagram]] of [[poset]]-[[homomorphisms]] ([[functors]]) [[commuting diagram|commutes]]:

   \[
     \label{RightBeckChevalleyDiagramForPreImages}
     \array{
       \mathcal{P}(D')
         &\overset{\beta_!}{\longrightarrow}&
       \mathcal{P}(D)
       \\
       \mathllap{{}^{\psi^{-1}}}\big\uparrow
       &&
       \big\uparrow\mathrlap{{}^{\phi^{-1}}}
       \\
       \mathcal{P}(C')
         &\underset{\alpha_!}{\longrightarrow}&
       \mathcal{P}(C)
     }
   \]
To see this it is sufficient to check commutativity on [[singleton]]-[[subsets]] (because every other subset is a [[union]] of singletons and, as just remarked, [[interactions of images and pre-images with unions and intersections|images and preimages preserve unions]])
and given a [[singleton]] set $\{c'\} \subset C'$ on an element $c' \,\in\, C'$, we directly compute as follows:
   \[
     \label{PreimagesInAPullbackDiagram}
     \begin{array}{l}
     \beta \circ \psi^{-1}\big(\{c'\}\big)
     \\
     \;\simeq\;
     \beta\Big(
       \big\{
         (c',d)
         \,\big\vert\,
         d \,\in\, D
         \,,
         \phi(d) = \alpha(c')
       \big\}
     \Big)
     \\
     \;=\;
     \big\{
       d \,\in\, D
       \,\big\vert\,
       \phi(d) = \alpha(c')
     \big\}
     \\
     \;=\;
     \phi^{-1}\big(
      \{\alpha(c')\}
     \big)
     \\
     \;=\;
     \phi^{-1} \circ \alpha\big(\{c'\}\big)
     \,.
     \end{array}
   \]
Here it is the first two steps which make use of the assumption that (eq:PullbackOfSets) is a [[pullback square]], namely which means that $D'$ is compatibly [[bijection|bijective]] to the [[fiber product]] of $C'$ with $D$ over $C$:

$$
  \array{
  D'
  &\simeq&
  \big\{
    (c', d) \in C' \times D
    \,\big\vert\,
    \phi(d)
      \,=\,
    \alpha(c')
  \big\}
  \\
  \mathllap{{}^{\beta}}
  \big\downarrow
    &&
  \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
  \big\downarrow\mathrlap{{}^{ (c',d) \mapsto d }}
  \\
  D
  &=&
  \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
  D
  }
$$
\end{example}

This elementary example \ref{BeckChevalleyForPreImageBetweenPowerSets} actually constitutes half of the proof of the archetypical example of Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations} below.

### Systems of functor categories
 {#PullbacksOfOpfibrations}

\begin{proposition}
\label{BCForPresheavesOnPullbacksOfOpfibrations}
Given

1. $\phi \colon D \to C$ an [[opfibration]] of [[small categories|small]] [[strict categories]],

1. a [[pullback]] [[diagram]] in the _[[1-category]]_ [[Cat]] of [[small category|small]] [[strict categories]] and [[functors]], of the form

   \[
     \label{StrictPullbackOfOpfibration}
     \array{
       D'
         &\overset{\beta}{\longrightarrow}&
       D
       \\
       \mathllap{{}^{\psi}}\big\downarrow
       &{}^{{}_{(pb)}}&
       \big\downarrow\mathrlap{{}^{\phi}}
       \\
       C'
         &\underset{\alpha}{\longrightarrow}&
       C
     }
   \]

   (hence presenting a [[2-pullback]]/[[homotopy pullback]], see Rem. \ref{PullbackOfOpfibrationModelsTwoPullback} below),

1. $\mathcal{C}$ any [[category]] with all [[small colimits]]

then the induced diagram of [[functor categories]]

$$
  \array{
    [D', \mathcal{C}]
      &\overset{\beta^*}{\longleftarrow}&
    [D, \mathcal{C}]
    \\
    \Big\downarrow\mathrlap{{}^{\psi_!}}
      &&
    \Big\downarrow\mathrlap{{}^{\phi_!}}
    \\
    [C', \mathcal{C}]
      &\underset{\alpha^*}{\longleftarrow}&
    [C,\mathcal{C}]
  }
$$

(for $(-)^\ast$ denoting [[precomposition]] and $(-)_!$ denoting [[left Kan extension]])

satisfies the Beck-Chevalley condition, in that there is a [[natural isomorphism]]

$$
  \psi_! \circ \beta^*
  \;\simeq\; \alpha^* \circ \phi_!
  \,.
$$
\end{proposition}
For this statement in the more general context of [[quasicategories]] see [Joyal (2008), prop. 11.6](#Joyal08).
\begin{proof}
The proof relies on two observations:

{#FirstObservationInProofForSystesmOfFunctorCategories} (1.) Since $\phi$ is [[opfibration|opfibered]], for every object $c \in C$ the inclusion functor

\begin{tikzcd}[sep=20pt]
  &
  \phi^{-1}(c)
  \ar[
    rrrr,
    "{ iota }"
  ]
  & && &
  \phi/c
  \\
  x
  \ar[
    rr,
    "{ f }"
  ]
  &&
  y
  &&
  x
  \ar[
    rr,
    "{ f }"
  ]
  &&
  y
  \\[-15pt]
  \phi(x)
    \ar[rr, equals]
    \ar[dr, equals]
    &&
  \phi(y)
  \ar[dl, equals]
  &&
  \phi(x)
    \ar[rr, "{ \phi(f) }"]
    \ar[dr, "{ p_x }"{swap}]
    &&
  \phi(y)
  \ar[dl, "{ p_y }"]
  \\
  & c
  & &&
  & c
\end{tikzcd}

(of the (strict) [[fiber]] $\phi^{-1}(c)$ into the [[comma category]] $\phi/c$) is a [[final functor]].

(This has a simple argument in the case that the categories $D$, $C$ are in fact [[groupoids]]: In this case, $\phi$ being an opfibration equivalently means that it is an [[isofibration]], by [this Exp.](isofibration#GrothIsIsoFibBetweenGroupoids), which evidently implies that the above inclusion $\iota$ is [[essentially surjective functor|essentially surjective]]. But since $\iota$ is also manifestly a [[full functor]], it follows that it is final, by [this Prop.](final+functor#FinalFunctorBetweenGroupoids).)

Therefore the [pointwise formula](Kan+extension#PointwiseByConicalLimits) for the [[left Kan extension]] $\phi_!$ is equivalently given by taking the [[colimit]] over the [[fiber]], instead of over the [[comma category]]:

  \[
    \label{PointwiseLeftKanExtensionInOpfiberedCase}
    \phi_!(X)_c
    \;\simeq\;
    \underset
      {\underset{\phi^{-1}\big(\{c\}\big)}{\longrightarrow}}
      {\lim}
     \,
      X
    \,.
  \]

(2.) Since (eq:StrictPullbackOfOpfibration) is a (strict) pullback of (strict) categories (i.e. an ordinary pullback of [[sets]] of [[objects]] and of [[sets]] of [[morphisms]]), the operations of taking [[images]] and [[preimages]] $(-)^{-1}$ ([[fibers]]) of the [[object]]-[[functions]] of the given functors satisfy the [[poset|posetal]] Beck-Chevalley condition from Exp. \ref{BeckChevalleyForPreImageBetweenPowerSets}:

   \[
     \label{PreimagesInAPullbackDiagram}
     \begin{array}{l}
     \beta \circ \psi^{-1}\big(\{c'\}\big)
     \\
     \;\simeq\;
     \beta\Big(
       \big\{
         (c',d)
         \,\big\vert\,
         d \,\in\, D
         \,,
         \phi(d) = \alpha(c')
       \big\}
     \Big)
     \\
     \;=\;
     \big\{
       d \,\in\, D
       \,\big\vert\,
       \phi(d) = \alpha(c')
     \big\}
     \\
     \;=\;
     \phi^{-1}\big(
      \{\alpha(c')\}
     \big)
     \\
     \;=\;
     \phi^{-1} \circ \alpha\big(\{c'\}\big)
     \,.
     \end{array}
   \]

   for all [[objects]] $c' \,\in\, Obj(C')$.

Now using first (eq:PointwiseLeftKanExtensionInOpfiberedCase) and then (eq:PreimagesInAPullbackDiagram) and then again (eq:PointwiseLeftKanExtensionInOpfiberedCase) yields the following sequence of [[isomorphisms]]

$$
  \begin{aligned}
    \big(\psi_! \beta^* (X)\big)(c')
    &
    \;\simeq\;
    \underset
      {\underset{
        \beta \circ \psi^{-1}\big(\{c'\}\big)
      }{\longrightarrow}}
      {\lim}
     \,
    X
    \\
    &
    \;\simeq\;
    \underset
      {\underset{
        \phi^{-1} \circ \alpha\big(\{c'\}\big)
      }{\longrightarrow}}
      {\lim}
    \,
    X
    \\
    &
    \;\simeq\;
    \big(\alpha^* \phi_! (X)\big)(c')
    \,,
  \end{aligned}
$$

all of them [[natural isomorphism|natural]] in $c' \,\in\, Obj(C')$.
\end{proof}

\begin{remark}
As a [[counter-example]] showing that in Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations} the condition for $\phi \colon D \to C$ to be an [[opfibration]] is necessary for the statement to hold, consider

* $C \coloneqq \{0\to 1\}$ the [[interval category]],

* $D \coloneqq \ast,\;\;C' \coloneqq \ast$ the [[terminal category]],

* $\phi\coloneqq const_0,\;\;\alpha \coloneqq const_1$.

Then the [[pullback]] $D' = \varnothing$ is the [[empty category]]. Therefore

* $[D',\mathcal{C}] = [\varnothing, \mathcal{C}]$ is the [[terminal category]], so that $\psi_!\circ \beta^\ast$ is a [[constant functor]]

  (concretely, $\psi_! \colon \ast \simeq [\varnothing, \mathcal{C}] \to [C',\mathcal{C}]$ is [[left adjoint]] to the projection functor to the [[terminal category]] and [hence](initial+object#AdjointsToConstantFunctors) picks the [[initial object]] of $[C',\mathcal{C}]$),

* while $\alpha^* \circ \phi_!$ is the [[identity functor]]

  (to see this observe first that $[C,\mathcal{C}]$ now may be identified with the [[arrow category]] of $\mathcal{C}$, and second that, under this identification, $\phi_! \colon X \mapsto id_X$ sends any [[object]] of $\mathcal{C}$ to its [[identity morphism]], as is verified by observing that this is clearly [[left adjoint]] to the [[domain]]-restriction $\phi^\ast$).

\end{remark}

\begin{remark}
\label{PullbackOfOpfibrationModelsTwoPullback}
  In Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations},
  the [[pullback]] (eq:StrictPullbackOfOpfibration) of an [[opfibration]] of [[strict categories]] formed in the [[1-category]] of [[strict categories]] may be understood as modelling the [[2-pullback]] (cf. [there](2-pullback#StrictPullback)) of the corresponding [[cospan]] in the actual [[2-category]] [[Cat]].

In particular, if the [[strict categories]] involved in the pullback diagram (eq:StrictPullbackOfOpfibration) are all [[groupoids]], then a functor such as $\phi$ being a [[Kan fibration]] is equivalent to its [[nerve]] being a [[Kan fibration]] (see at *[[right Kan fibration]]*, [this and](right%2Fleft+Kan+fibration#FibrationInGroupoidsAsRightKanFibrations) [this Prop](right%2Fleft+Kan+fibration#OverKanComplex)), in which case the pullback diagram (eq:StrictPullbackOfOpfibration) models the [[homotopy pullback]] (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)).
\end{remark}

### Enriched functor categories

Consider

* $(\mathbf{V}, \otimes, \mathbb{1})$ a [[cocomplete category|cocomplete]] [[symmetric monoidal category]], regarded as a [[cosmos for enrichment]],

* $\mathbf{C}$ a [[cocomplete category|cocomplete]] $\mathbf{V}$-[[enriched category]] which is also [[tensoring|tensored]] over $\mathbf{V}$, denoted:

  $$
    (-)\cdot(-)
    \;\colon\;
    \mathbf{V} \times \mathbf{C}
    \longrightarrow
    \mathbf{C}
  $$

and write

* for any [[small category|small]] $\mathbf{V}$-[[enriched category]] $\mathcal{X} \,\in\, \mathbf{V}Cat^{small}$:

  $$
    \mathbf{V}Func(\mathcal{X},\,\mathbf{C})
    \;\in\;
    Cat
  $$

  for the $\mathbf{V}$-[[enriched-functor category]] from $\mathcal{X}$ to $\mathbf{C}$,

* for any $\mathbf{V}$-enriched functor $f \,\colon\, \mathcal{X} \longrightarrow \mathcal{Y}$:

  $$
    \mathbf{V}Func(\mathcal{X},\,\mathbf{C})
    \underoverset
      {\underset{f^\ast}{\longleftarrow}}
      {\overset{f_!}{\longrightarrow}}
      {\;\;\; \bot \;\;\;}
    \mathbf{V}Func(\mathcal{Y},\,\mathbf{C})
  $$

  for the [[pair]] of [[adjoint functors]] given by [[precomposition]] and by [[left Kan extension]], the latter being expressible by the following [[coend]]-formula (see [there](Kan+extension#eq:LeftKanExtensionViaCoendFormula)):

\[
  \label{CoendFormulaForLeftKanExtension}
  \mathscr{V}_{(-)}
  \;\in\; \mathbf{V}Func(\mathcal{X},\,\mathbf{C})
  \;\;\;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;\;\;
  (f_! \mathscr{V})_y
  \;\simeq\;
  \int^{x \in \mathcal{X}}
  \mathcal{X}\big(f(x),\,y\big)
  \cdot
  \mathscr{V}_{x}
  \,.
\]

The following is a rather simplistic but maybe instructive example:

\begin{example}
\label{ForPushPullofEnrichedPresheavesThroughProjectionDiagram}
In the case of a *[[cartesian closed category|cartesian closed]]* [[cosmos]] $(\mathbf{V}, \times, \ast)$
consider for $\mathcal{X}, \mathcal{Y}, \mathcal{X}' \,\in\, \mathbf{V}Cat^{small}$
a commuting diagram of $\mathbf{V}$-[[enriched functors]] of the following form:
$$
  \array{
    \mathcal{X} \times \mathcal{X}'
    &\overset{f \times id}{\longrightarrow}&
    \mathcal{Y} \times \mathcal{X}'
    \\
    \mathllap{{}^{pr_{\mathcal{X}}}}
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \mathrlap{{}^{pr_{\mathcal{Y}}}}
    \\
    \mathcal{X}
    &\underset{\;\;\; f \;\;\;}{\longrightarrow}&
    \mathcal{Y}
  }
$$
(which is a [[pullback]] square in the [[1-category]] of [[strict category|strict]] $\mathbf{V}$-[[enriched categories]]).

Then the Beck-Chevalley condition on $\mathbf{C}$-valued $\mathbf{V}$-functors holds, in that the following [[commuting diagram|diagram commutes]]:
$$
  \array{
    \mathbf{V}Func(
      \mathcal{X} \times \mathcal{X}'
      ,\,
      \mathbf{C}
    \big)
    &\overset{(f \times id)_!}{\longrightarrow}&
    \mathbf{V}Func(
      \mathcal{Y} \times \mathcal{X}'
      ,\,
      \mathbf{C}
    )
    \\
    \mathllap{{}^{(pr_{\mathcal{X}})^\ast}}
    \big\uparrow
    &&
    \big\uparrow
    \mathrlap{{}^{(pr_{\mathcal{Y}})^\ast}}
    \\
    \mathbf{V}Func(\mathcal{X},\,\mathbf{C})
    &\underset{\;\;\; f_! \;\;\;}{\longrightarrow}&
    \mathbf{V}Func(\mathcal{Y},\,\mathbf{C})
  }
$$

To see this, we may check object-wise as follows:
$$
  \begin{array}{l}
  \big(
    (f \times id)_!
    (pr_{\mathcal{X}})^\ast
    \mathscr{V}
  \big)_{(y,x')}
  \\
  \;=\;
  \int^{(x,x'_0)}
  \mathcal{Y}\big(f(x),y\big)
  \times
  \mathcal{X}'\big(x'_0,x'\big)
  \cdot
  \big(
    (pr_{\mathcal{X}})^\ast\mathscr{V}_{x,x'_0}
  \big)
  \\
  \;=\;
  \int^{(x,x'_0)}
  \mathcal{Y}\big(f(x),y\big)
  \times
  \mathcal{X}'\big(x'_0,x'\big)
  \cdot
  \mathscr{V}_{x}
  \\
  \;=\;
  \int^{x'_0}
  \int^{x}
  \mathcal{Y}\big(f(x),y\big)
  \times
  \mathcal{X}'\big(x'_0,x'\big)
  \cdot
  \mathscr{V}_{x}
  \\
  \;\simeq\;
  \int^{x'_0}
  \mathcal{X}'\big(x'_0,x'\big)
  \cdot
  \int^{x}
  \mathcal{Y}\big(f(x),y\big)
  \cdot
  \mathscr{V}_{x}
  \\
  \;\simeq\;
  \int^{x}
  \mathcal{Y}\big(f(x), y\big)
  \cdot
  \mathscr{V}_{x}
  \\
  \;\simeq\;
  \big(
    f_!
    \mathscr{V}
  \big)_y
  \\
  \;\simeq\;
  \big(
  (pr_{\mathcal{Y}})^\ast
  (
    f_!
    \mathscr{V}
  )
  \big)_{y,x'}
  \,,
  \end{array}
$$
where we used (apart from the definition of [[precomposition]]), in order of appearance:

1. the [[coend]]-expression (eq:CoendFormulaForLeftKanExtension) for the [[left Kan extension]],

1. the [[Fubini theorem]] for coends ([here](end#Fubini)),

1. the fact that the [[closed monoidal category|closed tensor]] $(-) \times (-)$ [[preserves colimits]] and hence [[coends]] in each variable,

1. the [[co-Yoneda lemma]] in the coend-form [here](co-Yoneda+lemma#coYonedaLemma)

and finally the expression (eq:CoendFormulaForLeftKanExtension) again.
\end{example}

### Proper base change in étale cohomology

For [[coefficients]] of [[torsion group]], [[étale cohomology]]
satisfies Beck-Chevalley along [[proper morphisms]]. This is the statement of the  _[[proper base change theorem]]_. See there for more details.

### Grothendieck's yoga of six operations

A Beck-Chevalley condition is part of the [[axioms]] of  [[Grothendieck's yoga of six operations]], e.g. [Cisinski & Déglise (2009), §A.5](#CisinskiDéglise09), [Gallauer (2021), p. 16](#Gallauer21), [Scholze (2022), p. 11](#Scholze22).

## Related pages

* [[exact square]]
* [[Benabou-Roubaud theorem]]

## References
 {#References}

> The Beck-Chevalley condition has arisen in the theory of [[monadic descent|descent]] - as developed from [[Grothendieck]] 1959. [[Jon Beck]] and [[Claude Chevalley]] studied it independently from each another. [...] It is conspicuous that neither of them ever published anything on it. [[Pavlović 1991, §14](#Pavlović91)]

Original articles:

* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270**  (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]],   English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]

  > (proving the [[Bénabou-Roubaud theorem]])

* {#Lawvere70} [[William Lawvere]], p. 8 of: *[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]*, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [[pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf)]

  > (in terms of [[base change]] as the [[categorical semantics]] for [[universal quantification]]/[[dependent product]] and [[existential quantification]]/[[dependent sum]])

expanded on in

* {#Seely83} [[Robert A. G. Seely]], *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math. Logik und Grundlagen d. Math. **29** (1983) 505-542 [[doi:10.1002/malq.19830291005]( https://doi.org/10.1002/malq.19830291005), [pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)]

Review:

* {#Pavlovic1990} [[Duško Pavlović]], pp. 105 in: *Predicates and Fibrations*, PhD thesis, Utrecht (1990) [[pdf](https://dusko.org/wp-content/uploads/2020/03/1990-proefschrift-dusko.pdf), [[Pavlovic-PredicatesAndFibrations.pdf:file]]]

* {#Pavlović91} [[Duško Pavlović]], Section 1 of: *Categorical interpolation: Descent and the Beck-Chevalley condition without direct images*, in: *Category Theory*, Lecture Notes in Mathematics **1488** (1991) [[doi:10.1007/BFb0084229](https://doi.org/10.1007/BFb0084229), [pdf](http://dusko.org/wp-content/uploads/2020/03/1990-Como-BCDE.pdf), [[Pavlovic-CategoricalInterpolation.pdf:file]]]

* [[Simon John Ambler]], §5.5.1 in: *First Order Linear Logic in Symmetric Monoidal Closed Categories*, PhD thesis, Edinburgh (1991) [[ECS-LFCS-92-194](https://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194/ECS-LFCS-92-194.pdf), [[Ambler-FOLL.pdf:file]]]

> (in view of [[linear type theory]])

* {#Pavlović96} [[Duško Pavlović]], *Maps II: Chasing Diagrams in Categorical Proof Theory*, Logic Journal of the IGPL, **4** 2 (1996) 159–194 [[doi:10.1093/jigpal/4.2.159](https://doi.org/10.1093/jigpal/4.2.159), [pdf](http://dusko.org/wp-content/uploads/2020/03/00-95-IGPL-mapsII.pdf)]

* [[Paul Balmer]], around §7.5 of: *Stacks of group representations*, J. European Math. Soc. **17** 1 (2015) 189-228 [[arXiv:1302.6290](https://arxiv.org/abs/1302.6290), [doi:10.4171/jems/501](https://doi.org/10.4171/jems/501)]

Discussion for [[subobject lattices]]:

* [[Saunders MacLane]], [[Ieke Moerdijk]], chapter IV.9 (around p. 205) of: _[[Sheaves in Geometry and Logic]]_,   Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]

Review in the context of [[Grothendieck's yoga of six operations]]:

* {#CisinskiDéglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], section A.5 of: *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

* {#Gallauer21} [[Martin Gallauer]], pp. 16 of: *An introduction to six-functor formalism*, lecture at *[The Six-Functor Formalism and Motivic Homotopy Theory](https://sites.google.com/view/summer-school-motivic/home)*, Università degli Studi di Milano (Sept. 2021) [[arXiv:2112.10456](https://arxiv.org/abs/2112.10456), [pdf](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf)]

* {#Scholze22} [[Peter Scholze]], p. 11 of: *Six Functor Formalisms*, [lecture notes ](https://people.mpim-bonn.mpg.de/scholze/Course%20Winter%202223.html) (2022) [[pdf](https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf), [[Scholze-SixOperations.pdf:file]]]

Discussion for [[derived functors]] with focus on the example of [[base change]] of [[retractive spaces]] and [[homotopy categories]] of [[parameterized spectra]]:

* [[Michael Shulman]], *Comparing composites of left and right derived functors*, New York Journal of Mathematics **17** (2011) 75-125 [[arXiv:0706.2868](https://arxiv.org/abs/0706.2868), [eudml:229181](https://eudml.org/doc/229181)]

* [[Mike Shulman]], *Framed bicategories and monoidal fibrations*, Theory and Applications of Categories, **20** 18 (2008) 650–738 [[tac:2018](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html), [arXiv:0706.1286](https://arxiv.org/abs/0706.1286)]

Discussion in the generality of [[enriched category theory]]:

* [[Michael Shulman]], e.g. Thm. 9.3 in: *Enriched indexed categories*, Theory Appl. Categ. **28** (2013) 616-695 [[doi:1212.3914](https://arxiv.org/abs/1212.3914), [tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html)]

Discussion in the generality of [[(infinity,1)-categories|$\infty$-categories]]:

* {#HopkinsLurie13} [[Michael Hopkins]], [[Jacob Lurie]], Prop. 4.3.3 in: *[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]* (2013) [[pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)]

* [[Urs Schreiber]], around Def. 5.5 of: *[[schreiber:Quantization via Linear homotopy types]]* [[arXiv:1402.7041](https://arxiv.org/abs/1402.7041)]

* {#GaitsgoryLurie} [[Dennis Gaitsgory]], [[Jacob Lurie]], Section 2.4.1 of: _Weil's conjecture for function fields_ (2014-2017) [["first volume of expanded account" pdf](https://www.math.ias.edu/~lurie/papers/tamagawa-abridged.pdf)]

for [[infinity-cosmos|$\infty$-cosmoi]]:

* [[Emily Riehl]], [[Dominic Verity]], Prop. 5.3.9 in: *Kan extensions and the calculus of modules for ∞-categories*, Algebr. Geom. Topol. **17** (2017) 189-271 [[doi:10.2140/agt.2017.17.189](https://doi.org/10.2140/agt.2017.17.189), [arXiv:1507.01460](https://arxiv.org/abs/1507.01460)]

Discussion for presheaf categories in the context of [[quasicategories]] ([[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]]):

* {#Joyal08} [[André Joyal]], [p. 175](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=27) of: *[[The Theory of Quasi-Categories and its Applications]]*, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]

Discussion for [[Goursat categories]]:

* [[Marino Gran]], [[Diana Rodelo]], *Beck-Chevalley condition and Goursat categories*, Journal of Pure and Applied Algebra **221** 10 (2017) 2445-2457 [[arXiv:1512.04066](https://arxiv.org/abs/1512.04066), [doi:10.1016/j.jpaa.2016.12.031](https://doi.org/10.1016/j.jpaa.2016.12.031)]

[[!redirects Beck-Chevalley conditions]]

[[!redirects Beck-Chevalley_Condition]]
[[!redirects Beck-Chevalley Condition]]
[[!redirects Beck–Chevalley condition]]
[[!redirects Beck--Chevalley condition]]
[[!redirects Beck condition]]
[[!redirects Chevalley condition]]
[[!redirects Beck-Chevalley property]]
[[!redirects Beck–Chevalley property]]
[[!redirects Beck--Chevalley property]]

[[!redirects Beck-Chevalley transformation]]
[[!redirects Beck-Chevalley transformations]]

[[!redirects Beck-Chevalley relation]]
[[!redirects Beck-Chevalley relations]]

[[!redirects Beck-Chevalley]]


Becker-Gottlieb transfer

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The Becker-Gottlieb transfer is a variant of [[push-forward in generalized cohomology]] of [[cohomology theories]] along [[proper map|proper]] [[submersions]] of [[smooth manifolds]].

The Becker-Gottlieb transfer operation has been refined to [[differential cohomology]] in ([Bunke-Gepner 13](#BunkeGepner13)).

Its compatibility in [[differential algebraic K-theory]] with the differential refinement of the [[Borel regulator]] is the content of the _transfer index conjecture_ ([Bunke-Tamme 12, conjecture 1.1](#BunkeTamme12), [Bunke-Gepner 13, conjecture 5.3](#BunkeGepner13)).

For the moment see at _[regulator -- Becker-Gottlieb transfer](Beilinson+regulator#GottliebTransfer)_ for more.

## Definition

See e.g. ([Haugseng 13, def. 3.9](#Haugseng13)).

## Related concepts

* [[sheaf with transfer]]

## References

The original articles

* {#BeckerGottlieb75} [[James Becker]], [[Daniel Gottlieb]], _The transfer map and fiber bundles_, Topology , 14 (1975) ([pdf](https://www.math.purdue.edu/~gottlieb/Bibliography/Transfb.pdf),  doi:10.1016/0040-9383(75)90029-4)

  (which also gives a proof of the [[Adams conjecture]]).

* [[James Becker]], [[Daniel Gottlieb]], _Vector fields and transfers_ Manuscr. Math. , 72 (1991) pp. 111–130 ([pdf](https://www.math.purdue.edu/~gottlieb/Papers/manuscripta.pdf), [doi:10.1007/BF02568269](https://link.springer.com/article/10.1007/BF02568269))

Interpretation in terms of [[dualizable objects]]:

* [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]

* {#BeckerGottlieb} [[James Becker]], [[Daniel Gottlieb]]: *A History of Duality in Algebraic Topology* [[pdf](http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf), [[BeckerGottlieb-DualityHistory.pdf:file]]]

Review:

* [[Dai Tamaki]], [[Akira Kono]], Section 4.5 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* [[eom]], _[Becker-Gottlieb transfer](http://www.encyclopediaofmath.org/index.php/Becker-Gottlieb_transfer)_

Discussion in the context of [[differential algebraic K-theory]] is in

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))

In prop. 4.14 of

* {#Haugseng13} [[Rune Haugseng]], _The Becker-Gottlieb Transfer Is Functorial_ ([arXiv:1310.6321](http://arxiv.org/abs/1310.6321)) (withdrawn now)

Becker-Gottlieb transfer was identified with the [[Umkehr map]] induced from a [[Wirthmüller context]] in which in addition $f_\ast$ satisfies its [[projection formula]] (a "[[transfer context]]", def.4.9)

The article

* {#KleinMalkiewich16} [[John Klein]], [[Cary Malkiewich]], _The transfer is functorial_, [arXiv:1603.01872](http://arxiv.org/abs/1603.01872),

claimed to establish the functoriality of the Becker-Gottlieb transfer for fibrations with finitely dominated fibers on the level of [[homotopy categories]] (without higher coherences), but contained an unfixable mistake (cf. _[Corrigendum to 'The transfer is functorial'](https://www.sciencedirect.com/science/article/pii/S0001870822005692)_).

[[!redirects transfer index conjecture]]


becoming

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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to _[[Science of Logic]]_ ([SoL §152](Science+of+Logic#152), [Lawvere 91, p. 11](#Lawvere91)) "becoming" is the [[unity of opposites]] of [[nothing]] and [[being]]. The reverse is _[[ceasing]]_. The [[Aufhebung]] of both is [[Dasein]]:

| |  | [[Dasein]] | |  |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |

The universal factorization via $\emptyset$-[[unit of a monad|unit]] and $\ast$-[[counit of a comonad]]:

$$
  \array{
    \emptyset &\longrightarrow& X &\longrightarrow& \ast
    \\
    \\
    nothing && becoming && being
  }
$$

of the unique [[function]] from the [[empty type]] to the [[unit type]] through any other [[type]] $X$.

Indeed, this is the statement of [SoL §174](Science+of+Logic#174): _there is nothing which is not an intermediate state between being and nothing._

## References

* [[Hegel]], _[[Science of Logic]]_, Volume One: _The Objective Logic_, Book One: _The Doctrine of Being_, Chapter 1 _Being_, C _Becoming_

* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

* {#Lawvere91} [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_1991

[[!redirects Werden]]
[[!redirects werden]]


behavior

# Behaviours
* table of contents
{: toc}

## Idea

Presumably, this is some abstraction of the notion of behaviour?

## Defintions

A __behaviour__ (or __behavior__) is ....

Given two behaviour $B$ and $C$, a __behaviour morphism__ from $B$ to $C$ is ....

Behaviours and behaviour morphisms form a [[category]] $Beh$.

## References

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

[[!redirects behavior]]
[[!redirects behaviors]]
[[!redirects behaviour]]
[[!redirects behaviours]]
[[!redirects Beh]]


Behrang Noohi

Behrang Noohi is reader in mathematics at Queen Mary University, London.

* [website](https://www.qmul.ac.uk/maths/profiles/noohib.html)

## Selected writings

On [[transgression in group cohomology]] generalized to Real $\mathbb{Z}/2$-equivariant cohomology (as appropriate for twists of [[KR-theory]]):

* [[Behrang Noohi]], [[Matthew B. Young]], *Twisted loop transgression and higher Jandl gerbes over finite groupoids*, Algebr. Geom. Topol. **22** (2022) 1663-1712 [[arXiv:1910.01422](https://arxiv.org/abs/1910.01422), [doi:10.2140/agt.2022.22.1663](https://doi.org/10.2140/agt.2022.22.1663)]

## Related entries

* [[geometric stack]]

* [[butterfly]]

* [[nonabelian group cohomology]]

* [[string topology]]

category: people


Bei Zeng

* [institute page 1](https://uwaterloo.ca/institute-for-quantum-computing/about/people/bei-zeng)

* [institute page 2](https://physics.ust.hk/eng/people_detail.php?pplcat=1&id=555)

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

## Selected writings

On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

category: people


Beilinson conjecture

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###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Beilinson's conjectures_ ([Beilinson 85](#Beilinson85)) [[conjecture]] for [[arithmetic varieties]] over [[number fields]]

1. that the realification of the [[Beilinson regulator]] exhibits an [[isomorphism]] between the relevant [[algebraic K-theory]]/[[motivic cohomology]] groups and [[Deligne cohomology]] ([[ordinary differential cohomology]]) groups;

   (recalled e.g. as [Schneider 88, p. 30](#Schneider88), [Brylinski-Zucker 91, conjecture 5.20](#BrylinskiZucker91), [Deninger-Scholl (3.1.1)](#DeningerScholl), [Nekovar (6.1 (1))](#Nekovar)).

1. induced by this that [[special values of L-functions|special values]] of the ([[Hasse-Weil L-function|Hasse-Weil]]-type) [[L-function]] are proportional to the [[Beilinson regulator]], in analogy with the [[class number formula]] and the [[Birch and Swinnerton-Dyer conjecture]]

   (recalled e.g. as [Schneider 88, p. 31](#Schneider88) [Brylinski-Zucker 91, conjecture 5.21](#BrylinskiZucker91), [Deninger-Scholl (3.1.2)](#DeningerScholl), [Nekovar (6.1 (2))](#Nekovar))).

The Beilinson conjecture for [[special values of L-functions]] follows the [[Birch and Swinnerton-Dyer conjecture]] and [[Pierre Deligne]]'s conjecture on special value of L-functions.

## Related concepts

* [[L-function]]

* [[Beilinson regulator]]

* [[standard conjectures]]

## References

The original articles are

* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))

* {#Beilinson80} [[Alexander Beilinson]], _Higher regulators of curves_, Funct. Anal. Appl. 14 (1980), 116-118, [mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng).

* {#Beilinson87} [[Alexander Beilinson]], _Height pairing between algebraic cycles_, in _K-Theory, Arithmetic and Geometry_, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, [DOI](http://dx.doi.org/10.1007/BFb0078364).

Reviews include

* {#RapoportSchappacherSchneider88}
 [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_  Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

* {#Soule84} [[Christophe Soulé]], _Régulateurs_ Séminaire Bourbaki, 27 (1984-1985), Exp. No. 644, 17 p. ([Numdam](http://www.numdam.org/item?id=SB_1984-1985__27__237_0))

* {#Schneider88} [[Peter Schneider]], _Introduction to the Beilinson conjectures_, in [Rapoport-Schappacher-Schneider 88](#RapoportSchappacherSchneider88) ([[SchneiderBeilinsonConjectures.pdf:file]])

* {#Nekovar} [[Jan Nekovar]], section 3 of _Beilinson's Conjectures_ ([pdf](http://people.math.jussieu.fr/~nekovar/pu/mot.pdf))

* {#DeningerScholl} [[Christopher Deninger]], [[Anthony Scholl]], _The Beilinson conjectures_ ([pdf](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/d-s.pdf))

* {#BrylinskiZucker91} [[Jean-Luc Brylinski]], Steven Zucker, conjecture 5.20,5.21 in _An overview of recent advances in Hodge theory_,  English translation in _Several complex variables VI_, volume 69 of _Encyclopedia of Math. Sciences_, pages 39-142, 1990 (original in Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 1991,	 Volume 69, Pages 48–165  ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intf&paperid=177&option_lang=eng), [pdf (Russian original)](http://www.mathnet.ru/php/getFT.phtml?jrnid=intf&paperid=177&what=fullt&option_lang=eng)))

A [[noncommutative algebraic geometry|noncommutative]] analogue is considered in

* [[Dmitri Kaledin]], _Beilinson conjectures in the noncommutative setting_, [pdf](http://www.math.nyu.edu/~tschinke/books/finite-fields/final/03_kaledin.pdf).

[[!redirects Beilinson conjectures]]

[[!redirects Beilinson's conjectures]]


Beilinson monad

#Contents#
* table of contents
{:toc}

## Idea

A _monad_ in the sense of Beilinson is a [[chain complex]] whose [[chain homology]] is concentrated in degree 0. (For instance p. 21 [here](#ES)).
In other words, it is a certain [[Eilenberg-MacLane object]] in the context of [[homological algebra]].

(This is unrelated to [[monad (disambiguation)|other notions of monads]]).

## References

* [[David Eisenbud]], Frank-Olaf Schreyer, _Relative Beilinson Monad and Direct Image for Families of Coherent Sheaves_ ([arXiv:math/0506391](http://arxiv.org/abs/math/0506391))
  {#ES}

[[!redirects Beilinson monads]]


This entry is about the book

* [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.)

  \linebreak

  **Beilinson's Conjectures on Special Values of L-Functions**

  \linebreak

  Perspectives in Mathematics, Volume 4,

  Academic Press, Inc. 1988,

  Harcourt Brace Jovanovich, Publishers

  [ISBN:978-0-12-581120-0](https://www.sciencedirect.com/book/9780125811200/beilinsons-conjectures-on-special-values-of-l-functions)

  [Electronic version](https://web.archive.org/web/20190618180518/https://ivv5hpp.uni-muenster.de/u/pschnei/publ/beilinson-volume/)

on [[Deligne cohomology]] and [[Beilinson's conjectures]] on [[special values of L-functions]].

(The book is out of print and unavailable from [the publisher](https://www.sciencedirect.com/book/9780125811200/beilinsons-conjectures-on-special-values-of-l-functions), even electronically.)

#Contents#
* table of contents
{:toc}

## Preface

* pdf

Every Spring and Fall, in the woods of the Black Forest, mathematicians from
West-Germany and other countries gather at Oberwolfach for the “Arbeitsgemeinschaft Geyer-Harder” to teach themselves in a joint effort new theories or results. At each meeting the topic and organizers of the following Arbeitsgemeinschaft are chosen after an evening of discussions by democratic vote. The organizers then prepare a detailed program for the conference which is posted in the mathematical institutes.
Everyone interested in taking part in the meeting has to volunteer for one of the talks in the program. The organizers then choose the actual speakers.
The Arbeitsgemeinschaft of April 1986 was devoted to the work of Beilinson
and Bloch on regulators. The meeting centered around Beilinson's article “Higher regulators and values of L-functions”.
It was generally felt at the conference that the joint effort made to understand this fundamental paper should not be lost, and should be helpful to others. Hence this volume.
Most of its chapters are based on talks at the meeting; others have been added in order to give a coherent account of the conjectures and some of the known evidence for them. In the name of all participants we want to express our gratitude to the Mathematisches Forschungsinstitut Oberwolfach, which makes the Arbeitsgemeinschaft
possible.

1987

[[Michael Rapoport]]

[[Norbert Schappacher]]

[[Peter Schneider]]

## Program proposal

* pdf

## Introduction to the Beilinson Conjectures

[[Peter Schneider]].

* pdf

### §1 Complex L-functions

### §2 Deligne cohomology

### §3 Absolute cohomology

### §4 Chern classes

### §5 The conjectures

### §6 Further hints

## Deligne’s Conjecture

[[Maria Heep]], [[Uwe Weselmann]].

* pdf

### §1 Motives

### §2 Duality, functional equation, criticial values

### §3 Deligne’s periods

### §4 Beilinson’s periods

## Deligne-Beilinson Cohomology

[[Hélène Esnault]], [[Eckart Viehweg]].

* pdf

### §1 The Deligne cohomology

### §2 The Deligne-Beilinson complex

### §3 Products

### §4 Relative cohomology

### §5 Extensions and complements

### §6 The cycle map in the de Rham cohomology

### §7 The cycle map in the Deligne cohomology

### §8 Chern classes in the Deligne-Beilinson cohomology

## $\lambda$-Rings and Adams Operations in Algebraic K-Theory.

[[Wolfgang K. Seiler]].

* pdf

## The Theorem of Riemann-Roch

[[Günter Tamme]].

* pdf

### §1 The λ-ring structure and the Chern character for K-theory with supports

### §2 Riemann-Roch without denominators

### §3 The smooth Riemann-Roch theorem for K-theory with supports

### §4 The singular Riemann-Roch

### §5 Absolute cohomology and homology

## Comparison of the Regulators of Beilinson and of Borel.

[[Michael Rapoport]].

* pdf

## The Beilinson Conjecture for Algebraic Number Fields.

[[Jürgen Neukirch]].

### Part I: Regulators and values of Artin L-series

  * pdf

#### §1 Regulators for algebraic number fields

#### §2 Regulators for Artin motives

#### §3 L-series of Artin motives

#### §4 Dirichlet L-series

#### §5 Regulators of Dirichlet motives

### Part II: The regulator map for cyclotomic fields

  * pdf

#### §1 The main theorem

#### §2 Universal symbols

#### §3 Special symbols

#### §4 Reduction to the main lemma

#### §5 Proof of the main lemma for n = 1

#### §6 Proof of the main lemma for n ≥ 2

## On the Beilinson Conjectures for Elliptic Curves with Complex Multiplication

[[Christopher Deninger]], [[Kay Wingberg]].

  * pdf

### §1 A formula for the regulator of curves

### §2 A weakened version of the Beilinson conjecture for elliptic curves

### §3 Calculating $\langle\omega,[\alpha,\beta]_{\cal D}\rangle$ for elliptic curves over $\bf R$

### §4 Relations between the L-function of an elliptic curve over Q with complex multiplication and Eisenstein-Kronecker-Lerch series

### §5 The absolute cohomology

## Beilinson’s Theorem on Modular Curves.

[[Norbert Schappacher]], [[Anthony J. Scholl]].

* pdf

### §1 The Theorem

### §2 Transformation of L-values

### §3 Eisenstein series and modular units

### §4 Whittaker functions and L-factors

### §5 Evaluation of the regulator integral

### §6 Non-vanishing of the regulator integral

### §7 Integrality

## Deligne Homology, Hodge-D-Conjecture, and Motives

[[Uwe Jannsen]].

* pdf


Beilinson-Bernstein localization

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#Contents#
* table of contents
{:toc}

## Beĭlinson-Bernstein localization theorem

Consider a complex [[reductive group]] $G$ with [[Lie algebra]] $\mathfrak{g}$, [[Borel subgroup]] $B \subset G$, and [[flag variety]] $\mathcal{B} = G/B$. The localization theory of Beilinson-Bernstein identifies [[representations]] of $\mathfrak{g}$ with [[global sections]] of (twisted) [[D-modules]] on $\mathcal{B}$. In particular, highest weight representations are realized by $B$-equivariant $\mathcal{D}$-modules on $\mathcal{B}$, or in other words, by $\mathcal{D}$-modules on the [[quotient stack]] $B\backslash \mathcal{B}$.

Furthermore, given a [[subgroup]] $K \subset G$, it identifies [[modules]]
for the [[Harish-Chandra pair]] $(\mathfrak{g}, K)$ with global sections of $K$-equivariant twisted $\mathcal{D}$-modules on $\mathcal{B}$.

The case $K = G$ gives the [[Borel-Weil theorem|Borel-Weil]] description of irreducible algebraic (equivalently, finite-dimensional) representations of $G$ as sections of equivariant [[line bundles]] on $\mathcal{B}$.

([Ben-Zvi&Nadler 07](#BenZviNadler07))

## References

* A. Beilinson, J. Bernstein, _Localisation de $\mathfrak{g}$-modules_, C.R. Acad. Sci. Paris, 292 (1981), 15-18, [MR82k:14015](http://www.ams.org/mathscinet-getitem?mr=82k:14015),  Zbl 0476.14019

* [[A. Beilinson]], _Localization of representations of reductive Lie algebra_, Proc. of ICM 1982, (1983), 699-716.  Zbl 0571.20032

* [[Valery Lunts]], [[Alexander Rosenberg]], _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159, [MR2001f:17028](http://www.ams.org/mathscinet-getitem?mr=1694897), [doi](http://dx.doi.org/10.1007/s000290050044); _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3894), _II. D-Calculus in the braided case. The localization of quantized enveloping algebras_, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3916)

* [[Edward Frenkel]], [[Dennis Gaitsgory]], _Localization of $\mathfrak{g}$-modules on the affine Grassmannian_, Ann. of Math. (2) 170 (2009), no. 3, 1339–1381, [MR2600875](http://www.ams.org/mathscinet-getitem?mr=2600875), [doi](http://dx.doi.org/10.4007/annals.2009.170.1339)

* Toshiyuki Tanisaki, _The Beilinson-Bernstein correspondence for quantized enveloping algebras_, Math. Z. __250__ (2005), no. 2, 299–361, [MR2006h:17025](http://www.ams.org/mathscinet-getitem?mr=2178788), [doi](http://dx.doi.org/10.1007/s00209-004-0754-9) [math.QA/0309349](http://front.math.ucdavis.edu/0309.5349);

* H. Hecht, D. Miličić, W. Schmid, J. A. Wolf, _Localization and standard modules for real semisimple Lie groups, I: The Duality Theorem_
Zbl 0699.22022, [MR910203](http://www.ams.org/mathscinet-getitem?mr=910203), _II: Applications_,

* Hendrik Orem, Lecture notes: The Beilinson-Bernstein Localization
Theorem, [pdf](http://www.ma.utexas.edu/users/horem/bb-notes.pdf)

* [[David Ben-Zvi]], [[David Nadler]], _Loop Spaces and Langlands Parameters_ ([arXiv:0706.0322](http://arxiv.org/abs/0706.0322)))
 {#BenZviNadler07}

[[!redirects Beĭlinson-Bernstein localization]]
[[!redirects Beĭlinson-Bernstein localization theorem]]
[[!redirects Beilinson-Bernstein localization theorem]]


Beilinson-Drinfeld algebra

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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Beilinson-Drinfeld algebra_ (or _BD-algebra_ for short) is like a [[BV-algebra]] with nilpotent [[BV-operator]], but over [[formal power series]] in one formal parameter $\hbar$, and such that the [[Gerstenhaber algebra|Gerstenhaber bracket]] is proportional to that parameter.

This means that one may think of a BD-algebra as an $\hbar$-parameterized formal family of algebras which for $\hbar = 0$ are [[Poisson 0-algebras]] and for $\hbar \neq 0$ they look superficially like [[BV-algebras]]. But see remark \ref{RoleOfTheDifferential} below.

Such BD-algbras are used to formalize [[formal deformation quantization]] in the context of the [[BV-BRST formalism]] (in [Costello-Gwilliam](#CostelloGwilliam)).

## Definition

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

A **quantum BV complex** or **Beilinson-Drinfeld algebra** is

1. a [[differential graded-commutative algebra]] $A$ (whose [[differential]] we denote by $\Delta$) over the ring $\mathbb{R} [ [ \hbar ] ]$ of [[formal power series]] over the [[real numbers]] in a formal constant $\hbar$,

1. equipped with a [[Poisson 0-algebra|Poisson bracket]] $\{-,-\}$ of the same degree as the differential

such that

* the following equation holds for all elements $a,b \in A$ of homogeneous degree ${\vert a\vert}, {\vert b\vert} \in \mathbb{Z}$

  $$
    \Delta( a \cdot b) = (\Delta a) \cdot b  + (-1)^{\vert a\vert} a \Delta b + \hbar \{a,b\}$ for all homogenous elements $a, b \in A
    \,.
  $$

=--

(e.g. [Costello-Gwilliam, def. 1.4.0.1](#CostelloGwilliam), [Gwilliam 13, def. 2.2.5](#Gwilliam))

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

If in the equation in def. \ref{BeilinsonDrinfelAlgebra} one replaces $\hbar$ by 1, then it takes the form characteristic of a [[BV-algebra]]. However the differential $\Delta$ here is the differential in the underlying [[chain complex]] for an [[algebra over an operad]] in chain complexes, while in a BV-algebra $\Delta$ is the unary operation encoded in the [[BV-operad]], hence present already for algebras in plain modules/chain complexes over that operad. Accordingly, in the context of BD-algebra then for $\hbar \neq 0$ the bracket is actually trivial up to homotopy.

For more see at _[[relation between BV and BD]]_.

=--

## Related concepts

[[!include deformation quantization - table]]

## References

The notion was introduced in

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

A discussion is in section 2.4 of

* {#CostelloGwilliam} [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in quantum field theory Volume 2_ ([pdf](http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf))

See also

* {#Gwilliam13} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis (2013) ([[GwilliamThesis.pdf:file]])

* Martin Doubek, [[Branislav Jurčo]], Lada Peksová, Ján Pulmann, _Connected sum for modular operads and Beilinson-Drinfeld algebras_, [arXiv:2210.06517](https://arxiv.org/abs/2210.06517)
* Constantin-Cosmin Todea, _BD algebras and group cohomology_, Comptes Rendus. Mathématique __359__ (2021) no. 8, 925--937 [doi](https://doi.org/10.5802/crmath.246)

[[!redirects Beilinson-Drinfeld algebras]]

[[!redirects BD algebra]]
[[!redirects BD algebras]]

[[!redirects BD-algebra]]
[[!redirects BD-algebras]]


Beilinson–Deligne cup product

+-- {: .rightHandSide}
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### Context
#### Differential cohomology
+-- {: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Beilinson-Drinfeld cup product_ is an explicit presentation of the [[cup product]] in [[ordinary differential cohomology]] (see also at _[[cup product in differential cohomology]]_) for the case that the latter is modeled by the [[Cech cohomology|Cech]]-[[Deligne cohomology]]. It sends (see [[cup product in abelian Cech cohomology]])

$$
\cup: A[p]^\infty_D\otimes B[q]^\infty_D\to (A\otimes_{\mathbb{Z}} B)[p+q]^\infty_D,
$$
where $A$ and $B$ are lattices in $\mathbb{R}^n$, and $\mathbb{R}^m$ for some $n$ and $m$, respectively.
It is a morphism of complexes, so it induces a cup product in [[Deligne cohomology]].

For $A=B=\mathbb{Z}$, the Beilinson-Deligne cup product is associative and commutative up to homotopy, so it induces an associative and commutatvive cup product on [[ordinary differential cohomology|differential cohomology]]

## Definition

Let the [[Deligne complex]] $\mathbf{B}^n(\mathbb{R}//\mathbb{Z})_{conn}$ be given by

$$
  \array{
     &
     \mathbb{Z}
       &\hookrightarrow&
     C^\infty(-,\mathbb{R})
       &\stackrel{d_{dR}}{\to}&
     \cdots
       &\stackrel{d_{dR}}{\to}&
     \Omega^{n}(-)
     \\
     \\
     degree: & 0 && 1 && \cdots && (n+1)
  }
$$

where we refer to degrees as indicated in the bottom row.

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

The _Beilinson-Deligne product_ is the morphism of [[chain complexes]]
of [[sheaves]]

$$
  \cup
    :
   \mathbf{B}^p (\mathbb{R}//\mathbb{Z})_{conn}
   \otimes
   \mathbf{B}^q (\mathbb{R}//\mathbb{Z})_{conn}
   \to
   \mathbf{B}^{p+q+1} (\mathbb{R}//\mathbb{Z})_{conn}
$$

given on homogeneous elements $\alpha$, $\beta$ as follows:

$$
  \alpha \cup \beta :=
  \left\{
    \array{
    \alpha \wedge \beta = \alpha \beta & if\,deg(\alpha) = 0
    \\
    \alpha \wedge d_{dR}\beta & if\,deg(\alpha) \gt 0\,and\,deg(\beta) = q+1
    \\
    0 & otherwise
    }
  \right.
  \,.
$$

=--

## Applications

### In higher Chern-Simons theory

The [[action functional]] of abelian [[higher dimensional Chern-Simons theory]] is given by the [[fiber integration in ordinary differential cohomology]] over the BD cup product of differential cocycles

$$
  S_{CS} : H^{2k+2}(\Sigma)_diff  \to U(1)
$$

$$
  \hat C \mapsto \int_\Sigma \hat C \cup \hat C
  \,.
$$

For more on this see [[higher dimensional Chern-Simons theory]].

## Related concepts

* [[Deligne line bundle]]

* [[intersection pairing]]

* [[self-dual higher gauge theory]]

## References

The original references are

* [[Pierre Deligne]], _Théorie de Hodge II_ , IHES Pub. Math. (1971), no. 40, 5–57.

* [[Alexander Beilinson]], _Higher regulators and values of L-functions_ , J. Soviet Math. 30 (1985), 2036—2070

* [[Alexander Beilinson]], _Notes on absolute Hodge cohomology_ , Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.

A survey is for instance around prop. 1.5.8 of

* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_ Birkhäuser (1993)

and in section 3 of

* [[Helene Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_ in Rapoport, Schappacher, Schneider (eds.) _Beilinson's Conjectures on Special Values of L-Functions_ . Perspectives in Math. 4, Academic Press (1988) 43 - 91 ([pdf](http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf))

For the cup product of [[Cheeger-Simons differential character]]s see also

* [[Mike Hopkins]] and [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_ ([arXiv](http://arxiv.org/abs/math/0211216))

[[!redirects Deligne-Beilinson cup product]]
[[!redirects Deligne-Beilinson cup products]]
[[!redirects Deligne–Beilinson cup product]]
[[!redirects Deligne–Beilinson cup products]]
[[!redirects Deligne--Beilinson cup product]]
[[!redirects Deligne--Beilinson cup products]]
[[!redirects Beilinson-Deligne cup product]]
[[!redirects Beilinson-Deligne cup products]]
[[!redirects Beilinson–Deligne cup product]]
[[!redirects Beilinson–Deligne cup products]]
[[!redirects Beilinson--Deligne cup product]]
[[!redirects Beilinson--Deligne cup products]]
[[!redirects Beilinson-Deligne cup-product]]

[[!redirects differential cup-product]]
[[!redirects differential cup product]]
[[!redirects differential cup products]]

[[!redirects cup product in ordinary differential cohomology]]
[[!redirects cup-product in ordinary differential cohomology]]

[[!redirects cup product on differential cohomology]]

[[!redirects cup product in generalized differential cohomology]]
[[!redirects cup product on generalized differential cohomology]]

[[!redirects cup product in differential generalized cohomology]]
[[!redirects cup product on differential generalized cohomology]]


being

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to [[道德經]] (English translation following [[Xiao-Gang Wen]] [here](http://physics.stackexchange.com/a/94612/5603)):

> The nameless nonbeing is the origin of universe;

> The named being is the mother of all observed things.

> Within nonbeing, we enjoy the mystery of the universe.

> Among being, we observe the richness of the world.

> Nonbeing and being are two aspects of the same mystery.

> From nonbeing to being and from being to nonbeing is the gateway to all understanding.

Rather similarly, according to ([Hegel 12](#Hegel12)) _pure being_ is the [[unity of opposites|opposite]] of [[nothing]] whose unity is pure [[becoming]].

| |  | [[Dasein]] | |  |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |

According to the formalization of this proposed by ([Lawvere 91](#Lawvere91)), this is described by the [[adjoint modality]]

$$
  (\emptyset \dashv \ast)
$$

of the [[idempotent monad]] constant on a [[terminal object]] $\ast$ and its [[left adjoint]] $\emptyset$.

In this interpretation any other [[adjoint modality]] of the form $(\Box \dashv \bigcirc)$ characterizes a more "determinate" form of being ( _Dasein_ in the terminology of ([Hegel 12](#Hegel12))).

A [[category]] equipped with such a notion of being is, naturally, called a _[[category of being]]_.

## Related concepts

* [[ontology]], [[metaphysics]]

* [[category of being]]

* [[Aufhebung]]

* [[modality]]

* [[beable]]

* [[existential quantifier]]

## References

* {#Hegel12} [[Hegel]], _[[Science of Logic]]_, Volume One: _The Objective Logic_, Book One, _The Doctrine of Being_, Chapter 1 _Being_

* {#Lawvere91} [[Bill Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ (1991)

* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

* German Wikipedia, _[Sein](http://de.wikipedia.org/wiki/Sein)_

[[!redirects Sein]]
[[!redirects sein]]


This page collects material related to the article

* {#Cantor1895} [[Georg Cantor]],

  _Beiträge zur Begründung der transfiniten Mengenlehre_,

  Math. Ann. 46 (1895) pp.481-512,

  reprinted from p. 282 on in

  [[Ernst Zermelo]] (ed.),

  _Georg Cantor -- Gesammelte Abhandlungen mathematischen und philosophischen Inhalts_,

  Springer Berlin 1932

  ([online English translation](https://archive.org/details/contributionstof00cant))

which is the origin of the modern concept of [[cardinality]] of [[sets]]. (But see also the commentary in [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_).

#Contents#
* table of contents
{:toc}

## Section 1. The conception of Power or Cardinal Number**

By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung zu
einem Ganzen) $M$ of definite and separate objects of our intuition or our thought. These objects are called the "elements" of $M$.
In signs we express this thus :

(i) $M = \{m\}$.

We denote the uniting of many aggregates $M$, $N$,
$P$, $\cdots$, which have no common elements, into a
single aggregate by

(2) $(M, N, P, \cdots)$.

The elements of this aggregate are, therefore, the elements of $M$, of $N$, of $P$, . . ., taken together. We will call by the name "part" or "partial aggregate " of an aggregate M any other aggregate $M_1$ whose elements are also elements of $M$. If $M_2$ is a part of $M_1$ and $M_1$ is a part of $M$, then $M_2$ is a part of $M$.

Every aggregate $M$ has a definite "power", which
we will also call its "cardinal number".

We will call by the name "power" or "cardinal number" of $M$ the general concept which, by means
of our active faculty of thought, arises from the aggregate $M$ when we make abstraction of the
nature of its various elements $m$ and of the order
in which they are given.

[482] We denote the result of this double act of abstraction, the cardinal -number or power of M, by
(3) $\overline{\overline{M}}$

Since every single element $m$, if we abstract from its nature, becomes a "unit," the cardinal number $M$ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate $M$.
We say that two aggregates $M$ and $N$ are "equivalent,"
in signs

(4) $M \sim N$ or $N \sim M$

if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element
of the other. To every part $M_1$ of $M$ there corresponds, then, a definite equivalent part $N_1$ of $N$, and inversely.

$[$ ...$]$

Every aggregate is equivalent to itself

(5) $M \sim M$

If two aggregates are equivalent to a third, they are
equivalent to one another ; that is to say :

(6) from $M \sim P$ and $N \sim P$ follows $M \sim N$.

Of fundamental importance is the theorem that two aggregates $M$ and $N$ have the same cardinal number if, and only if, they are equivalent : thus,

(7) from $M \sim N$ we get $\overline{\overline{M}} = \overline{\overline{N}}$,

and

(8) from $\overline{\overline{M}} = \overline{\overline{N}}$ we get $M \sim N$.

Thus the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.

[483] In fact, according to the above definition of power, the cardinal number $M$ remains unaltered if in the place of each of one or many or even all
elements $m$ of $M$ other things are substituted. If, now, $M \sim N$, there is a law of co-ordination by means of which $M$ and $N$ are uniquely and reciprocally
referred to one another; and by it to the
element $m$ of $M$ corresponds the element $n$ of $N$.
Then we can imagine, in the place of every element
$m$ of $M$, the corresponding element $n$ of $N$ substituted,
and, in this way, $M$ transforms into $N$ without
alteration of cardinal number. Consequently

$\overline{\overline{M}} = \overline{\overline{N}}$.

The converse of the theorem results from the remark that between the elements of $M$ and the
different units of its cardinal number $M$ a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, $\overline{\overline{M}}$ grows, so to
speak, out of $M$ in such a way that from every element $w$ of $M$ a special unit of $M$ arises. Thus we can say that

(9) $M \sim \overline{\overline{M}}$

In the same way $N \sim \overline{\overline{N}}$. If then $\overline{\overline{M}} = \overline{\overline{N}}$, we have,
by (6), $M \sim N$.

We will mention the following theorem, which
results immediately from the conception of equival ence. If $M$, $N$, $P$, . . . are aggregates which have no common elements, $M'$, $N'$, $P'$, . . . are also aggregates
with the same property, and if

$M \sim M'$, $N \sim N'$, $P \sim P'$, ...,

then we always have

$(M, N, P, . . .) \sim (M', N', P', \cdots)$.

category: reference


Bekenstein-Hawking entropy

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## Idea

In [[gravity]],  __Bekenstein-Hawking entropy__ is an [[entropy]] assigned to [[black hole]], on the basis of laws of thermodynamics and observers outside black hole.  It striking property is that it is proportional to the surface area of the balck hole's [[horizon]].

## Properties

### Interpretation in AdS/CFT correspondence

In the context of [[string theory]] BH entropy is explained by a version of the [[AdS/CFT correspondence]]. Here every [[black brane]] solution in [[supergravity]] is the strong-coupling limit of a [[D-brane]] [[worldvolume]] [[QFT]]. After [[Kaluza-Klein mechanism|KK-reduction]] these black brane configurations become ordinary [[black hole]]s.  The [[entropy]] of the [[D-brane]] worldvolume theories on the [[event horizon]] turns out to coincide with the BH entropy of the corresponding black hole.

Detailed computations exist in particular for [[D1-brane]]/[[D5-brane]] systems. This is parts of the [[AdS/CFT correspondence]]. See ([AGMOO, chapter 5](#AGMOO)).

See also

* [[holographic entanglement entropy]]

* _[[string theory results applied elsewhere]]_.

### Interpretation by strong coupling limit of D-branes

Another way to derive Bekenstein-Hawking entropy in [[string theory]] is by computing the entropy of weakly coupled open strings on D-brane configurations in flat [[Minkowski space]] which transmute as the coupling constant is increased to given (supersymmetric) black hole configurations. More on this is at _[[black holes in string theory]]_.

## Related concepts

* gravitational entropy

  * [[black hole radiation]]

  * **Bekenstein-Hawking entropy**

  * [[generalized second law of thermodynamics]]

  * [[black holes in string theory]]

  * [[holographic entanglement entropy]]

* [[thermal quantum field theory]]

## References
 {#References}

### General
 {#ReferencesGeneral}

Textbook account:

* {#FrolovZelnikov11} [[Valeri Frolov]], Andrei Zelnikov, _Introduction to black hole physics_, Oxford 2011

Review:

* Dmitri V. Fursaev, *Black Hole Thermodynamics and Perturbative Quantum Gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.06081](https://arxiv.org/abs/2210.06081)]

Review form the point of view of [[thermal field theory]]:

* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), doi:10.1016/0370-1573(87)90136-0)

Basic introductory accounts:

* [[Robert Wald]], _The Thermodynamics of Black Holes_ ([arXiv:gr-qc/9912119](http://arxiv.org/abs/gr-qc/9912119))

* [[Jacob Bekenstein]], _[Bekenstein-Hawking entropy](http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)_, (2008), Scholarpedia, 3(10):7375

and further review:

* S. P. de Alwis, *Comments on Entropy Calculations in Gravitational Systems* [[arXiv:2304.07885](https://arxiv.org/abs/2304.07885)]

A more general discussion which identifies thermodynamic properties of all [[horizons]] appearing on gravity (not just [[black hole]] horizons) was given in

* {#Jacobson95} [[Ted Jacobson]], _Thermodynamics of Spacetime: The Einstein Equation of State_, Phys.Rev.Lett.75:1260-1263, 1995 ([arXiv:gr-qc/9504004](http://arxiv.org/abs/gr-qc/9504004))

This article showed that under some assumptions the [[Einstein equations]] can even be _derived_ from identifying gravitational horizon area with [[entropy]] and them imposing laws of [[thermodynamics]].

For more comments and more references on this observation see

* [[Thanu Padmanabhan]], _Thermodynamical Aspects of Gravity: New insights_, Rep. Prog. Phys. 73 (2010) 046901 ([arXiv:0911.5004](http://arxiv.org/abs/0911.5004))

(Later authors tried to argue that derivations like this show that gravity is not a fundamental force of nature such as [[electromagnetism]] or the [[strong nuclear force]], but rather an [[entropic force]] that arises only from more fundamental forces in a [[thermodynamic limit]]. This however remains at best unclear.)

Discussion of black hole entropy from entropy of [[conformal field theory]] associated with the horizon has been given in

* [[Steve Carlip]], _Entropy from Conformal Field Theory at Killing Horizons_, Class.Quant.Grav.16:3327-3348,1999 ([arXiv:gr-qc/9906126](http://xxx.lanl.gov/abs/gr-qc/9906126))

* [[Steve Carlip]], _Horizon Constraints and Black Hole Entropy_, Class.Quant.Grav.22:1303-1312, 2005 ([arXiv:hep-th/0408123](http://arxiv.org/abs/hep-th/0408123))

and reviewed in

* {#Carlip05} [[Steve Carlip]], _Horizon constraints and black hole entropy_ ([arXiv:gr-qc/0508071](http://arxiv.org/abs/gr-qc/0508071))

* {#Carlip07} [[Steve Carlip]], _Symmetries, Horizons, and Black Hole Entropy_, Gen.Rel.Grav.39:1519-1523,2007; Int.J.Mod.Phys.D17:659-664,2008 ([arXiv:0705.3024](http://arxiv.org/abs/0705.3024))

Further developments on black hole entropy are in

* [[Ashoke Sen]], _Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions_, JHEP04(2013)156 ([arXiv:arXiv:1205.0971](http://arxiv.org/abs/arXiv:1205.0971))

* Aitor Lewkowycz, [[Juan Maldacena]], _Generalized gravitational entropy_ ([arXiv:1304.4926](http://arxiv.org/abs/1304.4926))

A discussion of the "black hole firewall problem":

* [[Ahmed Almheiri]], [[Donald Marolf]], [[Joseph Polchinski]], James Sully, *Black holes: complementarity or firewalls?*, [[arXiv:1207.3123](http://arxiv.org/abs/arXiv:1207.3123)]

### Via Wick-rotated thermal field theory

Discussion via [[Wick rotation]] to [[Euclidean field theory]] on spacetimes with compact/periodic Euclidean time ([[thermal field theory]] on [[curved spacetimes]]) is in

* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), doi:10.1016/0370-1573(87)90136-0)

* {#GibbonsPerry78} [[Gary Gibbons]], Malcolm J. Perry, _Black Holes and Thermal Green Functions_, Vol. 358, No. 1695 (1978) ([jstor:79482](https://www.jstor.org/stable/79482))

### Interpretation in string theory

Microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:

* {#StromingerVafa96} [[Andrew Strominger]], [[Cumrun Vafa]], _Microscopic Origin of the Bekenstein-Hawking Entropy_, Phys. Lett. B379: 99-104, 1996 ([arXiv:hep-th/9601029](http://arxiv.org/abs/hep-th/9601029))

Review of interpretation of [[black holes in string theory]] includes

* {#AGMOO} [[Ofer Aharony]], S. S. Gubser, [[Juan Maldacena]], [[Hirosi Ooguri]], Y. Oz, Chapter 5 of _Large N field theories, string theory and gravity_, [arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111)

* [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_,  Lect. Notes Phys.755:193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))

* {#Sen07} [[Ashoke Sen]], _Black Hole Entropy Function, Attractors and Precision Counting of Microstates_, Gen. Rel. Grav. 40: 2249-2431, 2008 ([arXiv:0708.1270](http://arxiv.org/abs/0708.1270))

* [[Dieter Lüst]], Ward Vleeshouwers, sections 21-22 of _Black Hole Information and Thermodynamics_ ([arXiv:1809.01403](https://arxiv.org/abs/1809.01403))

* Sebastian De Haro, Jeroen van Dongen, Manus Visser, [[Jeremy Butterfield]], _Conceptual Analysis of Black Hole Entropy in String Theory_ ([arXiv:1904.03232](https://arxiv.org/abs/1904.03232))

* Jeroen van Dongen, Sebastian De Haro, Manus Visser, [[Jeremy Butterfield]], _Emergence and Correspondence for String Theory Black Holes_ ([arXiv:1904.03234](https://arxiv.org/abs/1904.03234))

Discussion in view of [[higher curvature corrections]] includes

* [[Thomas Mohaupt]], _Strings, higher curvature corrections, and black holes_ ([arXiv:hep-th/0512048](http://arxiv.org/abs/hep-th/0512048))

See also

* MO question, [statistical-physics-of-string-theory](http://mathoverflow.net/questions/31789/statistical-physics-of-string-theory)

### Interpretation as entanglement entropy

Discussions of the interpreation of BH entropy as [[holographic entanglement entropy]] include

* Alejandro Satz, [[Ted Jacobson]], _Black hole entropy and the renormalization group_ ([arXiv:1301.3171](http://arxiv.org/abs/1301.3171))

Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the [[ABJM theory]] for the [[M2-brane]] is discussed in

* Jun Nian, Xinyu Zhang, _Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole_ ([arXiv:1705.01896](https://arxiv.org/abs/1705.01896))

[[!include entanglement island proposal for black hole paradox -- references]]

### Relation to weight systems on chord diagrams

An argument that [[AdS/CFT duality|dual]] [[observables]] on [[black hole thermodynamics]] are generically given by [[single trace operators]] that evaluate to [[weight systems]] on [[chord diagrams]] (such as observables on [[SYK model]]-like systems)

(for more see at _[[weight systems on chord diagrams in physics]]_):

* [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

[[!redirects black hole entropy]]
[[!redirects BH entropy]]

[[!redirects black hole thermodynamics]]


Bell state

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## Idea

In [[quantum physics]], what are called *Bell states* or *EPR states* or *EPR pairs* are certain [[quantum states]] exhibiting maximal [[quantum entanglement]].

In [[qbit]]-based [[quantum computation]], the elementary Bell state is usually prepared via the following small [[quantum circuit]] consisting of a [[Hadamard gate]] followed by a [[quantum CNOT gate]]:

\begin{imagefromfile}
    "file_name": "BellStatePreparationCircuit-221026.jpg",
    "width": "680",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Related concepts

* [[quantum entanglement]]

* [[quantum information theory]]

* [[quantum teleportation]]

[[!include states and observables -- content]]

## References

The terminology "EPR state" is in honor of [Einstein, Podoldsky & Rosen 1935](EPR+paradox#EinsteinPodoldskyRosen35) and "Bell state" is in honor of [Bell 1964](Bell's+inequalities#Bell64).

Textbook accounts:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §1.3.2 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

In [[string diagram]]-calculus ([[finite quantum mechanics in terms of dagger-compact categories]]):

* {#Coecke10} [[Bob Coecke]], §4.2 of: *Quantum Picturalism*, Contemporary Physics **51** (2010) 59-83 [[arXiv:0908.1787](https://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]

See also:

* Wikipedia, *[Bell state](https://en.wikipedia.org/wiki/Bell_state)*

[[!redirects Bell states]]