Skyrme hadrodynamics with vector mesons -- references

### Skyrme hadrodynamics with vector mesons ($\pi$-$\omega$-$\rho$-model)
 {#ReferencesSkyrmeHadrodynamicsWithVectorMesons}

Inclusion of [[vector mesons]] ([[omega-meson]] and [[rho-meson]]/[[A1-meson]]) into the [[Skyrmion model]] of [[quantum hadrodynamics]], in addition to the [[pion]]:

First, on the equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:

* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))

See also

* Marcelo Ipinza, Patricio Salgado-Rebolledo, _Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model_ ([arXiv:2005.04920](https://arxiv.org/abs/2005.04920))

#### Inclusion of the $\omega$-meson
  {#SkyrmeModelWithOmegaMesonsReferences}

Original proposal for inclusion of the [[ω-meson]] in the [[Skyrme model]]:

* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), doi:10.1016/0370-2693(84)90239-9)

Relating to [[nucleon]]-[[scattering]]:

* J. M. Eisenberg, A. Erell, R. R. Silbar,  _Nucleon-nucleon force in a skyrmion model stabilized by omega exchange_, Phys. Rev. C 33, 1531 (1986) ([doi:10.1103/PhysRevC.33.1531](https://doi.org/10.1103/PhysRevC.33.1531))

Combination of the [[omega-meson]]-stabilized [[Skyrme model]] with the [[bag model for quark confinement|bag model]] for [[nucleons]]:

* [[Atsushi Hosaka]], _Omega stabilized chiral bag model with a surface $\omega q q$ coupling_, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (doi:10.1016/0375-9474(92)90544-T)

Discussion of [[nucleon]] [[phenomenology]] for the $\omega$-stabilized Skyrme model:

* {#GudnasonSpeight20} [[Sven Bjarke Gudnason]], James Martin Speight, _Realistic classical binding energies in the $\omega$-Skyrme model_ ([arXiv:2004.12862](https://arxiv.org/abs/2004.12862))

#### Inclusion of the $\rho$-meson

Original proposal for inclusion of the [[ρ-meson]]:

* Y. Igarashi, M. Johmura, A. Kobayashi, H. Otsu, T. Sato, S. Sawada, _Stabilization of Skyrmions via $\rho$-Mesons_, Nucl.Phys. B259 (1985) 721-729 ([spire:213451](http://inspirehep.net/record/213451), doi:10.1016/0550-3213(85)90010-0)

* {#Adkins86} [[Gregory Adkins]], _Rho mesons in the Skyrme model_, Phys. Rev. D 33, 193 (1986) ([spire:16895](http://inspirehep.net/record/16895),  [doi:10.1103/PhysRevD.33.193](https://doi.org/10.1103/PhysRevD.33.193))

Discussion for [[phenomenology]] of light [[atomic nuclei]]:

* {#NayaSutcliffe18} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions and clustering in light nuclei_, Phys. Rev. Lett. 121, 232002 (2018) ([arXiv:1811.02064](https://arxiv.org/abs/1811.02064))

* {#NayaSutcliffe18b} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions in models with pions and rho_, JHEP 05 (2018) 174 ([arXiv:1803.06098](https://arxiv.org/abs/1803.06098))

  APS Synopsis: _[Revamping the Skyrmion Model](https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002)_, 2018

#### Inclusion of the $\omega$- and $\rho$-meson

The resulting $\pi$-$\rho$-$\omega$ model:

* {#MeissnerZahed86} [[Ulf-G. Meissner]], [[Ismail Zahed]], _Skyrmions in the Presence of Vector Mesons_, Phys. Rev. Lett. 56, 1035 (1986) ([doi:10.1103/PhysRevLett.56.1035](https://doi.org/10.1103/PhysRevLett.56.1035))

  (includes also the [[A1-meson]])

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], Andreas Wirzba, [[Wolfram Weise]], _Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons_, Phys. Rev. Lett. 57, 1676 (1986) ([doi:10.1103/PhysRevLett.57.1676](https://doi.org/10.1103/PhysRevLett.57.1676))

* [[Ulf-G. Meissner]], _Low-energy hadron physics from effective chiral Lagrangians with vector mesons_, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

* L. Zhang, Nimai C. Mukhopadhyay, _Baryon physics from mesons: Leading order properties of the nucleon and $\Delta(1232)$ in the $\pi \rho\omega a_1(f_1)$ chiral soliton model_, Phys. Rev. D 50, 4668 (1994) ([doi:10.1103/PhysRevD.50.4668](https://doi.org/10.1103/PhysRevD.50.4668), [spire:384906](http://inspirehep.net/record/384906))

* Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, _Skyrmions with vector mesons in the hidden local symmetry approach_, Phys. Rev. D87:034023, 2013 ([arXiv:1209.3554](https://arxiv.org/abs/1209.3554))

  ([[hidden local symmetry]])

* Ju-Hyun Jung, Ulugbek T. Yakhshiev, Hyun-Chul Kim, _In-medium modified $\pi$-$\rho$-$\omega$ mesonic Lagrangian and properties of nuclear matter_, Physics Letters B Volume 723, Issues 4–5, 25 June 2013, Pages 442-447 ([arXiv:1212.4616](https://arxiv.org/abs/1212.4616), [doi:10.1016/j.physletb.2013.05.042](https://doi.org/10.1016/j.physletb.2013.05.042))

* Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim, Peter Schweitzerm, _In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a $\pi$-$\rho$-$\omgea$ soliton model_, Phys. Rev. D 89, 114021 (2014) ([arXiv:1402.0161](https://arxiv.org/abs/1402.0161))

* Yongseok Oh, _Skyrmions with vector mesons revisited_ ([arXiv:1402.2821](https://arxiv.org/abs/1402.2821))

See also

* Ki-Hoon Hong, Ulugbek Yakhshiev, Hyun-Chul Kim, _Modification of hyperon masses in nuclear matter_, Phys. Rev. C 99, 035212 (2019) ([arXiv:1806.06504](https://arxiv.org/abs/1806.06504))

Review:

* {#Kaiser00} Roland Kaiser, _Anomalies and WZW-term of two-flavour QCD_, Phys. Rev. D63:076010, 2001 ([arXiv:hep-ph/0011377](https://arxiv.org/abs/hep-ph/0011377), [spire:537600](https://inspirehep.net/literature/537600))

* {#Holzwarth05} Gottfried Holzwarth, Section 2.3 of: _Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models_ ([arXiv:hep-ph/0511194](https://arxiv.org/abs/hep-ph/0511194)), Chapter 2 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* Yongseok Oh, _Skyrmions with vector mesons: Single Skyrmion and baryonic matter_, 2013 ([[OhSkyrmionsWithVectorMesons.pdf:file]])

Combination of the [[omega-meson|omega]]-[[rho meson|rho]]-[[Skyrme model]] with the [[bag model of quark confinement]]:

* H. Takashita, S. Yoro, H. Toki, _Chiral bag plus skyrmion hybrid model with vector mesons for nucleon_, Nuclear Physics A
Volume 485, Issues 3–4, August 1988, Pages 589-605 (doi:10.1016/0375-9474(88)90555-6)

#### Inclusion of the $\sigma$-meson

Inclusion of the [[sigma-meson]]:

* Thomas D. Cohen, _Explicit $\sigma$ meson, topology, and the large-$N$ limit of the Skyrmion_, Phys. Rev. D 37 (1988) ([doi:10.1103/PhysRevD.37.3344](https://doi.org/10.1103/PhysRevD.37.3344))

For analysis of [[neutron star]] [[equation of state]]:

* David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, _Studying the parameters of the extended $\sigma$-$\omega$ model for neutron star matter_ ([arXiv:2006.03676](https://arxiv.org/abs/2006.03676))


skyrmion

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

> [[Tony Skyrme|Skyrme]] had studied with attention [[On Vortex Atoms|Kelvin's ideas on vortex atoms]].

> ([Ranada-Trueba 01, p. 200](On+Vortex+Atoms#RanadaTrueba01))

A _Skyrmion_ is a [[soliton]] in certain ([[flavour physics|flavour]]) [[gauge field theories]]. The concept exists quite generally (see [Rho-Zahed 16](#RhoZahed16)), but its original use ([Skyrme 62](#Skyrme62)), and still the most important one, realizes _[[baryons]]_ and [[atomic nuclei]] as [[solitons]] in the [[light meson]]-[[field (physics)|fields]] of [[chiral perturbation theory]], thus serving as a putative theory of [[non-perturbative quantum field theory|non-perturbative]] [[quantum chromodynamics]], the formulation of the latter being by and large an open problem (due to [[confinement]], see _[[mass gap problem]]_).

\begin{center}

\end{center}

> graphics grabbed from [Manton 11](#Manton11)

\begin{center}

\end{center}

> graphics grabbed form [FLM 12](#FLM12)

\begin{center}

\end{center}

> graphics grabbed from [Gudnason & Halcrow 2022](#GudnasonHalcrow22)

## Definition
 {#Definition}

For $G$ a [[simple Lie group]] with [[semisimple Lie algebra]] denoted $\mathfrak{g}$, with [[Lie bracket]] $[-,-]$ and with [[Killing form]] $\langle -,-\rangle$,
the Skyrme fields are [[smooth functions]]

$$
  U \;\colon\; \mathbb{R}^3 \longrightarrow G
$$

and the Skyrme [[Lagrangian density]] is

$$
  \mathbf{L}
  \;=\;
  -\tfrac{1}{2}
  \left\langle
    U^{-1}\mathbf{d}U \wedge \star U^{-1}\mathbf{d}U
  \right\rangle
  +
  \tfrac{1}{16}
  \Big(
    \big[
       U^{-1}\mathbf{d}U
       \wedge
       U^{-1}\mathbf{d}U
    \big]
    \wedge \star
    \big[
       U^{-1}\mathbf{d}U
       \wedge
       U^{-1}\mathbf{d}U
    \big]
  \Big)
$$

where $U^{-1} \mathbf{d}U = U^\ast \theta$ is the [[pullback of differential forms|pullback]] of the [[Maurer-Cartan form]] on $G$, and where $\ast$ denotes the standard [[Hodge star operator]] on [[Euclidean space]] $\mathbb{R}^3$.

(e.g. [Manton 11 (2.2)](#Manton11), [Cork 18b (1)](#Cork18b))

A _classical Skyrmion_ is a solution to the corresponding [[Euler-Lagrange equations]] which

1. is [[vanishing at infinity]] $U(r \to \infty) \to e \in G$

1. [[critical point|extremizes]] the [[energy]] implied by the above [[Lagrangian]].

## Properties

### Relation to chiral perturbation theory

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

### As a model for atomic nuclei
 {#AsAModelForBaryonsAndNuclei}

Skyrmions are candidate models for [[baryons]] and even some aspects of [[atomic nuclei]] ([Riska 93](#Riska93), [Battye-Manton-Sutcliffe 10](#BattyeMantonSutcliffe10), [Manton 16](#Manton16), [Naya-Sutcliffe 18a](#NayaSutcliffe18a), [Naya-Sutcliffe 18b](#NayaSutcliffe18b}).

For instance various resonances of the [[carbon]] [[nucleus]] are modeled well by a Skyrmion with baryon number 12 ([Lau-Manton 14](#LauManton14)):

\begin{center}

\end{center}

> graphics grabbed form [Lau-Manton 14](#LauManton14)

See also the animated computations in [Gudnason & Halcrow 2018](#GudnasonHalcrow18).

{#IncludingTowerOfMesons} For Skyrmion models of nuclei to match well to [[experiment]], not just the [[pion field]] but also the tower of [[vector mesons]] need to be included in the construction.

Including the [[rho meson]] gives good results for light nuclei ([Naya-Sutcliffe 18a](#NayaSutcliffe18a), [Naya-Sutcliffe 18b](#NayaSutcliffe18b))

\begin{center}

\end{center}

> graphics grabbed form [Naya-Sutcliffe 18a](#NayaSutcliffe18a)

An analogous discussion for inclusion of [[omega-mesons]] is in [Gudnason-Speight 20](#GudnasonSpeight20).

### As a holographic boundary theory
 {#AsBoundaryFieldTheory}

The Skyrmions in 4d spacetime [above](#Definition), with [[vector meson]]-contributions included, are the [[AdS-CFT|holographic]]/[[KK-theory]] reduction of [[instantons]] in [[D=5 Yang-Mills theory]] ([Sakai-Sugimoto 04, Section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, Section 3.3](#SakaiSugimoto05), reviewed in [Sugimoto 16, Section 15.3.4](#Sugimoto16), [Bartolini 17, Section 2](#Bartolini17).

This phenomenon is essentially the theorem of [Atiyah-Manton 89](#AtiyahManton89), this is highlighted and developed in [Sutcliffe 10](#Sutcliffe10), [Sutcliffe 15](#Sutcliffe15).

In this way Skyrmions (and hence [[baryons]] and [[atomic nuclei]], see [below](#AsAModelForBaryonsAndNuclei)) appear in the [[Witten-Sakai-Sugimoto model]], which realizes (something close to) [[non-perturbative quantum field theory|non-perturbative]] [[QCD]] as a [[D4-D8-brane intersection|D4/D8]]-[[intersecting D-brane model]] described by the [[AdS-QCD correspondence]] ("[[holographic QCD]]").

This way, via the equivalence between [[D4-D8-brane intersections]] with [[instantons]] in the [[D8-brane]]-[[worldvolume]], the Skyrme model becomes equivalent to a model of [[baryons]] by [[wrapped brane|wrapped]] [[D4-branes]] ([Sugimoto 16, 15.4.1](#Sugimoto16)):

\begin{center}

\end{center}

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

\linebreak

### As the ultimate bag model

> The [[Skyrmion]] is the ultimate [[soliton|topological]] [[quark bag model|bag model]] with zero size bag radius, lending further credence to the [[Cheshire cat principle]]. ([Nielsen-Zahed 09](Cheshire+cat+principle#NielsenZahed09))

## Related concepts

* [[soliton]], [[vortex]]

* [[instanton]], [[caloron]]

* [[baryon]]

* [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|non-perturbative]] [[QCD]]

[[!include effective field theories of nuclear physics -- contents]]

## History

From [Rho-Zahed 10, Preface](#RhoZahed10):

> Two path-breaking developments took place consecutively in physics in the years 1983 and 1984: First in nuclear physics with the rediscovery of Skyrme’s seminal idea on the structure of baryons and then a ["revolution" in string theory](string+theory#ReferencesHistory) in the following year.

> $[\cdots]$ at that time the most unconventional idea of Skyrme that [[fermion|fermionic]] [[baryons]] could emerge as topological [[solitons]] from [[π-meson]] cloud was confirmed in the context of [[quantum chromodynamics]] (QCD) in the large number-of-[[color charge|color]] ($N_c$) limit. It also confirmed how the solitonic structure of [[baryons]], in particular, the [[nucleons]], reconciled nuclear physics — which had been making an impressive progress [[phenomenology|phenomenologically]], aided mostly by [[experiments]] — with [[QCD]], the fundamental theory of [[strong nuclear force|strong interactions]]. Immediately after the rediscovery of what is now generically called "skyrmion" came the [first string theory revolution](string+theory#ReferencesHistory) which then took most of the principal actors who played the dominant role in reviving the skyrmion picture away from that problem and swept them into the mainstream of [[string theory]] reaching out to a much higher [[energy]] [[scale]]. This was in some sense unfortunate for the skyrmion model _per se_ but fortunate for nuclear physics, for it was then mostly nuclear theorists who picked up what was left behind in the wake of the celebrated string revolution and proceeded to uncover fascinating novel aspects of nuclear structure which otherwise would
have eluded physicists, notably concepts such as the ‘Cheshire Cat phenomenon’ in hadronic dynamics.

> What has taken place since 1983 is a beautiful story in [[physics]]. It has not only profoundly influenced nuclear physics — which was Skyrme’s original aim — but also brought to light hitherto unforseen phenomena in other areas of physics, such as [[condensed matter physics]], [[astrophysics]] and [[string theory]].

## References

### General

The original articles:

* {#Skyrme62} [[Tony Skyrme]], _A unified field theory of mesons and baryons_, Nuclear Physics Volume 31, March–April 1962, Pages 556-569 (doi:10.1016/0029-5582(62)90775-7)

* {#AdkinsNappiWitten83} [[Gregory Adkins]], [[Chiara Nappi]], [[Edward Witten]], _Static Properties of Nucleons in the Skyrme Model_, Nucl. Phys. B228 (1983) 552 ([spire:190174](http://inspirehep.net/record/190174), doi:10.1016/0550-3213(83)90559-X)

Textbook accounts:

* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific (2016) [[doi:10.1142/9710](https://doi.org/10.1142/9710)]

* {#Manton22} [[Nicholas Manton]], *Skyrmions – A Theory of Nuclei*, World Scientific (2022) [[doi:10.1142/q0368 ](https://doi.org/10.1142/q0368 )]

Other review:

* [[Edward Witten]], _Skyrmions and QCD_, in: Current Algebra and Anomalies, pp. 529-537, World Scientific (1985) ([doi:10.1142/9789814503044_0010](https://doi.org/10.1142/9789814503044_0010))

> If one assumes [[confinement]], then $[$[[SU(N)]]$]$-[[QCD]] is equivalent to a theory of [[mesons]] (and [[glueballs]]) in which the (quartic) [[meson]] [[coupling constant]] is $1/N$.

> $[..]$

> $[$here to stress that$]$ [[QCD]] is equivalent for any $N$, including $N=3$, to a meson theory in which $1/N$ is the coupling constant, [[large N limit|Large N]] is special only in that the meson coupling is only moderately weak.

> $[..]$

> These considerations remove all of the obstacles to interpreting baryons as the solitons of meson physics

> $[..]$

> There is every evidence that Skyrmions physics with varying couplings reproduces QCD baryons of varying $N$.

> $[..]$

> The [[large N limit|1/N expansion]] of [[QCD]], as understood from [[Feynman diagrams]], is the road map which makes the success of  [[Skyrmion]] physics rationally comprehensible.

* [[Eric D'Hoker]], Edward Farhi, _The Proton as a Topological Twist: A model of elementary particles, without reference to quarks, contains topological structures whose properties match experimental observations remarkably well_, American Scientist Vol. 73, No. 6 (November-December 1985), pp. 533-540 ([jstor:27853483](https://www.jstor.org/stable/27853483))

* Yossef Dothan, L. C. Biedenham, *Old models never die: the revival of the Skyrme model*, Comments on Nuclear and Particle Physics **17** 2 (1987) 63-91 ([spire:18567](https://inspirehep.net/literature/18567), [pdf](https://inspirehep.net/files/e03cc4f4c9fa76aa4a94436cba63443b))

* [[Ismail Zahed]], [[Gerald E. Brown]] *The Skyrme model*, Physics Reports **142** 1–2 (1986) 1-102 [doi:10.1016/0370-1573(86)90142-0]

* {#BrownRho88} [[Gerald E. Brown]], [[Mannque Rho]], *The Chiral Bag*, Comments Nucl. Part. Phys. **18** 1 (1988) 1-29 [[spire:18025](http://inspirehep.net/record/18025)]

  (in relation to the [[bag model of quark confinement]])

* [[Igor Klebanov]], _Strangeness in the Skyrme model_, in: D. Vauthrin, F. Lenz, J. W. Negele,  _Hadrons and Hadronic Matter_, Plenum Press 1989 ([doi:10.1007/978-1-4684-1336-6](https://link.springer.com/book/10.1007/978-1-4684-1336-6))

  (with emphasis on inclusion [[heavy mesons]])

* [[Robert E. Marshak]], Section 10.5 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], Chapter 7 of: _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

* {#Weigel96} [[Herbert Weigel]], _Baryons as Three Flavor Solitons_,  	Int. J. Mod. Phys. A11:2419-2544, 1996 ([arXiv:hep-ph/9509398](https://arxiv.org/abs/hep-ph/9509398), [cds:288541](http://cds.cern.ch/record/288541), [doi:10.1142/S0217751X96001218](https://doi.org/10.1142/S0217751X96001218))

* [[Ulrich Mosel]], Section 17.1 in: *Soliton Models of Hadrons* ([doi:10.1007/978-3-662-03841-3_17](https://doi.org/10.1007/978-3-662-03841-3_17)), which in turn is Chapter 17 in: *Fields, Symmetries, and Quarks*, Springer 1999 ([doi:10.1007/978-3-662-03841-3](https://link.springer.com/book/10.1007/978-3-662-03841-3))

* {#Weigel08} [[Herbert Weigel]], _Chiral Soliton Models for Baryons_,  Lecture Notes in Physics book series, volume 743, Springer 2008 ([doi:10.1007/978-3-540-75436-7](https://doi.org/10.1007/978-3-540-75436-7))

* Yong-Liang Ma, Masayasu Harada, _Lecture notes on the Skyrme model_ ([arXiv:1604.04850](https://arxiv.org/abs/1604.04850), [spire:1448311](https://inspirehep.net/literature/1448311))

* [[Nicholas Manton]], *Skyrmions as Models for Nuclei*, Ischia (2022) [[pdf](https://agenda.infn.it/event/19753/contributions/170544/attachments/90671/122280/SkyIschia.pdf)]

* [[Edward Witten]], from [34:05](https://youtu.be/TvIz-6YOdKs?t=2045) on in: *Some Milestones in the Study of Confinement*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023) [[web](https://www.ias.edu/video/some-milestones-study-confinement), [YT](https://youtu.be/TvIz-6YOdKs)]

See also

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

Further development:

* {#Manton11} [[Nicholas Manton]], _Classical Skyrmions -- Static Solutions and Dynamics_, Mathematical Methods in the applied Sciences, Volume35, Issue10, 2012, Pages 1188-1204 ([arXiv:1106.1298](https://arxiv.org/abs/1106.1298), [doi:10.1002/mma.2512]( https://doi.org/10.1002/mma.2512))

* {#NST11} Atsushi Nakamula, Shin Sasaki, Koki Takesue, _Atiyah-Manton Construction of Skyrmions in Eight Dimensions_, JHEP 03 (2017) 076 ([arXiv:1612.06957](https://arxiv.org/abs/1612.06957))

* {#FLM12} D. T. J. Feist, P. H. C. Lau, [[Nicholas Manton]], _Skyrmions up to Baryon Number 108_ ([arXiv:1210.1712](https://arxiv.org/abs/1210.1712))

* {#Manton17} [[Nicholas Manton]], _Lightly Bound Skyrmions, Tetrahedra and Magic Numbers_ ([arXiv:1707.04073](https://arxiv.org/abs/1707.04073))

* {#GudnasonHalcrow18} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *Vibrational modes of Skyrmions*, Phys. Rev. D **98** 125010 (2018) [[arXiv:1811.00562](https://arxiv.org/abs/1811.00562), [doi:10.1103/PhysRevD.98.125010](https://doi.org/10.1103/PhysRevD.98.125010)]

  *Database of Skyrmion Vibrations* [[www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/](http://www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/)]

* Avner Karasik, _Skyrmions, Quantum Hall Droplets, and one current to rule them all_ ([arXiv:2003.07893](https://arxiv.org/abs/2003.07893))

* [[Carlos Naya]], K. Oles, _Background fields and self-dual Skyrmions_ ([arXiv:2004.07069](https://arxiv.org/abs/2004.07069))

* C. Adam, K. Oles, A. Wereszczynski, _The Dielectric Skyrme model_ ([arXiv:2005.00018](https://arxiv.org/abs/2005.00018))

* [[Sven Bjarke Gudnason]], Marco Barsanti, [[Stefano Bolognesi]], _Near-BPS baby Skyrmions_ ([arXiv:2006.01726](https://arxiv.org/abs/2006.01726))

* [[Sven Bjarke Gudnason]], _Dielectric Skyrmions_ ([arXiv:2009.03082](https://arxiv.org/abs/2009.03082))

* Francisco Correa, Andreas Fring, Takanobu Taira, _Complex BPS Skyrmions with real energy_ ([arXiv:2102.05781](https://arxiv.org/abs/2102.05781))

* [[Chris Halcrow]], [[Thomas Winyard]], _A consistent two-skyrmion configuration space from instantons_ ([arXiv:2103.15669](https://arxiv.org/abs/2103.15669))

* [[Paul Sutcliffe]], *A Skyrme model with novel chiral symmetry breaking* [[arXiv:2301.03021](https://arxiv.org/abs/2301.03021)]

* Fabrizio Canfora, Scarlett C. Rebolledo-Caceres, *Skyrmions at Finite Density* [[arXiv:2306.10226](https://arxiv.org/abs/2306.10226)]

* Alberto García Martín-Caro, [[Chris Halcrow]],
*The charge density and neutron skin thickness of Skyrmions* [[https://arxiv.org/abs/2312.04335](https://arxiv.org/abs/2312.04335)]

Large computer search for Skyrmion solutions:

* {#GudnasonHalcrow22} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *A Smörgåsbord of Skyrmions* [[arXiv:2202.01792](https://arxiv.org/abs/2202.01792)]

Skyrmion [[scattering amplitudes]]:

* T. Gisiger, M. B. Paranjape, _Skyrmion-Skyrmion Scattering_ ([arXiv:hep-th/9310050](https://arxiv.org/abs/hep-th/9310050))

* David Foster, [[Steffen Krusch]], *Scattering of Skyrmions*, Nuclear Physics B 897, 697-716, 2015 ([arXiv:1412.8719](https://arxiv.org/abs/1412.8719))

[[nucleon]]$\,$[[interaction]]:

* [[Chris Halcrow]], [[Derek Harland]], _An attractive spin-orbit potential from the Skyrme model_ ([arXiv:2007.01304](https://arxiv.org/abs/2007.01304))

* [[Derek Harland]], [[Chris Halcrow]], _Nucleon-nucleon potential from skyrmion dipole interactions_ ([arXiv:2101.02633](https://arxiv.org/abs/2101.02633))

> The topic of this paper has a history of mistakes and
sign errors in the literature. For both these reasons, we present our calculation in painstaking detail. $[...]$.

> Compared with earlier attempts based on the Skyrme model, we obtain a very good match with the long-range parts of the Paris potential. Overall, these results provide an excellent starting point for describing the nucleon-nucleon interaction from the Skyrme model. Importantly, we can describe many features of the nucleon-nucleon interaction using a purely pionic theory. $[...]$

> our approach could be adapted to any model which treats nuclei as quantised solitons. This includes [[holographic QCD]], where nuclei are described as instantons on a curved spacetime

* [[Horatiu Nastase]], [[Jacob Sonnenschein]], _A $T \bar T$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering_ ([arXiv:2101.08232](https://arxiv.org/abs/2101.08232))

  > (via [[TT deformation]])

Gauged skyrmions with more realistic binding energies (via [[equivariant de Rham cohomology]] and maybe some kind of [[equivariant Hopf degree theorem]]):

* [[Josh Cork]], [[Derek Harland]], [[Thomas Winyard]], *A model for gauged skyrmions with low binding energies*, [arXiv:2109.06886](https://arxiv.org/abs/2109.06886)]

* [[Josh Cork]], [[Derek Harland]], *Geometry of Gauged Skyrmions*, SIGMA **19** o71 (2023) [[doi:10.3842/SIGMA.2023.071](https://doi.org/10.3842/SIGMA.2023.071), [arXiv:2303.02623](https://arxiv.org/abs/2303.02623)]

In strong [[magnetic fields]]:

* {#ChenQiuFukushima21} Shi Chen, Zebin Qiu, Kenji Fukushima, *Skyrmions in a magnetic field and $\pi^0$ domain wall formation in dense nuclear matter* ([arXiv:2104.11482](https://arxiv.org/abs/2104.11482))

Mathematical discussion in [[differential geometry]]:

* Christian Gross, _Differential Forms on the Skyrmion Bundle_, In: Antoine JP., Ali S.T., Lisiecki W., Mladenov I.M., Odzijewicz A. (eds.) _Quantization, Coherent States, and Complex Structures_, Springer 1995 ([doi:10.1007/978-1-4899-1060-8_7](https://doi.org/10.1007/978-1-4899-1060-8_7))

and in [[algebraic topology]]:

* Christian Gross, _Topology of the skyrmion bundle_, Journal of Mathematical Physics 36, 4406 (1995) ([doi:10.1063/1.530899](https://doi.org/10.1063/1.530899))

Relation to the [[complex Hopf fibration]]:

* {#GudnasonNitta20} [[Sven Bjarke Gudnason]], [[Muneto Nitta]], _Linking number of vortices as baryon number_ ([arXiv:2002.01762](https://arxiv.org/abs/2002.01762), [spire:1778698/](http://inspirehep.net/record/1778698/))

Generalization to [[SU(N)]]:

* Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora, [[Bianca Cerchiai]], _Analytic $SU(N)$ Skyrmions at finite Baryon density_ ([arXiv:2005.11301](https://arxiv.org/abs/2005.11301))

In higher dimensions:

* Emir Syahreza Fadhilla, Ardian Nata Atmaja, Bobby Eka Gunara,
*BPS Skyrmions of Generalized Skyrme Model In Higher Dimensions* ([arXiv:2108.08694](https://arxiv.org/abs/2108.08694))

For the [[electroweak field]]:

* Juan Carlos Criado, Valentin V. Khoze, Michael Spannowsky, *Electroweak Skyrmions in the HEFT* ([arXiv:2109.01596](https://arxiv.org/abs/2109.01596))

More in relation to [[hidden local symmetry]]:

* [[Mannque Rho]], *Skyrmions and Fractional Quantum Hall Droplets Unified by Hidden Symmetries in Dense Matter* ([arXiv:2109.10059](https://arxiv.org/abs/2109.10059))

Further resources:

* _[Geometric models of Nuclear Matter Conference 2014](https://www.kent.ac.uk/smsas/personal/skyrmions/conference.html)_

[[!include Skyrmions from rational maps -- references]]

[[!include Skyrme hadrodynamics with vector mesons -- references ]]

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

### As a model for atomic nculei
 {#ReferencesAsModelsForAtomicNuclei}

Skyrmions modelling [[atomic nuclei]]:

* {#Riska93} D. O. Riska, _Baryons and nuclei as skyrmions_,  Czech J Phys (1993) 43: 449 ([doi:10.1007/BF01589856](https://doi.org/10.1007/BF01589856))

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* {#Manton16} [[Nicholas Manton]], _Skyrmions and Nuclei_, talk at Brookhaven National Lab, November 2016 ([pdf](https://quark.phy.bnl.gov/~pisarski/talks/Manton_SkyBNL.pdf))

* {#NayaSutcliffe18a} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions and clustering in light nuclei_, Phys. Rev. Lett. 121, 232002 (2018) ([arXiv:1811.02064](https://arxiv.org/abs/1811.02064))

* {#NayaSutcliffe18b} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions in models with pions and rho_, JHEP 05 (2018) 174 ([arXiv:1803.06098](https://arxiv.org/abs/1803.06098))

  APS Synopsis: _[Revamping the Skyrmion Model](https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002)_, 2018

For [[carbon]]:

* {#LauManton14} P.H.C. Lau, [[Nicholas Manton]], _States of Carbon-12 in the Skyrme Model_,  Phys. Rev. Lett. 113, 232503 (2014) ([arXiv:1408.6680](https://arxiv.org/abs/1408.6680))

* Christoph Adam, [[Carlos Naya]], Andrzej Wereszczyński, *Carbon-12 in the generalized Skyrme model* [[arXiv:2401.08778](https://arxiv.org/abs/2401.08778)]

### As a models for neutron star

Discussion of [[model (in theoretical physics)|models]] of [[neutron stars]] by [[Skyrmions]]:

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _BPS Skyrmions as neutron stars_, Physics Letters B
Volume 742, 6 March 2015, Pages 136-142 ([arXiv:1407.3799](https://arxiv.org/abs/1407.3799))

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _Neutron stars in the BPS Skyrme model: mean-field limit vs. full field theory_, Phys. Rev. C 92, 025802 (2015) ([arXiv:1503.03095](https://arxiv.org/abs/1503.03095))

* Xiang-Hai Liu, Yong-Liang Ma, [[Mannque Rho]], _Topology change and nuclear symmetry energy in compact-star matter_, Phys. Rev. C 99, 055808 (2019) ([arXiv:1811.10012](https://arxiv.org/abs/1811.10012))

* [[Carlos Naya]], _Neutron stars within the Skyrme model_, 	Int. J. Mod. Phys. E 28, 1930006 (2019) ([arXiv:1910.01145](https://arxiv.org/abs/1910.01145))

* C. Adam, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _Adding crust to BPS Skyrme neutron stars_ ([arXiv:2004.03610](https://arxiv.org/abs/2004.03610))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro García, Ricardo Vázquez, Andrzej Wereszczynski, _A new consistent Neutron Star Equation of State from a Generalized Skyrme model_ ([arXiv:2006.07983](https://arxiv.org/abs/2006.07983))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, _Quasi-universal relations for generalized Skyrme stars_ ([arXiv:2011.08573](https://arxiv.org/abs/2011.08573))

With [[higher curvature corrections]] included ([[Starobinsky model of cosmic inflation|Starobinsky model]]):

* C. Adam, M. Huidobro, R. Vazquez, A. Wereszczynski, _BPS Skyrme neutron s tars in generalized gravity_ ([arXiv:2005.10834](https://arxiv.org/abs/2005.10834))

See also:

* [[Mannque Rho]], _Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter_ ([arXiv:2004.09082](https://arxiv.org/abs/2004.09082))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, *Kaon condensation in skyrmion matter and compact stars* [[arXiv:2212.00385](https://arxiv.org/abs/2212.00385)]

### Relation to instantons, calorons, solitons, monopoles

The construction of Skyrmions from [[instantons]] is due to

* {#AtiyahManton89} [[Michael Atiyah]], [[Nicholas Manton]], _Skyrmions from instantons_, Phys.  Lett.  B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

The relation between skyrmions, [[instantons]], [[calorons]], [[solitons]] and [[monopoles]] is usefully reviewed and further developed in

* {#Cork18a} [[Josh Cork]], _Calorons, symmetry, and the soliton trinity_, PhD thesis, University of Leeds 2018 ([web](http://etheses.whiterose.ac.uk/22097/))

* {#Cork18b} [[Josh Cork]], _Skyrmions from calorons_, J. High Energ. Phys. (2018) 2018: 137 ([arXiv:1810.04143](https://arxiv.org/abs/1810.04143))

based on

* {#Sutcliffe10} [[Paul Sutcliffe]], _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023))

which in turn relates to a [[Minkowski spacetime]]-version of the holographic realization of Skyrmions in the [[Sakai-Sugimoto model]] ([[AdS/QCD correspondence]]).

### In solid state physics
 {#ReferencesInSolidStatePhysics}

In [[solid state physics]] skyrmions in the magnetization of thin atomic layers are known as _magnetic skyrmions_.

See also:

* Wikipedia, _[Magnetic skyrmions](https://en.m.wikipedia.org/wiki/Magnetic_skyrmion)_

Realizations in [[experiment]] and application to [[computing]]:

* [[Andrew Kent]] et al. *Skyrmionics -- Computing and memory technologies based on topological excitations in magnets*, Journal of Applied Physics **130** (2021) 070908 [[arXiv:2101.09947](https://arxiv.org/abs/2101.09947), [doi:10.1063/5.0046950](https://doi.org/10.1063/5.0046950)]

Review and exposition:

* [[Andrew Kent]], *A new spin on magnetism with applications in information processing*, talk at *[[CQTS]]* (Nov 2022) [[[Kent-CQTS-Nov2022.pdf:file]], [video](https://drive.google.com/file/d/1LTz2y6KrJytps94rlS6o0BET6rmzPFU_/view)]

[[!redirects skyrmions]]

[[!redirects Skyrmion]]
[[!redirects Skyrmions]]

[[!redirects Skyrme model]]
[[!redirects skyrmion model]]
[[!redirects Skyrmion model]]


Skyrmions from rational maps -- references

### Skyrmions from rational maps
 {#SkyrmionsFromRationalMapsReferences}

The following is a list of references on the construction of [[Skyrmion]]-solutions of the [[Yang-Mills theory|Yang-Mills field]] via [[rational maps]] from the [[complex plane]], hence [[holomorphic maps]] from the [[Riemann sphere]], to itself, akin to the Donaldson-construction of [[Yang-Mills monopoles]].

The original idea:

* [[Conor J. Houghton]], [[Nicholas Manton]], [[Paul Sutcliffe]], *Rational Maps, Monopoles and Skyrmions*, Nucl. Phys. B510 (1998) 507-537 ([arXiv:hep-th/9705151](https://arxiv.org/abs/hep-th/9705151), doi:10.1016/S0550-3213(97)00619-6)

Further discussion:

* [[Steffen Krusch]], *$S^3$ Skyrmions and the Rational Map Ansatz*, Nonlinearity 13:2163, 2000 ([arXiv:hep-th/0006147](https://arxiv.org/abs/hep-th/0006147))

* [[Nicholas S. Manton]], [[Bernard M.A.G. Piette]], *Understanding Skyrmions using Rational Maps*, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 ([doi:10.1007/978-3-0348-8268-2_27](https://doi.org/10.1007/978-3-0348-8268-2_27), [arXiv:hep-th/0008110](https://arxiv.org/abs/hep-th/0008110))

* [[Richard Battye]], [[Paul Sutcliffe]], *Skyrmions, Fullerenes and Rational Maps*, 	Rev. Math. Phys. 14 (2002) 29-86 ([arXiv:hep-th/0103026](https://arxiv.org/abs/hep-th/0103026))

* W.T. Lin, [[Bernard M.A.G. Piette]], *Skyrmion Vibration Modes within the Rational Map Ansatz*,  	Phys. Rev. D77:125028, 2008 ([arXiv:0804.4786](https://arxiv.org/abs/0804.4786), [doi:10.1103/PhysRevD.77.125028](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.77.125028))

On [[quantization]] of [[Skyrmions]] informed by [[homotopy of rational maps]]:

* [[Steffen Krusch]], *Homotopy of rational maps and the quantization of Skyrmions*, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (doi:10.1016/S0003-4916(03)00014-9, [arXiv:hep-th/0210310](https://arxiv.org/abs/hep-th/0210310))

* [[Steffen Krusch]], *Skyrmions and Rational Maps*, talk at KIAS 2004  ([pdf](http://newton.kias.re.kr/KH04/talks/krusch.pdf), [[Krusch_SkyrmionsAndRationalMaps.pdf:file]])

* [[Steffen Krusch]], *Quantization of Skyrmions* ([arXiv:hep-th/0610176](https://arxiv.org/abs/hep-th/0610176))

the impact of which, on the computation of [[atomic nuclei]], is highlighted in:

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], p. 23 of: _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

See also:

* [[Derek Harland]], [[Paul Sutcliffe]], *Rational Skyrmions* $[$[arXiv:2307.09355](https://arxiv.org/abs/2307.09355)$]$


skyscraper sheaf

#Contents#
* table of contents
{:toc}

## Idea

A _skyscraper sheaf_ is a [[sheaf]] supported at a single point.

This is not unlike the Dirac $\delta$-distribution.

#Definition#

For $X$ a [[topological space]], $x \in X$ a point of $X$ and $S \in Set$ a [[set]], the **skyscraper sheaf** $skyscr_x(S) \in Sh(X)$ in the [[category of sheaves]] on the [[category of open subsets]] $Op(X)$ of $X$ supported at $x$ with value $S$ is the [[sheaf]] of sets given by the assignment

$$
  skysc_x(S) : (U \subset X) \mapsto
  \left\{
    \array{
      S & if \; x \in U
      \\
      {*} & otherwise
    }
  \right.
$$

## Remarks ##

* The skyscraper sheaf $skysc_x(S)$ is the [[direct image]] of $S$ under the [[geometric morphism]] $x : Set \to Sh(X)$ which defines the [[point of a topos]] given by $x \in X$ (see there for more details on this perspective).

## References

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_

category: sheaf theory

[[!redirects skyscraper sheaves]]


sl(2)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie algebra]] $\mathfrak{sl}(2,k)$ (over some [[ground field]] $k$) is the [[special linear Lie algebra]] $\mathfrak{sl}(n,k)$ for $n  = 2$.

Over [[ground field]] $k = \mathbb{R}$ ([[real numbers]]) or $k = \mathbb{C}$ ([[complex numbers]]) this is the [[Lie algebra]]  [[Lie differentiation|corresponding to]] the [[Lie group]] which is the [[special linear group]] $SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$, respectively.

## Properties

### As the complexification of $\mathfrak{su}(2)$

$\mathfrak{sl}(2,\mathbb{C})$ is the [[complexification]] of the [[special unitary Lie algebra]] $\mathfrak{su}(2)$ (see at [[su(2)]]) (...)

### Jacobson-Morozov theorem

See at _[[Jacobson-Morozov theorem]]_.

[[!redirects sl(2,C)]]


SL(2,H)

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

#Contents#
* table of contents
{:toc}

## Definition

By $SL(2,\mathbb{H})$ one denotes the [[special linear group]] of $2 \times 2$ [[matrices]] with [[coefficients]] in the [[quaternions]], where "special" refers to their [[Dieudonné determinant]] being unity:

$$
  SL(2,\mathbb{H})
  \;:=\;
  \left\{
     \left.
     \left[
       \array{
         a & b
         \\
         c & d
       }
     \right]
     \;\right\vert\;
     \left\Vert
       a d - a c a^{-1} b
     \right\Vert
     \;=\;
     1
  \right\}
$$

(here $\left\Vert q\right\Vert^2 \;:=\; q \bar q \in \mathbb{R}$ is the [[norm]](-square) on [[quaternions]]).

## Properties

### Relation to $Sp(2)$
 {#RelationToSp2}

Every quaternionic [[unitary matrix]] (hence in [[Sp(2)]]) happens to have unit [[Dieudonné determinant]] ([Cohen-De Leo 99, Cor. 6.4](Dieudonne+determinant#CohenDeLeo99)). Therefore we have a [[subgroup]] inclusion

$$
  Sp(2)
  \;=\;
  U(2,\mathbb{H})
  \;\subset\;
  SL(2,\mathbb{H})
  \,.
$$

### Relation to $Spin(5,1)$
 {#RelationToSpin51}

Under the [[conjugation action]] on $2 \times 2$ [[Hermitian matrices]] with [[coefficients]] in the [[quaternions]], $SL(2,\mathbb{H})$ is identified with [[Spin(5,1)]] and its canonical [[action]] on [[Minkowski spacetime]] $\mathbb{R}^{5,1}$.

For more on this see at _[[spin representation]]_, _[[supersymmetry and division algebras]]_ and _[[geometry of physics -- supersymmetry]]_.

## Related concepts

[[!include exceptional spinors and division algebras -- table]]

## References

* [[Joás Venâncio]], [[Carlos Batista]], Sections 2,3 of: _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_, Adv. Appl. Clifford Algebras **31** 71 (2021) ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296), [doi:10.1007/s00006-021-01172-1](https://doi.org/10.1007/s00006-021-01172-1))


SL(2,O)

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of a [[special linear group]] $SL(2,K)$ with $K = \mathbb{O}$ the [[octonions]] does not literally make sense, due to the failure of the [[octonions]] to be an [[associative algebra]].

Nevertheless, the pattern of the [[special linear groups]] of the other three [[real numbers|real]] [[normed division algebras]] suggests that for some suitably adjusted concept of "special linear group", it should make sense to speak of $SL(2,\mathbb{O})$, after all, and that it should essentially be the [[spin group]] $Spin(9,1)$:

[[!include exceptional spinors and division algebras -- table]]

## Realizations

### By Manogue--Schray

[Manogue and Schray](#MS93) have shown how to interpret $Spin(9,1)$ as $SL(2,\mathbb{O})$ as follows.  For a clear summary of results see also [Dray and Manogue](#DM10).

First, a warmup with the [[complex numbers]]. The space $\mathfrak{h}_2(\mathbb{C})$ of $2 \times 2$ [[hermitian matrices]] with entries in $\mathbb{C}$ may be identified with 4-dimensional [[Minkowski spacetime]], since the [[determinant]] is a [[quadratic form]] of [[signature of a quadratic form|signature]] $(1,3)$.  Any element $g \in \mathrm{SL}(2,\mathbb{C})$ acts as a linear transformation of $\mathfrak{h}_2(\mathbb{C})$ as follows:

$$  X \mapsto g X g^\ast , \qquad X \in \mathfrak{h}_2(\mathbb{C})  $$

and this action preserves the determinant on $\mathfrak{h}_2(\mathbb{C})$, so we obtain a [[group homomorphism]] from $\mathrm{SL}(2,\mathbb{C})$ to $\mathrm{O}(1,3)$.  In fact this is a 2-1 homomorphism from $\mathrm{SL}(2,\mathbb{C})$ onto $SO_0(1,3)$.   Since $\mathrm{SL}(2,\mathbb{C})$ is simply connected, this allows us to identify $\mathrm{SL}(2,\mathbb{C})$ with $\mathrm{Spin}(1,3) \cong \mathrm{Spin}(3,1)$.

Following this pattern, the space $\mathfrak{h}_2(\mathbb{O})$ of $2 \times 2$ [[hermitian matrices]] with entries in the [[octonions]] $\mathbb{O}$ may be identified with 10-dimensional [[Minkowski spacetime]], since the [[determinant]] is a [[quadratic form]] of [[signature of a quadratic form|signature]] $(1,9)$.   (The determinant of an element of $\mathfrak{h}_2(\mathbb{O})$ is well-defined using the usual formula for the determinant of a $2 \times 2$ matrix, because any octonion commutes with its conjugate.)  Let $\mathbb{M}_2(\mathbb{O})$ be the set of  $2 \times 2$ octonionic matrices all of whose entries lie in an arbitrary subalgebra of $\mathbb{O}$ [[isomorphism|isomorphic]] to $\mathbb{C}$.  Then the determinant of $g \in \mathbb{M}_2(\mathbb{O})$ is well-defined by the usual formula, and furthermore $(g X) g^\ast = g ( X g^\ast)$ for all $X \in \mathfrak{h}_2(\mathbb{O})$.

Let then $\mathrm{SL}(2,\mathbb{O})$ be the subgroup of linear transformations of $\mathbb{O}^2$ generated by those of the form

$$   \psi \mapsto g \psi $$

where $g \in \mathbb{M}_2(\mathbb{O})$ has $det(g) = 1$.  This group acts
by [[linear transformations]] of $\mathfrak{h}_2(\mathbb{O})$ in a unique way such that

$$  X \mapsto (g X) g^\ast , \qquad X \in \mathfrak{h}_2(\mathbb{O})  $$

when $g \in \mathbb{M}_2(\mathbb{O})$ has $det(g) = 1$.   This action
preserves the determinant on $\mathfrak{h}_2(\mathbb{O})$.  Thus, there is a [[group homomorphism]] from $\mathrm{SL}(2,\mathbb{O})$ to $\mathrm{O}(1,9)$, and in fact it maps to $\mathrm{SO}_0(1,9)$ in a 2-1 and onto way, which allows us to identify $\mathrm{SL}(2,\mathbb{O})$ with $\mathrm{Spin}(1,9) \cong \mathrm{Spin}(9,1)$.

### By Hitchin

Another proposal for making sense of $SL(2,\mathbb{O})$ is due to [Hitchin](#Hitchin18). In this approach, "$SL(2,\mathbb{O})$" is a submanifold of "$GL(2,\mathbb{O})$", which is an open orbit of $Spin(9,1) \times GL(2,\mathbb{R})$ on ${S} \otimes \mathbb{R}^2$.  Here ${S}$ is the 16-dimensional spin representation of $Spin(9,1)$; this may be identified with $\mathbb{O}^2$.

## References

* {#Baez} [[John Baez]], The octonions, Bull. Amer. Math. Soc. 39 (2002), 145--205.  Section 3.3: $\mathbb{O}\mathrm{P}^1$ and Lorentzian geometry.   ([html](https://math.ucr.edu/home/baez/octonions/node11.html))

* {#DM10} [[Tevian Dray]], [[Corinne Manogue]], _Octonionic Cayley spinors and $\mathrm{E}_6$_, Comment. Math. Univ. Carolin. 51 (2010), 193--207. ([arXiv:0911.2255](https://arxiv.org/abs/0911.2255))

* {#DHK14} [[Tevian Dray]], [[John Huerta]], Joshua Kincaid, _The magic square of Lie groups: the $2 \times 2$ case_, Lett. Math. Phys. 104 (2014), 1445--68. ([arXiv:2009.00390](https://arxiv.org/abs/2009.00390))

* {#Hitchin18} [[Nigel Hitchin]], $SL(2)$ over the octonions, Mathematical Proceedings of the Royal Irish Academy. Vol. 118. No. 1. Royal Irish Academy, 2018. ([arXiv:1805.02224](https://arxiv.org/abs/1805.02224))

* {#MS93} [[Corinne Manogue]], Jörg Schray, Finite Lorentz transformations, automorphisms, and division algebras.  Section 5: Lorentz transformations.  J. Math. Phys. 34 (1993), 3746-3767.  ([arXiv:hep-th/9302044](https://arxiv.org/abs/hep-th/9302044))

[[!redirects special linear group over the octonions]]

[[!redirects octonionic special linear group]]


SL(2,Z)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[special linear group]] in [[dimension]] 2 over the [[integers]].

## Related concepts

* [[Moebius group]]

* [[mapping class group]]

* [[S-duality]], [[U-duality]]

## References

* [[Keith Conrad]], *$SL_2(\mathbb{Z})$* [pdf, [[Conrad-SL2Z.pdf:file]]]

* [[Groupprops]], _[Special linear group:SL(2,Z)](https://groupprops.subwiki.org/wiki/Special_linear_group:SL(2,Z))_


slant product

In [[algebraic topology]], the **slant product in homology** is the following pairing between [[singular homology]] and [[singular cohomology]]:
$$
H_q(X,A)\otimes H^n(X\times Y,A')\to H^{n-q}(Y,A\otimes A').
$$

It is induced at the chains/cochains level by the [[cochain on a simplicial set|Eilenberg-Zilber chain map]]
$$
  Chains(X)\otimes Chains(Y)\to Chains(X\times Y).
$$

When the [[abelian group]] $A$ has a commutative ring structure, one can take $A'=A$ and postcompose with $A\otimes A\to A$ to obtain the pairing
$$
H_q(X,A)\otimes H^n(X\times Y,A)\to H^{n-q}(Y,A).
$$

In particular, for $Y=*$ one obtains the contraction
$$
H_q(X,A)\otimes H^n(X,A)\to H^{n-q}(*,A)
$$
taking values in the coefficient ring of the given cohomology theory.

Likewise, the **slant product in cohomology**
is a map of the form
$$H^i(Y,A)\otimes H_n(X\times Y,A')\to H_{n-i}(X,A\otimes A').$$
It is induced by the [[Alexander-Whitney map]].

There are also versions for [[relative homology]] and [[relative cohomology]].
See [Dold](#Dold), VII.11 and VII.13.

* {#Dold} Albrecht Dold, _Lectures on Algebraic Topology_, Springer, 1980.


Slater determinant

+-- {: .rightHandSide}
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### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] and especially in [[condensed matter theory]], *Slater determinants* are certain [[wavefunctions]] expressing the joint [[quantum state]] of multiple [[electrons]] (or possiby other [[fermions]]) as skew-symmetrized products of given single-particle wavefunctions.

Concretely, given a [[linear basis]]

$$
  \big( \psi_{I} \big)_{I = 1}^\infty
$$

for the [[Hilbert space]] of single-electron [[wavefunctions]], being [[square-integrable functions]] on the ambient [[Euclidean space|Euclidean $d$-space]]

$$
  \psi_i \;\in\; L^2\big( \mathbb{R}^d; \mathbb{C}\big)
  \,.
$$

a *Slater determinant* for $N \in \mathbb{N}$ particles is a function on the $N$-fold [[product space]] $\big( \mathbb{R}^d\big)^N$ of the following [[determinant]]-form

$$
  \Psi_{\mathcal{I}}
  (\vec x_1, \cdots, \vec x_N)
  \;\coloneqq\;
  det
  \big(
    \psi_{I_i}(\vec x_j)
  \big)
  \;=\;
  \underset{
    \sigma \in Sym(N)
  }{\sum}
  sgn(\sigma)
  \cdot
  \psi_{I_1}(\vec x_{\sigma(1)})
  \cdot
  \psi_{I_2}(\vec x_{\sigma(2)})
  \cdots
  \psi_{I_N}(\vec x_{\sigma(N)})
  \,,
$$

where $\mathcal{I} = \big( I_1, \cdots, I_N\big)$ is an [[n-tuple|$N$-tuple]] of indices, with $\big( \psi_{I_1}, \cdots \psi_{I_N} \big)$ the corresponding [[n-tuple|$N$-tuple]] of 1-electron wavefunctions.

(Here $Sym(N)$ denotes the [[symmetric group]] of [[permutations]] of $N$ ordered elements, and $sgn(\sigma) \in \{\pm 1\}$ denotes the [[signature of a permutation|signature]] of a given permutation $\sigma$.)

In fact, for actual electrons the wavefunctions are also functions of their [[spin]], which means, in the non-relativistic case,  that the $\psi_I$ depend also on an argument in $\{\uparrow, \downarrow\}$, in addition to their dependence on $\vec x$, and the corresponding Slater determinant states are obtained by skew-symmetrizing over all of these degrees of freedom.

The point of this construction is that it enforces the skew-symmetry under permutation of position of electrons, which is their characteristic property as *[[fermions]]*. As the multi-index set $\mathcal{I}$ ranges, the corresponding Slater determinants span the Hilbert space of $N$-electron quantum states.

In the practice of computing [[ground states]] etc. in [[solid state physics]], one tries to use as few multi-indices $\mathcal{I}$ as possible:

In the extreme case, the *[[Hartree-Fock method]]* tries to approximate a multi-electron system by the clever choice of a *single* Slater determinant. More accurate approximation methods use linear combinations of more and more Slater determinants, as the multi-index set $\mathcal{I}$ ranges. If, in principle, the full space of Slater determinants is used, one speaks of the *[[configuration interaction method]]*.

## Related concepts

* [[Pauli exclusion principle]]

## References

The construction was maybe first made explicit as eq. (15) in

* [[Paul A. M. Dirac]], *On the theory of quantum mechanics*, Proceedings of the Royal Society **112** 762 (1926) $[$[di:10.1098/rspa.1926.0133](https://doi.org/10.1098/rspa.1926.0133)$]$

It is named after:

* [[John C. Slater]], *The Theory of Complex Spectra*, Phys. Rev. **34** (1929) 1293 $[$[doi:10.1103/PhysRev.34.1293](https://doi.org/10.1103/PhysRev.34.1293)$]$

Review:

* [[Attila Szabo]], [[Neil S. Ostlund]], Sec. 2.2.3 of: *Modern Quantum Chemistry -- Introduction to Advanced Electronic Structure Theory*, Macmillan (1982), McGraw-Hill (1989), Dover (1996) $[$[pdf](https://chemistlibrary.files.wordpress.com/2015/02/modern-quantum-chemistry.pdf)$]$

* C. Lanczos, R. C. Clark, G. H. Derrick (eds.), p. 196 in: *Mathematical Methods in Solid State and Superfluid Theory*, Springer (1986) $[$[doi:10.1007/978-1-4899-6435-9](https://doi.org/10.1007/978-1-4899-6435-9)$]$

* Pablo Echenique, J. L. Alonso, around (33) in:  *A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry*, Molecular Physics **105** (2007) 3057-3098 $[$[doi:10.1080/00268970701757875](https://doi.org/10.1080/00268970701757875)$]$

See also:

* Wikipedia, *[Slater determinant](https://en.wikipedia.org/wiki/Slater_determinant)*

[[!redirects Slater determinants]]

[[!redirects Hartree-Fock method]]
[[!redirects Hartree-Fock methods]]

[[!redirects configuration interaction method]]
[[!redirects configuration interaction methods]]


Slava Rychkov

* [webpage](https://sites.google.com/site/slavarychkov/home)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Vyacheslav_Rychkov)

## Selected writings

On [[AdS/CFT duality]] in the perspective of [[gauge/string duality]]:

* [[Alexander Polyakov]], [[Vyacheslav Rychkov]] *Gauge fields-strings duality and the loop equation*, Nucl. Phys. B **581** (2000) 116-134 $[$[arXiv:hep-th/0002106](https://arxiv.org/abs/hep-th/0002106), doi:10.1016/S0550-3213(00)00183-8$]$

On the [[conformal bootstrap]]:

* [[David Poland]], [[Slava Rychkov]], [[Alessandro Vichi]], _The Conformal Bootstrap: Numerical Techniques and Applications_, Rev. Mod. Phys. 91, 15002 (2019) ([arXiv:1805.04405](https://arxiv.org/abs/1805.04405))

Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

category: people

[[!redirects Vyacheslav Rychkov]]


Slawomir Kwasik

* [institute page](https://sse.tulane.edu/math/faculty/kwasik)

## Selected writings

On the [[equivariant Whitehead theorem]]:

* {#Kwasik81} [[Slawomir Kwasik]], _On the Equivariant Homotopy Type of $G$-ANR's_, Proceedings of the American Mathematical Society
Vol. 83, No. 1 (Sep., 1981), pp. 193-194 (2 pages) ([jstor:2043921](https://www.jstor.org/stable/2043921))

category: people


Slawomir Lasota

* [personal page](https://www.mimuw.edu.pl/~sl/)

* [GoogleScholar page](https://scholar.google.pl/citations?user=j_k7K5kAAAAJ)

## Selected writings

On [[monads in computer science]] (as [[strong monads]] and [[commutative monad|commutative]] [[monoidal monads]]):

* {#GLLN08} [[Jean Goubault-Larrecq]], [[Slawomir Lasota]], [[David Nowak]], *Logical Relations for Monadic Types*, Mathematical Structures in Computer Science, **18** 6 (2008) 1169-1217 [[arXiv:cs/0511006](https://arxiv.org/abs/cs/0511006), [doi:10.1017/S0960129508007172](https://doi.org/10.1017/S0960129508007172)]

category: people


slepton

[[!redirects sleptons]]

+-- {: .rightHandSide}
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

## Idea

The (hypothetical) [[superpartners]] of [[leptons]] are called _sleptons_.

See at _sfermion_.

## Related concepts

* [[split supersymmetry]]


slice

* [[slice category]]


slice 2-category

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of a [[slice category]] to [[2-categories]].

## Definition

Let $C$ be a (non-strict) [[2-category]], and $X\in C$ an [[object]].  Then the **slice 2-category** $C/X$ has:

* as objects, the 1-[[morphisms]] $a\colon A\to X$ in $C$;

* as 1-morphisms from $a\colon A\to X$ to $b\colon B\to X$, the pairs $(f,\phi)$ where $f\colon A\to B$ is a 1-morphism in $C$ and $\phi\colon a \cong b f$ is a 2-isomorphism in $C$.

* as 2-morphisms from $(f,\phi)$ to $(g,\psi)$, the 2-morphisms $\xi\colon f \to g$ such that $(b \xi) . \phi = \psi$.

If $C$ is a [[strict 2-category]], then so is $C/X$.  Moreover, in this case, we can also define the **strict-slice 2-category** to be the subcategory $C/_s X$ of $C/X$ containing all the objects, only those morphisms $(f,\phi)$ such that $\phi$ is an identity, and all 2-morphisms between these.

If, on the other hand, we do not require $\phi : a \to b f$ to be invertible, then we obtain the **lax-slice 2-category** $C\sslash X$.  This can be regarded as an [[F-category]] whose tight morphisms are those where $\phi$ *is* invertible (or an identity).

Dually, the morphisms in the **colax-slice 2-category** involve a $\phi : b f \to a$.  It is not clear whether there is a universally accepted convention as to which is the lax-slice and which is the colax-slice; the one adopted here is that used by [Johnson-Yau](#JohnsonYau) and is such that for the lax-slice $C\sslash X$, the canonical transformation
\begin{tikzcd} C/\!/ X \ar[r] \ar[d] \ar[dr,phantom,"\Downarrow"] & C \ar[d,equals] \\ 1 \ar[r,"X"'] & C \end{tikzcd}
is a [[lax natural transformation]].

Finally, the subcategory of $C/X$ whose objects are the [[fibration in a 2-category|fibrations]] and whose morphisms are the maps of fibrations is denoted $Fib(X) = Fib_C(X)  = Fib_C/X$, and may be called the **fibrational-slice 2-category**.  Similarly we have the **opfibrational-slice** $Opf(X)$.

## Remarks

When $C$ is a 1-category, the slice, strict-slice, lax- and colax-slice, and fibrational- and opfibrational-slice 2-categories all coincide with the usual [[slice category]].  When $C$ is a [[(2,1)-category]], then all of them coincide except the strict one.  Thus, when generalizing a concept involving slice categories from categories to 2-categories, it can sometimes be a little subtle to decide on the correct version of slice category to use.

## Related concepts

* [[michaelshulman:fibrational slice]]

* [[slice category]]

* [[slice (∞,1)-category]]

## References

* {#JohnsonYau} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_ ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055))

On [[2-colimits]] in slice 2-categories:

* [[Luca Mesiti]], *Colimits in 2-dimensional slices* [[arXiv:2305.01494](https://arxiv.org/abs/2305.01494)]

[[!redirects slice 2-category]]
[[!redirects slice 2-categories]]
[[!redirects over 2-category]]
[[!redirects over 2-categories]]
[[!redirects strict slice 2-category]]
[[!redirects strict slice 2-categories]]
[[!redirects strict over 2-category]]
[[!redirects strict over 2-categories]]
[[!redirects lax slice 2-category]]
[[!redirects lax slice 2-categories]]
[[!redirects colax slice 2-category]]
[[!redirects colax slice 2-categories]]
[[!redirects oplax slice 2-category]]
[[!redirects oplax slice 2-categories]]
[[!redirects lax over 2-category]]
[[!redirects lax over 2-categories]]
[[!redirects colax slice 2-category]]
[[!redirects colax slice 2-categories]]
[[!redirects colax over 2-category]]
[[!redirects colax over 2-categories]]
[[!redirects strict-slice 2-category]]
[[!redirects strict-slice 2-categories]]
[[!redirects strict-over 2-category]]
[[!redirects strict-over 2-categories]]
[[!redirects lax-slice 2-category]]
[[!redirects lax-slice 2-categories]]
[[!redirects lax-over 2-category]]
[[!redirects lax-over 2-categories]]
[[!redirects colax-slice 2-category]]
[[!redirects colax-slice 2-categories]]
[[!redirects colax-over 2-category]]
[[!redirects colax-over 2-categories]]
[[!redirects fibrational-slice 2-category]]
[[!redirects fibrational-slice 2-categories]]
[[!redirects opfibrational-slice 2-category]]
[[!redirects opfibrational-slice 2-categories]]
[[!redirects fibrational slice]]
[[!redirects opfibrational slice]]


slice automorphism groups in higher prequantum geometry - table

**[[slice (∞,1)-topos|slice]]-[[automorphism ∞-groups]] in [[higher prequantum geometry]]**

| [[cohesive]] [[∞-groups]]: | [[Heisenberg ∞-group]] | $\hookrightarrow$ | [[quantomorphism ∞-group]] | $\hookrightarrow$ | [[∞-bisections]] of [[higher Courant groupoid]] | $\hookrightarrow$ | [[∞-bisections]] of [[higher Atiyah groupoid]] |
|--|--|--|--|--|--|--|--|
| [[L-∞ algebras]]: | [[Heisenberg L-∞ algebra]] | $\hookrightarrow$ | [[Poisson L-∞ algebra]] | $\hookrightarrow$ | [[Courant L-∞ algebra]] | $\hookrightarrow$ | twisted vector fields |


slice category > history

< [[slice category]]
[[!redirects slice category -- hstory]]


slice filtration

#Contents#
* table of contents
{:toc}

## Idea

A [[filtration]] leading to a [[slice spectral sequence]]...

## Related concepts

* [[Lurie spectral sequences -- table]]

[[!redirects slice filtrations]]


slice model structure

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

#Contents#
* table of contents
{:toc}

## Definition ##

\begin{proposition}\label{SliceModelStructure}
**(slice model structure)** \linebreak
For $\mathcal{C}$ a [[model category]] and $X \in \mathcal{C}$ any [[object]],
the [[slice category]] $\mathcal{C}_{/X}$
as well as the [[coslice category]] $\mathcal{C}^{X/}$
inherit themselves structures of [[model categories]], whose
[[fibrations]], [[cofibrations]] and [[weak equivalences]] are precisely the morphism
whose images under the [[forgetful functors]] $\mathcal{C}_{/X} \to \mathcal{C}$ or $\mathcal{C}^{X/} \to \mathcal{C}$
are fibrations, cofibrations or weak equivalences, respectively, in $\mathcal{C}$.
\end{proposition}

([Hirschhorn 2002, Thm. 7.6.5](#Hirschhorn02), [May & Ponto 2012, Th. 15.3.6](#MayPonto12))

## Properties ##

### Cofibrant generation, properness, combinatoriality

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

If $\mathcal{C}$ is

* a [[cofibrantly generated model category]]

* or a [[proper model category]]

* or a [[cellular model category]]

then so are $\mathcal{C}_{/X}$ and $\mathcal{C}^{X/}$.

More in detail, if $I,J \subset Mor(\mathcal{C})$ are the classes of [[generating cofibrations]] and of generating acylic cofibrations of $\mathcal{C}$, respectively, then

* the generating (acyclic) cofibrations of $\mathcal{C}^{X/}$ are the image under $X \sqcup(-)$ of those of $\mathcal{C}$.

=--

([Hirschhorn (2021)](#Hirschhorn21); [May & Ponto (2012, Th. 15.3.6)](#MayPonto12)).

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

If $\mathcal{C}$ is a [[combinatorial model category]], then so is $\mathcal{C}_{/X}$.

=--

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

By [basic properties](locally+presentable+category#Properties) of [[locally presentable categories]] they are stable under slicing. Hence with $\mathcal{C}$ locally presentable also $\mathcal{C}_{/X}$ is,
and by prop. \ref{ModelStructureInheritsGoodProperties} with $\mathcal{C}$ cofibrantly generated also $\mathcal{C}_{/X}$ is.

=--

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

If $\mathcal{C}$ is an [[enriched model category]] over a [[cartesian closed model category]], then so is its [[enriched slice category]] $\mathcal{C}_{/X}$.

=--

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

By basic properties of [[enriched categories]] over [[cartesian closed categories]] they are stable under slicing, where tensoring is computed in $\mathcal{C}$ (see at *[[enriched slice category]]*).
Hence with $\mathcal{C}$ enriched also $\mathcal{C}_{/X}$ is.
The [[pushout product axiom]] now follows from the fact
that in overcategories pushouts are reflected in the underlying category $\mathcal{C}$ (by [this Prop.](over+category#LimitsAndColimits)).
The [[unit axiom]] follows from the unit axiom of $\mathcal{C}$
using the fact that tensorings are computed in $\mathcal{C}$.

=--

### As presentations for over (∞,1)-categories

When restricted to [[fibrant objects]], the operation of forming the model structure on an overcategory presents the operation of forming the [[over (∞,1)-category]] of an [[(∞,1)-category]].

More explicitly, for any [[model category]] $\mathcal{C}$, let

\[
  \label{LocalizationFunctor}
  \gamma \colon \mathcal{C} \longrightarrow L_W \mathcal{C}
\]

denote the [[localization of an (infinity,1)-category|localization]] (as an [[(∞,1)-category]]) inverting the [[weak equivalences]] (as e.g. given by [[simplicial localization]]; see also the [[model structure on relative categories]]). Then:

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

If $\mathcal{C}$ is a [[model category]] and $X \in \mathcal{C}$ is [[fibrant object|fibrant]], then $\gamma$ (eq:LocalizationFunctor) induces an  [[(∞,1)-functor]] $\mathcal{C}/X \to L_W(\mathcal{C})/\gamma(X)$, which in turn induces an [[equivalence of (∞,1)-categories]]

$$
  L_W(\mathcal{C}/X)
    \overset{\simeq}{\longrightarrow}
  L_W(\mathcal{C})/\gamma(X)
  \,.
$$

=--

This main result is corollary 7.6.13 of [Cisinski 20](#Cisinski20). Model categories are (∞,1)-categories with weak equivalences and fibrations as defined in [Cisinski Def. 7.4.12](#Cisinski20).

We spell out a proof for the special case that $\mathcal{C}$ carries the [[extra structure]] of a [[simplicial model category]] (this proof was written [in 2011](https://ncatlab.org/nlab/revision/model+structure+on+an+over+category/7) when no comparable statement seemed to be available in the literature):

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

If $\mathcal{C}$ is a [[simplicial model category]] and $X \in \mathcal{C}$ is [[fibrant object|fibrant]], then the [[overcategory]] $\mathcal{C}/X$ with the above slice model structure is a [[presentable (infinity,1)-category|presentation]] of the
[[over-(∞,1)-category]] $L_W \mathcal{C} / \gamma(X)$:
we have an [[equivalence of (∞,1)-categories]]

$$
  L_W(\mathcal{C}/X) \simeq  (L_W\mathcal{C}) / \gamma(X)
  \,.
$$

=--

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

We write equivalently $(-)^\circ \coloneqq L_W(-)$.

It is clear that we have an [[essentially surjective (∞,1)-functor]] $\mathcal{C}^\circ/X \to (\mathcal{C}/X)^\circ$. What has to be shown is that this is a [[full and faithful (∞,1)-functor]] in that it is an [[equivalence in an (∞,1)-category|equivalence]] on all [[hom-object|hom]]-[[∞-groupoid]]s $\mathcal{C}^\circ/X(a,b) \simeq (\mathcal{C}/X)^\circ(a,b)$.

To see this, notice that the hom-space in an [[over-(∞,1)-category]] $\mathcal{C}^\circ/X$ between objects $a \colon A \to X$ and $b \colon B \to X$ is given (as discussed there) by the  [[(∞,1)-pullback]]

$$
  \array{
    \mathcal{C}^\circ/X(A \stackrel{a}{\to} X, B \stackrel{b}{\to} X)
     &\to& \mathcal{C}^\circ(A,B)
    \\
    \big\downarrow
      &&
    \big\downarrow^{\mathrlap{b_*}}
    \\
    {*} &\stackrel{a}{\to}& \mathcal{C}^\circ(A,X)
  }
$$

in [[∞Grpd]].

Let $A \in C$ be a cofibrant representative and $b \colon B \to X$ be a
fibration representative in $C$ (which always exists) of the objects of these names in $C^\circ$, respectively. In terms of these we have a cofibration

$$
  \array{
    \emptyset &&\hookrightarrow&& A
    \\
    & \searrow && \swarrow_{\mathrlap{a}}
    \\
    && X
  }
$$

in $\mathcal{C}/X$, exhibiting $a$ as a cofibrant object of $\mathcal{C}/X$;  and a fibration

$$
  \array{
    B &&\stackrel{b}{\to}&& X
    \\
    & {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{Id}}
    \\
    && X
  }
$$

in $\mathcal{C}/X$, exhibiting $b$ as a fibrant object in $\mathcal{C}/X$.

Moreover, the diagram in [[sSet]] given by

$$
  \array{
    \mathcal{C}/X(a, b) &\longrightarrow& \mathcal{C}(A,B)
    \\
    \big\downarrow && \big\downarrow^{\mathrlap{b_*}}
    \\
    {*} &\overset{a}{\longrightarrow}& \mathcal{C}(A,X)
  }
$$

is

1. a [[pullback]] diagram in [[sSet]] (by the definition of morphism in an ordinary [[overcategory]]);

1. a [[homotopy pullback]] in the [[model structure on simplicial sets]], because  by the [[pullback power axiom]] on the [[sSet]]${}_{Quillen}$ [[enriched model category]] $C$ and the above (co)fibrancy assumptions, all objects are [[Kan complexes]] and the right vertical morphism is a [[Kan fibration]];

1. has in the top left the correct [[derived hom-space]] in $C/X$ (since $a$ is cofibrant and $b$ fibrant).

This means that this correct hom-space $\mathcal{C}/X(a,b) \simeq (\mathcal{C}/X)^\circ(a,b) \in sSet$ is indeed a model for $\mathcal{C}^\circ/X(a,b) \in \infty Grpd$.

=--

### Quillen adjunctions
 {#PropertiesQuillenAdjunctions}

#### Base change Quillen adjunction
 {#BaseChangeQuillenAdjunction}

\begin{proposition}\label{LeftBaseChangeQuillenAdjunction}
**(left base change Quillen adjunction)** \linebreak
  For $\mathcal{C}$ a [[model category]] and $c_1 \xrightarrow{f} c_2$ any [[morphism]] in $\mathcal{C}$, the left [[base change]] adjunction $(f_! \dashv f^\ast)$ along $f$ (where $f_!$ is [[postcomposition]] with and $f^\ast$ is [[pullback]] along $f$) is a [[Quillen adjunction]] between the slice model structures (from Prop. \ref{SliceModelStructure}):
  $$
    \mathcal{C}_{/c_1}
      \underoverset
        {\underset{f^\ast}{\longleftarrow}}
        {\overset{f_!}{\longrightarrow}}
        {\;\;\;\;\bot_{\mathrlap{Qu}}\;\;\;\;}
    \mathcal{C}_{/c_2}
    \,.
  $$
\end{proposition}
\begin{proof}
  Since the [[left adjoint]] $f_!$ is the [[postcomposition]] operation, it manifestly preserves the classes of [[underlying]] morphisms, hence in particular preserves the classes of (acyclic) cofibrations in the slice model structure (by Prop. \ref{SliceModelStructure}), hence is a [[left Quillen functor]].
\end{proof}

\begin{prop}\label{LeftBaseChangeQuillenEquivalence}
**([[left base change Quillen equivalence]])** \linebreak

Let $\mathcal{C}$ be a [[model category]],
and $\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T$ be a [[weak equivalence]] in $\mathcal{C}$.

Then the left base change Quillen adjunction along $\phi$ (Prop. \ref{LeftBaseChangeQuillenAdjunction}) is a [[Quillen equivalence]]

$$
  \mathcal{C}_{/T}
    \underoverset
      {\underset{\phi^*}{\longrightarrow}}
      {\overset{\phi_!}{\longleftarrow}}
      {\phantom{{}_{Qu}}\simeq_{Qu}}
  \mathcal{C}_{/S}
$$

if and only if $\phi$ has this property:

$(\ast)$ *The [[pullback]] ([[base change]]) of $\phi$ along any [[fibration]] is still a [[weak equivalence]].*
\end{prop}
Notice that the property $(\ast)$ of $\phi$ is implied as soon as either:

* $\mathcal{C}$ is [[right proper model category|right proper]], or

* $\phi$ is an [[acyclic fibration]], or

* both $S$ and $T$ are [[fibrant objects]]

(for the first this follows by definition; for the second by the fact that $\phi^\ast$ is a [[right Quillen functor]] by Prop. \ref{LeftBaseChangeQuillenAdjunction}; for the third by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback) on recognizing [[homotopy pullbacks]]).
\begin{proof}
Using the characterization of Quillen equivalences by derived adjuncts ([here](Quillen+equivalence#AdjunctOfWeakEquivalence)), the base change adjunction is a Quillen equivalence iff for

* any [[cofibrant object]] $X \to S$ in the slice over $S$ (i.e. $X$ is cofibrant in $\mathcal{C}$)

* and a [[fibrant object]] $p \colon Y \to T$ in the slice over $T$ (i.e. $p$ is a [[fibration]] in $\mathcal{C}$),

we have that

(1) $X \to \phi^*(Y) = S \times_T Y$ is a weak equivalence

iff

(2) $\phi_!(X) \to Y$ is a weak equivalence.

But the latter morphism is the top composite in the following [[commuting diagram]]:

$$
  \array{
    X
      &\longrightarrow&
    S \times_T Y
      &\overset{p^\ast \phi}{\longrightarrow}&
    Y
    \\
    &\searrow&
    \big\downarrow
    &{}^{_{(pb)}}&
    \big\downarrow {}^{\mathrlap{p \in Fib}}
    \\
    && S &\underset{\phi \in \mathrm{W} }{\longrightarrow}& T
  }
$$

Hence the [[two-out-of-three]]-property says that (1) is equivalent to (2) if $p^\ast \phi$ is a weak equivalence.

Conversely, taking $X \to \phi^\ast(X)$ to be a weak equivalence (hence a [[cofibrant resolution]] of $\phi^\ast(X)$), [[two-out-of-three]] implies that if $(\phi_! \dashv \phi^\ast)$ is a Quillen equivalence, then $p^\ast \phi$ is a weak equivalence.
\end{proof}

In particular:

\begin{proposition}\label{RightPropernessAndLeftQuillenBaseChange}
The following are equivalent:

1. $\mathcal{C}$ is a [[right proper model category]].

1. If $f \colon c_1 \to c_2$ is any [[weak equivalence]] in $\mathcal{C}$, then the left base change Quillen adjunction $(f_! \dashv f^\ast)$ (from Prop. \ref{LeftBaseChangeQuillenAdjunction}) is a [[Quillen equivalence]].

\end{proposition}

This is due to [Rezk 02, Prop. 2.5](#Rezk02).

#### Sliced Quillen adjunctions

\begin{proposition}\label{SlicedQuillenAdjunction}
**(slice Quillen adjunctions)** \linebreak
Given a [[Quillen adjunction]]
$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\bot_{\mathrlap{Qu}}\;\;\;\;}
  \mathcal{C}
  \,,
$$

then:

1. for any [[object]] $b \in \mathcal{C}$ the [sliced adjunction](over+category#OnSlices) over $b$ is a  [[Quillen adjunction]] between the corresponding slice model categories (Prop. \ref{SliceModelStructure}):

   \[
     \label{SlicedQuillenAdjointFunctorsOverLb}
     \mathcal{D}_{/L(b)}
     \underoverset
       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
       {\bot_{\mathrlap{Qu}}}
     \mathcal{C}_{/b}
     \mathrlap{\,,}
   \]

1. for any [[object]] $b \in \mathcal{D}$ the [sliced adjunction](over+category#OnSlices) over $b$ is a  [[Quillen adjunction]] between the corresponding slice model categories (Prop. \ref{SliceModelStructure}):

   \[
     \label{SlicedQuillenAdjointFunctorsOverRb}
     \mathcal{D}_{/b}
     \underoverset
       {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
       {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
       {\bot_{\mathrlap{Qu}}}
     \mathcal{C}_{/R(b)}
     \mathrlap{\,,}
   \]

\end{proposition}

(e.g. [Li 2016, Prop. 2.5 (2)](#Li16))

\begin{proof}
Consider the first case:
By the nature of the [sliced adjunction](over+category#OnSlices),
its [[left adjoint]] $L_{/b}$ acts as $L$ on [[underlying]] [[morphisms]].
But since $L$ is assumed to be a [[left Quillen functor]]
and since the (acyclic) cofibrations of the slice model structure are those of [[underlying]] morphisms (Prop. \ref{SliceModelStructure}),
$L_{/b}$ preserves them and is hence itself a [[left Quillen functor]].

The second case is directly analogous: Here it is evident that $R_{/b}$ is a [[right Quillen functor]], since it acts via $R$ on [[underlying]] morphisms, and $R$ is right Quillen by assumption.
\end{proof}

\begin{proposition}\label{SlicedQuillenQuivalences}
**(sliced Quillen equivalences)** \linebreak
Consider a [[Quillen equivalence]]
$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;}
  \mathcal{C}
  \,.
$$

Then:

1. for a [[cofibrant object]] $b \in \mathcal{C}$ such that $L(b)$ is a [[fibrant object]], the sliced Quillen adjunction (eq:SlicedQuillenAdjointFunctorsOverLb) from Prop. \ref{SlicedQuillenAdjunction} is itself a [[Quillen equivalence]]:

$$
  \mathcal{D}_{/L(b)}
  \underoverset
    {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
    {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
    {\simeq_{\mathrlap{Qu}}}
  \mathcal{C}_{/b}
$$

1. for a [[fibrant object]] $b \in \mathcal{D}$, the sliced Quillen adjunction (eq:SlicedQuillenAdjointFunctorsOverRb) from Prop. \ref{SlicedQuillenAdjunction} is itself a [[Quillen equivalence]]:

$$
  \mathcal{D}_{/b}
  \underoverset
    {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}}
    {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}}
    {\simeq_{\mathrlap{Qu}}}
  \mathcal{C}_{/R(b)}
$$

\end{proposition}
(e.g. [Li 16, Prop. 3.1](#Li16))
\begin{proof}
  It is sufficient to check that the [[derived adjunction unit]] and [[derived adjunction counit]] are [[weak equivalences]] (...)
\end{proof}

## Examples

### Pointed objects

\begin{example}\label{ModelCategoriesOfPointedObjects}
**([[model categories of pointed objects]])**
\linebreak
For every [[model category]] $\mathcal{C}$, its [[category of pointed objects]], hence the [[undercategory|category under]] the [[terminal object]] $\mathcal{C}^{\ast/}$ carries the under-category model structure: the canonical *[[model structure on pointed objects]]*.

For instance, the [[classical model structure on pointed topological spaces]] is the model structure on the [[undercategory]] under the point (the [[category of pointed objects]]) of the [[classical model structure on topological spaces]].
\end{example}

\begin{prop}\label{InducedQuilleAdjunctionsOnModelCategoriesOfPointedObjects}
**(induced [[Quillen adjunction]] on [[model categories of pointed objects]])** \linebreak
Given a [[Quillen adjunction]] between [[model categories]]
$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{C}
  \,,
$$
there is induced a [[Quillen adjunction]] between the corresponding [[model categories of pointed objects]]
$$
  \mathcal{D}^{\ast\!/}
    \underoverset
      {\underset{R^{\ast\!/}}{\longrightarrow}}
      {\overset{L^{\ast\!/}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{C}^{\ast\!/}
  \,,
$$
where

* the [[right adjoint]] acts directly as $R$ on the triangular [[commuting diagrams]] in $\mathcal{C}$ that define the morphisms in $\mathcal{C}^{\ast\!/}$;

* the [[left adjoint]] is the [[composition|composite]] of the corresponding direct application of $L$ followed by [[pushout]] along the [[adjunction counit]] $L(\ast) \simeq L \circ R(\ast) \xrightarrow{ \;\epsilon_\ast \; } \ast$ (using that $R(\ast) \simeq \ast$ since [[right adjoints preserve limits]] and hence [[terminal objects]]):

  $$
    L^{\ast\!/}
    \;\colon\;
    \mathcal{C}^{\ast\!/}
    \xrightarrow{ \;\; L  \;\; }
    \mathcal{D}^{L(\ast)\!/}
    \;\simeq\;
    \mathcal{D}^{L\circ R(\ast)\!/}
    \xrightarrow{ \;\; (-) \sqcup \epsilon_\ast \;\; }
    \mathcal{D}^{\ast\!/}
    \,.
  $$

\end{prop}
\begin{proof}
  It is fairly straightforward to check this directly (e.g. [Hovey 1999, Prop. 1.3.5](model+structure+on+pointed+objects#Hovey99)), but it is also a special case of Prop. \ref{SlicedQuillenAdjunction} --- to make this explicit, notice that passing to [[opposite categories]] with their [[opposite model structures]] turns the original Quillen adjunction into the [[opposite Quillen adjunction]]:

$$
  \mathcal{C}^{op}
    \underoverset
      {\underset{L^{op}}{\longrightarrow}}
      {\overset{R^{op}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{D}^{op}
  \,.
$$

Now the passage to [[pointed objects]] corresponds to [[slice category|slicing]] (instead of [[coslice category|co-slicing]]), since

\[
  \label{PointedObjectsIsOppositeOfCopointedObjectsInOpposite}
  \mathcal{C}^{\ast\!/}
  \;\;
  \simeq
  \big(
    \mathcal{C}^{op}_{/\ast}
  \big)^{op}
  \,,
  \;\;\;\;\;\;\;
  \text{similarly}
  \;\;\;
  \mathcal{D}^{\ast\!/}
  \;\;
  \simeq
  \big(
    \mathcal{D}^{op}_{/R(\ast)}
  \big)^{op}
  \,,
\]

whence item (1) in Prop. \ref{SlicedQuillenAdjunction} says that there is a Quillen adjunction of the form

$$
  \mathcal{C}^{op}_{/R(\ast)}
    \underoverset
      {\underset{L^{op}_{/\ast}}{\longrightarrow}}
      {\overset{R^{op}_{/\ast}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{D}^{op}_{/\ast}
  \,,
$$

hence with [[opposite Quillen adjunction]] of the required form

$$
  \mathcal{D}^{\ast\!/}
  \simeq
  \big(\mathcal{D}^{op}_{/R(\ast)}\big)^{op}
    \underoverset
      {\underset{
        R^{\ast\!/} \,\coloneqq\, \big( R^{op}_{/\ast} \big)^{op}
      }{\longrightarrow}}
      {\overset{
        L^{\ast\!/} \,\coloneqq\, \big( L^{op}_{/\ast} \big)^{op}
      }{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \big(\mathcal{C}^{op}_{/\ast}\big)^{op}
  \simeq
  \mathcal{C}^{\ast\!/}
  \,,
$$

with $R^{op}$ acting directly as $R$ on underlying diagrams, and with $L^{op}$ acting as the composite of $L$ following by [[pullback]] -- in $\mathcal{C}^{op}$ -- along the [[adjunction unit]] of $(R^{op} \dashv L^{op})$. Since the component morphism of the [[adjunction unit|unit]] of the [[opposite adjunction]] $(R^{op} \dashv L^{op})$ is that of the adjunction unit of $(L \dashv R)$, and since [[pullback]] in an [[opposite category]] is [[pushout]] in the original category, this implies the claim.
\end{proof}

### Slices over simplicial sets
 {#SlicesOverSimplicialSets}

\begin{example}\label{BorelModelStructure}
**([[Borel model structure]])**
\linebreak
  For $G \,\in\, Grp(sSet) $ a [[simplicial group]] and $\overline{W}G \,\in\, sSet$ its [[simplicial classifying space]], the slice model structure (Prop. \ref{SliceModelStructure}) over $\overline{W}G$ of the [[classical model structure on simplicial sets]] is [[Quillen equivalence|Quillen equivalent]] to the [[Borel model structure]] of $G$-[[actions]] (see [this Prop.](Borel+model+structure#{QuillenEquivalenceToSliceOverSimplicialClassifyingSpace)):
$$
    G Act\big(sSet_{Qu}\big)_{proj}
      \underoverset
        {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}}
        {\overset{ (-) \times_{\overline{W}G} W G  }{\longleftarrow}}
        {\bot}
    \big(sSet_{Qu}\big)_{/\overline{W}G}
$$
For more on this see also at *[[infinity-action|$\infty$-action]]*.
\end{example}

\begin{lemma}\label{SimplicialWeakEquivalencesInSliceDetectedFiberwise}
  For any [[simplicial set]] $B \,\in$ [[sSet]] and any [[pair]] of [[Kan fibrations]] $X \underoverset{\in Fib}{p}{\longrightarrow} B$ and $X' \underoverset{\in Fib}{p'}{\longrightarrow} B$, a morphism
$$
  \array{
    X
    &&
    \xrightarrow{\;\; f \;\;}
    &&
    X
    \\
    &
    {}_{\mathllap{p}}\searrow
    &&
    \swarrow{}_{\mathrlap{p'}}
    \\
    && B
  }
  \;\;\;\;\;\;
  \in
  (sSet_{Qu})_{/B}
$$
is a [[simplicial weak homotopy equivalence]] (hence a [[weak equivalence]] in the slice model structure, from Prop. \ref{SliceModelStructure}, over $B$ of the [[classical model structure on simplicial sets]]) if and only if so are all its restrictions (all its [[base changes]] by [[pullback]])
to all ([[homotopy fiber|homotopy]]) [[fibers]] $X_b$

$$
  b^\ast(f)
  \;\colon\;
  X_b \xrightarrow{\;\; f_b\;\;} X'_b
  \,,
$$

for all points $b \,\in\, B_0$.
\end{lemma}
\begin{proof}
  In one direction, assume that $f$ is a weak equivalence.
By Prop. \ref{LeftBaseChangeQuillenEquivalence} the pullback operation $b^\ast$ is a [[right Quillen functor]]. Therefore [[Ken Brown's lemma]] ([here](factorization+lemma#KenBrownLemma)) implies that it preserves weak equivalences between [[fibrant objects]]. Since $p$ and $p'$ are fibrant by assumption, this means that $b^\ast(f)$ is a weak equivalence.

In the other direction,
assume that $b^\ast(f)$ is a weak equivalence for all $b \,\in\, B_0$.
Then for any $x \,\in\, X_0$ let $b \,\coloneqq\, p(x) \,\in\, B_0$ and consider the resulting [[morphism]] of [[homotopy fiber]]-[[diagrams]]:

\begin{tikzcd}
  X_b
  \ar[
    r,
    "{ \mathrm{fib}_b(p) }"
  ]
  \ar[
    d,
    "{ f_b }"{left},
    "{\in \mathrm{W}}"{right}
  ]
  &
  X
  \ar[
    r,
    "{p}"{above}.
    "\in \mathrm{Fib}"{below}
  ]
  \ar[
    d,
    "{ f }"
  ]
  &
  B
  \ar[
    d,-,
    shift left=1pt
  ]
  \ar[
    d,-,
    shift right=1pt
  ]
  \\
  X'_b
  \ar[
    r,
    "{ \mathrm{fib}_b(p') }"
  ]
  &
  X'
  \ar[
    r,
    "{p'}"{above}.
    "\in \mathrm{Fib}"{below}
  ]
  &
  B
\end{tikzcd}

and, in turn, the induced morphim of [[long exact sequences of homotopy groups]], which has the following segments, for all $n \,\in\, \mathbb{N}_+$ (where $x' \,\coloneqq\, f(x)$):

\begin{tikzcd}
  \pi_{n+1}(B,b)
  \ar[r]
  \ar[d,-,shift left=1pt]
  \ar[d,-,shift right=1pt]
  &
  \pi_n(X_b, x)
  \ar[d, "{\sim}"{xshift=-6pt, yshift=-3.7pt, sloped}]
  \ar[r]
  &
  \pi_n(X,x)
  \ar[
    d,
    "{  \pi_n(f,x)  }"
  ]
  \ar[r]
  &
  \pi_n(B,b)
  \ar[r]
  \ar[d,-,shift left=1pt]
  \ar[d,-,shift right=1pt]
  &
  \pi_{n-1}(X_b, x)
  \ar[d, "{\sim}"{xshift=-6pt, yshift=-3.7pt, sloped}]
  \\
  \pi_{n+1}(B,b)
  \ar[r]
  &
  \pi_n(X'_b, x')
  \ar[r]
  &
  \pi_n(X',x')
  \ar[r]
  &
  \pi_n(B,b)
  \ar[r]
  &
  \pi_{n-1}(X_b, x)
\end{tikzcd}

Now the ([non-abelian](five+lemma#InHomologicalCategories)) [[five lemma]] implies that $\pi_n(f,x)$ is an [[isomorphism]], for all $n \in \mathbb{N}_+$ and all $x \in X$.

It only remains to see that $\pi_0(f)$ is an isomorphism.
This follows by the same argument after replacing $B$ by the [[connected components]] $\tilde B \xhookrightarrow{\;} B = B \sqcup (B \setminus \tilde B)$ which are, under $\pi_0$, in the image of $p$. This yields a morphism of exact sequences of the above form by replacing the rightmost item by the [[singleton]]; and the conclusion follows.
\end{proof}

## Related concepts

* [[parametrized homotopy theory]]

* [[over-category]]

  * [[slice category]]

  * [[under category]]

  * [[over topos]]

* [[over (∞,1)-category]],

  * **model structure on an over-category**

  * [[over-(∞,1)-topos]]

* [[opposite model structure]]

## References

* {#Hirschhorn02} [[Philip Hirschhorn]], Theorem 7.6.5 of: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* {#Hirschhorn21} [[Philip Hirschhorn]], *Overcategories and undercategories of model categories*, J. Homotopy Relat. Struct. **16**  (2021) 753–768 [[arXiv:1507.01624](https://arxiv.org/abs/1507.01624), [doi:10.1007/s40062-021-00294-4](https://doi.org/10.1007/s40062-021-00294-4)]

* {#MayPonto12} [[Peter May]], [[Kate Ponto]], Thm. 15.3.6 in: _[[More concise algebraic topology]]_,   University of Chicago Press (2012) ([ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

* {#Li16} Zhi-Wei Li, *A note on the model (co-)slice categories*, Chinese Annals of Mathematics, Series B volume 37, pages 95–102 (2016) ([arXiv:1402.2033](https://arxiv.org/abs/1402.2033), [doi:10.1007/s11401-015-0955-z](https://doi.org/10.1007/s11401-015-0955-z))

See also:

* {#Rezk02} [[Charles Rezk]], _Every homotopy theory of simplicial algebras admits a proper model_, Topology and its Applications, Volume 119, Issue 1, 2002, Pages 65-94, (<a href="https://doi.org/10.1016/S0166-8641(01)00057-8">doi:10.1016/S0166-8641(01)00057-8</a>, [arXiv:math/0003065](http://arxiv.org/abs/math/0003065))

* {#Cisinski20} [[Denis-Charles Cisinski]], _Higher category theory and homotopical algebra_, 2020 ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

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