interacting vacuum

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#### Vacua
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#### Algebraic Qunantum Field Theory
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#Contents#
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## Idea

A [[vacuum state]] for an [[interacting field theory]] is sometimes called, for emphasis, an _interacting vacuum_.

In [[perturbative quantum field theory]] one starts with a [[free field theory]] and a [[vacuum state]] (or [[Hadamard vacuum state]]) on that, the "free vacuum"; and then finds, perturbatively, the [[interacting field algebra of observables]]. The free vacuum is typically still a [[state on a star-algebra|state]] on the [[interacting field algebra of observables]], but it is in general no longer a _vacuum_ state with respect to the interacting field algebra. To highlight the difference to an actual vacuum state for the interacting field algebra, one speaks of the _interacting vacuum_ (e.g. [Rafelski 90](#Rafelski90)).

Issues related to the [[adiabatic limit]] and [[infrared divergences]] in [[perturbative QFT]] are argued to be related to the need to pass to the correct interacting vacuum ([Duch 17, p. 113-114](#Duch17)).

This is understood to some extent in [[quantum electrodynamics]] but remains a major open problem in [[quantum chromodynamics]] ([below](#InQCD)).

## Examples

### In QCD
 {#InQCD}

There are arguments that the phenomenon of [[confinement]] in [[QCD]] will be explained by finding the properly interacting vacuum of the perturbative theory ([Rafelski 90, around pages 12-16](#Rafelski90)). Possibly this is a "[[theta-vacuum]]" reflecting [[instanton in QCD|QCD instanton]] contributions ([Schäfer-Shuryak 98, section III D](#SchaeferShuryak98)).

## Related concepts

* [[non-perturbative effect]]

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

## References

* {#Rafelski90} [[Johann Rafelski]], _Vacuum structure -- An Essay_, in pages 1-29 of H. Fried, Berndt Müller (eds.) _Vacuum Structure in Intense Fields_, Plenum Press 1990 ([GBooks](https://books.google.de/books?id=5uXcBwAAQBAJ&pg=PA14&lpg=PA14&dq=confinement+%22interacting+vacuum%22&source=bl&ots=xPGYJ-JOc-&sig=AoYbqWQeNRMg6hMSRZjJ3nzq8B0&hl=en&sa=X&ved=0ahUKEwjIgK68-tnYAhVCESwKHThMDykQ6AEIKTAA#v=onepage&q=confinement%20%22interacting%20vacuum%22&f=false))

* {#SchaeferShuryak98} [[Thomas Schaefer]], [[Edward Shuryak]], _Instantons in QCD_, Rev. Mod. Phys.70:323-426,1998 ([arXiv:hep-ph/9610451](http://arxiv.org/abs/hep-ph/9610451))

* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))

[[!redirects interacting vacua]]

[[!redirects interacting vacuum state]]
[[!redirects interacting vacuum states]]


interaction

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#### Physics
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#Contents#
* table of contents
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## Idea

Given a [[local Lagrangian density]] $L$ defining the [[action functional]] of a [[local field theory]], typically it splits as a sum

$$
  L = L_{free} + L_{int}
$$

of a term $L_{free}$ that defines a [[free field theory]] ([[kinetic energy]]) and a remainder $L_{int}$ defining the interactions of a corresponding [[interacting field theory]]. That remainder then describes the _interactions_ that the otherwise free fields undergo.

The _[[interaction picture]]_ of [[quantum physics]] serves to decompose the [[quantization]] of [[local field theories]] such that the interaction is seen as a [[perturbation]] of the quantization of the [[free field theory]]. This underlies most constructions of [[perturbative quantum field theory]], including the mathematically rigorous formulation via [[causal perturbation theory]].

## Examples

* [[phi^n interaction]]

* [[electron-photon interaction]]

*  The [[holonomy]] of a [[circle bundle with connection]] gives the _[[gauge theory|gauge]] interaction_ term for a [[particle]] [[sigma-model]] charged under an [[electromagnetic field]] given by a [[line bundle with connection]].

* More generally the [[volume holonomy]] of a [[circle n-bundle with connection]] gives the _[[higher gauge theory|higher gauge]]_ interaction of an $(n-1)$-[[brane]] [[sigma model]], for instance the [[string]] sigma-model charged under the [[B-field]] or the [[membrane]] sigma-model charged under the [[supergravity C-field]].

## Related concepts

* [[renormalizable interaction]]

* [[interacting field theory]]

* [[interacting field algebra]]

* [[perturbative quantum field theory]]

* [[scattering]], [[scattering matrix]]

* [[hidden sector]]

[[!include action (physics) - table]]

[[!redirects interactions]]

[[!redirects interaction term]]
[[!redirects interaction terms]]

[[!redirects gauge interaction]]


interaction vertex redefinition

[[!redirects vertex redefinitions]]

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#Contents#
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## Idea

In [[perturbative quantum field theory]] an ([[adiabatic switching|adiabatically switch]]) [[interaction]] [[action functional]] is a [[local observable]] of the form

$$
  g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
  \,.
$$

In the [[Feynman perturbation series]] this is interpreted as the label of [[vertices]] in [[multigraphs]] underlying [[Feynman diagrams]] which encode [[field (physics)|field]] [[interactions]] at that point.

In the process of [[renormalization|(re-)normalization]] of [[perturbative QFT]] one considers redefinitions of such "interaction vertices" by terms of higher order in the [[coupling constant]] $g$  ad [[source field]] $j$:

$$
  \mathcal{Z}
  \;\colon\;
  g S_{int} + j A
  \;\mapsto\;
  g S_{int} + j A
  +
  \underset{
    \mathcal{O}(g^2, j^2, g j)
  }{
  \underbrace{
    S_{counter}
  }}
  \,.
$$

In the context of [[effective QFT]] via [[UV cutoffs]], the correction term $S_{counter}$ is called a _[[counterterm]]_.

Under mild assumptions, such _interaction vertex redefinitions_ form a [[group]], called the _[[Stückelberg-Petermann renormalization group]]_. See there for more.

## Definition

+-- {: .num_defn #InteractionVertexRedefinition}
###### Definition
**([[perturbative interaction vertex redefinition]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] ([this def.](S-matrix#VacuumFree)).

A _[[perturbative interaction vertex redefinition]]_ (or just _[[vertex redefinition]]_, for short) is an [[endofunction]]

$$
  \mathcal{Z}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$

on [[local observables]] with formal parameters adjoined ([this def.](S-matrix#FormalParameters)) such that there exists a sequence $\{Z_k\}_{k \in \mathbb{N}}$ of [[continuous linear functionals]], symmetric in their arguments, of the form

$$
  \left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}}
  \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$

such that for all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$ the following conditions hold:

1. (perturbation)

   1. $Z_0(g S_{int + j A}) = 0$

   1. $Z_1(g S_{int} + j A) = g S_{int} + j A$

   1. and

      $$
        \begin{aligned}
          \mathcal{Z}(g S_{int} + j A)
          & =
          Z \exp_\otimes( g S_{int} + j A )
          \\
          &
          \coloneqq
          \underset{k \in \mathbb{N}}{\sum}
          \frac{1}{k!}
          Z_k(
            \underset{
              k \, \text{args}
            }{
              \underbrace{
                g S_{int} + j A , \cdots, g S_{int} + j A
              }
            }
          )
        \end{aligned}
      $$

1. (field independence) The [[local observable]] $\mathcal{Z}(g S_{int} + j A)$ depends on the [[field histories]] only through its argument $g S_{int} + j A $, hence by the [[chain rule]]:

   $$
     \label{FieldIndependenceVertexRedefinition}
     \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
     \mathcal{Z}(g S_{int} + j A)
     \;=\;
     \mathcal{Z}'_{g S_{int} + j A}
     \left(
         \frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A)
     \right)
   $$

=--

## Properties

The following proposition should be compared to the axiom of _[[causal additivity]]_ of the [[S-matrix]] scheme ([this equation](S-matrix#eq:CausalAdditivity)):

+-- {: .num_prop #InteractionVertexRedefinitionAdditivity}
###### Proposition
**(local additivity of [[vertex redefinitions]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] ([this def.](S-matrix#VacuumFree)) and let $\mathcal{Z}$ be a [[vertex redefinition]] (def. \ref{InteractionVertexRedefinition}).

Then for all [[local observables]] $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle$ with spacetime support denoted $supp(O_i) \subset \Sigma$ ([this def.](A+first+idea+of+quantum+field+theory#SpacetimeSupport)) we have

1. (local additivity)

   $$
     \begin{aligned}
       &
       \left(
         supp(O_1)
         \cap
         supp(O_2)
         =
         \emptyset
       \right)
       \\
       &
         \Rightarrow
       \phantom{AA}
       \mathcal{Z}( O_0 + O_1 + O_2)
       =
       \mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2)
     \end{aligned}
     \,.
   $$

1. (preservation of spacetime support)

   $$
     supp
     \left(
       {\, \atop \,}
         \mathcal{Z}(O_0 + O_1)
         -
         \mathcal{Z}(O_0)
       {\, \atop \,}
     \right)
     \;\subset\;
     supp(O_1)
   $$

   hence in particular

   $$
     supp
     \left(
       {\, \atop \,}
       \mathcal{Z}(O_1)
       {\, \atop \,}
     \right) = supp(O_1)
   $$

=--

([Dütsch 18, exercise 3.98](#Duetsch18))

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

Under the inclusion

$$
  LocObs(E_{\text{BV-BRST}})
    \hookrightarrow
  PolyObs(E_{\text{BV-BRST}})
$$

of [[local observables]] into [[polynomial observables]] we may think of each $Z_k$ as a [[generalized function]], as for [[time-ordered products]] in [this remark](S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions).

Hence if

$$
  O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j )
$$

is the [[transgression of variational differential forms|transgression]] of a [[Lagrangian density]] $\mathbf{L}$ we get

$$
  Z_k(
    (O_1 + O_2 + O_3)
    ,
    \cdots
    ,
    (O_1 + O_2 + O_3)
  )
  =
  \underset{
    j_1, \cdots, j_k \in \{0,1,2\}
  }{\sum}
  \underset{\Sigma^{k}}{\int}
    Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
  \,.
$$

Now by definition $Z_k(\cdots)$ is in the subspace of [[local observables]], i.e. those [[polynomial observables]] whose [[coefficient]] [[distributions]] are [[support of a distribution|supported]] on the [[diagonal]], which means that

$$
  \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
  \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
  Z_{k}(\cdots)
  =
  0
  \phantom{AA}
  \text{for}
  \phantom{AA}
  x \neq y
$$

Together with the axiom "field independence" (eq:FieldIndependenceVertexRedefinition) this means that the support of these generalized functions in the [[integrand]] here must be on the [[diagonal]], where $x_1 = \cdots = x_k$.

By the assumption that the spacetime supports of $O_1$ and $O_2$ are disjoint, this means that only the summands with $j_1, \cdots, j_k \in \{0,1\}$ and those with $j_1, \cdots, j_k \in \{0,2\}$ contribute to the above sum. Removing the overcounting of those summands where all $j_1, \cdots, j_k \in \{0\}$ we get

$$
  \begin{aligned}
    & Z_k\left(
      {\, \atop \,}
      (O_1 + O_2 + O_3)
      ,
      \cdots
      ,
      (O_1 + O_2 + O_3)
      {\, \atop \,}
    \right)
    \\
    & =
    \underset{
      j_1, \cdots, j_k \in \{0,1\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & \phantom{=}
    -
    \underset{
      j_1, \cdots, j_k \in \{0\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & \phantom{=}
    -
    \underset{
      j_1, \cdots, j_k \in \{0,2\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & =
    Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right)
    -
    Z_k\left( {\, \atop \,} O_0, \cdots,  O_0 {\, \atop \,} \right)
    +
    Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right)
  \end{aligned}
  \,.
$$

This directly implies the claim.

=--

## References

The original discussion is due to

* {#StueckelbergPetermann53} [[Ernst Stückelberg]], [[André Petermann]], _La normalisation des constantes dans la theorie des quanta_, Helv. Phys. Acta 26 (1953), 499–520

in the context of the [[Stückelberg-Petermann renormalization group]].

For more see the references there and those at _[[main theorem of perturbative renormalization]]_.

[[!redirects perturbative interaction vertex redefinition]]
[[!redirects perturbative interaction vertex redefinitions]]

[[!redirects interaction vertex redefinition]]
[[!redirects interaction vertex redefinitions]]

[[!redirects vertex redefinition]]
[[!redirects vertex redefinitions]]


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#### Type theory
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#Contents#
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{:toc}

## Introduction

There is a battery of little [[set theory|set-theoretic]] [[lemmas]] which inevitably recur when laying the [[foundations]] of core [[mathematics]], as when teaching the standard curriculum that includes basic [[topology]], real [[analysis]], [[algebra]], and so forth. In this article we collect some of those core results having to do with the properties of [[image]] and [[pre-image]] operators, especially regarding their preservation (or not) of set-theoretic operations like [[unions]] and [[intersections]].

The attitude and approach of the mathematics professor toward such routine bread-and-butter issues is a matter of some interest. A professional, upon being presented with any one of these "naive set theory" [[propositions]], will probably be able to verify it on the spot using ordinary follow-your-nose logic, driven by the definitions. While being able to produce such verifications is part of basic training in mathematics, it is not quite the same as giving an *explanation*, and in fact even good mathematicians trained this way may struggle to remember "now which is it that direct images preserve, unions or intersections?"[^1]

[^1]: There is a famous story of how [[Mikhail Yakovlevich Suslin|Suslin]] discovered an error in an argument of [[Lebesgue]], concerning just this type of routine set-theoretic detail (that Lebesgue perhaps just misremembered). In response, [[Mikhail Yakovlevich Suslin|Suslin]] was led to develop some of the basics of [[descriptive set theory]]. Details of the story have been told by Dave Renfro, [here](http://mathforum.org/kb/message.jspa?messageID=4966721) and [here](http://mathforum.org/kb/message.jspa?messageID=4967733)

A strong pedagogy would not only instill this sort of basic training, but make the battery of routine results more memorable by concentrating their essence in one or two basic principles that provide an explanatory basis for the rest. According to [Lawvere 69](#Lawvere69), [Lawvere 05](#Lawvere05), logic is an interlocking system of certain types of [[adjoint functors]], and we believe putting those [[adjunction|adjoint]] relationships front and center and seeing how the rest flows from them is an effective way to arrange the facts.

## Statements

+-- {: .num_prop #PreservationOfUnionsAndIntersectionsOfSets}
###### Proposition
**([[images]] preserve [[unions]] but not in general [[intersections]])**

Let $f \colon X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $\{S_i \subset X\}_{i \in I}$ be an [[indexed set]] of [[subsets]] of $X$. Then

1. $im_f\left( \underset{i \in I}{\cup}  S_i\right) = \left(\underset{i \in I}{\cup} im_f(S_i)\right)$ (the [[image]] under $f$ of a [[union]] of subsets is the union of the images)

1. $im_f\left( \underset{i \in I}{\cap}  S_i\right) \subset \left(\underset{i \in I}{\cap} im_f(S_i)\right)$ (the [[image]] under $f$ of the [[intersection]] of the subsets is contained in the intersection of the images).

The [[injective function|injection]] in the second item is in general proper. If $f$ is an [[injective function]] then this is a [[bijection]]:

* $(f\,\text{injective}) \Rightarrow \left(im_f\left( \underset{i \in I}{\cap}  S_i\right) = \left(\underset{i \in I}{\cap} im_f(S_i)\right)\right)$

*provided* that $I$ is [[inhabited set|inhabited]].

=--

+-- {: .num_prop #PreImagePreservesUnionsAndIntersections}
###### Proposition
**(pre-images preserve unions and intersections)**

Let $f \colon X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $\{T_i \subset Y\}_{i \in I}$ be an [[indexed set]] of [[subsets]] of $Y$. Then

1. $f^{-1}\left( \underset{i \in I}{\cup}  T_i\right) = \left(\underset{i \in I}{\cup} f^{-1}(T_i)\right)$ (the [[pre-image]] under $f$ of a [[union]] of subsets is the union of the pre-images),

1. $f^{-1}\left( \underset{i \in I}{\cap}  T_i\right) = \left(\underset{i \in I}{\cap} f^{-1}(T_i)\right)$ (the [[pre-image]] under $f$ of the [[intersection]] of the subsets is the intersection of the pre-images).
=--

+-- {: .num_prop}
###### Proposition
**([[projection formula]])**
Let $f: X \longrightarrow Y$ be a [[function]] between [[sets]]. Let $S \subset X$ be a [[subset]] of $X$, and $T \subseteq Y$ be a subset of $Y$. Then

$$f(S) \cap T = f(S \cap f^{-1}(T)).$$
=--

## Proofs via adjoints

The properties 1., 2. of Proposition \ref{PreservationOfUnionsAndIntersectionsOfSets} may be proved by appeal to fundamental relationships between [[direct image]] and [[inverse image]] and the like, which [[category theory|category theorists]] call [[adjunctions]] (similar in form to adjoints in linear algebra). The advantage of this type of proof is that, despite its utter simplicity, it generalizes to much wider contexts (beyond elementary classical set theory).

### Direct images preserve unions, inverse images preserve intersections

The first such relationship is:

\begin{proposition}\label{PreimageRightAdjointToImage}
**([[preimage]] is [[right adjoint]] to [[image]])**
\linebreak
For all [[subsets]] $S \subseteq X$ and $T \subseteq Y$, we have

$$
  f(S) \subseteq T \qquad iff \qquad S \subseteq f^{-1}(T)
  \,.
$$
\end{proposition}

One says in this situation that the direct [[image]] mapping $f(-)$ is *[[left adjoint]]* to the [[preimage|inverse image]] mapping $f^{-1}(-)$ (with respect to the [[posets]] of subsets, regarded as [[(0,1)-categories]]), or that inverse image is *[[right adjoint]]* to direct image. This terminology is guided by the formally similar usage in [[linear algebra]] where [[linear mappings]] $A$ and $B$ are adjoint if one has an equation

$$\langle A x, y \rangle = \langle x, B y \rangle$$

for all $x, y$ in suitable [[inner product spaces]] (and in fact the analogy is not idle; see for instance [Baez 1997](#JB97)).

In that case, we have the following elementary inferences:

$$\array{
\bigcup_{i \in I} f(S_i) \subseteq T & iff & (\forall_{i \in I})\; f(S_i) \subseteq T \\
 & iff & (\forall_{i \in I})\; S_i \subseteq f^{-1}(T) \\
 & iff & \bigcup_{i \in I} S_i \subseteq f^{-1}(T) \\
 & iff & f\left(\bigcup_{i \in I} S_i\right) \subseteq T
}$$

where the first and third lines use the defining [[universal property|property]] of unions. Then, putting $T = \bigcup_{i \in I} f(S_i)$ and reasoning forward, we get $f\left(\bigcup_{i \in I} S_i\right) \subseteq \bigcup_{i \in I} f(S_i)$. On the other hand, putting $T = f\left(\bigcup_{i \in I} S_i\right)$ and reasoning backward, we get $\bigcup_{i \in I} f(S_i) \subseteq f\left(\bigcup_{i \in I} S_i\right)$. These two inclusions together give $\bigcup_{i \in I} f(S_i) = f\left(\bigcup_{i \in I} S_i\right)$.

The [[duality|dual]] of this proof immediately gives the fact that inverse images preserve arbitrary intersections, but let us spell it out directly:

$$\array{
S \subseteq \bigcap_{i \in I} f^{-1}(T_i) & iff & (\forall_{i \in I})\; S \subseteq f^{-1}(T_i) \\
 & iff & (\forall_{i \in I})\; f(S) \subseteq T_i \\
 & iff & f(S) \subseteq \bigcap_{i \in I} T_i \\
 & iff & S \subseteq f^{-1}\left(\bigcap_{i \in I} T_i \right)
}$$

and again by using the reasoning forward/backward trick, we infer $\bigcap_{i \in I} f^{-1}(T_i) = f^{-1}\left(\bigcap_{i \in I} T_i\right)$.

+-- {: .num_remark}
###### Remark
The "reasoning forward/backward trick" may be summarized by saying that in a [[poset]], we have $A = B$ iff ($A \leq T \Leftrightarrow B \leq T$), or dually that $A = B$ iff ($S \leq A \Leftrightarrow S \leq B$). This trick is vastly extrapolated by the [[Yoneda lemma]].
=--

+-- {: .num_remark #Lawvere}
###### Remark
People who work with categorial forms of logic denote direct images $f(S)$ by $\exists_f(S)$. The suggestion is to view [[existential quantifier|existential quantification]] as corresponding to taking of a [[direct image]]. For example, given a property $P$ of pairs $(x, y)$, i.e. a subset $P \subseteq X \times Y$ where we write $P(x, y)$ to say $(x, y) \in P$ holds, the set of $y$ such that $(\exists_{x \in X})\; P(x, y)$ holds is exactly the direct image $\pi_2(P)$ under the [[projection map]] $\pi_2: X \times Y \to Y$. This suggests furthermore a useful extended meaning of existential quantification, by considering direct images along more general maps, not just projection maps.  With this in mind, category theorists often denote $f(S) = \{y \in Y: (\exists_{x: f(x) = y})\; x \in S\}$ by $\exists_f(S)$, giving an operator $\exists_f: P(X) \to P(Y)$ between [[power sets]]. This is [[left adjoint]] to the taking of [[inverse images]] $T \mapsto f^{-1}(T)$ as an operator $f^{-1}: P(Y) \to P(X)$, also denoted by $f^\ast: P(Y) \to P(X)$. The adjointness relationship between direct image and inverse image, denoted by $\exists_f \dashv f^\ast$, is an example of a famous slogan due to [[Lawvere]]: "Logical quantification is adjoint to substitution" (with resonances far beyond the purview of logic as ordinarily conceived; see Remark \ref{MoreJargon}).
=--

As for direct images and intersections, we have

$$f\left(\bigcap_{i \in I} S_i\right) \subseteq \bigcap_{i \in I} f(S_i)$$

because this is equivalent to $(\forall_{i \in I})\; f\left(\bigcap_{i \in I} S_i\right) \subseteq f(S_i)$, which is obvious because for all $i \in I$, we have $\bigcap_{i \in I} S_i \subseteq S_i$, and $f(A) \subseteq f(B)$ if $A \subseteq B$.

This inclusion is not usually an equality. For example, in the [[cartesian space]] $\mathbb{R}^2$ with $x y$-coordinates, the [[projection]] $(x, y) \mapsto x$ does not preserve the intersection of the lines $y = 1$ and $y = 2$.

However, if $f: X \to Y$ is injective, then the direct image operator $S \mapsto f(S)$ *does* preserve intersections of subsets $(S_i)_{i \in I}$ indexed over *nonempty* sets $I$. This is not difficult to check once we verify that

* For any map $f: X \to Y$ (not necessarily injective) and any subset $T \subseteq Y$, we have $f(X) \cap T = f(f^{-1}(T))$.

* If $f$ is injective, then for any subset $S \subseteq X$ we have $S = f^{-1}(f(S))$.

For then, if $T = \bigcap_{i \in I} f(S_i)$ and $I$ is inhabited, we have for some $i$ that $T \subseteq f(S_i) \subseteq f(X)$, and then

$$\array{
T = f(X) \cap T & = & f(f^{-1}(T)) \\
 & = & f(f^{-1}(\bigcap_i f(S_i))) \\
 & = & f\left(\bigcap_{i \in I} f^{-1}(f(S_i))\right) \\
 & = & f\left(\bigcap_{i \in I} S_i\right)
}$$

which was to be shown.

### Inverse images preserve unions, codirect images preserve intersections

As for the statement that inverse images of unions are unions of inverse images, it turns out this follows from a second fundamental logical adjunction. For a function $f: X \to Y$ and a subset $U \subseteq X$, define

$$\forall_f(U) = \{y \in Y: (\forall_{x: f(x) = y})\; x \in U\}$$

which is analogous to the formula $\exists_f(S) = f(S) = \{y \in Y: (\exists_{x: f(x) = y})\; x \in S\}$ in Remark \ref{Lawvere}. (Alternatively, in [[classical logic]] where the [[negation]] operator $\neg$ on power sets means complementation, we have $\forall_f = \neg \exists_f \neg$.) Then it is easy to verify a second adjunction

$$f^{-1}(T) \subseteq U \qquad iff \qquad T \subseteq \forall_f (U)$$

The same adjointness proof as used to prove property 1., then shows that the left adjoint part which here is the inverse image operator $f^{-1} = f^\ast: P(Y) \to P(X)$ preserves unions, and that right adjoint part which here is the "codirect image" operator $\forall_f: P(X) \to P(Y)$ preserves intersections.

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

In terms of some more [[category theory]] jargon the above situation may be phrased as follows:

Subsets of some set $S$ are the [[(-1)-truncated objects]] in the [[slice category]] $Set_{/S}$ of [[Set]] over $S$. Since [[Set]] is a [[topos]], every [[morphism]], i.e. [[function]] $X \longrightarrow Y$ induces a [[base change]] [[adjoint triple]]

$$
  Set_{/X}
     \underoverset
       {\overset{f_\ast}{\longrightarrow}}
       {\overset{f_!}{\longrightarrow}}
       {\overset{f^\ast}{\longleftarrow}}
  Set_{/Y}
  \,.
$$

Here in the language of [[type theory]]

1. the [[left adjoint]] $f_!$ is also called the [[dependent sum]]

1. the middle morphism is also called [[context extension]]

1. the [[right adjoint]] $f_\ast$ is also called the [[dependent product]].

All this restricts to the (-1)-truncated objects by composing the left adjoint with [[(-1)-truncation]]. Then one also says that we are looking at a morphism in a [[hyperdoctrine]] and (under "[[propositions as types]]") may think of

1. the left adjoint $f_!$ as the [[existential quantifier]] $\exists_f$

1. the right adjoint $f_\ast$ as the [[universal quantifier]] $\forall_f$.

Now the fact that [[left adjoint]] $\exists_f$ preserves [[unions]] is the fact that [[left adjoints preserve colimits]], and the fact that the left- and [[right adjoint]] $f^\ast$ preserves unions and intersections is that in addition [[right adjoints preserve limits]].

=--

## Projection formula

Another useful formula for interactions between images and pre-images and intersections is the [[projection formula]]:

\begin{proposition}
\label{ProjectionFormulaForFunctionsAndInverseImages}
Let $f \,\colon\, X \to Y$ be a [[function]], and let $S \subseteq X,\; T \subseteq Y$ be [[subsets]]. Then

$$
  f\big(
    S \cap f^{-1}(T)
  \big)
  \;=\;
  f(S) \cap T
  \,.
$$

\end{proposition}
(e.g. [Lee 2000, Ex. A.4(k)](#Lee00))
\begin{proof}
First observe that we have an inclusion$f(S \cap f^{-1}(T)) \subseteq f(S) \cap T$:

By the [[universal property|defining property]] of [[intersections]], it suffices to show (i) $f\big(S \cap f^{-1}(T)\big) \subseteq f(S)$ and (ii) $f\big(S \cap f^{-1}(T)\big) \subset T$. Here (i) follows since $S \cap f^{-1}(T) \subseteq S$, and $A \subseteq B$ implies $f(A) \subseteq f(B)$. and (ii) is seen as

$$
  f\big(
    S \cap f^{-1}(T)
  \big)
  \;\subseteq\;
  f\big(
    f^{-1}(T)
  \big)
  \;\subseteq\;
  T
  \,,
$$

where the first step is the functoriality of images with respect to subset inclusions, and the second step is the [[adjunct]] of $f^{-1}(T) \subseteq f^{-1}(T)$ with respect to the [[adjunction]] between direct and inverse image.

Now to see that this inclusion is actually an equality...
\end{proof}

## Beck-Chevalley condition

The two ways of consecutively forming [[images]] and [[preimages]] along maps that form a [[pullback diagram]] coincide according to Prop. \ref{BeckChevalleyForPreImages}  below -- a simple example of what is known as a "[[Beck-Chevalley condition]]".

First to recall our notation: As above, for a [[set]] $S \,\in\, Set$, we 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 follows 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]].

As discussed above, these three functors constitute an [[adjoint triple]]

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

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]].

\begin{proposition}\label{BeckChevalleyForPreImages}
**([[Beck-Chevalley condition]] for pre-/images)**
\linebreak
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 consecutively forming (pre)images along these maps satisfies a *[[Beck-Chevalley condition]]* in that the following [[diagram]] of [[poset]]-[[homomorphisms]] ([[functors]]) [[commuting diagram|commutes]]:

   \[
     \label{RightBeckChevalleyDiagramForPreImages}
     \array{
       \mathcal{P}(D')
         &\overset{\beta_!}{\longrightarrow}&
       \mathcal{P}(D)
       \\
       \mathllap{{}^{\psi^{-1}}}\big\uparrow
       &&
       \big\uparrow\mathrlap{{}^{\phi^{-1}}}
       \\
       \mathcal{P}(C')
         &\underset{\alpha_!}{\longrightarrow}&
       \mathcal{P}(C)
     }
   \]
\end{proposition}
\begin{proof}
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{proof}

## Related statements

* [[countable unions of countable sets are countable]]

* [[limits and colimits by example]]

## References

Textbook accounts include:

* [[Tom Apostol]], [p. 44](http://www.ru.ac.bd/wp-content/uploads/sites/25/2019/03/205_04_Apostol-Mathematical-Analysis-1973.pdf#page=57) in: *Mathematical Analysis* 1973 ([pdf](http://www.ru.ac.bd/wp-content/uploads/sites/25/2019/03/205_04_Apostol-Mathematical-Analysis-1973.pdf))

* {#Lee00} [[John M. Lee]], Exercise A.4 in: _Introduction to topological manifolds_. Graduate Texts in Mathematics 202 (2000), Springer.  ISBN: 0-387-98759-2, 0-387-95026-5.
Second edition: Springer, 2011.  ISBN: 978-1-4419-7939-1 ([doi:10.1007/978-1-4419-7940-7](https://doi.org/10.1007/978-1-4419-7940-7), errata [pdf](https://sites.math.washington.edu/~lee/Books/ITM/errata.pdf))

The slogan of Lawvere on logic and adjoint functors appears in

* {#Lawvere05} [[F. William Lawvere]], _An elementary theory of the category of sets_, Proceedings of the National Academy of Science of the U.S.A 52 (1965), 1506-1511. Reprinted in an expanded version, with commentary by Colin McLarty and by Lawvere, in Reprints in Theory and Applications of Categories, No. 11 (2005), pp. 1-35. ([link](http://www.tac.mta.ca/tac/reprints/articles/11/tr11abs.html))

based on

* {#Lawvere69} [[William Lawvere]], _[[Adjointness in Foundations]]_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

The story of Lebesgue's slip that projections commute with intersections, in attempting to prove that [[projections]] of [[Borel sets]] are Borel, is well explained here:

* Dave L. Renfro, *Mikhail Y. Suslin and Lebesgue's error*, two Mathforum posts dated July 29, 2006 (first post [here](http://mathforum.org/kb/message.jspa?messageID=4966721)).

The analogy between adjoint functors and adjoints in linear algebra becomes very strong when jacking up from Hilbert spaces to [[2-Hilbert spaces]]. See

* {#JB97} [[John C. Baez]], *Higher-Dimensional Algebra II. 2-Hilbert Spaces*, Advances in Mathematics, Volume 127 Issue 2 (10 May 1997), 125-189. [https://doi.org/10.1006/aima.1997.1617](https://doi.org/10.1006/aima.1997.1617) ([Elsevier link](http://www.sciencedirect.com/science/article/pii/S0001870897916170))

[[!redirects images preserve unions]]

[[!redirects the image of an intersection is contained in the intersection of the images]]
[[!redirects images of unions are unions of images]]


intercategory

# Intercategories

* table of contents
{:toc}

## Idea

An *intercategory* is a particular sort of [[triple category]] that is pseudo in some ways and lax in others.  Many different kinds of higher-categorical structures can be viewed as, or give rise to, intercategories.  In particular, an intercategory can be viewed as a [[double category]] with "extra strictness/weakness information" given by the third direction.

## Definition

The data of an intercategory is like that of a triple category: it has

* objects
* three kinds of arrows, called *transversal*, *vertical*, and *horizontal*
* three kinds of square 2-cells, called *vertical* (those having transversal and vertical arrows on their boundaries), *horizontal* (those having transversal and horizontal arrows on their boundaries), and *basic* (those having horizontal and vertical arrows on their boundaries)
* cubes

All three kinds of arrows can be composed in the obvious way, all three kinds of 2-cell can be composed like in a double category, and cubes can be composed like in a triple category.  Composition of transversal arrows, and transversal composition of horizontal and vertical squares and cubes, is strictly associative and unital.  Horizontal and vertical composition is associative and unital up to coherent transversal isomorphism, as in a bicategory.  In particular, the vertical cells and the horizontal cells each form the cells in a pseudo double category of the usual sort.  Finally, composition of basic cells in the vertical and horizontal directions only satisfies the "interchange law" laxly (up to a not-necessarily-invertible comparison cell).

For a precise definition, see the references.

## Examples

* Of course, any strict [[triple category]] can be regarded as an intercategory.

* A Verity [[double bicategory]] can be regarded as an intercategory in which the only transversal arrows are identities, and the horizontal and vertical cells have basic [[companions]] and [[conjoints]].  This includes in particular the case of [[quintets]] in a [[bicategory]].

* A [[locally cubical bicategory]] (a bicategory enriched over pseudo double categories) can be regarded as an intercategory with only identity transversal arrows, vertical arrows, and vertical cells.  The horizontal arrows are the objects of the hom-double-categories, the horizontal and basic cells are their arrows, and the cubes are their squares.  This includes in particular [[monoidal double categories]] as the special case where there is only one object.

* A [[duoidal category]] can be regarded as an intercategory with one object, one arrow of each sort, and one vertical and one horizontal cell.  The objects of the duoidal category are the basic cells and its morphisms are the cubes; the two tensor products are the horizontal and vertical compositions.

* A [[Gray-category]] gives rise to an intercategory of [[quintets]], whose objects are the objects of the Gray-category, only identity transversal arrows, vertical cells, and horizontal cells, whose horizontal *and* vertical arrows are those of the Gray-category, whose basic cells are "quintet" 2-cells in the Gray-category, and whose cubes are the 3-cells.  Because interchange is lax, this includes even categories enriched over Gray's lax" tensor product, in addition to the "pseudo" one that is usually used to define Gray-categories.

## References

* [[Marco Grandis]], [[Robert Paré]], *Intercategories*, [arXiv](http://arxiv.org/abs/1412.0144), [TAC](http://tac.mta.ca/tac/volumes/30/38/30-38abs.html)

* [[Marco Grandis]], [[Robert Paré]], *Intercategories: A framework for three-dimensional category theory*, [doi](https://doi.org/10.1016/j.jpaa.2016.08.002), [arXiv](http://arxiv.org/abs/1412.0212)

[[!redirects intercategories]]


interior

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

# Contents
* table of contents
{:toc}

## Definition

### For topological spaces

For $S \subset X$ a [[subset]] of a [[topological space]] $X$, a, **interior point** of $S$ is a point $x \in S$ which has a [[neighbourhood]] in $X$ that is contained in $S$. The union of all interior points is the **interior** $S^\circ$ of $S$. It can be defined as the largest open set contained in $S$.

In general, we have $S^\circ \subseteq S$.  $S$ is [[open subset|open]] if and only if $S^\circ = S$, so that in particular $S^{\circ\circ} = S^\circ$. This makes the interior operator $P(X) \to P(X): S \mapsto S^\circ$ a [[co-closure operator]]. It also satisfies the equations $(S \cap T)^\circ = S^\circ \cap T^\circ$ and $X^\circ = X$. Moreover, any co-closure operator $c$ on $P(X)$ that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of $c$; this gives one of many equivalent ways to define a topological space.

Compare the [[topological closure]] $\bar{S}$ and [[frontier]] $\partial S = \bar{S} \setminus S^\circ$.

### For toposes

The _interior of a subtopos_ $\mathcal{E}_j$ of a [[Grothendieck topos]] $\mathcal{E}$, as well as the [[dense subtopos|exterior]], were defined in an exercise in [SGA4](#SGA4): $Int(\mathcal{E}_j)$ as the largest [[open subtopos]] contained in $\mathcal{E}_j$. The boundary of a subtopos is then naturally defined as the subtopos complementary to the (open) join of the exterior and interior subtoposes in the [[lattice of subtoposes]].

## Properties

### Relation to topological closure

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

Let $(X,\tau)$ be a [[topological space]] and let $S \subset X$ be a [[subset]]. Then the topological interior of $S$ equals the [[complement]] of the [[topological closure]] $Cl(X\backslash S)$ of the complement of $S$:

$$
  Int(S)
    =
  X \backslash Cl\left(
    X \backslash S
  \right)
  \,.
$$

=--

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

By taking [[complements]] once more, the statement is equivalent to

$$
  X \backslash Int(S)
  =
  Cl( X \backslash S )
  \,.
$$

Now we compute:

$$
  \begin{aligned}
    X \backslash Int(S)
    & =
    X \backslash \left(  \underset{{U \, open} \atop {U \subset S}}{\cup}U \right)
    \\
    & = \underset{U \subset S}{\cap} X \backslash U
    \\
    & = \underset{{C\, closed} \atop {C \supset X \backslash S}}{\cap} C
    \\
    & = Cl(X \backslash S)
  \end{aligned}
$$

=--

## Related entries

* [[interior point]]

* [[ionad]]

## Reference

* [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462) {#SGA4}

[[!redirects interior]]
[[!redirects interiors]]
[[!redirects topological interior]]
[[!redirects topological interiors]]


interior point

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

#Contents#
* table of contents
{:toc}

## Definition

Given a [[subset]] $A$ of a [[topological space]] $X$, a point $x \in A$ is an *interior point* of $A$ it it is contained in the [[interior]] of $A$, hence if $A$ is a [[neighborhood]] of $x$ in $X$, i.e., if there is some [[open subset]] containing $x$ that is included in $A$.

## Related concepts

* [[open point]]

* [[closed point]]

[[!redirects interior points]]


intermediate Jacobian

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Traditionally the $(k+1)$st _intermediate Jacobian variety_ $J^{k+1}(\Sigma)$ of a [[complex analytic space]] $\Sigma$ is the [[quotient]] of its [[ordinary cohomology]] in degree $2k+1$ with [[real number]] [[coefficients]] by that with [[integer]] coefficients

$$
  J^{k+1}(\Sigma) \coloneqq H^{2k+1}(\Sigma, \mathbb{R}) / H^{2k+1}(\Sigma, \mathbb{Z})
  \,.
$$

This space naturally carries the structure of a [[complex manifold]] (in fact two such structures, named after Griffiths and after Weil) and this complex analytic space, which is in fact a _[[complex torus]]_, is properly what is called the $(k+1)$st _intermediate Jacobian variety_ of $\Sigma$. This terminology derives from the term _[[Jacobian variety]]_ which is the (historically earlier) special case for $k = 0$ and $dim_{\mathbb{C}}(\Sigma) = 1$.

Notice that, conceptually, we may (cf. at *[[Deligne cohomology]]* the [exact sequences](Deligne+cohomology#ExactSequenceForCurvatureAndCharacteristicClass) and generally at *[[differential cohomology hexagon]]*):

* think of $H^{2k+1}(\Sigma,\mathbb{R})$ as the space of those [[circle n-bundle with connection|$n$-form connections]] on $\Sigma$ which are both [[flat infinity-connection|flat]] and have trivial [[underlying]] [[line n-bundle|line $n$-bundle]];

* think of $H^{2k + 1}(\Sigma,\mathbb{Z})$ as the group of "large" (i.e.: not connected to the identity) [[higher gauge field|higher]] [[gauge transformations]] acting on these gauge fields;

* hence understand $J^{k+1}(\Sigma)$ as the [[moduli space]] of flat $n$-form connections on trivial underlying line $n$-bundles.

This turns out to be a natural and  useful perspective on intermediate Jacobians: _Deligne's theorem_ (discussed as theorem \ref{DeligneTheorem} below) characterizes the intermediate Jacobians as subgroups of the relevant [[Deligne cohomology]] group of $\Sigma$, where [[Deligne cohomology]] is a model for [[ordinary differential cohomology]] that classifies these [[circle n-bundle with connection|line n-bundles with connection]]. Moreover, as discussed in Rem. \ref{AsFiberOf0TruncationOnFractureSquare} below, Deligne's theorem in the formulation of [Esnault & Viehweg (1988)](#EsnaultViehweg88) may be rephrased such as to manifestly give a formal incarnation of the the statement that $J^{k+1}(\Sigma)$ is just that subgroup of the Deligne complex given by [[circle n-bundle with connection|line $n$-connections]] with trivial [[curvature]] and trivial [[underlying]] [[line n-bundle|line $n$-bundle]].

This formulation, in turn, has an evident generalization from [[ordinary differential cohomology]] to general (namely [[Whitehead-generalized cohomology|Whitehead-generalized]]) [[differential cohomology]], which we discuss further [below](#GeneralDescripion).

This perspective on intermediate Jacobians from [[higher gauge theory]] also faithfully reflects their role in fundamental [[physics]] (in [[quantum field theory]] and [[string theory]]) [[Witten (1996)](#Witten96), [Hopkins & Singer (2002)](#HopkinsSinger02)]. Here [[higher dimensional Chern-Simons theory]] has as [[field (physics)|fields]] certain [[higher gauge fields]] specified by some type of [[differential cohomology]], and the [[connected components]] of its [[phase space]] (of solutions to the [[equations of motion]]) is precisely the corresponding intermediate Jacobian. Moreover the [[transgression]] of the higher Chern-Simons [[action functional]] produces a [[line bundle]] on the intermediate Jacobian, which is the [[prequantum line bundle]] of the theory. By [[geometric quantization]] one has to choose a [[Kähler polarization]] for this line bundle and the Weil complex structure on $J^{k+1}(\Sigma)$ is precisely that.
In terms of [[complex geometry]] this state of affairs directly translates into the statement that the Weil intermediate Jacobians are [[polarized varieties]]. In fact they are [[principally polarized variety|principally polarized]], which on the physics side corresponds to the [[metaplectic correction]] of the [[Kähler polarization]] used for [[geometric quantization]]. The [[holomorphic section]] of the resulting [[Theta characteristic]] on the intermediate Jacobian is physically the [[partition function]] of [[self-dual higher gauge theory]] on $\Sigma$ (see there for more) which mathematically is the corresponding [[theta function]].

By way of these deep relations intermediate Jacobians play an important role in ([[higher geometry|higher]]) [[geometry]].

## Definition

### For differential ordinary cohomology
 {#InComplexGeometry}

#### The underlying real manifold

Let $\Sigma$ be a [[projective variety|projective]] [[smooth variety|smooth]] [[complex variety]] (see at _[[GAGA]]_).

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

For $k \in \mathbb{N}$ the _$k$th intermediate Jacobian_ of $\Sigma$ is, as a [[real manifold]], the [[quotient]]

$$
  J^{k+1}(\Sigma)
    \coloneqq
  H^{2k+1}(\Sigma,\mathbb{R})/H^{2k+1}(\Sigma,\mathbb{Z})
$$

of the [[ordinary cohomology|ordinary]] [[cohomology groups]] of $X$ with [[coefficients]] in the [[abelian groups]] of [[real numbers]] and of [[integers]], respectively, induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$.

=--

Here $H^{2k+1}(\Sigma,\mathbb{R})$ is naturally a [[vector space]] over the [[real numbers]] and this is what induces the [[smooth manifold]]-structure on the quotient.

For the purpose of eventually equipping this with the [[structure]] of a [[complex manifold]] one may realizes it as the quotient of the [[complex vector space]] of [[complex cohomology|complex]] [[ordinary cohomology]], as follows:

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

A real [[differential form]]

$$
  \alpha
  \;\in\;
  \Omega^{2k+1}_{\mathbb{R}}(\Sigma)
$$

is, by the [[Hodge theorem]], a sum of complex differential forms in homogeneous [[Dolbeault complex|Dolbeault bidegree]] of the form

$$
  \alpha
  \;=\;
   \alpha^{2k+1,0}+ \alpha^{2k,1}
     +
     \cdots
      +
     \alpha^{k+1,k}
      +
     \overline{\alpha^{k+1,k}}
     +
     \cdots
     +
     \overline{\alpha^{2k,1}} + \overline{\alpha^{2k+1,0}}
  \,,
$$

where

$$
  \overline{(-)}
  \;\colon\;
  \Omega^{p,q}(\Sigma)\longrightarrow \Omega^{q,p}(\Sigma)
$$

is the [[antilinear function]] on complex differential forms given by [[complex conjugation]].

=--

It follows with the [[de Rham theorem]] that:

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

There is a canonical [[isomorphism]] of [[real vector spaces]]

$$
  H^{2k+1}(\Sigma, \mathbb{R})
   \;\simeq\;
  H^{2k+1}(\Sigma,\mathbb{C})/(F^{k+1} H^{2k+1}(\Sigma,\mathbb{C}))
 \,,
$$

where

$$
  F^{k+1} H^{2k+1}(\Sigma,\mathbb{C})
  \;\coloneqq\;
  \underset{p \geq k+1}{\oplus} H^{p,2k+1-p}(\Sigma)
$$

is the $(k+1)$st stage in the [[Hodge filtration]] of $H^{2k+1}(\Sigma,\mathbb{C})$.

=--

Hence an equivalent way of writing the intermediate Jacobian (still as a real manifold) is as the [[quotient space]] of the real manifold underlying a complex vector space, as follows:

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

The intermediate Jacobian of def. \ref{PlainIJacobian} is equivalently

$$
  J^{k+1}(\Sigma)
  \;\simeq\;
   H^{2k+1}(\Sigma,\mathbb{C})
   /
  \big(
    F^{k+1} H^{2k+1}(\Sigma, \mathbb{C})
    \,\oplus\,
    H^{2k+1}(\Sigma,\mathbb{Z})
  \big)
  \,.
$$

=--

Yet one more reformulation is useful when properly working in [[complex analytic geometry]]/[[GAGA]]:

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

Write $\big(\mathbf{B}^k \mathbb{G}_a\big)_{conn}$ for the [[abelian sheaf]] of [[chain complexes]] on [[site]] of [[complex manifolds]] which assigns the truncated [[de Rham complexes]] of [[holomorphic differential forms]]:

$$
  \big(\mathbf{B}^k \mathbb{G}_a\big)_{conn}
  \;\coloneqq\;
  \left(
    \mathcal{O}
     \stackrel{\partial }{\to}
    \Omega^{1}
      \stackrel{\partial}{\to}
      \cdots
      \stackrel{\partial}{\to}
    \Omega^{k}
  \right)
$$

regarded as sitting in degrees $k$ to 0.

=--

(e.g. [Esnault & Viehweg (1988), top of p. 14](#EsnaultViehweg88))

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

The quotient in prop. \ref{AsQuotientByHodgeFiltering}
is equivalently the [[abelian sheaf cohomology|abelian sheaf]]
[[hypercohomology]] with [[coefficients]]
in $\mathbf{B}^{2k}\mathbb{G}_a$ of def. \ref{ModuliOfHolomorphickConnections}:

$$
  H^{2k+1}\big(
    \Sigma
    ,\,
    \mathbb{C}
  \big)
  /
  F^{k+1}
  H^{2k+1}(\Sigma, \mathbb{C})
  \;\simeq\;
  \Big[
    \Sigma
    ,\,
    \mathbf{B}^k\big(\mathbf{B}^k \mathbb{G}_a\big)_conn
  \Big]
  \,.
$$

=--

(e.g. [Esnault & Viehweg (1988), 2.5 b)](#EsnaultViehweg88)).

There are two canonical ways of equipping $H^{2k+1}(\Sigma,\mathbb{C})$, and hence the above quotient, with the structure of a [[complex manifold]]. Sometimes these agree, but in general they do not, and hence they go by different names:

1. *[Griffith intermediate Jacobian](#GriffithIntermediateJacobian)*,

1. *[Weil intermediate Jacobian](#WeilIntermediateJacobian)*.

#### Characterization as Hodge-trivial Deligne cohomology
 {#CharacterizationAsHodgeTrivialDeligneCohomology}

A theorem due to [[Pierre Deligne]] says that the intermediate Jacobian $J^k(\Sigma)$ is characterised as being the [[fiber]] of a canonical map from ([[complex analytic geometry|complex analytic]]) [[Deligne cohomology]] to the $k$th [[Hodge filtration]] of [[integral cohomology]].

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

The group $Hdg^{k+1}(\Sigma)$ of _[[Hodge cocycle|Hodge cohomology classes]]_ is the subgroup of $\mathbb{Z}(k+1)$-cohomology classes whose image in complex cohomology is in the $(k+1)$st stage of the [[Hodge filtration]], hence the group sitting in the following [[pullback]] diagram

$$
  \array{
     Hdg^{k+1}(\Sigma)
       &\longrightarrow&
     F^{k+1} H^{2k+2}(\Sigma;\,\mathbb{C})
     \\
     \Big\downarrow
     &{}^{{}_{(pb)}}&
     \Big\downarrow
     \\
     H^{2k+2}\big(\Sigma;\,\mathbb{Z}(k+1)\big)
       &\longrightarrow&
     H^{2k+2}(\Sigma;\,\mathbb{C})
     \mathrlap{\,.}
  }
$$

=--

The following says this in a [[complex analytic geometry|complex analytic]]-way that generalizes:

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

Equivalently, the Hodge cohomology classes of def. \ref{HodgeCohomologyClasses} are given by the [[pullback]]

$$
  \array{
     Hdg^{k+1}(\Sigma)
       &\longrightarrow&
     H^{2k+2}\big(\Sigma;\, \Omega^{\bullet \geq k+1}\big)
     \\
     \Big\downarrow && \Big\downarrow
     \\
     H^{2k+2}\big(\Sigma;\,\mathbb{Z}(k+1)\big)
       &\longrightarrow&
     H^{2k+2}(\Sigma;\,\mathbb{C})
  }
  \,,
$$

where now in the top right we have the [[abelian sheaf cohomology|abelian sheaf]] [[hypercohomology]] with [[coefficients]] in the [[holomorphic de Rham complex]], truncated (but otherwise unshifted) as indicated.

=--

([Esnault & Viehweg (1988), section 7.8](#EsnaultViehweg88))

+-- {: .num_theorem #DeligneTheorem}
###### Theorem
**(Deligne)**

As an [[abelian group]] the intermediate Jacobian $J^k(\Sigma)$, def. \ref{PlainIJacobian}, is the [[fiber]] of the canonical map from [[Deligne cohomology]] to Hodge cohomology classes, has as fitting into a [[short exact sequence]] of the following form:

$$
  0
    \to
  J^{k+1}(\Sigma)
    \longrightarrow
  H^{2k+2}\big(\Sigma;\, \mathbb{Z}(k+1)_{D}\big)
    \longrightarrow
  Hdg^{k+1}(\Sigma)
    \to
  0
  \,.
$$

=--

(e.g. [Esnault & Viehweg (1988), (7.9)](#EsnaultViehweg88); [Peters & Steenbrink (2008), lemma 7.20](#PetersSteenbrink08))

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

Use prop. \ref{ReplacingHodgeFiltrationQuotientByHolomorphicConnections} and the [characteristic long exact sequences](ordinary%20differential%20cohomology#CurvatureAndCharClass) of [[ordinary differential cohomology]].

=--

+-- {: .num_remark #AsFiberOf0TruncationOnFractureSquare}
###### Remark

The [[fiber product]]-incarnation of $Hdg^{k+1}(\Sigma)$ in prop. \ref{HodgeCyclesAsPullbackOfHolomorphic} is noteworthy in that it
is analogous to the [[homotopy fiber]]-characterization of the
holomorphic [[Deligne complex]] itself.

Consider the following [[diagram]] of [[sheaves of chain complexes]]
on the site $SteinSp$ of [[Stein manifolds]] (see at _[[complex analytic ∞-groupoid]]_ for more on this):

$$
  \array{
    \mathbb{Z}(p)[-2k-2]
     && &&
     \Omega^{\bullet \geq k+1}[-2k-2]
     \\
     & \searrow && \swarrow
     \\
     && \mathbb{C}[-2k-2]
  }
  \,.
$$

This is just of the form as discussed in some detail at _[[circle n-bundle with connection]]_ and also at _[[differential cohomology diagram]]_ in the section on _[Deligne coefficients](differential+cohomology%20diagram#DeligneCoefficients)_. In particular the [[homotopy limit]] over this diagram -- hence the [[homotopy fiber]] of the two maps -- is a version of the [[Deligne complex]]:

$$
  \array{
     &&\mathbb{Z}(p)_D[-2k-2]
     \\
     && =
     \\
     &&(\mathbb{Z}(p) \to \mathcal{O} \to \Omega^1 \to \cdots \to \Omega^{k} \to 0 \to \cdots)[-2k-2]
     \\
     & \swarrow && \searrow
    \\
    \mathbb{Z}(p)[-2k-2]
     && (hpb) &&
     \Omega^{\bullet \geq k+1}[-2k-2]
     \\
     & \searrow && \swarrow
     \\
     && \mathbb{C}[-2k-2]
  }
  \,.
$$

Since [[homotopy pullbacks]] are preserved by foming [[mapping spaces]] into them, this statement holds true after evaluating on $\Sigma$ (which produces the [[Cech cohomology|Cech-]][[Deligne complexes]]). Forming the [[n-truncated object of an (infinity,1)-category|0-truncation]] $\tau_0$ of the result gives the differential cohomology group $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_{D})$
appearing in theorem \ref{DeligneTheorem}.

Alternatively, first passing to the 0-truncation of the diagram and then producing the pullback yields the Hodge cocycle group of prop. \ref{HodgeCyclesAsPullbackOfHolomorphic}.

Accordingly, the statement of theorem \ref{DeligneTheorem} may equivalently be rephrased in the following more suggestive way:

the intermediate Jacobian $J^{k+1}(\Sigma)$ is the [[fiber]] in

$$
  J^{k+1}(\Sigma)
   \longrightarrow
  \tau_0\left(
     [\Sigma,\mathbb{Z}(p)[-2k-2]]
     \underset{[\Sigma, \mathbb{C}[-2k-2]]}{\times}
     \Omega^{\bullet \geq k+1}[-2k-2]
  \right)
  \longrightarrow
  \tau_0 [\Sigma,\mathbb{Z}(p)[-2k-2]]
   \underset{\tau_0 [\Sigma, \mathbb{C}[-2k-2]]}{\times}
  \tau_0 [\Sigma, \Omega^{\bullet \geq k+1}[-2k-2]]
  \,.
$$

=--

This formulation of the intermediate Jacobian has a
straightforward generalization from [[ordinary differential cohomology]] to [[differential cohomology|differential]] [[Whitehead-generalized cohomology]]. This we turn to [below](#GeneralDescripion).

#### The Griffith complex structure
 {#GriffithIntermediateJacobian}

The [[isomorphism]]

$$
  H^{2k+1}(\Sigma , \mathbb{C})
  \simeq
  H^{2k+1}(\Sigma , \mathbb{R})\otimes_{\mathbb{R}} (\mathbb{C})
$$

induces a [[complex manifold]] structure on $H^{2k+1}(\Sigma , \mathbb{C})$ and hence the structure of a [[complex torus]] on $k$th intermediate Jacobian as defined above. This is the structure originally defined in ([Griffiths 68a](#Griffiths68a), [Griffiths 68b](#Griffiths68b)) and hence called the _Griffith intermediate Jacobian_.  Reviews include ([Walls (2012)](#Walls12), [Esnault & Viehweg (1988), section 7.8](#EsnaultViehweg88)).

#### The Weil complex structure
 {#WeilIntermediateJacobian}

There is another natural [[complex structure]] on $H^{2k-1}(X, \mathbb{R})/H^{2k-1}(X, \mathbb{Z})$, equipped with that it is called the _Weil intermediate Jacobian_.

Let as before $n \,\coloneqq\, dim_{\mathbb{C}}(\Sigma)$.
Choose a [[Hermitian manifold]] structure on $\Sigma$. Then [[Serre duality]] on forms of total odd degree

$$
  \bar \star \;\colon\; \Omega^{p,2k+1-p}(\Sigma) \longrightarrow \Omega^{n-p-2k-1,p}(\Sigma)
$$

is an [[antilinear function]] which squares to -1. Therefore

$$
 i \bar \star \;\colon\;  H^{2k+1}(\Sigma,\mathbb{C}) \to  H^{2k+1}(\Sigma,\mathbb{C})
$$

is a [[real structure]] on $H^{2k+1}(\Sigma,\mathbb{C})$. This hence defines a [[complex manifold]] structure on $H^{2k+1}(\Sigma,\mathbb{C})$ and hence on the above [[quotient]] which is the intermediate Jacobian $J^{k+1}(\Sigma)$. As such this is the _Weil intermediate Jacobian_.

#### The polarized mid-dimensional Weil (Lazzeri) intermediate Jacobian
 {#MidDimensionalWeilIntermediateJacobian}

The Weil intermediate Jacobian is particularly interesting in mid degree, hence if

$$
  n =dim_{\mathbb{C}}(\Sigma) = 2k+1
$$

then for $J^{k+1}(\Sigma)$. This case is also known as _Lazzeri's Jacobians_ see ([Rubei 98](#Rubei98)).

In this case the [[intersection pairing]]

$$
  (\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \beta
$$

defines a [[symplectic form]], for which the [[Hodge star]] operator is a compatible [[complex structure]] and hence the [[Serre duality]]-pairing

$$
  (\alpha, \beta) \mapsto \int_{\Sigma}\alpha \wedge \star \beta
$$

is the corresponding [[Kähler manifold|Kähler]]. This makes the Weil intermediate Jacobian a [[polarized variety]].

Notice that the holomorphic coordinates in

$$
  ker \tfrac{1}{2}( 1 + i \bar \star  ) \in H^{2k+1}(\Sigma, \mathbb{C})
$$

may be thought of as the mid-degree [[self-dual higher gauge fields]] on $\Sigma$. From this point of view the above is the [[Kähler polarization]] of the [[prequantum line bundle]] on [[higher dimensional Chern-Simons theory]] in dimension $4k+3$.

#### The intermediate Jacobian of a Hodge structure

By prop. \ref{AsQuotientByHodgeFiltering} above
the intermediate Jacobian is defined by the canonical
[[Hodge filtering]] on complex [[ordinary cohomology]].
The definition obtained this way directly generalizes to
other [[Hodge structures]] $H$ and hence one
speaks more generally of the intermediate Jacobian

$$
  J(H)= H_{\mathbb{C}}/(H_\mathbb{Z}\oplus F^{k+1})
$$

if $H$ has weight $2k+1$.

(e.g. [Peters-Steenbrink 08, example 3.30, section 7.1.2](#PetersSteenbrink08))

### For differential generalized cohomology
 {#GeneralDescripion}

#### General construction

> under construction, see also ([Hopkins-Quick 12](#HopkinsQuick12))...

The formulation of the traditional intermediate
Jacobian by remark \ref{AsFiberOf0TruncationOnFractureSquare}
above suggest the following generalization.

(We use notation from _[[differential cohomology hexagon]]_).

Given any [[differential cohomology]] [[spectrum]] $\hat E$ (hence a [[spectrum object]]) in a [[cohesive (∞,1)-topos]] $\mathbf{H}$, it sits in its [[differential cohomology hexagon]] part of which is the [[homotopy pullback]]

$$
  \array{
    && \flat_{dR} \hat E
    \\
    & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow
    \\
    \hat E && && \Pi \flat_{dR} \hat E
    \\
    & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}}
    \\
    && \Pi \hat E
  }
  \,.
$$

Hence for any $\Sigma$ also the [[mapping spectra]]

$$
  \array{
    && [\Sigma, \flat_{dR} \hat E]
    \\
    & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow
    \\
    [\Sigma, \hat E] && && [\Sigma, \Pi \flat_{dR} \hat E]
    \\
    & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}}
    \\
    && [\Sigma, \Pi \hat E]
  }
  \,.
$$

Write $\tau_0 \colon \mathbf{H}\to \mathbf{H}$ for the [[n-truncated object in an (infinity,1)-category|0-truncation]] map and consider the [[fiber product]] $Hdg(\Sigma,E)$ in

$$
  \array{
    && \tau_0 [\Sigma, \flat_{dR} \hat E]
    \\
    & {}^{\mathllap{\theta_{\hat E}}}\nearrow && \searrow
    \\
    Hdg(\Sigma,\hat E) && && \tau_0 [\Sigma, \Pi \flat_{dR} \hat E]
    \\
    & \searrow && \nearrow_{\mathrlap{ch_{\hat E}}}
    \\
    && \tau_0[\Sigma, \Pi \hat E]
  }
  \,.
$$

We may call this the _Hodge cohomology_ of $\Sigma$ with coefficients in $\hat E$.
The evident morphism of diagrams induces a morphism

$$
  [\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E)
$$

and the [[homotopy fiber]] of that

$$
  \mathbf{J}(\Sigma,\hat E)
    \longrightarrow
   [\Sigma, \hat E]\longrightarrow Hdg(\Sigma, \hat E)
$$

we may call the _intermediate Jacobian $\infty$-stack_ of $\Sigma$ with coefficients in $\hat E$.

Notice that by commutativity of [[homotopy pullbacks]] with [[homotopy fibers]], this is equivalently the homotopy pullback in

$$
  \array{
    && ker([\Sigma,\flat_{dR}\hat E] \to \tau_0[\Sigma,\flat_{dR}\hat E] )
    \\
    & \nearrow && \searrow
    \\
    \mathbf{J}(\Sigma,\hat E) && && ker([\Sigma,\Pi \flat_{dR}\hat E] \to \tau_0[\Sigma,\Pi \flat_{dR}\hat E] )

    \\
    & \searrow && \nearrow
    \\
    && ker([\Sigma,\Pi\hat E] \to \tau_0[\Sigma,\Pi\hat E] )
  }
  \,.
$$

In this form this manifestly says that $\mathbf{J}(\Sigma,\hat E)$ is precisely the differential cohomology theory of $\Sigma$ obtained from $\hat E$ by restricting to trivial curvature and trivial underlying $\Pi(E)$-cohomology.

#### For complex K-theory
 {#ForComplexKTheory}

Intermediate Jacobians of [[K-theory]] classes were considered in the [[physics]]-style literature in ([Witten 99, section 4.3](#Witten99), [Moore-Witten 99, section 3](#MooreWitten99), [DMW 00, section 7.1](#MW00), [Belov-Moore 06b, section 5](#BelovMooreII)) as a means for [[quantization]] of the [[RR-field]] in [[type II superstring theory]] as a [[self-dual higher gauge theory]] (see there at _[Examples -- RR-fields in 10d](self-dual%20higher%20gauge%20theory#RRFieldsin10d)_). A mathematical discussion inspired by this is in ([MPS 11](#MPS11)).

## Properties

### Relation between the Griffiths and the Weil complex structure
 {#RelationBetweenGriffithsAndWeilStructure}

While the Griffiths complex structure on the intermediate Jacobian is not Kähler/not an [[polarized variety|algebraic polarization]] as the Weil complex structure is, it still has an [[p-convex polarization]] and there is a [[symplectomorphism]] which is an [[isomorphism]] between the Griffiths and the Weil intermediate Jacobians as real symplectic manifolds

$$
  (J^{k+1}(X), \omega_{Griffiths})
  \simeq
  (J^{k+1}(X), \omega_{Weil})
  \,.
$$

This is due to ([Griffiths 68b](#Griffiths68b)), recalled as [Griffiths 12 (2.6)](#Griffiths12)

(...)

### Cycle map / Abel-Jacobi map

The intermediate Jacobians receive canonical maps from cycles (...)
See at _[[Abel-Jacobi map]]_.

### Polarization

For a [[Hodge manifold]] the intermediate Jacobian canonically inherits the structure of a [[polarized variety]]. (...)

### Theta-characteristics

A certain [[square root]] of the [[canonical bundle]] on intermediate Jacobians -- hence a [[Theta characteristic]] -- in dimension $2k+1$ thought of as [[moduli spaces]] of (flat) [[circle n-bundles with connection|circle (2k+1)-bundles with connection]] yields the [[partition function]] of [[self-dual higher gauge theory]].  ([Witten 96](#Witten96), [Hopkins-Singer 02](#HopkinsSinger02)).

### Relation to Artin-Mazur formal groups
 {#RelationToArtinMazurFormalGroups}

By theorem \ref{DeligneTheorem} the [[formal geometry]]
of intermediate Jacobians around their canonical point is equivalently the [[deformation theory]] of [[Deligne cohomology]]/[[line n-bundles with connection]]. This is given by (when it exists) the [[Artin-Mazur formal group]]
for [deformations of Deligne cohomology](Artin-Mazur+formal+group#DeformationsOfDeligneCohomology) (see there).

## Examples

The two extreme cases of intermediate Jacobians $J^k(\Sigma)$ with minimal $k = 0$ and maximal $k = dim_{\mathbb{C}}(\Sigma)= 1$ go by special names, the

* [Picard variety](#ExamplePicard)

* [Albanese variety](#ExampleAlbanese)

respectively.

Of special interest are also the intermediate Jacobian

* [of Calabi-Yau varieties](#ExampleCY).

### $k = 0$: the Picard variety $J^1(\Sigma)$
 {#ExamplePicard}

+-- {: .num_prop #J1IsPicard}
###### Proposition

The intermediate Jacobian $J^1(\Sigma)$, def. \ref{PlainIJacobian}, of a [[complex curve]] ($dim_{\mathbb{C}}(\Sigma) = 1$) coincides with the connected component $Pic_0(\Sigma)$ of the [[Picard variety]] $Pic(\Sigma)$ of $\Sigma$, hence with the [[Jacobian variety]] $Jac(\Sigma)$:

$$
  J^1(\Sigma) = Pic_0(\Sigma) = Jac(\Sigma)
  \,.
$$

=--

First consider the elementary proof by direct inspection (e.g. [Polishchuk 03, section 16.4](#Polishchuk03)):

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

Notice that the canonical map

$$
  H^1(\Sigma,\mathbb{R})
  \hookrightarrow
  H^1(\Sigma, \mathbb{C})
  \to
  H^{0,1}(\Sigma)
  \stackrel{\simeq}{\to} H^1(\Sigma, \mathcal{O}_{\Sigma})
$$

is an [[isomorphism]]. The first map is induced by the splitting $H^1(\Sigma, \mathbb{C}) \simeq H^1(\Sigma,\mathbb{R})\oplus i H^1(\Sigma,\mathbb{R})$ given by [[complexification]] and the second by the splitting $H^1(\Sigma,\mathbb{C}) \simeq H^{0,1}(\Sigma)\oplus H^{1,0}(\Sigma)$ of [[Dolbeault cohomology]], the last map is the [[Dolbeault isomorphism]].

Therefore by the [[long exact sequence in cohomology]] of the [[exponential exact sequence]] we have that

$$
  \begin{aligned}
  J^1(\Sigma)
  & \coloneqq
  H^1(\Sigma, \mathbb{R})/H^1(\Sigma, \mathbb{Z})
  \\
  & \simeq
  H^1(\Sigma,\mathcal{O}_{\Sigma})/H^1(\Sigma, \mathbb{Z})
  \\
  & \simeq
  ker(H^1(\Sigma, \mathcal{O}^\times_{\Sigma})\to H^2(\Sigma, \mathbb{Z}))
  \\
  & =
  Pic_0(\Sigma)
  \end{aligned}
$$

is the connected component of the [[Picard variety]] of $\Sigma$.

=--

Alternatively, prop. \ref{J1IsPicard}
derives from theorem \ref{DeligneTheorem} as follows:

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

Since $k = 0$ then $\mathbf{B}^2\mathbb{Z}(k+1)_D\simeq \mathbf{B}\mathbb{G}_m$ is just the universal moduli stack of line bundles without connection and so $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D ) \simeq H(\Sigma,\mathbf{B}\mathbb{G}_m)$ is the full [[Picard variety]]. The fiber in the exact sequence in theorem \ref{DeligneTheorem} then restricts this to the elements which have trivial [[first Chern class]], hence the [[Jacobian variety]].

=--

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

There is a [[non-abelian cohomology|non-abelian]] generalization of this statement that the [[moduli space of flat connections|moduli space of]] real bundles with flat connections is equivalently a moduli space of complex-analytic bundles, but without connection. This is a corollary of the [[Narasimhan-Seshadri theorem]] (for $dim_{\mathbb{C}}\Sigma = 1$) or of the [[Donaldson-Uhlenbeck-Yau theorem]] (for [[Kähler manifolds]] $\Sigma$) and generally of the [[Kobayashi-Hitchin correspondence]] (for arbitrary complex $\Sigma$), stated for instance as ([Scheinost-Schottenloher 96, corollary 1.16](#ScheinostSchottenloher96)):

the moduli space of [[flat connection|flat]] [[special unitary group|SU(n)]]-[[principal connections]] on $\Sigma$ is equivalently the moduli space of [[special linear group|SL(n,C)]]-[[holomorphic vector bundles]] which have vanishing [[Chern classes]] and are [[stable vector bundle|semi-stable]].

=--

### $k = n-1$: Albanese variety
 {#ExampleAlbanese}

For $\Sigma$ any space of complex dimension
$n \coloneqq dim_{\mathbb{C}}(\Sigma)$ then
with $k = n-1$ the $(k+1)$st intermediate Jacobian
is built from cohomology in degree one less than the real dimension of $\Sigma$:

$$
  J^{n-1}(\Sigma) = H^{2k-1}(\Sigma,\mathbb{R})/H^{2k-1}(\Sigma, \mathbb{Z})
  \,.
$$

This $(n-1)$st intermediate Jacobian is known as the _[[Albanese variety]]_ of $\Sigma$.

### Of Calabi-Yau varieties
 {#ExampleCY}

A review of intermediate Jacobians of [[Calabi-Yau varieties]] of ([[complex manifold|complex]]) [[dimension]] 3 is in ([Baarsma 11, section 2](#Baarsma11)).

The (real) dimensional of the intermediate Jacobian of a CY3 $X$ is

$$
  dim (J(X)) = 2(1+ h^{1,2})
$$

(e.g. [Baarsma 11, (2.21)](#Baarsma11))

Hence the intermediate Jacobian of a rigid CY3 (with $h^{1,2} = 0$) is an [[elliptic curve]] (e.g. [BKNPP 09, (1.8)](#BKNPP09)).

### $n = 3$: supergravity C-field

For the moment see at _[[7d Chern-Simons theory]]_ and at _[[M5-brane]]_.

### $n = 3$: type II 3-form

For the [[RR-field]] component in degree 4 of [[type IIA superstring theory]]: ([Morrison 95](#Morrison95))

## Related concepts

[[!include moduli of higher lines -- table]]

## References

### In ordinary differential cohomology

#### General

The definition of the Griffith intermediate Jacobian is due to

* {#Griffiths68a} [[Phillip Griffiths]], _Periods of integrals on algebraic manifolds. I_ Construction and properties of the modular varieties", American Journal of Mathematics 90 (2): 568–626, (1968) [doi:10.2307/2373545, ISSN 0002-9327, JSTOR 2373545, MR 0229641]

* {#Griffiths68b} [[Phillip Griffiths]], _Periods of integrals on algebraic manifolds. II_ Local study of the period mapping", American Journal of Mathematics 90 (3): 805–865 (1968) [doi:10.2307/2373485, ISSN 0002-9327, JSTOR 2373485, summary:[[GriffithsPeriodsSummary.pdf:file]], MR 0233825 ]

Review:

* {#Griffiths12} [[Phillip Griffiths]], section 1 of _Some results on algebraic cycles on algebraic manifolds_, Proceedings of the International Conference on Algebraic Geometry, Tata Institute (Bombay), 2012 ([web](http://publications.ias.edu/node/181), [pdf](http://publications.ias.edu/sites/default/files/someresultsonalg.pdf))

* [Haus (2022), Ch 1](#Haus22).

For $k = 0$ but with generalization to non-abelian [[moduli space of flat connections]] the Grifiths-like follows also with the [[Donaldson-Uhlenbeck-Yau theorem]] as discussed in

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], pp. 154 (11 of 76) of _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))

The mid-dimensional case was discussed in unpublished work by Lazzeri, see

* {#Rubei98} Elena Rubei, _Lazzeri's Jacobian of oriented compact riemannian manifolds_ ([arXiv:math/9812110](http://arxiv.org/abs/math/9812110))

The relation of the intermediate Jacobian to [[Deligne cohomology]] (Deligne's theorem) due to [[Pierre Deligne]] is discussed in

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], Section 7 of: _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_  Perspectives in Mathematic **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

Reviews and surveys include

* [[eom]], _[Intermediate Jacobian](http://www.encyclopediaofmath.org/index.php/Intermediate_Jacobian)_

* Wikipedia, _[Intermediate Jacobian](http://en.wikipedia.org/wiki/Intermediate_Jacobian)_

* {#Walls12} [[Patrick Walls]], _Intermediate Jacobians and Abel-Jacobi maps_, 2012 ([[WallsJacobian.pdf:file]])

* {#Brylinski07} [[Jean-Luc Brylinski]], around theorem I 1.5.11 of _Loop Spaces, Characteristic Classes and Geometric Quantization_ Springer, 2007

* [[Arnaud Beauville]], _Variétés de Prym et jacobiennes intermédiaire_,  Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 3 (1977), p. 309-391   ([pdf](http://math.unice.fr/~beauvill/pubs/prym.pdf)) ([jstor]( http://www.numdam.org/item?id=ASENS_1977_4_10_3_309_0))

* Valentin Zakharevich, _Mixed Intermediate Jacobians_, 2012 ([pdf](http://www.algant.eu/documents/theses/zakharevich.pdf))

* {#Polishchuk03} [[Alexander Polishchuk]], _Abelian varieties, Theta functions and the Fourier transform_, Cambridge University Press (2003) ([pdf](http://math1.unice.fr/~beauvill/pubs/poli.pdf))

Discussion of the generalization to [[Hodge structures]] includes

* {#PetersSteenbrink08} [[Chris Peters]], [[Jozef Steenbrink]], _[[Mixed Hodge Structures]]_, Ergebisse der Mathematik (2008) ([pdf](http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck))

* [pdf](http://www-fourier.ujf-grenoble.fr/~peters/world.f/Torino.pdf)

#### For Calabi-Yau 3-folds

Discussion of intermediate Jacobians of [[Calabi-Yau 3-folds]] includes

* C. Herbert Clemens, [[Phillip Griffith]], _The intermediate Jacobian of the cubic threefold_, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 ([JSTOR](http://www.jstor.org/stable/1970801))

* [[Claire Voisin]] ([pdf](http://www.math.polytechnique.fr/~voisin/Articlesweb/griffithsgroup.pdf))

* [[Andreas Höring]], _Minimal classes on the intermediate Jacobian of a generic cubic threefold_, 2008 ([pdf](http://math.unice.fr/~hoering/articles/a5-intermediate.pdf))

In [[positive number|positive]] [[characteristic]]:

* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_ ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))

Applications in [[string theory]]:

* {#Morrison95} [[David Morrison]], section 4 of _Mirror Symmetry and the Type II String_, Nucl.Phys.Proc.Suppl. 46 (1996) 146-155 ([arXiv:hep-th/9512016](http://arxiv.org/abs/hep-th/9512016))

* {#DDP06} Diaconescu, [[Ron Donagi]], [[Tony Pantev]], _Intermediate Jacobians and ADE Hitchin Systems_ ([arXiv:hep-th/0607159](http://arxiv.org/abs/hep-th/0607159))

* {#BKNPP09} Ling Bao, Axel Kleinschmidt, Bengt E. W. Nilsson, Daniel Persson, Boris Pioline, _Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1)_, Commun. Num. Theor. Phys. 4 (1), 187-266 (2010) ([arXiv:0909.4299](http://arxiv.org/abs/0909.4299))

* {#Baarsma11} A. Baarsma, _The hypermultiplet moduli space of compactified type IIA string theory_, Master Thesis, Utrecht 2011 ([web](http://dspace.library.uu.nl/handle/1874/203182))

The relation of Theta characteristics on [[intermediate Jacobians]] to [[self-dual higher gauge theory]] was first recognized in

* {#Witten96} [[Edward Witten]], *Five-Brane Effective Action In M-Theory*, J. Geom. Phys. **22** (1997) 103-133 [[arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234)]

and the argument there was made rigorous in

* {#HopkinsSinger02} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_ (2005)

### For generalized cohomology

Intermediate Jacobians of [[K-theory]] classes were discussed in the [[physics]] literature context of [[self-dual higher gauge theory]] for [[RR-fields]] in

* {#Witten99} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031,2000 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032,2000 ([arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279))

* {#MW00} D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134,2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))

* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))

A mathematical discussion inspired by this is in

* {#MPS11} [[Stefan Müller-Stach]], Chris Peters, Vasudevan Srinivas, _Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory_ ([arXiv:1105.4108](http://arxiv.org/abs/1105.4108), [pdf slides](http://www-fourier.ujf-grenoble.fr/~peters/world.f/Torino.pdf))

Discussion of intermediate Jacobians in generalized [[Hodge-filtered differential cohomology]],:

* {#HopkinsQuick12} [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_ ([arXiv:1212.2173](http://arxiv.org/abs/1212.2173))

specifically in Hodge-filtered [[differential cobordism cohomology]]:

* {#Quick16} [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*, Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]

* {#Haus22} [[Knut Bjarte Haus]], *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

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[[!redirects Weil intermediate Jacobians]]


intermediate model structure

# Intermediate model structures

* table of contents
{: toc}

## Definition

Let $M$ be a [[category]] with two [[model structures]] $(C_1,F_1,W)$ and $(C_2,F_2,W)$ having the same class of [[weak equivalences]].  Moreover, let $(L,R)$ be a [[weak factorization system]] such that $C_1 \subseteq L \subseteq C_2$.

The following theorem is essentially due to [Jardine](#Jardine).

+-- {: .num_theorem #IMS}
###### Theorem
There is a (necessarily unique) **intermediate model structure** $(L,F_i,W)$ with the same weak equivalences and $L$ as the cofibrations.
=--
+-- {: .proof}
###### Proof
Define $F_i$ to be the class of morphisms having the right lifting property with respect to $L\cap W$.  Now we observe the following:

* The assumption $C_1 \subseteq L \subseteq C_2$ implies $F_2 \cap W \subseteq R \subseteq F_2\cap W$.  In particular, $R\subseteq W$.
* We also have $C_1\cap W \subseteq L\cap W \subseteq C_2\cap W$, hence $F_2 \subseteq F_i \subseteq F_1$.
* Finally, $L\cap W\subseteq L$ implies $R\subseteq F_i$.  Thus, $R\subseteq F_i \cap W$.

Now one of the weak factorization systems in the desired model structure is just $(L,R)$.  The other will be $(L\cap W, F_i)$, for which it suffices to verify the existence of factorizations.  Given $f$, we first factor it as $p i$ with $p\in F_2$ (hence also $p\in F_i$) and $i\in C_2\cap W$.  Now factor $i$ as $q j$ with $j\in L$ and $q\in R$, hence also $q\in F_i \cap W$.  Thus, by the 2-out-of-3 property, $j\in W$, so we have $f = (p q) j$ with $j\in L\cap W$ and $p q\in F_i$.

It remains only to show that $F_i \cap W \subseteq R$, and this follows by a standard retract argument.  Given $f\in F_i \cap W$, factor it as $p i$ with $p\in R$ and $i\in L$.  Then $p\in W$, so by 2-out-of-3 $i\in L\cap W$ as well.  Hence $f$ has the right lifting property with respect to $i$, so it is a retract of $p$, hence lies in $R$.
=--

We also have:

+-- {: .num_theorem #CofGen}
###### Theorem
If in the above situation $M$ is a [[locally presentable category]] and either $(C_1,F_1,W)$ or $(C_2,F_2,W)$ is [[cofibrantly generated model category|cofibrantly generated]] (hence [[combinatorial model category|combinatorial]]), and $(L,R)$ is cofibrantly generated, then $(L,F_i,W)$ is also combinatorial.
=--
+-- {: .proof}
###### Proof
The assumption about $(C_1,F_1,W)$ or $(C_2,F_2,W)$ ensures that $Arr_W(C)$ is an accessibly embedded accessible full subcategory of $Arr(C)$.  The rest of the hypotheses of Smith's theorem about [[combinatorial model categories]] are automatically satisfied because we already know that $(L,F_i,W)$ is a model category.
=--

## Examples

### Model structures on diagram categories

Let $C$ be a small category and $M$ a [[combinatorial model category]].  Then $M^C$ has a [[projective model structure]] $(C_{proj},F_{proj},W)$ with $F_{proj}$ the objectwise fibrations, and an [[injective model structure]] $(C_{inj},F_{inj},W)$ with $C_{inj}$ the objectwise cofibrations, and $W$ in both cases the objectwise weak equivalences.

Thus, from any weak factorization system $(L,R)$ on $M^C$ in which (1) every $L$-map is an objectwise cofibration and (2) every $R$-map is an objectwise acyclic fibration, we obtain a new model structure $(L,F_i,W)$ on $M^C$, and dually.

Specific examples include:

* If $S$ is a set of cofibrations in $M^C$ containing the generating projective cofibrations, then by the small object argument it generates a weak factorization system $(L,R)$ with $C_{proj} \subseteq L \subseteq C_{inj}$.  This is the original example.

* If $C$ is a [[Reedy category]] (or a [[generalized Reedy category]]), then we have a weak factorization system consisting of Reedy cofibrations and Reedy trivial fibrations, giving rise to the Reedy model structure.  The dual of the theorem, applied to the weak factorization system of Reedy trivial cofibrations and Reedy fibrations, produces the same Reedy model structure.

* By the [[algebraic small object argument]], we can enhance the two weak factorization systems of $M$ to [[algebraic weak factorization systems]], and then simply apply them objectwise to obtain two weak factorization systems on $M^C$ to which the theorem and its dual can be applied.

## See also

* [[mixed model structure]]

## References

* [[Rick Jardine]], [Intermediate model structures for simplicial presheaves](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/01840A79C7F09F37ACCFA110BD0C81EB/S0008439500005427a.pdf/intermediate-model-structures-for-simplicial-presheaves.pdf), Canad. Math. Bull. 49(3) (2006), 407–413.
 {#Jardine}

* a review is on [p. 12](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf#page=12) of [Field Lectures: Simplicial presheaves](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf)

[[!redirects intermediate model structures]]


intermediate value theorem

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

# The intermediate value theorem
* table of contents
{: toc}

## Idea

The _intermediate value theorem_ (IVT) is a fundamental principle of [[analysis]] which allows one to find a desired value by [[interpolation]].  It says that a [[continuous function]] $f \colon [0,1] \to \mathbb{R}$ from an [[interval]] to the [[real numbers]] (all with its [[Euclidean space|Euclidean]] [[topological space|topology]]) takes all values in between $f(0)$ and $f(1)$.

The IVT in its general form was not used by [[Euclid]].  Although it is hard to doubt that Euclid believed that, for any given angle, there was an angle with one-third the measure, this angle cannot be constructed by the methods available to Euclid, so he would never refer to it (see at _[[Euclidean geometry]]_).  In contrast, [[Archimedes]] made general arguments in which a quantity is approached from above and below, allowing him not only to trisect the angle but also to calculate [[pi|π]].

As normally stated, the IVT is not valid in [[constructive mathematics]], although there are constructively valid versions. These versions either weaken the conclusion to an *approximate* zero, or to strengthen the hypothesis to require the functions satisfy additional properties or have other structures, such as locally nonconstancy and lifting to locators in functions in the given example below. Even interpreted classically, these are prima facie weaker results.

## Statements and Proofs

### Classical IVT

+-- {: .num_theorem #classical}
###### Theorem
(classical IVT, assuming [[excluded middle]])

Let $f\colon [a,b] \to \mathbb{R}$ be a [[continuous function]] from a [[compact space|compact]] [[closed interval]] to the [[real line]], and suppose that $f(a) \lt 0$ while $f(b) \gt 0$. Then there exists a point $c$ in the unit interval such that $f(c) = 0$.
=--

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

Let $g:\mathbb{R} \to \mathbb{R}$ be defined as $g(x) \coloneqq (b - a) x + a$. Then we can define a function $h:[0,1] \to \mathbb{R}$ by $h \coloneqq f \circ g$.

By [this example](connected+topological+space) the interval $[0,1]$ is a [[connected topological space]] (this is where [[excluded middle]] is used).

By [this prop.](connected+topological+space) also its [[image]] $f([0,1]) \subset \mathbb{R}$ is connected. By [this example](connected+topological+space) that image is itself an interval. This implies the claim is true for $h$. Since linear functions preserve the properties of an interval being compact and closed, if the claim is true for $h$, it is true for $f$.

=--

### Constructive IVT with weakened conclusion

+-- {: .num_theorem #conclusion}
###### Theorem
(constructive IVT with weakened conclusion)

For real numbers $a$ and $b$, let $f\colon [a,b] \to \mathbb{R}$ be a [[pointwise continuous function]] from the [[closed interval]] $[a, b]$ to the [[real line]], and supposed that $f(a) \lt 0$ and $f(b) \gt 0$. Then for every positive number $\epsilon$ there exists a point $c_\epsilon$ in the unit interval such that ${|f(c_\epsilon)|} \lt \epsilon$.
=--

+-- {: .proof}
###### Proof of Theorem \ref{conclusion}

This proof originally appeared in [Frank 2020](#Frank20).

Let us inductively define the following [[sequences]]:

$$a_0 \coloneqq a$$
$$b_0 \coloneqq b$$
$$c_n \coloneqq \frac{a_n + b_n}{2}$$
$$d_n \coloneqq \max\left(0, \min\left(\frac{1}{2} + \frac{f(c_\epsilon)}{\epsilon}, 1\right)\right)$$
$$a_{n + 1} = c_n - \frac{d_n (b - a)}{2^{n + 1}}$$
$$a_{n + 1} = b_n - \frac{d_n (b - a)}{2^{n + 1}}$$

Then
$$b_n - a_n = \frac{b - a}{2^n}$$

and the sequence $c_n$ is a [[Cauchy sequence]], because for [[natural numbers]] $m \lt n$,

$$\vert c_m - c_n \vert \leq \frac{b - a}{2^m}$$

Lemma: For every natural number $m$, either 1. there exists a $j \leq m$ such that $\vert f(c_j) \lt \epsilon$, or 2. $f(a_m) \lt 0$ and $f(b_m) \gt 0$. This could be proved by induction on natural numbers:

When $m = 0$, $f(a_0) = f(a) \lt 0$ and $f(b_0) = f(b) \gt 0$.

Now, assume that the above lemma is true for a particular $m$. If there exists a $j \leq m$ such that $\vert f(c_j) \lt \epsilon$, then there exists a $j \leq m + 1$ such that $\vert f(c_j) \lt \epsilon$. Otherwise, either $2 f(c_m) \lt -\epsilon$, $2 f(c_m) \gt \epsilon$, or $\vert f(c_m) \vert \gt \epsilon$.

* If $2 f(c_m) \lt -\epsilon$, then we define
$$d_m \coloneqq 1$$
$$a_{m + 1} \coloneqq a_m$$
$$b_{m + 1} \coloneqq c_m$$
so that $a_{m + 1} \lt 0$ and $b_{m + 1} \gt 0$.

* If $2 f(c_m) \gt \epsilon$, then we define
$$d_m \coloneqq 1$$
$$a_{m + 1} \coloneqq c_m$$
$$b_{m + 1} \coloneqq b_m$$
so that $a_{m + 1} \lt 0$ and $b_{m + 1} \gt 0$.

* If $\vert f(c_m) \vert \gt \epsilon$, then there exists a $j \leq m + 1$ such that $\vert f(c_j) \lt \epsilon$.

Thus, the above lemma is true.

Now, by pointwise continuity at $c$, let $\delta$ be such that $\vert x - y \vert \lt \delta$ implies $\vert f(x) - f(y)\vert  \lt \epsilon$. Choose a natural number $m$ such that

$$\vert c - c_m \vert \lt \frac{\delta}{2}$$
$$\frac{\vert b - a \vert}{2^{m + 1}} \lt \frac{\delta}{2}$$

If there exists a $j \leq m$ such that $\vert f(c_j) \lt \epsilon$, then the intermediate value is true. Otherwise, $f(a_m) \lt 0$ and $f(b_m) \gt 0$, and so

$$\vert c - c_m \vert \lt \vert c - c_m \vert + \vert c_m - a_m \vert \lt \delta$$
$$\vert c - c_m \vert \lt \vert c - c_m \vert + \vert c_m - b_m \vert \lt \delta$$

and so that means that

$$\vert f(c) - f(a_m) \vert \lt \epsilon$$
$$\vert f(c) - f(b_m) \vert \lt \epsilon$$

which means that $\vert f(c) \vert \lt \epsilon$.

=--

If [[excluded middle]] is true, then the classical IVT follows from the above theorem:

+-- {: .proof}
###### Proof of Theorem \ref{classical}

By way of contradiction (applying the [[double negation]] law of [[classical logic]]), suppose that ${|f(c)|} \gt 0$ for every $c$ in $[0,1]$.  Then the extra hypothesis of Theorem \ref{hypothesis} is certainly satisfied, so there exists some $c$ such that $f(c) = 0$ after all.  (Constructively, this is enough to show that the classical theorem has no counterexample.)
=--

There is also another constructively valid version of the IVT, where the conclusion is classically the contrapositive of the classical IVT, but which is weaker constructively since the reverse contrapositive rule does not hold constructively:

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

Let $f\colon [a,b] \to \mathbb{R}$ be a [[continuous function]] from a [[compact space|compact]] [[closed interval]] to the [[real line]], and suppose that $f(x) \lt 0$ or $f(x) \gt 0$ for all $x \in [a,b]$. Then $f$ is either everywhere positive or everywhere negative.
=--

This appeared in [Bauer 2016](#Bauer16).

### Constructive IVT with strengthened hypothesis

+-- {: .num_theorem #hypothesis}
###### Theorem
(constructive IVT with strengthened hypothesis, assuming the interval endpoints and the zero have [[locators]] and that the function is [[locally nonzero]])

For real numbers $a$ and $b$ with [[locators]], let $f\colon [a,b] \to \mathbb{R}$ be a [[pointwise continuous function]] from the [[closed interval]] $[a, b]$ to the [[real line]] that is a [[locally nonzero function]] and that [[lifts to locators]], and suppose that $f(a) \leq 0$ and $f(b) \geq 0$. Then, there exists a point $c$ in $[a, b]$ with a locator such that $f(c) = 0$.
=--

+-- {: .proof}
###### Proof of Theorem \ref{hypothesis}

This proof originally appeared in [Booij 2018](#Booij18)

...

=--

+-- {: .num_theorem #hypothesis}
###### Theorem
(constructive IVT with strengthened hypothesis, assuming [[weak countable choice]] and that the function is [[locally nonzero]])

Assuming [[weak countable choice]], for real numbers $a$ and $b$, let $f\colon [a,b] \to \mathbb{R}$ be a [[pointwise continuous function]] from the [[closed interval]] $[a, b]$ to the [[real line]] that is a [[locally nonzero function]], and suppose that $f(a) \leq 0$ and $f(b) \geq 0$. Then, there exists a point $c$ in $[a, b]$ with a locator such that $f(c) = 0$.
=--

+-- {: .num_theorem #hypothesis}
###### Theorem
(constructive IVT with strengthened hypothesis, assuming [[weak countable choice]] and that the function is [[uniformly continuous]])

Let $f\colon [0,1] \to \mathbb{R}$ be a [[uniformly continuous map|uniformly continuous function]] from the [[unit interval]] to the [[real line]], and suppose that $f(0) \lt 0$ while $f(1) \gt 0$.  Suppose further that, for any points $a,b$ in the unit interval with $a \lt b$, there exists a point $c_{a,b}$ such that $a \lt c_{a,b} \lt b$ and ${|f(c_{a,b})|} \gt 0$.  (In other words, the non-[[zero set]] $\{ c : {|f(c)|} \gt 0 \}$ is [[dense subspace|dense]].)  Then there exists a point $c$ in the unit interval such that $f(c) = 0$.
=--

+-- {: .num_theorem #hypothesis}
###### Theorem
(constructive IVT with strengthened hypothesis, assuming that the function is [[strictly monotonic]] with $0$ strictly between $f(0)$ and $f(1)$)

Let $f\colon [0,1] \to \mathbb{R}$ be a [[strictly monotonic]] from the [[unit interval]] to the [[real line]], and suppose that $f(0) \lt 0$ while $f(1) \gt 0$. Then there exists a point $c$ in the unit interval such that $f(c) = 0$.
=--

\begin{proof}
We construct [[Dedekind cuts]] $L = \{y \in [0,1] \vert f(y) \lt 0\}$ and $U = \{z \in [0,1] \vert f(z) \gt 0\}$ and by the Dedekind completeness of the real numbers, there is a unique $x \in [0, 1]$ such that $y \lt x \lt z$.
\end{proof}

## See also

* [[extreme value theorem]]
* [[fundamental theorem of algebra]]

## References

* [[Peter Schuster]]; Unique existence, approximate solutions,
and countable choice; [doi](http://dx.doi.org/10.1016/S0304-3975%2802%2900707-7).

* Matt F.; answer to Approximate intermediate value theorem in pure constructive mathematics; MathOverflow; [web](http://mathoverflow.net/q/255371).

* {#Frank20} [[Matthew Frank]], _Interpolating Between Choices for the Approximate Intermediate Value Theorem_ ([arxiv:1701.02227](https://arxiv.org/abs/1701.02227)), Logical Methods in Computer Science, July 14, 2020, Volume 16, Issue 3 - [doi:10.23638/LMCS-16(3:5)2020](https://doi.org/10.23638/LMCS-16%283:5%292020)

* [[Paul Taylor]]; The intermediate value theorem; A lambda calculus for real analysis, 14; [web](http://www.paultaylor.eu/ASD/lamcra/asdivt).

* {#Booij18} [[Auke Booij]], *Extensional constructive real analysis via locators*, ([abs:1805.06781](https://arxiv.org/abs/1805.06781))

* {#Bauer16} [[Andrej Bauer]], *Five Stages of Accepting Constructive Mathematics*, Bulletin of the American Mathematical Society, Volume 54, Number 3, July 2017, Pages 481–498. ([doi:10.1090/bull/1556](http://dx.doi.org/10.1090/bull/1556), [pdf](https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf))

On the intermediate value theorem in [[cohesive homotopy type theory]], see:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

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