integral Hodge conjecture

## Statement

Let $X$ be a smooth projective [[complex manifold]].
The _integral Hodge conjecture_ states that every integral [[Hodge cycle]], i.e. $2k$-degree [[cohomology class]] of $X$ which lies in the $(k,k)$-piece of the [[Hodge decomposition]], is _algebraic_, i.e. the class of an [[algebraic cycle]].

This conjecture is known to be false, hence the refinement of the [[Hodge conjecture]] to rational cohomology classes.  However it is true for $k=1$ by the [[Lefschetz theorem on (1,1)-classes]].

## Counterexamples

The integral Hodge conjecture can fail in two ways:

* There are [[torsion]] [[cohomology classes]] which are not [[algebraic cycle|algebraic]].
* There are [[cohomology classes]] of infinite [[order of a group|order]] which are not [[algebraic cycle|algebraic]], but for which some multiple is algebraic.

Counterexamples of the first type were given first by [Atiyah-Hirzebruch 61](#AtiyahHirzebruch61).
They were later re-interpreted in terms of [[complex cobordism]] by [Totaro 97](#Totaro97).
These examples were so-called [[Godeaux-Serre varieties]], constructed in  ([Serre 58](#Serre58), section 20).

Counterexamples of the second type were first given by [Kollar 90](#Kollar90).

## Related concepts

* [[Hodge conjecture]]
* [[Tate conjecture]]

## References

* {#Serre58} [[J.-P. Serre]], _Sur la topologie des varietes algebriques en caracteristique p_, Symposium internacional de topologia algebraica, Mexico (1958), 24-53; in Oeuvres vol. 1, 501-530.

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Analytic cycles on complex manifolds_, 1961, Topology Vol. 1, pp. 25-45, [pdf](http://hirzebruch.mpim-bonn.mpg.de/151/1/29_Analytic%20cycles%20on%20complex%20manifolds.pdf).

* {#Totaro97} [[Burt Totaro]], _Torsion algebraic cycles and complex cobordism_, J. Amer.
Math. Soc. 10 (1997), no. 2, 467–493, [pdf](http://www.math.ucla.edu/~totaro/papers/public_html/bord.pdf).

* {#Kollar90} [[J. Kollar]], _Trento examples_, in Classification of irregular varieties,
edited by E. Ballico, F. Catanese, C. Ciliberto, Lecture Notes in Math.
1515, Springer (1990).

* [[C. Soulé]], [[C. Voisin]], _Torsion cohomology classes and algebraic cycles on complex projective  manifolds_, Adv. Math. 198 (2005), no. 1, 107–127, [arXiv:math/0403254](http://arxiv.org/abs/math/0403254v2).

These are summarized in section 2 of the following notes

* [[Claire Voisin]], _Some aspects of the Hodge conjecture_, notes from Takagi lectures, Kyoto 2006, [pdf](http://www.math.polytechnique.fr/~voisin/Articlesweb/takagifinal.pdf).

and section 4 of

* [[Arnaud Beauville]], _The Hodge conjecture_, [pdf](http://math.unice.fr/~beauvill/pubs/Hodge.pdf).

A [[motivic homotopy theory|motivic]] reinterpretation is discussed in

* [[Alena Pirutka]], [[Nobuaki Yagita]], _Note on the counterexamples for the integral Tate conjecture over finite  fields_, [arXiv:1401.1620](http://arxiv.org/abs/1401.1620).

The ideas of Atiyah-Hirzebruch and Totaro are shown to extend to [[positive characteristic]], using [[etale cohomology]] and [[etale homotopy theory]], to give counterexamples to the integral [[Tate conjecture]], in

* [[Jean-Louis Colliot-Thélène]], [[Tamás Szamuely]], _Autour de la conjecture de Tate `a coefficients Z_l pour les vari'et'es  sur les corps finis_, [arXiv:0902.1666](http://arxiv.org/abs/0902.1666), [pdf](http://www.math.u-psud.fr/~colliot/CTSz21dec2009.pdf).

and

* [[Gereon Quick]], _Torsion algebraic cycles and etale cobordism_, [arXiv:0911.0584](http://arxiv.org/abs/0911.0584v3).

New examples are in

* [[Benjamin Antieau]], _On the integral Tate conjecture for finite fields and representation  theory_, [arXiv:1504.04879](http://arxiv.org/abs/1504.04879).


integral homotopy theory

[[!redirects integral chains homotopy theory]]

#Contents#
* table of contents
{:toc}

## Idea

_Integral homotopy theory_ is the refinement of [[rational homotopy theory]] to integer coefficients.

Just as there are two established approaches in rational homotopy theory for encoding rational homotopy types, those of Quillen and of Sullivan, so there are analog approaches for integral homotopy types. In ([Blomquist & Harper 16](#BlomquitHarper16)) chains in [[ordinary cohomology]] with [[rational number]] [[coefficients]] are lifted to chains with [[integer]] coefficients. While ([Horel 22](#Horel22)) and ([Yuan 23](#Yuan23)) both employ [[cochains]].

## References

* {#BlomquitHarper16} [[Jacobson Blomquist]], [[John Harper]], _An integral chains analog of Quillen's rational homotopy theory equivalence_ ([arXiv:1611.04157](https://arxiv.org/abs/1611.04157))

Analogs of Sullivan's rational homotopy theory equivalence are in

* {#Horel22} Geoffroy Horel, _Binomial rings and homotopy theory_ ([arXiv:2211.02349](https://arxiv.org/abs/2211.02349))

* {#Yuan23} Allen Yuan, _Integral models for spaces via the higher Frobenius_. Journal of the American Mathematical Society, 36(1):107–175, 2023, ([arXiv:1910.00999](https://arxiv.org/abs/1910.00999)).


integral lattice

#Contents#
* table of contents
{:toc}

## Idea

A _[[lattice in a vector space]]_ is called _integral_ if the canonical [[inner product]] of all pairs of lattice vectors is an [[integer]] number.

## Related concepts

* [[modular integral lattice]]

## References

* A. E. Brouwer, _Lattices_, Course notes (2002) ([pdf](http://www.win.tue.nl/~aeb/latt/lattices.pdf))

[[!redirects integral lattices]]


integral monoid

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# Contents
* table of contents
{: toc}

## Idea

One should be able to abstract away the additive structure of an [[integral domain]] and just focus on the multiplicative structure, yielding the concept of _integral monoid_.

## Definition

An [[absorption monoid]] $M$ is an __integral monoid__ if it is nontrivial and the submonoid $M \backslash \{0\}$ is a [[cancellative monoid]] (i.e., $1 \neq 0$ and left and right multiplication by $c$ is injective if $c \neq 0$, which may be combined as left and right multiplication by $c$ is injective if and only if $c \neq 0$).

### In constructive mathematics

in constructive mathematics, there are different inequivalent ways to define an integral monoid

#### Discrete integral monoids

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

If we replace "left and right multiplication by $c$ is injective iff $c$ is nonzero" in the above definition by "left and right multiplication by $c$ is injective [[xor]] $c$ is zero" (which is equivalent in [[classical logic]] but stronger in [[constructive logic]]), then we obtain the notion of **discrete integral monoid**.  This condition implies that $0 \neq 1$.
=--

Such an integral monoid $M$ is 'discrete' in that it decomposes as a coproduct $M = \{0\} \sqcup M^\times$ (where $M^\times$ is the submomoid of $M$ that is cancellative). An advantage is that this is a [[coherent logic|coherent theory]] and hence also a [[geometric theory]].  A disadvantage is that this axiom is not satisfied (constructively) by the multiplicative monoid of the [[real numbers]] (however these are defined), although it is satisfied by the multiplicative monoid of the [[integers]] and the multiplicative monoid of the [[rational number|rationals]].

#### Heyting integral monoids

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

If we interpret $\neq$  as a [[tight apartness relation]], such that the [[absorption monoid]] becomes [[strong extensionality|strongly extensional]], then we obtain the notion of **Heyting integral monoid**.
=--

This is how 'practising' constructive analysts of the Bishop school would define the simple word 'integral monoid'.

An advantage is that the multiplicative monoid of the (located Dedekind) [[real numbers]] form a Heyting integral monoid. A disadvantage is that this is not a coherent axiom and so cannot be [[internalization|internalized]] in as many categories.

Of course, if the underlying set of the monoid has [[decidable equality]] ---as is true of the multiplicative monoids of $\mathbb{N}$, $\mathbf{Z}$, $\mathbf{Q}$, $\mathbf{Z}/n$, [[finite fields]], etc--- then a Heyting integral monoid is a discrete integral monoid.

## Examples

* An integral monoid whose largest submonoid not containing $0$ is a [[GCD monoid]] is called an __GCD integral monoid__.

* An commutative integral monoid whose largest submonoid not containing $0$ is a [[unique factorization monoid]] is called an __unique factorization integral monoid__.

* An integral monoid whose largest submonoid not containing $0$ is a [[Bézout monoid]] is called an __Bézout integral monoid__.

* An integral monoid whose largest submonoid not containing $0$ is a [[principal ideal monoid]] is called an __principal ideal integral monoid__.

* An integral monoid whose largest submonoid not containing $0$ is a [[group]] is called an __division monoid__.

## Related concepts

* [[monoid]]

* [[absorption monoid]]

* [[cancellative monoid]]

* [[integral domain]], [[field]]

[[!redirects integral monoid]]
[[!redirects integral monoids]]

[[!redirects discrete integral monoid]]
[[!redirects discrete integral monoids]]

[[!redirects Heyting integral monoid]]
[[!redirects Heyting integral monoids]]
[[!redirects heyting integral monoid]]
[[!redirects heyting integral monoids]]

[[!redirects GCD integral monoid]]
[[!redirects GCD integral monoids]]
[[!redirects unique factorization integral monoid]]
[[!redirects unique factorization integral monoids]]
[[!redirects Bézout integral monoid]]
[[!redirects Bézout integral monoids]]
[[!redirects principal ideal integral monoid]]
[[!redirects principal ideal integral monoids]]
[[!redirects division monoid]]
[[!redirects division monoids]]


integral Morava K-theory

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## Idea

A localization of [[complex bordism]] that maps to ordinary [[Morava K-theory]].

## Definition

For instance ([Kriz-Sati 04, p. 41/385](#SatiKriz04), [Buhné, def. 4.2.4](#Buhne11)).

## Properties

### Orientation

The [[orientation in generalized cohomology|orientation]] of [[integral Morava K-theory]] is discussed in ([Sati-Kriz 04](#SatiKriz04), [Buhné 11](#Buhne11)). It is essentially given by the vanishing of the seventh [[integral Stiefel-Whitney class]] $W_7$.

Notice that this is in analogy to how [[orientation in generalized cohomology|orientation]] in [[complex K-theory]] is given by the vanishing third [[integral Stiefel-Whitney class]] $W_3$ ([[spin^c-structure]]).

Precisely:

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

Let $X$ be a [[connected topological space|connected]] [[closed manifold]] of [[dimension]] 10 with [[spin structure]]. This is [[orientation in generalized cohomology|generalized oriented]] in second integral Morava K-theory $\tilde K(2)$ (for $p = 2$) precisely if its seventh [[integral Stiefel-Whitney class]] vanishes, $W_7(X) = 0$.

=--

This is ([Buhné 11, prop. 8.1.13](#Buhne11)), following ([Sati-Kriz 04](#SatiKriz04)).

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

Let $X$ be a [[connected topological space|connected]] [[closed manifold]] of [[dimension]] 10 with [[spin structure]]. This is [[orientation in generalized cohomology|generalized oriented]] in second [[Johnson-Wilson cohomology theory]] ([[Morava E-theory]]) $E(2)$ (for $p = 2$) precisely if its seventh [[integral Stiefel-Whitney class]] vanishes, $W_7(X) = 0$.

=--

This is ([Buhné 11, cor. 8.1.14](#Buhne11)), following ([Sati-Kriz 04](#SatiKriz04)).

### Role in string theory / M-theory

See at _[[Diaconescu-Moore-Witten anomaly]]_.

## Related concepts

* [[chromatic homotopy theory]]

[[!include orientations in higher quantization - table]]

## References
 {#References}

* [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor.
Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))
 {#SatiKriz04}

* [[Lukas Buhné]], _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_, Diploma thesis, Hamburg (2011)
 {#Buhne11}


integral scheme

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* table of contents
{:toc}

## Definition

An algebraic [[scheme]] $X$ is __integral__ if for any [[Zariski topology|Zariski]] [[open subset]] $U\subset X$ the ring of sections $\mathcal{O}_X(U)$ of the [[structure sheaf]] over $U$ is an [[integral domain]].

## Properties

### Equivalent characterizations

A scheme is integral iff it is both [[reduced scheme|reduced]] and [[irreducible topological space|irreducible]]. Integral schemes of finite type over the [[spectrum]] of an [[algebraically closed field]] correspond (in the sense of [[equivalence of categories]]) to classical [[algebraic varieties]].

## Related concepts

* [[algebraic cycle]]

[[!redirects integral schemes]]


integral Steenrod square

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## Definition

+-- {: .num_defn #IntegralSteenrodSquares}
###### Definition
**([[integral Steenrod squares]])**

For [[odd natural numbers|odd]] $2n + 1 \in \mathbb{N}$ the _integral Steenrod square_ $Sq^{2n + 1}_{\mathbb{Z}}$ is the [[composition]] of the mod-2 [[Steenrod square]] $Sq^{2n}$ with the [[Bockstein homomorphism]] $\beta$ associated with the sequence $\mathbb{Z} \overset{\cdot 2}{\to} \mathbb{Z} \overset{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z}$:

$$
  Sq^{2n + 1}_{\mathbb{Z}}
  \;\coloneqq\;
  \beta \circ Sq^{2n}
  \,.
$$

=--

## Properties

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

The odd-degree integral Steenrod squares from def. \ref{IntegralSteenrodSquares} are indeed integral lifts of the mod-2 Steenrod squares in that

$$
  (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}}
  \;=\;
  Sq^{2n+1}
  \,,
$$

=--

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

This follows from the relation of the [[Bockstein homomorphism]] to the first [[Steenrod square]]

$$
  (mod, 2) \circ  \beta = Sq^1
$$

([this example](Bockstein+homomorphism#Mod2BocksteinIntoMod2Cohomology)) together with the first [[Adem relation]]

$$
  Sq^1 \circ Sq^{2n} = Sq^{2n+1}
$$

([this example](Steenrod+square#CompositionWithSq1)):

$$
  \array{
    Sq^{2n+1}_{\mathbb{Z}}
    &\colon&
    B^{\bullet} (\mathbb{Z}/2\mathbb{Z})
      &\overset{Sk^{2n}}{\longrightarrow}&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{ \beta }{\longrightarrow}&
    B^{\bullet + 2n + 1} \mathbb{Z}
    \\
    &&
    \downarrow^{ id }
    &&
    \downarrow^{ id }
      &&
    \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}}
    \\
    Sq^{2n+1}
    &\colon&
    B^{\bullet} (\mathbb{Z}/2\mathbb{Z})
      &\underset{Sk^{2n}}{\longrightarrow}&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
       &\underset{  Sq^1 }{\longrightarrow}&
    B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z})
  }
$$

=--

+-- {: .num_prop #IntegralSteenrodSquareInTermsOfBocksteinForExponentialSequence}
###### Proposition
**([[integral Steenrod square]] in terms of [[Bockstein homomorphism]] for [[exponential sequence]])**

The integral Steenrod squares (def. \ref{IntegralSteenrodSquares}) may equivalently be written in terms of the [[Bockstein homomorphism]] $\delta$ of the [[exponential sequence]] $\mathbb{Z} \overset{\cdot 2\pi}{\longrightarrow} \mathbb{R} \overset{mod\, 2 \pi}{\longrightarrow} U(1)$ as

$$
  \label{IntegralSteenrodSquareViaBocksteinOfExponentialSequence}
  Sq^{2n+1}_{\mathbb{Z}}
  \;\colon\;
  B^{\bullet} (\mathbb{Z}/2\mathbb{Z})
    \overset{
      Sq^{2 n}
    }{\longrightarrow}
  B^{\bullet + 2n } (\mathbb{Z}/2\mathbb{Z})
  \overset{
    \iota
  }{\longrightarrow}
  B^{\bullet + 2n} U(1)
   \overset{\delta}{\longrightarrow}
  B^{\bullet + 2n + 1} \mathbb{Z}
  \,.
$$

=--

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

Since $\beta = \delta \circ \iota$, by [this example](Bockstein+homomorphism#Mod2BocksteinAndExponentialExactSequence).

=--

## Examples
 {#Examples}

+-- {: .num_example #IntegralSteenrodSquareRefinedToOrdinaryDifferentialCohomology}
###### Example
**([[integral Steenrod square]] refined to [[ordinary differential cohomology]])**

Let $\hat G_{2n+2} \colon X \to \mathbf{B}^{2n+1} U(1)_{conn}$ be a [[cocycle]] in [[ordinary differential cohomology]] of degree $2n + 2$.

By inserting the bottom triangle of the ordinary [[differential cohomology hexagon]] ([this diagram](differential+cohomology+diagram#eq:OrdinarCohomologyHexagon)) into the factorization in (eq:IntegralSteenrodSquareViaBocksteinOfExponentialSequence) we obtain a canonical refinement of the [[integral Steenrod square]] $Sq^{2n+1}_{\mathbb{Z}} [G_{2n + 2}]$ to a cocycle $\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n+2}$ in [[ordinary differential cohomology]], which happens to be [[flat infinity-bundle|flat]]

$$
  \array{
     && && \mathbf{B}^{4n+2} \flat U(1)
     &\longrightarrow&
     \mathbf{B}^{4n+2}U(1)_{conn}
     \\
     &&
     & {}^{\mathllap{ \iota \circ Sq^{2n} }}\nearrow
     &
     &
     {}_{\mathllap{\delta}}\searrow
     &
     \downarrow^{\chi}
     \\
     X
     &\underset{G_{2n + 2}}{\longrightarrow}&
     B^{2n+2} \mathbb{Z}
     &&
       \underset{ Sq^{2n+1}_{\mathbb{Z}} }{\longrightarrow}
     &&
     B^{4n+3} \mathbb{Z}
  }
  \,.
$$

If one moreover asks that the integral Steenrod square vanishes

$$
  [ Sq^{2n+1}_{\mathbb{Z}} G_{2n+2}]
  \;=\;
  0
  \;\in\;
  H^{4n+3}(X,\mathbb{Z})
$$

(as in [Diaconescu-Moore-Witten 00, around (6.9)](#DiaconescuMooreWitten00) for $n = 1$) then the curvature exact sequence and characteristic class exact sequence in [[ordinary differential cohomology]] ([this prop.](ordinary+differential+cohomology#ExactSequencesForOrdDiffCohomology)) imply that the class of $\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n + 2}$ is identified with a class in [[de Rham cohomology]] in degree $4n+3$:

$$
  H^{2n+2}_{diff}(X)|_{Sq^{2n+1}_{\mathbb{Z}} = 0}
    \overset{\widehat{Sq}_{\mathbb{Z}}^{2n+1}}{\longrightarrow}
  H^{4n+3}_{dR}(X)
  \,.
$$

=--

## Related concepts

* [[integral Stiefel-Whitney classes]]

* [[integral Wu structure]]

## References

The third integral Steenrod square $Sq^3_{\mathbb{Z}}$ plays a central role in the discussion of the [[supergravity C-field]] in

* {#DiaconescuMooreWitten00} [[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](https://arxiv.org/abs/hep-th/0005090))

[[!redirects integral Steenrod squares]]


integral Stiefel-Whitney class

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## Definition

The [[short exact sequence]] of [[abelian group]]s

$$
0\to \mathbb{Z}\stackrel{\cdot2}{\to} \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to 0
$$

induces a [[fiber sequence]]

$$
\cdots\to\mathbf{B}^n \mathbb{Z}\to \mathbf{B}^n\mathbb{Z}\to \mathbf{B}^n\mathbb{Z}/2\mathbb{Z}\to \mathbf{B}^{n+1}\mathbb{Z}\to \cdots
$$

and so, for any object $X$, a [[fiber sequence]]

$$
\cdots\to\mathbf{H}(X,\mathbf{B}^n \mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z})\stackrel{\beta_2}{\to} \mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})\to \cdots
$$

of [[cocycle]] [[∞-groupoid]] (with respect to any ambient [[(∞,1)-topos]] $\mathbf{H}$, such as [[Top]] $\simeq$ [[∞Grpd]]), where $\beta_2$ is the [[Bockstein homomorphism|Bockstein morphism]] asociated with the multiplication by 2.

The image via $\beta_2$ of the $n$-th [[Stiefel-Whitney class|Stiefel-Whitney map]] $w_n\in \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z})$ in $\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})$ is called the $(n+1)$st **integral Stiefel-Whithey map** and is denoted by $W_{n+1}$.

One usually uses the same symbol to denote the image of this characteristic map in [[cohomology]] (on [[0-truncated|connected components]] ) of $W_{n+1}$ in $H^{n+1}(X;\mathbb{Z})=\pi_0\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})$, and calls this the $(n+1)$-th **integral Stiefel-Whitney class**.

## Examples

### Third integral SW class

The third integral Stiefel-Whitney class $W_3(T X)$ of the [[tangent bundle]] of an [[oriented]] $n$-dimensional [[manifold]] $X$ vanishes if and only if the second [[Stiefel-Whitney class]] $w_2(T X)$ is in the image of the reduction mod 2 morphism

$$
  H^2(X;\mathbb{Z})\to H^2(X;\mathbb{Z}/2\mathbb{Z})
  \,.
$$

Since $H^2(X;\mathbb{Z})$ classifies [[isomorphism classes]] of $U(1)$-[[principal bundles]] over $X$ and $W_3(T X)$ is the obstruction to the existence of a [[spin^c structure]] on $X$, we see that $X$ has a $spin^c$ structure if and only if there exists a principal $U(1)$-bundle on $X$ "killing" the second Stiefel-Whitney class of $X$.

In particular, when $w_2(T X)$ is killed by the trivial $U(1)$-bundle, i.e., when $w_2(T X)=0$, then $X$ has a [[spin structure]].

The vanishing of the third integral SW class, hence [[spin^c-structure]] is the [[orientation in generalized cohomology|orientation condition]] in [[complex K-theory]] $KU$ over oriented manifolds. In the context of [[string theory]] this is also known as the [[Freed-Witten anomaly]] cancellation condition.

### Seventh integral SW class

Analogously, the vanishing of the seventh integral SW class is essentially the condition for [[orientation in generalized cohomology|orientation]] in second [[integral Morava K-theory]].

In the context of [[string theory]] this is also known as the [[Diaconescu-Moore-Witten anomaly]] cancellation condition.

## Related concepts

* [[integral Steenrod square]]

* [[integral Wu structure]]

[[!include orientations in higher quantization - table]]

\linebreak

## References

* {#RudolhSchmidt} Gerd Rudolph, Matthias Schmidt, around Def. 4.2.20 of _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

[[!redirects integral Stiefel-Whitney classes]]


integral transform

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## Idea

An _integral transform_ on [[functions]] is a [[linear map]] between functions on [[spaces]] $X$, $Y$ encoded by a function $K$ (or [[generalized function]], e.g. [[distribution of two variables]]) on the [[product]] space $X \times Y$ and given by a formula of the type

$$
  (function\;f\;on\;X) \mapsto
  \left(y \mapsto
  \int_{X} f(x) K(x,y) \right)
  \,,
$$

where on the right we have some kind of [[integration]] over $X$.

Here $K$ is called the integral **kernel** of the transformation.

Typically the definition of an integral transform on functions involves some delicate technical issues concerning the precise nature of the function space, the [[measure]] with respect to which the integral is defined, etc.

On the other hand, one may understand the general form of an integral transform as the [[decategorification]] of a very natural general abstract construction in [[higher category theory]]: that of [[integral transforms on sheaves]] given by spans of [[base change geometric morphism]]s.

Special cases of such [[categorification|categorified]] integral transforms are discussed at

* [[Schwartz kernel]] ([[distribution of two variables]])

* [[Fourier-Mukai transform]], [[topological T-duality]]

* [[groupoidification]]

* [[geometric infinity-function theory]].

## Examples

* The simplest example is [[matrix multiplication]], which corresponds to the case where $X$ and $Y$ are [[discrete space]]s.

* Standard examples involving genuine [[functional analysis]] are for instance the [[Fourier transform]] and related [[Laplace transform]], [[Mellin transform]], [[Hankel transform]], [[Hilbert transform]].

* Also the [[path integral]] in [[quantum mechanics]] and [[quantum field theory]] is supposed to be a class of examples of a (secondary) integral kernel.

* [[Fourier-Mukai transform]]

* [[Penrose transform]]

* [[Hecke transform]]

* [[Harish Chandra transform]]

## History
 {#History}

In [[noncommutative algebraic geometry]], one of the most important results is [[Dmitri Orlov]]'s representability theorem (1997) which states that every fully faithful [[triangulated functor]] between the [[derived categories]] of [[coherent sheaves]] on two [[smooth scheme|smooth]] [[projective varieties]] is representable by some "[[integral kernel]]", i.e., a coherent complex on the [[product]].  One would like to remove the "fully faithful" assumption, but this has proved extremely difficult so far.  In the context of [[dg-categories]] the analogous result, discovered by [[Bertrand Toen]] in 2004, does have the clean form one would like it to.  Along with other problems with [[triangulated categories]], this has been one of the motivations for people like [[Maxim Kontsevich]] and [[Goncalo Tabuada]] to start doing [[noncommutative geometry]] with [[pretriangulated dg-categories]] instead of triangulated categories.

On the other hand [[pretriangulated dg-categories]] are known to provide a model for linear [[stable (infinity,1)-categories]].  Using a different model, like [[quasi-categories]], would be more convenient for extending Toen's theorem from (smooth proper) schemes to (smooth proper) [[derived stacks]].  This was done in [Ben-Zvi & Francis & Nadler 08](#BZFN08).  In the followup ([Ben-Zvi & Nadler & Preygel 13](#BZNP13)) the authors have also extended these results to the non-smooth case.

## Related concepts

* [[polynomial functor]]

* [[Schwartz kernel]]

* [[heat kernel]]

* [[kernel method]]

## References
 {#References}

Discussion of integral kernels in the sense of [[functional analysis]] (as in the [[Schwartz kernel theorem]]) is in

* {#Treves67} [[François Trèves]], _Topological Vector Spaces, Distributions and Kernels_ (Academic Press, New York, 1967)

Discussion of integral transforms in [[derived algebraic geometry]] (see also at _[[geometric infinity-function theory]]_) is in

* {#BZFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_ ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

* {#BZNP13} [[David Ben-Zvi]], [[David Nadler]], Anatoly Preygel, _Integral transforms for coherent sheaves_ ([arXiv:1312.7164](http://arxiv.org/abs/1312.7164))

Comments on the formalization of integral transforms and [[quantization]] in [[dependent linear type theory]] are at

* _[[schreiber:Type-semantics for quantization]]_

category: analysis, geometry
[[!redirects integral transforms]]
[[!redirects integral kernel]]
[[!redirects integral kernels]]


integral transforms on sheaves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There is a sense in which a [[sheaf]] $F$ is like a [[categorification]] of a [[function]]: We may think of the [[stalk]]-map from [[point of a topos|topos point]]s to [[set]]s $(x^* \dashv x_*) \mapsto F(x) := x^* F \in Set$ under [[decategorification]] as a [[cardinality]]-valued [[function]].

Under this interpretation, many constructions in [[category theory]] have analogs in [[linear algebra]]: for instance products of numbers correspond to categorical [[product]]s (more generally to [[limit]]s) and addition of numbers to [[coproduct]]s (more generally to [[colimit]]s). Accordingly a [[colimit]]-preserving [[functor]] between [[sheaf topos]]es is analogous to a [[linear map]] or to a [[distribution]]: one also speaks of [[Lawvere distribution]]s.

This [[categorification]] of [[linear algebra]] becomes even better behaved if we pass all the way to [[(∞,1)-sheaf (∞,1)-topos]]es. Under [[∞-groupoid cardinality]] their [[stalk]]s take values also in [[integer]]s, in [[rational number]]s, and in [[real number]]s. See also the discussion at [[Goodwillie calculus]].

A [[span]] of [[base change geometric morphism]]s between toposes behaves under this interpretation like the [[linear map]] given by a [[matrix]]. Such categorified [[integral transform]]s turn out to be of considerable interest in their own right: they include operations such as the [[Fourier-Mukai transform]] which categorifies the [[Fourier transform]].

These analogies have been noticed and exploited at various places in the literature. See for instance the entries [[groupoidification]] or [[geometric ∞-function theory]]. Here we try to give a general abstract [[(∞,1)-topos theory|(∞,1)-topos theoretic]] description with examples from ordinary [[topos theory]] to motivate  the constructions.

## Linear bases

Every [[(∞,1)-topos]] is a [[locally presentable (∞,1)-category]]. More generally we may think of arbitrary locally presentable [[(∞,1)-categories]] as being analogous to [[vector space]]s of [[linear functional]]s.

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

Every [[locally presentable (∞,1)-category]] is a [[reflective sub-(∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]].

=--

See [[locally presentable (∞,1)-category]] for details.

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

For $C$ a small [[(∞,1)-category]] the [[(∞,1)-category of (∞,1)-presheaves]]

$$
  \hat C := Func(C^{op}, \infty Grpd)
$$

is the [[free cocompletion|free (∞,1)-cocompletion]] of $C$, hence the free completion under [[(∞,1)-colimit]]s. Under the interpretation of colimits as sums, this means that it is analogous to the vector spaces _spanned_ by the [[basis]] $C$.

Accordingly an arbitrary locally presentable $(\infty,1)$-category is analogous in this sense to a sub-space of a vector space spanned by a basis.

=--

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

For $\hat C, \hat D$ two [[(∞,1)-categories of (∞,1)-presheaves]], a morphism $\hat C \to \hat D$ in [[Pr(∞,1)Cat]] is equivalently a [[(∞,1)-profunctor]] $C  ⇸ D$.

=--

See [[profunctor]] for details.

## Hom-spaces

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

For $C, D \in $ [[Pr(∞,1)Cat]] we have that $Func^L(C,D)$ is itself locally presentable.

=--

See [[Pr(∞,1)Cat]] for details.

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

This means that to the extent that we may think of $C, D$ as analogous to vector spaces, also the space of linear maps between them is analogous to a vector space.
=--

## Tensor products

+-- {: .un_prop}
###### Fact

For $C$ and $D$ two [[locally presentable (∞,1)-categories]] there is locally presentable $(\infty,1)$-category $C \otimes D$ and an [[(∞,1)-functor]]

$$
  C \times D \to C \otimes D
$$

which is [[universal property|universal]] with respect to the property that it preserves [[(∞,1)-colimit]]s in both arguments.

=--

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

This means that in as far as $C, D \in $ [[Pr(∞,1)Cat]] are analogous to vector spaces, $C \otimes D$ is analogous to their [[tensor product]].

=--

## Function spaces {#FunctionSpaces}

We consider from now on some fixed ambient [[(∞,1)-topos]] $\mathbf{H}$.

Notice that for each [[object]] $X \in \mathbf{H}$ the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is the [[little topos]] of $(\infty,1)$-sheaves on $X$. So to the extent that we think of these as **function objects**, and of locally presentable $(\infty,1)$-categories as linear spaces, we may think of $\mathbf{H}/X$ as the $\infty$-vector space of $\infty$-functions on $X$

+-- {: .un_remark #EtaleIdentificationRemark}
###### Remark

The [[over-(∞,1)-topos]]es $\mathbf{H}/X$ sit by an [[etale geometric morphism]] over $\mathbf{H}$ and are characterized up to equivalence by this property.

Moreover, we have an equivalence of the ambient $(\infty,1)$-topos $\mathbf{H}$ with the $(\infty,1)$-category of [[etale geometric morphism]]s into it.

$$
  ((\infty,1)Topos/\mathbf{H})_{et} \simeq
  \mathbf{H}
  \,.
$$

=--

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

Let $\mathbf{H} =$ [[FinSet]] be the ordinary [[topos]] of [[finite set]]s. Then for $X \in FinSet$ a finite set, a _function object_ on $X$ is a morphism
$\psi : \Psi \to X$ of sets. Under the [[cardinality]] [[decategorification]]

$$
  |-| : FinSet \to \mathbb{N}
$$

we think of this as the [[function]]

$$
  |\psi| : X \to \mathbb{N}
$$

given by

$$
  x \mapsto |\Psi_x|
  \,,
$$

where $\Psi_x \in FinSet$ is the [[fiber]] of $\psi$ over $X$.

=--

+-- {: .un_example #PresheavesOnGroupoids}
###### Example

Let $\mathbf{H} = $ [[∞Grpd]]. By the [[(∞,1)-Grothendieck construction]] we have for $X \in \infty Grpd$ an [[∞-groupoid]] an [[equivalence of (∞,1)-categories]]

$$
  \infty Grpd/X \simeq PSh_{(\infty,1)}(X) \simeq  Func_{(\infty,1)}(X,\infty Grpd)
$$

of the [[over-(∞,1)-category]] of all $\infty$-groupoids over $X$  with the [[(∞,1)-category of (∞,1)-presheaves]] on $X$. And since the $\infty$-groupoid $C$ is equivalent to its [[opposite (∞,1)-category]] this is also equivalent to the [[(∞,1)-category of (∞,1)-functors]] from $C$ to [[∞Grpd]].

=--

## Products of function objects

For $\psi : \Psi \to X$ and $\phi : \Phi \to X$ in $\mathbf{H}/X$ two function objects on $X$, their [[product]] $\psi \times \phi$ in $\mathbf{H}/X$ we call the **product of function objects**.

This is computed in $\mathbf{H}$ as the [[fiber product]]

$$
 \psi \times^{\mathbf{H}/X} \phi = \Psi \times^{\mathbf{H}}_X \Phi
$$

and the morphism down to $X$ is the evident projection

$$
  \array{
    && \Psi \times_{X}^{\mathbf{H}} \Phi
    \\
    & \swarrow && \searrow
    \\
    \Psi &&\downarrow^{\psi \times^{\mathbf{H}/X} \phi}&& \Phi
    \\
    & {}_{\mathllap{\psi}}\searrow && \swarrow_{\mathrlap{\phi}}
    \\
    && X
  }
  \,.
$$

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

In $\mathbf{H} = $ [[FinSet]] we have that the $\mathbb{N}$-valued function underlying the product function object is the usual pointwise product of functions

$$
  |\psi \cdot \phi| : x \mapsto |\psi|(x) \cdot |\phi|(x)
  \,.
$$

=--

## Fiber integration

For every [[morphism]] $v : X \to Y$ in the ambient [[(∞,1)-topos]] $\mathbf{H}$ there is the corresponding [[base change geometric morphism]]

$$
  (v_! \dashv v^* \dashv v_*)
    :
  \mathbf{H}/X
   \stackrel{\overset{v_!}{\to}}{\stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}}}
   \mathbf{H}/Y
$$

between the corresponding [[over-(∞,1)-topos]]es. Here $v_!$ acts simply by postcomposition with $v$:

$$
  v_! : (\Psi \stackrel{\psi}{\to} X) \mapsto (\Psi \stackrel{\psi}{\to} X \stackrel{v}{\to} Y)
$$

while $v^*$ acts by [[(∞,1)-pullback]] along $v$:

$$
  v^* : (\Phi \stackrel{\phi}{\to} Y) \mapsto (X \times_Y \Phi)
  \,.
$$

There is a further [[right adjoint]] $v_*$. For the present purpose the relevance of its existence is that it implies that both $v_!$ as well as $v^*$ are [[left adjoint]]s and hence both preserve [[(∞,1)-colimit]]s. Therefore these are morphism in [[Pr(∞,1)Cat]] and hence behave like linear maps on our function spaces $\mathbf{H}/X$ and $\mathbf{H}/Y$.

When we think of [[base change]] in the context of linear algebra on sheaves, we shall write $\int_{X/Y}  := v_!$

$$
  (\int_{X/Y} \dashv v^*) :
  \mathbf{H}/X
  \stackrel{\overset{\int_{X/Y}}{\to}}{\underset{v^*}{\leftarrow}}
  \mathbf{H}/Y
$$

and call $\int_{X/Y} \psi$ the **[[fiber integration]]** of $F$ over the fibers of $v$. In particular when $Y = *$ is the [[terminal object]] we write simply

$$
  \int_X \psi \in \mathbf{H}
$$

for the **integral** of $\psi$ with values in the ambient $(\infty,1)$-topos. (See also the notation for [[Lawvere distribution]]s).

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

Consider the ordinary [[topos]] $\mathbf{H} = $  [[FinSet]] and for $X \in \mathbf{H}$ any set the unique [[morphism]] $v : X \to *$ to the [[terminal object]].

For $\psi : \Psi \to X$ a function object with underlying function $\psi : x \mapsto |\Psi_x|$ we have that the integral

$$
  \int_X \psi : \Psi \to *
$$

has as underlying function the constant

$$
  |\int_X \psi| = \sum_{x \in X} |\psi|(x)
  \,.
$$

=--

## Integral transforms

If we are given an oriented [[span]] or [[correspondence]]

$$
  \left(
  \array{
     && A
     \\
     & {}^{\mathllap{i}}\swarrow && \searrow^{\mathrlap{o}}
     \\
     X &&&& Y
  }
  \right)
$$

in $\mathbf{H}$ it induces by composition of pullback and fiber integration operations a colimit-preserving $(\infty,1)$-functor

$$
  \underline{A}
  :
  \mathbf{H}/X
   \stackrel{i^*}{\to}
  \mathbf{H}/A
   \stackrel{\int_{A/Y}}{\to}
  \mathbf{H}/Y
  \,.
$$

We may always factor $(i,o)$ through the [[(∞,1)-limit|(∞,1)-product]]

$$
  \left(
  \array{
     && A
     \\
     && \downarrow^{\mathrlap{(i,o)}}
     \\
     && X \times Y
     \\
     & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
     \\
     X &&&& Y
  }
  \right)
  \,.
$$

We call the function object

$$
  ((i,o) : A \to X \times Y) \in \mathbf{H}/(X\times Y)
$$

on $X \times Y$ the **integral kernel** of $\underline{A}$.

+-- {: .un_lemma}
###### Observation

We have the **pull-tensor-push formula** for $\underline{A}$:

$$
  \underline{A} F = \int_{A/Y} i^* F =  (p_2)_!(A \times (p_1^* F) )
  \,.
$$

=--

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

This follows from the pasting law for pullbacks in $\mathbf{H}$:

$$
  \array{
    i^* \Psi &\to& p_1^* \Psi &\to& \Psi
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\psi}}
    \\
    A &\stackrel{(i,o)}{\to}& X \times Y &\stackrel{p_1}{\to}& X
  }
  \,.
$$

=--

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

By the [above remark](#EtaleIdentificationRemark) on [[etale geometric morphism]]s we have that we can recover the span $X \stackrel{i}{\leftarrow} A \stackrel{o}{\to} Y$ in $\mathbf{H}$ from the span

$$
  \array{
     \mathbf{H}/X
    &\stackrel{\overset{i^*}{\to}}{\underset{i_*}{\leftarrow}}&
      \mathbf{H}/A
    &\stackrel{\overset{o^*}{\leftarrow}}{\underset{o_*}{\to}}&
    \mathbf{H}/Y
    \\
    & \searrow\nwarrow & \downarrow\uparrow & \swarrow\nearrow
    \\
    && \mathbf{H}
  }
$$

in $((\infty,1)Topos/\mathbf{H})_{et}$.
=--

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

In $\mathbf{H} = $ [[FinSet]] we have that
$(|A_{x,y}|)$ is a $|X|$-by-$ |Y|$-[[matrix]] with entries in [[natural number]]s and the function

$$
  |A \psi | : y \mapsto | (i^* \Psi)_y | = \sum_{x \in X} |A_{x,y}| \cdot |\psi|(x)
$$

is the result of applying the familiar [[linear map]] given by usual [[matrix calculus]] on $|\psi|$.

=--

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

In the case $\mathbf{H} = $ [[∞Grpd]] we have -- as in the [above example](#PresheavesOnGroupoids) -- by the [[(∞,1)-Grothendieck construction]] an equivalence

$$
  \infty Grpd / (X \times Y) \simeq PSh_{(\infty,1)}(X \times Y)
  \,.
$$

Since the $\infty$-groupoid $Y$ is equivalent to its [[opposite (∞,1)-category]] this may also be written as

$$
  \infty Grpd / (X \times Y) \simeq  Func_{(\infty,1)}(X \times Y^{op}, \infty Grpd)
  \,.
$$

The objects on the right we may again think of as $(\infty,1)$-[[profunctor]]s $X ⇸ Y$. So in particular the kernel $(A \to X \times Y) \in \infty Grpd/(X \times Y)$ is under this equivalence on the right hand identified with an $(\infty,1)$-profunctor

$$
  \tilde A : X ⇸ Y
  \,.
$$

=--

## Related concepts

* [[integral transform]]

  * [[Fourier-Mukai transform]], [[Hecke transform]]

* [[polynomial functor]]

* [[pure motives]], [[motives in physics]], [[motivic quantization]]

* [[path integral as a pull-push transform]]


integral Wu structure

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

#Contents#
* table of contents
{:toc}

## Defintion

A _differential integral Wu structure_ in degree $2k$ on an [[orientation|oriented]] [[smooth manifold]] $X$  is a refinement of the [[Wu class]] $\nu_{2k} \in H^{2k}(X, \mathbb{Z}_2)$ $\nu_{2k}$ by a [[cocycle]] $\phi$ in degree $2k$ [[ordinary differential cohomology]] $H^{2k}_{diff}(X)$, hence a [[circle n-bundle with connection|circle (2k-1)-bundle with connection]] $\nabla_{2k-1}$ whose underlying higher [[Dixmier-Douady class]] $DD(\nabla_{2k-1})$ equals $\nu_{2k}$ modulo 2-reduction

$$
  DD(\nabla_{2k-1}) mod 2 = \nu_{2k} \in H^{2k}(X, \mathbb{Z}_2)
  \,.
$$

These are the [[characteristic element of a bilinear form|characteristic elements]] of the [[intersection product]] on [[ordinary cohomology]]/[[ordinary differential cohomology]], inducing its [[quadratic refinements]].

## Related concepts

* [[Wu class]], [[twisted Wu structure]]

* [[canonical bundle]]

* [[Spin Chern-Simons theory]]

* [[integral Steenrod square]]

* [[integral Stiefel-Whitney class]]

[[!include quadratic invariants - table]]

[[!include square roots of line bundles - table]]

## References

The notion was introduced in def. 2.12 of

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

motivated by considerations about abelian [[7d Chern-Simons theory]] in

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

* [[Edward Witten]], _On flux quantization in M-theory and the effective action_ ([arXiv:hep-th/9609122v2](http://arxiv.org/abs/hep-th/9609122v2))

A [[smooth infinity-groupoid|smooth stack]] refinement is considered in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] _[[schreiber:The moduli 3-stack of the C-field]]_

[[!redirects integral Wu structures]]
[[!redirects differential integral Wu structure]]
[[!redirects differential integral Wu structures]]


integrand

The argument of an [[integral]] is called the _integrand_.

For instance if $\omega \in \Omega^p(X)$ is a [[differential form]] then it is the integral in the expression $\int_X \omega$ for the [[integration of differential forms]].

[[!redirects integrands]]


integration axiom

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

\tableofcontents

## Idea

In the axiomatic formulation of [[differential geometry]] given by [[synthetic differential geometry]] the standard [[Kock-Lawvere axiom]] provides a notion of [[differentiation]]. In general this need not come with an inverse operation of [[integration]]. The additional _integration axiom_ on a [[smooth topos]] does ensure this.

## Definition

+-- {: .un_defn}
###### Definition
**(integration axiom)**

Let $(\mathcal{T}, R)$ be a [[smooth topos]] and let the line object $R$ be equipped with the structure of a [[partial order]] $(R, \leq)$ compatible with its [[ring]] structure $(R, +, \cdot)$ in the obvious way.

Then for any  $a, b\in R$ write

$$
  [a,b] := \{x \in R | a \leq x \leq b\}
$$

We say that $(\mathcal{T},(R,+,\cdot,\leq))$ satisfies the **integration axiom** if for all such intervals, all functions on the interval arise uniquely as derivatives on functions on the interval that vanish at the left boundary:

$$
  \forall f \in R^{[a,b]} : \exists ! \int_a^{-} f \in R^{[a,b]} :
  (\int_a^{-} f)(a) = 0 \wedge (\int_a^{-} f)' = f
  \,.
$$

=--

> ... need to say more ...

## Examples

The axiom holds for all the [[smooth topos]] presented in [[Models for Smooth Infinitesimal Analysis|MSIA]], listed in appendix 2 there. See appendix 3 for the proof.

## References

page 49 of

* [[Anders Kock]], _Synthetic differential geometry_ (page 61 of [pdf](http://home.imf.au.dk/kock/sdg99.pdf))

appendix 3 of

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], [[Models for Smooth Infinitesimal Analysis]]


integration by parts

## Related concepts

* [[integration]]

* [[formally adjoint differential operator]]

* [[derivative of a distribution]]

## References

See also

* Wikipedia, _[Integration by parts](https://en.wikipedia.org/wiki/Integration_by_parts)_

[[!redirects integrations by parts]]

[[!redirects partial integration]]
[[!redirects partial integrations]]


integration of differential forms

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Integration theory
+-- {: .hide}
[[!include integration theory - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Here we discuss the [[integration]] of a [[differential form]] (possibly twisted in some way) on a [[topological manifold]] (possibly with additional structure) over an appropriately structured [[submanifold]] (or [[formal linear combination]] thereof).

## Description
 {#Description}

See at _[[differential form]]_ for basic definitions.

### Integration of top-dimension pseudoforms (pseudoforms to measures)

Let $X$ be an $n$-dimensional [[topological manifold]], and let $\omega$ be a [[continuous map|continuous]] $n$-pseudoform on $X$.  Suppose that $X$ is [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]], so that we may find a [[locally finite cover]] of $X$ with a subordinate [[partition of unity]] and a continuous coordinate chart on each patch.  (When $X$ is [[differentiable manifold|differentiable]], or even [[smooth manifold|smooth]], then these may also be chosen to be differentiable or smooth, which may be convenient but is not necessary.)  Then $\omega$ defines a [[measure space|measure]] on $X$ as follows:

*  On each coordinate patch $U$, fix the orientation given by the coordinates to turn $\omega$ into an untwisted $n$-form $\hat{\omega}$; then write $\hat{\omega}$ in coordinates as
   $$ \hat{\omega} = \omega_U \wedge \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n .$$
   In this situation, it is convenient also to write
   $$ \omega = \omega_U \mathrm{d}x^1 \cdots \mathrm{d}x^n ;$$
   in other words, we interpret $\mathrm{d}x^1 \cdots \mathrm{d}x^n$ as the absolute value of $\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n$.

*  The coordinates on $U$ define a [[diffeomorphism]] between $U$ and an open subset of $\mathbf{R}^n$ that we\'ll also call $U$; so use the latter formula to interpret
   \[ \label{absvalposs} \int_U \omega = \int_U \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,\]
   where the right-hand side is now interpreted in the usual way as an integral with respect to [[Lebesgue measure]].

*  Using the partition of unity, write
   $$ \omega = \sum_U w_U \omega_U ,$$
   where $w_U$ is a weight function defined on $U$ and $\omega_U$ is the [[restriction]] of $\omega$ to $U$.
   Then we have
   $$ \textstyle{\int}_X \omega = \sum_U \textstyle{\int}_U w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,$$
   or more generally,
   $$ \textstyle{\int}_E \omega = \sum_U \int_U \chi_E(x^1,\ldots,x^n) w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n $$
   for $E$ a measurable subset of $X$ and $\chi_E$ the [[characteristic function]] of $E$.
   This makes $\omega$ into a [[measure]] $\omega \colon E \mapsto \int_E \omega$ on $X$.

_A priori_, this definition depends not only on the particular coordinate patches chosen but also on the partition of unity chosen to go with them.  Furthermore, the definition could be done just as easily (perhaps even more easily) for something other than an $n$-pseudoform.  But the (perhaps surprising) fact that justifies it all is this:

+-- {: .num_theorem #IntegrationTheorem}
###### Theorem
When $\omega$ is an $n$-pseudoform, the definition of $\int_E \omega$ is independent of the coordinates and partition chosen.  Furthermore, the map from $n$-pseudoforms to measures is linear.
=--

Note that, if $\omega$ were an $n$-form instead of a pseudoform, then the definition would depend on the orientation of the coordinates chosen.  We could fix that by using the absolute value ${|\omega_U|}$ in place of $\omega_U$ in (eq:absvalposs) and the following equations, but then the map from forms to measures would not be linear.

### Measures to pseudoforms

It may also be enlightening to consider how to go back from a measure to an $n$-pseudoform.  If $\omega$ is an [[absolutely continuous measure|absolutely continuous]] [[Radon measure]] on $X$, then it defines an $n$-pseudoform (which we may also call $\omega$) as follows:

*  Given a point $a$, choose one of the two local orientations at $a$.
*  Given $n$ linearly independent vectors $(v_1,\ldots,v_n)$ at $a$, develop them into a coordinate system on a neighbourhood $U$ of $a$.
*  For sufficiently large natural number $k$, the coordinate cube $C_k$ of points with coordinates in $[0,1/k]^n$ exists (lies within $U$).
*  Let $L$ be $\lim_{k \to \infty} k^n \omega(C_k)$.
*  If the coordinate system on $U$ is positively oriented at $a$, then let $\omega(v_1,\ldots,v_n)$ be $L$; if the coordinate system on $U$ is negatively oriented at $a$, then let $\omega(v_1,\ldots,v_n)$ be $-L$.
*  Extend the definition to $n$ arbitrary vectors by continuity (which necessarily maps a linearly dependent tuple of vectors to zero).

Again, this definition is independent of the coordinate system chosen (as long as it extends the given vectors); or if that\'s not true, then we messed up and need to add further restrictions to the absolutely continuous Radon measure $\omega$.  The definition is *not* independent of the orientation chosen, of course; thus we get a pseudoform rather than an untwisted form.  You might try to ignore the orientation and take $\omega(v_1,\ldots,v_n)$ to be $L$ always, but that does not define an exterior form, as is most easily seen if two vectors are switched (which does not change $L$).  Instead, this would define an [[absolute differential form]] (which is equivalent to a pseudoform when, as here, the degree equals the dimension).

### Integration of more general forms

One can integrate forms other than $n$-pseudoforms, of course, but only over certain structures within the manifold $X$.  Specifically, if $R$ is a $p$-dimensional [[submanifold]] of $X$ (that is a $p$-dimensional manifold $U$ equipped with a map $R\colon U \to X$), then we would like to integrate $p$-forms or $p$-pseudoforms (defined on $X$) over $R$.  Here is how we do this:

*  We may integrate a $p$-form $\eta$ over $R$ if $R$ is _oriented_, that is if $U$ is oriented.  We pull back $\eta$ from $X$ to $U$, then use the orientation on $U$ to turn $\eta$ into a $p$-pseudoform, which we can then integrate on the $p$-dimensional manifold $U$.

*  We may integrate a $p$-pseudoform $\eta$ over $R$ if $R$ is _pseudooriented_, that is if it is equipped with a map that, for each point $a$ on $U$, takes a local orientation of $X$ at $R(a)$ to a local orientation of $U$ at $a$, continuously in $a$ and taking opposite orientations to opposite orientations.  Then locally, we turn $\eta$ into a $p$-form on $X$ using a local orientation on $X$, pull that back to $U$, and use the corresponding local orientation on $U$ to turn that back into a $p$-pseudoform, which we can then integrate on $U$.

Thus, while integration of $n$-pseudoforms is the most basic, integration of general $p$-forms is actually a bit simpler than integration of general $p$-pseudoforms.  Integration of other twisted or vector-valued forms can also be done, again given appropriate structure on $R$.

Note that, if $X$ is thought of a submanifold of itself, then it has a natural pseudoorientation that takes each local orientation to itself, and so we recover the original definition of integration of $n$-pseudoforms on $X$.  Also, if $X$ is an oriented manifold, then it may be viewed as an oriented submanifold of itself, giving a definition of integration of $n$-forms on $X$.

To integrate on *unoriented* submanifolds of arbitrary dimension, use [[absolute differential forms]].  To integrate *nonlinear* differential forms, use [[cogerm differential forms]].

### Integration on other domains

One often sees the definition of integration given for *parametrised* submanifolds, that is submanifolds where $U$ is an open subset of $\mathbf{R}^p$.  This amounts to a combination of the concepts above, with the two uses of $U$ (as a coordinate patch in $X$ or as the source of a submanifold of $X$) identified.  The theorem that the integral of an $n$-pseudoform on $X$ is independent of the coordinates chosen now becomes a theorem that the integral of a parametrised submanifold is independent of the parametrisation (up to some details about orientation), which in the end returns the result that one can integrate forms over arbitrary submanifolds (given an orientation or pseudoorientation as above).

We can also integrate on a [[formal linear combination]] of submanifolds, which is handy since these are the [[chains]] of [[de Rham homology]].  This is straightforward:
$$ \int_{\sum_i \alpha_i U_i} \omega \coloneqq \sum_i \alpha_i \int_{U_i} \omega .$$

## In cohesive homotopy-type theory
 {#InCohesiveHomotopyTypeTheory}

We discuss here a [[general abstract]] formulation of [[differential forms]], their [[integration]] and the [[Stokes theorem]] in [[cohesive (∞,1)-toposes]], hence the axiomatics of [[cohesive homotopy type theory]] (following [Bunke-Nikolaus-Völkl 13, theorem 3.2](#BunkeNikolausVoelkl13)).

### General definition

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]]. We write $(\Pi \dashv \flat \dashv \sharp)$ for the defining triple of [[adjoint modalities]] ([[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]).

As usual (see at [structures in a cohesive ∞-topos -- de Rham cohomology](cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology)) we deduce from this the secondary cohesive modalities $\Pi_{dR}$ and $\flat_{dR}$: for a pointed object with [[loop space object]] $G$ (for instance the [[delooping]] $\mathbf{B}G$) $\flat_{dR}G$ is defined by the long [[homotopy fiber sequence]]

$$
  G \stackrel{\theta_G}{\longrightarrow} \flat_{dR} G \stackrel{}{\longrightarrow} \flat \mathbf{B}G \longrightarrow \mathbf{B}G
$$

where the morphism on the right is the $\flat$-[[counit of a comonad|counit]]. Here $\theta_G$ is the _[[Maurer-Cartan form]]_ on $G$.

> (Beware that elsewhere we write $\flat_{dR}\mathbf{B}G$ for what here we write $\flat_{dR}G$.)

Dually $\Pi_{dR}$ is defined.

Write now

$$
  Stab(\mathbf{H}) = T_\ast \mathbf{H} \hookrightarrow T \mathbf{H} \to \mathbf{H}
$$

for [[tangent cohesive (∞,1)-topos]] of $\mathbf{H}$, extending it by its [[stabilization]] $Stab(\mathbf{H})$ given by [[spectrum objects]] in $\mathbf{H}$.

We assume that there is an [[interval object]]

$$
  \ast \cup \ast \stackrel{(i_0, i_1)}{\longrightarrow}  \Delta^1
$$

"exhibiting the cohesion" (see at _[[continuum]]_) in that there is a (chosen) [[equivalence]] between the [[shape modality]] $\Pi$ and the [[localization of an (∞,1)-category|localization]] $L_{\Delta^1}$ at the the [[projection]] maps out of [[Cartesian products]] with this line $\Delta^1\times (-) \to (-)$

$$
  \Pi \simeq L_{\Delta^1}
  \,.
$$

This is the case for instance for the "standard [[continuum]]", the [[real line]] in $\mathbf{H} = $ [[Smooth∞Grpd]].

It follows in particular that there is a chosen [[equivalence of (∞,1)-categories]]

$$
  \flat(\mathbf{H})\simeq L_{\Delta^1}\mathbf{H}
$$

between the [[flat modality|flat]] [[modal types|modal homotopy-types]] and the $\Delta^1$-homotopy invariant homotopy-types.

Given a [[stable homotopy type]] $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ [[cohesion]] provides two objects

$$
  \Pi_{dR} \hat E \,,\;\;  \flat_{dR}\hat E
  \;\;
  \in Stab(\mathbf{H})
$$

which may be interpreted as [[de Rham complexes]] with [[coefficients]] in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map

$$
  \array{
    \Pi_{dR}\hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\hat E
    \\
    & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}}
    \\
    &&
    \hat E
  }
$$

which interprets as the [[de Rham differential]] $\mathbf{d}$. See at _[[differential cohomology diagram]]_ for details.

Throughout in the following we leave the "inclusion" $\iota$ of "differential forms regarded as $\hat E$-connections on trivial $E$-bundles" implicit.

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

_Integration of differential_ forms is the morphism

$$
  \int_{\Delta^1} \;\colon\;  [\Delta^1, \flat_{dR}\hat E] \longrightarrow \Pi_{dR}\hat E
$$

which is induced via the [[homotopy cofiber]] property of $\flat_{dR}\Omega \hat E$ from the [[counit of a comonad|counit]] naturality square of the [[flat modality]] on $[(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -]$, using that this square exhibits a [[null homotopy]] due to the $\Delta^1$-homotopy invariance of $\flat \hat E$.

=--

([Bunke-Nikolaus-Völkl 13, (9), (10)](#BunkeNikolausVoelkl13)

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

The [[Stokes theorem]] holds:

$$
  \int_0^1 \circ \mathbf{d} \simeq i_1^\ast - i_0^\ast
  \,.
$$

=--

([Bunke-Nikolaus-Völkl 13, theorem 3.2](#BunkeNikolausVoelkl13)

### Recovering traditional integration of differential forms

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

In $\mathbf{H} = $ [[Smooth∞Grpd]] let $\hat E \in Stab(\mathbf{H})$ be given, under the [[stable Dold-Kan correspondence]], by the traditional truncated [[de Rham complex]] $\hat E \coloneqq \Omega^{\bullet \geq n }$.

Then on this object the general cohesive integration map of def. \ref{CohesiveIntegrationOfDifferentialForms} reduces on the $-n$th [[homotopy group]] to the tradition [[fiber integration]] of differential forms as [above](#Description):

$$
  \int_{(\Delta^1 \times X)/X}
  \;\colon\;
  \Omega(\Delta^1 \times X)_{cl}^n
  \longrightarrow
  \Omega^{n-1}(X)/im(\mathbf{d})
  \,.
$$

=--

([Bunke-Nikolaus-Völkl 13, lemma 4.2 (7)](#BunkeNikolausVoelkl13)

## Properties

* _[[Stokes theorem]]_

* [[kernel of integration is the exact differential forms]]

## Related concepts

* [[Lie integration]]

* [[cohomological integration]]

* [[iterated integral]], [[parallel transport]], [[holonomy]]

* [[absolute differential form]], [[cogerm differential form]]

* [[motivic integration]], [[p-adic integration]]

## References

* [[Raoul Bott]], [[Loring Tu]], §3 in: *[[Differential Forms in Algebraic Topology]]*, Graduate Texts in Mathematics **82** Springer (1982) [[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)]

Exposition with an eye towards applications in [[physics]]

* [[Theodore Frankel]], §3 in: *[[The Geometry of Physics - An Introduction]]*

Discussion in the abstract context of [[cohesion]] and [[differential cohomology]] is in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

[[!redirects integral of differential forms]]
[[!redirects integrals of differential forms]]
[[!redirects integration of forms]]


integration over infinite-dimensional manifolds

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

#Contents#
* table of contents
{:toc}

## Idea

[[integration]] over [[infinite-dimensional manifolds]].

For instance [[path integrals]], [[Wiener integrals]]

## References

* [[Irving Segal]], _Algebraic integration theory_, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489  ([Euclid](http://projecteuclid.org/euclid.bams/1183526903))

* Hui-Hsiung Kuo, _Integration theory on infinite-dimensional manifolds_, Transactions of the American Mathematical Society Vol. 159, (Sep., 1971), pp. 57-78 ([JSTOR]( http://www.jstor.org/stable/1995998))

* David Shale, _Invariant integration over the infinite dimensional orthogonal group and related spaces_,  Transactions of the American Mathematical Society
Vol. 124, No. 1 (Jul., 1966), pp. 148-157 ([JSTOR]( http://www.jstor.org/stable/1994441))


integration over supermanifolds

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are different ways to define a [[volume form|differential volume element]] on a [[smooth manifold]]. Some of these definitions can be carried over to [[supergeometry]], others cannot. The possibly most familiar way of talking about differential volume elements, in terms of top-degree [[differential form]]s, does _not_ carry over to [[supermanifold]]s.

### No top-degree forms in supergeometry

In supergeometry the notion of _top-degree form_ does not in general make sense, since there are no top-degree wedge powers of "[[superdifferential form|odd 1-forms]]": if for instance $\theta_1$ and $\theta_2$ are odd functions on some [[super Cartesian space]] $\mathbb{R}^{p|q}$ and $d \theta_1$ and $d \theta_2$ are their [[differential 1-forms]], then the wedge product of these is symmetric in that

$$
  d\theta_1 \wedge d \theta_2 = +
  d\theta_2 \wedge d \theta_1
  \,.
$$

Notice the plus sign on the right, which is the product of one minus sign for interchanging $\theta_1$ and $\theta_2$, and another minus sign for interchanging the two differentials. See at _[[signs in supergeometry]]_ for more on this.

Accordingly, the wedge product of the differential of an odd function $\theta$ with itself does not in general vanish:

$$
  (d \theta \wedge d\theta = 0)
  \Leftrightarrow
  (\theta = 0)
  \,.
$$

On the cartesian supermanifold $\mathbb{R}^{n|m}$ with canonical even coordinate functions $\{x^i\}_1^n$ and canonical odd coordinate functions $\{\theta^j\}_1^m$ the differential form which one would want to regard as the canonical [[volume form]] is

$$
  \omega :=
  d x^1 \wedge \cdots \wedge d x^n \wedge
  d\theta^1 \wedge \cdots \wedge d\theta^m
  \,.
$$

Due to the above, this is not a _top_ form, since for instance

$$
  \omega \wedge d\theta^1 \neq 0
  \,.
$$

But this example also indicates the solution: apparently for integration it is not really essential that a form is a top power, what is rather essential is that it is, locally, the wedge product of a _basis_ of 1-forms. This perspective then does lead to a sensible definition of volume forms (and more generally "integrable forms") on supermanifolds, described below.

### Solution

Therefore the naïve identification of differential volume measures with top degree forms has to be refined. The idea is to characterize a volume form by other means, in particular as an equivalence class of choices of [[bases]] for the space of 1-forms, and then to define **integrable forms** to be pairs consisting of such a generalized volume form and a **multivector**: this pair is supposed to represent the differential form one would obtain could one contract the multivector with the volume form, as in ordinary differential geometry.

The definition of integration of integrable forms in supergeometry in terms of multivector fields leads, in the case that the supermanifold in an [[NQ-supermanifold]] to the [[BV theory|BV formalism]].

## Related concepts

* [[integration]]

* [[Berezin integral]]

## References

An exposition of the standard lore is in

* [[Urs Schreiber]], [_Integration over supermanifolds_](http://www.math.uni-hamburg.de/home/schreiber/sin.pdf)

A general abstract discussion in terms of [[D-module]] theory is in

* [[Frédéric Paugam]], _Homotopical Poisson Reduction of gauge theories_ ([pdf](http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf))

Geometric discussion of [[picture number]] appearing in the context of [[integration over supermanifolds]] (and originally seen in the [[quantization]] of the [[NSR superstring]], crucial in [[superstring field theory]]) is due to

* {#Belopolsky97b} [[Alexander Belopolsky]], _Picture changing operators in supergeometry and superstring theory_ ([arXiv:hep-th/9706033](https://arxiv.org/abs/hep-th/9706033))

and further amplified in

* {#Witten12} [[Edward Witten]], appendix D of _Notes On Super Riemann Surfaces And Their Moduli_ ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459))

In this perspective picture number is an extra grading on [[differential forms on supermanifolds]] induced from a choice of _integral top-form_ needed to define [[integration over supermanifolds]]:

* {#CatenacciGrassiNoja18} [[Roberto Catenacci]], [[Pietro Grassi]], [[Simone Noja]], _Superstring Field Theory, Superforms and Supergeometry_, Journal of Geometry and Physics Volume 148, February 2020, 103559 ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

  (with an eye towards [[superstring field theory]])

* [[C. A. Cremonini]], [[Pietro Grassi]], _Pictures from Super Chern-Simons Theory_ ([arXiv:1907.07152](https://arxiv.org/abs/1907.07152))

  (with an eye towards [[super Chern-Simons theory]])

* [[C. A. Cremonini]], [[Pietro Grassi]], S. Penati, _Supersymmetric Wilson Loops via Integral Forms_ ([arXiv:2003.01729](https://arxiv.org/abs/2003.01729))

  (for super-[[Wilson lines]])

Review:

* [[Pietro Grassi]], *Integral Forms and Applications*, Sestri Levante 2015 ([pdf](https://agenda.infn.it/event/8823/attachments/55101/64989/Grassi-SL.pdf), [[GrassiIntegralForms2015.pdf:file]])

See also:

* Sergio L. Cacciatori, Simone Noja, Riccardo Re, _The Unifying Double Complex on Supermanifolds_ ([arXiv:2004.10906](https://arxiv.org/abs/2004.10906))


integration theory - contents

**[[integration]]**

| analytic integration | cohomological integration |
|--|--|
| [[measurable space]] | [[Poincaré duality]] |
| [[measure]] | [[orientation in generalized cohomology]] |
| [[volume form]] | ([[virtual fundamental class|virtual]]) [[fundamental class]] |
| [[Riemann integration|Riemann]]/[[Lebesgue integration]] [[integration of differential forms|of differential forms]] | [[push-forward in generalized cohomology]]/[[fiber integration in differential cohomology|in differential cohomology]] |

## Analytic integration

[[integral calculus]]

[[Riemann integration]], [[Lebesgue integration]]

[[line integral]]/[[contour integration]]

[[integration of differential forms]]

[[integration over supermanifolds]], [[Berezin integral]], [[fermionic path integral]]

[[Kontsevich integral]], [[Selberg integral]], [[elliptic Selberg integral]]

## Cohomological integration

[[integration in ordinary differential cohomology]]

[[integration in differential K-theory]]

[[Riemann-Roch theorem]]

## Variants

[[Lie integration]]

[[path integral]]

[[Batalin-Vilkovisky integral]]

[[!redirects integration theory  - contents]]


intensional type theory

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

#Contents#
* table of contents
{:toc}

## Idea

By _intensional type theory_ one means (but see Rem. \ref{RemarkOnTerminology}) the flavor of [[type theory]] in which [[identity types]] are not forced to be [[mere propositions]] (that is: not necessarily [[(-1)-truncated]]).

The original definition of [[identity types]] in [[Martin-Löf dependent type theory]] [[Martin-Löf (1975, §1.7 and p. 94)](#Martin-Löf75)] is by default *intensional*: It takes an additional [[axiom]] or rule --- the *identity reflection rule*, cf. [Streicher (1993, p. 4, 13 )](#Streicher93); [Hofmann (1995, p. 16)](#Hofmann95) --- to turn intensional into [[extensional type theory]] (in which identity types are then [[truncated type|truncated]] to [[mere propositions]]).

In particular, [[homotopy type theory]] is intensional: the [[categorical semantics]] of its [[identity types]] are [[path space objects]].

\begin{remark}
\label{RemarkOnTerminology}
**(terminology)**
\linebreak
Beware that some authors take the words "intensional type theory" to refer to [[type theories]] which *fail to satisfy [[function extensionality]]*.  This is in general a requirement orthogonal to the (-1)-truncation of identity types.

Generally, the origin of the names "extensional" and "intensional" is somewhat confusing. Even where the term refers to the behaviour of [[equality]], it in fact refers not to the [[typal equality]] expressed by [[identity types]], but to *[[judgmental equality]]*:

The idea is that the [[identity type]] is *always* an "[[extension (semantics)|extensional]]" notion of [[equality]] (although it can be more or less extensional, depending on whether further extensionality principles like [[function extensionality]] and [[univalence]] hold). Thus, if the judgmental equality *coincides* with the identity type, then the former is also extensional, whence the type theory is called "extensional" --- while if the two equalities do *not* coincide, then the judgmental equality has room to be more intensional than the identity type, making the type theory "intensional" (but see also [Hofmann (1995, ftn. 3)](#Hofmann95), [p. 17](https://ncatlab.org/nlab/files/HofmannExtensionalIntensionalTypeTheory.pdf#page=17)).

(On this historical reading, "[[extensional type theory]]" should refer only to *judgmentally* extensional type theory with the reflection rule, which is much stronger than merely requiring all types to be [[h-sets]].)
\end{remark}

## Properties

### Decidability
 {#Decidability}

Only the intensional but not the [[extensional type theory|extensional]] [[Martin-Löf type theory]] has [[decidability|decidable]] type checking. ([Martin-Löf 75](#MartinLof75), [Hofmann 95](#Hofmann95)).

## Examples

* [[calculus of inductive constructions]]

## Related concepts

* [[homotopy type theory]]

* [[extensional type theory]]

* [[observational type theory]], [[higher observational type theory]]

* [[function extensionality]]

## References

* {#Martin-Löf75} [[Per Martin-Löf]], _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80** Pages 73-118,  Elsevier 1975 (doi:10.1016/S0049-237X(08)71945-1, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926))

* {#Streicher93} [[Thomas Streicher]], *Investigations into Intensional Type Theory*, Habilitation Thesis, Darmstadt (1993) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf), [[Streicher-IntensionalTT.pdf:file]]]

* {#Hofmann95} [[Martin Hofmann]], _Extensional concepts in intensional type theory_, Ph.D. thesis, University of
Edinburgh, (1995) ([ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]])

[[!redirects intensional type theories]]


intensive or extensive quantity

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

# Contents
* table of contents
{: toc}

## Idea

### In physics

In [[physics]] and especially in [[continuum mechanics]] and [[thermodynamics]], a physical quantity associated with a [[physical system]] extended in [[space]] is called

* _intensive_ if it is a [[function]] on (the physical system extended in) space;

* _extensive_ if it is a [[density]] or [[linear distribution]] on (the physical system extended in) space.

For instance, for a [[solid body]] its [[temperature]] is intensive, but its [[mass]] is extensive: there is a temperature assigned to every point of the body (in the idealization of [[classical mechanics|classical]] [[continuum mechanics]] anyway) but a mass is assigned only to every little "extended" piece of the body, not to a single point.

This terminology in [[physics]] appears vaguely in ([Hegel 1812](#Hegel1812)), [Hegel 1817](#Hegel1817)), more precisely in ([Grassmann 1844](#Grassmann1844)) and in its fully modern form is maybe due to [[Richard Tolman]] ([1917](#Tolman1917)).

If we take into account that a physical system may or may not have a particular kind of quantity (for example, a thermodynamic system may or may not have a temperature), a system has a value of an intensive quantity if and only if all of its subsystems have a value of that quantity, and then all of these values are equal.  However, it's generally safe to assume that every system has a value of every extensive quantity.

To be precise, given a system $S_0$ made up of the subsystems $S_1$ and $S_2$:

*  For an intensive quantity $q$, $S_0$ has a value $q_0$ of $q$ if and only if $S_1$ and $S_2$ also have values of $q$, and then we have $q_1 = q_0$ and $q_2 = q_0$.

*  For an extensive quantity $Q$, all three systems have values of $Q$, and we have $Q_0 = Q_1 + Q_2$.

### In geometry and algebra

In ([Lawvere 86](#Lawvere86)) it is amplified that this [[duality]] is generally a fundamental one also in [[mathematics]]: given a [[topos]] $\mathbf{H}$ with a [[commutative ring|commutative]] [[ring object]] $R \in CRing(\mathbf{H})$, then

* the space of _intensive quantities_ on an [[object]] $X \in \mathbf{H}$ is the [[mapping space]] $[X,R]_{\mathbf{H}} \in CRing(\mathbf{H})$ formed in $\mathbf{H}$;

* the space of _extensive quantities_ on $X$ is the $R$-linear dual, namely the mapping space $[X,R]^\ast \coloneqq [[X,R], R]_{R Mod}$ formed in $R$-[[modules]] in $\mathbf{H}$.

  (here the operation $[[-,R],R]_{R Mod}$ happens to be what is also called the "[[continuation monad]]" for $R$)

* the [[integration]] map is the canonical [[evaluation]] pairing

  $$
    \int_X \;\colon\; [X,R] \times [X,R]^\ast  \longrightarrow R
    \,.
  $$

$$ (x, f) \mapsto f(x) $$

See also at _[[Lawvere distribution]]_.

The elusive connection between integration and (co)ends can here be explained:

**Proposition:** Let $V$ be a monoidal closed category with tensor $\otimes_V$ and internal hom $[-, -]_V$. Suppose that the Kan extension $\text{Lan}_{\text{Id}_V} (\text{Id}_V)$ exists and is pointwise. Then
 $$ X \cong \text{Id}_V (X) \cong \text{Lan}_{\text{Id}_V} (\text{Id}_V)(X) \cong \int^{Y \in V} [Y, X]_V \otimes_V Y$$
and the structure maps are
$$ \epsilon_Y : [Y, X]_V \otimes_V Y \rightarrow X$$
Occuring as counits in the supposed monoidal closed adjunction.

This is an instance of the [coend formula for a Kan extension](https://ncatlab.org/nlab/show/Kan+extension#PointwiseByCoEnds).

We may think of $V$ as the category of $R$-module objects in a topos $\mathbf{H}$, or, more generally, some category $\mathbf{H}$ resembling spaces. In that case, we get:

**Corollary:** Let $\mathbf{H}$ be a category with internal hom $[-, -]_{\mathbf{H}}$, and let $R$ be a commutative ring object in $\mathbf{H}$. Let $R \text{-mod}$ be the category of $R$-module objects in $\mathbf{H}$, with a chosen monoidal closed structure $\otimes_R$ (tensor product) and $[-, -]_R$ (internal Hom). Then, taking $V = R \text{-mod}$ above, and supposing that $\text{Lan}_{\text{Id}_{R \text{-mod}}} (\text{Id}_{R \text{-mod}})$ we have

$$ R \cong  \int^{X \in R \text{-mod}} [X, R]_R  \otimes_R X  $$

with structure maps:

$$ \epsilon_X : [X, R]_R \otimes_R X \rightarrow R $$

In the above notation, $ \epsilon_X = \int_X$. To make the analogy quite clear, consider the following example, which could naturally be called the example of "Lawvere Banach spaces", in direct analogy with "Lawvere metric spaces":

**Example:** Let $\mathbf{H}$ be the category of [Lawvere metric spaces](https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace) (Note: this category is not a topos, but the corollary above will still apply).  Maps are short maps. $\mathbf{H}$ has an internal Hom consisting of bounded maps. Let $\mathbb{R}$ be the real numbers, with $d(x, y) = ||x - y||$ as Lawvere metric. $\mathbb{R} \text{-mod}$ is here the category of normed $\mathbb{R}$-modules, with short maps as maps, with the exception that $||x|| = 0$ does not imply that $x = 0$, and that $||x|| = \infty$ is possible. (This temporary notation is not to be confused with the category of real vector spaces over the real numbers, in which norms play no role).

There is a natural notion of [completion](https://ncatlab.org/nlab/show/Cauchy+complete+category) on such spaces. The category of complete objects is essentially the category of Banach spaces over $\mathbb{R}$, except for the previously mentioned differences that $||x|| = 0$ does not imply that $x = 0$, and $||x|| = \infty$ is possible. Completion makes this category into a reflective subcategory of $\mathbb{R}$-mod. Write $\text{LawvBan}$ for the this category - it is a natural continuation of Lawvere metric spaces.

There is a natural choice of monoidal closed structure on $\text{LawvBan}$; the tensor product is [projective tensor product](https://ncatlab.org/nlab/show/tensor+product+of+Banach+spaces) (note: slight tweaking is necessary to account for the differences between Lawvere Banach spaces and Banach spaces). The internal Hom consists of bounded maps. The corollary above applies to this setup, so that

$$ R \cong  \int^{X \in R \text{-mod}} [X, R]_R  \otimes_R X  $$

where $\otimes_R$ is projective tensor product and $[X, R]_R$ consists of bounded maps from $X$ to $R$. There are structure maps

$$ \epsilon_X : [X, R]_R \otimes_R X \rightarrow R $$

sending $f \otimes x$ to $f(x)$, for which we suggestively write

$$ \int_X : [X, R]_R \otimes_R X \rightarrow R $$

### In higher geometry and higher algebra

Viewed this way, this naturally generalizes to the case where $\mathbf{H}$ is in fact an [[(∞,1)-topos]] and $R \in CRing(\mathbf{H})$ an [[E-∞ ring]]. In this case $[X,R]$ is called the  $R$-[[cohomology]] [[spectrum]] of $X$ and $[X,R]^\ast$ is the corresponding [[generalized homology]] spectrum. In this form intensive and extensive properties appear in [[physics]] in the context of [[motivic quantization]] of [[local prequantum field theory]].

More generally, for $\chi$ an $R$-[[(∞,1)-line bundle]] over $X$ then the corresponding extensive object is the $\chi$-twisted [[Thom spectrum]] $R_{\bullet + \chi}(X)$ and the intensive object is the $\chi$-[[twisted cohomology]] [[spectrum]] $R^{\bullet + \chi}(X) = [R_{\bullet+ \chi}(X),R]_{R Mod}$. See at _[[motivic quantization]]_ for how this appears in [[physics]].

## In modal homotopy type theory

Assume we are working in the context of a [[cohesive (∞,1)-topos]], $\mathbf{H}$, with the three [[adjoint modalities]], [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]] $\int \dashv \flat \dashv \sharp$.

We may characterize the codomains of those [[functions]] which are _intensive_ or _extensive quantities_ in terms of $\sharp$.

* **Intensive**: functions whose value is genuinely given by their restriction to all possible points have as codomains types $X$ that are fully determined by their moment of continuity, that is those for which $X \to \sharp X$ is a [[monomorphism]]. In [[categorical semantics]] these are the [[concrete objects]] or equivalently the [[separated presheaves]] for $\sharp$: they are determined by their global points.

* **Extensive**: objects which have purely the [[negative moment]] of continuity $\overline{\sharp}$, or, in other words, which are maximally  non-concrete, form codomains for "functions" which vanish on points and receive their contribution only from regions that _extend_ beyond a single point. For example, the smooth moduli space of differential $n$-forms is maximally non-concrete. This concept of _extension_ is precisely that which gave the name to [[Hermann Grassmann]]'s _[[Ausdehnungslehre]]_ that introduced the concept of [[exterior algebra|exterior]] [[differential form]].

So, the adjunction $(\flat \dashv \sharp)$ expresses _quantity_, discrete quantity and continuous quantity, and the latter is further subdivided into intensive and extensive quantity.

## References

The concepts of intensive and extensive quantity are highlighted in

* {#Grassmann1814} [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

which states (p. xxiv, xxv) that intensive quantity is the topic of [[differential calculus]] and [[integration]] theory, while extensive quantity is the topic of this very _Ausdehnungslehre_.

General discussion includes

* Wikipedia, _[Intensive and extensive properties](http://en.wikipedia.org/wiki/Intensive_and_extensive_properties)_

A formalization in [[categorical logic]]/[[topos theory]] is proposed in

* {#Lawvere86} [[William Lawvere]], Introduction to _[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

See also the exposition of his ideas at [Higher toposes of laws of motion](Higher+toposes+of+laws+of+motion#iii_extensiveintensive_duality_and_cohomological_quantization).

Lawvere's terminology is probably (see at _[[objective logic]]_) inspired by

* {#Hegel1812} [[Georg Hegel]], section _[Extensives und Intensives Quantum](Science+of+Logic#ExtensiveAndIntensiveQuantity)_, in _[[Science of Logic]]_, 1812

* {#Hegel1817} [[Georg Hegel]], part II "Philosophy of Nature", second section "Physics", B, [§298b](Science+of+Logic#PN298b) in _[[Encyclopedia of the Philosophical Sciences]]_

and meant to be a formalization of this part of the "[[objective logic]]", see also at _[[Science of Logic]]_.

The first use of the terms 'intensive' and 'extensive' appears to be

* [[Richard Tolman]] (1917). The Measurable Quantities of Physics. Physical Review 9 (3): 237--253.

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inter universal Teichmüller theory


inter-universal Teichmüller theory

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## Idea

The generalization of [[Teichmüller theory]] to [[arithmetic geometry]] has been called _inter-universal Teichmüller theory_ (often abbreviated _IUTT_) by [[Shinichi Mochizuki]]. It is a part of [[anabelian geometry]].

{#InterUniversal} The term "inter-universal" apparently refers to the fact that the theory is meant to formulated explicitly in a way that respects [[universe enlargement]], hence that it is [[universe polymorphism|universe polymorphic]] ([IUTT IV, remark 3.1.4](#IUTTIV), [Yamashita 13](#Yamashita13)).

It is claimed ([IUTT IV](#IUTTIV)) that a proof of the [[abc conjecture]] can be given in IUTT. In 2018, a document [[why_abc_is_still_a_conjecture.pdf:file]] was written by [[Peter Scholze]] and [[Jakob Stix]] raising objections to the argument.

## Details

### Pilot objects

Important to the arguments in the latter parts of the IUTT series are what Mochizuki refers to as pilot objects. Though the terminology in [IUTT III](#IUTTIII) is very dense, Mochizuki explains in Example 3.6, Remark 3.6.1, and subsequently that these can be understood in quite elementary terms, which we now describe.

Let $X$, $F$, $F_{mod}$, and $\underline{\mathbb{V}}$ be as at [[initial Θ-data]]: $X$ is a once-puncturing of an [[elliptic curve]] $E$, $F$ is a [[number field]], $F_{mod}$ is the [[field of moduli]] of $X$ with respect to $F$, $\underline{\mathbb{V}}$ is a certain set of [[valuation|valuations]], and all of $X$, $F$, $\overline{F}$, and $\underline{\mathbb{V}}$ satisfy certain conditions which will be referred to here only as needed. Let $F^{\times}_{mod}$ denote the [[group of units|multiplicative group]] of $F_{mod}$.

Let $v \in \underline{\mathbb{V}}$. By definition, $v$ is a valuation on $K_E$, where $K_E$ is as defined at [[initial Θ-data]]. Denote the completion of $K_E$ with respect to $v$ by $K_{v}$, denote the [[ring of integers]] of $K_{v}$ by $\mathcal{O}_{K_{v}}$, and denote the [[group of units]] of these two rings by $K^{\times}_{v}$ and $\mathcal{O}_{v}^{\times}$ respectively.

Let $\beta_{v}: F_{mod}^{\times} \rightarrow K_{v}^{\times} / \mathcal{O}^{\times}_{v}$ denote the [[homomorphism|group homomorphism]] induced by the composition of the field inclusion $F_{mod} \hookrightarrow K_{E}$ with the canonical ring homomorphisms $K_{E} \hookrightarrow K_{v} \twoheadrightarrow K_{v} / \mathcal{O}_{v}$.

\begin{defn} \label{DefinitionFStarMOD} We denote by $F^{*}_{MOD}$ the category whose objects consist of:

1. An $F^{\times}_{mod}$-[[torsor]] $T$, where [[torsor]] here is understood with respect to the [[category of sets]].

1. For every $v \in \underline{\mathbb{V}}$, a trivialisation $t_{v}$ of the $K_{v}^{\times} / \mathcal{O}_{K_v}^{\times}$-torsor $T_{v}$ obtained from $T$ by change of structure group with respect to $\beta_{v}$. We require that there is a $t \in T$ such that, for all but finitely many $v$, $t_{v}$ is equal to the trivialisation of $T_{v}$ determined by $\beta_{v}(t)$. (See [[torsor]] for the details of trivialisation of a torsor, and for the notion of change of structure group of a torsor.)

TODO: FINISH

\end{defn}

\begin{rmk} Definition \ref{DefinitionFStarMOD} is Example 3.6 in [IUTT III](#IUTTIII), and is also discussed in Remark 3.6.1. Remark 3.1.5 of [IUTT I](#IUTTI) is also relevant. \end{rmk}

## Related concepts

* [[anabelian geometry]]

* [[étale theta function]]

* [[Frobenioid]]

* [[initial Θ-data]]

* [[Mochizuki's corollary 3.12]]

* [[universe polymorphism]]

* [[poly-morphism]] (not to be be confused with [[polymorphism]])

## References

* {#IUTTI} [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory I, Construction of Hodge theaters_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf))

* {#IUTTII} [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory II, Hodge-Arakelov-theoretic evaluation_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf))

* {#IUTTIII} [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory III, Canonical splittings of the Log-theta-lattice_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf))

* {#IUTTIV} [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory IV, Log-volume computations and set-theoretic foundations_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf))

Surveys include

* [[Shinichi Mochizuki]], _Panoramic overview of inter-universal Teichmuller theory_, [pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf)

* {#Yamashita13} Go Yamashita, _FAQ on 'Inter-Universality'_ ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf))

* {#Fesenko15} [[Ivan Fesenko]], _Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions_, European Journal of Mathematics September 2015, Volume 1, Issue 3, pp 405-440 ([publisher](http://link.springer.com/article/10.1007/s40879-015-0066-0), [pdf](https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf))

* [[Minhyong Kim]], _Brief superficial remarks on Shinichi Mochizuki's Interuniversal Teichmueller Theory (IUTT), version 1_, 10/11/2015, ([pdf](http://people.maths.ox.ac.uk/kimm/papers/pre-iutt.pdf)).

* [[Taylor Dupuy]], Hodge Theaters: _[A First Look at the Big Hodge Theater](https://www.youtube.com/watch?v=V_tWUswSUoo)_,  _[ Confused Groups and Torsors](https://www.youtube.com/watch?v=8yacRMVIIHs&list=PLJmfLfPx1OecDVF-VgXNeCfAlisy74PHt)_

*  [RIMS/Symmetries and Correspondences workshop](https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html): _[Inter-universal Teichmüller Theory Summit 2016](https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-kyoto-pr.html)_

* {#Morrow17} [[Jackson Morrow]], _Kummer classes and Anabelian geometry_, notes from Super [QVNTS](http://www.dms.umontreal.ca/%7Eqvnts/): [Kummer Classes and Anabelian Geometry](http://www.uvm.edu/%7Etdupuy/anabelian.html) 2017 ([pdf](https://www.dropbox.com/s/kiszgkrrsjrqu4p/VermontNotes_Final.pdf?dl=0))

The following two papers give a _statement_ of [[Mochizuki's Corollary 3.12]] in plain language (not using the elaborate setup in the IUT papers), and **assuming** the inequality in it as given, derive concrete versions of (weakenings of) Szpiro's conjecture, again, using conventional terminology and techniques.

* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_, arXiv:[2004.13228](https://arxiv.org/abs/2004.13228)

* [[Taylor Dupuy]], [[Anton Hilado]], _Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12_, arXiv:[2004.13108](https://arxiv.org/abs/2004.13108)

[[!redirects IUTT]]

[[!redirects inter-universal Teichmueller theory]]


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* table of contents
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## Idea

In [[perturbative quantum field theory]] the [[algebra of observables]] of an [[interacting field theory]] constructed as a [[perturbation]] of the [[Wick algebra]] [[algebra of observables|of observables]] of a [[free field theory]] is called, for emphasis, the _interacting field algebra of observables_, often just "interacting field algebra", for short.

In terms of [[causal perturbation theory]], the interacting field algebra  is obtained from the free field [[Wick algebra]] of observables and the [[perturbative S-matrix]] by [[differentiation|differentiating]] _[[Bogoliubov's formula]]_, yielding a _[[Møller operator]]_.

More abstractly, the algebra of observables is the [[formal deformation quantization]] (specifically [[Fedosov deformation quantization]]) of the [[interacting field theory]] ([Collini 16](#Collini16), [Hawkins-Rejzner 16](#HawkinsRejzner16)).

$\,$

## Properties

### Causal locality of interacting field quantum observables

+-- {: .num_prop #CausalLocalityOfBogoliubovFormula}
###### Proposition
**([[causal locality]])**

As the spacetime support varies, the algebras of [[interacting field quantum observables]] spanned via the Bogoliubov formula consistitute a [[causally local net of observables]], hence an instance of [[perturbative AQFT]].

=--

([Dütsch-Fredenhagen 00, section 3](#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](#BrunettiFredenhagen99),
[Il'in-Slavnov 78](#IlinSlavnov78))

For **[[proof]]** see [this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet) at _[[S-matrix]]_.

## Related concepts

* [[Bogoliubov's formula]]

* [[quantum Møller operator]]

[[!include products in pQFT -- table]]

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

## References

The observation that the pertruabtive interacting field quantum observables form a [[causally local net of quantum observables]] is due to

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32. ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

then rediscovered in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], *Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds*, Commun. Math. Phys. **208** (2000) 623-661 [[math-ph/9903028](https://arxiv.org/abs/math-ph/9903028), [doi:10.1007/s002200050004](https://doi.org/10.1007/s002200050004)]

and made more explicit in

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

The observation that these algebras are the [[formal deformation quantization]] of the interacting field theory is due to

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

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interacting field theory

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#Contents#
* table of contents
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## Idea

A _[[Lagrangian field theory]]_ is _interacting_ if it is not a _[[free field theory]]_. Just as in the discussion at _[[free field theory]]_, there is some room for making this precise, but at the very least it should mean that the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] are _not [[linear differential equation|linear]]_.

In [[perturbative quantum field theory]] the [[algebras of observables]] of interacting field theories are [[quantization|quantized]] as [[formal power series]] in [[Planck's constant]] and in the [[coupling constant]] ([[interacting field algebra]]), as a formal expansion around the quantizaton of a [[free field theory]] ([[Wick algebra]]).

To date [[perturbative quantum field theory]] is the only way known to approach the [[quantization]] of [[interacting field theories]] in [[spacetime]] [[dimension]] $\geq 4$. In [[constructive field theory]] examples of [[non-perturbative quantum field theory|non-perturbative]] interacting quantum field theores in [[dimension]] 3 have been constructed and examples in dimension 2 are common, such as [[2d CFTs]].

## Examples

Key examples of interacting field theories are [[phi^n theory]] ([[scalar field theory]] with [[interaction]] term given by the power $(\mathbf{\Phi}(x))$ of the [[field observable]]), [[quantum electrodynamics]] (with its [[electron-photon interaction]]), pure [[Yang-Mills theory]] for [[nonabelian group|nonabelian]] [[gauge group]], [[quantum chromodynamics]], and [[gravity]].

The [[quantization of Yang-Mills theory]], as a [[non-perturbative quantum field theory]] is a famous open problem.
Yang-Mills theory with [[abelian group|abelian]] [[gauge group]] the [[circle group]] is a [[free field theory]], but coupled to a [[Dirac field]] there is a canonical [[interaction]] term that makes this the interacting field theory called _[[quantum electrodynamics]]_ (QED).  Similarly [[quantum chromodynamics]] (QCD) is an interacting field theories. These interacting theories (QED, QCD with [[Higgs field]]-coupling) make up the [[standard model of particle physics]].

## Related concepts

* [[interaction]]

* [[interacting field algebra]]

* [[interaction picture]]

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interacting quantum observables > history

> see [[interacting field algebra]]