Sp(n).Sp(1)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie group]] denoted $Sp(n).Sp(1)$
([Alekseevskii 68](#Alekseevskii68), [Gray 69](#Gray69))
or just $Sp(n)Sp(1)$ is the [[quotient group]] of the [[direct product group]] of the given [[quaternion unitary groups]] by their [[diagonal]] [[center]] [[cyclic group of order 2]].

A [[smooth manifold]] of [[dimension]] $4n$ with [[G-structure]] for this group $G = Sp(n).Sp(1)$ is a [[quaternion-Kähler manifold]].

Similarly, for $Spin(n_1)$, $Spin(n_2)$ [[spin groups]] in some dimension, the group denoted $Spin(n_1) \cdot Spin(n_2)$ or just $Spin(n_1)Spin(n_2)$ is the [[quotient group]] of the [[direct product group]] $Spin(n_1) \times Spin(n_2)$ by the [[diagonal]] [[center]] [[cyclic group of order 2]].

These products $G_1 \cdot G_2$ are examples of [[central products of groups]].

\linebreak

## Definition

+-- {: .num_defn #SpnSp1}
###### Definition

For $n \in \mathbb{N}$ with $n \geq 2$, the [[Lie group]] denoted $Sp(n).Sp(1)$ or just $Sp(n)Sp(1)$ is the [[quotient group]] of the [[direct product group]] $Sp(n) \times Sp(1)$ of [[quaternion unitary groups]] $Sp(n)$ (in particular $Sp(1) \simeq$ [[Spin(3)]]) by the [[diagonal]] [[center]] [[cyclic group of order 2]] $\mathbb{Z}_2$:

$$
  Sp(n).Sp(1)
  \;\coloneqq\;
  \big(
    Sp(n)
     \times
    Sp(1)
  \big)/_{diag}\mathbb{Z}_2
$$

hence the [[quotient group]] by the [[subgroup]]

\[
  \label{DiagonalCenter}
  \mathbb{Z}_2
  \;\simeq\;
  \big\{
    (1,1), (-1,-1)
  \big\}
  \hookrightarrow
  Sp(n) \times Sp(1)
  \,.
\]

=--

(e.g. [Čadek-Vanžura 97, Sec. 2](#CadekVanzura97))

A similar definition yields

+-- {: .num_defn #Spinn1Spinn2}
###### Definition

Write

$$
  Spin(n_1) \cdot Spin(n_2)
  \;\coloneqq\;
  \big(
    Spin(n_1)
    \times
    Spin(n_2)
  \big)/\mathbb{Z}_2
$$

for the [[quotient group]] of the [[direct product group]] of [[spin groups]] by their [[diagonal]] [[subgroup]]

$$
  \mathbb{Z}_2
  \;\simeq\;
  \big\{
    (1,1), (-1,-1)
  \big\}
  \;\hookrightarrow\;
  Spin(n_1) \times Spin(n_1)
  \,.
$$

=--

([McInnes 99a, p. 9](#McInnes99a), [Hilgert-Neeb 12, Prop. 17.3.1](#HilgertNeeb12))

Sometimes one sees the notation further generalized to include cases such as

* $Spin(n) \cdot U(1) \simeq Spin(n)\cdot Spin(2) \simeq$ [[Spin^c]],

see Example \ref{SpinnSpin2IsSpinc} below.

\linebreak

## Properties

### As the effective quotient of $Sp(n)\times Sp(1)$ acting on $\mathbb{H}^n$

The [[direct product group]] $Sp(n) \times Sp(1)$ has a canonical [[action]] on the [[quaternion]] [[vector space]] $\mathbb{H}^n$, where the factor [[Sp(n)]] acts as $n \times n$ [[quaternion unitary group|quaternion unitary]] [[matrix multiplication]] from the left, and $Sp(1)$ acts by [[diagonal]] $1 \times 1$ matrix action on each $\mathbb{H}$-summand from the right.

For instance for $n = 2$ this action controls the [[quaternionic Hopf fibration]] and its $Sp(2)$ [[equivariant function|equivariance]] (see [there](quaternionic+Hopf+fibration#EquivariantStructure)).

But this action is not an [[effective group action]]: Precisely the diagonal center (eq:DiagonalCenter) acts trivially.

There is then a [[commuting diagram]] of [[Lie groups]]

\[
  \label{CompatibilityDiagram}
  \array{
    Sp(2) \times Sp(1)
    &\longrightarrow&
    Spin(8)
    \\
    \big\downarrow && \big\downarrow
    \\
    Sp(2) \cdot Sp(1)
    &\longrightarrow&
    SO(8)
  }
\]
with the horizontal maps being [[group homomorphisms]] to [[Spin(8)]] and [[SO(8)]], respectively, the left morphism being the defining [[quotient]] projection and the right morphism the [[double cover]] morphism that defines the [[spin group]].

(e.g. [Čadek-Vanžura 97, p. 4](#CadekVanzura97))

### Lift to $Sp(n) \times Sp(1)$

([Marchiafava-Romani 76](#MarchiafavaRomani76), [Salamon 82, around Def. 2.1](#Salamon82))

(...)

\linebreak

## Examples

### $Sp(1)\cdot Sp(1)$ is $SO(4)$
 {#SO4}

The case of $Sp(n)\cdot Sp(1)$ for $n = 1$ is special, as in this case the canonical inclusion $Sp(n)\cdot Sp(1) \hookrightarrow SO(4n)$ becomes an [[isomorphism]]

$$
  Sp(1)\cdot Sp(1)
  \;\simeq\;
  SO(4)
$$

with the [[special orthogonal group]] [[SO(4)]],
and hence the compatibility diagram (eq:CompatibilityDiagram) now exhibits at the top the [[exceptional isomorphism]] $Sp(1) \times Sp(1) \simeq $ [[Spin(4)]] (see [there](Spin4#ExceptionalIsomorphisms))

In summary:

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

There is a [[commuting diagram]] of [[Lie groups]] of the form

$$
  \array{
    ( q_1, q_2 )
    &\mapsto&
    (x \mapsto q_1 \cdot x \cdot \overline{q}_2)
    \\
    Sp(1) \times Sp(1)
    &\overset{\simeq}{\longrightarrow}&
    Spin(4)
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    Sp(1)\cdot Sp(1)
    &\overset{\simeq}{\longrightarrow}&
    SO(4)
  }
$$

where

1. in the top left we have [[Sp(1)]] = [[Spin(3)]],

1. in the top right we have [[Spin(4)]],

1. in the bottom left we have [[Sp(n).Sp(1)|Sp(1).Sp(1)]]

1. in the bottom right we have [[SO(4)]]

1. the horizontal morphism assigns the [[conjugation action]] of unit [[quaternions]], as indicated,

1. the right vertical morphism is the defining [[double cover]],

1. the left vertical morphism is the defining [[quotient group]]-projection.

=--

\linebreak

### $Spin(n)\cdot Spin(2)$ is $Spin^c(n)$
  {#Spinc}

+-- {: .num_example #SpinnSpin2IsSpinc}
###### Example

For $n \in \mathbb{N}$,  group $Sp(n) \cdot Sp(2)$ in Def. \ref{Spinn1Spinn2} is the group otherwise known as [[spin^c|spin^c(n)]]:

$$
  Spin(n)\cdot Spin(2)
  \;\simeq\;
  Spin^c(n)
  \,.
$$

This is due to the identification of the [[double cover]] by [[Spin(2)]] of [[SO(2)]] with the [[real Hopf fibration]] ([this Prop](Spin2#Spin2OverSO2IsRealHopfFibration)), which identifies $Spin(2) \simeq U(1)$ compatible with the subgroupinclusion of $\mathbb{Z}_2$.

=--

(See also e.g. [Gompf 97, p. 2](#Gompf97))

### Triality
 {#Triality}

+-- {: .num_prop #Spin3DotSpin5SubgroupsInSO8}
###### Proposition
**([[Spin(5).Spin(3)]]-[[subgroups]] in [[SO(8)]])

The [[direct product group]] [[SO(3)]] $\times$ [[SO(5)]] together with the groups [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[SO(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[SO(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      Sp(2)\cdot Sp(1)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      &
      SO(8)
      &
      SO(3) \times SO(5)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      Sp(1) \cdot Sp(2)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Kollross 02, Prop. 3.3 (3)](#Kollross02))

Similarly:

+-- {: .num_prop #Spin3DotSpin5SubgroupsInSpin8}
###### Proposition
**([[Spin(5).Spin(3)]]-[[subgroups]] in [[Spin(8)]])

The groups [[Spin(5).Spin(3)]], [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[Spin(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      Sp(2)\cdot Sp(1)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      &
      Spin(8)
      &
      Spin(3) \cdot Spin(5)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      Sp(1) \cdot Sp(2)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Čadek-Vanžura 97, Sec. 2](#CadekVanzura97))

In summary:

\begin{xymatrix}
    \mathrm{Sp}(1) {\cdot} \mathrm{Sp}(2)
    \ar@{=}[dddddr]
    \ar@{_{(}->}[ddrr]|-{ \iota' }
    \\
    \\
    &&
    \mathrm{Spin}(8)
    \ar@{->>}[dddddr]
    &&
    \mathrm{Spin}(3) {\cdot} \mathrm{Spin}(5)
    \ar@{->>}[dddddr]
    \ar@{_{(}->}[ll]|-{\iota}
    \\
    \\
    \mathrm{Sp}(2) {\cdot} \mathrm{Sp}(1)
     \ar@{=}[dddddr]
     \ar@{^{(}->}[uurr]
       |<<<<<<<<<<<<<<<<<<<<<<<<<<{ \phantom{AA \atop AA} }
       |>>>>>>>>>>>>>>>>>>>>>>>{ \iota'' }
    \\
    &
    \mathrm{Sp}(1) {\cdot} \mathrm{Sp}(2)
    \ar@{_{(}->}[ddrr]|-{ \iota' }
    \\
    \\
    &&&
    \mathrm{SO}(8)
    &&
    \mathrm{SO}(3) \times \mathrm{SO}(5)
    \ar@{_{(}->}[ll]|-{\iota}
    \\
    \\
    &
    \mathrm{Sp}(2) {\cdot} \mathrm{Sp}(1)
    \ar@{^{(}->}[uurr]|-{ \iota'' }
\end{xymatrix}

### $Sp(1)Sp(1)Sp(1) = Spin(4)\cdot Spin(3)$
 {#Sp1Sp1Sp1}

+-- {: .num_example #Spin4Spin3}
###### Example
**([[Spin(4).Spin(3)]])**

The group

$$
  Spin(4)\cdot Spin(3)
  \;\coloneqq\;
  \big(
    Spin(4) \times Spin(3)
  \big)/\mathbb{Z}_2
$$

is the [[quotient group]] of the [[direct product group]] of [[Spin(4)]] with [[Spin(3)]] by the [[subgroup]]

\[
  \label{Spin4Spin3DiagonalCenter}
  \mathbb{Z}_2
  \;\simeq\;
  \big\{
    (1,1), (-1,-1)
  \big\}
  \hookrightarrow
  Spin(4) \times Spin(3)
  \,.
\]

Due to the exception [[isomorphism]] [[Spin(4)]] $\simeq$ [[Spin(3)]] $\times$ [[Spin(3)]] ([this Prop.](Spin4#ExceptionalIsoWithSpin3TimesSpin3)) this is [[isomorphism|isomorphic]] to the [[quotient group]] of the [[direct product group|direct product]] of 3 copies of [[Sp(1)]] $\simeq$ [[Spin(3)]] with itself

$$
  Spin(4)\cdot Spin(3)
  \;\simeq\;
  Sp(1)Sp(1)Sp(1)
  \;\coloneqq\;
  \big( Spin(3) \times Spin(3) \times Spin(3)\big)/_{diag} \mathbb{Z}_2
$$

by the triple diagonal center

\[
  \label{Spin3Spin3Spin3DiagonalCenter}
  \mathbb{Z}_2
  \;\simeq\;
  \big\{
    (1,1,1), (-1,-1,-1)
  \big\}
  \hookrightarrow
  Spin(3) \times Spin(3) \times Spin(3)
  \,.
\]

=--

See the references [below](#ReferencesSpin4Spin3).

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

The [[coset space]] of [[Sp(2).Sp(1)]] (Def. \ref{SpnSp1}) by [[Sp(1)Sp(1)Sp(1)]] (Def. \ref{Spin4Spin3}) is the [[4-sphere]]:

$$
  \frac{
    Sp(2)\cdot Sp(1)
  }
  {
    Sp(1)Sp(1)Sp(1)
  }
  \;\simeq\;
  S^4
  \,.
$$

This follows essentially from the [[quaternionic Hopf fibration]] and its $Sp(2)$-[[equivariant function|equivariance]]...

=--

(e.g. [Bettiol-Mendes 15, (3.1), (3.2), (3.3)](#BettiolMendes15))

### Spin-Grassmannians
 {#SpinGrassmannians}

We have the following [[coset spaces]] of [[spin groups]] by dot-products of Spin groups as above:

$$
  Spin(7)/ \big(  Spin(4)\cdot Spin(3) \big)
  \;\simeq\;
  SO(7) / \big( SO(4) \times SO(3)  \big)
  \;\simeq\;
  Gr(4, 7)
$$

is the space of [[Cayley 4-planes]] ([[Cayley 4-form]]-[[calibrated submanifolds]] in 8d [[Euclidean space]]). This happens to also be [[homeomorphism|homeomorphic]] to just the plain [[Grassmannian]] of 4-planes in 7d (recalled e.g. in [Ornea-Piccini 00, p. 1](#OrneaPiccini00)).

Similarly,

$$
  Spin(6)/ \big(  Spin(3)\cdot Spin(3) \big)
  \;\simeq\;
  SU(6)/ SO(4)
$$

is the Grassmannian of those [[Cayley 4-planes]] that are also [[special Lagrangian submanifolds]] ([BBMOOY 96, p. 7 (8 of 17)](Cayley+form#BBMOOY96)).

Moreover,

$$
  Spin(8)/ \big(  Spin(5)\cdot Spin(3) \big)
  \;\simeq\;
  Gr(3, 8)
$$

is the [[Grassmannian]] of 3-planes in 8d. ([Cadek-Vanzura 97, Lemma 2.6](#CadekVanzura97)).

\linebreak

## Related concepts

[[!include low dimensional rotation groups -- table]]

## References
 {#References}

### $Sp(n)\cdot Sp(1)$

Very early appearances of the notation $Sp(n)\cdot Sp(1)$ are mostly in discussions of [[Berger's theorem]] for exceptional [[holonomy]]:

* {#Gray69} [[Alfred Gray]], [_A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) $\cdot$ Sp(1)_](https://projecteuclid.org/euclid.mmj/1029000212), Michigan Math. J. Volume 16, Issue 2 (1969), 125-128.

* {#Alekseevskii68} [[Dmitri Alekseevskii]], [_Riemannian spaces with exceptional holonomy groups_](https://link.springer.com/article/10.1007%2FBF01075943), Functional Analalysis and its Applications (1968) 2: 97.

However, the even earlier paper:

* Joseph Wolff, _Complex homogeneous contact manifolds and quaternionic symmetric spaces_, Journal of Mathematics and Mechanics, vol. 14 (1965), pp. 1033-1048.

describes this construction as a "local [[direct product]]" of [[topological groups]] and applies it to the classification of [[quaternionic manifolds]]. The notation in the classical paper of [Bonan](#) for this group is $V_{4n} [Sp(n) \otimes_\mathbf{H} Sp(1)]$.

Of early algebraic interest is the structure theory article:

* [[Stefano Marchiafava]], Giuliano Romani, _Sul classificante del gruppo $Sp(n) \cdot Sp(1)$_, Annali di Matematica Pura ed Applicata December 1976, Volume 110, Issue 1, pp 295–319 ([doi:10.1007/BF02418010](https://doi.org/10.1007/BF02418010))

More on the [[cohomology]] of $Sp(n)\cdot Sp(1)$ and its [[classifying space]]:

* Stefano Marchiafava, Giuliano Romani, _Alcune osservazioni sui sottogruppi abeliani del gruppo $Sp(n)\cdot Sp(1)$_, Annali di Matematica 1977 ([doi:10.1007/BF02413792](https://doi.org/10.1007/BF02413792))

* [[Paolo Piccinni]], Giuliano Romani, _A generalization of symplectic Pontrjagin classes to vector bundles with structure group $Sp(n)\cdot Sp(1)$_,  Annali di Matematica pura ed applicata (1983) 133: 1 ([doi:10.1007/BF01766008](https://doi.org/10.1007/BF01766008))

* [[Paolo Piccinni]], _Vector fields and characteristic numbers on hyperkàhler and quaternion Kâhler manifolds_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992) Volume: 3, Issue: 4, page 295-298 ([dml:244204](https://eudml.org/doc/244204))

* [[Dmitri Alekseevskii]] S. Marchiafava, _Quaternionic structures on a manifold and subordinated structures_, Annali di Matematica pura ed applicata (1996) 171: 205 ([doi:10.1007/BF01759388](https://doi.org/10.1007/BF01759388))

Discussion of the lift to $Sp(n) \times Sp(1)$ appears in

* {#MarchiafavaRomani76} S. Marchiafava, G. Romani, _Sui fibrati con struttura quaternionale generalizzata_, Ann. Mat. Pura Appl. (IV) CVII, 131-157 (1976) ([doi:10.1007/BF02416470](https://doi.org/10.1007/BF02416470))

* {#Salamon82} [[Simon Salamon]], around Def. 2.1 in _Quaternionic Kähler manifolds_, Invent Math (1982) 67: 143. ([doi:10.1007/BF01393378](https://doi.org/10.1007/BF01393378))

### $Sp(2)\cdot Sp(1)$
 {#ReferencesSp2Sp1}

Articles dealing specifically with the group $Sp(2)\cdot Sp(1)$:

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], Section 2 of _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures  in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))

* {#CadekVanzura98b} [[Martin Čadek]], [[Jiří Vanžura]], _Almost quaternionic structures on eight-manifolds_, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 ([euclid:1200787905](https://projecteuclid.org/euclid.ojm/1200787905))

* {#Kollross02} Andreas Kollross, Prop. 3.3 of _A Classification of Hyperpolar and Cohomogeneity One Actions_, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 ([jstor:2693761](https://www.jstor.org/stable/2693761))

See also the references at _[[quaternion-Kähler manifold]]_.

### $Spin(n_1)\cdot Spin(n_2)$

A textbook occurrence of dot notation for general spin groups, $Spin(n_1)\cdot Spin(n_2)$, appears in

* {#HilgertNeeb12} Joachim Hilgert, [[Karl-Hermann Neeb]], Prop. 17.3.1 _Structure and Geometry of Lie Groups_, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 ([doi:10.1007/978-0-387-84794-8](https://link.springer.com/book/10.1007/978-0-387-84794-8))

The identification of $Spin \cdot Spin(2)$ with [[Spin^c]] appears for instance in

* {#Gompf97} [[Robert Gompf]], _$Spin^c$ structures and homotopy equivalences_, Geom. Topol. 1 (1997) 41-50 ([arXiv:math/9705218](https://arxiv.org/abs/math/9705218))

Discussion of central product spin groups as [[subgroups]] of [[semi-spin groups]] (motivated by analysis of the [[gauge groups]] and [[Green-Schwarz anomaly cancellation]] of [[heterotic string theory]]) is in

* {#McInnes99a} [[Brett McInnes]], p. 9 of _The Semispin Groups in String Theory_, J. Math. Phys. 40:4699-4712, 1999 ([arXiv:hep-th/9906059](https://arxiv.org/abs/hep-th/9906059))

* {#McInnes99b} [[Brett McInnes]], _Gauge Spinors and String Duality_,  Nucl. Phys. B577:439-460, 2000 ([arXiv:hep-th/9910100](https://arxiv.org/abs/hep-th/9910100))

As such these also appear as [[U-duality groups]] and their [[subgroups]], e.g.

* [[Arjan Keurentjes]], p. 10 of _The topology of U-duality (sub-)groups_, Class.Quant.Grav. 21 (2004) 1695-1708 ([arXiv:hep-th/0309106](https://arxiv.org/abs/hep-th/0309106))

### $Sp(1)Sp(1)Sp(1) \simeq Spin(4)\cdot Spin(3) $
 {#ReferencesSpin4Spin3}

The group $Spin(4)\cdot Spin(3) \simeq (Spin(3))^3/_{diag} \mathbb{Z}_2$ (Example \ref{Spin4Spin3}) is discussed in the following, largely in describing the Grassmannian of [[Cayley 4-planes]], see [there](Cayley+form#GrassmannianOfCayley4Planes):

* Wu-Chung Hsiang, Wu-Yi Hsiang, Tables A of _Differentiable Actions of Compact Connected Classical Groups: II_, Annals of Mathematics
Second Series, Vol. 92, No. 2 (1970), pp. 189-223 ([jstor:1970834](https://www.jstor.org/stable/1970834))

* {#HarveyLawson82} [[Reese Harvey]], [[H. Blaine Lawson]], theorem 1.38 of _Calibrated geometries_, Acta Math. Volume 148 (1982), 47-157 ([Euclid:1485890157](https://projecteuclid.org/euclid.acta/1485890157))

* {#BryantHarvey89} [[Robert Bryant]], [[Reese Harvey]], (3.19) in _Submanifolds in Hyper-Kähler Geometry_, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 ([jstor:1990911](https://www.jstor.org/stable/1990911))

* {#GluckMackenzieMorgan95} Herman Gluck, Dana Mackenzie, Frank Morgan, (5.20) in _Volume-minimizing cycles in Grassmann manifolds_, Duke Math. J. Volume 79, Number 2 (1995), 335-404 ([euclid:1077285156](https://projecteuclid.org/euclid.dmj/1077285156))

* {#Kerr96} Megan M. Kerr, Lemma 6.2 of _Some New Homogeneous Einstein Metrics on Symmetric Spaces_, Transactions of the American Mathematical Society, Vol. 348, No. 1 (1996), pp. 153-171 ([jstor:2155169](https://www.jstor.org/stable/2155169))

* {#BBMOOY96} [[Katrin Becker]], [[Melanie Becker]], [[David Morrison]], [[Hirosi Ooguri]], Y. Oz, Z. Yin, (3.5) of _Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds_, Nucl. Phys. B480:225-238, 1996 ([arXiv:hep-th/9608116](https://arxiv.org/abs/hep-th/9608116))

* {#KacSmilga00} [[Victor Kac]], A.V. Smilga, around (1.10) in _Vacuum structure in supersymmetric Yang-Mills theories with any gauge group_, in _[[The Many Faces of the Superworld]], pp. 185-234 World Scientific (2000)_ ([arXiv:hep-th/9902029](https://arxiv.org/abs/hep-th/9902029), [doi:10.1142/9789812793850_0014](https://doi.org/10.1142/9789812793850_0014))

* {#OrneaPiccini00} Liviu Ornea, [[Paolo Piccinni]], _Cayley 4-frames and a quaternion-Kähler reduction related to Spin(7)_, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 ([arXiv:math/0106116](https://arxiv.org/abs/math/0106116))

* {#GroveWilkingZiller} Karsten Grove, Burkhard Wilking, Wolfgang Ziller, p. 30 of _Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry_ ([arXiv:math/0511464](https://arxiv.org/abs/math/0511464))

* {#BettiolMendes15} Renato G. Bettiol, Ricardo A. E. Mendes, _Flag manifolds with strongly positive curvature_, Math. Z. 280 (2015), no. 3-4, 1031-1046 ([arXiv:1412.0039](https://arxiv.org/abs/1412.0039))

* Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Prop. 2.2 in _Clifford systems in octonionic geometry_ ([arXiv:1511.06239](https://arxiv.org/abs/1511.06239))

Discussion of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ in the context of [[super Lie algebras]] and [[superconformal geometry|superconformal symmetry]] is in:

* {#Freund78} [[Peter Freund]], p. 634 of _World topology and gauged internal symmetries_, Proc. 19th Int. Conf. High Energy Physics, Tokyo 1978 ([spire:137780](http://inspirehep.net/record/137780/), [pdf](https://cds.cern.ch/record/870701/files/c78-08-23-p617.pdf))

and possibly with the $\mathbb{Z}_2$-quotient not made explicit:

* [[Peter Goddard]] (auth.), [[Peter Freund]], K. T. Mahanthappa, p. 128 of _Superstrings_, NATO ASI Series 175, Springer 1988

* Kazuo Hosomichi, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], slide 13 of _New  Superconformal Chern-Simons Theories_ ([pdf](http://www3.ic.ac.uk/pls/portallive/docs/1/46083696.PDF))

[[!redirects Sp(2).Sp(1)]]
[[!redirects Spin(5).Spin(3)]]

[[!redirects Spn.Sp1]]

[[!redirects SpnSp1]]

[[!redirects Sp(1)Sp(1)Sp(1)]]
[[!redirects Sp(1).Sp(1).Sp(1)]]

[[!redirects central product spin group]]
[[!redirects central product spin groups]]

[[!redirects Spin(n).Spin(m)]]

[[!redirects Spin(n1).Spin(n2)]]

[[!redirects Spin(3)Spin(3)Spin(3)]]
[[!redirects Spin(3).Spin(3).Spin(3)]]

[[!redirects Spin(4).Spin(3)]]


space

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

**space** $\leftarrow$ [[Isbell duality]]  $\rightarrow$ [[algebra]]

***

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

## Idea

The term 'space' is quite generic and can mean many different things depending on context.  Often we will link it because we want to include many of these contexts at once!

Probably the default spaces are [[topological spaces]] (as defined by [[Bourbaki]]), with [[continuous maps]] between them.  There are many variants, from [[convergence spaces]] (general) to [[metric space|metric (metrisable) spaces]] (specific), all of which fall under the topic of [[topology]].  An important variation gives [[locales]]; many of the theorems from topology that require the [[axiom of choice]] for Bourbaki spaces become [[constructive mathematics|constructively]] valid for locales.

Often one restricts to [[nice topological spaces]] or to a [[nice category of spaces]]; these are not always the same kind of restriction!  There are many examples under those two headings; [[m-cofibrant spaces]] in particular allow one to identify [[homotopy equivalence]] with [[weak homotopy equivalence]].  More generally in [[homotopy theory]], one uses the [[homotopy hypothesis]] to identify spaces with $\infty$-[[infinity-groupoid|groupoids]], of which there are several models.  Since one common model is [[simplicial sets]], some homotopy theorists use "space" to mean "simplicial set."

Often one is also interested in spaces with additional structure.  For instance, smooth [[manifolds]] and [[generalized smooth spaces]] are spaces with smooth structure, and [[algebraic varieties]] and [[schemes]] are spaces with algebraic structure.  One can also speak more generally of spaces "modeled" on any suitable starting category; in this way one often comes to consider [[sheaves]] on a suitable [[site]] (and variations such as [[presheaves]], [[copresheaf|copresheaves]], or objects of an [[Isbell envelope]]) as spaces.  Generalized smooth spaces and schemes are two commonly encountered examples of spaces "modeled" on a starting category in this way.

One can also [[vertical categorification|categorify]] the concept of space.  As [[space and quantity|space is dual to quantity]], space and quantity can be [[categorification|categorified]] together; see $\infty$-[[infinity-space|space]] and compare $\infty$-[[infinity-quantity|quantity]].  Other types of categorified spaces include [[Grothendieck toposes]] (which are categorified locales) and [[stacks]] (which are categorified [[sheaves]]) and even their $\infty$-versions ($(\infty,1)$-[[(infinity,1)-topos|toposes]] and $\infty$-[[infinity-stack|stacks]]).

## Notions of Space {#NotionsOfSpace}

In [[Structured Spaces]], [[Jacob Lurie]] provides a coherent general picture of generalized notions of space in the context of
[[higher geometry]] (often called "derived geometry"). Here is an outline of the central aspects.

The **central ingredient** which we choose at the beginning to get a theory of [[higher geometry]] going is an [[(∞,1)-category]] $\mathcal{T}$ whose objects we think of as **model spaces** : the simplest objects exhibiting the geometric structures that we mean to consider.

+-- {: .standout}

**Examples for categories of model spaces**

* with smooth structure

  * $\mathcal{T} = $ [[Diff]], the category of smooth [[manifold]]s;

  * $\mathcal{T} = \mathbb{L}$, the category of [[smooth locus|smooth loci]];

* without smooth structure

  * $\mathcal{T} = (C Ring^{fin})^{op}$, the formal dual of [[CRing]]: the category of (finitely generated) algebraic [[affine scheme]]s;

  * $\mathcal{T} = (sC Ring^{fin})^{op}$, the formal dual of [[simplicial object]]s in  [[CRing]];

  * $\mathcal{T} = (E_\infty Ring^{fin})^{op}$, the formal dual of [[E-∞ ring]]s: the category of (finitely generated) algebraic derived [[affine scheme]]s.

=--

These [[(∞,1)-category|(∞,1)-categories]] $\mathcal{T}$ are naturally equipped with the structure of a [[site]] (and a bit more, which we won't make explicit for the present purpose). Following [[Jacob Lurie]] we call such a $\mathcal{T}$ a **(pre-)[[geometry (for structured (infinity,1)-toposes)|geometry]]** .

Every pregeometry $\mathcal{T}$ gives rise to a [[geometry (for structured (infinity,1)-toposes)|geometry]] $\mathcal{G}$ -- it's _geometric envelope_ . Roughly speaking, this contains not just the original test spaces but also their [[higher geometry|derived version]]. More on this below.

We want to be talking about generalized spaces _modeled on_ the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:

+-- {: .standout}

**Hierarchy of generalized spaces modeled on $\mathcal{G}$**

$$
  \array{
    \mathcal{G}
    &\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}&
    Sch(\mathcal{G})
    &\hookrightarrow&
    \mathcal{L}Top(\mathcal{G})^{op}
    &\hookrightarrow&
    Sh_{(\infty,1)}(Pro(\mathcal{G}))
    \\
    \\
    model spaces
    &&
    spaces locally like model spaces
    &&
    concrete spaces coprobeable by model spaces
    &&
    spaces probeable by model spaces
    \\
    \\
    affine\;\mathcal{G}-schemes
    &&
    \mathcal{G}-schemes
    &&
    \mathcal{G}-structured\;(\infty,1)-toposes
    &&
    \infty-stacks\;on\;\mathcal{G}
    \\
    \stackrel{tame\;but\;restrictive}{\leftarrow} &
    &&&&
    &
    \stackrel{versatile\;but\;possibly\;wild}{\to}
  }
$$

=--

We explain what this means from right to left.

### spaces probeable by model spaces: $\infty$-stacks

An object $X$ _probeable_ by objects of $\mathcal{G}$ should come with an assignment

$$
  X : (U \in \mathcal{G}) \mapsto (X(U) \in \infty Grpd)
$$

of an [[∞-groupoid]] of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{G}$

$$
   X \in PSh(\mathcal{G})
   =
   Funct(\mathcal{G}^{op}, \infty Grpd)
$$

But for $X$ to be _consistently_ probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the [[coverage|topology]] of $\mathcal{G}$. More precisely, this should mean that the [[(∞,1)-presheaf]] $X$ is actually an object in an [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$

$$
  X \in
  Sh(\mathcal{G}) \stackrel{}{\hookrightarrow}
  PSh(\mathcal{G})
  \,.
$$

Such objects are called [[∞-stack]]s on $\mathcal{G}$. The [[(∞,1)-category]] $Sh(\mathcal{G})$ is called an [[∞-stack]] [[(∞,1)-topos]].

A supposedly pedagogical discussion of the general philosophy of [[∞-stacks]] as probeable spaces is at [[motivation for sheaves, cohomology and higher stacks]].

The [[∞-stack]]s on $\mathcal{G}$ that are used in the following are those that satisfy [[descent]] on [[?ech cover]]s. But we will see [[(∞,1)-topos]]es of [[∞-stack]]s that may satisfy different descent conditions, in particular with respect to [[hypercover]]s. Every [[∞-stack]] [[(∞,1)-topos]] has a [[hypercompletion]] to one of this form.

For concretely working with [[hypercomplete (∞,1)-topos]]es it is often useful to use [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].

+-- {: .standout}

$$
  \array{
    Sh^{hc}_{(\infty,1)}(C)
   &\stackrel{\stackrel{\;\;\;\;\;lex\;\;\;\;\;\;}{\leftarrow}}
  {\hookrightarrow}&
     PSh_{(\infty,1)}(C)
    && \text{general abstract def of (\infty,1)-topos}
    \\
    \uparrow^{\simeq} &&
    \uparrow^{\simeq}
    && \text{Lurie's theorem}
    \\
    ([C^{op}, SSet]_{loc})^\circ
    &\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}}{\to}&
    ([C^{op}, SSet]_{glob})^\circ
    &&
    \text{model category of simplicial presheaves}
  }
$$

=--

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

This discussion here is glossing over all set-theoretic size issues. See [[Structured Spaces|StSp, warning 2.4.5]].

=--

### concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes

Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a _concrete underlying space_ , even for general definitions of what _that_ might mean.

**(Counter-)Example** For $\mathcal{G} = $ [[Diff]], for every $n \in \mathbb{N}$ we have the [[∞-stack]] $\Omega_{cl}^n(-)$ (which happens to be an ordinary [[sheaf]]) that assigns to each manifold $U$ the set of closed [[differential form|n-form]]s on $U$. This is important as a generalized space: it is something like the rational version of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}, n)$. But at the same time this is a "wild" space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.

In the classical theory for instance of [[ringed space]]s or [[diffeological space]]s a _concrete underlying space_ is taken to be a [[topological space]]. But this in turn is a bit _too_ restrictive for general purposes: a topological space is the same as a [[localic topos]]: a [[category of sheaves]] on a [[category of open subsets]] of a [[topological space]]. The obvious generalization of this to [[higher geometry]] is: an [[n-localic (∞,1)-topos]] $\mathcal{X}$.

This makes us want to say and make precise the statement that

+-- {: .standout}

A **concrete [[∞-stack]]** $X$ is one which has an _underlying_ [[(∞,1)-topos]] $\mathcal{X}$:

the collection of $U$-probes of $X$ is a [[subobject]] of the collection of [[(∞,1)-topos]]-morphisms from $U$ to $\mathcal{X}$:

$$
  X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})
$$

=--

We think of $\mathcal{X}$ as the [[(∞,1)-topos]] of [[∞-stack]]s on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the [[terminal object]] ${*}_{\mathcal{X}} \in \mathcal{X}$.

To say that $\mathcal{X}$ is _modeled on $\mathcal{G}$_ means that among all the [[∞-stack]]s on the would-be space a [[structure sheaf]] of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a [[structure sheaf]] $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.

This yields an [[(∞,1)-functor]]

$$
  \mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
$$

We think of $X$ as being a concrete space _co-probebale_ by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).

Such an $\mathcal{O}$ is a _consistent_ collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.

More precisely:

+-- {: .un_def}
###### Definition
**($\mathcal{G}$-structure, [[Structured Spaces|StSp, def. 1.2.8]])**

An [[(∞,1)-functor]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$  is a **$\mathcal{G}$-valued structure sheaf** on the [[(∞,1)-topos]] if

* it preserves finite [[limit]]s

* and sends covering coproducts $(\coprod_i U_i) \to U$ to [[effective epimorphism]]s.

A pair $(\mathcal{X}, \mathcal{O})$ of an [[(∞,1)-topos]] $\mathcal{X}$ equipped with $\mathcal{G}$-valued [[structure sheaf]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a [[structured (∞,1)-topos]].

=--

In summary:

+-- {: .standout}

A **concrete [[∞-stack]] $X$ modeled on $\mathcal{G}$** is

* an [[(∞,1)-topos]] $\mathcal{X}$ ("of $\infty$-stacks on $X$")

* equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.

=--

The fundamental **example** for [[structured (∞,1)-topos]]es
are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.

+-- {: .un_theorem}
###### Theorem
**([[Structured Spaces|StSp, thm. 2.1.1]])**

Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of [[geometry (for structured (infinity,1)-toposes)|geometries]],
then the obvious [[(∞,1)-functor]] $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a [[left adjoint]]

$$
  f^*
  :
  \mathcal{L}Top(\mathcal{G}')
   \stackrel{\leftarrow}{\to}
  \mathcal{L}Top(\mathcal{G})
  :
  Spec_{\mathcal{G}}^{\mathcal{G}'}
$$

called the **relative spectrum functor**.

=--

For $\mathcal{G}$ any [[geometry (for structured (infinity,1)-toposes)|geometry]], write $\mathcal{G}_{disc}$ for the [[geometry (for structured (infinity,1)-toposes)|geometry]] obtained from this by forgetting its [[coverage|Grothendieck topology]] and instead using the discrete topology where only equivalences cover.

Notice
that we may identify $\mathcal{G}_{disc}$-structures on
the archetypical [[(∞,1)-topos]] [[∞Grpd]], being finite [[limit]]-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$
with [[ind-object]]s in $\mathcal{G}^{op}$, hence with the
opposite of [[pro-object]]s in $\mathcal{G}$. This gives a canonical inclusion

$$
  Pro(\mathcal{G})
    \hookrightarrow
  \mathcal{L}Top(\mathcal{G})^{op}
  \,.
$$

+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.1.2]])**

The composite [[(∞,1)-functor]]

$$
  Spec^{\mathcal{G}}
  :
  Pro(\mathcal{G})^{op}
  \hookrightarrow
  \mathcal{L}Top(\mathcal{G}_{disc})
  \stackrel{Spec_{\mathcal{G}}^{\mathcal{G}_{disc}}}{\to}
  \mathcal{L}Top(\mathcal{G})
$$

we call the **absolute spectrum functor**

=--

This [[category theory|general abstract]] description is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:

+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.2.9]])**

For every $U \in \mathcal{G}$ there is naturally induced a [[coverage|topology]] on the [[over category]] $Pro(\mathcal{G})/U$. Define the [[(∞,1)-topos]]

$$
  Spec U := Sh_{(\infty,1)}(Pro(\mathcal{G})/U)
  \,,
$$

naturally to be thought of as the [[(∞,1)-topos]] of [[∞-stack]]s _on $U$_ .

This is canonically equipped with a [[(∞,1)-functor]]

$$
  \mathcal{O}_{Spec X}
  :
  \mathcal{G} \to Spec X
  \,.
$$

=--

And this is indeed the concrete underlying space produced by the absolute spectrum functor:

+-- {: .un_theorem}
###### Theorem
**[[Structured Spaces|StSp, prop. 2.2.11, thm. 2.2.12]])**

For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a [[structured (∞,1)-topos]] and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.

=--

**Example** For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard [[coverage|topology]] we have that 0-localic $\mathcal{G}$-structured spaces are _[[locally ringed space]]s_ , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary [[ringed space]]s. Applying the above machinery to this situaton gives a spectrum functor that takes a [[ring]] $R$ first to the [[ringed space]] $({*,R})$ and this then to the [[locally ringed space]] $(Spec R, R)$.

### Spaces locally like model spaces: generalized schemes

We have seen that $\mathcal{G}$-[[structured (∞,1)-topos]]es are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an "underlying topological structure" in the form of an underlying [[(∞,1)-topos]]. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.

In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.

But in many situations one would like to regard $\mathcal{G}$-[[structured (∞,1)-topos]]es that are not globally but _locally_ equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.

+-- {: .un_def}
###### Definition
**[[Structured Spaces|StSp, def. 2.3.9]]

A [[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O})$ is a **$\mathcal{G}$-generalized scheme** if

* there exists a collection $\{V_i \in \mathcal{X}\}$

* such that

  * this covers $\mathcal{X}$ in that the canonical morphism

    $$
      (\coprod_i V_i) \to {*}_{\mathcal{X}}
    $$

    to the [[terminal object]] in $\mathcal{X}$
    is an [[effective epimorphism]]

  * the [[structured (∞,1)-topos]]es
    $(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by
    the $V_i$ are model spaces in that there exists
    $\{U_i \in \mathcal{G}\}$ and equivalences

    $$
      (\mathcal{X}/V_i, \mathcal{O}|_{V_i})
      \simeq
      Spec^{\mathcal{G}} U_i
    $$

=--

**Examples**

> **warning** the following statements really pertain to pregeometries, not geometries. For the moment, this here is glossing over the difference between the two. See [[geometry (for structured (∞,1)-toposes)]] for the details.

* ordinary smooth [[manifold]]s are [[n-localic (infinity,1)-topos|0-localic]] [[Diff]]-[[generalized scheme]]s ([Structured Spaces|StSp, ex. 4.5.2]])

* ordinary [[schemes]] are those $(CRing^{fin})^{op}$-[[generalized scheme]]s whose underlying [[(∞,1)-topos]] is
[[n-localic (infinity,1)-topos|0-localic]] and whose [[structure sheaf]] is [[n-truncated object of an (infinity,1)-category|0-truncated]]  ([Structured Spaces|StSp, prop. 4.2.9]])

* [[Deligne-Mumford stack]]s are 1-localic $(CRing^{fin})_{et}^{op}$-[[generalized scheme]]s ([Structured Spaces|StSp, prop. 4.2.9]])

* This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-[[generalized scheme]] a **derived Deligne-Mumford stack

* Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in [[Spectral Schemes]] a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by [[E-∞ ring]]s) is gonna be a 1-localic  $(E_\infty Ring^{fin})^{op}$-[[generalized scheme]].

## Related concepts

* [[space form]]

* [[gros topos]],

* [[geometric homotopy type theory]], [[cohesive homotopy type theory]]

* [[spacetime]] ([[physics]])

[[!redirects spaces]]


space and quantity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Consider a [[category]] $C$ whose [[objects]] are thought of as _[[spaces]]_ of sorts ("test spaces"), and whose [[morphisms]] are regarded as [[homomorphisms]] between these spaces.

There is then a general notion of

* **spaces** modeled on $C$ that are _testable_ or _probe-able_ by objects of $C$;

* **quantities** with values in $C$.

Very generally, following [Lawvere 86](#Lawvere86):

* a generalized [[space]] modeled on the objects of $C$ is a [[presheaf]] on $C$, i.e. a [[functor]] of the form

  $$X \;\colon\; C^{op} \to Set.$$

  We think of each such presheaf as being a rule that assigns to each test space $U \in C$ the set $X(U)$ of allowed maps from $U$ _into_ the would-be space $X$ (this is really the perspective of [[functorial geometry]], originally due to [Grothendieck 65](functorial+geometry#Grothendieck65));

* a generalized [[quantity]] modeled on $C$ is a [[copresheaf]] on $C$, i.e. a [[functor]] of the form

  $$A \;\colon\; C \to Set.$$

  We think of each such copresheaf $A$ as a rule that assigns to each test space $U \in C$ the set $A(U)$ of allowed maps _from_ the would-be space $A$ into $U$, hence as the collection of $U$-valued _functions_ on $A$. Since a function on a point is a "quantity", these are generalized quantities.

One may view the _[[Yoneda lemma]]_ and the resulting _[[Yoneda embedding]]_ as expressing consistency conditions on this perspective: The [[Yoneda lemma]] says that the prescribed rule for how to test a generalized space $X$ by a test space $U$ turns out to coincide with the actual maps from $U$ to $X$, when $U$ is itself regarded as a generalized space, and the [[Yoneda embedding]] says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.

Beyond this _automatic_ consistency condition, guaranteed by [[category theory]] itself, typically the admissible (co)presheaves that are regarded as generalized spaces and quantities are required to respect one more consistency condition:

* If $C$ carries the structure of a [[site]], one asks a generalized space to be a presheaf $X = PSh(C) = [C^{op},Set]$ that respects the way objects in $C$ are [[covering|covered]] by other objects. These are the _[[sheaves]]_. The [[category of sheaves]]

  $$Sh(C) \hookrightarrow PSh(C)$$

  is the [[topos]] of spaces modeled on objects in $C$. More details on how to think of sheaves as generalized spaces is at [[motivation for sheaves, cohomology and higher stacks]].

* Given any generalized spaces, functions out of it are expected to respect [[product | products]] of coefficient objects, in that a function with values in $U \times V$ is the same as a pair of functions, one with values in $U$, one with values in $V$. Hence one is typically interested in copresheaves that preserve at least [[product]]

  $CoSh(C) \hookrightarrow CoPSh(C)$.

### Details

As indicated in [Lawvere 86, from p. 17 on](#Lawvere86)

* (generalized) spaces;

* (generalized) quantities (e.g. [[algebras of functions]]);

* the [[duality]] between the two;

which underlies much of mathematics is at its heart controlled by the following elementary [[category theory|category theoretic]] reasoning:

Let $C$ be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance $C = \Delta$, the [[simplicial category]], or $C = $ [[CartSp]], the category of $\mathbb{R}^n$ and smooth maps between them.

An ordinary [[manifold]], for instance, is a space required to be _locally isomorphic_ to an object in $C = CartSp$. But more generally, a space $X$ modeled on $C$ need only be _probeable_ by objects of $C$, giving a rule with which, to each test object $U \in C$, we assign the set of probing maps from $U$ to $X$,  such that this assignment is well-behaved with respect to morphisms in $C$. Such an assignment is nothing but a [[presheaf]] on $C$, i.e. a contravariant functor
$$
  X : C^{op} \to Set
  \,.
$$
Therefore general spaces modeled on $C$ are nothing but presheaves on $C$:
$$
  Spaces_C := PSh(C)
  \,.
$$
Of course this is an extremely general notion of spaces modeled on $C$.

For example, any smooth manifold $M$ is a presheaf on CartSp by $F_M:= Hom_{Man}(-, M): CartSp^{op} \to Set$, where we consider CartSp as a full subcategory of Man, the category of [[smooth manifold|smooth manifolds]] and smooth maps between them. $F_M$ maps each $\mathbb{R}^n$ to the set of all the smooth ways that $\mathbb{R}^n$ can probe $M$.

In particular, every object in $C$ is a space modeled on $C$, by the [[Yoneda lemma|Yoneda embedding]] $C \hookrightarrow Spaces_C$, whereby every object $c$ in $C$ is embedded as $Hom_C(-, c)$. That is, any object in $C$ is nothing but a consistent way to be probed by all the objects in $S$.

Now take a space $X$ modeled on $S$, and consider the set of _quantities_ on $X$ with values in $U \in S$. It should be
$$
  C(X,U) := Hom_{Spaces_S}(X,U)
  \,.
$$
This defines a covariant functor $C(X) := Hom_{Spaces_S}(X,-): S \to Set$. More generally, we can consider the S-valued quantities on $X$ to be a copresheaf on $S$, namely a covariant functor
$$
  C(X): S \to Set
  \,.
$$
One can think of $C(X)$ as a generalized quantity which may be _co-probed_ by objects of $S$.

In this vein, one can say, generally, that copresheaves on $S$ are generalized quantities modeled on $S$, and we write
$$
  Quantities_S := CoPSh(S)
  \,.
$$

Given any such generalized quantity $A \in Quantities_S$, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called $Spec(A)$ and can be defined as a presheaf by the assignment
$$
  Spec(A)
  :
  U \mapsto Hom_{Quantities_S}(A, C(U))
  \,.
$$

In  total this yields an adjoint pair of contravariant functors between generalized spaces and generalized quantities:

$$
  Spaces_S^{op}
\stackrel{\stackrel{C(-)}{\to}}{\stackrel{Spec(-)}{\leftarrow}}
  Quantities_S
  \,.
$$

(That this is an adjunction can be understood as a special case of [[abstract Stone duality]] induced by a [[dualizing object]].)

Lawvere refers to this [[adjoint pair]] as **[[Isbell conjugation]]**.

In conclusion, the grand [[duality]] between spaces and quantities is a consequence of the [[formal duality]] which reverses the arrows in the category $S$ of test spaces.

This story generalizes straightforwardly from [[presheaf|presheaves]] with values in [[Set]] to presheaves with values in other categories. Of relevance are in particular presheaves with values in the category [[Top]] of [[topological space | topological spaces]] and presheaves with values in the category of [[spectrum|spectra]]. See the examples below.

## Isbell duality: global functions and spectrum {#Isbell}

we describe the [[duality]] between space and quantity induced by forming

* functions on spaces;

* spectra of function algebras.

Let $V$ be a [[symmetric monoidal category]] and $C$ a $V$-[[enriched category]]. Write $[C^{op},V]$ for the [[enriched functor category]] and $j : C \to [C^{op},V]$ for the [[Yoneda embedding]].

There is canonically a $V$-[[adjunction]]

$$
  (\mathcal{O} \dashv Spec)  \;\;: \;\; [C, V]^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op},V]
$$

the **[[Isbell adjunction]]**. Here

* $\mathcal{O} := [C^{op},V](j(-), -)$ sends a presheaf $X$ to the copresheaf $U \mapsto [C^{op},V](X,j(U))$;

* $Spec := [C,V]^{op}(j(-),-)$ sends a copresheaf $A$ to the presheaf $U \mapsto [C,V](A, j^{op}(U))$.

If we assume that $C$ is [[copower|tensored]] over $V$, then that this is an adjunction may be seen in [[end]]/[[coend]]-calculus to express the [[hom-object | hom-objects]] in the [[enriched functor category]] as follows. We compute

$$
  \begin{aligned}
    [C,V]^{op}(\mathcal{O}(X),A)
    & =
    \int_{u \in C} V(A(u), [C^{op},V](X,j(u)))
    \\
    & \simeq
    \int_{u \in C} V(A(u), [C^{op},V](\int^{v \in C} j(v) \cdot X(v),j(u)))
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), [C^{op},V](j(v),j(u)))
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u))
  \end{aligned}
  \,,
$$

where we used the [[Yoneda lemma]] $[C^{op},V](j(v),j(u)) \simeq V(v,u)$ and the [[co-Yoneda lemma]] $X \simeq \int^{v \in V} j(v) \cdot X(v)$ and the fact that the $V$-enriched hom sends colimits and coends in the first argument to limits and ends.

Analogously we find

$$
  \begin{aligned}
    [C^{op},V](X,Spec A)
    & =
    \int_{v \in C} V(X(v),[C,V](A, j^{op}(v)))
    \\
    & \simeq
    \int_{v \in C} V(X(v), [C,V](\int^{u \in C} j^{op}(u) \cdot X(v),j^{op}(u)))
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), [C,V](j^{op}(v),j^{op}(u)))
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u))
  \end{aligned}
  \,,
$$

## Examples

### Cartesian test spaces: diffeological spaces and smooth algebras

Consider the category of test spaces $C = $ [[CartSp]].

Then

* spaces modeled on $C$ are [[generalized smooth space | generalized smooth spaces]] such as [[diffeological space | diffeological spaces]];

* quantities modeled on $C$ are [[smooth algebra | smooth algebras]] ($C^\infty$-rings).

The adjunction $(\mathcal{O} \dashv Spec)$ sends a smooth space to its smooth algebra of functions and a smooth algebra of functions to its "spectrum".

## Higher space and higher quantity

There are various specializations of interest on this

* [[higher category theory|higher categorical]] version

  * [[∞-space]] modeled on $C$ is a [[simplicial presheaf]] on $C$, i.e. a functor $X : C^{op} \to $ [[SSet]].

  * [[∞-quantity]] modeled on $C$ is a cosimplicial copresheaf on $C$, i.e. a functor $X : C \to CoSSet$ .

With the advent of [[Higher Topos Theory]] abstract concepts of _space and quantity_ have been realized fully in the context of [[(∞,1)-topos | (∞,1)-toposes]] in terms of [[structured (∞,1)-topos | structured (∞,1)-toposes]] and [[generalized scheme | generalized schemes]]. For a summary see the tables at [notions of space](http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+the+derived+moduli+stack+of+derived+elliptic+curves#NotionsOfSpace).

## Related entries

* [[Isbell duality]]

* [[space]]

* [[generalized smooth space]]

* [[geometry of physics -- categories and toposes]]

## References

The general perspective is due to

* {#Lawvere86} [[William Lawvere]], _Taking categories seriously_, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted in: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html))

* {#Lawvere92} [[William Lawvere]], _Categories of space and quantity_, in: J. Echeverria et al (eds.), _The Space of mathematics_, de Gruyter, Berlin, New York (1992) ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1992-categories-of-space-and-quantity.pdf))

See also

* [[William Lawvere]], _John Isbell's Adequate Subcategories_, TopCom **11** no.1 2006. ([link](http://at.yorku.ca/t/o/p/d/65.htm))

[[!redirects quantity]]
[[!redirects quantities]]

[[!redirects space and quantity]]
[[!redirects duality between space and quantity]]

[[!redirects generalized space]]
[[!redirects generalized spaces]]


space attachment

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

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]], the result of a _space attachment_ (sometimes called an _attaching space_ or _adjunction space_) is a [[topological space]], denoted $X \cup_{f} Y$, which is constructed by "attaching" or "gluing" two topological spaces $X$ and $Y$ along a [[topological subspace]] $A \subset X$ by means of a [[continuous function]] $f \colon A \to Y$. The function $f$ is then called the _attaching map_.

> (graphics taken from [AGP08, §3.1](#AGP08))

More [[category theory|abstractly]], space attachments are [[pushouts]] along [[monomorphisms]] in the [[category]] [[Top]] of all [[topological spaces]]. The [[formal dual|formally dual]] concept is that of [[fiber]] spaces or more generally of [[fiber products]] of topological spaces.

## Definition

Let $X,Y \in Top$ be [[topological spaces]], let $A \subset X$ be a [[topological subspace]] and let $f \colon A \to Y$ be a [[continuous function]].

Then the _attaching space_ $X \cup_f Y \in Top$ may be realized as the [[quotient topological space]] of the [[disjoint union space]] $X \sqcup Y$ by the [[equivalence relation]] which identifies a point $x \in A \subset X$ with its [[image]] $f(x) \in Y$:

$$
  X \cup_f Y
  \;\simeq\;
  \left(
    X \sqcup Y
  \right)/\sim
  \,.
$$

More [[category theory|category theoretically]], the attaching space is the [[pushout]] in the [[category]] [[Top]] of [[topological spaces]] of the subspace inclusion $i \colon A \hookrightarrow X$ along $f$, i.e. the topological space which is [[universal property|universal]] with the property that it makes the following [[commuting square|square commute]]:

$$
  \array{
    A &\overset{\phantom{A}i\phantom{A}}{\hookrightarrow}& X
    \\
    {}^{\mathllap{f}}\downarrow &(po)& \downarrow
    \\
    Y &\longrightarrow& X \cup_f Y
  }
  \,.
$$

For more on this see at _[Top -- Universal constructions](Top#UniversalConstructions)_.

## Examples

* In forming topological [[cell complexes]] such as [[CW-complexes]], one consecutively forms space attachments along [[n-sphere|(n-1)-sphere]] inclusions as [[boundaries]] of [[n-disks]]. These are called _cell attachments_.

## Related concepts

[[!include universal constructions of topological spaces -- table]]

* [[Whitehead product]]

## References

* {#AGP08} [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], §3.1 in: *Algebraic topology from a homotopical viewpoint*, Springer (2008) [[doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586)]

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Chapter 17, p. 217 of:  _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics **82**, Springer (1982) [[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)]

See also

* Wikipedia, _[Adjunction space](https://en.wikipedia.org/wiki/Adjunction_space)_

[[!redirects space attachments]]

[[!redirects attaching space]]
[[!redirects attaching spaces]]

[[!redirects attaching map]]
[[!redirects attaching maps]]

[[!redirects adjunction space]]
[[!redirects adjunction spaces]]

[[!redirects cell attachment]]
[[!redirects cell attachments]]

[[!redirects attachment space]]
[[!redirects attachment spaces]]


space of field histories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], given a [[physical system]] then its _space of [[field histories]] (e.g. _[[trajectories]]_ for [[sigma models]]) is the [[space]] of all ways that the system may be configured in space and _[[time]]_, hence in _[[spacetime]]_.

For a [[field theory]] over a [[spacetime]] $\Sigma$ and defined by a [[field bundle]] $E \overset{fb}{\to} \Sigma$, the space of trajectories is the space of [[field (physics)|field]] configurations over spacetimes, which is the [[space of sections]] of the field bundle.

For [[Lagrangian field theories]] this space of trajectories carries a canonical [[presymplectic form]] and it comes with an [[equation of motion]] that picks some of the trajectories as being the "physically realizable ones". The subspace of trajectories solving the equations of motion (the "[[shell]]") equipped with this [[presymplectic form]] is called the _[[covariant phase space]]_ of the system.

For the moment see at _[[field theory]]_ for more details.

## Related concepts

* [[configuration space (physics)]]

## References

Discussion in the convenient context of [[smooth sets]]:

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §1 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects space of trajectories]]
[[!redirects spaces of trajectories]]

[[!redirects space of field histories]]
[[!redirects spaces of field histories]]

[[!redirects space of histories]]
[[!redirects spaces of histories]]


space of finite subsets

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

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[topological space]] and $n \in \mathbb{N}$ a [[natural number]], the space of [[finite set|finite]] [[subsets]] of [[cardinality]] $\leq n$ in $X$ is the suitably topologized [[set]] of [[finite set|finite]] [[subsets]] $S \subset X$ of [[cardinality]] $\left\vert S\right\vert \leq n$, often denoted $\exp^n X$ or similar (e.g. [Félix-Tanré 10](#FelixTanre10)).

## Properties

### Relation to unordered configuration space of points

The [[topological subspace]]  of finite subsets of cardinality exactly equal to $n$ is the [[unordered configuration space of points]] in $X$

\[
  \label{InjectionOfUnorderedConfigurationSpaceOfPoints}
  Conf_n(X)
  \hookrightarrow
  \exp^n(X)
\]

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

Let $X$ be an [[inhabited set|non-empty]] [[regular topological space]] and $n \geq 2 \in \mathbb{N}$.

Then the injection (eq:InjectionOfUnorderedConfigurationSpaceOfPoints)

\[
  \label{UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets}
  Conf_n(X)
  \hookrightarrow
  \exp^n(X)/\exp^{n-1}(X)
\]

of the [[unordered configuration space of n points]] of $X$ into the [[quotient space]] of the [[space of finite subsets]] of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an [[open subspace]]-inclusion.

Moreover, if $X$ is [[compact topological space|compact]], then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion (eq:UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets) exhibits the [[one-point compactification]] $\big(  Conf_n(X) \big)^{+} $ of the configuration space:

$$
  \big(
    Conf_n(X)
  \big)^{+}
  \;\simeq\;
  \exp^n(X)/\exp^{n-1}(X)
  \,.
$$

=--

([Handel 00, Prop. 2.23](#Handel00), see also [Félix-Tanré 10](#FelixTanre10))

## References

* {#Handel00} David Handel, _Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces_, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (pdf[hjm:Vol26-4](https://www.math.uh.edu/~hjm/Vol26-4.html))

* {#FelixTanre2010} [[Yves Félix]], [[Daniel Tanré]] _Rational homotopy of symmetric products and Spaces of finite subsets_, Contemp. Math 519 (2010): 77-92 ([pdf](http://tanre.org/Pro/Articles_files/SpnFiniteDef.pdf))

* Jacob Mostovoy, Rustam Sadykov, _On the connectivity of finite subset spaces_ ([arXiv:1203.5180](https://arxiv.org/abs/1203.5180))


space of knots

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[manifold]] $\Sigma$, such as the [[3-sphere]], its _space of knots_ is a [[topological space]] parametrizing [[knots]] in $\Sigma$, hence something like the [[topological subspace]]

$$
  Emb(S^1, \Sigma)
  \;\subset\;
  Maps(S^1,\Sigma)
$$

of the [[space of maps]] $S^1 \to \Sigma$ on those maps which are [[embedding of smooth manifolds|embeddings]]. This space is often denoted just $\mathcal{K}$.

A [[knot invariant]] is then equivalently a [[locally constant function]] on the space of knots.

## References

### General

* [[Allen Hatcher]], _Spaces of Knots_ ([arXiv:math/9909095](https://arxiv.org/abs/math/9909095))

* Ryan Budney, _Topology of spaces of knots in dimension 3_,  	Proceedings of the London Mathematical Society Vol 101 pt. 2 (Sept. 2010) ([arXiv:math/0506524](https://arxiv.org/abs/math/0506524))

### Using graph complexes

Discussion of the [[rational homotopy type]] of knot spaces in terms of [[graph complexes]]:

* [[Dev Sinha]], _The Topology of Spaces of Knots: Cosimplicial Models_, American Journal of Mathematics, Vol. 131, No. 4 (Aug., 2009), pp. 945-980 (36 pages) ([arXiv:math/0202287](https://arxiv.org/abs/math/0202287), [jstor:40263760 ](https://www.jstor.org/stable/40263760))

* {#CattaneoCottaRamusinoLongoni02} [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))

* {#CattaneoCottaRamusinoLongoni05} [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218), [doi:10.1142/S0218216505004019](http://dx.doi.org/10.1142/S0218216505004019), [math.GT/0307218](http://arxiv.org/abs/math/0307218), [MR2006g:58021](http://www.ams.org/mathscinet-getitem?mr=2006g:58021))

* [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))

* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))

[[!redirects spaces of knots]]

[[!redirects knot space]]
[[!redirects knot spaces]]


space of sections

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[bundle]] $E \overset{}{\to} \Sigma$, then its _space of [[sections]]_ is like a [[mapping space]], but relative to the base space $\Sigma$.

Formally this is given by the _[[dependent product]]_ construction. See at _[section -- In terms of dependent product](section#InTermsOfDependentProduct)_ and at _[dependent product -- In terms of spaces of sections](dependent%20product#RelationToSpacesOfSections)_.

## Definition

Let $\mathbf{H}$ be a [[topos]] (for instance $\mathbf{H} =$[[SmoothSet]]) or [[(∞,1)-topos]] (for instance $\mathbf{H} = $ [[Smooth∞Grpd]]) and consider

$$
  \left[
    \,\,
    \array{
       E
       \\
       {}^{\mathllap{p}}\downarrow
       \\
       \Sigma
    }
  \right]
  \;\colon\;
  \mathbf{H}_{/\Sigma}
$$

a [[bundle]] in $\mathbf{H}$, regarded as an object in the [[slice topos]]/[[slice (∞,1)-topos]].

Then the _space of sections_ $\Gamma_\Sigma(E)$ of this bundle is the [[dependent product]]

$$
  \Gamma_\Sigma E
    \coloneqq
  \prod_{\Sigma} E
  \;\in\; \mathbf{H}
$$

hence the image of the bundle under the [[right adjoint]] $\Sigma_\ast$ in the [[base change]] [[adjoint triple]]

$$
  \mathbf{H}_{/\Sigma}
    \underoverset
      {\underset{\Sigma_\ast}{\longrightarrow}}
      {\overset{\Sigma_!}{\longrightarrow}}
      {\overset{\Sigma^\ast}{\longleftarrow}}
  \mathbf{H}
  ,.
$$

By [[adjunction]] this means that for $U \in \mathbf{H}$ a test object, then a $U$-parameterized family of sections of $E$, hence a morphism in $\mathbf{H}$ of the form

$$
  U \longrightarrow \Gamma_\Sigma(E)
$$

is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form

$$
  \Sigma \times U = \Sigma^\ast U \longrightarrow E
  \,.
$$

This is equivalently a [[diagram]] in $\mathbf{H}$ of the form

$$
  \array{
    && E
    \\
    & {}^{\mathllap{\phi_U}}\nearrow & \downarrow^{\mathrlap{p}}
    \\
    U \times \Sigma
    &\underset{pr_2}{\longrightarrow}&
    \Sigma
  }
  \,,
$$

where the right and bottom morphisms are fixed, and where $\phi_U$ (and  the [[2-cell]] filling the diagram) is, manifestly, the $U$-parameterized family of [[sections]].

## Properties

### Topological vector space structure

+-- {: .num_defn #TVSStructureOnSpacesOfSmoothSections}
###### Definition
**([[Fréchet space|Fréchet]] [[topological vector space]] on spaces of smooth sections of a [[smooth vector bundle]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]]. On its [[real vector space]] $\Gamma_\Sigma(E)$ [[space of sections|of smooth sections]] consider the [[seminorms]] indexed by a [[compact subset]] $K \subset \Sigma$ and a [[natural number]] $N \in \mathbb{N}$ and given by

$$
  \array{
    \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}&
    [0,\infty)
    \\
    \Phi &\mapsto&  \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,,
  }
$$

where on the right we have the [[absolute values]] of the [[covariant derivatives]] of $\Phi$ for any fixed choice of [[connection on a bundle|connection]] on $E$ and [[norm]] on the [[tensor product of vector bundles]] $(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E $.

This makes $\Gamma_\Sigma(E)$ a [[Fréchet space|Fréchet]] [[topological vector space]].

For $K \subset \Sigma$ any [[closed subset]] then the sub-space of sections

$$
  \Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)
$$

of sections whose [[support]] is inside $K$ becomes a [[Fréchet space|Fréchet]] [[topological vector spaces]] with the induced [[subspace topology]], which makes these be [[closed subspaces]].

=--

([Bär 14, 2.1, 2.2](#Baer14))

### Extension of sections

See at _[[Whitney extension theorem]]_ ([Roberts-Schmediung 18](Whitney+extension+theorem#RobertsSchmeding18)).

## Examples

* The space of section of an universal [[associated infinity-bundle]] is the space of [[homotopy coinvariants]] of the corresponding [[infinity-action]] (see there for more).

## Related concepts

* [[dependent product]]

* [[twisted cohomology]]

* [[space of states (in geometric quantization)]]

## References

The [[topological vector space]] on spaces of smooth sections is discussed in

* {#BrunettiFredenhagenRibeiro12} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Ribeiro]], around remark 2.2.1 in _Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields_ ([arXiv:1209.2148](https://arxiv.org/abs/1209.2148), [spire](http://inspirehep.net/record/1185123/))

* {#Baer14} [[Christian Bär]], _Green-hyperbolic operators on globally hyperbolic spacetimes_, Communications in Mathematical Physics 333, 1585-1615 (2014) ([doi](http://dx.doi.org/10.1007/s00220-014-2097-7), [arXiv:1310.0738](https://arxiv.org/abs/1310.0738))

[[!redirects spaces of sections]]

[[!redirects space of smooth sections]]
[[!redirects spaces of smooth sections]]

[[!redirects ∞-groupoid of sections]]
[[!redirects ∞-groupoids of sections]]
[[!redirects infinity-groupoid of sections]]
[[!redirects infinity-groupoids of sections]]


space of states (in geometric quantization)

+-- {: .rightHandSide}
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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are at least two formalizations of [[quantization]], one of them is _[[geometric quantization]]_. In this context a [[quantum state]] is identified with a certain [[section]] (a [[polarization|polarized]] section) of a certain [[complex line bundle]]: the [[prequantum line bundle]].

Here a [[section]] of the [[prequantum line bundle]] is what is physics is called a **[[wave function]]** or _[[probability amplitude]]_ on the space of [[field (physics)|field]] configurations. A choice of [[polarization]] on this space is a choice of "[[canonical coordinates]]" and "[[canonical momenta]]". Hence a polarized section, and hence a quantum state in the sense of geometric quantization, is, in [[physics]] language, a _wave function of the canonical coordinates_.

So the space of states is the subspace of the [[space of sections]] of the [[prequantum line bundle]] on those that are [[polarization|polarized]].

For details see at _[geometric quantization -- Space of quantum states](geometric+quantization#QuantumStates)_.

## Related concepts

* [[Laughlin wavefunction]]

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

\linebreak

[[!include Isbell duality - table]]

[[!include holographic principle -- table]]

[[!redirects prequantum spaces of states]]
[[!redirects prequantum space of states]]

[[!redirects space of quantum states]]
[[!redirects spaces of quantum states]]

[[!redirects wave function]]
[[!redirects wave functions]]

[[!redirects wavefunction]]
[[!redirects wavefunctions]]


spaceoid

## Idea

By [[Gelfand duality]], spaces are formal duals of commutative $C^*$-[[C-star-algebra|algebras]]. The purpose of a spaceoid is to act as the formal dual of a commutative $C^*$-[[C-star-category|category]].

## Definition

...

#References#

[Spaceoids](http://golem.ph.utexas.edu/category/2008/01/spaceoids.html)

[[!redirects spaceoids]]


spaceoids > history

< [[spaceoids]]
[[!redirects spaceoids -- history]]


spacetime

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

### General

A _spacetime_ is a [[manifold]] that models [[space]] and [[time]] in [[physics]].

This is formalized by saying that a **spacetime** is a [[smooth Lorentzian space]] $(X,\mu)$ equipped with a [[time orientation]] (see there).

Hence a point in a spacetime is called an _[[event]]_.

In the context of classical [[general relativity]] a **spacetime** is usually in addition assumed to be connected and four-dimensional. A connected Lorentzian manifold is either time orientable or it has a two-sheeted covering which
is time orientable.

In [[classical physics]], notably in [[special relativity]] and [[general relativity]] points in $X$ model coordinates where _events_ can take place from the viewpoint of an observer ("points in space and time") while the metric $\mu$ models the field of [[gravity]] in [[general relativity]].

### Intermingling of space and time
The noun "spacetime" is used in both [[special relativity]] and [[general relativity]], but is best motivated from the viewpoint of [[general relativity]]. [[special relativity|Special relativity]] deals with the [[Minkowski spacetime]] only. The [[Minkowski spacetime]] allows a _canonical_ choice of _global_ coordinates such that the metric tensor has in every point the form diag(-1, 1, 1, 1), which identifies the first coordinate as representing the time coordinate and the others as representing space coordinates.

Given a general spacetime, there is not necessarily a globally defined coordinate system, and therefore not necessarily a globally defined _canonical_ time coordinate. More specifically, there are spacetimes that admit coordinates defined on subsets where the _physical_ interpretation of the coordinates as modelling time and space coordinates _changes_ over the domain of definition.

(TODO: references and explanations).

## Examples

* [[Minkowski spacetime]]

[[!include charged and rotating black holes -- table]]

* [[asymptotically flat spacetime]]

* [[Kerr-Schild spacetime]]

* [[piecewise flat spacetime]]

* [[gravitational wave]]

* [[anti de Sitter spacetime]]

* [[FRW spacetime]]

* [[Taub-NUT spacetime]]

* [[Einstein manifold]]

* [[pp-wave spacetime]]

* [[KK-monopole]]

* [[Malament–Hogarth spacetime]]

* [[super spacetime]]

  * [[super Minkowski spacetime]]

* [[black brane]]

## Related concepts

* [[spacetime signature]]

* [[static spacetime]], [[stationary spacetime]]

* [[universal spacetime]]

* [[general covariance]]

* [[gravity]], [[cosmology]], [[black hole]]

* [[particle physics]], [[matter]], [[sigma-model]]

* [[force]], [[gauge theory]]

* [[mass]]

## References

### Books

* S. W. Hawking, G. F. R. Ellis, _The large scale structure of space-time_, Cambridge Univ. Press
* John Beem, Paul Ehrlich, _Global Lorentzian geometry_, Marcel Dekker 1981 (and Russian, updated translation, Mir 1985)

### Articles

* L. Markus, _Line element fields and Lorentz structures on differentiable manifolds_, Ann. of Math. (2) 62 (1955), 411–417, [MR0073169](http://www.ams.org/mathscinet-getitem?mr=0073169) [jstor](http://www.jstor.org/stable/1970071)

* [[Roger Penrose]], _Gravitational collapse and space-time singularities_, Phys. Rev. Lett. __14__, 57--59

* Paul Ernest, The Mathematics of Space-Time and the Limits of Human Understanding, Philosophy of Mathematics Education Journal, No. 30 (October 2016) ISSN:1465-2978

category: physics, geometry

[[!redirects space-time]]
[[!redirects space time]]
[[!redirects spacetimes]]

[[!redirects spacetime manifold]]
[[!redirects spacetime manifolds]]


spacetime support

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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[local field theory]] with [[field (physics)|fields]] on a given [[spacetime]] $X$, the _spacetime support_ of an [[observable]] $A$ is the maximal region in spacetime such that $A$ depends on ("observes") the values of fields at the points in this region.

In a [[local field theory]] spacetime support of observables is typically required to be a [[compact subset]] of spacetime, which, under the [[Heine-Borel theorem]], reflects the intuition that every [[experiment]] (every "observation" of [[physics]]) is necessarily [[bounded subset|bounded]] in [[spacetime]].

For more see at _[[geometry of physics -- perturbative quantum field theory]]_ the chapter _[7. Observables](geometry+of+physics+--+perturbative+quantum+field+theory#Observables)_

## Definition

+-- {: .num_defn #SpacetimeSupport}
###### Definition
**([[spacetime support]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{FieldsAndFieldBundles}),
with induced [[jet bundle]] $J^\infty_\Sigma(E)$

For every [[subset]] $S \subset \Sigma$ let

$$
  \array{
    J^\infty_\Sigma(E)\vert_S
      &\overset{\iota_S}{\hookrightarrow}&
    J^\infty_\Sigma(E)
    \\
    \downarrow &(pb)& \downarrow
    \\
    S &\hookrightarrow& \Sigma
  }
$$

be the corresponding restriction of the [[jet bundle]] of $E$.

The _spacetime support  _ $supp_\Sigma(A)$ of a [[differential form]] $A \in \Omega^\bullet(J^\infty_\Sigma(E))$
on the [[jet bundle]] of $E$ is the [[topological closure]] of the maximal subset $S \subset \Sigma$
such that the restriction of $A$ to the jet bundle restrited to this subset does not vanishes:

$$
  supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma |  \iota_{\{x\}}^\ast A \neq 0 \} )
$$

We write

$$
  \Omega^{r,s}_{\Sigma,cp}(E)
  \coloneqq
  \left\{
    A \in \Omega^{r,s}_\Sigma(E)
    \;\vert\;
    supp_\Sigma(A) \, \text{is compact}
  \right\}
    \;\hookrightarrow\;
  \Omega^{r,s}_\Sigma(E)
$$

for the subspace of differential forms on the jet bundle whose spacetime support is a [[compact subspace]].

=--

## Related concepts

* [[support]], [[support of a distribution]]

* [[compact support]]

* [[local observable]]

## References

* _[[geometry of physics -- perturbative quantum field theory]]_ the chapter _[7. Observables](geometry+of+physics+--+perturbative+quantum+field+theory#Observables)_

[[!redirects spacetime supports]]


Span

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###Context###
#### 2-Category theory
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[[!include 2-category theory - contents]]
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# Span

* table of contents
{: toc}

## Definition

$Span$ is the [[bicategory]], or [[double category]], whose

* objects are [[sets]]
* (horizontal) morphisms are [[spans]]
* 2-cells are morphisms of spans
* (in the double category case) vertical morphisms are [[functions]].

More generally, we have $Span(C)$ consists of spans in $C$ for any [[category]] $C$ with [[pullbacks]].

## Properties

$Span$ is a sort of classifying object for arbitrary functors; see _[[displayed category]]_.

## Categorifications

There is an [[(infinity,n)-category of correspondences]].


span

> This entry is about the notion of spans/correspondences which generalizes that of _[[relations]]_. For spans in [[vector spaces]] or [[modules]], see _[[linear span]]_.

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### Context
#### Relations
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[[!include relations - contents]]
=--
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
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#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
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#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
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# Contents
* table of contents
{: toc}

## Definition

### In set theory

In [[set theory]], a **span** or **correspondence** between [[sets]] $A$ and $B$ is a set $C$ with a [[function]] $R:C \to A \times B$ to the [[product set]] $A \times B$. A span between a set $A$ and $A$ itself is a [[directed pseudograph]], which is used to define [[categories]] in set theory.

### In dependent type theory

In [[dependent type theory]], there is a distinction between a *span*, a *multivalued partial function*, and a *correspondence*:

* A **span** between types $A$ and $B$ is a type $C$ with families of elements $x:C \vdash g(x):A$ and $x:C \vdash h(x):B$

* A **[[multivalued partial function]]** from type $A$ to type $B$ is a type family $x:A \vdash P(x)$ with a family of elements $x:A, p:P(x) \vdash f(x, p):B$

* A **correspondence** between types $A$ and $B$ is a type family $x:A, y:B \vdash R(x, y)$.

However, from any one of the above structures, one could get the other two structures, provided one has [[identity types]] and [[dependent pair types]] in the dependent type theory. Given a type family $x:A \vdash P(x)$, let $z:\sum_{x:A} P(x) \vdash \pi_1(z):A$ and $z:\sum_{x:A} P(x) \vdash \pi_2(z):P(\pi_1(z))$ be the dependent pair projections for the [[dependent pair type]] $\sum_{x:A} P(x)$.

* From every span one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:C} g(y) =_A x$ and the family of elements $x:A, p:P(x) \vdash f(x, p):B$ as $f(x, p) \coloneqq h(\pi_1(x))$.

* From every multivalued partial function one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} P(x)$ and the family of elements $x:C \vdash g(x):A$ as $g(x) \coloneqq \pi_1(x)$.

* From every multivalued partial function one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$.

* From every correspondence one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:B} R(x, y)$, and the family of elements $x:A, p:P(x) \vdash h(x, p):B$ as $h(x, p) \coloneqq \pi_1(p)$

* From every span one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{z:C} (g(z) =_A x) \times (h(z) =_B y)$.

* From every correspondence one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} \sum_{y:B} R(x, y)$, the family of elements $z:C \vdash g(z):A$ as $g(z) \coloneqq \pi_1(z)$, and the function $z:C \vdash h(z):B$ as $h(z) \coloneqq \pi_1(\pi_2(z))$

Given types $A$, $B$, and $C$ and spans $(D, x:D \vdash g_D(x):A, x:D \vdash h_D(x):B)$ between $A$ and $B$ and $(E, y:E \vdash g_E(y):B, y:E \vdash h_E(y):C)$ between $B$ and $C$, there is a span
$$(E \circ D, z:E \circ D \vdash g_{E \circ D}(z):A, z:E \circ D \vdash h_{E \circ D}(z):C)$$
defined by
$$E \circ D \coloneqq \sum_{x:D} \sum_{y:E} h_D(x) =_B g_E(y)$$
$$z:E \circ D \vdash g_{E \circ D}(z) \coloneqq g_D(\pi_1(z))$$
$$z:E \circ D \vdash h_{E \circ D}(z) \coloneqq h_E(\pi_1(\pi_2(\pi_1(z))(z)))$$

Given types $A$, $B$, and $C$ and correspondences $x:A, y:B \vdash R(x, y)$ and $y:B, z:C \vdash S(y, z)$, there is a correspondence $x:A, z:C \vdash (S \circ R)(x, z)$ defined by
$$(S \circ R)(x, z) \coloneqq \sum_{y:B} R(x, y) \times S(y, z)$$

### In category theory

In any [[category]] $C$, a __span__, or __roof__, or __correspondence__, from an [[object]] $x$ to an object $y$ is a [[diagram]] of the form

$$
  \array{
     && s
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
     \\
     x
     &&&&
     y
  }
$$

where $s$ is some other object of the category.  (The word "correspondence" is also sometimes used for a [[profunctor]].)

This [[diagram]] is also called a 'span' because it looks like a little bridge; 'roof' is similar.  The term 'correspondence' is prevalent in geometry and related areas; it comes about because a correspondence is a generalisation of a binary [[relation]].

Note that a span with $f = 1$ is just a morphism from $x$ to $y$, while a span with $g = 1$ is a morphism from $y$ to $x$.  So, a span can be thought of as a generalization of a morphism in which there is no longer any asymmetry between source and target.

A span in the [[opposite category]] $C^op$ is called a [[co-span]] in $C$.

A span that has a [[cocone]] is called a [[coquadrable span]].

### Categories of spans

If the category $C$ has [[pullback|pullbacks]], we can compose spans.  Namely, given a span from $x$ to $y$ and a span from $y$ to $z$:
$$
  \array{
     && s &&&& t
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
       &
       & {}^{h}\swarrow
      && \searrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
$$

we can take a pullback in the middle:

$$
  \array{
     &&&& s \times_y t
     \\&
    &&
      {}^{p_s}\swarrow
      && \searrow^{p_t}
    \\
     && s &&&& t
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
       &
       & {}^{h}\swarrow
      && \searrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
$$

and obtain a span from $x$ to $z$:

$$
  \array{
     && s \times_y t
      \\
      & {}^{f p_s}\swarrow
      && \searrow^{i p_t}
     \\
     x
     &&&&
     z
  }
$$

This way of composing spans lets us define a [[bicategory]] [[Span]]$(C)$ with:

* objects of $C$ as objects
* spans as morphisms
* maps between spans as 2-morphisms

This is a weak 2-category: it has a nontrivial [[associator]]: composition of spans is not strictly associative, because pullbacks are defined only up to canonical isomorphism.  A [[Lack's coherence theorem|naturally defined]] [[strict 2-category]] which is equivalent to $Span(C)$ is the strict 2-category of [[linear polynomial functors]] between [[slice categories]] of $C$.

(Note that we must choose a specific pullback when defining the composite of a pair of morphisms in $Span(C)$, if we want to obtain a [[bicategory]] as traditionally defined; this requires the [[axiom of choice]]. Otherwise we obtain a bicategory with 'composites of morphisms defined only up to canonical iso-2-morphism'; such a structure can be modeled by an [[anabicategory]] or an [[opetopic bicategory]].)

By including functions as well, instead of a bicategory we obtain a [[pseudo-double category]].

## Properties

### The 1-category of spans

Let $C$ be a category with [[pullback]]s and let $Span_1(C) := (Span(C))_{\sim 1}$ be the 1-category of objects of $C$ and isomorphism classes of spans between them as morphisms.

Then

* $Span_1(C)$ is a [[dagger category]].

Next assume that $C$ is a [[cartesian monoidal category]]. Then clearly $Span_1(C)$ naturally becomes a [[monoidal category]] itself, but more: then

* $Span_1(C)$ is a [[dagger compact category]].

### Universal property of the bicategory of spans
 {#UniversalPropertyOfTheBicategoryOfSpans}

We discuss the [[universal property]] that characterizes 2-categories of spans.

For $C$ be a [[category]] with [[pullbacks]], write

1. $Span_2(C) \coloneqq (Span(C))_{\sim2}$ for the [[weak 2-category]] of objects of $C$, spans as morphisms, and maps between spans as 2-morphisms,

1. $\eta_C: C \rightarrow Span_2(C)$ for the functor given by:

\begin{centre}
\begin{tikzcd}
  & & & &
  x
  \arrow[ld, "1_x"{above}]
  \arrow[rd, "f"]
  &
  \\
  x
  \arrow[r, "f"]
  &
  y
  &
  \mapsto
  &
  x
  & &
  y
\end{tikzcd}
\end{centre}

Now let

1. $K$ be any [[bicategory]]

1. $F, G \,\colon\, C \rightarrow K$ be [[functors]] such that every map in $C$ is sent to a map in $K$ possessing a [[right adjoint]] and satisfying the [[Beck-Chevalley Condition]] for any [[commutative square]] in $K$,

1. $\alpha \,\colon\, F \rightarrow G$ be a [[natural transformation]].

Then:

\begin{proposition}
**(universal property of the bicategory of spans)**
\linebreak
The following holds:

1. $\eta_C$ is [[universal]] among such functors $F$, i.e. $F$ as above factors as $F = \hat{F} \circ \eta_C$ for a functor $\hat{F} \,\colon\, Span_2(C) \rightarrow K$ which is unique up to isomorphism.

1. There exists a unique [[lax natural transformation]]: $\hat{\alpha} \,\colon\, \hat{F} \rightarrow \hat{G}$ such that $\hat{\alpha} \eta_C = \alpha$.

1. Let $x, y$ be objects in $C$ and $f: x \rightarrow y$ be a morphism in $C$. If $(\alpha_x, \alpha_y)$ induce a pseudo-map of adjoints $F(f) \dashv (Ff)^* \rightarrow G(f) \dashv (Gf)^*$, then $\hat{\alpha}$ is a [[pseudonatural transformation]]

Furthermore, if we denote $Pbk$ as the [[2-category]] of categories with pullbacks, [[pullback]]-preserving [[functor]]s, and [[equifibered natural transformation]]s and $BiCat$ as the [[tricategory]] of [[bicategories]], $Span(-): Pbk \rightarrow BiCat$ is well-defined as a functor.
\end{proposition}
This is due to [Hermida 1999](#Hermida99).

### Limits and colimits
 {#LimitsAndColimits}

Since a category of spans/correspondences $Corr(\mathcal{C})$ is evidently [[equivalence of categories|equivalent]] to its [[opposite category]], it follows that to the extent that [[limits]] exists they are also [[colimits]] and vice versa.

If the underlying category $\mathcal{C}$ is an [[extensive category]], then the [[coproduct]]/[[product]] in $Corr(\mathcal{C})$ is given by the [[disjoint union]] in $\mathcal{C}$. (See also [this](http://mathoverflow.net/questions/32526/what-is-the-product-in-the-2-category-of-spans) MO discussion).

More generally, every [[van Kampen colimit]] in $\mathcal{C}$ is a (co)limit in $Corr(\mathcal{C})$ — and conversely, this property characterizes van Kampen colimits. ([Sobocinski-Heindel 11](SobocinskiHeindel11)).

### Relation to relations
 {#RelationToRelations}

Correspondences may be seen as generalizations of [[relations]]. A relation is a [[correspondence]] which is [[(-1)-truncated]] as a [[morphism]] into the [[cartesian product]]. See at _[[relation]]_ and at _[[Rel]]_ for more on this.

## Examples

* Spans in [[FinSet]] behave like the [[categorification]] of  [[matrices]] with entries in the [[natural number]]s: for $X_1 \leftarrow N \to X_2$ a span of finite sets, the [[cardinality]] of the [[fiber]] $X_{x_1, x_2}$ over any two elements $x_1 \in X_1$ and $x_2 \in X_2$ plays the role of the corresponding matrix entry. Under this identification composition of spans indeed corresponds to matrix multiplication. This implies that the category of spans of finite sets is equivalent to the [[Lawvere theory]] of commutative monoids, that is, to the [[PROP]] for the free bicommutative [[bialgebra]].  Spans over finite sets is a [[rig category]] with respect to the tensor products induced by the coproduct and product in FinSet.  The coproduct in FinSet remains the coproduct, but the product becomes the bilinear [[tensor product of modules]].

* The [[Burnside category]] is essentially the category of correspondences in [[G-sets]] for $G$ a [[finite group]].

* A [[cobordism]] $\Sigma$ from $\Sigma_{in}$ to $\Sigma_{out}$ is an example of a cospan $\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$ in the category of [[smooth manifold]]s. However, composition of cobordisms is not quite the pushout-composition of these cospans: to make the composition be a [[smooth manifold]] again some extra technical aspects must be added ("[[collars]]").

* In [[prequantum field theory]] (see there for details), spans of [[stacks]] model [[trajectories]] of [[field (physics)|fields]].

* The [[category of Chow motives]] has as [[morphisms]] [[equivalence classes]] of [[linear combinations]] of spans of [[smooth variety|smooth]] [[projective varieties]].

* The [[Weinstein symplectic category]] has as morphisms [[Lagrangian correspondences]] between [[symplectic manifolds]].

  More generally [[symplectic dual pairs]] are correspondences between [[Poisson manifolds]].

* Cospans of homomorphisms of [[C*-algebras]] represent morphisms in [[KK-theory]] (by Cuntz' result).

* Correspondences of [[flag manifolds]] play a role as [[twistor correspondences]], see at _[Schubert calculus -- Correspondences](Schubert+calculus#Correspondences)_ and at _[[horocycle correspondence]]_

* The [[Fourier-Mukai transform]] is a pull-push operation through correspondences equipped with objects in a cocycle given by an object in a [[derived category of quasi-coherent sheaves]].

* [[Hecke correspondence]]

* A [[hypergraph]] is a span from a set of vertices to a set of (hyper)edges.

## Related concepts

* [[(∞,n)-category of spans]].

* [[sink]], [[cosink]]

* [[product]], [[coproduct]]

* [[multispan]]

* [[correspondence type]]

* [[reflexive graph]]

A category of correspondences is a refinement of a category [[Rel]] of relations. See there for more.

## References ##

The terminology "span" appears on page 533 of:

* [[Nobuo Yoneda]], *On ext and exact sequences* (PhD thesis), Journal of the Faculty of Science, Section I. **8** University of Tokyo (1960) 507–576 [[pdf](http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf), [CiNii:naid/500000325773](https://ci.nii.ac.jp/naid/500000325773)]

The $Span(C)$ construction was introduced by Jean Bénabou (as an example of a bicategory) in

* [[Jean Bénabou]], Introduction to Bicategories, _Lecture Notes in Mathematics 47_, Springer (1967), pp.1-77. ([doi](http://dx.doi.org/10.1007/BFb0074299))

Bénabou cites an article by Yoneda (1954) for introducing the concept of span (in the category of categories).

An exposition discussing the role of spans in [[quantum theory]]:

* [[John Baez]], _Spans in quantum Theory_ ([web](http://math.ucr.edu/home/baez/span/), [pdf](http://math.ucr.edu/home/baez/span/span.pdf), [blog](http://golem.ph.utexas.edu/category/2007/10/spans_in_quantum_theory.html))

The relationship between spans and [[bimodules]] is briefly discussed in

* [[John Baez]], _Bimodules versus spans_ ([blog](http://golem.ph.utexas.edu/category/2008/08/bimodules_versus_spans.html))

The relation to [[van Kampen colimits]] is discussed in

* {#SobocinskiHeindel11} [[Pawel Sobocinski]], Tobias Heindel, _Being Van Kampen is a universal property_, ([arXiv:1101.4594](http://arxiv.org/abs/1101.4594))

The [[universal property]] of categories of spans is due to

* {#Hermida99} [[Claudio Hermida]], _Representable Multicategories_, Advances in Mathematics, __151__ (2000), No. 2, 164-225, ([doi:10.1006/aima.1999.1877](https://doi.org/10.1006/aima.1999.1877))

and further discussed in:

* {#DawsonParePronk04} R. Dawson, [[Robert Paré]], [[Dorette Pronk]], _Universal properties of Span_, Theory and Appl. of Categories __13__, 2004, No. 4, 61-85, [TAC](http://www.tac.mta.ca/tac/volumes/13/4/13-04abs.html), [MR2005m:18002](http://www.ams.org/mathscinet-getitem?mr=2116323)

*  {#DawsonParePronk10} R. Dawson, [[Robert Paré]], [[Dorette Pronk]], _The span construction_, Theory Appl. Categ. __24__ (2010), No. 13, 302–377, [TAC](http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html) [MR2720187](http://www.ams.org/mathscinet-getitem?mr=2720187)

The structure of a [[k-tuply monoidal (n,r)-category|monoidal]] [[tricategory]] on spans in [[2-categories]] is discussed in

* {#Hoffnung} [[Alex Hoffnung]], _Spans in 2-Categories: A monoidal tricategory_ ([arXiv:1112.0560](http://arxiv.org/abs/1112.0560))

Generally, an [[(∞,n)-category of spans]] is indicated in section 3.2 of

* {#Lurie} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

[[!redirects spans]]

[[!redirects category of spans]]
[[!redirects categories of spans]]
[[!redirects bicategory of spans]]

[[!redirects correspondence]]
[[!redirects correspondences]]

[[!redirects category of correspondences]]
[[!redirects categories of correspondences]]

[[!redirects correspondence space]]
[[!redirects correspondence spaces]]


span rewriting

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Span rewriting

* table of contents
{: toc}

## Idea

Span rewriting is a collection of abstract [[category theory|category-theoretic]] methods of using [[spans]] to "rewrite" [[objects]] in a [[category]] by "deleting" part of them and replacing it with a substitute part that retains some of the same "connections" between the deleted part and the rest of the object.

The most common application is to the category of [[quivers]] (called [[directed graphs|(directed) graphs]] in the relevant literature) or related categories; thus span rewriting is often called "graph rewriting".

There is probably not much connection to algebraic [[rewriting]].

## Definitions

All the definitions below have the following context in common.  A **production** or **rule** in a category (often the category [[Quiv]] of [[quivers]]) is a [[span]]

$$ L \xleftarrow{l} K \xrightarrow{r} R. $$

A **match** for this production is a morphism $f:L\to C$ for some object $C$.  A **derivation** of a match along a production is supposed to be a new object obtained by "deleting the image of $L$ and replacing it with $R$".

A production is **left-linear** if $l$ is a [[monomorphism]], and **linear** if in addition $r$ is a monomorphism.

### Double pushout rewriting

For this definition, we work in an [[adhesive category]] (which includes all [[toposes]], hence in particular $Quiv$).  Given a left-linear production $ L \xleftarrow{l} K \xrightarrow{r} R$, a match $f:L\to C$ satisfies the **gluing condition** if the pair $(l,f)$ has a [[pushout complement]] consisting of $g:K\to E$ and $v:E\to C$.  In this case the **(double-pushout) derivation** associated to the match is the pushout of $g$ along $r$, yielding a pair of pushout squares

$$\array{ L & \xleftarrow{l} & K & \xrightarrow{r} & R \\
^f\downarrow && \downarrow^{\mathrlap{g}} && \downarrow \\
C & \xleftarrow{v}  & E & \to & D.}$$

The restriction of $l$ to be monic is necessary to ensure that pushout complements are essentially unique when they exist.  Often we further restrict $r$ to be monic to ensure overall good behavior, obtaining the notion of **linear production**.

[Lack and Sobocinski](#LS) describe the intuition in this way:

> in order to perform the rewrite, we need to match $L$ as a substructure of a redex $C$. The structure $K$, thought of as a substructure of $L$, is exactly the part of $L$ which is to remain invariant as we apply the rule to $C$. Finally, parts of $R$ which are not in $K$ should be added to produce the final result of the rewrite.

> Thus, an application of a rewrite rule consists of three steps. First we must match $L$ as a substructure of the redex C; secondly, we delete all of parts of the redex matched by $L$ which are not included in $K$. Thirdly, we add all of $R$ which is not contained in $K$, thereby producing a new structure $D$.  The deletion and addition of structure is handled, respectively, by finding a pushout complement and constructing a pushout.

### Sesqui-pushout rewriting

In an adhesive category, the pushout of a monomorphism is also a pullback.  Thus, the pushout complements involved in double-pushout rewriting are also "pullback complements".  Pullback complements are not in general unique, even in an adhesive category, but those arising as pushout complements do satisfy a [[universal property]]: they are [[final pullback complements]].

This suggests the following generalization.  In an arbitrary category, given an arbitrary production $ L \xleftarrow{l} K \xrightarrow{r} R $, the **(sesqui-pushout) derivation** associated to a match $f:L\to C$ is a diagram

$$\array{ L & \xleftarrow{l} & K & \xrightarrow{r} & R \\
^f\downarrow && \downarrow^{\mathrlap{g}} && \downarrow \\
C & \xleftarrow{v}  & E & \to & D.}$$

in which the left square is a final pullback complement and the right square is a pushout.  Such a derivation may or may not exist, but if it does then it is essentially unique.  Moreover, if we are in an adhesive category and $l$ is monic, then any double-pushout derivation is also a sesqui-pushout derivation.

More generally, given the construction of final pullback complements using [[exponential objects]], if the exponential $\Pi_f l$ exists and the [[counit]] $f^* \Pi_f l \to l$ is an isomorphism, then such a derivation exists with $E = \Pi_f l$.  In particular, this happens in any [[locally cartesian closed category]] if $f$ and $l$ are both monomorphisms.

### Single pushout rewriting

We can also regard a left-linear production as a [[partial morphism]], in which case the universal property of a final pullback complement suggests the following different viewpoint.  Given a left-linear production $ L \xleftarrow{l} K \xrightarrow{r} R $, the **(single-pushout) derivation** associated to a match $f:L\to C$ is the [[pushout]] of $(l,r)$ and $f$ in the category of partial morphisms (if it exists).

It can be shown (see [CHHK](#CHHK)) that in a certain abstract context, any sesqui-pushout derivation of a left-linear production is also a single-pushout derivation; and that the converse holds if and only if the match is "conflict-free", and if and only if the induced partial morphism from $R$ to the target object $D$ is total.

### General gluing constructions

In [Lowe10](#Lowe10) and [Lowe12](#Lowe12) a yet more general construction is considered for span rewriting, involving $3\times 3$ diagrams consisting of a pullback, two final pullback complements, and a pushout.

## References

* H. Ehrig, M. Pfender, and H.J. Schneider, _Graph-grammars: an algebraic approach_, In IEEE Conf. on Automata and Switching Theory, pages 167–180, 1973.

* [[Steve Lack]] and [[Pawel Sobocinski]], *Adhesive categories*, [PDF](http://users.ecs.soton.ac.uk/ps/papers/adhesive.pdf)
 {#LS}

* Andrea Corradini, Tobias Heindel, Frank Hermann, and Barbara König, *Sesqui-pushout rewriting*, 2006, [springerlink](https://link.springer.com/chapter/10.1007/11841883_4), [pdf](http://www.ti.inf.uni-due.de/publications/koenig/icgt06b.pdf).
 {#CHHK}

* Michael Löwe, *Graph rewriting in Span-categories*, 2010, [springerlink](https://link.springer.com/chapter/10.1007/978-3-642-15928-2_15)
 {#Lowe10}

* Michael Löwe, *Refined graph rewriting in Span-categories*, 2012, [springerlink](https://link.springer.com/chapter/10.1007/978-3-642-33654-6_8)
 {#Lowe12}

* Nicolas Behr, Pawel Sobocinski, *Rule Algebras for Adhesive Categories*, [arXiv:1807.00785](https://arxiv.org/abs/1807.00785)

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