SO orientation of elliptic cohomology

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#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
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## Idea

The [[Ochanine elliptic genus]] $\Omega^{SO}_\bullet \to M_\bullet = \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$ lifts to a map of [[ring spectra]]

$$
  M SO \longrightarrow Ell
$$

([Landweber-Ravenel-Stong 93](#LandweberRavenelStong93)). Here $Ell[\tfrac{1}{6}]$ is equivalently [[tmf0(2)]] ([Behrens 05](#Behrens05)) and as such this lift is analogous to the [[string orientation of tmf]] $M String \to tmf$.

If this map of [[ring spectra]] could be shown to be "highly structured" in that it preserves [[E-∞ ring]] structure, then it would equivalently be a universal [[orientation in generalized cohomology|orientation]]  (see at [relation between orientations and genera](orientation+in+generalized+cohomology#RelationToGenera)).

## Definition

After inversion of the [[prime number]] 2, the oriented [[cobordism ring]] is a [[polynomial ring]] over $\mathbb{Z}[\tfrac{1}{2}]$ on generators in degrees $4k$

$$
  \Omega^{SO}_\bullet[\tfrac{1}{2}]
  \simeq
  \mathbb{Z}[\tfrac{1}{2}] [x_4, x_8, x_{12}, \cdots  ]
$$

where $x_4$ is the class of the complex [[projective space]] $\mathbb{C}P^2$ and  $x_8$ that of $\mathbb{H}P^2$ and where all [[elliptic genera]] vanish on all the other generators ([Landweber-Ravenel-Stong 93, prop. 3.2](#LandweberRavenelStong93)).

From this one gets that the [[quotient]] by the [[ideal]] generated by these higher elements is

$$
  \Omega^{SO}_\bullet[\tfrac{1}{2}]/(x_{4(k \geq 3)})
  \simeq
  MF_0(2)_\bullet \coloneqq \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]
$$

where the right hand side here is naturally identified as the ring of those [[modular forms]] for the [[congruence subgroup]] $\Gamma_0(2)$ which have half-integral [[coefficients]] in their $q$-expansion at the [[nodal curve]] ([Landweber-Ravenel-Stong 93, theorem 1.5](#LandweberRavenelStong93)).

Now by a general construction due to ([Baas 73](#Baas73)) this induces a [[generalized homology theory]]
$$
  \Omega^{SO}_\bullet[\tfrac{1}{2}](-)
$$

[[Brown representability theorem|represented]] by some [[spectrum]] $Ell$,whose coefficient ring is as above

$$
  Ell_\bullet \simeq MF_0(2)_\bullet
  \,.
$$

By construction, this comes with a quotient map

$$
  M SO[\tfrac{1}{2}] \longrightarrow Ell
$$

which is a map of [[ring spectra]] by ([Mironov 78](#Mironov78)). This maplifts the universal [[elliptic genus]] (in that it reproduces it on [[homotopy groups]]) ([Landweber-Ravenel-Stong 93, section 4.6, 4.7](#LandweberRavenelStong93))

## Properties

### Induced relation between cobordism and homology

The SO orientation of elliptic cohomology makes it expressible in terms of the [[cobordism cohomology theory]], see at _[[cobordism theory determining homology theory]]_ ([Landweber-Ravenel-Stong 93, theorem 1.2](#LandweberRavenelStong93)).

### Relation to the Atiyah-Bott-Shapiro Spin orientation of KO

There are maps of spectra

$$
  Ell [\epsilon^{-1}] \longrightarrow KO[\tfrac{1}{2}]
$$

and

$$
  Ell [(\delta^2- \epsilon)^{-1}] \longrightarrow KO[\tfrac{1}{2}]
$$

such that postcomposition of the above SO-orientation with reproduces the [[signature genus]] and the [[Atiyah-Bott-Shapiro orientation]] of [[KO]], respective, hence the [[A-hat genus]] ([Landweber-Ravenel-Stong 93, prop. 4.9](#LandweberRavenelStong93)).

Notice that here the second localization correponds again to including the [[nodal curve]]:

[[!include moduli stack of curves -- table]]

### Homotopy type of the spectrum $Ell$

After suitable localization the spectrum Ell is a wedge sum of suspensions of the [[Morava E-theory]] $E(2)$ ([Baker 97](#Baker97)).

Specifically after [[K(n)-local spectrum|K(2)-localization]] and inversion of 6 it coincides with [[tmf0(2)|TMF0(2)]]

$$
  L_{K(2)} TMF_0(2) \simeq L_{K(2)}(E(2) \vee \Sigma^8 E(2))
  \,.
$$

## Related concepts

[[!include genera and partition functions - table]]

## References

The construction is due to

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et. al. (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

based on general constructions of multiplicative homology theories from cobordism theories due to

* {#Baas73} [[Nils Baas]], _On bordism theory of manifolds with singularities_, Math. Scand. 33 (1973), 279–302.

* {#Mironov78} O. K. Mironov, _Multiplications in cobordism theories with singularities_, and Steenrod-tom Dieck operations_, Izv. Akad. Nauk SSSR, Ser. Mat. 42 (1978), 789–806; English transl. in Math. USSR Izvestiya 13 (1979), 89–106.

Analysis of $Ell$ is in

* {#Baker97} [[Andrew Baker]], The homotopy type of the spectrum representing elliptic cohomology,  Proceedings of the American Mathematical Society 107.2 (1989): 537-548. ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/homell.pdf))

* [[Andrew Baker]], _On the Adams $E_2$-term for elliptic cohomology_, 1997 ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/ell-ext.pdf))

The interpretation of $Ell$ in terms of [[tmf0(2)|TMF0(2)]] is discussed in

* {#Behrens05} [[Mark Behrens]], _A modular description of the K(2)-local sphere at the prime 3_ ([arXiv:math/0507184](http://arxiv.org/abs/math/0507184))

More is in

* [[Dylan Wilson]], _Orientations and Topological Modular Forms with Level Structure_ ([arXiv:1507.05116](http://arxiv.org/abs/1507.05116))


SO(10)

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## Idea

$SO(10)$ is the [[special orthogonal group]] in [[dimension]] 10.

$Spin(10)$ is the [[spin group]] in [[dimension]] 10.

In the [[classification of simple Lie groups]] this is the entry [[D5]].

## Related concepts

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

\linebreak

## References

### General

(...)

### As a grand unified gauge group

Discussion as a [[gauge group]] in [[grand unified theory]] (see [there](GUT#TheGUTKnownAsSO10)):

* {#BLM09} Stefano Bertolini, Luca Di Luzio, [[Michal Malinský]], _Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal_, Phys. Rev. D80:015013, 2009 ([arXiv:0903.4049](https://arxiv.org/abs/0903.4049))

review:

* [[Howard Georgi]], §24 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

* {#Malinsky09} [[Michal Malinský]], _35 years of GUTs - where do we stand?_, 2009 ([pdf](https://www.mpi-hd.mpg.de/lin/seminar_theory/talks/Malinsky.pdf))

for non-superymmetric [[model (physics)|models]]:

* {#LavouraWolfenstein10} L. Lavoura and Lincoln Wolfenstein, _Resuscitation of minimal $SO(10)$ grand unification_, Phys. Rev. D 48, 264 ([doi:10.1103/PhysRevD.48.264](https://doi.org/10.1103/PhysRevD.48.264))

* {#AltarelliMeloni13} Guido Altarelli, Davide Meloni, _A non Supersymmetric SO(10) Grand Unified Model for All the Physics below $M_{GUT}$_ ([arXiv:1305.1001](https://arxiv.org/abs/1305.1001))

* {#DueckRodejohann13} Alexander Dueck, Werner Rodejohann, _Fits to $SO(10)$ Grand Unified Models_ ([arXiv:1306.4468](http://arxiv.org/abs/1306.4468))

* {#FongMeloni14} Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, _Leptogenesis in $SO(10)$_ ([arXiv:1412.4776](http://arxiv.org/abs/1412.4776))

  (in view of [[leptogenesis]])

* {#OhlssonPernow19} Tommy Ohlsson, Marcus Pernow, _Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution_ ([arXiv:1903.08241](https://arxiv.org/abs/1903.08241))

* {#CPS19} Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, _Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings_ ([arXiv:1906.05601](https://arxiv.org/abs/1906.05601))

  > Results indicating non-SUSY $SO(10)$ as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.

* Nobuchika Okada, Digesh Raut, Qaisar Shafi, _Inflation, Proton Decay, and Higgs-Portal Dark Matter in $SO(10) \times U(1)_\pri$_ ([arXiv:1906.06869](https://arxiv.org/abs/1906.06869))

for [[supersymmetry|supersymmetric]] [[model (physics)|models]]:

* Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, _LHC Phenomenology of $SO(10)$ Models with Yukawa Unification II_ ([arXiv:1404.5628](http://arxiv.org/abs/1404.5628))

* Ila Garg, _New minimal supersymmetric $SO(10)$ GUT phenomenology and its cosmological implications_ ([arXiv:1506.05204](http://arxiv.org/abs/1506.05204))

On [[symmetry breaking]] in [[Spin(10)]]-[[grand unified theory]] to the exact [[standard model of particle physics|standard model]] [[gauge group]], via [[real normed division algebra]] ([[octonions]], [[quaternions]]):

* [[Kirill Krasnov]], *Geometry of $Spin(10)$ Symmetry Breaking* [[arXiv:2209.05088](https://arxiv.org/abs/2209.05088)]

* [[Nichol Furey]], M. J. Hughes, *Division algebraic symmetry breaking* [[arXiv:2210.10126](https://arxiv.org/abs/2210.10126)]

[[!redirects Spin(10)]]


SO(11)

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#### Group Theory
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#Contents#
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## Idea

$SO(11)$ is the [[special orthogonal group]] in [[dimension]] 11.

$Spin(11)$ is the [[spin group]] in [[dimension]] 11.

In the [[classification of simple Lie groups]] this is the entry B5.

## Related concepts

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

\linebreak

## References

### General

(...)

### As a grand unified gauge group

Discussion as a [[gauge group]] in [[grand unified theory]] (see [there](GUT#TheGUTKnownAsSO11)):

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([doi:10.1093/ptep/ptv153](https://doi.org/10.1093/ptep/ptv153), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

[[!redirects Spin(11)]]


SO(12)

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#Contents#
* table of contents
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## Idea

$SO(12)$ is the [[special orthogonal group]] in [[dimension]] 12.

$Spin(12)$ is the [[spin group]] in [[dimension]] 12.

In the [[classification of simple Lie groups]] this is the entry [[D6]].

## Related concepts

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

\linebreak

## References

### General

(...)

### As a grand unified gauge group

Discussion as a [[gauge group]] in [[grand unified theory]] (see [there](GUT#TheGUTKnownAsSO12)):

* S. Rajpoot and P. Sithikong, _Implications of the $SO(12)$ gauge symmetry for grand unification_, Phys. Rev. D 23, 1649 (1981) ([doi:10.1103/PhysRevD.23.1649](https://doi.org/10.1103/PhysRevD.23.1649))

* Takaaki Nomura and Joe Sato, _Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space_, Nucl.Phys.B811:109-122, 2009 ([arXiv:0810.0898](https://arxiv.org/abs/0810.0898))

* Takaaki Nomura, _Physics beyond the standard model with $S^2$ extra-space_, 2009 ([pdf](http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf), [[NomuraSO12GUT.pdf:file]])

* Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, _Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry_ ([arXiv:1109.5835](https://arxiv.org/abs/1109.5835))

[[!redirects Spin(12)]]


SO(16)

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#### Group Theory
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## Idea

The [[special orthogonal group]] in [[dimension]] 16.

## Related concepts

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

* [[type I' string theory]]


SO(3)

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* table of contents
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## Idea

The [[special orthogonal group]] in [[dimension]] 3.

## Properties

### Cohomology

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

The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] $B SO(3)$ is

$$
  H^\bullet\big(
    B SO(3), \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3)
  \,,
$$

where

* $p_1 \in H^4\big(B SO(3), \mathbb{Z}\big)$ is the [[universal characteristic class|universal]] [[first Pontryagin class]];

* $W_3 = \beta(w_2) \in H^3\big( B SO(3), \mathbb{Z}\big)$ is the [[universal characteristic class|universal]] degree-3 [[integral Stiefel-Whitney class]].

=--

This is a special case of [Brown 82, theorem 1.5](#Brown82), which is also reviewed as [Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25](#RudolphSchmidt17).

\linebreak

### Finite subgroups

+-- {: .num_theorem #ClassificationOfFiniteSubgroupsOfSO3}
###### Theorem
**([[ADE classification]] of [[finite group|finite]] [[subgroups]] of [[SO(3)]] and [[spin group|Spin(3)]]$\simeq$ [[SU(2)]])**

The [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ as well as the [[finite group|finite]] [[subgroups]] of the [[special unitary group]] [[SU(2)]] are, up to [[conjugation]], given by the following classification:

[[!include ADE -- table]]

Here under the [[double cover]] projection (using the [exceptional isomorphism](spin+group#ExceptionalIsomorphisms) $SU(2) \simeq Spin(3)$)

$$
  SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)
$$

all the finite subgroups of $SU(2)$ except the [[odd number|odd]]-[[order of a group|order]] [[cyclic groups]] are the [[preimages]] of the corresponding finite subgroups of $SO(3)$, in that we have [[pullback]] [[diagrams]]

$$
  \array{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n}
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n}
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/n
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

exhibiting the [[even number|even]] [[order of a group|order]] [[cyclic groups]] as subgroups of [[Spin(2)]], including the the minimal case of the [[group of order 2]]

$$
  \array{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i}
     \right)
     =
     -1
    \right\rangle
    &
    =
    &
    \mathbb{Z}/2
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i}
     \right)
     }
     =
     e
    \right\rangle
    &
    =
    &
    1
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

as well as

$$
  \array{
    \left\langle
      \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j}
    \right\rangle
    &=&
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
      Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n}  \right) },
      \,
      Ad_{\mathrm{j}}
    \right\rangle
    &&
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

exhibiting the [[binary dihedral groups]] as sitting inside the [[Pin(2)]]-[[subgroup]] of [[Spin(3)]],

but only [[commuting diagrams]]

$$
  \array{
    \left\langle
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{{2n+1}}
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &&\overset{\phantom{AA}}{\hookrightarrow}&&
    Spin(3)
    \\
    &&
    \big\downarrow
      &&
      &&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{2n+1}
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

for the [[odd number|odd]] [[order of a group|order]] [[cyclic group|cyclic]] [[subgroups]].

=--

This goes back to ([Klein 1884, chapter I](finite+rotation+group#Klein1884)).
Full proof for $SO(3)$ is spelled out for instance in ([Rees 05, theorem 11](finite+rotation+group#Rees05), [De Visscher 11](finite+rotation+group#DeVisscher11)). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in ([Miller-Blichfeldt-Dickson 16](finite+rotation+group#MillerBlichfeldtDickson16)) reviewed in ([Serrano 14, section 2](finite+rotation+group#Serrano14)). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in [Keenan 03, theorem 4](finite+rotation+group#Keenan03).

\linebreak

## Related concepts

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

* [[Euclidean group]]

* [[rigid body dynamics]]

\linebreak

## References

* Jason Hanson, _Rotations in three, four, and five dimensions_ ([arXiv:1103.5263](https://arxiv.org/abs/1103.5263))

See also

* Wikipedia, _[3D rotation group](https://en.wikipedia.org/wiki/3D_rotation_group)_

On the [[integral cohomology]] of the [[classifying space]]:

* {#Brown82} [[Edgar H. Brown]], _The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients_, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 ([jstor:2044298](https://www.jstor.org/stable/2044298))

reviewed in

* {#RudolphSchmidt17} Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))


SO(32)

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* table of contents
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## Idea

The [[special orthogonal group]] in [[dimension]] 32.

## Properties

### In string theory

In [[type I string theory]], and more generally in [[type II string theory]] on [[toroidal orbifold|toroidal]] [[orientifolds]], [[RR-field tadpole cancellation]] implies that the [[gauge group]] is [[SO(32)]] (see [there](orientifold+plane#OPlaneChargeForFlatOrientifolds)).

Under the [[duality between type I and heterotic string theory]] this translates to the [[semi-spin group|semi-spin]] [[gauge group]] [[SemiSpin(32)]] of [[heterotic string theory]].

Discussion of type-I [[string phenomenology]] and [[grand unified theory]] based on [[SO(32)]] type-I strings: ([MMRB 86](#MMRB86), [Ibanez-Munoz-Rigolin 98](#IbanezMunozRigolin98), [Yamatsu 17](#Yamatsu17)).

## Related concepts

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

## References

Type I [[string phenomenology]] and discussion of [[GUT]]s based on [[SO(32)]] type I strings:

* {#MMRB86} H.S. Mani, A. Mukherjee, R. Ramachandran, A.P. Balachandran, _Embedding of $SU(5)$ GUT in $SO(32)$ superstring theories_, Nuclear Physics B Volume 263, Issues 3–4, 27 January 1986, Pages 621-628 (doi:10.1016/0550-3213(86)90277-4)

* {#IbanezMunozRigolin98} [[Luis Ibáñez]], C. Muñoz, S. Rigolin, _Aspects of Type I String Phenomenology_, Nucl.Phys. B553 (1999) 43-80 ([arXiv:hep-ph/9812397](https://arxiv.org/abs/hep-ph/9812397))

* {#Dudas00} Emilian Dudas, _Theory and Phenomenology of Type I strings and M-theory_, Class. Quant. Grav.17:R41-R116, 2000 ([arXiv:hep-ph/0006190](https://arxiv.org/abs/hep-ph/0006190))

* {#Yamatsu17} Naoki Yamatsu, _String-Inspired Special Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1  ([arXiv:1708.02078](https://arxiv.org/abs/1708.02078), [doi:10.1093/ptep/ptx135](https://doi.org/10.1093/ptep/ptx135))


SO(4)

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#Contents#
* table of contents
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## Idea

The [[special orthogonal group]] in [[dimension]] 4.

## Properties

### Exceptional isomorphisms

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

=--

### Cohomology

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

The [[integral cohomology]] [[cohomology ring|ring]] of the [[classifying space]] $B SO(4)$ is

$$
  H^\bullet
  \big(
    p_1, \chi, W_3
  \big)
  /
  \big(
    2 W_3
  \big)
$$

where

* $p_1$ is the [[first Pontryagin class]]

* $\chi$ is the [[Euler class]],

* $W_3$ is the [[integral Stiefel-Whitney class]].

Notice that the [[cup product]] of the [[Euler class]] with itself is the [[second Pontryagin class]]

$$
  \chi \smile \chi \;=\; p_2
  \,,
$$

which therefore, while present, does not appear as a separate generator.

=--

This is a special case of [Brown 82, Theorem 1.5](#Brown82), reviewed for instance as [Rudolph-Schmidt 17, Thm. 4.2.23 with Rmk. 4.2.25](#RudolphSchmidt17).

\linebreak

### Homotopy groups

The [[homotopy groups]] of $SO(4)$ in [[low-dimensional topology|low degrees]] are

| $G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $SO(4)$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{3}^{\oplus 2}$ | $\mathbb{Z}_{15}^{\oplus 2}$|$\mathbb{Z}_{2}^{\oplus 2}$ |$\mathbb{Z}_{2}^{\oplus 4}$  | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}$ |

\linebreak

## Related concepts

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

## References

* {#Berger87} [[Marcel Berger]], Section 8.9.8 of: *Geometry I*, Springer 1987 ([doi:10.1007/978-3-540-93815-6](https://doi.org/10.1007/978-3-540-93815-6))

* Jason Hanson, _Rotations in three, four, and five dimensions_ ([arXiv:1103.5263](https://arxiv.org/abs/1103.5263))

See also

* Wikipedia, _[Rotations in 4-dimensional Euclidean space](https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space)_

On the [[integral cohomology]] of the [[classifying space]]:

* {#Brown82} [[Edgar H. Brown]], _The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients_, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 ([jstor:2044298](https://www.jstor.org/stable/2044298))

reviewed in

* {#RudolphSchmidt17} Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

[[!redirects SO4]]


SO(5)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[special orthogonal group]] in [[dimension]] 5.

## Related concepts

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

## References

* Jason Hanson, _Rotations in three, four, and five dimensions_ ([arXiv:1103.5263](https://arxiv.org/abs/1103.5263))

[[!redirects SO5]]


SO(6)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[special orthogonal group]] in [[dimension]] 6.

## Related concepts

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

[[!redirects SO6]]


SO(7)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[special orthogonal group]] in [[dimension]] 7.

## Related concepts

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

[[!redirects SO7]]


SO(8)

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

/clissifying
#Contents#
* table of contents
{:toc}

## Idea

Among all [[special orthogonal groups]] $SO(n)$, the case of $SO(8)$ is special, since in the [[ADE classification]] of [[simple Lie groups]] it corresponds to [[D4]], which makes its [[representation theory]] enjoy _[[triality]]_.

## Properties

### Subgroup lattice
 {#SubgroupLattice}

+-- {: .num_prop #Spin7SubgroupsInSpin8}
###### Proposition
**([[Spin(7)]]-[[subgroups]] in [[Spin(8)]])

There are precisely 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[Spin(7)]]-[[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      \mathrm{Spin}(7)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      & \mathrm{Spin}(8)
      &
      \mathrm{Spin}(7)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      \mathrm{Spin}(7)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 6](#Varadarajan01), see also [Kollross 02, Prop. 3.3 (1)](#Kollross02))

+-- {: .num_prop #G2IsIntersectionOfSpin7SubgroupsInSpin8}
###### Proposition
**([[G2]] is [[intersection]] of [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])**

The [[intersection]] inside [[Spin(8)]] of any two [[Spin(7)]]-subgroups from distinct [[conjugacy classes of subgroups]] (according to Prop. \ref{Spin7SubgroupsInSpin8})
is the [[exceptional Lie group]] [[G2]], hence we have [[pullback squares]] of the form

\begin{center}
\begin{xymatrix}
  G_2
   \ar@{^{(}->}[r]
   \ar@{^{(}->}[d]
  &
  \mathrm{Spin}(7)
  \ar@{^{(}->}[d]^-{\iota^\prime}
  \\
  \mathrm{Spin}(7)
   \ar@{^{(}->}[r]_-{\iota}
  &
  \mathrm{Spin}(8)
 \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 13](#Varadarajan01))

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

We have the following [[commuting diagram]] of [[subgroup]] inclusions, where each [[square]] exhibits a [[pullback]]/[[fiber product]], hence an [[intersection]] of subgroups:

\begin{center}
\begin{xymatrix}
    \mathrm{SU}(2)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{SU}(3)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{G}_2
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[d]^{\iota'}
    \\
    \mathrm{Spin}(5)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(6)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[r]_-{\iota}
    &
    \mathrm{Spin}(8)
\end{xymatrix}
\end{center}

Here in the bottom row we have the [[Lie groups]]

[[Spin(5)]] $\hookrightarrow$ [[Spin(6)]] $\hookrightarrow$ [[Spin(7)]] $\hookrightarrow$ [[Spin(8)]]

with their canonical [[subgroup]]-inclusions, while in the top row we have

[[SU(2)]] $\hookrightarrow$ [[SU(3)]] $\hookrightarrow$ [[G2]] $\hookrightarrow$ [[Spin(7)]]

and the right vertical inclusion $\iota'$ is one of the two non-standard inclusions, according to Prop. \ref{Spin7SubgroupsInSpin8}.

=--

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

The square on the right is that from Prop. \ref{G2IsIntersectionOfSpin7SubgroupsInSpin8}.

The square in the middle is [Varadarajan 01, Lemma 9 on p. 10](#Varadarajan01).

The statement also follows with
[Onishchik 93, Table 2, p. 144](#Onishchik93):

\begin{center}
\begin{imagefromfile}
  "file_name": "SO8SubgroupIntersection.jpg",
  "width": 690
\end{imagefromfile}
\end{center}

=--

+-- {: .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))

{#Sp2DotSp1SubgroupInclusionsInSummary} In summary we have these subgroup inclusions

\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}

permuted by triality:

\begin{center}
\begin{imagefromfile}
  "file_name": "TrialityOnSp2Sp1.jpg",
  "width": 680
\end{imagefromfile}
\end{center}

> graphics grabbed from [FSS 19, Sec. 3.3](#FSS19)

\linebreak

### Homotopy groups

The [[homotopy groups]] of $SO(8)$ in [[low-dimensional topology|low degrees]]:

| $G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $SO(8)$ | $\mathbb{Z}_2$ | $0$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 3}$ | $\mathbb{Z}_{2}^{\oplus 3}$ | $\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}$ | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$ | $\mathbb{Z}_2^{\oplus 7}$ |

\linebreak

### Cohomology of classifying spaces

+-- {: .num_prop #CohomologyRingOfSpin8}
###### Proposition

The [[ordinary cohomology|ordinary]] [[cohomology ring]] of the [[classifying space]] $B Spin(8)$ is:

**1)** with [[coefficients]] in the [[cyclic group of order 2]]:

$$
  H^\bullet
  \big(
    B Spin(8), \mathbb{Z}_2
  \big)
  \;\simeq\;
  \mathbb{Z}
  \big[
    w_4, w_6, w_7, w_8,
    \;
    \rho_2
    \left(
    \tfrac{1}{4}
    \left(
      p_2
      -
      \big(\tfrac{1}{2}p_1\big)^2
      -
      2 \chi
    \right)
    \right)
  \big]
$$

where $w_i$ are the [[universal characteristic class|universal]] [[Stiefel-Whitney classes]],

and where

$$
  \rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)
$$

is [[cyclic group|mod 2 reduction]]

**2)** with [[coefficients]] in the [[integers]]:

$$
  H^\bullet
  \big(
    B Spin(8), \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}
  \Big[
    \tfrac{1}{2}p_1,
    \;
    \tfrac{1}{4}
    \left(
      p_2
      -
      \big(\tfrac{1}{2}p_1\big)^2
    \right)
    -
    \tfrac{1}{2}\chi
    ,
    \;
    \chi,
    \;
    \beta(w_6)
  \Big]
  /
  \big\langle 2 \beta(w_6)\big\rangle
  \,,
$$

where $p_1$ is the [[first fractional Pontryagin class]], $p_2$ is the [[second Pontryagin class]], $\chi$ is the [[Euler class]], and

$$
  \beta
  \;\colon\;
  H^\bullet
  \big(
    B Spin(8),
    \mathbb{Z}_2
  \big)
  \longrightarrow
  H^{\bullet + 1}
  \big(
    B Spin(8),
    \mathbb{Z}
  \big)
$$

is the [[Bockstein homomorphism]].

Moreover, we have the following relations:

$$
  \begin{aligned}
    \rho_2\left(
      \tfrac{1}{2}p_1
    \right)
    & =
    w_4
    \\
    \rho_2\big( \chi\big)
    & =
    w_8
  \end{aligned}
$$

=--

This is due to [Quillen 71](#Quillen71), [Čadek-Vanžura 95](#CadekVanzura95), see [Čadek-Vanžura 97, Lemma 4.1](#CadekVanzura97).

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

Consider the [[looping and delooping|delooping]] of the [[triality]] automorphism relating [[Sp(2).Sp(1)]] with [[Spin(5).Spin(3)]] (Prop. \ref{Spin3DotSpin5SubgroupsInSpin8}) on [[classifying spaces]]

$$
  \array{
    B
    \big(
      Spin(5) \cdot Spin(3)
    \big)
    &\hookrightarrow&
    B Spin(8)
    \\
    \big\downarrow
    &&
    \big\downarrow^{ B \mathrlap{tri} }
    \\
    B
    \big(
      Sp(2) \cdot Sp(1)
    \big)
    &\hookrightarrow&
    B Spin(8)
  }
$$

Then the pullback of the [[universal characteristic classes]] of $B Spin(8)$ (from Prop. \ref{CohomologyRingOfSpin8}) along $B tri$ is as follows:

$$
  \big(
    B tri
  \big)^\ast
  \;\colon\;
  \begin{aligned}
    \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1
    \\
    \chi
      & \mapsto
      -
      \tfrac{1}{4}
      \big(
        p_2
        - \big(\tfrac{1}{2}p_1\big)^2
      \big)
      +
      \tfrac{1}{2}\chi
    \\
      \tfrac{1}{4}
      \big(
        p_2
        -
        \big(\tfrac{1}{2}p_1\big)^2
      \big)
      -
      \tfrac{1}{2}\chi
    & \mapsto
      -
      \chi
  \end{aligned}
$$

=--

([Čadek-Vanžura 97, Lemma 4.2](#CadekVanzura97))

In fact $tri^{-1} = tri$.

Hence, in rational cohomology:

$$
  \begin{aligned}
    \big(
      B tri
    \big)^\ast
    \big(
      \tfrac{1}{4}p_2
    \big)
    & =
    \big(
      B tri
    \big)^\ast
    \Big(
      \big(
        \tfrac{1}{4}p_2
        - \big(\tfrac{1}{4}p_1\big)^2
        - \tfrac{1}{2}\chi
      \big)
      + \big(\tfrac{1}{4}p_1\big)^2
      + \tfrac{1}{2}\chi
    \Big)
    \\
    & =
    -\chi
    +
    \big(\tfrac{1}{4}p_1\big)^2
    -
    \tfrac{1}{2}
    \big(
      \tfrac{1}{4}p_2
      - \big(\tfrac{1}{4}p_1\big)^2
      - \tfrac{1}{2}\chi
    \big)
  \end{aligned}
$$

\linebreak

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

\linebreak

## Related concepts

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

\linebreak

## References

### General

* [[Robert Bryant]], _Remarks on Spinors in Low Dimension_ ([pdf](https://services.math.duke.edu/~bryant/Spinors.pdf), [[BryantRemarksOnSpinorsInLowDimension.pdf:file]])

See also

* Wikipedia, _SO(8)_

### Subgroup lattice

On the [[subgroup lattice]] of [[Spin(8)]]

* {#Onishchik93} A. L. Onishchik (ed.) _Lie Groups and Lie Algebras_

  * *I.* A. L. Onishchik, E. B.  Vinberg, _Foundations of Lie Theory_,

  * *II.* V. V. Gorbatsevich, A. L. Onishchik, _Lie Transformation Groups_

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, [pdf](https://core.ac.uk/download/pdf/81114499.pdf))

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _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))

* Megan M. Kerr, _New examples of homogeneous Einstein metrics_, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 ([euclid:1030132086](https://projecteuclid.org/euclid.mmj/1030132086))

* {#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))

Discussion with an eye towards foundations of [[M-theory]]:

* {#FSS19} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

### Cohomology

The [[integral cohomology]] of the [[classifying spaces]] $B SO(8)$ and $B Spin(8)$ and the [[action]] of [[triality]] on these is discussed in

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

* {#Quillen71} [[Daniel Quillen]], _The mod 2 cohomology rings of extra-special 2-groups and the spinor groups_, Math. Ann . 194 (1971), 19

* {#CadekVanzura95} [[Martin Čadek]], [[Jiří Vanžura]], _On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes_, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 ([dml-cz:118764](https://dml.cz/handle/10338.dmlcz/118764))

* {#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))

[[!redirects Spin(8)]]

[[!redirects SO8]]

[[!redirects Spin8]]


sober topological space

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

# Contents
* table of contents
{: toc}

## Idea

If $\mathcal{O}(X)$ is the topology on a [[topological space]] $X$ (i.e. its [[frame of opens]]), and if a map $\mathcal{O}(X) \to \mathcal{O}(1)$ that preserves finite [[meets]] and arbitrary [[joins]] (a [[homomorphism]] of [[frames]]) is considered an instance of "seeing a point $1 \to X$", then $X$ is _sober_ precisely if every point we see is really there (i.e., is induced from a [[continuous function]] $1 \to X$), and if we never see double.

The condition that a [[topological space]] be _sober_ is an extra condition akin to a [[separation axiom]]. In fact with [[classical logic]] it is a condition implied by the $T_2$ [[separation axiom]] ([[Hausdorff implies sober]]) and implying $T_0$.

| [[separation axioms]]                 |
|---------------------------------------|
| $\array{\\ &&& T_2 = \text{Hausdorff}  \\ && \swArrow && \seArrow \\ \, & T_1 && && \text{sober} & \, \\ && \seArrow && \swArrow \\ &&& T_0 = \text{Kolmogorov} \\ }  $ |

(Note that this diagram is not a pullback -- there are [[Hausdorff implies sober|$T_1$ sober spaces which are not Hausdorff]].)

But the sobriety condition on a topological space has deeper meaning. It means that [[continuous functions]] between sober topological spaces are entirely determined by their [[inverse image]] functions on the [[frames of opens]], disregarding the underlying sets of points. Technically this means that the sober topological spaces are precisely the [[locales]] among the topological spaces.

## Definition

A [[topological space]] $X$ is **sober** if its [[points]] are exactly determined by its [[lattice of open subsets]].  Different equivalent ways to say this are:

* The [[continuous map]] from $X$ to the space of points of the [[locale]] that it gives rise to (see there for details) is a [[homeomorphism]].

* The [[function]] from points of $X$ to the _[[completely prime filter|completely prime filters]]_ of its open-set lattice is a [[bijection]].

* (Assuming classical logic) every [[irreducible closed set]] (non-empty closed set that is not the [[union]] of any two proper closed subsets) is the [[topological closure|closure]] of a unique point.

In each case, half of the definition is that $X$ is [[T0]], the other half states that $X$ has __enough points__:

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

A [[topological space]] $X$ has _enough points_ if the following equivalent conditions hold:

* The [[continuous map]] from $X$ to the space of points of the [[locale]] that it gives rise to (see there for details) is a [[quotient space|quotient map]].

* The [[function]] from points of $X$ to the _[[completely prime filter|completely prime filters]] of its [[lattice of open subsets|open-set lattice]] is a [[surjection]].

* (Assuming classical logic) every [[irreducible closed set]] (non-empty closed set that is not the [[union]] of any two proper closed subsets) is the [[topological closure|closure]] of a point.

=--

## Properties

### Separation

* Sobriety is a [[separation property]] that is stronger than [[T0]], but incomparable with [[T1]].  With [[classical logic]], every [[Hausdorff space]] is sober (see at _[[Hausdorff implies sober]]_), but this can fail [[constructive mathematics|constructively]].

$$
  Hausdorff = T_2 \Rightarrow sober \Rightarrow T_0
$$

### As locales with enough points
 {#AsLocalesWithEnoughPoints}

What makes the concept of sober topological spaces special is that
for them the concept of [[continuous functions]] may be expressed entirely in terms
of the relations between their [[open subsets]], disregarding the underlying set of points of which these open are in fact subsets.
In order to express this
property (proposition \ref{FrameMorphismsBetweenOpensOfSoberSpaces} below), we first introduce the following terminology:

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

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be [[topological spaces]].
Then a function

$$
  \tau_X \longleftarrow \tau_Y \;\colon\; \phi
$$

between their [[frame of opens|sets of open subsets]] is called a _[[frame]] [[homomorphism]]_
if it preserves

1. arbitrary [[unions]];

1. [[finite number|finite]] [[intersections]].

In other words, $\phi$ is a frame homomorphism if

1. for every [[set]] $I$ and every $I$-indexed set $\{U_i \in \tau_Y\}_{i \in I}$ of elements of $\tau_Y$, then

   $$
     \phi\left(\underset{i \in I}{\cup} U_i\right) \;=\; \underset{i \in I}{\cup} \phi(U_i)\;\;\;\;\in \tau_X
     \,,
   $$

1. for every [[finite set]] $J$ and every $J$-indexed set $\{U_j \in \tau_Y\}$ of elements in $\tau_Y$, then

   $$
     \phi\left(\underset{j \in J}{\cap} U_j\right)
       \;=\;
     \underset{j \in J}{\cap} \phi(U_j)
     \;\;\;\;\in \tau_X
     \,.
   $$

=--

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

A [[frame]] [[homomorphism]] $\phi$ as in def. \ref{HomomorphismOfFramesOfOpens}
necessarily also preserves inclusions in that

* for every inclusion $U_1 \subset U_2$ with $U_1, U_2 \in \tau_Y \subset P(Y)$ then

  $$
    \phi(U_1) \subset \phi(U_2) \;\;\;\;\;\;\; \in \tau_X
    \,.
  $$

This is because inclusions are witnessed by unions

$$
  (U_1 \subset U_2)
    \;\Leftrightarrow\;
  \left( U_1 \cup U_2 = U_2 \right)
$$

and by finite intersections:

$$
  (U_1 \subset U_2)
    \;\Leftrightarrow\;
  \left(
    U_1 \cap U_2 = U_1
  \right)
  \,.
$$

=--

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

For

$$
  f \;\colon\; (X,\tau_X) \longrightarrow (Y, \tau_Y)
$$

a [[continuous function]], then its function of [[pre-images]]

$$
  \tau_X \longleftarrow \tau_Y \;\colon\; f^{-1}
$$

is a frame homomorphism according to def. \ref{HomomorphismOfFramesOfOpens}.

=--

For sober topological spaces the converse holds:

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

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are [[sober topological spaces]], then
for every frame homomorphism (def. \ref{HomomorphismOfFramesOfOpens})

$$
  \tau_X \longleftarrow \tau_Y \;\colon\; \phi
$$

there is a unique [[continuous function]] $f \colon X \to Y$ such that $\phi$ is the function
of forming [[pre-images]] under $f$:

$$
  \phi = f^{-1}
  \,.
$$

=--

We prove this [below](#ProofOfFrameMorphismsBetweenOpensOfSoberSpaces), after the following lemma.

Let $\ast = (\{1\}, \tau_\ast = \left\{\emptyset, \{1\}\right\})$ be the [[point]] [[topological space]].

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

For $(X,\tau)$ a [[topological space]], then there is a [[bijection]] between
the [[irreducible closed subspaces]] of $(X,\tau)$ and the
[[frame]] [[homomorphisms]] from $\tau_X$ to $\tau_\ast$, given bys

$$
  \array{
    Hom_{Frame}(\tau_X, \tau_\ast)
      &\underoverset{\simeq}{}{\longrightarrow}&
    IrrClSub(X)
    \\
    \phi
      &\mapsto&
    X \backslash U_\emptyset(\phi)
  }
$$

where $U_\emptyset(\phi)$ is the [[union]] of all elements $U \in \tau_x$ such that $\phi(U) = \emptyset$:

$$
  U_{\emptyset}(\phi)
    \coloneqq
  \underset{{U \in \tau_X} \atop {\phi(U) = \emptyset} }{\cup}
   U
  \,.
$$

=--

See also ([[Stone Spaces|Johnstone 82, II 1.3]]).

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

First we need to show that the function is well defined in that given
a frame homomorphism $\phi \colon \tau_X \to \tau_\ast$ then  $X \backslash U_\emptyset(\phi)$
is indeed an irreducible closed subspace.

To that end observe that:

$(\ast)$ _If there are two elements $U_1, U_2 \in \tau_X$ with $U_1 \cap U_2 \subset U_{\emptyset}(\phi)$
then $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$._

This is because

$$
  \begin{aligned}
    \phi(U_1) \cap \phi(U_2)
    & =
    \phi(U_1 \cap U_2)
    \\
    & \subset \phi(U_{\emptyset}(\phi))
    \\
    & =
    \emptyset
  \end{aligned}
  \,,
$$

where the first equality holds because $\phi$ preserves finite intersections by def. \ref{HomomorphismOfFramesOfOpens}, the inclusion holds because $\phi$ respects
inclusions by remark \ref{PreservationOfInclusionsByFrameHomomorphism}, and the second equality holds because $\phi$ preserves arbitrary unions
by def. \ref{HomomorphismOfFramesOfOpens}.
But in $\tau_\ast = \{\emptyset, \{1\}\}$ the intersection of two open subsets is empty precisely if at least one of them is empty,
hence $\phi(U_1) = \emptyset$ or $\phi(U_2) = \emptyset$. But this means that $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$, as claimed.

Now according to [this prop.](irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible),
the condition $(\ast)$ identifies the [[complement]]
$X \backslash U_{\emptyset}(\phi)$ as an [[irreducible closed subspace]] of $(X,\tau)$.

Conversely, given an irreducible closed subset $X \backslash U_0$, define $\phi$ by

$$
  \phi
    \;\colon\;
  U
    \mapsto
  \left\{
    \array{
      \emptyset & \vert \, \text{if} \, U \subset U_0
      \\
      \{1\} & \vert \, \text{otherwise}
    }
  \right.
  \,.
$$

This does preserve

1. arbitrary unions

   because $\phi(\underset{i}{\cup} U_i) = \{\emptyset\}$ precisely if $\underset{i}{\cup}U_i \subset U_0$ which is the
   case precisely if all $U_i \subset U_0$, which means that all $\phi(U_i) = \emptyset$ and because $\underset{i}{\cup}\emptyset = \emptyset$;

   while $\phi(\underset{i}{\cup}U_1) = \{1\}$ as soon as one of the $U_i$ is not contained in $U_0$, which means that
   one of the $\phi(U_i) = \{1\}$ which means that $\underset{i}{\cup} \phi(U_i) = \{1\}$;

1. finite intersections

   because if $U_1 \cap U_2 \subset U_0$, then by $(\ast)$ $U_1 \in U_0$ or $U_2 \in U_0$, whence $\phi(U_1) = \emptyset$
   or $\phi(U_2) = \emptyset$, whence with $\phi(U_1 \cap U_2) = \emptyset$ also $\phi(U_1) \cap \phi(U_2) = \emptyset$;

   while if $U_1 \cap U_2$ is not contained in $U_0$ then neither $U_1$ nor $U_2$ is contained in $U_0$ and hence with
   $\phi(U_1 \cap U_2) = \{1\}$ also $\phi(U_1) \cap \phi(U_2) = \{1\} \cap \{1\} = \{1\}$.

Hence this is indeed a frame homomorphism $\tau_X \to \tau_\ast$.

Finally, it is clear that these two operations are inverse to each other.

=--

+-- {: .proof #ProofOfFrameMorphismsBetweenOpensOfSoberSpaces}
###### Proof
of prop. \ref{FrameMorphismsBetweenOpensOfSoberSpaces}

We first consider the special case of frame homomorphisms of the form

$$
  \tau_\ast \longleftarrow \tau_X \;\colon\; \phi
$$

and show that these are in bijection to the underlying set $X$, identified with the continuous functions
$\ast \to (X,\tau)$.

By lemma \ref{FrameHomomorphismsToPointAreIrrClSub}, the frame homomorphisms $\phi \colon \tau_X \to \tau_\ast$
are identified with the irreducible closed subspaces $X \backslash U_\emptyset(\phi)$ of $(X,\tau_X)$.
Therefore by assumption of [[sober topological space|sobriety]] of $(X,\tau)$ there is a unique point
$x \in X$ with $X \backslash U_{\emptyset} = Cl(\{x\})$. In particular this means that for $U_x$ an open
neighbourhood of $x$, then $U_x$ is not a subset of $U_\emptyset(\phi)$, and so it follows that $\phi(U_x) = \{1\}$.
In conclusion we have found a unique $x \in X$ such that

$$
  \phi
    \;\colon\;
  U \mapsto
  \left\{
    \array{
      \{1\} & \vert \,\text{if}\, x \in U
      \\
      \emptyset & \vert \, \text{otherwise}
    }
  \right.
  \,.
$$

This is precisely the [[inverse image]] function of the continuous function $\ast \to X$ which sends $1 \mapsto x$.

Hence this establishes the bijection between frame homomorphisms of the form $\tau_\ast \longleftarrow \tau_X$
and continuous functions of the form $\ast \to (X,\tau)$.

With this it follows that a general frame homomorphism of the form $\tau_X \overset{\phi}{\longleftarrow} \tau_Y$
defines a function of sets $X \overset{f}{\longrightarrow} Y$ by [[composition]]:

$$
  \array{
    X &\overset{f}{\longrightarrow}& Y
    \\
    (\tau_\ast \leftarrow \tau_X)
    &\mapsto&
    (\tau_\ast \leftarrow \tau_X \overset{\phi}{\longleftarrow} \tau_Y)
  }
  \,.
$$

By the previous analysis, an element $U_Y \in \tau_Y$ is sent to $\{1\}$ under this composite precisely if
the corresponding point $\ast \to X \overset{f}{\longrightarrow} Y$ is in $U_Y$, and similarly for
an element $U_X \in \tau_X$. It follows that $\phi(U_Y) \in \tau_X$ is precisely that subset of
points in $X$ which are sent by $f$ to elements of $U_Y$, hence that $\phi = f^{-1}$ is the [[pre-image]]
function of $f$. Since $\phi$ by definition sends open subsets of $Y$ to open subsets of $X$, it follows
that $f$ is indeed a continuous function. This proves the claim in generality.

=--

### Soberification reflection
 {#Reflection}

The [[category]] of sober spaces is [[reflective subcategory|reflective]] in the category of all topological spaces; the [[left adjoint]] is called the **soberification**.

This reflection is also induced by the [[idempotent adjunction]] between spaces and [[locales]]; thus sober spaces are precisely those spaces that are the spaces of points of some [[locale]], and the [[category]] of sober spaces is [[equivalence of categories|equivalent]] to the category of [[locales with enough points]].

We now say this in detail.

Recall again the [[point]] topological space $\ast \coloneqq ( \{1\}, \tau_\ast = \left\{ \emptyset, \{1\}\right\} )$.

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

Let $(X,\tau)$ be a [[topological space]].

Define $S X$ to be the set

$$
  S X \coloneqq Hom_{Frame}( \tau_X, \tau_\ast )
$$

of [[frame]] [[homomorphisms]] from the [[frame of opens]] of $X$ to that of the point. Define a [[topological space|topology]] $\tau_{S X} \subset P(S X)$ on this set by declaring it to have one element $\tilde U$ for each element $U \in \tau_X$ and given by

$$
  \tilde U
    \;\coloneqq\;
  \left\{
    \phi \in S X
    \,\vert\,
    \phi(U) = \{1\}
  \right\}
  \,.
$$

Consider the function

$$
  \array{
    X &\overset{s_X}{\longrightarrow}& S X
    \\
    x &\mapsto& (const_x)^{-1}
  }
$$

which sends an element $x \in X$ to the function which assigns [[inverse images]] of the [[constant function]] $const_x \;\colon\; \{1\} \to X$ on that element.

=--

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

The construction $(S X, \tau_{S X})$ in def. \ref{SoberificationConstruction} is a [[topological space]], and the function $s_X \colon X \to S X$ is a [[continuous function]]

$$
  s_X \colon (X, \tau_X) \longrightarrow (S X, \tau_{S X})
$$

=--

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

To see that $\tau_{S X} \subset P(S X)$ is closed under arbitrary unions and finite intersections, observe that the function

$$
  \array{
    \tau_X &\overset{\widetilde{(-)}}{\longrightarrow}& \tau_{S X}
    \\
    U &\mapsto& \tilde U
  }
$$

in fact preserves arbitrary unions and finite intersections. Whith this the statement follows by the fact that $\tau_X$ is closed under these operations.

To see that $\widetilde{(-)}$ indeed preserves unions, observe that (e.g. [Johnstone 82, II 1.3 Lemma](#Johnstone82))

$$
  \begin{aligned}
    p \in \underset{i \in I}{\cup} \widetilde{U_i} \;
    & \Leftrightarrow
    \underset{i \in I}{\exists} p(U_i) = \{1\}
    \\
    & \Leftrightarrow \underset{i \in I}{\cup} p(U_i) = \{1\}
    \\
    & \Leftrightarrow p\left(\underset{i \in I}{\cup} U_i\right) = \{1\}
    \\
    & \Leftrightarrow p \in \widetilde{ \underset{i \in I}{\cup} U_i }
  \end{aligned}
  \,,
$$

where we used that the frame homomorphism $p \colon \tau_X \to \tau_\ast$ preserves unions. Similarly for intersections, now with $I$ a [[finite set]]:

$$
  \begin{aligned}
    p \in \underset{i \in I}{\cap} \widetilde{U_i} \;
    & \Leftrightarrow
    \underset{i \in I}{\forall} p(U_i) = \{1\}
    \\
    & \Leftrightarrow \underset{i \in I}{\cap} p(U_i) = \{1\}
    \\
    & \Leftrightarrow p\left(\underset{i \in I}{\cap} U_i\right) = \{1\}
    \\
    & \Leftrightarrow p \in \widetilde{ \underset{i \in I}{\cap} U_i }
  \end{aligned}
  \,,
$$

where now we used that the frame homomorphism $p$ preserves finite intersections.

To see that $s_X$ is continuous, observe that $s_X^{-1}(\tilde U) = U$, by construction.

=--

+-- {: .num_lemma #UnitIntoSXDetectsT0AndSoberity}
###### Lemma

For $(X, \tau_X)$ a [[topological space]],
the function $s_X \colon X \to S X$ from def. \ref{SoberificationConstruction} is

1. an [[injection]] precisely if $X$ is [[separation axiom|T0]];

1. a [[bijection]] precisely if $X$ is sober.

   In this case $s_X$ is in fact a [[homeomorphism]].

=--

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

By lemma \ref{FrameHomomorphismsToPointAreIrrClSub} there is an identification $S X \simeq IrrClSub(X)$ and via this $s_X$ is identified with the map $x \mapsto Cl(\{x\})$.

Hence the second statement follows by definition, and the first statement by [this prop.](separation+axioms#T0InTermsOfClosureOfPoints).

That in the second case $s_X$ is in fact a homeomorphism follows from the definition of the opens $\tilde U$: they are identified with the opens $U$ in this case (...expand...).

=--

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

For $(X,\tau)$ a [[topological space]], then the topological space $(S X, \tau_{S X})$ from def. \ref{SoberificationConstruction}, lemma \ref{SoberificationConstructionWellDefined} is sober.

=--

(e.g. [Johnstone 82, lemma II 1.7](#Johnstone82))

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

Let $S X \backslash \tilde U$ be an [[irreducible closed subspace]] of $(S X, \tau_{S X})$. We need to show that it is the [[topological closure]] of a unique element $\phi \in S X$.

Observe first that also $X \backslash U$ is irreducible.

To see this use [this prop.](irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible), saying that irreducibility of $X \backslash U$ is equivalent to $U_1 \cap U_2 \subset U \Rightarrow (U_1 \subset U) or (U_2 \subset U)$. But if $U_1 \cap U_2 \subset U$ then also $\tilde U_1 \cap \tilde U_2 \subset \tilde U$ (as in the [proof](#ProofOfSoberificationConstructionWellDefined) of lemma \ref{SoberificationConstructionWellDefined}) and hence by assumption on $\tilde U$ it follows that $\tilde U_1 \subset \tilde U$ or $\tilde U_2 \subset \tilde U$. By lemma \ref{FrameHomomorphismsToPointAreIrrClSub} this in turn implies $U_1 \subset U$ or $U_2 \subset U$.  In conclusion, this shows that also $X \backslash U$ is irreducible .

By lemma \ref{FrameHomomorphismsToPointAreIrrClSub} this irreducible closed subspace corresponds to a point $p \in S X$. By that same lemma, this frame homomorphism $p \colon \tau_X \to \tau_\ast$ takes the value $\emptyset$ on all those opens which are inside $U$. This means that the [[topological closure]] of this point is just $S X \backslash \tilde U$.

This shows that there exists at least one point of which $X \backslash \tilde U$ is the topological closure. It remains to see that there is no other such point.

So let $p_1 \neq p_2 \in S X$ be two distinct points. This means that there exists $U \in \tau_X$ with $p_1(U) \neq p_2(U)$. Equivalently this says that $\tilde U$ contains one of the two points, but not the other. This means that $(S X, \tau_{S X})$ is [[separation axiom|T0]]. By [this prop.](separation+axioms#T0InTermsOfClosureOfPoints) this is equivalent to there being no two points with the same topological closure.

=--

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

For $(X, \tau_X)$ any [[topological space]], for $(Y,\tau_Y^{sob})$ a sober topological space, and for $f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y)$ a [[continuous function]], then it factors uniquely through the soberification $s_X \colon (X, \tau_X) \longrightarrow(S X, \tau_{S X})$ from def. \ref{SoberificationConstruction}, lemma \ref{SoberificationConstructionWellDefined}

$$
  \array{
    (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau_Y^{sob})
    \\
    {}^{\mathllap{s_X}}\downarrow & \nearrow_{\exists !}
    \\
    (S X , \tau_{S X})
  }
  \,.
$$

=--

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

By the construction in def. \ref{SoberificationConstruction}, we find that the outer part of the following square [[commuting square|commutes]]:

$$
  \array{
    (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau^{sob}_Y)
    \\
    {}^{\mathllap{s_X}}\downarrow & \nearrow& \downarrow^{\mathrlap{s_{S X}}}
    \\
    (S X, \tau_{S X})
     &\underset{S f}{\longrightarrow}&
    (S S X, \tau_{S S X})
  }
  \,.
$$

By lemma \ref{SoberificationIsIndeedSober} and lemma \ref{UnitIntoSXDetectsT0AndSoberity}, the right vertical morphism $s_{S X}$ is an isomorphism (a [[homeomorphism]]), hence has an [[inverse morphism]]. This defines the diagonal morphism, which is the desired factorization.

To see that this factorization is unique, consider two factorizations $\tilde f, \overline{f} \colon \colon (S X, \tau_{S X}) \to (Y, \tau_Y^{sob})$ and apply the soberification construction once more to the triangles

$$
  \array{
    (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau_Y^{sob})
    \\
    {}^{\mathllap{s_X}}\downarrow & \nearrow_{ \tilde f, \overline{f}}
    \\
    (S X , \tau_{S X})
  }
  \phantom{AAA}
    \mapsto
  \phantom{AAA}
  \array{
    (S X, \tau_{S X}) &\overset{S f}{\longrightarrow}& (Y, \tau_Y^{sob})
    \\
    {}^{\simeq}\downarrow & \nearrow_{ \tilde f, \overline{f}}
    \\
    (S X , \tau_{S X})
  }
  \,.
$$

Here on the right we used again lemma \ref{UnitIntoSXDetectsT0AndSoberity} to find that the vertical morphism is an isomorphism, and that $\tilde f$ and $\overline{f}$ do not change under soberification, as they already map between sober spaces. But now that the left vertical morphism is an isomorphism, the commutativity of this triangle for both $\tilde f$ and $\overline{f}$ implies that $\tilde f = \overline{f}$.

=--

### Enough points

A topological space has enough points in the sense of def. \ref{EnoughPointsOfATopologicalSpace} if and only if its $T_0$ quotient is sober.  The category of topological spaces with enough points is a [[reflective subcategory]] of the category [[Top]] of all topological spaces, and a topological space is [[T0]] iff this reflection is sober.

## Examples and Non-examples

* With [[classical logic]], every [[Hausdorff space]] is sober, but this can fail [[constructive mathematics|constructively]]. See at _[[Hausdorff implies sober]]_.

  {#NonComparisonToT1} Since the Hausdorff condition implies the $T_1$ [[separation axiom]], this means that ([[classical logic|classically]]) there is a large [[intersection]] of the [[classes]] of $T_1$-topological spaces and sober topological spaces. But neither class is contained in the other, as the following counter-examples show:

  * The [[Sierpinski space]] is sober, but not $T_1$.

  * The [[cofinite topology]] on a non-[[finite set]] is $T_1$ but not sober.

* The topological space underlying any [[scheme]] is sober. See at _[[schemes are sober]]_.

Further examples of spaces that are _not_ sober includes the following:

* Any nontrivial [[indiscrete space]] is not sober, since it is not $T_0$.

More interestingly:

* For $R$ a [[commutative ring]], then the space $R^2$ with the [[Zariski topology]] is $T_1$ but not sober, since every subvariety is an irreducible closed set which is not the [[topological closure|closure]] of a point.  Its _soberification_ is, unsurprisingly, the [[scheme]] $Spec(R[x,y])$, which contains "generic points" whose closures are the subvarieties.

The following non-example shows that sobriety is not a hereditary separation property, i.e., [[topological subspaces]] of sober spaces need not be sober:

* The [[Alexandroff topology]] on a [[poset]] is also not, in general, sober.  For instance, if $P$ is the infinite binary tree (the set of finite $\{0,1\}$-words ([[lists]]) with the "extends" preorder), then the soberification of its Alexandroff topology is [[Wilson space]], the space of finite or infinite $\{0,1\}$-words ([[streams]]).

## Related concepts

* [[nice topological space]]

## References

* [[Francis Borceux]], volume 3, section 1.9  of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* {#Johnstone82} [[Peter Johnstone]], section II.1, from II.1 on in _[[Stone Spaces]]_, Cambridge Studies in Advanced Mathematics __3__, Cambridge University Press 1982. xxi+370 pp. [MR85f:54002](http://www.ams.org/mathscinet-getitem?mr=698074), reprinted 1986.

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], around definition IX.3.2 in _[[Sheaves in Geometry and Logic]]_

* Sober spaces in the [$\pi$-base](http://topology.jdabbs.com/properties/73).

A [[model category]] of [[locales]] which makes the [[reflective subcategory|reflection]] of [[sober topological spaces]] a [[Quillen adjunction]] to the sober-restriction of the [[classical model structure on topological spaces]]:

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], Thm. 4.45 in: *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

[[!redirects sober]]
[[!redirects sober topological space]]
[[!redirects sober topological spaces]]
[[!redirects sober topology]]
[[!redirects sober topologies]]
[[!redirects sober space]]
[[!redirects sober spaces]]

[[!redirects sobrification]]
[[!redirects sobrifications]]

[[!redirects soberification]]
[[!redirects soberifications]]

[[!redirects sober reflection]]
[[!redirects sober reflections]]

[[!redirects soberification reflection]]
[[!redirects soberification reflections]]

[[!redirects topological space with enough points]]
[[!redirects topological spaces with enough points]]


Sobolev space

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

#Contents#
* table of contents
{:toc}

## Idea

If $1\leq p \lt \infty$ and $\Omega$ is a [[domain]] (in a $n$-dimensional real space with easy generalization to [[manifolds]]), one first considers the [[Lebesgue spaces]] $L_p = L_p(\Omega)$ ([wikipedia](http://en.wikipedia.org/wiki/Lp_space)) of (equivalence classes of) [[measurable function|measurable]] (complex- or real-valued) functions $f$ whose (absolute values of) $p$-th powers are [[Lebesgue integral|Lebesgue integrable]]; i.e. whose norm

$$\| f\|_{L_p} = \left(\int_\Omega |f|^p d\mu\right)^{1/p}$$

is finite. For $p = \infty$, one looks at the [[essential supremum]] norm $\|f\|_{L_\infty}$ instead.

For $1\leq p \leq \infty$, and $k\geq 1$ the **Sobolev space** $W^k_p = W^k_p(\Omega)$ or $W^{k,p}(\Omega)$ is the [[Banach space]] of measurable functions $f$ on $\Omega$ such that its generalized [[partial derivative]]s $\partial_1^{i_1}\ldots\partial_n^{i_n} f$ (e.g. in the sense of [[generalized function]]s) for all multiindices $i = (i_1,\ldots, i_n)\in\mathbb{Z}^n_{\geq 0}$ with $i_1+\ldots +i_n\leq k$ are in $L_p(\Omega)$. The most important case is the case of the Sobolev spaces $H^k(\Omega) := W^k_2(\Omega)$. Sobolev spaces are particularly important in the theory of partial differential equations.

## References

* L. C. Evans, _Partial Differential Equations_, Amer. Math. Soc. 1998.

* R.A. Adams, _Sobolev spaces_, Acad. Press 1975.

* wikipedia: [Sobolev space](http://en.wikipedia.org/wiki/Sobolev_space).

* Springer Encyclopaedia of Mathematics: [Sobolev space](http://eom.springer.de/S/s085980.htm)

* G. Wilkin, _Sobolev spaces on Euclidean space_, Liber Mathematicae 2011,  [link](http://math.colorado.edu/libermath/repositorium/article.php?doc=4)

category: analysis
[[!redirects Sobolev spaces]]


socle

Given a [[ring]] $R$, the __socle__ $Soc(M)$ of a left $R$-[[module]] $M$ is the (internal) sum of all [[simple object|simple]] [[subobject|submodules]] of $M$. The correspondence $M\to Soc(M)$ is clearly a subfunctor of the identity functor ${}_R Mod\to {}_R Mod$. It is moreover left exact (but not a [[kernel functor]] in the sense of Goldman).

By the definition, the socle is a [[semisimple module|semisimple]] $R$-module. If we assume the axiom of choice, then the socle of $M$ can be presented as a direct sum of some subfamily of all simple submodules of $M$.

The notion of socle is important in [[representation theory]].

Notice that the notion is dual to the notion of the radical $Rad(M)$ which is the intersection of all maximal submodules of $M$.

* [[Joachim Lambek]], _Lectures on rings and modules_, Waltham Mass. 1966
* wikipedia [socle (mathematics)](http://en.wikipedia.org/wiki/Socle_%28mathematics%29)


Socrates

## Related entries

* [[Plato]]

## References

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

category: people


soft graviton theorem

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## Idea

In [[perturbative quantum gravity]], the  _soft graviton theorem_ ([Weinberg 65](#Weinberg65)) re-formulates [[scattering amplitudes]] of a set of finite energy external particles with one or more low energy external [[gravitons]], in terms of the amplitude without the low energy gravitons.

In the [[classical limit]], there is a different manifestation of the same theorem ([Saha-Sahoo-Sen 19](#SahaSahooSen19)): here it determines the low frequency component of the [[gravitational wave]]-form produced during a scattering process in terms of the momenta and spin of the incoming and outgoing objects, without any reference to the interactions responsible for the scattering.

## References

Original reference:

* {#Weinberg65} [[Steven Weinberg]], _Infrared photons and gravitons_, Phys. Rev. 140, B516 (1965) ([doi:10.1103/PhysRev.140.B516](https://doi.org/10.1103/PhysRev.140.B516))

Review with focus on [[photons]] in [[electromagnetism]]:

* Noah Miller, *From Noether's Theorem to Bremsstrahlung: a pedagogical introduction to large gauge transformations and classical soft theorems* ([arXiv:2112.05289](https://arxiv.org/abs/2112.05289))

See also:

* [[Martin Beneke]], Patrick Hager, Robert Szafron, *Soft-Collinear Gravity and Soft Theorems*, in section *Effective Quantum Gravity* of *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09336](https://arxiv.org/abs/2210.09336)]

Discussion of application to [[gravitational waves]] in the [[classical limit]]:

* {#SahaSahooSen19} Arnab Priya Saha, Biswajit Sahoo, [[Ashoke Sen]], _Proof of the Classical Soft Graviton Theorem in $D=4$_ ([arXiv:1912.06413](https://arxiv.org/abs/1912.06413))

Discussion in the [[BFSS matrix model]]:

* Noah Miller, [[Andrew Strominger]], Adam Tropper, Tianli Wang, *Soft Gravitons in the BFSS Matrix Model* [[arXiv:2208.14547](https://arxiv.org/abs/2208.14547)]

* Adam Tropper, Tianli Wang, *Lorentz Symmetry and IR Structure of The BFSS Matrix Model* [[arXiv:2303.14200](https://arxiv.org/abs/2303.14200)]

* [[Aidan Herderschee]], [[Juan Maldacena]], *Soft Theorems in Matrix Theory* [[arXiv:2312.15111](https://arxiv.org/abs/2312.15111)]

Analogous discussion for [[gluons]]:

* Peng Cheng, Pujian Mao, *Soft gluon theorems in curved spacetime* [[arXiv:2211.00031](https://arxiv.org/abs/2211.00031)]

Analogous discussion for [[phonons]]:

* Clifford Cheung, Maria Derda, Andreas Helset, Julio Parra-Martinez, *Soft Phonon Theorems* [[arXiv:2301.11363](https://arxiv.org/abs/2301.11363)]

[[!redirects soft graviton theorems]]


soft group manifold

In the literature on the [[D'Auria-Fre formulation of supergravity]] (also called the "FDA-formalism") what is called a (extended) "soft group manifold" is precisely a collection of [[∞-Lie algebra valued differential forms]] on a [[smooth manifold]].

See these links for further details.

[[!redirects soft group manifold]]


soft linear logic

# Soft linear logic

* table of contents
{: toc}

## Idea

Soft linear logic is a version of [[linear logic]] in which the [[exponential modality]] is weakened.  This has as a particular consequence that the only "definable functions" (in various senses) are computable in [[polynomial time]].

## Definition

The usual rules of the exponential modality $!$ in (intuitionistic) linear logic are promotion, dereliction, contraction, and weakening:

$$\frac{!\Gamma\vdash A}{!\Gamma\vdash !A}(prom) \qquad \frac{\Gamma,A \vdash C}{\Gamma,!A\vdash C}(der) \qquad \frac{\Gamma,!A,!A\vdash C}{\Gamma,!A\vdash C}(ctr) \qquad \frac{\Gamma\vdash C}{\Gamma,!A\vdash C}(wk)$$

Soft linear logic is obtained by observing that these rules are equivalent to soft promotion, multiplexing, and digging:

$$\frac{\Gamma\vdash A}{!\Gamma\vdash !A}(sp) \qquad \frac{\Gamma,\overbrace{A,\dots,A}^n\vdash C}{\Gamma,!A\vdash C}(mpx) \qquad \frac{\Gamma,!!A\vdash C}{\Gamma,!A\vdash C}(dig)$$

and then removing digging.  Thus, the only rules governing $!$ in soft linear logic are soft promotion and multiplexing.

## Categorical semantics

Inspecting the rules for the soft $!$, we see that soft promotion says that $!$ is a [[lax monoidal functor]] (or equivalently a functor between [[multicategories]]), while multiplexing says that we have a (presumably natural) transformation $!A \to A^{\otimes n}$ for all $n$.  On a non-[[thin category]] one would presumably ask for some coherence axioms on these transformations.

Intriguingly, however, the naive formula for a "cofree comonoid" on an object $A$ in a symmetric monoidal category with countable products:

$$ !A = \prod_{n\in\mathbb{N}} A^{\otimes n},$$

though it does not in general define a comonoid at all, *is* a lax monoidal functor with transformations $!A \to A^{\otimes n}$.  The latter are just the product projections, while the lax monoidal structure $!A \otimes !B \to !(A\otimes B) = \prod_n (A\otimes B)^{\otimes n}$ has components

$$ !A \otimes !B = \left(\prod_n A^{\otimes n}\right)\otimes \left(\prod_n B^{\otimes n}\right) \to A^{\otimes n} \otimes B^{\otimes n} \xrightarrow{\cong} (A\otimes B)^{\otimes n}. $$

In particular, therefore, any symmetric monoidal poset with countable meets admits a modality of this form satisfying the rules of the soft exponential.  For a non-thin category, one could take a further limit in an attempt to impose whatever axioms are desired of the structure on $!$.  For instance, if instead of $A^{\otimes n}$ we use an appropriate [[equalizer]], we could force permutation-invariance of the transformations $!A \to A^{\otimes n}$.

## Related pages

* [[computational complexity]]
  * [[implicit computational complexity]]
  * [[polynomial time]]
* [[light linear logic]]
* [[naive set theory]]
* [[ultrafinitism]]

## References

* Yves Lafont, Soft linear logic and polynomial time, Theoretical Computer Science, Volume 318, Issues 1–2, 2004, Pages 163-180, [doi](https://doi.org/10.1016/j.tcs.2003.10.018), [ps](http://iml.univ-mrs.fr/~lafont/pub/soft.ps)

* Marco Gaboardi, Jean-Yves Marion, Simona Ronchi Della Rocca, Soft Linear Logic and Polynomial Complexity Classes, Electronic Notes in Theoretical Computer Science, Volume 205, 2008, Pages 67-87, [doi](https://doi.org/10.1016/j.entcs.2008.03.066)

* Richard McKinley, Soft Linear Set Theory, The Journal of Logic and Algebraic Programming, Volume 76, Issue 2, 2008, [doi](https://doi.org/10.1016/j.jlap.2008.02.010)

[[!redirects soft exponential]]
[[!redirects soft promotion]]
[[!redirects multiplexing]]


soft sheaf

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## Definition

Given a [[sheaf]] $F$ of [[sets]] over a [[topological space]] $X$, the [[section]] over an *arbitrary* (= not necessarily open) subset $V\subset X$ is a continuous section of the corresponding [[etale space]] restricted to $V$.

A [[sheaf]] $F$ of sets (or of [[abelian groups]]) over a [[paracompact space|paracompact]] [[Hausdorff space]] $X$ is __soft__ if for any [[closed subset]] $V\subset X$, every section of $F$ over $V$ can be extended to the whole $X$.

## Properties

The local extension of germs to the open neighborhoods of points by paracompactness gives rise to an extension of the section to an [[open neighborhood]] of the whole set $V$. Therefore every [[flabby sheaf]] is soft, because flabbiness gives the extension from open subsets. [[fine sheaf|Fine sheaves]] are always soft.

## Related concepts

* [[sheaf]]

* [[abelian sheaf cohomology]]

  * **soft sheaf**

  * [[fine sheaf]]

  * [[flabby sheaf]]

## References

Standard references:

* [[Tohoku]]

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

category: sheaf theory


soft supersymmetry breaking term

[[!redirects soft supersymmetry breaking terms]]

## Idea

A [[supersymmetry breaking]]-term in a [[Lagrangian density]] is called _soft_ if it does not affect the [[ultraviolet divergences]].

## Related concepts

* [[hidden sector]]

## References

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 2.2 and section 2.6.5 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012


Soichiro Fujii

* [website](https://sites.google.com/view/soichirofujii/)

category: people


sol > history

< [[sol]]


solar system

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## Related concepts

* [[observable universe]]

* [[The Music of the Spheres]]

## References

* Wikipedia, _[Solar System](https://en.wikipedia.org/wiki/Solar_System)_

A visualization of the scales in the solar system is in

* [A tediously accurate scale model of the solar system](http://joshworth.com/dev/pixelspace/pixelspace_solarsystem.html)


solenoid

# Solenoids
* table of contents
{: toc}

## Idea

A simple example of a solenoid is the dyadic solenoid.  This is the [[codirected limit|inverse limit]] of the inverse sequence, $(X(n),p_n)$, in which each $X(n)$ is a copy of the [[circle]] $S^1$ and each 'structure map' $p_n\colon X(n) \to X(n-1)$ is given by the squaring map on the circle, that is $e^{i\theta}\mapsto e^{2i\theta}$ (viewing $S^1$ as the unit circle in the [[complex plane]]).

We will restrict attention, for the moment, to the class of solenoids defined by inverse sequences of circles:

## General definition

Let $P = (r_1,\ldots, r_n,\ldots)$ be a sequence of [[prime numbers]].  The _$P$-adic-solenoid_ is the space,

$S_P = Lim (X(n),p_n,\mathbb{N})$

where for each $n$, $X(n)= S^1$ and (considering $S^1$ as the unit circle in $\mathbb{C}$), $p_n(z) = z^{r_n}$.

The $P$-adic solenoid is a [[compact space|compact]], [[connected space|connected]] [[topological group]].

## Solenoids and shape

In [[shape theory]], the solenoids provide good examples of _non-stable spaces_. If one takes the [[Čech fundamental group]] of $S_P$, then it is the limit of the inverse sequence of the [[fundamental groups]] of the $X(n)$ together with the induced homomorphisms. It thus is trivial as it contains all integers that are divisible by all the primes in the sequence, and that an infinite number of times! The Strong Shape fundamental group of $S_P$ is not trivial as it involves the first derived functor $Lim^{(1)}$ of the inverse sequence of groups as well.

## Related concepts

* [[decimal number#solenoid]]

## References

For solenoids in many settings, see the [Wikipedia page](http://en.wikipedia.org/wiki/Solenoid_%28mathematics%29), which also has some very neat graphics!

For solenoids and shape, references include:

* [[K. Borsuk]], _Theory of Shape_, Monografie Matematyczne Tom 59,Warszawa 1975.

and

* [[Jean-Marc Cordier|J.-M. Cordier]] and [[Tim Porter|T. Porter]], (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).

The subject is treated in others of the sources listed under [[shape theory]].

[[!redirects solenoid]]
[[!redirects solenoids]]
[[!redirects p-adic solenoid]]
[[!redirects p-adic solenoids]]


solid abelian group

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## Idea

Solid abelian groups are objects which form the basis of [[non-archimedean analysis]] in certain approaches, such as in [[condensed mathematics]].

## Definition

### As a functor

A ***solid abelian group*** is an [[additive functor]] from the category of [[free abelian groups]] to the category of [[abelian groups]].

This definition is due to [[Dustin Clausen]] [here](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057781).

### In terms of condensed abelian groups

Let $S = \underset{\leftarrow}{\lim}_i S_i$ be a [[profinite set]], and define the [[condensed abelian group]] $\mathbb{Z}[S]^\square \coloneqq \underset{\leftarrow}{\lim}_i \mathbb{Z}[S_i]$, where $\mathbb{Z}[T]$ is the [[free abelian group]] on the set $T$. There is a natural map $m:S \to \mathbb{Z}[S]^\square$ which induces a map $\mathbb{Z}[S] \to \mathbb{Z}[S]^\square$.

A **solid abelian group** is a condensed abelian group $A$ such that for all profinite sets $S$ and all maps $f:S \to A$, there is a unique map $g:\mathbb{Z}[S]^\square \to A$ such that $f = g \circ m$.

## Properties

According to [[Peter Scholze]] in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057780) in the absense of the [[presentation axiom]], the category of solid abelian groups is not a condensed category.

## See also

* [[condensed abelian group]]
* [[solidification functor]]
* [[solid spectrum]]

## References

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)


solid functor

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## Idea

A *solid functor* (also called a *semi-topological functor*) is a [[forgetful functor]] $U\colon A\to X$ for which the structure of an $A$-object can be universally lifted along [[sinks]].  One can also say that $U$ has not just a [[left adjoint]] but all possible "relative" left adjoints.

When $U$ is solid, [[colimit]]s in $A$ can be constructed in a natural way out of colimits in $X$, and $A$ inherits strong cocompleteness properties from $X$.

## Definition

Let $U\colon A\to X$ be a [[faithful functor]].  A **$U$-structured sink** is a [[sink]] in $X$ of the form $(U a_i \overset{f_i}{\to} x)$.  Note that the indexing family $i\in I$ need not be a [[set]], it can be a [[proper class]].  A **[[semi-final lift]]** of such a $U$-structured sink consists of a morphism $x\overset{g}{\to} U b$ in $X$ such that

1. Every composite $g \circ f_i\colon U a_i \to U b$ is in the image of $U$, i.e. is of the form $U(\tilde{g})$ for some $\tilde{g}\colon a_i\to b$ (necessarily unique, since $U$ is faithful), and

1. $g$ is [[initial object|initial]] with this property, i.e. for any other morphism $x \overset{g'}{\to} U b'$ such that each $g' \circ f_i$ is in the image of $U$, there exists a unique $h\colon b\to b'$ in $A$ such that $g' = U(h) \circ g$.

Finally, $U$ is called **solid** if every $U$-structured sink has a semi-final lift.

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=--

## Examples

* Any [[topological functor]] is solid.  Indeed, a functor $U$ is topological just when it has final lifts for all $U$-structured sinks, where a *final lift* is a semi-final lift for which $g$ is an isomorphism.

* Any [[monadic functor]] into $Set$ is solid.

* A [[fully faithful functor]] is solid if and only if it has a left adjoint.

* If $U\colon A\to X$ is faithful and has a left adjoint, and moreover $A$ is [[cocomplete category|cocomplete]] and [[well-powered category|well-copowered]], then $U$ is solid.

* For $C$ a [[cofibrantly generated model category]] with monic generating cofibrations, the forgetful functor from [[model structure on algebraic fibrant objects|algebraic fibrant objects]] to $C$ is solid. See there for details.

## Properties

### Left adjoint

For any $x\in X$, the empty family of morphisms into $x$ is a $U$-structured sink, and a semi-final lift for this family is a universal arrow $x\to U b$.  Therefore, if $U$ is solid, then it has a [[left adjoint]].

### Lifting of colimits

Suppose that $U\colon A\to X$ is solid and let $F\colon D\to A$ be a [[diagram]] such that $U F$ has a [[colimit]] in $X$, consisting of a [[cocone]] $U F d_i \to c$.  Let $c \to U e$ be a semi-final lift of this $U$-structured sink, for which we have induced morphisms $F d_i \to e$ in $A$.  Since $U$ is faithful, these morphisms are a cocone under $F$, and the semi-finality makes it into a colimit in $A$.

Therefore, if $A$ is solid over $X$, then it admits all colimits which $X$ does.  Moreover, if we understand colimits in $X$, and we understand the semi-final lifts, then we understand colimits in $A$.

In particular, if $X$ is [[cocomplete category|cocomplete]], then so is $A$.  In fact, more is true: if $X$ is [[total category|total]], then so is $A$.

### Lifting of model structures

The standard *[[transferred model structure|transfer theorem]]* for [[model category|model structures]] states that if $U\colon A\to X$ is a functor such that

1. $U$ has a left adjoint $F$,
1. $U$ is [[accessible functor|accessible]], i.e. preserves $\kappa$-[[filtered colimits]] for sufficiently large $\kappa$,
1. $X$ has a [[cofibrantly generated model category|cofibrantly generated model structure]], and
1. [[transfinite composition|Transfinite composites]] of [[pushouts]] of images under $F$ of generating acyclic cofibrations in $X$ become weak equivalences after applying $U$ (the *acyclicity condition*),

then $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$.  Using an argument of ([Nikolaus](#Nikolaus)) we can show:

+-- {: .un_theorem #LiftingTheorem}
###### Theorem

Let $U\colon A\to X$ be an accessible solid functor, and assume that $X$ has a cofibrantly generated model structure and the following acyclicity condition:

* If $F\colon D\to A$ is a [[filtered diagram]] and $U(F(d_i)) \to x$ is a cocone under $U\circ F$, each of whose legs is an acyclic cofibration in $X$, then the semifinal lift $x\to U(b)$ of this $U$-structured sink is also an acyclic cofibration.

Then the transfer theorem applies, so that $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$.

=--
+-- {: .proof}
###### Proof
We have remarked above that $U$ has a left adjoint, and we assumed it to be accessible, so it remains to show that the given acyclicity condition implies the standard one.

We first show that [[pushout]]s in $A$ of images under $F$ of generating acyclic cofibrations become acyclic cofibrations (not just weak equivalences) upon applying $U$.  Let $i\colon x\to y$ be a generating acyclic cofibration, and
$$\array{F(x) & \overset{f}{\to} & a\\
  ^{F (i)}\downarrow \\
  F(y)}$$
a diagram in $A$ of which we would like to take the pushout.  Consider the pushout of the corresponding diagram in $X$:
$$\array{x & \overset{\bar{f}}{\to} & U(a)\\
  ^{i}\downarrow && \downarrow^g\\
  y& \underset{h}{\to} & U(a) \sqcup_{x} y.}$$
Since $X$ is a model category, $g$ is an acyclic cofibration.  Therefore, if $U(a) \sqcup_x y \overset{k}{\to} U(b)$ is a semifinal lift of the singleton sink $\{g\}$, by assumption, $k$ is also an acyclic cofibration and thus so is the composite $U(a)\to U(b)$.  But it is straightforward to verify that in fact, the map $a\to b$ of which this is the image (which exists by assumption) gives a pushout diagram in $A$:
$$\array{F(x) & \overset{f}{\to} & a\\
  ^{F(i)}\downarrow && \downarrow\\
  F(y) & \underset{\bar{h}}{\to} & b.}$$

If $U$ is not just accessible but [[finitary functor|finitary]], then it preserves all transfinite composites, so any transfinite composite of such pushouts in $A$ maps to a transfinite composite in $X$, and we know that transfinite composites of acyclic cofibrations in $X$ are acyclic cofibrations, so the desired acyclicity condition follows.  In general, we can argue as follows: given a transfinite sequence $a_0\to a_1\to\dots$ in $A$ of such pushouts, its colimit (= composition) can be constructed as above by forgetting down to $X$, taking the colimit there, and then taking the semifinal lift.  But since acyclic cofibrations in $X$ are closed under transfinite composites, the legs of the colimiting cocone in $X$ are acyclic cofibrations.  Hence by the assumed acyclicity condition, so is the semifinal lift, and hence (by composition) so are the images in $X$ of the legs of the colimiting cone in $A$.  This completes the proof.
=--

## References

* [[Walter Tholen]], _Semitopological functors. I_

* and others...

The example of algebraic fibrant objects and the argument entering the above [lifting theorem](#LiftingTheorem) appears in

* [[Thomas Nikolaus]], _Algebraic models for higher categories_ ([arXiv](http://arxiv.org/abs/1003.1342))
{#Nikolaus}

See also [[model structure on algebraic fibrant objects]].

[[!redirects solid functors]]
[[!redirects semi-topological functor]]
[[!redirects semitopological functor]]
[[!redirects semi-topological functors]]
[[!redirects semitopological functors]]