double dimensional reduction

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

## Idea

What is called _double dimensional reduction_ is a variant of [[Kaluza-Klein mechanism]] combined with [[fiber integration]] in the presence of [[branes]]: given a [[spacetime]] of [[dimension]] $d+1$ in which a $p+1$-[[brane]] propagates, its KK-reduction results in a $d$-dimensional effective spacetime containing a $p+1$-brane together with a "doubly reduced" $p$-brane, which is the reduction of those original $(p+1)$-brane configurations that [[wrapped brane|wrapped]] the cycle along which the KK-reduction takes place.

## Definition
 {#Definition}

### Via fiber integration in ordinary differential cohomology

Let $\mathbf{H}$ be the [[smooth infinity-groupoid|smooth topos]]. For $p+1 \in \mathbb{N}$ write $\mathbf{B}^{p+1}U(1)_{conn} \in \mathbf{H}$ for the universal [[moduli stack]] of [[circle n-bundles with connection]] (given by the [[Deligne complex]]).

Notice that [[fiber integration in ordinary differential cohomology]] has the following stacky incarnation (see [here](fiber+integration+in+ordinary+differential+cohomology#ViaSmoothHomotopyType)):

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

For $\Sigma$ an [[orientation|oriented]] [[closed manifold]] of [[dimension]] $k \leq p+1$, then [[fiber integration]] in [[ordinary differential cohomology]] is reflected by a morphism of the form

$$
  \array{
    [\Sigma, \mathbf{B}^{p+1}U(1)_{conn}]
    &\stackrel{\int_\Sigma}{\longrightarrow}&
    \mathbf{B}^{p+1-k}U(1)_{conn}
    \\
    \downarrow^{[\Sigma,curv]} && \downarrow^{curv}
    \\
    [\Sigma,\mathbf{\Omega}^{p+2}]
    &\stackrel{\int_\Sigma}{\longrightarrow}&
    \mathbf{\Omega}^{p+2-k}
  }
  \,,
$$

where the vertical morphisms are the [[curvature]] maps and the bottom morphisms reflects ordinary [[fiber integration]] of [[differential forms]].

=--

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

Given a [[cocycle]]

$$
  \nabla \;\colon\; X \times \Sigma \longrightarrow \mathbf{B}^{p+1}U(1)_{conn}
  \,,
$$

on the [[Cartesian product]] of some [[smooth space]] $X$ with $\Sigma$, then its _double dimensional reduction_ is the cocycle on $X$ which is given by the composite

$$
  X \longrightarrow [\Sigma, X \times \Sigma]
  \stackrel{[\Sigma,\nabla]}{\longrightarrow}
  [\Sigma,\mathbf{B}^{p+1}U(1)_{conn}]
  \stackrel{\int_\Sigma}{\longrightarrow}
  \mathbf{B}^{p+1-k}U(1)_{conn}
  \,,
$$

where the first morphism is the [[unit of an adjunction|unit]] of the ([[Cartesian product]] $\dashv$ [[internal hom]])-[[adjunction]].

=--

### Via cyclic loop spaces
 {#ViaCyclicLoopSpaces}

We discuss here a formalization of double dimensional reduction via cyclification adjunction ([FSS 16, section 3](#FSS16), [BMSS 18, section 2.2](#BMSS18)). For more see at _[[geometry of physics -- fundamental super p-branes]]_ the _[section on double dimensional reduction](geometry+of+physics+--+fundamental+super+p-branes#DoubleDimensionalReduction)_.

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

Let $\mathbf{H}$ be any  [[(∞,1)-topos]] and let $G$ be an [[∞-group]] in $\mathbf{H}$. Then the right [[base change]]/[[dependent product]] along the canonical point inclusion $\ast \to \mathbf{B}G$ into the [[delooping]] of $G$ takes the following form: There is a pair of [[adjoint ∞-functors]] of the form

$$
  \mathbf{H}
    \underoverset
      {\underset{[G,-]/G}{\longrightarrow}}
      {\overset{hofib}{\longleftarrow}}
      {\bot}
  \mathbf{H}_{/\mathbf{B}G}
  \,,
$$

where

* $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$,

* $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|conjugation]] [[∞-action]] for $G$ equipped with its canonical [[∞-action]] by left multiplication and the argument
regarded as equipped with its trivial $G$-$\infty$-action (for $G = S^1$ the [[circle group]] this is the [[cyclic loop space]] construction).

Hence for

* $\hat X \to X$ a $G$ [[principal ∞-bundle]]

* $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]

then there is a [[natural equivalence]]

$$
  \underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    \mathbf{H}(\hat X\;,\; A)
  }
  }
    \;\;
    \underoverset
      {\underset{oxidation}{\longleftarrow}}
      {\overset{reduction}{\longrightarrow}}
      {\simeq}
    \;\;
  \underset{
     \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    \mathbf{H}_{/\mathbf{B}G}(X \;,\; [G,A]/G)
  }
  }
$$

given by

$$
  \left(
     \hat X \longrightarrow A
  \right)
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left(
    \array{
       X && \longrightarrow && [G,A]/G
       \\
       & \searrow && \swarrow
       \\
       && \mathbf{B}G
    }
  \right)
$$

=--

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

First observe that the [[conjugation action]] on $[G,X]$ is the [[internal hom]] in
the [[(∞,1)-category]] of $G$-[[∞-actions]] $Act_G(\mathbf{H})$.
Under the [[equivalence of (∞,1)-categories]]

$$
  Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
$$

(from [NSS 12](geometry+of+physics+-+fundamental+super+p-branes#NSS12)) then $G$ with its canonical [[∞-action]] is $(\ast \to \mathbf{B}G)$
and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.

Hence

$$
  [G,X]/G
    \simeq
  [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
  \;\;\;\;\;
  \in
   \mathbf{H}_{/\mathbf{B}G}
  \,.
$$

Actually, this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place, abstractly.

But now since the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ is itself [[cartesian closed (infinity,1)-category|cartesian closed]], via

$$
  E \times_{\mathbf{B}G}(-)
  \;\;\;
   \dashv
  \;\;\;
  [E,-]_{\mathbf{B}G}
$$

it is immediate that
there is the following sequence of [[natural equivalences]]

$$
  \begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} \ast,
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}(
     \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
     X
  )
  \\
  & \simeq
  \mathbf{H}(hofib(Y),X)
  \end{aligned}
$$

Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$
denotes the [[base change]] along it.

=--

## Examples

### Reduction on the circle
 {#ReductionOnTheCircle}

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

When $\Sigma = S^1$ is the [[circle]], and we think of $X \times S^1$ as a [[spacetime]] of [[11-dimensional supergravity]], then $\nabla \colon X \times S^1 \to \mathbf{B}^3 U(1)_{conn}$ may represent the [[supergravity C-field]] as a cocycle in ordinary differential cohomology. Then its double dimensional reduction in the sense of def. \ref{DoubleDimensionalReductionForDifferentialCocycles} is the differential cocycle representing the [[B-field]] on $X$, in the sense of [[string theory]].

=--

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

For $\Sigma = S^1$ a circle as in example \ref{ReductionOnTheCircle}, then the morphism $X \longrightarrow [\Sigma, X \times \Sigma]$ in def. \ref{DoubleDimensionalReductionForDifferentialCocycles} sends each point of $X$ to the loop in $X\times S^1$ that winds identically around the copy of $S^1$ at that point. Hence in this case it would make sense to consider, more generally, for each $p \in \mathbb{Z}$ the "order $p$" double dimensional reduction, given by the operation where one instead considers the map that lets the loop wind $p$ times around the $S^1$.

The resulting double dimensional reduction is just $p$-times the original one, so in a sense nothing much is changed, but maybe it is suggestive that now we are looking at the space of $C_p$-[[fixed points]] of the [[free loop space]] (for $C_p$ the [[cyclic group]] of order $p$). In [[E-infinity geometry]] this fixed-point structure on the [[free loop spaces]] makes the derived function algebras -- the [[topological Hochschild homology]] of the original function algebras -- be [[cyclotomic spectra]].

=--

### For general super $p$-branes

Double dimensional reduction for the super-$p$-[[branes]] in $D$ dimensions which are described by the [[Green-Schwarz action functional]] corresponds to moving down and left the diagonals in the brane scan table of consistent such branes:

[[branescan.gif:pic]]

In particular

* the [[superstring]] of [[type IIA string theory]] appears as the double dimensional reduction of the [[M2-brane]] in the [[KK-compactification]] from [[11-dimensional supergravity]]/[[M-theory]] down to 10-dimensional [[type II supergravity]]/[[type II string theory]].

* the [[D4-brane]] appears as the double dimensional reduction of the [[M5-brane]] under this process;

* the double dimensional reduction of the [[super 2-brane in 4d]] is [[super 1-brane in 3d]] (see there).

### From M-branes to F-branes

* [[duality between M-theory and type IIA string theory]]

[[!include F-branes -- table]]

## Related concepts

* [[cyclic loop space]], [[cyclic loop stack]]

* [[caloron correspondence]]

## References
 {#References}

Formalization of double dimensional reduction is discussed in [[rational homotopy theory]] in

* {#FSS16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of  _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, ATMP Volume 22 (2018) Number 5 ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536))

and in full [[homotopy theory]] in

* {#BMSS18} [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], Section 2.2 of:  _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics 371: 197 (2019) ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4))

Exposition is in

* {#Schreiber16} [[Urs Schreiber]], [Section 4](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction) of _[[schreiber:Super Lie n-algebra of Super p-branes]]_, talks at [Fields, Strings, and Geometry Seminar](http://www.surrey.ac.uk/maths/news/seminars/fsg/), Surrey, 5th-9th Dec. 2016

and further discussion in

* {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortsch. d. Phys. 2020 ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010))

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

### Reduction of membrane to string

The concept of double dimensional reduction was introduced, for the case of the reduction of the [[supermembrane]] in 11d to the [[Green-Schwarz superstring]] in 10d, in

* {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) [[spire:245249](http://inspirehep.net/record/245249), doi:10.1016/0370-2693(87)91323-2]

The above "brane scan" table showing the double dimensional reduction pattern of the super-$p$-branes given by the [[Green-Schwarz action functional]] (see there for more references on this) is taken from

* {#Duff} [[Michael Duff]], _Supermembranes: the first fifteen weeks_ CERN-TH.4797/87 (1987) ([scan](http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198708425))

### Reduction of M5-brane to D4-brane

The [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]:

* [[Paul Townsend]], _D-branes from M-branes_, Phys. Lett. B373 (1996) 68-75 ([arXiv:hep-th/9512062](https://arxiv.org/abs/hep-th/9512062))

* {#APPS97a} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of _World-Volume Action of the M Theory Five-Brane_, Nucl.Phys. B496 (1997) 191-214 ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

* [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, _M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills_, JHEP 1101:083 (2011) ([arXiv:1012.2882](http://arxiv.org/abs/1012.2882))

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

### Reduction of black M2s and black M5s

The [[differential geometry]] of the double dimensional reduction of the [[M2-brane]]- and [[M5-brane]]-charges was maybe first clearly written out in:

* [[Varghese Mathai]], [[Hisham Sati]], §4 in: *Some Relations between Twisted K-theory and $E_8$ Gauge Theory*, J. High Energ. Phys. **2004** 03 (2004) 016 [[arXiv:hep-th/0312033](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://doi.org/10.1088/1126-6708/2004/03/016)]

For their [[black brane]]-reductions see:

* [[José Figueroa-O'Farrill]], [[Joan Simón]], _Supersymmetric Kaluza-Klein reductions of M2 and M5-branes_, Adv. Theor. Math. Phys. 6:703-793, 2003 ([arXiv:hep-th/0208107](https://arxiv.org/abs/hep-th/0208107))

### Reduction of M-Waves and MK6s

* [[José Figueroa-O'Farrill]], [[Joan Simón]], _Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles_, Class. Quant. Grav.19:6147-6174, 2002 ([arXiv:hep-th/0208108](https://arxiv.org/abs/hep-th/0208108))

[[!redirects double dimensional reductions]]

[[!redirects double dimensional reduction]]


double factorial

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###Context###
#### Arithmetic
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[[!include arithmetic geometry - contents]]
=--
#### Combinatorics
+--{: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $n$ a [[natural number]], the double factorial $(2n+1)!!$ is defined to be the product $(2n+1)(2n-1)\ldots 1$. Alternatively, in terms of the ordinary [[factorial]],

\[(2n-1)!! = \frac1{2^n} \frac{(2n)!}{n!}\]

so that in particular, $(-1)!!$ is defined to be $1$. Relatedly, we may define $(2n)!! = 2^n n!$.

Double-factorials have a number of applications in [[enumerative combinatorics]]. They are particularly prone to appear whenever dealing with [[binomial coefficients]]

\[\binom{x}{n} = \frac{x(x-1)\ldots (x-n+1)}{n!}\]

in the case $x = 1/2$ or $x = -1/2$, or when dealing with middle binomial coefficients $\binom{2n}{n}$, or when dealing with the values of the [[Gamma function]] at half-integers.

According to the combinatorial interpretation below, the exponential generating function of the sequence $a_n$ defined by $a_{2n} = (2n-1)!!$ and $a_{2n+1} = 0$ is

\[\sum_{n \geq 0} \frac{a_n x^n}{n!} = \exp(x^2/2).\]

This is related to the fact that the double-factorials also crop up in calculations dealing with [[Gaussian integrals]] such as

\[\int_0^\infty p(x) e^{-x^2/2}\; d x\]

for even polynomials $p$, with consequent applications in [[quantum mechanics]], for example calculations surrounding the [[quantum harmonic oscillator]]. See the section below on moments of Gaussian distributions.

## Combinatorial interpretation

The double-factorials $(2n-1)!!$ count the number of [[involutions]] without [[fixed points]] on a set with $2n$ elements, or the number of [[partitions]] of a $(2n)$-element set into $2$-element sets, or the number of isomorphism classes of [[rooted chord diagrams]] with $n$ chords. This follows readily from the exponential generating function expression above, and follows readily by considering the [[species]] composition of the exponential species $\exp$ (the terminal object in the category of species) with the species $x^2/2$, defined to be terminal at $2$-element sets and empty at others.

## Moments of the standard Gaussian distribution

The computation begins with a famous observation

\[\frac1{\sqrt{2\pi}} \int_{-\infty}^\infty e^{-x^2/2}\; d x = 1\]

which says that $x \mapsto \frac1{\sqrt{2\pi}} e^{-x^2/2}$ is a probability distribution on $\mathbb{R}$ with Lebesgue measure.

+-- {: .num_prop}
###### Proposition
For each $n \geq 0$,

\[\frac1{\sqrt{2\pi}} \int_{-\infty}^\infty x^n e^{-x^2/2}\; d x = a_n\]

where $a_n$ are the MacLaurin coefficients in $e^{y^2/2} = \sum_{n \geq 0} \frac{a_n y^n}{n!}$, namely, $a_n = 0$ if $n$ is odd, and $a_{2n} = (2n-1)!!$.
=--

+-- {: .proof}
###### Proof
By matching MacLaurin coefficients, it is enough to show

\[\sum_{n \geq 0} \left(\frac1{\sqrt{2\pi}} \int_{-\infty}^\infty x^n e^{-x^2/2} \; d x\right) \frac{y^n}{n!} = e^{y^2/2}\]

However, the left side equals

\[\array{
\frac1{\sqrt{2\pi}} \displaystyle\int_{-\infty}^\infty \sum_{n \geq 0} \frac{(x y)^n}{n!} e^{-x^2/2}\; d x & = & \frac1{\sqrt{2\pi}} \displaystyle\int_{-\infty}^\infty e^{x y} e^{-x^2/2} \;d x \\
 & = & e^{y^2/2} \cdot \frac1{\sqrt{2\pi}} \displaystyle\int_{-\infty}^\infty e^{-(x-y)^2/2} \;d x \\
 & = & e^{y^2/2}
}\]

where the last line results from the famous observation by substituting $x-y$ for $x$.
=--

## References

See also

* Wikipedia, _[Double factorial](https://en.wikipedia.org/wiki/Double_factorial)_

[[!redirects double factorials]]

[[!redirects double-factorial]]
[[!redirects double-factorials]]


double fibration

# Double fibrations

* toc
{: toc}

## Idea

A version of a [[Grothendieck fibration]] for [[double categories]].  There are several possible definitions; see the references.

## References

* Michael Lambert, _Discrete double Fibrations_ ([arXiv:2101.06734](https://arxiv.org/abs/2101.06734)), 2021.

* [[Geoffrey Cruttwell]], Michael Lambert, [[Dorette Pronk]], Martin Szyld, _Double Fibrations_ ([arXiv:2205.15240](https://arxiv.org/abs/2205.15240)), 2022.

* Benedikt Fröhlich, Lyne Moser, _Yoneda lemma and representation theorem for double categories_, ([arXiv:2402.10640](https://arxiv.org/abs/2402.10640)), 2024

[[!redirects double fibrations]]


double field theory

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###Context###
#### String theory
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[[!include string theory - contents]]
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#### Duality
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[[!include duality - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The term _double field theory_ has come to be used for [[field theory]] ([[prequantum field theory]]/[[quantum field theory]]) on [[spacetimes]] which are [[T-folds]] ([[doubled geometries]]) hence for "[[T-duality]]-[[equivariance|equivariant]] field theory".

## Proposal of formalization in para-Hermitian geometry

The most popular proposal of formalization of double field theory is in the context of para-Hermitian geometry.

### Doubled space as para-Hermitian manifold
The use of para-hermitian geometry in Double Field Theory was introduced by Izu Vaisman in ([Vai 12](#Vaisman)), then by David Svoboda ([Svo 18](#Svoboda)).

\begin{defn} An _almost para-complex manifold_ is a [[manifold]] $M$ equipped with a vector bundle endomorphism $F\in\mathrm{End}(T M)$ such that $F^2=1$ and its $\pm 1$ eigenbundles $T^\pm M$ have same rank. \end{defn}

\begin{defn} The _para-complex projectors_ are the canonical projectors onto $T^\pm M$ defined by $P_\pm = \frac{1}{2}(1\pm F)$ \end{defn}

\begin{defn} A $\pm$_-para-complex manifold_ is an almost para-complex manifold $(M,F)$ such that $T^\pm M$ of $F$ is Frobenius integrable as a distribution. A _para-complex_ manifold is a manifold that is both $+$-para-complex and $-$-para-complex. \end{defn}

\begin{rmk} A doubled manifold $M$ equipped with the $O(d,d)$-structure $\eta$ carries a natural almost para-hermitian structure. On patches $U$ with coordinates $(x^\mu,\tilde{x}_\mu)$ we have the canonical para-complex structure
$$F\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial x^\mu},\,\, F\frac{\partial}{\partial \tilde{x}_\mu} = - \frac{\partial}{\partial \tilde{x}_\mu}$$
with eigenbundles

* $T^+U=\big\langle\frac{\partial}{\partial x^\mu}\big\rangle$,

* $T^-U=\big\langle\frac{\partial}{\partial \tilde{x}_\mu}\big\rangle$,

with associated foliations

* $\mathcal{F}_+=\{x^\mu = \mathrm{const}\}$,

* $\mathcal{F}_-=\{\tilde{x}_\mu = \mathrm{const}\}$.

In analogy with [[complex geometry]], if we define $\Omega^{r,s}(M)$ as the space of sections of $\Lambda^r(T^+M)\wedge\Lambda^s(T^-M)$, we have the decomposition
$$
    \Omega^k(M)=\bigoplus_{k=r+s}\Omega^{r,s}(M)
$$
Let us call $\pi^{r,s}:\Omega^{r+s}(M)\rightarrow\Omega^{r,s}(M)$ the canonical projector induced by $P_\pm$.

In analogy with complex geometry, there are _para-Dolbeault operators_:

* $\mathrm{d}^+=\pi^{r+1,s}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r+1,s}(M)$,

* $\mathrm{d}^-=\pi^{r,s+1}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r,s+1}(M)$.

with properties:

* $\mathrm{d} = \mathrm{d}^+ + \mathrm{d}^-$,

* $(\mathrm{d}^\pm)^2= 0$,

* $\mathrm{d}^+\mathrm{d}^-+\mathrm{d}^-\mathrm{d}^+ =0$.

We can also define [[Lie derivatives]] on the eigenbundles $T^\pm M$ by
$$
    \mathcal{L}^\pm_{X_\pm}\xi = (\mathrm{d}^\pm\iota_{X_\pm} + \iota_{X_\pm}\mathrm{d}^\pm)\xi
$$
for any vector $X_\pm\in\Gamma(T^\pm M)$ and $\xi\in\Omega^{r,s}(M)$.
\end{rmk}

\begin{defn} An _almost para-hermitian manifold_ is an almost para-complex manifold $M$ equipped with a compatible metric, i.e a symmetric tensor $\eta\in\mathrm{Sym}^2(T M)$ such that $\eta(F\cdot\,,F\cdot\,)=-\eta(\,\cdot\,,\,\cdot\,)$. \end{defn}

\begin{rmk} The contraction with the metric $\eta$ defines two isomorphisms

$$
  \phi^\pm : T^\pm M \rightarrow (T^\mp M)^\ast
$$

by

* $\phi^\pm(X_\pm) =  X_\pm^\flat$ and

* $(\phi^\pm)^{-1}(\xi_\pm)=\xi^\sharp$

that map a vector in $T^\pm M$ to a $1$-form in $(T^\mp M)^\ast$ and vice-versa. We used the notation $\flat,\sharp$ for the musical isomorphisms induced by the metric $\eta$ between $T M$ and $T^\ast M$.

This can be used to define a new couple of isomorphisms
$$
    \Phi^\pm : T M \longrightarrow T^\pm M\oplus(T^\pm M)^\ast
$$
that maps a vector a vector $X=X_++X_-$ with $X_\pm\in T^\pm M$ into $X_+ + X_-^\flat$. \end{rmk}

\begin{defn} A _para-hermitian manifold_ is an almost para-hermitian manifold $(M,\eta,F)$ such that $(M,F)$ is a para-complex manifold. \end{defn}

###Foliated Courant algebroid and spacetime
Given a $+$-para-hermitian manifold $(M,\eta,F)$, consider the triple $\big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+M}\big)$ where

* the skew-symmetric bracket $[\,\cdot\,,\,\cdot\,]_+:=[\,\cdot\,,\,\cdot\,]|_{T^+ M}$

* the anchor $1_{T^\pm M}:=1_{T M}|_{T^\pm M}$
are the restrictions of Lie bracket and identity of $T M$.

Since $M$ is $+$-para-hermitian we have that $T^+M$ is integrable and therefore it is the tangent bundle $T^+M = T\mathcal{F}_\pm$ of a foliation $\mathcal{F}_+$. This means that this triple is just the tangent Lie algebroid of the foliation $\mathcal{F}_+$:
$$
    \big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+ M}\big) = \big(T\mathcal{F}_+,[\,\cdot\,,\,\cdot\,]_{T\mathcal{F}_+},1_{T\mathcal{F}_+}\big).
$$
There is a natural Courant algebroid structure on the bundle $T^+ M\oplus(T^+ M)^\ast$ with skew-symmetric pairing
$$
    [X+\alpha,Y+\beta]_{+} = [X,Y]_+ + \mathcal{L}_X^+\beta -\mathcal{L}_Y^+\alpha +\mathrm{d}_+(\iota_Y\alpha)
$$
and symmetric pairing
$$
    \langle X+\alpha,Y+\beta\rangle_+ = \iota_X\beta + \iota_Y\alpha.
$$

The isomorphism $\Phi^+:T^+ M\oplus (T^+ M)^\ast\rightarrow T^+ M\oplus T^- M = T M$ previously defined induces a Courant algebroid isomorphism and hence a Courant algebroid structure on $T M$. This induces a metric $\eta$ on $M$ and a skew-symmetric pairing on $T M$ by
$$
   [[X_++X_-,Y_++Y_-]]_+ := [X_+,Y_+] + \Big[\mathcal{L}_{X_+}^+ Y_-^\flat -\mathcal{L}_{Y_+}^+ X_-^\flat +\mathrm{d}_+\big(\eta(X_-,Y_+)\big)\Big]^\sharp,
$$
where $X\in T M$ is split in $X_\pm=P_\pm X$.

\begin{rmk} Since $M$ is assumed $+$-para-hermitian, $T^+ M\oplus (T^+ M)^\ast$ can be written as $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$. Therefore we constructed an isomorphism between the Courant algebroid on the whole $T M$ and the generalized tangent bundle $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$ of the foliation. In other terms para-hermitian geometry of the doubled manifold $M$ reduces to Generalized Geometry of physical spacetime. \end{rmk}

\begin{rmk} The same argument can be clearly applied to $T^-M$ too. \end{rmk}

###C-bracket
In previous section we assumed that the $+1$-eigenbundle $T^+M$ is integrable. This is equivalent to assuming that there exists a well defined foliation that can be interpreted as the physical spacetime. However it is possible to construct a more general bracket that does not require such an assumption, but only an almost para-complex structure. Therefore it works even when a global physical spacetime foliation is not defined. This is achieved by Vaisman with the definition of _C-bracket_ by using a generalization of the notion of Levi-Civita connection (look ([Vai 2012](#Vaisman))).

In the special case of an (integrable) para-hermitian manifold C-bracket is given by

$$
    [[ X+\alpha,Y+\beta ]]_{\mathrm{C}} = [X,Y] + \mathcal{L}_X\beta -\mathcal{L}_Y\alpha + \mathrm{d}\big(\eta(X+\alpha,Y+\beta)\big) +
    [\alpha,\beta]^\ast + \mathcal{L}^\ast_\alpha Y -\mathcal{L}^\ast_\beta X - \mathrm{d}^\ast\big(\eta(X+\alpha,Y+\beta)\big)
$$

where $[\,\cdot\,,\,\cdot\,]$, $\mathcal{L}_X$ and $\mathrm{d}$ are usual Lie bracket, Lie derivative and differential on $T\mathcal{F}_+$. On the other hand $[\,\cdot\,,\,\cdot\,]^\ast$, $\mathcal{L}^\ast_\alpha$ and $\mathrm{d}^\ast$ are Lie bracket, Lie derivative and differential on $T\mathcal{F}_+^\ast$ induced by the former ones.

### Analogy with geometric quantization
The analogy between [[geometric quantization]] and DFT was firstly noticed by [[David Berman]].

Given a symplectic manifold $(M,\omega)$ there exist couples of lagrangian foliations $\mathcal{F}_+,\mathcal{F}_-$ of $M$ defined by
$$
\forall X,Y\in T\mathcal{F}_\pm, \: \omega(X,Y)=0.
$$
For example for a symplectic space $(\mathbb{R}^{2d},\omega=\mathrm{d}x^\mu\wedge\mathrm{d}p_\mu)$ we can have $\mathcal{F}_+ = \{x^\mu = \mathrm{const} \}$ and $\mathcal{F}_-= \{p_\mu = \mathrm{const} \}$. But notice that any symplectic rotation of this choice is a couple of lagrangian foliations that works fine.

Heuristically, in geometrical quantization we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to "select" a physical spacetime $\mathcal{F}_+$ from the whole symplectic-covariant theory on $M$.

Similarly in DFT, when $M$ is an (integrable) para-hermitian manifold we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to "select" a physical spacetime $\mathcal{F}_+$ from the whole T-duality-covariant theory on $M$.

## Relation between para-Hermitian geometry and gerbes

Let us start from a [[bundle gerbe]] $\pi:P\rightarrow M$ on a $d$-dimensional [[smooth manifold]] $M$. This is locally isomorphic to $P|_U \cong U\times \mathbf{B}U(1)$, where $\mathbf{B}U(1)$ is the [[circle 2-group]] and $U$ is an open set. If we consider an [[atlas]] $\mathbb{R}^d\rightarrow U$, we immediately have an [[atlas]] $\mathbb{R}^d\times \mathbf{B}U(1)\rightarrow U\times \mathbf{B}U(1)$ which extends the former. In other words the [[bundle gerbe]] is locally modelled on the [[Lie 2-group]] $\mathbb{R}^d\times \mathbf{B}U(1)$.

The [[Lie 2-algebra]] of $\mathbb{R}^d\times \mathbf{B}U(1)$ is
 $\mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1)$. An [[atlas]] for this [[Lie 2-algebra]] will be just an ordinary [[Lie algebra]] $\mathfrak{a}$ and a [[homomorphism]] of [[Lie n-algebras]]

$$ f: \mathfrak{a} \longrightarrow \mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1) $$

which is surjective in the lowest degree. What choice of $\mathfrak{a}$ can we make?

To find a working atlas we can consider the dual [[homomorphism]] of [[Chevalley-Eilenberg algebras]], i.e. the injective [[homomorphism]]

$$ f^\ast: \mathrm{CE}(\mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1)) \longrightarrow \mathrm{CE}(\mathfrak{a}). $$

The [[dg-algebra]] $\mathrm{CE}(\mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1))$ has a degree $2$ generator $b$ which is annihilated by the differential, i.e. $\mathrm{d} b=0$. Therefore its image $\omega := f^\ast b$ in $\mathfrak{a}$ is a of degree $2$ element which satisfies the equation

$$ \mathrm{d} \omega = 0 .$$

Since $\mathbb{R}^d$ must be a subalgebra of $\mathfrak{a}$ and $\omega$ must be rotation-invariant, we have no other choice than $\mathfrak{a}=\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast $ and

$$ \omega = \mathrm{d}\tilde{x}_\mu \wedge \mathrm{d}x^\mu. $$

where $\mathrm{d}\tilde{x}_\mu$ and $\mathrm{d}x^\mu$ are respectively generators of $\mathrm{CE}((\mathbb{R}^d)^\ast)$ and $\mathrm{CE}(\mathbb{R}^d)$.

The vector space $\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast$ equipped with $\omega = \mathrm{d}\tilde{x}_\mu \wedge \mathrm{d}x^\mu \in \Omega_{\mathrm{LI}}^1(\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast)$ is nothing but the local space on which para-Hermitian geometry is modelled. On the other side of the [[atlas]] map $f: \mathfrak{a} \longrightarrow \mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1)$ we have an [[extended Minkowski spacetime]], on which [[bundle gerbes]] are modelled.

This relation is introduced in ([Alf20](#Alfonsi20)).

## Relation with non-geometry

Double field theory is supposed to formalize the [[non-geometric backgrounds]] of [[type II string theory]].

### Examples

* [[T-fold]],

* [[Exotic branes]] of [[type II string theory]].

## Related concepts

* [[type II supergravity]]

* [[doubled geometry]], [[generalized geometry]]

* [[non-geometry]]

  * [[T-fold]]

  * [[exotic brane]]

* [[exceptional field theory]], [[exceptional generalized geometry]]

* [[generalized Scherk-Schwarz reduction]], [[gauged supergravity]], [[tensor hierarchy]]

## References

### Foundational papers

The idea of "doubled spacetime geometry" is a variant of the idea of [[T-folds]], due to

* {#Hull06} [[Chris Hull]], _Doubled geometry and T-folds_ JHEP0707:080,2007  ([arXiv:hep-th/0605149](http://arxiv.org/abs/hep-th/0605149))

The coinage of the term "double field theory" for field theory on such doubled geometry goes back to

* [[Chris Hull]], [[Barton Zwiebach]], _Double Field Theory_, JHEP 0909:099,2009 ([arXiv:0904.4664](http://arxiv.org/abs/0904.4664))

### Formalization in para-Hermitian geometry

Discussion about para-Hermitian formalism started in

* {#Vaisman} Izu Vaisman, _On the geometry of double field theory_ Journal of Mathematical Physics 53, 033509 (2012)  ([arXiv:1203.0836](https://arxiv.org/abs/1203.0836v1))

Para-Hermitian formalism further developed and generalized in

* {#Svoboda} David Svoboda, _Algebroid structures on para-Hermitian manifolds_ Journal of Mathematical Physics 59, 122302 (2018)  ([arxiv:1802.08180](https://arxiv.org/abs/1802.08180))

### Formalization in higher geometry
  {#ReferencesFormalizationInHigherGeometry}

Discussion of [[double field theory]] using [[higher differential geometry]]:

Discussion in the context of [[L-infinity algebra]] includes

* [[Andreas Deser]], [[Jim Stasheff]], _Even symplectic supermanifolds and double field theory_, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 ([arXiv:1406.3601](http://arxiv.org/abs/1406.3601))

Discussion of an extended version of Riemannian geometry suitable for the description of double field theory

* [[Andreas Deser]], [[Christian Saemann]], _Extended Riemannian Geometry I: Local Double Field Theory_, ([arXiv:1611.02772](https://arxiv.org/abs/1611.02772))

Comprehensive discussion in [[higher differential geometry]]:
{#ReferencesInHigherDifferentialGeometry}

* {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortsch. d. Phys. 2020 [[arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010)]

  (relating [[Kaluza-Klein compactification]] on [[principal ∞-bundles]] to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]], ...)

* {#Alfonsi20} [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

* [[Luigi Alfonsi]], _Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory_ ([arXiv:2102.10970](https://arxiv.org/abs/2102.10970))


double functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **double functor** is a [[functor]] between [[double categories]].  Since if $C$ and $D$ are double categories, the only possible sort of functor $C\to D$ is a double functor, it is unambiguous to leave off the adjective and simply speak about "functors" between double categories.

However, just like [[2-functors]], double functors do come in different flavors: strict, pseudo/strong, lax, and oplax.  Moreover, these various flavors can be chosen more or less independently in the two directions of a double category (vertical and horizontal).  Thus we can have functors which are strict in both directions, strict in one direction and pseudo in the other, pseudo in both directions, strict in one direction and lax in the other, and so on.  (It's not clear whether lax+lax or lax+oplax are sensible, though.)

## Definitions

If $\mathcal{C}$ and $\mathcal{D}$ are strict double categories, i.e. [[internal categories]] in $\mathbf{Cat}$, then a strict double functor $\mathcal{C} \to \mathcal{D}$ is by definition an internal functor in $\mathbf{Cat}$.  Thus, it takes objects, arrows of both sorts, and squares in $\mathcal{C}$ to the same structures in $\mathcal{D}$, preserving sources and targets and also preserving all identities and composites.

In explicit terms:

\begin{definition}
   Let $\mathcal{C}$ and $\mathcal{D}$ be double categories. Denote by $v\mathcal{C}$ (resp. $v\mathcal{D}$) and $h\mathcal{C}$ (resp. $h\mathcal{D}$) the horizontal and vertical categories, respectively, underlying $\mathcal{C}$ (resp. $\mathcal{D}$). Moreover, let us denote by $\vert$ the horizontal composition of squares and by $\frac{\phantom{xx}}{\phantom{xx}}$ the vertical one.

A double functor $F: \mathcal{C} \to \mathcal{D}$ consists of the following data:

* a function $F : Ob \mathcal{C} \to Ob \mathcal{D}$, called the *object part*,
* a functor $vF : v\mathcal{C} \to v\mathcal{D}$, called the *vertical part*, coinciding with $F$ on objects,
* a functor $hF : h\mathcal{C} \to h\mathcal{D}$, called the *horizontal part*, coinciding with $F$ on objects,
* for every square
$$\array{& X & \overset{f}\rightarrow & Z & \\
          g & \downarrow & \alpha &\downarrow & g'\\
          &Y & \underset{f'}\rightarrow& W & \\
}$$
in $\mathcal{C}$, a square
$$\array{& FX & \overset{hF(f)}\rightarrow & FZ & \\
          vF(g) & \downarrow & \square F\alpha &\downarrow & vF(g')\\
          &FY & \underset{hF(f')}\rightarrow& FW & \\
}$$
in $\mathcal{D}$,

obeying the following axioms:

* $\square F(\alpha \vert \beta) = \square F(\alpha) \vert \square F(\beta)$,
* $\square F\left(\frac{\alpha}{\beta}\right) = \frac{\square F(\alpha)}{\square F(\beta)}$,
* For each horizontal map $f : X \to Z$, $\square F(1^h_f) = 1^h_{hF(f)}$ ($\square F$ preserves horizontal identity squares),
* For each vertical map $g : X \to Y$, $\square F(1^v_g) = 1^v_{vF(g)}$ ($\square F$ preserves vertical identity squares)

\end{definition}

In practice, $vF$, $hF$ and $\square F$ are all denoted with the same symbol $F$ and the correct assignment is deduced from the type of object it is applied to.

## References

Definitions of double categories, double functors and more can be found in

* {#Myers21} [[David Jaz Myers]], _Categorical Systems Theory_, book in preparation, [github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book)

Double pseudofunctors are discussed in

* {#Shulman07} [[Michael Shulman]], section 6 of _Comparing composites of left and right derived functors_, New York Journal of Mathematics 17 (2011), 75-125 ([arXiv:0706.2868](https://arxiv.org/abs/0706.2868))

[[!redirects double pseudofunctor]]
[[!redirects double pseudofunctors]]

[[!redirects pseudo double functor]]
[[!redirects lax double functor]]
[[!redirects oplax double functor]]
[[!redirects double functors]]
[[!redirects pseudo double functors]]
[[!redirects lax double functors]]
[[!redirects oplax double functors]]


double gluing

# Double gluing

* table of contents
{: toc}

## Idea

**Double gluing**, coupled with orthogonality, is a method for creating new models of [[linear logic]] such as [[star-polycategories]] and [[star-autonomous categories]]. It is a variation of gluing methods (such as [[Artin gluing]]) that works for classical rather than intuitionistic (linear) logic. It is a general method that encompases many existing spaces such as [[phase spaces]], [[coherence spaces]], [[finiteness spaces]], [[totality spaces]], [[probabilistic coherence spaces]].

Although the name presumably refers to gluing along "two functors at once", it turns out to also be closely connected to [[double categories]], and in particular to [[comma double categories]].

## Double gluing along hom-sets with orthogonality

### Coherence spaces via double gluing

We begin with the definition of [[coherence spaces]], presented in a way that is amenable to generalization.

If $u\subset X$ and $v\subseteq X$, write $u\perp v$ if $|u\cap v|\leq 1$. Given $U\subseteq P(X)$, let ${U}^\perp=\{v \subseteq X \mid \forall u \in U. u \perp v\}$. Now a [[coherence space]] can be given by a set $X$ together with a set $U\subseteq P(X)$ of cliques and a set $V\subseteq P(X)$ of co-cliques that are orthogonal in the sense that $U=V^\perp$ and $V=U^\perp$.

In this presentation of coherence spaces, we can regard $u$ as a relation $1\to X$, and $v$ as a relation $X\to 1$. The relation $\perp$ is then a family of subsets $\perp_X\subseteq [1\to X]\times [X\to 1]$.

### General definition of double gluing along hom-sets with tight orthogonality

Let $C$ be a [[symmetric monoidal category]] with an object $I^*$, and let a family $\perp_X\subseteq C(I,X)\times C(X,I^*)$ of relations be given. (For coherence spaces, let $C=Rel$ and $I=I^*$ and $\perp$ be as above.

Let $X$ be an object of $C$. The relation $\perp$ defines a [[Galois connection]] between $C(I,X)$ and $C(X,I^*)$ in the usual way: for ${U}\subseteq C(I,X)$ we have ${U}^\perp = \{ v \in C(X,I^*) \mid \forall u\in {U}. u\perp v \}$, and likewise for ${V}\subseteq C(X,I^*)$ we have ${V}^\perp = \{ u \in C(I,X) \mid \forall v\in {V}. u\perp v \}$.

We define a **tight orthogonality space** $(X,U,V)$ to be an object $X\in C$ together with ${U}\subseteq C(I,X)$ and ${V}\subseteq C(X,I^*)$ that is a fixed point of this Galois connection, ${U}^\perp = {V}$ and ${V}^\perp= U$.
A **morphism** between tight orthogonality spaces $(X_1,{U_1},{V_1})\to (X_2,{U_2},{V_2})$ is a morphism $f\in C(X_1,X_2)$ such that

1. if $u\in {U_1}$, then $f\circ u \in {U_2}$;
2. if $v\in{V_2}$ then $v\circ f\in {V_1}$.

**Theorem:** (Hyland-Schalk): If an orthogonality $\bot_X\subseteq C(I,X)\times C(X,I^*)$ on a star-autonomous category satisfies four conditions for orthogonalities, a symmetry condition, and a precision condition, then the tight orthogonality spaces form a [[star-polycategory]].

One source of good orthogonality relations $\bot_X\subseteq C(I,X)\times C(X,I^*)$ are the focuses $F\subseteq C(I,I^*)$, from which we can let
$u\bot_X v$ if $(v\circ u)\in F$. However, not all the useful orthogonalities arise from focuses.

### Examples

* [[coherence spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|\leq 1$.
* [[totality spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|=1$.
* [[finiteness spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|\lt\omega$.
* [[phase semantics]]: Let $M$ be a commutative monoid, and let $F\subseteq M$ be given. Regarding $M$ as a one-object category, with object $I$, we put $u\bot_I v$ if $vu\in F$. Then an object of the orthogonality category $(I,U,V)$ is determined by a "fact" $U\subseteq M$. However, to get the poset version of phase semantics we restrict the morphisms of the orthogonality category to the ones that come from the identity $I\to I$.
* probabilistic coherence spaces: Let $C$ be the category of weighted relations, i.e. countable sets and $[0,\infty]$-valued matrices between them. Let $u\bot_X v$ if $(v\circ u)\in [0,1]$. In other words, the focus $F\subseteq C(I,I^*)$ comprises the scalars in $[0,1]$. One then cuts down further to impose a bounded completeness condition.
* quantum causal structure: Let $C=CP$, the category of finite dimensional Hilbert spaces and completely positive maps between them (see [[quantum operation]], although here we do not begin by insisting that they are trace preserving). Let $u\bot_X v$ if $v\circ u=1$. In other words, the focus $F\subseteq CP(\mathbb{C},\mathbb{C})$ is the singleton map $\{1\}$. Now, for any finite dimensional Hilbert space $X$, $\{trace_X\}^\perp$ comprises "states" of $X$, and a morphism
$(X,\{trace_X\}^\perp,\{trace_X\}^{\perp\perp})\to (Y,\{trace_Y\}^\perp,\{trace_Y\}^{\perp\perp})$ is the same thing as a [[quantum operation]], i.e. a completely positive trace preserving map.
* [[Frölicher spaces]]: Let $C=Set$, let $I=I^*=\mathbb{R}$, and let $u\bot_X v$ if $(v\circ u):\mathbb{R}\to\mathbb{R}$ is smooth. In other words, the focus $F\subseteq C(\mathbb{R},\mathbb{R})$ comprises the smooth maps.

## General double gluing

Let $C$ be any (symmetric) [[polycategory]] and $E$ a co-subunary polycategory (i.e. all morphisms have codomain arity 0 or 1), and let $L:C\to Chu(E)$ be a polycategory functor.  We consider $C$ as a vertically discrete [[double polycategory]] and $L$ as a functor $C\to \mathbb{C}hu(E)$ into the [[double Chu construction]].  Similarly, consider $Chu(E)$ as a vertically discrete double polycategory, including into $\mathbb{C}hu(E)$ as the horizontal polycategory.

The **double gluing** polycategory is then the [[comma double category|comma double]] polycategory (i.e. the [[comma object]] in the 2-category of polycategories)
\begin{tikzcd}
  Gl(L) \ar[d] \ar[r] \ar[dr,phantom,"\Downarrow"] & Chu(E) \ar[d] \\
  C \ar[r,"L"'] & \mathbb{C}hu(E)
\end{tikzcd}

Note that:

1. If $C$ is a multicategory (i.e. a co-unary polycategory), then so is $Gl(L)$.

2. If $E$ has a counit that is terminal so that $Chu(E) = Chu(E,1)$, $C$ is a representable multicategory (i.e. a symmetric monoidal category), and $E$ is representable and closed with products so that $Chu(E,1)$ is a $\ast$-autonomous category, then polycategory functors $C\to Chu(E)$ are equivalent to pairs of a lax symmetric monoidal functor $L:C\to E$ and a functor $K:C\to E^{op}$ together with a "contraction" $L(R) \otimes K(R\otimes S) \to K(S)$ satisfying a few axioms.  This is how the definition is phrased in [Hyland and Schalk](#HylandSchalk).

3. If $C$ is a $\ast$-polycategory, then $\ast$-polycategory functors $L:C\to Chu(E)$ are equivalent to functors $U_{\le 1} C\to E$ where $U_{\le 1} C$ is the underlying co-subunary polycategory of $C$.  If $E$ has a counit that is terminal so that $Chu(E) = Chu(E,1)$, then these are equivalent to multicategory functors $U_{=1} C \to E$.  In particular, we can double-glue along any lax symmetric monoidal functor with $\ast$-autonomous domain.

If $C$ and $E$ are closed and representable with sufficient limits and colimits, then one can show (similarly to the conditions for a Chu construction to be representable) that $Gl(L)$ is also representable (as a multicategory or a polycategory, respectively) and closed.  Thus, double gluing can produce closed symmetric monoidal and $\ast$-autonomous categories.

The case of double gluing along hom-functors, discussed above, is the case when $E=Set$ so that $Chu(E,1)$ is $\ast$-autonomous, as above, and $L(X) =C(I,X)$ with $K(X) = C(X,I^*)$, together with a restriction that the gluing morphisms be monic (and the additional "orthogonality" restriction.

## References

* [[Martin Hyland]] and Andrea Schalk, _Glueing and orthogonality for models of linear logic_, Theoretical Computer Science 294 (2003) 183–231 ([pdf](https://core.ac.uk/download/pdf/21173316.pdf))
 {#HylandSchalk}

* [[Aleks Kissinger]] and Sander Uijlen, _A  categorical semantics for causal structure_. Arxiv [1701.04732](https://arxiv.org/abs/1701.04732).


double groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _double groupoid_ is, equivalently,

* a [[double category]] in which all [[1-morphisms]] and [[2-morphisms]] are [[isomorphisms]];

* a [[groupoid]] object [[internalization|internal]] to the [[1-category]] [[Grpd]];

* an [[n-fold groupoid]] for $n = 2$.

Equipped with the relevant extra [[stuff, structure, property]] one obtains notions of double [[topological groupoids]], [[double Lie groupoids]], etc.

## Definition

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

Let $\mathbf{H}$ be a [[(2,1)-topos]], hence a [[(2,1)-category]] whose [[objects]] may be thought of as [[groupoids]] equipped with some geometric structure ([[stacks]]). Then a **double groupoid** with that geometric structure is a [[groupoid object in an (∞,1)-category|groupoid object in an (2,1)-category]] in $\mathbf{H}$, hence a [[simplicial object]]

$$
  \mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}
$$

which satisfies the groupoidal [[Segal conditions]].

=--

In the literature, the following special cases of def. \ref{DoubleGroupoidInStacks} are often taken to be the default notion of "double groupoid".

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

The archetypical special case of def. \ref{DoubleGroupoidInStacks} is that where $\mathbf{H} = $ [[Grpd]] is the [[(2,1)-category]] of bare (geometrically discrete) [[groupoids]].

=--

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

A special case of example \ref{BareDoubleGroupoids}, in turn, are bare double groupoids in the image of the embedding $Grpd_1^{\Delta^{op}} \to Grpd^{\Delta^{op}}$, where $Grpd_1$ is the [[1-category]] of groupoids (suppressing the [[2-morphisms]] given by [[natural isomorphisms]]). A [[groupoid object]] in $Grpd_{1}$ is equivalently a pair of groupoids $\mathcal{G}_1$ and $\mathcal{G}_0$ equipped with [[functors]] $s,t \colon \mathcal{G}_1 \to \mathcal{G}_0$, $i \colon \mathcal{G}_0 \to \mathcal{G}_1$ and $\circ \colon \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 \to \mathcal{G}_1$ that satisfy the usual axioms of a [[small category]] [[groupoid]] without any non-trivial [[natural isomorphisms]] weakening them. This is called a **strict double groupoid**.

=--

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

If one writes out the structure functors

$$
  \array{
    \mathcal{G}_1
    \\
    {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}}
    \\
    \mathcal{G}_0
  }
$$

of a double groupoid $\mathcal{G}_\bullet$ themselves in components, one obtains a square diagram of the form

$$
  \array{
    \mathcal{G}_{1,1}
    &
    \stackrel{\to}{\to}
    &
    \mathcal{G}_{1,0}
    \\
    \downarrow \downarrow && \downarrow \downarrow
    \\
    \mathcal{G}_{0,1}
    &
    \stackrel{\to}{\to}
    &
    \mathcal{G}_{0,0}
  }
$$

(where now we are notationally suppressing the degeneracy maps/identity assigning maps, for readability). In this form double groupoid are presented in traditional literature.

=--

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

For $\mathbf{H} = $ [[SmoothGrpd]], double groupoids in $\mathbf{H}$ which are in the inclusion of [[LieGrpd]]${\Delta}^{op} \to$ [[SmoothGrpd]]${}^{\Delta^{op}}$ are called **[[double Lie groupoids]]**.

=--

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

More generally, one can consider double groupoids in an arbitrary [[(∞,1)-topos]] $\mathbf{H}$, to be a 3-coskeletal [[groupoid object in an (∞,1)-category]] consisting degreewise of [[1-truncated]] objects. The [[realization]] map

$$
 \underset{\to}{\lim} \colon \mathbf{H}^{\Delta^{op}} \to \mathbf{H}
$$

restricted to such double groupoids is a presentation of [[2-truncated]] objects in $\mathbf{H}$.

=--

## Related concepts

* [[2-group]]

* [[double Lie algebroid]]

* [[2-category equipped with proarrows]]

* [[groupoid object in an (infinity,1)-category]]

## References

Double [[Lie groupoids]] are discussed (usually for the strict case) in

* [[Ronnie Brown]], [[Kirill Mackenzie]], _Determination of a double Lie groupoid by its core diagram_. J.
Pure Appl. Algebra 80 (1992), no. 3, 237–272

* [[Kirill Mackenzie]], _General theory of Lie groupoids and Lie algebroids_ Cambridge Univ. Press, Cambridge (2005)

Some homotopical aspects of double groupoids and their relationship to  [[homotopy 2-types]] are explored in

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

[[!redirects double groupoids]]


double Lie algebroid

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

#Contents#
* table of contents
{:toc}

## Idea

A double Lie algebroid is the [[Lie algebroid]]-analog of a [[Lie groupoid]]-[[double groupoid]]; essentially a Lie algebroid object [[internalization|internal]] to the category of Lie algebroids.

## Examples

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

For $(A \stackrel{\rho}{\to} X, [-,-] : \wedge^2\Gamma(A) \to \Gamma)$ a [[Lie algebroid]], its **tangent double Lie algebroid** $T A$ is the degreewise [[tangent Lie algebroid]] of the base space $X$ and of the total space $A$, respectively

$$
  \array{
    T A &\stackrel{d \rho}{\to}& T X
    \\
    \downarrow && \downarrow
    \\
    A &\stackrel{\rho}{\to}& X
  }
  \,.
$$

=--

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

For

$$
  \array{
    \mathcal{Mor}_1 &\stackrel{\to}{\to}& \mathcal{Mor}_0
    \\
    \downarrow && \downarrow
    \\
    \mathcal{Obj}_1 &\stackrel{\to}{\to}& \mathcal{Obj}_0
  }
$$

a [[double Lie groupoid]], applying [[Lie differentiation]] degreewise yields a double Lie algebroid

$$
  \array{
    Lie(\mathcal{Mor})
    \\
    \downarrow
    \\
    Lie(\mathcal{Obj})
  }
  \,.
$$

=--

## Related concepts

* [[double Lie algebroid]], [[Lie algebroid-groupoid]], [[double Lie groupoid]]

* [[foliation of a Lie algebroid]]

## References

The notion originates somewhere around

* [[Kirill Mackenzie]], _Double Lie algebroids and the double of a Lie bialgebroid_ ([arXiv:math/9808081](http://arxiv.org/abs/math/9808081))

Further discussion is in

* [[Kirill Mackenzie]], _Double Lie algebroids and second-order geometry. I_. Adv. Math. 94 (1992), no. 2, 180–239

  _Double Lie algebroids and second-order geometry. II_. Adv.Math. 154 (2000), no. 1, 46–75. [dg-ga/9712013]([arXiv:dg-ga/9712013](http://arxiv.org/abs/dg-ga/9712013)).

A textbook account is in

* [[Kirill Mackenzie]], _General theory of Lie groupoids and Lie algebroids_ Cambridge Univ. Press, Cambridge, 2005.MR2157566.

[[!redirects double Lie algebroids]]


double Lie groupoid

[[!redirects double Lie groupoids]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The combination of the notions of [[double groupoid]] and [[Lie groupoid]].

## Related concepts

* [[double Lie algebroid]], [[Lie algebroid-groupoid]], **double Lie groupoid**

* [[Lie 2-groupoid]]

* [[higher differential geometry]]

## References

* [[Ronnie Brown]], [[Kirill Mackenzie]], _Determination of a double Lie groupoid by its core diagram_. J.
Pure Appl. Algebra 80 (1992), no. 3, 237–272

* [[Kirill Mackenzie]], _General theory of Lie groupoids and Lie algebroids_ Cambridge Univ. Press, Cambridge (2005)

[[!redirects double Lie groupoids]]


double limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **double limit** is the appropriate notion of [[limit]] for a [[double category]]. Just as double categories generalise [[2-categories]], double limits generalise [[2-limits]].

## Definition

For now, see [Grandis and Paré (1999)](#GP).

## Examples

* [[tabulator]]

## Properties

* A double category has all small double limits if and only if it has small double products, double equalisers, and [[tabulators]].

* A 2-category admits all [[weighted 2-limits]] if and only if it admits all double limits. Double limits thus provide a "[[conical colimits|conical]]" approach to 2-limits that is an alternative to [[marked 2-limits]].

* A 2-category admits all [[flexible limits]] if and only if it admits all **persistent** double limits.

## Variants

* A **pseudo double limit** is analogous to a [[pseudo limit]] in 2-category theory.

* A **persistent double limit** is a limit that is equivalent to a pseudo double limit. $\mathbb{I}$-indexed double limits are persistent if and only if each connected component of the underlying 2-category of $\mathbb{I}$ has a [[weak initial object]].

## Related concepts

* [[2-limit]]
* [[marked 2-limit]]

## References

* {#BastianiEhresmann74} A. Bastiani, [[Charles Ehresmann]], pages 272-273 of  _Multiple functors. I. Limits relative to double categories_, Cah. Top. Géom. Différ. Catég. 15 (1974) 215–292

* [[Robert Paré]], _Double limits_, Talk given at [[Category Theory conference|Category Theory 1989]], ([pdf](https://www.mscs.dal.ca/~pare/July1989.pdf))

* [[Dominic Verity]], _Enriched categories, internal categories and change of base_ Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html))

* {#GP} [[Marco Grandis]] and [[Robert Paré]], _Limits in double categories_, [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques, Vol. 40, No. 3, 1999, pp. 162-220. [[cahiers](http://archive.numdam.org/item/CTGDC_1999__40_3_162_0/)]

* [[Marco Grandis]] and [[Robert Paré]], _Adjoint for double categories_, [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques, Vol. 45, No. 3, 2004, pp. 193-240. [[cahiers](http://archive.numdam.org/item/CTGDC_2004__45_3_193_0/)]

A notion of **pseudo double limit** is introduced in:

* [[Marco Grandis]] and [[Robert Paré]], _Persistent double limits_, [[Cahiers]] de topologie et géométrie différentielle catégoriques, Vol. 60, No. 3, 2019, pp. 255-297. [[cahiers](http://cahierstgdc.com/index.php/volume-lx/)]

The connection between persistent double limits and flexible limits is proven in:

* [[Marco Grandis]] and [[Robert Paré]], _Persistent double limits and flexible weighted limits_, 2019, [pdf](https://www.mathstat.dal.ca/~pare/DblPrs2.pdf)

[[!redirects double limits]]


double loop group

## References

* Sergey Arkhipov, [[Kobi Kremnizer]], _2-gerbes and 2-Tate spaces_ ([arXiv:0708.4401](http://arxiv.org/abs/0708.4401))

[[!redirects double loop groups]]


double negation

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

# Double negation
* table of contents
{: toc}

## Idea

In [[logic]], double negation is the operation that takes $P$ to $\neg{\neg{P}}$, where $\neg$ is [[negation]].  In other words, double negation is the [[composite]] of [[negation]] with itself. This is a [[closure operator]]/[[modality]] and as such a special case of a [[continuation monad]].

The _[[double negation translation]]_ says that a [[proposition]] $P$ is provable in [[classical logic]] precisely if its double negation $\not \not P$ is provable in [[constructive logic]].

The _double negation topology_ closes morphisms under double negation (def. \ref{DoubleNegationTopology} below), and its [[category of sheaves]] forms a [[boolean topos]] (prop. \ref{DoubleNegationSheavesFormBooleanTopos} below). This notably serves to capture the _[[forcing]]_ of [[set theory]] in terms of [[topos theory]] ([[classifying topoi]]), see also remark \ref{RelationToForcing} below.

## In logic

In [[classical logic]], the double negation of any [[truth value]] or [[proposition]] is itself.  More abstractly, double negation is the [[identity function]] on any [[boolean algebra]].

In [[intuitionistic logic]], double negation is weaker than the identity.  That is, we have $P \Rightarrow \neg{\neg{P}}$ but not conversely.  In [[paraconsistent logic]], it is the other way around.  More abstractly, this holds in any [[Heyting algebra]] (intuitionistic) or its dual (paraconsistent).

In [[linear logic]], double negation is the identity again, although linear logic also has notions of intuitionistic negation and paraconsistent negation which act as above.

## In locale theory{#double_negation_locale}

Even in [[classical mathematics]], a [[frame]] is a Heyting algebra but not a boolean algebra.  Accordingly, double negation is usually not the identity on a frame.  However, the operation of double negation is a [[nucleus]] on any frame.

Thus, every [[locale]] $L$ has a [[sublocale]] given by that nucleus, called the __double negation sublocale__ and denoted $L_{\neg\neg}$.  This sublocale is [[dense sublocale|dense]], and in fact it is the smallest dense sublocale of $L$, the [[intersection]] of all dense sublocales.

In [[classical mathematics]], if $L$ is a [[spatial locale]], then we have $L = L_{\neg\neg}$ if and only if $L$ is the [[discrete locale]] on some [[set]] $S$ of points.  In [[constructive mathematics]], the same holds except that $S$ must also have [[decidable equality]].

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

Let $\mathcal{O}_X$ be the sheaf of [[continuous map|continuous]] (or [[smooth map|smooth]], or [[holomorphic map|holomorphic]], or [[regular map|regular]]) functions on a [[topological space]] (or [[smooth manifold]], or [[complex manifold]], or [[reduced scheme]]) $X$. Then the [[direct image|pushforward]] of the [[inverse image|pullback]] of $\mathcal{O}_X$ to the smallest dense sublocale of $X$ is the sheaf of [[meromorphic functions]] on $X$ (i.e. sections over an open subset $U$ are given by sections of $\mathcal{O}_X$ defined on some dense open subset $V \subseteq U$).
=--

## In topos theory {#topology}

The notion of double negation sublocale may be [[categorification|categorified]] from locales to [[toposes]].

If $E$ is a [[topos]] with [[subobject classifier]] $\Omega$, there is a [[negation]] operator $\neg \colon \Omega \to \Omega$, defined by virtue of the fact that $\Omega$ is an [[internalisation|internal]] [[Heyting algebra]].

### Definition

+-- {: .num_defn #DoubleNegationTopology}
###### Definition/Proposition
**(double negation topology)**

The double negation morphism

$$
  \not \not \colon \Omega \to \Omega
$$

constitutes a [[Lawvere-Tierney topology]] on $\mathcal{E}$.

This is called the **double negation topology**.
=--

+-- {: .proof}
###### Proof
The topology axioms can be formulated in purely equational form, i.e., as equations between operations of the form $\Omega^n \to \Omega$. By the Yoneda lemma, it suffices to verify the corresponding equations between transformations $Hom(-, \Omega)^n \to Hom(-, \Omega)$, which boils the problem down to checking the equations for ordinary Heyting algebras in $Set$. For ordinary Heyting algebras, proofs may be found [here](Heyting+algebra#ToBooleanAlgebras).
=--

### Example

* {#quiv}In case of $Set^{\rightrightarrows}$, the presheaf topos of directed graphs, the action of $\neg\neg$ as a [[closure operator]] on subgraphs $X\subseteq Y$ amounts to adding to the edge set of $X$ all the edges in $Y$ between vertices that are in $X$. The patient reader will find further details on the $\neg\neg$-topology for this elementary example spelled out at length at [[Quiv]].

### Properties
  {#Properties}

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

The sheaf topos $\mathcal{E}_{\not \not} \hookrightarrow \mathcal{E}$ corresponding to the [double negation topology](#DoubleNegationTopology) (def. \ref{DoubleNegationTopology}) is a [[Boolean topos]].

=--

This appears for [[sheaf toposes]] $\mathcal{E}$ as ([MacLane-Moerdijk, theorem VI 3](#MacLaneMoerdijk)), and in the general case ([[elementary toposes]]) as a special case of ([Johnstone, Lemma A4.5.21](#Johnstone)).

The following says that $\mathcal{E}_{\not \not}$ is the smallest subtopos $\mathcal{E}_j$ such that $\emptyset$ is a $j$-sheaf. This property looks innocent but when thinking of $\mathcal{E}$ as a generalized (topological) space becomes, as in the case of locales, rather remarkable.

+-- {: .num_prop #smallest_dense_subtopos}
###### Proposition
$\mathcal{E}_{\not \not} \hookrightarrow \mathcal{E}$ is the smallest [[dense subtopos]].
=--

([Johnstone 2002, below Corollary A4.5.20](#Johnstone), or [Johnstone 1977](#Johnstone77), p.140)

+-- {: .num_prop #smallest_j-dense}
###### Proposition
$\not \not$ is the smallest topology $j$ on $\mathcal{E}$ such that the canonical mono $(\top,\bot):2=1\coprod 1\rightarrowtail\Omega$ is $j$-dense.
=--

This is theorem 1.4. in [Caramello (2009)](#Caramello09).

+--{: .num_prop #Boolean_mono}
###### Proposition
$\neg\neg$ is the smallest [[Lawvere-Tierney topology|topology]] $j$ on $\mathcal{E}$ such that all monomorphisms of the form $A\vee\neg A\rightarrowtail E$ for subobjects $A\rightarrowtail E$ in $\mathcal{E}$ are $j$-dense.
=--

This appears as proposition 6.2 in [Caramello (2012a)](#Caramello12a).

From the above we have that $\mathcal{E}_{\not\not}$ is a [[Boolean topos|Boolean]] and [[dense subtopos]]. In fact, even better, we have

+-- {: .num_prop #boolean_subtopos}
###### Proposition
$\not\not$ is the unique topology $j$ such that (1) $j$ is [[dense subtopos|dense]], i.e. $j(0)=0$, and (2) the sheaf topos $\mathcal{E}_j$ is [[Boolean topos|Boolean]].
=--
+-- {: .proof}
###### Proof
It remains to show that (1) and (2) imply that $j=\not\not$.  First note that the dense monos corresponding to $j$ are classified by the subobject classifier $\Omega_j$ of $\mathcal{E}_j$.  Since (2) implies that $\Omega_j$ is an internal [[Boolean algebra]], it follows that the dense subobjects of any object $X$ form a Boolean algebra.

This Boolean algebra is a reflective sub-poset of the [[Heyting algebra]] of all subobjects of $X$, whose reflector is lex, i.e. preserves finite [[meets]].  Thus, it will suffice to show that if $B$ is a Boolean algebra that is a lex-reflective sub-poset of a Heyting algebra $H$ and if $0\in B$, then $B = \{ U | U = \neg\neg U \}$.

To show this, first note that the Boolean negation in $B$ is the restriction of the Heyting negation in $H$.  Thus, Booleanness of $B$ implies $U=\neg\neg U$ for all $U\in B$.  Thus, it remains to show that if $U=\neg\neg U$ then $U\in B$.  But since $0\in B$ and $B$ is an [[exponential ideal]], by the definition $\neg U = (U\Rightarrow 0)$ it follows that $\neg\neg U\in B$ for any $U$.  Thus, if $U=\neg\neg U$ then $U\in B$ as well.
=--

Another, slightly more general, way to state this is is the following (cf. [Blass-Scedrov 1983](#BlassScedrov83), p.19, [Caramello 2012](#Caramello12), p.9):

+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be a topos. A topology $j$ satisfies $j\le\neg\neg$, i.e. $j$ is [[dense subtopos|dense]], iff $(\mathcal{E}_j)_{\not\not}=\mathcal{E}_{\not\not}$.
=--

Notice that, though these results prevent a topos from having more than one _dense_ Boolean subtopos, nothing prevents a topos from having more than one _Boolean_ subtopos e.g. the [[Sierpinski topos]] $Set^{\to}$ has two non trivial ones that complement each other in the [[lattice of subtoposes]]. This example, incidentally, also shows that in the [above proposition](#negdense) just $(\mathcal{E}_j)_{\neg\neg}\cong\mathcal{E}_{\neg\neg}$ wouldn't do.

As the smallest dense subtopos, $\mathcal{E}_{\not\not}$ becomes important for Lawvere's calculus of [[Aufhebung]]:

+-- {: .num_prop #RelationToAufhebungof01}
###### Proposition

The smallest  _essential_ subtopos $\mathcal{E}_j$ that is dense (in other words the [[Aufhebung]] of $0 \dashv 1$) has the property that $\mathcal{E}_{\neg\neg}\le \mathcal{E}_j$ and concides with $\mathcal{E}_{\neg\neg}$ in case the latter is an [[essential geometric morphism|essential]] subtopos: $\mathcal{E}_j=\mathcal{E}_{\neg\neg}$ (cf. [Lawvere-Menni 2015](#LawvereMenni15)).[^law]
=--

For further discussion of this relation see at [[dense subtopos]].

[^law]: [Lawvere (1991)](#Lawvere91) says around p.8: "_The base in fact seems in examples to be determined by the given category of Being itself, either as the latter's QD reflection with the extra localness condition supplying the right adjoint pure Becoming inclusion, or else (for example simplicial sets) as the double-negation sheaves with the extra essentialness condition supplying the left adjoint inclusion (in the latter case it is in [[Aufhebung|Hegelian fashion]] always the smallest level for which both 0,1 are sheaves)"_.

The double negation topology is closely related to the class of [[skeletal geometric morphism|skeletal geometric morphims]] i.e. $f:\mathcal{F}\to\mathcal {E}$ that restrict to a geometric morphism $\mathcal{F}_{\neg\neg}\to\mathcal{E}_{\neg\neg}$ e.g. skeletal geometric morphisms are the 1-cells in 2-category of toposes in which Boolean toposes are co-reflective (cf. Johnstone (2002, p.1008)).

The next propositions consider the important special case of $\neg\neg$ on [[presheaf topos|presheaf toposes]]:

+-- {: .num_prop}
###### Proposition
Let $C$ be a small category admitting a [[calculus of fractions|right calculus of fractions]] with respect to the set $\Sigma$ of _all_ morphisms and let $G$ be the free groupoid generated by $C$, i.e. $G\cong C\Sigma^{-1}$. Then the following holds:
$$
Sh_{\neg\neg}(Set^{C^{op}})\cong Set^{G^{op}}\quad .
$$
=--

This appears as ex.5.2 in [Johnstone (1977, p.162)](#Johnstone77). It applies e.g. to $C$ a commutative monoid.

+-- {: .num_prop}
###### Proposition
For every [[presheaf topos]] $[C^{op}, Set]$ the double negation topology coincides with the [[dense topology]].
=--

This appears as [MacLaneMoerdijk, corollary VI 5](#MacLaneMoerdijk).

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

Let $C$ be a [[poset]]. Then the double negation sheaf topos
$ Sh_{\not \not}(C) \hookrightarrow [C^{op}, Set]$
satisfies the [[axiom of choice]].
=--

This appears as [MacLaneMoerdijk, corollary VI 9](#MacLaneMoerdijk).

+-- {: .num_remark #RelationToForcing}
###### Remark
**(relation to [[forcing]])**

Essentially because of prop. \ref{DoubleNegationSheavesSatisfyAxiomOfChoice}, double-negation sheaves on posets are the basic context for _[[forcing]]_ in [[set theory]] (since set theorists generally want the [[axiom of choice]] to be preserved in forcing models). For such a use of double negation in the so called **Cohen topos** see at _[[continuum hypothesis]]_.

=--

###Remarks

* A topos $\mathcal{E}$ such that $\mathcal{E}_{\neg\neg}$ is an [[open subtopos]] is called _$\bot$-scattered_. They play a role in the modeling of [[provability logic]] (cf. [[scattered topos]]).

* The booleanization $\mathcal{E}_{\neg\neg}$ of a topos $\mathcal{E}$ has a close relative: the [[De Morganization]] $\mathcal{E}_m$.

## In higher topos theory
 {#InHigherToposTheory}

[[classical logic|Classically]] the double negation modality is equivalent to the [[n-truncation modality]] for $n = -1$ (the [[bracket type]]).  In general, it\'s still true that double negation takes any type (object in the higher topos) to a $(-1)$-type, but the bracket type ${\|A\|_{-1}}$ only entails the double negation $\neg(\neg{A})$:

there is a canonical [[function]]

$$
  {\|A\|_{-1}} \longrightarrow \neg(\neg{A})
$$

and this is a [[1-epimorphism]] precisely if the [[law of excluded middle]] holds.

## Related entries

* [[law of double negation]]

* [[Heyting algebra]]

* [[Boolean algebra]]

* [[double negation translation]]

* [[dense subtopos]]

* [[logical topology]], [[synthetic differential topology]]

* [[skeletal geometric morphism]]

* [[de Morganization]]

* [[scattered topos]]

[[!include logic symbols -- table]]

## References

Informal exposition of double negation with an eye towards [[physics]] is in

* [[Andrej Bauer]], _[Intuitionistic mathematics for physics](http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/)_, 2008

In [[topos theory]]:

* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_ , Academic Press 1977 (Dover reprint 2014). (pp.139-140)

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vols. I,II_, Oxford UP 2002. (pp.211,219-220,1008)

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (chap. VI, in particular sec.VI.1)

In [[homotopy type theory]]:

* [[Univalent Foundations Project]], section 3.4 of  _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

Discussion in relation to [[cohesion]] and the [[sharp modality]] is in

* {#Lawvere91} [[F. W. Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ , pp.1-13 in LNM **1488** Springer Heidelberg 1991.

More detailed discussion of this is in

* {#LawvereMenni15} [[William Lawvere]], [[Matías Menni]], _Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness_, TAC **30** no. 26 (2015) pp.909-932. ([abstract](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html))

* {#Menni18} [[Matías Menni]], _The unity and identity of opposites between decidable objects and double-negation sheaves_ , JSL to appear. ([preprint](https://drive.google.com/file/d/18P2qsihtZG7_eCT5VdjvQ1I7T-U_S7y9/view))

Other useful references include

* {#BlassScedrov83}[[Andreas Blass]], Andrej Scedrov, _Boolean Classifying Topoi_ , JPAA **28** (1983) pp.15-30.

* {#Caramello09}[[Olivia Caramello]], _De Morgan classifying toposes_ , Advances in Mathematics **222** no.6 (2009) pp.2117-2144. ([arXiv:0808.1519](http://arxiv.org/abs/0808.1519))

* {#Caramello12a}[[Olivia Caramello]], _Universal models and definability_ , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. ([arXiv:0906.3061](http://arxiv.org/abs/0906.3061))

* {#Caramello12}[[Olivia Caramello]], _Topologies for intermediate logics_ , arXiv:1205.2547 (2012). ([abstract](http://arxiv.org/abs/1205.2547))

* D. S. Mcnab, _Some applications of double-negation sheafification_ , Proc. Edinburgh Math. Soc. **20** (1977) pp.279-285.

[[!redirects double negation]]
[[!redirects double negations]]
[[!redirects double-negation]]

[[!redirects not not]]
[[!redirects not-not]]

[[!redirects double negation sublocale]]
[[!redirects double negation sublocales]]
[[!redirects double-negation sublocale]]
[[!redirects double-negation sublocales]]
[[!redirects double negation nucleus]]
[[!redirects double negation nuclei]]

[[!redirects double-negation nucleus]]
[[!redirects double-negation nuclei]]

[[!redirects double negation topology]]
[[!redirects double negation topologies]]
[[!redirects double-negation topology]]
[[!redirects double-negation topologies]]

[[!redirects double negation modality]]
[[!redirects double negation modalities]]

[[!redirects double negation monad]]
[[!redirects double negation monads]]

[[!redirects boolean truncation]]


double negation translation

# Contents
* table of contents
{: toc}

## Idea

**Double negation translation** is a method for converting [[propositions]] [[valid]] in [[classical logic]] into propositions valid in [[intuitionistic logic]]. It can be used to establish relative consistency results. For example, it may be possible to show that the proof of a [[contradiction]] in a classical theory could be translated to the proof of a contradiction in a constructive theory. [[Kurt Gödel]] used this technique to show that [[Peano arithmetic]] and [[Heyting arithmetic]] are equiconsistent.

Double negation translation was discovered independently by a number of mathematicians including [[Kurt Gödel]], [[Gerhard Gentzen]], and [[Andrey Kolmogorov]], and is also called the __Gödel--Gentzen negative translation__.

## Construction

The traditional descriptions are highly syntactic, but can be motivated by recalling some conceptual relationships between [[Boolean algebras]] (which are algebras for classical propositional logic) and [[Heyting algebras]] (which are algebras for intuitionistic propositional logic).

### Propositional case

Recall from the article [[Heyting algebra]] that the [[left adjoint]] $F$ to the [[forgetful functor]]

$$U \colon Bool \to Heyt$$

is given objectwise by taking a Heyting algebra $L$ to the poset $L_{\neg\neg}$ of [[regular elements]] $x \in L$, i.e., those that are fixed by [[double negation]]: $x = \neg \neg x$. It was shown that $L_{\neg\neg}$ is a Boolean algebra, and $\neg \neg \colon L \to L_{\neg\neg}$ is a Heyting algebra homomorphism which is universal among Heyting algebra maps $L \to U B$ into Boolean algebras.

In particular, if $Heyt(S)$ is the free Heyting algebra on a set of [[variables]] $S$, it follows by composing left adjoints

$$Set \stackrel{free}{\to} Heyt \stackrel{F}{\to} Bool$$

that $Heyt(S)_{\neg\neg}$ is the free Boolean algebra on $S$.

Furthermore, it was shown in [[Heyting algebra]] that for Heyting algebras $L$,

* $\neg\neg \colon L \to L$ preserves finite [[meets]];

* $\neg\neg \colon L \to L$ preserves the [[implication]] operation $\Rightarrow$.

Consequently, the inclusion of regular elements $L_{\neg\neg} \to L$ also preserves meets and implications, __strictly__. This gives the following result.

+-- {: .num_thm}
###### Glivenko's Theorem
If $p \Rightarrow q$ is a [[tautology]] in classical propositional logic, then $(\neg \neg p) \Rightarrow (\neg \neg q)$ is a tautology in intuitionistic propositional logic, and conversely.
=--

+-- {: .proof}
###### Proof
Here $p$ and $q$ are [[term]] expressions in variables $x_i \in S$ over the signature $(0, 1, \vee, \wedge, \Rightarrow)$. To say $p \Rightarrow q$ is a classical tautology means that $1 \leq (p \Rightarrow q)$ holds when interpreted in $Bool(S)$. But since $Bool(S) = Heyt(S)_{\neg\neg}$, this is equivalent to saying that

$$1 \leq \neg\neg(p \Rightarrow q) = (\neg\neg p) \Rightarrow (\neg\neg q)$$

when interpreted in $Heyt(S)$, which is to say that $(\neg \neg p) \Rightarrow (\neg\neg q)$ is an intuitionistic tautology.
=--

Continuing this thought: the join $\vee$ in $Bool(S) = Heyt_{\neg\neg}$ is computed as

$$a \vee_{Bool} b = \neg\neg(a \vee_{Heyt} b)$$

since $\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}$ preserves joins (it is a left adjoint). Putting all this together, because $\neg\neg \colon Heyt(S) \to Heyt(S)_{\neg\neg}$ preserves Heyting algebra structure, we arrive at the following syntactic translation.

+-- {: .un_def #defn}
###### Definition
The double-negation translation $p \mapsto p^{N}$ on term expressions in the theory of Heyting algebras $L$ is defined by induction as follows.

* $x^N = \neg\neg x$ for variables $x$.

* $0^N = 0$ (since $\neg\neg \colon L \to L$ preserves $0$)

* $1^N = 1$ (since $\neg\neg \colon L \to L$ preserves $1$)

* $(p \wedge q)^N = p^N \wedge q^N$ (since $\neg\neg \colon L \to L$ preserves the meet operation)

* $(p \Rightarrow q)^N = p^N \Rightarrow q^N$ (since $\neg\neg \colon L \to L$ preserves the implication operation)

* $(p \vee q)^N = \neg\neg(p^N \vee q^N)$.
=--

Thus, by Glivenko's theorem, $p$ is a classical tautology if and only if $p^N$ is an intuitionistic tautology. This result may be extended to theories as well: suppose $L$ is an intuitionistic theory or Heyting algebra, given by a presentation as a coequalizer in $Heyt$:

$$Heyt(T) \stackrel{\to}{\to} Heyt(S) \to L.$$

Then, since the functor $L \mapsto L_{\neg\neg}$ is a left adjoint, it takes this coequalizer to a coequalizer

$$Bool(T) \stackrel{\to}{\to} Bool(S) \to L_{\neg\neg}$$

so that an term expression $p$ is a theorem in the classical theory $L_{\neg\neg}$ if and only if $p^N$ is a theorem in the intuitionistic theory $L$.

### First-order case

Double negation translation for a formula, $\phi$, of a [[first-order language]] is defined inductively by the following clauses:

* $\phi^N = \neg \neg \phi$, for atomic $\phi$.

* $(\phi \wedge \psi)^N = \phi^N \wedge \psi^N$.

* $(\phi \vee \psi)^N = \neg(\neg \phi^N \wedge \neg \psi^N)$ (or equivalently, $\neg \neg (\phi^N \vee \psi^N)$)

* $(\phi \to \psi)^N = \phi^N \to \psi^N$.

* $(\neg \phi)^N = \neg \phi^N$.

* $(\forall x \phi)^N = \forall x \phi^N$.

* $(\exists x \phi)^N = \neg \forall x \neg \phi^N$ (or equivalently, $\neg \neg \exists x \phi^N$).

### Higher-order case

The basic idea here is that any [[topos]] $E$ gives rise to a [[Boolean topos]] $E_{\neg\neg}$.

## References

Informal exposition of a tiny aspect of this double negation business is in

* [[Andrej Bauer]], _[Intuitionistic mathematics for physics](http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/)_, 2008

[[!redirects double negation translation]]
[[!redirects double negation translations]]
[[!redirects double-negation translation]]
[[!redirects double-negation translations]]
[[!redirects negative translation]]
[[!redirects negative translations]]


double nerve

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

## Idea ##

The _double nerve_ of a [[2-category]] $C$ is a [[bisimplicial set]] obtained by regarding $C$ as a [[Cat]]-[[enriched category]] or a [[Cat]]-[[internal category]] and then applying the ordinary [[nerve]] operation to each [[hom-object]] [[category]] to obtain first a [[simplicially enriched category]] and then to apply the obvious nerve operation again to obtain a [[bisimplicial set]].

## References ##

See for instance page 227 of

* M. Bullejos, E. Faro, V. Blanco, _A full and faithful nerve for 2-categories_ ([arXiv](http://arxiv.org/abs/math.CT/0406615))

or page 39 of

* A. Cegarra and J. Remedios, _The relationship between the diagonal and the bar constructions on a bisimplicial set_ Topology and its Applications
Volume 153, Issue 1, 1 August 2005, Pages 21--51


double Poisson structure

Related notions include [[double derivation]], [[shifted symplectic structure]], [[Poisson algebra]], [[noncommutative geometry]], [[Kontsevich-Rosenberg principle]]

* [[Michel Van den Bergh]], _Double Poisson algebras_,  Trans. of Amer. Math. Soc. (2008) 5711--5769
* [[J. P. Pridham]], _Shifted bisymplectic and double Poisson structures on noncommutative derived prestacks_, [arXiv:2008.11698](https://arxiv.org/abs/2008.11698)
* [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935)
*  [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Parameter-dependent associative Yang–Baxter equations and Poisson brackets_, Int. J. Geom. Meth. Mod. Phys. __11__:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 [doi](https://doi.org/10.1142/S0219887814600366)
* A. Pichereau, G. Van de Weyer, _Double Poisson cohomology of path algebras of quivers_, J. Algebra __319__:5 (2008) 2166--2208.

Double Poisson vertex algebra generalize double Poisson algebras:

* Alberto De Sole, [[Victor G. Kac]], Daniele Valeri, _Double Poisson vertex algebras and non-commutative Hamiltonian equations_, Adv. Math. __281__ (2015) 1025--1099 [doi](https://doi.org/10.1016/j.aim.2015.05.011)
* Luis Álvarez-Cónsul, David Fernández, Reimundo Heluani, _Noncommutative Poisson vertex algebras and Courant-Dorfman algebras_, [arXiv:2106.00270](https://arxiv.org/abs/2106.00270)

There is also a viewpoint of double algebras

* M.E.Goncharov, P.S.Kolesnikov, _Simple finite-dimensional double algebras_, J. Algebra __500__ (2018) 425--438 [doi](https://doi.org/10.1016/j.jalgebra.2017.04.020)

There are some restrictions and no-go theorems for commutative algebras

* Geoffrey Powell, _On double Poisson structures on commutative algebras_, J. Geom. Phys. __110__ (2016) 1--8 [doi](https://doi.org/10.1016/j.geomphys.2016.07.003)

category: algebra

[[!redirects double Poisson algebra]]
[[!redirects double Poisson bracket]]


double polycategory

A **double polycategory** is an [[internal category]] in the category of [[polycategories]].


double profunctor

# Contents
* tic
{:toc}

## Idea

A **double profunctor** is an appropriate notion of [[profunctor]] between [[double categories]].

Thinking of a profunctor $H : C ⇸ D$ as representing some notion of [[heteromorphism]], double profunctors either represent a notion of _vertical_ heteromorphism or _horizontal_ heteromorphism. For clarity, this article focuses on the horizontal double profunctors. Vertical double profunctors can be defined by taking the transpose.

That is, $H$ defines for each $c\in C, d \in D$ a set $H(d,c)$ of horizontal heteromorphisms from $d$ to $c$, so that a heteromorphism $h \in H(d,c)$ can be composed with horizontal morphisms $f : d' \to d$  in $D$  and $g : c \to c'$ in $C$ to yield a heteromorphism $(f;h;g) \in H(d',c')$.

Furthermore, $H$ defines, for each heteromorphisms $h \in H(d,c), h' \in H(d',c')$ and _vertical_ arrows $\alpha : d \to d'$ in $D$ and $\beta : c \to c'$ in $C$, a notion of hetero-2-cell:

$$
  \begin{matrix}
    d
    &\stackrel{h}{\to} &
    c
    \\
    {}^{\mathllap{\alpha}}\downarrow
    &\Downarrow^{\mathrlap{\phi}}&
    \downarrow^{\mathrlap{\beta}}
    \\
    d'
    &\underset{h'}{\to} &
    c'
 \end{matrix}
$$

These can be composed vertically with hetero-2-cells along $h,h'$, and horizontally with 2-cells in $D, C$ of the appropriate shape along $\alpha, \beta$.

## Definition

A double profunctor can be defined in several equivalent ways.  In each case, however, we must choose whether to regard the "vertical" or the "horizontal" direction as primary; applying transposes results in a different notion of double profunctor.

### As internal profunctors in Cat

Since a (strict) double category is an [[internal category]] in [[Cat]], it makes sense to define a **double profunctor** to be an [[internal profunctor]] in $Cat$.  Thus, if $C$ and $D$ are double categories, a double profunctor $C ⇸ D$ consists of a span $C_0 \leftarrow H \to D_0$ in $Cat$, together with actions $C_1 \times_{C_0} H \to H$ and $H\times_{D_0} D_1 \to H$ which are associative and commute with each other.

### As collages in DblCat

Another natural way to define a double profunctor is in terms of its [[collage]].  From this perspective, a double profunctor $C ⇸ D$ consists of a double category $H$, together with double functors $C\to H$ and $D\to H$ such that:

1. The images of $C$ and $D$ in $H$ are disjoint.
1. The induced functor $C\sqcup D\to H$ is bijective on objects and vertical arrows.
1. Each functor $C\to H$ and $D\to H$ is fully faithful on horizontal arrows and squares.
1. There are no horizontal arrows in $H$ from an object of $C$ to an object of $D$, and therefore also no squares whose horizontal source is a vertical arrow in $C$ and whose horizontal target is a vertical arrow in $D$.

In order to identify this with the previous definition as written, for a double category $C = C_1 \;\rightrightarrows\; C_0$ written as an internal category in $Cat$, we have to write the arrows of $C_0$ vertically and the objects of $C_1$ horizontally.

Defined in this way, double profunctors can be seen to be precisely the two-sided [[codiscrete cofibration|codiscrete cofibrations]] in the 2-category $DblCat$ of double categories, double functors, and horizontal transformations.

### As Span-valued double functors

A third natural way to define double profunctors is as functors defined on $D^{op}\times C$.  In order to make this agree with the previous two definitions, we have to consider *lax* double functors from $D^{op}\times C$ to $Span$, where $D^{op}$ denotes horizontal reversal, and $Span$ denotes the (pseudo) double category whose objects are sets, whose horizontal arrows are functions, and whose vertical arrows are spans.

### Explicit definition

Each of these definitions can be unraveled to give a more explicit definition, and thereby shown to be the same.  Explicitly, a double profunctor $H\colon C ⇸ D$ consists of the following.

* An ordinary [[profunctor]] $H^0\colon C^0 ⇸ D^0$, where now $C^0$ denotes the *horizontal* category of $C$.

* An ordinary profunctor $H^1 \colon C^1  ⇸ D^1$.

* Each $r \in H^1(q,p)$ is assigned a vertical source $s(r) \in H^0(s(q),s(p))$ and target $t(r) \in H^0(t(q),t(p))$.

* Each $h \in H^0(d,c)$ is assigned an identity $1_h \in H^1(1_d,1_c)$.

* If $r_1 \in H^1(q_1,p_1)$ and $r_2 \in H^1(q_2,p_2)$ and $t(r_1) = s(r_2)$ (hence $t(q_1)=s(q_2)$ and $t(p_1)=s(p_2)$), then there is a vertical composite $r_2 \boxminus r_1 \in H^1(q_2 q_1,p_2 p_1)$ with $s(r_2\boxminus r_1) = s(r_1)$ and $t(r_2\boxminus r_1) = t(r_2)$.

* The composition $\boxminus$ is associative and unital (with $1_h$ as identities), and interchanges with the action of the profunctor $H^1$ and the vertical composition of squares in $C$ and $D$.

## Examples

* As with any notion of profunctor, the appropriate [[hom functor|hom]] should be a double profunctor. In fact, since there are 2 kinds of morphism, there are 2 different hom profunctors. Following the convention above, the _horizontal_ hom is a profunctor $C ⇸ C$ and is given by the canonical span $C_0 \stackrel{s}{\leftarrow} C_1 \stackrel{t}{\to} C_0$ and the vertical hom is just the horizontal hom on the transpose.

* If $C$ and $D$ are [[strict 2-categories]] regarded as vertically-discrete double categories, then a double profunctor $C ⇸ D$ is the same as a $Cat$-[[enriched category|enriched]] [[profunctor]], i.e. a strict 2-functor $D^{op}\times C \to Cat$.  There is a corresponding statement in the pseudo case (see below).

* On the other hand, if $C$ and $D$ are 2-categories regarded as *horizontally*-discrete double categories, then a double profunctor $C ⇸ D$ is the same as a lax 2-functor $D^{co}\times C \to Span$, or equivalently a normal lax 2-functor $D^{co}\times C \to Prof$.

## Weakenings

The definition can be weakened to the case when the composition of $C$ and $D$ is weak in one direction or the other, or both (modulo a suitable definition of the latter situation, see [[double category]] for several possibilities).

* The internal definition can easily be weakened in the case when $C$ and $D$ are *horizontally* pseudo, since then they are simply pseudo internal categories in $Cat$.

* The collage definition, on the other hand, can easily be weakened to the case when $C$ and $D$ are *vertically* pseudo, by looking at codiscrete cofibrations in the 2-category of vertically-pseudo double categories, pseudo double functors, and horizontal transformations.  The explicit description of the collages could also be given for horizontally or doubly pseudo double categories, although in those cases it would be harder to identify them with codiscrete cofibrations, since horizontal transformations would no longer be the 2-cells in a 2-category (only a [[tricategory]] of some sort).

* The Span-valued double functor definition makes perfect sense as written if $C$ and $D$ are vertically pseudo, and also if they are horizontally or doubly pseudo, modulo a suitable definition of "vertically lax functor" in the latter two cases.

* The explicit definition can also easily be weakened, with an insertion of coherence isomorphisms for $C$ and $D$ in appropriate places.

There is also a natural notion of profunctor between [[virtual double categories]], namely the proarrows in the [[virtual equipment]] $vDblProf = KMod(Span,fc)$ whose objects are virtual double categories (here $fc$ is the free-category monad on $Span = Span(Set)$).  If $C$ and $D$ are (possibly vertically pseudo) double categories, regarded as vertically-virtual double categories, then any double profunctor $C ⇸ D$ in the sense of the above definitions can also be considered as a "virtual-double profunctor" in a straightforward way.  However, not every virtual-double profunctor between double categories is a double profunctor; those that are are those satisfying a "representability" property.

## Composites

Composition of double profunctors is, unfortunately, hard to define and not well-behaved.  Several of the above definitions suggest a possible way to compose them.  However, not all of these ideas are well-defined, and those that are well-defined are not associative.  In particular, there is not a double category of double categories, double functors, and double profunctors (although there are various weaker structures, such as a [[virtual double category]] and a [[lax double category]]).

### Using internal profunctors

Internal profunctors can be composed, using a coequalizer, in any category which has coequalizers preserved by pullback.  However, while $Cat$ has coequalizers, they are not preserved by pullback, so this does not work -- it is not even possible to define actions making the putative "composite" into a double profunctor.  (This failure has nothing to do with strictness or weakness; [[2-colimits]] in $Cat$ are also not preserved by [[2-pullbacks]].)

This definition does, however, suggest a replacement for the nonexistent double category of double profunctors.  Namely, internal categories, functors, and profunctors in *any* category with pullbacks always form a [[virtual double category]], and in fact a [[virtual equipment]].   Thus, in particular, there is a virtual equipment $DblProf$ of double categories, double functors, and double profunctors.  In terms of the explicit definition, a cell in $DblProf$ with some given boundary consists of:

* An ordinary cell in $Prof$, whose boundary is obtained by applying $(-)^0$, and

* Another ordinary cell in $Prof$, whose boundary is obtained by applying $(-)^1$, such that

* Vertical sources, targets, identities, and composites are respected.

For many purposes, having a virtual equipment of double profunctors is sufficient.  For instance, this is all we need in order to talk about [[weighted limits]], and in order to define [[generalized multicategories]].  We can also use this as a setting in which to ask whether some particular pair (or string) of double profunctors might have a composite, or a "weak composite," as described at [[tensor product]].

The virtual equipment $DblProf$ is also contained as a non-full sub-virtual-equipment of the virtual equipment $vDblProf$ of virtual double categories, functors, and virtual-double profunctors mentioned above.

### Using collages

The standard way to compose codiscrete cofibrations $C\to H \leftarrow D$ and $D\to K\leftarrow E$ is to take the pushout (or 2-pushout) $H\sqcup_D K$ and then factor $C\sqcup E \to H\sqcup_D K$ as $C\sqcup E \to H\circ K \to H\sqcup_D K$ in some way such as to make $C\sqcup E \to H\circ K$ codiscrete.  We can do this for codiscrete cofibrations in $DblCat$ and thereby obtain composites of double profunctors.  However, the resulting composition operation is not associative, since pushouts in $DblCat$ do not preserve the requisite factorizations.

I believe that the binary composite of double profunctors defined in this way is a "weak composite" in the virtual double category $DblProf$, i.e. it has a universal property relative to cells with source of length 2 only.  We can likewise define basic $n$-ary composites of codiscrete cofibrations by taking an $n$-ary pushout and then a factorization, but for $n\gt 2$, as far as I can tell, there is little relation between these composites and $DblProf$.  In particular, these composites form a "lax double category", whereas a virtual double category with all weak composites can be identified with an "oplax double category."  I also don't know whether the cells in $DblProf$ with source of length $\gt 2$ can be seen from the perspective of collages.

### Failure of associativity

Here is a concrete example of the failure of associativity, for any reasonable notion of "composite" for double profunctors.

* Let $A$ be the free double category on two composable vertical arrows $a_0 \xrightarrow {\alpha_0} a_1 \xrightarrow{\alpha_1} a_2$.
* Let $B$ have four objects, vertical arrows $\beta_0 : b_0\to b_1$ and $\beta_1 : b_1'\to b_2$, and a horizontal arrow $\beta' : b_1\to b_1'$ (plus identities).
* Let $C=A$, named instead $c_0 \xrightarrow{\gamma_0} c_1 \xrightarrow{\gamma_1} c_2$.
* Let $D$ be the free double category on one vertical arrow, $d_0 \xrightarrow{\delta} d_2$.

* Let $H:A\to B$ be the double profunctor with
  * Four heteromorphisms $h_0 :a_0 \to b_0$, $h_1 : a_1 \to b_1$, $h_1' : a_1 \to b_1'$, and $h_2 : a_2 \to b_2$.
  * Composition action $h_1 ; \beta' = h_1'$
  * Two hetero-2-cells
    $$
  \begin{matrix}
    a_0
    &\stackrel{h_0}{\to} &
    b_0
    \\
    {}^{\mathllap{\alpha_0}}\downarrow
    &\Downarrow^{\mathrlap{\theta_0}}&
    \downarrow^{\mathrlap{\beta_0}}
    \\
    a_1
    &\underset{h_1}{\to} &
    b_1
 \end{matrix}
\qquad\text{and}\qquad
  \begin{matrix}
    a_1
    &\stackrel{h_1'}{\to} &
    b_1'
    \\
    {}^{\mathllap{\alpha_1}}\downarrow
    &\Downarrow^{\mathrlap{\theta_1}}&
    \downarrow^{\mathrlap{\beta_1}}
    \\
    a_2
    &\underset{h_2}{\to} &
    b_2
 \end{matrix}
    $$
    (note that they are not vertically composable!)

* Let $K:B\to C$ be the double profunctor with
  * Four heteromorphisms $k_0 :b_0 \to c_0$, $k_1 : b_1 \to c_1$, $k_1' : b_1' \to c_1'$, and $k_2 : b_2 \to c_2$.
  * Composition action $\beta' ; k_1' = k_1$
  * Two hetero-2-cells
    $$
  \begin{matrix}
    b_0
    &\stackrel{k_0}{\to} &
    c_0
    \\
    {}^{\mathllap{\beta_0}}\downarrow
    &\Downarrow^{\mathrlap{\kappa_0}}&
    \downarrow^{\mathrlap{\gamma_0}}
    \\
    b_1
    &\underset{k_1}{\to} &
    c_1
 \end{matrix}
\qquad\text{and}\qquad
  \begin{matrix}
    b_1'
    &\stackrel{k_1'}{\to} &
    c_1
    \\
    {}^{\mathllap{\beta_1}}\downarrow
    &\Downarrow^{\mathrlap{\kappa_1}}&
    \downarrow^{\mathrlap{\gamma_1}}
    \\
    b_2
    &\underset{k_2}{\to} &
    c_2
 \end{matrix}
    $$
    (also not vertically composable!)

* Let $L:C\to D$ be the double profunctor with
  * Two heteromorphisms $\ell_0 : c_0 \to d_0$ and $\ell_2 : c_2 \to d_2$.
  * One hetero-2-cell
    $$
  \begin{matrix}
    c_0
    &\stackrel{\ell_0}{\to} &
    d_0
    \\
    {}^{\mathllap{\gamma_0;\gamma_1}}\downarrow
    &\Downarrow^{\mathrlap{\lambda}}&
    \downarrow^{\mathrlap{\delta}}
    \\
    c_2
    &\underset{\ell_2}{\to} &
    d_2
 \end{matrix}
    $$

Then in the composite $(H;K) : A\to C$, the composite hetero-2-cell $(\theta_0, \kappa_0)$ has vertical codomain $(h_1,k_1)$, while the composite hetero-2-cell $(\theta_1,\kappa_1)$ has vertical domain $(h_1',k_1')$.  But

$$(h_1,k_1) = (h_1, (\beta' ; h_1)) = ((h_1;\beta'),h_1) = (h_1',k_1').$$

Thus, $(\theta_0, \kappa_0)$ and $(\theta_1,\kappa_1)$ are vertically composable in $(H;K)$, producing a hetero-cell from $\alpha_0;\alpha_1$ to $\gamma_0;\gamma_1$.  This can then be hetero-composed with $\lambda$ to produce a hetero-2-cell in $((H;K);L)$ from $\alpha_0;\alpha_1$ to $\delta$.

However, there are no hetero-2-cells in $K$ that can be composed with $\lambda$ in $K;L$.  Thus, in $(H;(K;L))$ there is no hetero-2-cell from $\alpha_0;\alpha_1$ to $\delta$.  Therefore

$$ ((H;K);L) \neq (H;(K;L)). $$

## References

- [[Robert Paré]]. _Yoneda theory for double categories_. Theory and Applications of Categories 25.17 (2011): 436-489.

[[!redirects double profunctors]]
[[!redirects DblProf]]
[[!redirects vDblProf]]
[[!redirects virtual double profunctor]]
[[!redirects virtual double profunctors]]
[[!redirects double distributor]]
[[!redirects double distributors]]