S-module

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#Contents#
* table of contents
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## Idea
 {#Idea}

*S-modules* is the name (due to [EKMM 97](#EKMM97)) of one model of [[highly structured spectra]] that supports a [[symmetric monoidal smash product of spectra]].

Here "$S$" stands for the *[[sphere spectrum]]* (other traditional notation being "$Q S^0$" for the [[suspension spectrum]] $Q$ of the [[0-sphere]]) regarded as a [[ring spectrum]]. Namely, in [[categorification|homotopification]] of how general [[abelian groups]] may equivalently be understood as [[integers|$\mathbb{Z}$]]-[[modules]], so general [[spectra]] may (once their [[brave new algebra|brave new algebraic]] [[stable homotopy theory]] is set up) equivalently be understood as [[module spectra]] over the [[sphere spectrum]].

Concretely, the [[1-category]] of _S-modules_ as set up in [EKMM 97](#EKMM97) is a *[[presentable (∞,1)-category|presentation]]* (see at *[[model structure on S-modules]]*) of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the _[[model structure on symmetric spectra]]_. This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] S-modules.

Other presentations sharing this property are _[[symmetric spectra]]_ and _[[orthogonal spectra]]_.

## Construction

The construction of [[S-modules]] by ([EKMM 97](#EKMM97)) begins with the notion of [[coordinate-free spectrum|coordinate free Lewis-May spectra]].  Using the [[linear isometries operad]], one can construct a [[monad]] $\mathbb{L}$ on the category $\mathcal{S}$ of such spectra, and the category of $\mathbb{L}$-algebras is a well-behaved model for the stable homotopy category, and moreover admits a smash product which is associative up to isomorphism, but unital only up to weak equivalence.  However, the subcategory of the $\mathbb{L}$-algebras for which the unit transformations are isomorphisms is again a well-behaved model for $Ho \mathbb{S}$, which is moreover symmetric monoidal.

Since the unit transformation is of the form $S\wedge E \to E$, where $S$ is the [[sphere spectrum]], and this map looks like the action of a ring on a module, the objects of this subcategory are called **$S$-modules** and the category is called $Mod_S$.  The intuition is that just as an abelian group is a [[module]] over the archetypical ring $\mathbb{Z}$ of [[integer|integers]], a spectrum should be regarded as a module over the archetypal ring spectrum, namely the sphere spectrum.

Similarly, just as an ordinary [[ring]] is a [[monoid]] in the category $Mod_\mathbb{Z}$ of $\mathbb{Z}$-[[module]]s, i.e. a $\mathbb{Z}$-algebra, an $A_\infty$ or $E_\infty$ ring spectrum is a (possibly commutative) monoid in the category of $S$-modules, and thus referred to as an **$S$-algebra**.  More generally, for any $A_\infty$-[[A-infinity-ring|ring spectrum]] $R$, there is a notion of $R$-[[module]] spectra forming a category $Mod_R$, which in turn carries an associative and commutative smash product $\wedge_R$ and a [[model category]] structure on $Mod_R$ such that $\wedge_R$ becomes unital in the [[homotopy category]].  All this is such that an $A_\infty$-[[A-infinity-algebra|algebra]] over $R$ is a [[monoid object]] in $(Mod_R, \wedge_R)$.  Similarly $E_\infty$-[[E-infinity-algebra|algebras]] are commutative monoid objects in $(Mod_R, \wedge_R)$.

## Related concepts

[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* **S-module**, [[model structure on S-modules]]

## References

The construction originates in

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], *[[Rings, modules and algebras in stable homotopy theory]]*, AMS 1997 (2014) [[ISBN:978-0-8218-4303-1](https://bookstore.ams.org/surv-47-s), [pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf)]

Review:

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], sections 2 and 3 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

[[!redirects S-modules]]


s-rule

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#### String theory
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#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]], specifically for [[intersecting D-brane models]], what has come to be known as the _s-rule_ is the conjecture that the configuration of [[Dp-D(p+2)-brane bound states]] with the [[D-brane|Dp-branes]] stretching from the [[D-brane|D(p+2)-branes]] to  [[NS5-branes]], can be [[supersymmetry|supersymmetric]] only if at most one D$p$-brane ends on any one D$(p+2)$-brane ([Hanany-Witten 97](#HananyWitten97)).

## Examples

### D3-D5 brane intersections

For [[D3-D5 brane intersections]]:

[Hanay-Witten 97](#HanayWitten97)

### D4-D6 brane intersections

For [[D4-D6 brane intersections]]:

> graphics grabbed from [Fazzi 17](#Fazzi17)

### D6-D8 brane intersections

For [[D6-D8 brane intersections]]:

> graphics grabbed from [Gaiotto-Tomasiello 14](#GaiottoTomasiello14)

> graphics grabbed from [Fazzi 17](#Fazzi17)

> graphics grabbed from [GKSTY 02](NS5-brane#GKSTY02)

## Related concepts

[[!include brane bound states -- table]]

## References

The s-rule is attributed to

* {#HananyWitten97} [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B 492:152-190, 1997 ([arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

which considers [[D3-D5 brane intersections]], speaks of "s-configurations" as those with _more_ than one [[D3-brane]] ending on any [[D5-brane]] and gives plausibility arguments that these s-configurations are not supersymmetric.

Review:

* [[Amit Giveon]], [[David Kutasov]], p. 83 onwards in: _Brane Dynamics and Gauge Theory_, Rev. Mod. Phys. 71:983-1084, 1999 ([arXiv:hep-th/9802067](https://arxiv.org/abs/hep-th/9802067))

Relation to the [[Pauli exclusion principle]]:

* [[Constantin Bachas]], [[Michael Green]], _A Classical Manifestation of the Pauli Exclusion Principle_, JHEP 9801 (1998) 015 ([arXiv:hep-th/9712187](https://arxiv.org/abs/hep-th/9712187))

See also

* Wikipedia, _[Hanany-Witten transition -- The S-rule](https://en.wikipedia.org/wiki/Hanany%E2%80%93Witten_transition#The_S-rule)_

For [[D2-D4 brane intersections]]:

* Minoru Eto, Youichi Isozumi, Muneto Nitta, Keisuke Ohashi, Kazutoshi Ohta, Norisuke Sakai, Section 3 of: _D-brane Construction for Non-Abelian Walls_, Phys. Rev. D71 (2005) 125006 ([arXiv:hep-th/0412024](https://arxiv.org/abs/hep-th/0412024))

For [[D3-D5 brane intersections]]:

* [[Constantin Bachas]], [[Michael Green]], Adam Schwimmer, Section 2.3 of: _$(8,0)$ Quantum mechanics and symmetry enhancement in type I' superstrings_, JHEP 9801 :006, 1998 ([arXiv:hep-th/9712086](https://arxiv.org/abs/hep-th/9712086))

* Akikazu Hashimoto, Peter Ouyang, Masahito Yamazaki, Section 3.6 of: _Boundaries and Defects of N=4 SYM with 4 Supercharges, Part II: Brane Constructions and 3d N=2 Field Theories_, JHEP 1410 : 108, 2014 ([arXiv:1406.5501](https://arxiv.org/abs/1406.5501))

* Federico Carta, Hirotaka Hayashi, _Hilbert series and mixed branches of $T[SU(N)]$ theory_, JHEP 02 (2017) 037 ([arXiv:1609.08034](https://arxiv.org/abs/1609.08034))

For [[D4-D6 brane intersections]]:

* {#OoguriVafa97} [[Hirosi Ooguri]], [[Cumrun Vafa]], Section 6.2 of: _Geometry of $N = 1$ Dualities in Four Dimensions_, Nucl. Phys. B 500:62-74, 1997 ([arXiv:hep-th/9702180](https://arxiv.org/abs/hep-th/9702180))

* [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 3 of: _Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane_, Adv. Theor. Math. Phys.1:1-52, 1998 ([arXiv:hep-th/9706082](https://arxiv.org/abs/hep-th/9706082))

For [[D6-D8 brane intersections]]:

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], p. 24-25 in: _Holography for $(1,0)$ theories in six dimensions_, JHEP 12 (2014) 003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

Review includes:

* {#Fazzi17} [[Marco Fazzi]], around Figure 3.5 and Figure 3.13 in: _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

[[!redirects S-rule]]


S-theory

#Contents#
* table of contents
{:toc}

## Idea

Aspects of [[duality]] in [[stable homotopy theory]] have been referred to as _S-theory_ ([Spanier](#Spanier))

## Related concepts

* [[Spanier-Whitehead duality]]

## References

* [[Edwin Spanier]], _Duality and S-theory_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/spanier.pdf))
 {#Spanier}

[[!redirects S theory]]


S. C. Steenkamp

* [personal page](https://www.cl.cam.ac.uk/~scs62/)

## Selected writings

On [[quotient types]], [[inductive types]] and [[inductive-inductive types]]:

* [[Marcelo P. Fiore]], [[Andrew M. Pitts]], [[S. C. Steenkamp]], *Quotients, inductive types, and quotient inductive types*, Logical Methods in Computer Science **18** 2 (2022) lmcs:7076  [[arXiv:2101.02994](https://arxiv.org/abs/2101.02994), doi:10.46298/lmcs-18(2:15)2022]

category: people


S. Carnahan

* [website](http://member.ipmu.jp/scott.carnahan/)

category: people


S. L. Woronowicz

__Stanisław Lech Woronowicz__ is a mathematical physicist from Warsaw, a specialist in operator algebras and one of the founders of [[quantum group]] theory.

* [homepage](http://www.fuw.edu.pl/~slworono)
* [articles](http://www.fuw.edu.pl/~slworono/Prace.html)
* [Wikipedia entry](http://en.wikipedia.org/wiki/S._L._Woronowicz)
* _Compact matrix pseudogroups_, Commun. Math. Phys. __111__, 613--665 (1987) [euclid](http://projecteuclid.org/euclid.cmp/1104159726) [MR0901157](http://www.ams.org/mathscinet-getitem?mr=0901157)
* _Differential calculus on compact matrix pseudogroups (quantum groups)_ Commun. Math. Phys. __122__, 125--170 (1989) [euclid](http://projecteuclid.org/euclid.cmp/1104178320) [MR0994499](http://www.ams.org/mathscinet-getitem?mr=0994499)

* _Tannaka-Krein duality for compact matrix pseudogroups_, Twisted SU(N) groups, Inventiones Mathematicae __93__, No. 1, 35-76, [doi](http://dx.doi.org/10.1007/BF01393687)

* _Unbounded elements affiliated with $C^*$-algebras and non-compact quantum groups_, Commun. Math. Phys. __136__, 399--432 (1991) [euclid](http://projecteuclid.org/euclid.cmp/1104202358) [MR1096123](http://www.ams.org/mathscinet-getitem?mr=1096123)
* _$C^\ast$-algebras generated by unbounded elements_,  [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/GENER.pdf)
* [[S. L. Woronowicz]], K. Napiórkowski, _Operator theory in $C^\ast$-framework_,  Reports on Mathematical Physics __31__, Issue 3 (1992), 353-371,
doi, [pdf](http:www.fuw.edu.pl/~slworono/PDF-y/OP.pdf)
* W. Pusz, S. L. Woronowicz, _Passive states and KMS states for general quantum systems_, Commun. Math. Phys. __58__ (1978), no. 3, 273–290, [MR471796](http://www.ams.org/mathscinet-getitem?mr=471796), [euclid](http://projecteuclid.org/euclid.cmp/1103901491)
* _On the existence of KMS states_, Letters on Math. Phys. __10__, no 1, 29--31 (1985) [MR86m:46066](http://www.ams.org/mathscinet-getitem?mr=796996), [euclid](http://projecteuclid.org/euclid.cmp/1103901491), [doi](http://dx.doi.org/10.1007/BF00704583)
* W. Pusz, S. L. Woronowicz, _Twisted second quantization_, Rep. Math. Phys. __27__ (1989), no. 2, 231–257, [MR93c:81082](http://www.ams.org/mathscinet-getitem?mr=1067498), doi
* _A noncommutative Gelfand Naimark theorem_, Journal of Operator Theory 8, 361--389 (1982)

### Related $n$Lab entries

* [[quantum group]]
* [[bicovariant differential calculus]]
* [[Operator theory in the C*-algebra framework]]

category: people

[[!redirects Stanisław Lech Woronowicz]]
[[!redirects Stanisław L. Woronowicz]]
[[!redirects Stanislaw Woronowicz]]
[[!redirects Stanisław Woronowicz]]


S4 modal logic

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#### Type theory
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#### Modalities
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[[!include modalities - contents]]
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# Contents #
* table of contents
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## Idea

The flavor of (classical) _[[modal logic]]_ called $S4$ is (classical) [[propositional logic]] equipped with a single [[modality]] usually written "$\Box$" subject to the rules that for all [[propositions]] $p, q \colon Prop$ we have

1. $\Box K \colon \Box(p \to q) \to (\Box p \to \Box q)$ ([[K modal logic]])

1. $\Box T \colon \Box p \to p$ ([[T modal logic]])

1. $\Box 4 \colon \Box p \to \Box \Box p$. (S4 modal logic)

1. (in addition in [[S5 modal logic]] one adds: $\lozenge p \to \Box \lozenge p$).

Traditionally the canonical interpretation of the Box operator is that $\Box p$ is the statement that "$p$ is _[[necessity|necessarily]]_ true." (The [[de Morgan duality|de Morgan dual]] to the necessity modality is the "[[possibility]]" modality.) Then the interpretation of $\Box $ is that "If $p$ is necessarily true then it is necessarily necessarily true."
S4 modal logic appears in many [[temporal logics]].

If instead of a single Box operator one considers $n \in \mathbb{N}$ box operators $\Box_i$ of this form, the resulting modal logic is denoted $S4(n)$. Here $\Box_i p$ is sometimes interpreted as "the $i$th agent knows $p$."

In intuitionistic (or constructive) Modal Logic, one does not expect the possibility modality to be dual to the necessity modality, necessarily. [[Alex Simpson]] has written in his thesis the most used constructive modal logic system, but other systems exist.

## Properties

### Relation to Kripke frames

The [[models]] for [[T modal logic]] corresponded to [[Kripke frames]] where each [[relation]] $R_i$ is [[reflexive relation|reflexive]].
For $S4$ modal logic they are furthermore [[transitive relation|transitive]].

+-- {: .un_theorem}
###### Theorem (Soundness of $S4_{(m)}$)

$S4_{(m)}\vdash \phi \Rightarrow \mathcal{S}4(m)\models \phi.$
=--

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

(We show this for $m = 1$.)

Suppose that $\mathfrak{M}= ((W,R),V)$ where $R$ is a reflexive transitive relation on $W$.  We have to check that $\mathfrak{M}\models (4)$.

Suppose $\mathfrak{M},w\models K p$, then, for every  $t$ with $R w t$, we have $\mathfrak{M},t\models p$.  Now suppose we seek to check that $\mathfrak{M},w\models K K p$ so we have $t$ with $R w t$ and want $\mathfrak{M},t\models K p$, so we look at all $u$ with $R t u$ and have to see if $\mathfrak{M},u\models p$, but as $R w t$ and $R t u$ hold then $R w u$ holds, since $R$ is transitive, and we then _know_ that $\mathfrak{M},u\models p$.
=--

## Related concepts

* [[modal logic]], [[S5 modal logic]]

* [[epistemic modal logic]], [[necessity and possibility]]

* [[K modal logic]]

* [[T modal logic]]

* [[Kripke frame]]

* [[Löb's axiom]]

## References

The terminology "S4" in modal logic originates with:

* [[Clarence I. Lewis]], [[Cooper H. Langford]], App. II, p 501 of: *Symbolic Logic* (1932), Dover (2000) [App. II [pdf](https://www.hist-analytic.com/Lewisstrict.pdf)]

The "[[possible worlds]]" [[Kripke semantics]] of S4 originates with:

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

See also:

* {#Kripke80} [[Saul Kripke]], _[[Naming and Necessity]]_ (1980)

A [[natural deduction]]-formulation and making explicit the [[modal operator]] as a [[comonad]]:

* {#BiermanDePaiva96} [[Gavin M. Bierman]], [[Valeria de Paiva]], *Intuitionistic necessity revisited*, School of Computer Science research reports-University of Birmingham CSR (1996) [[researchgate](https://www.researchgate.net/publication/2810611_Intuitionistic_Necessity_Revisited), [[Bierman-dePaiva-NecessityRevisited.pdf:file]]]

* [[Gavin M. Bierman]], [[Valeria de Paiva]], _On an Intuitionistic Modal Logic_ Studia Logica **65** (2000) 383–416 [[doi:10.1023/A:1005291931660](https://doi.org/10.1023/A:1005291931660), [pdf] (https://www.researchgate.net/profile/Valeria-De-Paiva/publication/226515897_On_An_Intuitionistic_Modal_Logic/links/00b4951ed416906ccc000000/On-An-Intuitionistic-Modal-Logic.pdf)]

* {#Kobayashi97} [[Satoshi Kobayashi]], *Monad as modality*, Theoretical Computer Science **175** 1 (1997) 29-74 [doi:10.1016/S0304-3975(96)00169-7]

* Natasha Alechina, Michael Mendler, [[Valeria de Paiva]], Eike Ritter: *Categorical and Kripke Semantics for Constructive S4 Modal Logic*, in *Computer Science Logic. CSL 2001* Lecture Notes in Computer Science **2142**, Springer (2001) 292 [[doi:10.1007/3-540-44802-0_21](https://doi.org/10.1007/3-540-44802-0_21)]

See also:

* [[Alex Simpson]], _The proof theory and semantics of intuitionistic modal logic_ (1994) [[pdf] (https://era.ed.ac.uk/handle/1842/407), [handle:1842/407](http://hdl.handle.net/1842/407)]

More on $S4$, $S5_{(m)}$ and their applications in Artificial Intelligence can be found in

* J.- J. Ch. Meyer and W. Van der Hoek, _Epistemic logic for AI and Computer Science_, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995.

General books on modal logics which treat  these logics thoroughly in the general context include

* [[Patrick Blackburn]], M. de Rijke and [[Yde Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

* [[Marcus Kracht]], _Tools and Techniques in Modal Logic,_ Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.

[[!redirects the logic S4(m)]]
[[!redirects S4(m)]]
[[!redirects S4 modal logic]]

[[!redirects S4]]


S5 modal logic

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#### $(0,1)$-Category theory
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#### Type theory
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[[!include type theory - contents]]
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#### Modalities
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[[!include modalities - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[modal logic]] the moniker "S5" refers to the logic obtained from [[S4 modal logic]] by adjoining the [[axiom scheme]] that there is the [[implication]]

$$
  \lozenge p \to \Box \lozenge p
  \,.
$$

In the [[possible worlds semantics]] for modal logic, S5 corresponds  precisely to those [[Kripke frames]] $\big(W \colon Set, \, R \colon W \times W \to Prop\big)$ where the [[relation]] $R$ is an *[[equivalence relation]]* [[Kripke (1959)](#Kripke59) [(1964)](#Kripke63), cf. [Fagin, Halpern, Moses & Vardi (1995) Thm. 3.1.5 (c)](#FaginHalpernMosesVardi95)].

## Via dependent type theory
 {#ViaDependentTypeTheory}

In mild generalization of the discussion at *[necessity & possibility -- Via dependent types](necessity+and+possibility#InFirstOrderLogicAndTypeTheory)*, we may observe that the [[modal operators]] in S5 Kripke semantics

> {#SubtypeNotation} (Here we are writing --- following notation for *[[subtypes]]* --- $P \,\colon\,W \to Prop$ for a $W$-[[dependent type|dependent]] [[proposition]], equivalently a proposition about the elements of $W$, equivalently the [[subset]] of $W$ where the proposition holds.)

are precisely those arising as [[adjoint pairs]] of ([[comonad|co]])[[monads]] induced on the left from a [[base change]] [[adjoint triple]] (the [[categorical semantics of dependent types|categorical semantics]] in [[Set]] of [[dependent pair type]] $\dashv$ [[context extension]] $\dashv$ [[dependent function type]]-[[type formation|formation]]) along [[display maps]] $W \to \Gamma \equiv W_{/R}$ identified with the [[quotient]] [[coprojection]] of the Kripke frame:

(Obvious as this may be, the only existing reflection of this fact in the literature may be a side remark in [Awodey 2006, p. 279](#Awodey06), albeit not relating to dependent type formation.)

\begin{remark}
This suggests that one may equivalently regard [[dependent type theory]] as a universal form of [[epistemic logic|epistemic]] [[modal type theory]], in generalization of how [[modal logics]] may be viewed as just another perspective on ([[fragments]] of) [[first-order logic]] (as in [Blackburn, van Benthem & Wolter (2007) pp. xiii](modal+logic#BlackburnvanBenthemWolter07)): In both cases one switches perspective from type formation by [[base change]] [[adjoint triples]] (of [[quantification]] or dependent (co)product formation, respectively) to the induced [[adjoint pair]] of ([[comonad|co]])[[monads]]. (Another example of a change in perspective of this form occurs when describing [[descent]] as *[[monadic descent]]*).
\end{remark}

\begin{remark}
  \label{HaxagonOfEpistemicEntailments}
  The above (co)monadic formulation of modal logic makes manifest another basic point not usually discussed in the subject, namely that the canonical epistemic entailments

are *[[natural transformations]]*.

This means that the following [[commuting square|squares commute]]. Their [[pasting diagram|pasting]] to the hexagonal [[commuting diagram]] as shown may not have received attention (?):

\end{remark}

\linebreak

## Related concepts

[[!include flavors of modal logics and relations -- table]]

* [[modal logic]], [[S4 modal logic]]

* [[epistemic modal logic]], [[necessity and possibility]]

* [[K modal logic]]

* [[T modal logic]]

* [[Kripke frame]]

* [[Löb's axiom]]

## References

### General

The terminology "S5" in modal logic originates with:

* [[Clarence I. Lewis]], [[Cooper H. Langford]], App. II, p 501 of: *Symbolic Logic* (1932), Dover (2000) [App. II [pdf](https://www.hist-analytic.com/Lewisstrict.pdf)]

The [[geometric model for modal logics|geometric model]] for S5 modal logic via [[Kripke frames]] originates with:

* {#Kripke59} [[Saul A. Kripke]], *A Completeness Theorem in Modal Logic*, The Journal of Symbolic Logic **24** 1 (1959) 1-14 [[doi:10.2307/2964568](https://doi.org/10.2307/2964568), [jstor:2964568](https://www.jstor.org/stable/2964568), [pdf](https://www.filosoficas.unam.mx/~morado/Cursos/17Modal/Kripke1959.pdf)]

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

See also:

* Wikipedia, *[S5 (modal logic)](http://en.wikipedia.org/wiki/S5_%28modal_logic%29)*

A [[natural deduction]]-formulation and making explicit the [[modal operator]] as a [[comonad]]:

* {#BiermanDePaiva96} [[Gavin M. Bierman]], [[Valeria de Paiva]], *Intuitionistic necessity revisited*, School of Computer Science research reports-University of Birmingham CSR (1996) [[researchgate](https://www.researchgate.net/publication/2810611_Intuitionistic_Necessity_Revisited), [[Bierman-dePaiva-NecessityRevisited.pdf:file]]]

* [[Gavin M. Bierman]], [[Valeria de Paiva]], _On an Intuitionistic Modal Logic_ Studia Logica **65** (2000) 383–416 [[doi:10.1023/A:1005291931660](https://doi.org/10.1023/A:1005291931660), [pdf] (https://www.researchgate.net/profile/Valeria-De-Paiva/publication/226515897_On_An_Intuitionistic_Modal_Logic/links/00b4951ed416906ccc000000/On-An-Intuitionistic-Modal-Logic.pdf)]

* {#Kobayashi97} [[Satoshi Kobayashi]], *Monad as modality*, Theoretical Computer Science **175** 1 (1997) 29-74 [doi:10.1016/S0304-3975(96)00169-7]

* Natasha Alechina, Michael Mendler, [[Valeria de Paiva]], Eike Ritter: *Categorical and Kripke Semantics for Constructive S4 Modal Logic*, in *Computer Science Logic. CSL 2001* Lecture Notes in Computer Science **2142**, Springer (2001) 292 [[doi:10.1007/3-540-44802-0_21](https://doi.org/10.1007/3-540-44802-0_21)]

Understanding of S5 [[Kripke semantics]] as [[base change]] along the lines discussed at *[necessity and possibility -- via dependent types](necessity+and+possibility#InFirstOrderLogicAndTypeTheory)*:

* {#Awodey06} [[Steve Awodey]], p. 279 in: *Category theory*, Oxford University Press (2006, 2010) [[doi:10.1093/acprof:oso/9780198568612.001.0001](https://doi.org/10.1093/acprof:oso/9780198568612.001.0001), [ISBN:9780199237180](https://global.oup.com/ukhe/product/category-theory-9780199237180), [pdf](http://englishonlineclub.com/pdf/Category%20Teory%20%5BEnglishOnlineClub.com%5D.pdf)]

[[!include S5 modal logic as epistemic logic -- references]]

[[!redirects S5 modal logics]]

[[!redirects S5]]

[[!redirects S5-modal logic]]
[[!redirects S5-modal logics]]


S5 modal logic as epistemic logic -- references

### S5 modal logic as epistemic logic

Discussion of [[S5 modal logic]] as [[epistemic logic]]:

* [[Robert J. Aumann]], *Interactive epistemology I: Knowledge*, International Journal of Game Theory **28** (1999) 263–300 $[$[doi:10.1007/s001820050111](https://doi.org/10.1007/s001820050111)$]$

> $[$[Ditmarsch, Hoek & Kooi (2008)](#DitmarschHoekKooi08):$]$ "in Aumann’s survey paper on interactive epistemology the reader will immediately recognise the system S5."

* [[Joseph Y. Halpern]], [[Yoram Moses]], §2.3 in: *A guide to completeness and complexity for modal logics of knowledge and belief*, Artificial Intelligence **54** 3 (1992) 319-379 $[$doi:10.1016/0004-3702(92)90049-4$]$

* {#FaginHalpernMosesVardi95} Ronald Fagin, [[Joseph Y. Halpern]], [[Yoram Moses]], Moshe Y. Vardi, *Reasoning About Knowledge*, The MIT Press (1995) $[$[ISBN:9780262562003](https://mitpress.mit.edu/9780262562003/reasoning-about-knowledge/)$]$

> p. 35: "in a precise sense the S5 properties completely characterize our definition of knowledge"

* [[Joseph Y. Halpern]], *Should knowledge entail belief?*, Journal of Philosophical Logic **25** 5 (1996) 483-494  $[$[jstor:30226583](https://www.jstor.org/stable/30226583), [philpapers:HALSKE](https://philpapers.org/rec/HALSKE)$]$

* {#Fitting07} [[Melvin Fitting]], *Logics of Knowlege*, Sec. 9 in: *Modal proof theory*, Ch. 2 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$doi:10.1016/S1570-2464(07)80005-X, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$

> pp. 121: "What standard
logics of knowledge capture is not actual knowledge, but potential knowledge --- what one
is entitled to know. The switch to potential knowledge means we drop all considerations
of complexity $[...]$ It is easy to see that, under such an assumption, a knowledge modality should be a normal modal
operator. But, what else should be required? $[...]$ All these together make a
knowledge operator obey the S5 conditions."

* [[Wiebe van der Hoek]], [[Marc Pauly]], *Epistemic Logic*, Sec 4 in: *Modal logic for games and information*, Ch. 20 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$doi:10.1016/S1570-2464(07)80023-1, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$

> p. 198: "Modal epistemic logic, the logic of knowledge, provides a very natural interpretation to the accessibility relation in Kripke models. For an agent $i$, two worlds $w$ and $v$ are connected (written $R_i w v$), if the agent cannot (epistemically) distinguish them. In other words, we have $R_i v w$ if, according to $i$’s information at $w$, the world might as well be in state $v$, or that $v$ is compatible with i’s information at w. Using this interpretation of access, $R_i$ is obviously an equivalence relation. $[...]$ Thus, we are in the realm of the multi-modal logic $S5_m$."

* {#DitmarschHoekKooi08} [[Hans Ditmarsch]], [[Wiebe Hoek]], Barteld Kooi, *Epistemic Logic*, chapter 2 in: *Dynamic Epistemic Logic*, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) **337**, Springer (2008) $[$[doi:10.1007/978-1-4020-5839-4_2](https://doi.org/10.1007/978-1-4020-5839-4_2), [pdf](http://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Epistemic-logic.pdf)$]$

> p. 11: "The logical system S5 is by far the most popular and accepted epistemic logic"

* [[Dov Samet]], *S5 knowledge without partitions*, Synthese **172** (2010) 145–155 $[$[doi:10.1007/s11229-009-9469-0](https://doi.org/10.1007/s11229-009-9469-0)$]$

* Meghyn Bienvenu, Hélène Fargier, Pierre Marquis, *Knowledge Compilation in the Modal Logic S5*, Proceedings of the AAAI Conference on Artificial Intelligence **24** 1 (2010) $[$[doi:10.1609/aaai.v24i1.7587](https://doi.org/10.1609/aaai.v24i1.7587), [pdf](https://ojs.aaai.org/index.php/AAAI/article/view/7587/7448)$]$

  > p. 1: "Propositional epistemic logic S5 is a well-known modal logic which is suitable for representing and reasoning about the knowledge of a single agent"

* Rasmus Rendsvig, John Symons, *Epistemic Logic*, The Stanford Encyclopedia of Philosophy (2011) $[$[web](https://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=logic-epistemic)$]$

> "[Fagin, Halpern, Moses, and Vardi (1995)](FaginHalpernMosesVardi95) and many others use S5 for knowledge"

* Rachel Boddy, *Epistemic Issues and Group Knowledge*, Amsterdam (2014) $[$[pdf](https://eprints.illc.uva.nl/id/eprint/921/1/MoL-2014-03.text.pdf)$]$

> p. 13: "Formal approaches to epistemology – such as game
theory and computer science – typically assume the S5 conditions for knowledge, which is (partly) explained by the convenient formal properties of the logic. Philosophers typically opt for a weaker notion. Hintikka (1962), for instance, argues that the proper logic for knowledge is the modal system S4"

* Yakoub Salhi and Michael Sioutis, *A Resolution Method for Modal Logic S5*, EPiC Series in Computer Science
**36** (2015) 252–262 $[$[pdf](https://easychair.org/publications/open/3XXQ)$]$

* Ronald de Haan, Iris van de Pol, *On the Computational Complexity of Model Checking for Dynamic Epistemic Logic with S5 Models* $[$[arXiv:1805.09880](https://arxiv.org/abs/1805.09880)$]$


S5(m) modal logic

[[!redirects the logic S5(m)]]

# Contents #
* table of contents
{:toc}

## Idea

This entry is meant to be about the flavor of [[modal logic]] known as [[S5 modal logic]] with several ($m \in \mathbb{N}$) [[modal operators]] and their duals.

## The axioms $B_i$

The axiom denoted $B_i$, (see [Kracht 1999](#Kracht99) for indication of possible reasons) is

* $p \to K_i M_i p$.

The interpretation is the 'if $p$ is true then agent $i$ knows that it is possible that $p$ is true.'

Another axiom that is relevant here is:

## The axioms $(5)_i$

* $\neg K_i p \to K_i \neg K_i p$

Axiom (5) interprets as saying 'If agent $i$ does not know that $p$ holds, then (s)he knows that (s)he does not know'. This is termed 'negative introspection'.

## The logics $S5$ and $S5_{(m)}$

* $S5$ - this logic is obtained from $S4$ by adding the axiom $B$.

* $S5_{(m)}$ starting from $S4_{(m)}$ [[the logic S4(m)]], add, for each $i = 1,\ldots, m$ the axiom $B_i$.

In either case the same result can be obtained by adding in $5$ or $(5)_i$ in place of the corresponding $B$.

## Equivalence frames

With $T_{(m)}$, the models corresponded to frames where each relation $R_i$ was reflexive. With $S4_{(m)}$, the frames needed to be transitive as well.  Here we consider the class $\mathcal{S}5(m)$ of models with [[Kripke frames]], where each $R_i$ is an [[equivalence relation]].  These are sometimes called _equivalence frames_.

\begin{theorem}
**(Soundness of $S5_{(m)}$)**

$S5_{(m)}\vdash \phi \Rightarrow \mathcal{S}5(m)\models \phi.$

\end{theorem}

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

(We show this for $m = 1$.)
We have already shown (here in [[the logic S4(m)]]) that the older axioms $T$ and $(4)$ hold so it remains to show if we have a frame, $\mathfrak{M}= ((W,R),V)$, where $R$ is an equivalence relation on $W$ then $\mathfrak{M}\models B$.

We suppose the we have a state $w$ so that $\mathfrak{W},w \models p$.  Now we need to find out if $\mathfrak{W},w \models K M p$, so we note that

1. $\mathfrak{W},w \models K M p$ if and only if $\forall u$ with $R w u$, $\mathfrak{W},u \models M p$, but

2. that holds if $\forall u$ with $R w u$, there is some $v$  with $R u v$ and $\mathfrak{W},v \models   p$.

However whatever $u$ we have with $R w u$, we have $R u w$ as $R$ is symmetric, and we know, by assumption, that  $\mathfrak{W},w \models p$, so we have what we need.

=--

## References

General books on modal logics which treat  these logics thoroughly in the general context include e

* [[Patrick Blackburn]], M. de Rijke and [[Yde Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

* {#Kracht99} [[Marcus Kracht]], _Tools and Techniques in Modal Logic,_ Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.

Application to artificial intelligence:

* J.- J. Ch. Meyer and W. Van der Hoek, _Epistemic logic for AI and Computer Science_, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995.

[[!redirects the logic S5(m)]]
[[!redirects S5(m)]]


sAb

# Contents
* table of contents
{: toc}

## Idea

The category $sAb$ of [[simplicial abelian groups]]. Equivalently that of [[simplicial objects]] in [[Ab]], and also equivalently that of [[abelian group|abelian]] [[group objects]] in [[sSet]].

## Properties

* By the [[Dold-Kan correspondence]], $sAb$ is [[equivalence of categories|equivalent]] to the [[category of chain complexes]] $Ch_\bullet(Ab)$ (of [[abelian groups]] and in non-negative degree).

## Related concepts

* [[sGrp]]

* [[KanCplx]]

category: category


SAb > history

< [[SAb]]
[[!redirects SAb -- history]]


Sabine Hossenfelder

Sabine Hossenfelder is a physicist in Frankfurt focused on foundation questions, [[dark matter]], and [[MOND]].

* [web](http://sabinehossenfelder.com/index.php)

* Sabine Hossenfelder, _Lost in Math: How Beauty Leads Physics Astray_, Basic Books 2018. German translation: _Das hässliche Universum: Warum unsere Suche nach Schönheit die Physik in die Sackgasse führt_, Fischer 2018

* Sabine Hossenfelder, _Was läuft falsch in der gegenwärtigen Physik?_, talk at University of Stuttgart, 2019 ([video](https://www.youtube.com/watch?v=99hVAu1k6G8))

## On the foundations of quantum mechanics

* J. R. Hance, S. Hossenfelder, _The wave function as a true ensemble_, Proc. R. Soc. A 478: 20210705 (2022)
[doi](https://doi.org/10.1098/rspa.2021.0705)

## On dark matter and MOND

* [[Sabine Hossenfelder]], [[Tobias Mistele]], *Strong lensing with superfluid dark matter*. Journal of Cosmology and Astroparticle Physics, Volume 2019, February 2019. ([doi:10.1088/1475-7516/2019/02/001](https://doi.org/10.1088/1475-7516/2019/02/001), [arXiv:1809.00840](https://arxiv.org/abs/1809.00840))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Galactic mass-to-light ratios with superfluid dark matter*, Astronomy and Astrophysics, Volume 664, Article 40, August 2022 ([doi:10.1051/0004-6361/202243216](https://doi.org/10.1051/0004-6361/202243216), [arXiv:2201.07282](https://arxiv.org/abs/2201.07282))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Aether Scalar Tensor theory confronted with weak lensing data at small accelerations*, ([arXiv:2301.03499](https://arxiv.org/abs/2301.03499))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Superfluid dark matter in tension with weak gravitational lensing data* ([arXiv:2303.08560](https://arxiv.org/abs/2303.08560))

* [[Mariia Khelashvili]], [[Anton Rudakovskyi]], [[Sabine Hossenfelder]], *SPARC galaxies prefer Dark Matter over MOND* ([arXiv:2401.10202](https://arxiv.org/abs/2401.10202))

## Related pages

Related pages include [[empiricism]].

category: people


Sabinin algebra

A __Sabinin algebra__ (earlier defined as a hyperalgebra, not to mix with older notion of a hyperalgebra of an affine algebraic group in prime characteristic, obtained by certain completion) is a vector space with a family of operations...
(under construction)
satisfying axioms
...

Applications: tangent algebra to local smooth multiplicative systems.

* José M. Pérez-Izquierdo, _Algebras, hyperalgebras, nonassociative bialgebras and loops_, Adv. Math. __208__:2 (2007) 834–876 [doi](http://dx.doi.org/10.1016/j.aim.2006.04.001)


Sabrina Pasterski

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

* [Google Scholar page](https://scholar.google.com/citations?user=EunS6JEAAAAJ&hl=en)

## Selected writings

On [[celestial amplitudes]]:

* [[Sabrina Pasterski]], *Lectures on Celestial Amplitudes*, The European Physical Journal C **81** 1062 (2021) [[arXiv:2108.04801](https://arxiv.org/abs/2108.04801), [doi:10.1140/epjc/s10052-021-09846-7](https://doi.org/10.1140/epjc/s10052-021-09846-7)]

* [[Sabrina Pasterski]], *Celestial Amplitudes*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021) [slides: [pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/07/Sabrina-G.-Pasterski.pdf), video: [YT](https://youtu.be/cXSv_SWQsTs)]

* [[Sabrina Pasterski]], *A Chapter on Celestial Holography*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, [[arXiv:2310.04932](https://arxiv.org/abs/2310.04932)]

category: people


Sachdev-Ye-Kitaev model

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

#Contents#
* table of contents
{:toc}

## Idea

From [Maldacena-Stanford 16](#MaldacenaStanford16):

> Studies of [[AdS-CFT|holography]] have been hampered by the lack of a simple solvable model that can capture features of [[Einstein gravity]].  The simplest model, which is a single matrix [[quantum mechanics]], does not appear to lead to [[black holes]]. $\mathcal{N} = 4$ [[N=4 D=4 super Yang-Mills theory|super Yang Mills]] at strong ’t Hooft [[coupling constant|coupling]] certainly leads to [[black holes]], and exact results are known at [[large N limit|large N]] for many anomalous dimensions and some [[vacuum amplitude|vacuum]] [[correlation functions]], but at [[thermal quantum field theory|finite temperature]] the theory is difficult to study.

> A system that reproduces some of the dynamics of [[black holes]] should be [[interacting quantum field theory|interacting]], but we might hope for a model with interactions that are simple enough that it is still reasonable solvable.

> [[Alexei Kitaev|Kitaev]] has proposed to study a quantum mechanical model of $N$ [[Majorana spinor|Majorana]] [[fermions]] [[interaction|interacting]]  with  random  interactions ([Kitaev 15](#Kitaev15)).   It is a simple variant of a model introduced by [[Subir Sachdev|Sachdev]] and Ye ([Sachdev-Ye 93](#SachdevYe93)), which was first  discussed in relation to [[AdS-CFT|holography]] in ([Sachdev 10](#Sachdev10)).

From [Maldacena 18](#Maldacena18):

> The SYK model gives us a glimpse into the interior of an [[extremal black hole]]... That's the feature of SYK that I find most interesting... It is a feature this model has, that I think no other model has

## Definition
 {#Definition}

Let $\mathcal{J}_{i j k l}$ be [[random variables]] with [[expectation values]] $E[\mathcal{J}_{i j k l}]=0$ and $E[\mathcal{J}_{i j k l}^2]=\frac{6J^2}{N^3}$.

The [[Lagrangian density]] defining the SYK model is this:

$$
L = \frac{1}{2} \sum_{i=1}^N \chi_i \partial_t \chi_i - \frac{1}{4!} \sum_{i,j,k,l=1}^N \mathcal{J}_{i j k l} \chi_i \chi_j \chi_k \chi_l
$$

## Related concepts

* [[strange metal]]

* [[Jackiw-Teitelboim gravity]]

* [[AdS-CFT in condensed matter physics]]

* [[AdS-CFT correspondence]]

* [[BFSS matrix model]]

## References

### General

The model is due to

* {#SachdevYe93} [[Subir Sachdev]], Jinwu Ye, _Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet_, Phys. Rev. Lett. 70:3339, 1993 ([arXiv:cond-mat/9212030](https://arxiv.org/abs/cond-mat/9212030))

* {#Kitaev15} [[Alexei Kitaev]], _A simple model of quantum holography_, Talks at KITP, April 7, 2015 and May 27, 2015. ([part I](#http://online.kitp.ucsb.edu/online/entangled15/kitaev/), [part II](http://online.kitp.ucsb.edu/online/entangled15/kitaev2/))

Textbook accounts:

* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], Section 5.11 of: _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))

Further review:

* {#Sachdev18} [[Subir Sachdev]], _The SYK model_, talk at [Aspen Center for Physics](https://www.aspenphys.org/), 2018 ([pdf](http://qpt.physics.harvard.edu/talks/aspen_tutorial18.pdf))

* {#Rosenhaus19} Vladimir Rosenhaus, _An introduction to the SYK model_,  Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 32 ([arXiv:1807.03334](https://arxiv.org/abs/1807.03334), [doi:10.1088/1751-8121/ab2ce1](https://iopscience.iop.org/article/10.1088/1751-8121/ab2ce1))

* Matteo Laudonio, Romain Pascalie, Adrian Tanasa, _Combinatorial aspects of the Sachdev-Ye-Kitaev model_ ([arXiv:2001.11849](https://arxiv.org/abs/2001.11849))

See also

* Wikipedia, _[SYK model of non-Fermi liquids and black holes](https://en.wikipedia.org/wiki/Subir_Sachdev#SYK_model_of_non-Fermi_liquids_and_black_holes)_

Further developments:

* Biao Lian, S. L. Sondhi, [[Zhenbin Yang]], _The chiral SYK model_ ([arXiv:1906.03308](https://arxiv.org/abs/1906.03308))

* Alexandre Streicher, _SYK Correlators for All Energies_ ([arxiv:1911.10171](https://arxiv.org/abs/1911.10171))

Relation to [[random matrix theory]]:

* Yiyang Jia, Jacobus J. M. Verbaarschot, _Spectral Fluctuations in the Sachdev-Ye-Kitaev Model_ ([arXiv:1912.11923](https://arxiv.org/abs/1912.11923))

Discussion of possible realization of the [[SYK-model]] in [[condensed matter physics]]:

* D. I. Pikulin, M. Franz, _Black hole on a chip: proposal for a physical realization of the SYK model in a solid-state system_, Phys. Rev. X 7, 031006 (2017) ([arXiv:1702.04426](https://arxiv.org/abs/1702.04426))

Relation of the SYK-model to the [[strange metals]] via [[AdS/CMT]]:

* [[Subir Sachdev]], *Statistical mechanics of strange metals and black holes* ([arXiv:2205.02285](https://arxiv.org/abs/2205.02285))

[[!include SYK-model and AdS2-CFT1 -- references]]

### In terms of chord diagrams
 {#ReferencesViaWeightSystems}

Discussion of ([[Lie algebra weight system|Lie algebra]]-)[[weight systems]] on [[chord diagrams]] encoding [[SYK model]] [[single trace observables]]:

(for more see at [[weight systems on chord diagrams in physics]])

* Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, _Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$_, JHEP 04 (2018) 146 ([arXiv:1801.02696](https://arxiv.org/abs/1801.02696))

* Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: _Large $N$ expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs_, JHEP 11 (2018) 031 ([arXiv:1806.03271](https://arxiv.org/abs/1806.03271))

* [[Micha Berkooz]], Prithvi Narayan, [[Joan Simón]], _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

following:

* László Erdős, Dominik Schröder, _Phase Transition in the Density of States of Quantum Spin Glasses_, D. Math Phys Anal Geom (2014) 17: 9164 ([arXiv:1407.1552](https://arxiv.org/abs/1407.1552))

which in turn follows

* Philippe Flajolet, Marc Noy, _Analytic Combinatorics of Chord Diagrams_, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), _Formal Power Series and Algebraic Combinatorics_, Springer 2000 ([doi:10.1007/978-3-662-04166-6_17](https://doi.org/10.1007/978-3-662-04166-6_17))

With emphasis on [[AdS-CFT|holographic content]]:

* [[Micha Berkooz]], Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: _Towards a full solution of the large $N$ double-scaled SYK model_, JHEP 03 (2019) 079 ([arxiv:1811.02584](https://arxiv.org/abs/1811.02584))

* Vladimir Narovlansky, Slide 23 (of 28) of: _Towards a Solution of Large $N$ Double-Scaled SYK_, 2019 ([[NarovlanskySYK19.pdf:file]])

and specifically in relation to [[Jackiw-Teitelboim gravity]]:

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))

[[!redirects Sachdev-Ye-Kitaev model]]

[[!redirects SYK model]]
[[!redirects SYK models]]

[[!redirects SYK-model]]
[[!redirects SYK-models]]


Sachin Grover

* [institute page](https://www.hri.res.in/people/Physics/sachingrover)

* [inspire page](https://inspirehep.net/authors/2139519)

## Selected writings

On the [[su(2)]]-[[Wess-Zumino-Witten model]] at [fractional level](Wess-Zumino-Witten+model#FractionalLevelWZWModelReferences):

* [[Sachin Grover]], *Quasi-Characters in $\widehat{\mathfrak{su}}(2)$ Current Algebra at Fractional Levels*, SciPost Phys. Core **6** 068 (2023)  [[arXiv:2208.09037](https://arxiv.org/abs/2208.09037), [doi:10.21468/SciPostPhysCore.6.4.068](https://scipost.org/10.21468/SciPostPhysCore.6.4.068)]

category: people


Sachin J. Valera

member of [[CQTS]]

* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/sachin-valera.html) at NYU Abu Dhabi

* [personal page](https://sites.google.com/view/sachinvalera/home)

## Selected writings

Emphasis that the description of [[anyons]] by [[braided fusion categories]] had remained [[folklore]], and a list of sufficient conditions that would have to be demonstrated:

* [[Sachin J. Valera]], *Fusion Structure from Exchange Symmetry in (2+1)-Dimensions*, Annals of Physics **429** (2021) [
[arXiv:2004.06282](https://arxiv.org/abs/2004.06282), [doi:10.1016/j.aop.2021.168471](https://doi.org/10.1016/j.aop.2021.168471)]

On [[braided fusion categories]] and [[skein relations]]:

* Anup Poudel, [[Sachin J. Valera]], *Skein-Theoretic Methods for Unitary Fusion Categories* [[arXiv:2008.07129](https://arxiv.org/abs/2008.07129)]

These make up the thesis:

* [[Sachin J. Valera]], *Topological Quantum and Skein-Theoretic Aspects of Braided Fusion Categories* [[hdl:11250/2770245](https://hdl.handle.net/11250/2770245)]

Exposition and review:

* [[Sachin Valera]], *A Quick Introduction to the Algebraic Theory of Anyons*, talk at *[[CQTS]] Inituial Researcher Meeting* (Sep 2022) [[[CQTS-InitialResearcherMeeting-Valera-220914.pdf:file]]]

On [[quantum teleportation]] with [[anyons]]:

* {#Valera23} [[Sachin J. Valera]], *Topological Quantum Teleportation and Superdense Coding Without Braiding* [[arXiv:2303.17700](https://arxiv.org/abs/2303.17700)]

category: people

[[!redirects Sachin Valera]]


Sadok Kallel

* [webpage Lille](http://math.univ-lille1.fr/~kallel/)

* [webpage Sharjah](https://www.aus.edu/faculty/dr-sadok-kallel)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18622)

## Selected writings

On the [[rational cohomology]] of [[iterated loop spaces]] of [[n-spheres]] ([[cocycle spaces]] in [[rational Cohomotopy]] on [[n-spheres]]):

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structureof Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

On [[configuration spaces of points]]:

* [[Sadok Kallel]], _Spaces of particles on manifolds and Generalized Poincaré Dualities_, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 ([doi:10.1093/qjmath/52.1.45](https://doi.org/10.1093/qjmath/52.1.45))

* [[Sadok Kallel]], Ines Saihi, _Homotopy Groups of Diagonal Complements_, Algebr. Geom. Topol. 16 (2016) 2949-2980 ([arXiv:1306.6272](https://arxiv.org/abs/1306.6272))

On [[graph-indexed configuration spaces of points]]:

* [[Sadok Kallel]], *The Homotopy Type of Graph Configuration Spaces*, [talk at](Center+for+Quantum+and+Topological+Systems#KallelOct2023) [[CQTS]] (Oct 2023) [slides:[[Kallel-GraphConfiguration.pdf:file]], [YT](https://youtu.be/eNdPBkjA-eQ)]

## Related entries

* [[configuration space of points]]

* [[iterated loop space]]

* [[rational Cohomotopy]]

category: people


Sadri Hassani

* [Institute page](https://physics.illinois.edu/people/directory/profile/hassani)

## Selected writings

On [[mathematical physics]]:

* {#Hassani99} [[Sadri Hassani]], *Mathematical Physics -- A Modern Introduction to Its Foundations* Springer 1999, 2002, 20013 ([ISBN:9783319011943](https://www.springer.com/de/book/9783319011943), [pdf](http://202.38.64.11/~jmy/documents/ebooks/Hassani%20Mathematical%20Physics%20A%20Modem%20Introduction%20to%20Its%20Foundations%20-%20S.%20Hassani%20%5B0-387-98579-4%5D.pdf))

category: people


Sage

#Contents#
* table of contents
{:toc}

## Idea

__Sage__ (or SageMath) is an open-source _mathematics_ software package. It is interoperable with SINGULAR which comes in a standard package. It uses a version of _Python_ for the user interface language. It is possible to have a local installation, or use it via an account on the cloud (the latter from a private company called CoCalc and via the SageCell server).

## Related entries

* [[list of mathematics software]]

  * [[Mathematica]]

  * [[geogebra]]

## References

* [sagemath.org](https://www.sagemath.org)

* [Sage](http://www.azimuthproject.org/azimuth/show/Sage) at Azimuth Project

* Gregg Musiker, Christian Stump, _A compendium on the cluster algebra and quiver package_, Séminaire Lotharingien de Combinatoire 65, B65d, 2011, 67 pages [link](http://www.emis.de/journals/SLC/wpapers/s65mustump.html)

* [CoCalc](https://cocalc.com), see also [Wikipedia](https://en.wikipedia.org/wiki/CoCalc)

* [SageCell](https://sagecell.sagemath.org/)

*  A. Casamayou, N. Cohen, G. Connan, T. Dumont, L. Fousse, F. Maltey, M. Meulien, M. Mezzarobba, C. Pernet, N. M. Thiéry, P. Zimmermann, _Calcul mathématique avec Sage_, [pdf](http://dl.lateralis.org/public/sagebook/sagebook-web-20130530.pdf) 468 pp. (2013) ISBN 9781481191043

category: reference
category: software

[[!redirects sage]]
[[!redirects Sage]]

[[!redirects SageMath]]


Saharon Shelah

__Saharon Shelah__ is a leading [[model theory|model theorist]]. He developed the classification theory, introduced [[stability theory]], [[abstract elementary class]]es, and [[Galois type]]s in particular. He is also the creator of [[pcf theory]], among other things.

Shelah's articles are numerated and often referred to according to this numeration.

* [wikipedia](http://en.wikipedia.org/wiki/Saharon_Shelah)
* [Shelah archive](http://shelah.logic.at)
* [Shelah's books](http://shelah.logic.at/books.html)

__Shelah's categoricity conjecture.__ Let $T$ be a countable theory in $L_{\omega_1,\omega}$ (cf. [[infinitary logic]]). If there exists $\lambda\geq\beth_{\omega_1}$ such that the number of isomorphism classes of models in cardinality $\lambda$ is $I(\lambda,T) = 1$ then $I(\mu,T) = 1$ holds for every
$\mu\geq\beth_{\omega_1}$. More generally, let  $K = (K,\lt)$ be an [[abstract elementary class]]. If there exists $\lambda\geq|2^{LS(K)}|$ such that the number of isomorphism classes of models in cardinality $\lambda$ is $I(\lambda,K) = 1$ then $I(\mu,K) = 1$ holds for every
$\mu\geq|2^{LS(K)}|$.

[[!redirects Shelah]]
[[!redirects S. Shelah]]


Sakura Schäfer-Nameki

[[!redirects Sakura Schafer-Nameki]]

## Selected writings

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation operator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schäfer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

On [[D-branes]] in [[Gepner models]] via [[boundary conformal field theory]]:

* {#BraunSchaeferNameki05} [[Volker Braun]], [[Sakura Schäfer-Nameki]], _D-Brane Charges in Gepner Models_, J. Math. Phys. 47 (2006) 092304 ([arXiv:hep-th/0511100](https://arxiv.org/abs/hep-th/0511100))

On the [[3d-3d correspondence]]:

* Julius Eckhard, Heeyeon Kim, [[Sakura Schäfer-Nameki]], Brian Willett, _Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence_ ([arxiv:1910.14086](https://arxiv.org/abs/1910.14086))

On relating the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]] (and the [[moduli space of monopoles]]):

* Benjamin Assel, [[Sakura Schäfer-Nameki]], Jin-Mann Wong, _M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models_, J. High Energ. Phys. (2016) 2016: 120 ([arxiv:1604.03606](https://arxiv.org/abs/1604.03606))

On [[Higgs bundles]] and [[M-theory on G2-manifolds]]:

* {#BCHS19} [[Andreas Braun]], Sebastjan Cizel, Max Hubner, [[Sakura Schäfer-Nameki]], _Higgs Bundles for M-theory on G2-Manifolds_ ([arXiv:1812.06072](https://arxiv.org/abs/1812.06072))

On [[F-theory on Spin(7)-manifolds]]/[[M-theory on Spin(7)-manifolds]] and variant of the [[compact twisted connected sum G2-manifolds]]:

* [[Andreas Braun]], [[Sakura Schäfer-Nameki]], _Spin(7)-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories_, JHEP06(2018)103 ([arXiv:1803.10755](https://arxiv.org/abs/1803.10755))

On [[duality in string theory|duality]] of [[M-theory on G2-manifolds]] with [[heterotic string theory on CY3-manifolds]]:

* {#BraunSchaeferNameki17} [[Andreas Braun]], [[Sakura Schäfer-Nameki]], _Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality_, JHEP04(2018)126 ([arXiv:1708.07215](https://arxiv.org/abs/1708.07215))

On [[F/M-theory on elliptically fibered Calabi-Yau 5-folds]], relating to [[D=2 N=(2,0) SCFT]]:

* [[Sakura Schäfer-Nameki]], [[Timo Weigand]], _F-theory and 2d (0,2) Theories_, J. High Energ. Phys. 2016, 59 (2016) ([arXiv:1601.02015](https://arxiv.org/abs/1601.02015))

* [[Sakura Schäfer-Nameki]], _F-theory and $2d$ $(0,2)$ Theories_, 2016 ([[SchaeferNamekiFTheoryAnd2DQFT.pdf:file]])

category: people

[[!redirects Sakura Schäfer-Nameki]]
[[!redirects Sakura Schaefer-Nameki]]
[[!redirects Sakura Schafer-Nameki]]
[[!redirects Sakura Schäfer-Nameki]]


salamander lemma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Diagram chasing lemmas
+-- {: .hide}
[[!include diagram chasing lemmas - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _salamander lemma_ is a fundamental [[lemma]] in [[homological algebra]] providing information on the relation between [[homology groups]] at different positions in a [[double complex]].

By a simple consequence illustrated in remark \ref{ExtramuralMapsAsDiagonals} below, all the standard [[diagram chasing lemmas]] of [[homological algebra]] are direct and transparent consequences of the salamander lemma, such as the [[3x3 lemma]], the [[four lemma]], hence the [[five lemma]], the [[snake lemma]], and also the [[long exact sequence in cohomology]] corresponding to a [[short exact sequence]].

These lemmas are all classical, but their traditional [[proofs]] are, while elementary, not very illuminating. The Salamander lemma serves to make the mechanism behind these lemmas more transparent and also to make evident a host of further lemmas of this kind not traditionally considered, such as an [nxn lemma](#nxnLemma) for all $n \in \mathbb{N}$.

## The Salamander lemma
 {#TheSalamanderLemma}

The Salamander lemma, prop. \ref{SalamanderLemma} below, is a statement about the [[exact sequence|exactness]] of a sequence naturally associated with any [[morphism]] in a [[double complex]]. In the _[Preliminaries](#Preliminaries)_ we first introduce this sequence itself.

### Preliminaries
 {#Preliminaries}

+-- {: .num_remark}
###### Remark
**(general assumption/convention)**

As always in [[homological algebra]], when we consider _[[elements]]_ of [[objects]] in the ambient [[abelian category]] $\mathcal{A}$ it is either assumed that $\mathcal{A}$ is of the form $R$[[Mod]] for some [[ring]] $R$, or that one of the _[embedding theorems](abelian+category#EmbeddingTheorems)_ has been used to embed it into such, by a [[faithful functor|faithful]] and [[exact functor]], so that these elements are actual elements of the [[sets]] underlying these [[modules]].

In the following, many of the [[proofs]] are spelled out in terms of elements this way, and we will not always repeat this assumption. This should help to amplify how utterly elementary the salamander lemma is. But explicitly element-free/general abstract proofs can of course be given without much more effort, too, see [Wise](#Wise).

=--

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

Let  $A_{\bullet \bullet}$ be a [[double complex]] in some [[abelian category]] $\mathcal{A}$, hence a [[chain complex]] of chain complexes $A_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$, hence a [[diagram]] of the form

$$
  \array{
   && \vdots && \vdots
   \\
   & & \downarrow^{\mathrlap{\partial^{vert}}} && \downarrow^{\mathrlap{\partial^{vert}}}
   \\
    \cdots &\to &
    A_{n,m} &\stackrel{\partial^{hor}}{\to}& A_{n-1,m}
    & \to & \cdots
   \\
   & & \downarrow^{\mathrlap{\partial^{vert}}} && \downarrow^{\mathrlap{\partial^{vert}}}
   \\
    \cdots &\to &
    A_{n,m-1} &\stackrel{\partial^{hor}}{\to}& A_{n-1,m-1}
    & \to & \cdots
   \\
   & & \downarrow^{\mathrlap{\partial^{vert}}} && \downarrow^{\mathrlap{\partial^{vert}}}
   \\
   && \vdots && \vdots
  }
$$

where $\partial^{hor} \circ \partial^{hor} = 0$, where $\partial^{vert} \circ \partial^{vert} = 0$ and where all squares [[commuting diagram|commute]], $\partial^{hor} \circ \partial^{vert} = \partial^{vert} \circ \partial^{hor}$.

Let $A \coloneqq A_{k l}$ be any [[object]] in the double complex at any position $(k,l)$. This is the [[source]] and [[target]] of horizontal, vertical and diagonal (unique composite of a horizontal and a vertical) [[morphisms]] to be denoted as follows:

$$
  \array{
     ^\mathllap{\partial_{in}^{diag}} \searrow
     & \downarrow^{\mathrlap{\partial_{in}^{vert}}}
     \\
     \stackrel{\partial_{in}^{hor}}{\to} & A &
     \stackrel{\partial_{out}^{hor}}{\to}
     \\
     & ^\mathllap{\partial_{out}^{vert}} \downarrow
     &
     \searrow^{\mathrlap{\partial_{out}^{diag}}}
  }
  \,.
$$

Define

* $A^{hor} \coloneqq ker (\partial^{hor}_{out}) / im (\partial^{hor}_{in}) \in \mathcal{A}$ -- the horizontal [[chain homology]] at $X$;

* $A^{vert} \coloneqq ker (\partial^{vert}_{out}) / im (\partial^{vert}_{in}) \in \mathcal{A}$ -- the vertical [[chain homology]] at $X$;

* ${}^{\Box}A \coloneqq \frac{ker (\partial^{hor}_{out}) \cap ker(\partial^{vert}_{out})}{im(\partial^{diag}_{in})} \in \mathcal{A}$ -- the "receptor" at $A$;

* $A_{\Box}\coloneqq \frac{ker (\partial^{diag}_{out}) }{ im(\partial^{hor}_{in}) + im(\partial^{vert}_{in})}$ -- the "donor" at $A$;

where

* $ker(-)$ denotes the [[kernel]] of a map,

* $im(-)$ the [[image]] of a map,

* $\frac{N_1}{N_2}$ the [[quotient module]] of the module $N_1$ by a [[submodule]] $N_2 \hookrightarrow N_1$,

*  $N_1 + N_2$ the sum of two submodules (i.e., their [[join]] in the [[lattice]] of submodules under inclusion).

=--

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

The [[identity]] on representatives in $A$ induces a [[commuting diagram]] of [[homomorphisms]] from the donor of $A$ to the receptor of $A$, def. \ref{DonorReceptor}, via the horizontal and vertical [[homology groups]] at $A$:

$$
  \array{
    && {}^\Box A
    \\
    & \swarrow && \searrow
    \\
    A^{vert} &&&& A^{hor}
    \\
    & \searrow && \swarrow
    \\
    && A_\Box
    \,.
  }
$$

These morphisms are to be called the **intramural maps** of $A$.

=--

This is immediate, here is one way to make it fully explicit:

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

The statement that the top two morphisms exist and are given by the identity on representatives is that if $[a] \in {}^\Box A$ is represented by $a \in A$, then $a$ also represents an element in $A^{hor}$ and $A^{vert}$.
But this is the very definition of ${}^\Box A$: being a [[quotient module]] of $ker(\partial^{vert}_{out}) \cap ker(\partial^{hor}_{out})$ means that its elements are represented by elements of $A$ that are annihiliated by both $\partial^{vert}_{out}$ and $\partial^{hor}_{out}$. Moreover if $[a] \in {}^\Box A$ is 0 then $a$ also represents the 0-element in $A^{hor}$ and in $A^{vert}$, because by definition of ${}^\Box A$ it is then in the [[image]] of $\partial^{hor} \circ \partial^{vert} = \partial^{vert} \circ \partial^{hor}$. Moreover it is clear that everything respects addition of module elements and the [[action]] by the ring $R$, hence the top morphisms are well defined [[module]] [[homomorphisms]].

Similarily the bottom two morphisms exist and are given by the identity on representatives by the very definition of $A_{\Box}$: this being a quotient of the kernel of $\partial^{vert} \partial^{hor} = \partial^{hor} \partial^{vert}$ it contains in particular the elements that are represented in $A$ by elements in the kernel of $\partial^{hor}$ and in the kernel of of $\partial^{vert}$ separately. And if $[a]$ is the 0-element in $A^{hor}$ or $A^{vert}$ then $a$ is in the image of $\partial^{hor}$ or of $\partial^{vert}$, respectively, and since these two images are quotiented out to obtain $A_{\Box}$, such $a$ also represent the 0-element there.

=--

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

In lemma \ref{Intramural} "intramural" is meant to allude to the fact that these morphisms go between objects that all come from $A$ and hence remain "in the context of $A$". The "extramural" maps to follow in lemma \ref{Extramural} below instead go from objects associated to some $A$ to objects associated to some $B$, so they go "out of the context of $A$".

=--

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

Any vertical or horizontal morphism $\partial : A \to B$ in the double complex induces a homomorphism

$$
  A_\Box \to {}^\Box B
$$

from the donor of $A$ to the receptor of $B$, def. \ref{DonorReceptor}, called the **extramural map** associated with $\partial$.

=--

Again this is immediate, here is one way to make it explicit:

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

We discuss the case that $\partial = \partial^{hor}$ is a horizontal [[differential]]. The other case works verbatim the same way, only with the roles of $\partial^{hor}$ and $\partial^{vert}$ interchanged.

By definition \ref{DonorReceptor}, an $[a] \in A_{\Box}$ is represented by an $a \in A$ for which $\partial^{vert}\partial^{hor} a = 0$. The claim is that $\partial^{hor} a$ then represents an element in ${}^{\Box}B$ such that this is a module homomorphism.

*  We have $\partial^{vert} (\partial^{hor} a) = 0$ by assumption on $a$ and $\partial^{hor} (\partial^{hor} a) = 0$ by the chain complex property. Hence $\partial^{hor} a$ represents an element in ${}^{\Box} B$.

* If $[a] = 0$ then there is $c$ such that $a = \partial^{hor} c$ or $d$ such that $a = \partial^{vert} d$. In the first case the chain complex property gives that $\partial^{hor}a = \partial^{hor}\partial^{hor}c = 0$ and hence $[\partial^{hor} a] = 0$ in ${}^\Box B$, in the second $\partial^{hor}a = \partial^{hor}\partial^{vert} d$ which is also 0 in ${}^\Box B$ since this is the quotient by $im(\partial^{hor} \partial^{vert})$.

=--

+-- {: .num_remark #ExtramuralMapsAsDiagonals}
###### Remark
**(central idea on diagram chasing)**

It is useful in computations, such as those shown below in _[Implications - The diagram chasing lemmas](#Implications)_, to draw the extramural morphisms of lemma \ref{Extramural} as follows.

1. For a horizontal $\partial^{hor} \colon A \to B$ we draw the induced extramural map as

   $$
     \array{
        &&& \Box
        \\
        A && \nearrow & & B
        \\
        & \Box
     }
     \,.
   $$

1. For a vertical $\partial^{vert} \colon A \to B$ we draw the induced extramural map as

   $$
     \array{
       & A
       \\
       && \Box
       \\
       & \swarrow
       \\
       \Box
       \\
       & B
     }
   $$

This notation makes it manifest that in every [[double complex]] $X_{\bullet, \bullet}$ the extramural maps form long diagonal [[zigzags]] between donors and receptors

$$
  \array{
     && &&&&&&&& \udots
     \\
     && &&&&& X_{k, l+1} && \nearrow
     \\
     && &&&&&& \Box
     \\
     && &&&&& \swarrow
     \\
     && &&&& \Box
     \\
     && &X_{k+1,l} && \nearrow & & X_{k,l}
     \\
     && && \Box
     \\
     && &\swarrow
     \\
     && \Box
     \\
     & \nearrow & & X_{k+1, l-1}
     \\
     \udots
  }
  \,.
$$

But moreover, the intramural maps relate the donors and receptors in particular at the far end of these zigzags back to the actual homology groups of interest:

$$
  \array{
     && && && &&&&&&&& & \Box
     \\
     && && && &&&&&            &&           & \udots && B
     \\
     && && && &&&&& X_{k, l+1} && \nearrow  &        &&    & \searrow
     \\
     && && && &&&&&            & \Box      &&        &&    & & B^{hor}
     \\
     && & & && &&&&& \swarrow
     \\
     && & & && &&&& \Box
     \\
     && & & && &X_{k+1,l} && \nearrow & & X_{k,l}
     \\
     && & & && && \Box
     \\
     && & & && &\swarrow
     \\
     A^{hor} && & & && \Box
     \\
     &\searrow & & & & \nearrow & & X_{k+1, l-1}
     \\
     && A & & \udots
     \\
     && & \Box
  }
  \,.
$$

This means that in order to get "far diagonal identifications" of homology groups in a double complex, all one needs is sufficient conditions that all the intramural and extramural maps in a "long salamander" like this are [[isomorphisms]].

These turn out to be certain exactness conditions to be checked/imposed locally at each of the positions involved in a long salamander like this discussed in  _[Intramural and extramural isomorphism](#IntraExtramuralIsomorphisms)_ below.
All the long diagonal identifications of the standard [[diagram chasing lemmas]] follows by piecing together such long salamanders. This is discussed in the  _[Implications](#Implications)_ below.

=--

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

Moreover, it is useful to combine the extramural notation of remark \ref{ExtramuralMapsAsDiagonals} with the evident diagonal notation for the intramural maps, lemma \ref{Intramural}, which allow extramural maps to "enter at the receptor" and "exit at the donor" of a given entry in the double complex. Together these two notations yield for every piece of a [[double complex]] of the form

$$
  \array{
    C
    \\
    \downarrow^{\mathrlap{\partial^{vert}}}
    \\
    A &\stackrel{\partial^{hor}}{\to}& B
    \\
    && \downarrow^{\mathrlap{\partial^{vert}}}
    \\
    && D
  }
$$

the **salamander**-shaped diagrams of mural maps

$$
  \array{
      & C
      \\
      && \Box
      \\
      & \swarrow
      \\
      \Box &&& & \Box
      \\
      & A && \nearrow & & B
      \\
      && \Box & && & \Box
      \\
      &&&&& \swarrow
      \\
      &&&& \Box
      \\
      &&&&& D
  }
  \,.
$$

These give the salamander lemma, prop. \ref{SalamanderLemma} below, its name.

=--

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

For $\partial^{hor} : A \to B$ any horizontal morphism in the double complex, the canonically induced morphism $A^{vert} \to B^{vert}$ on vertical homology is the composite of the above intramural and extramural maps:

$$
  A^{vert} \to A_{\Box} \to {}^\Box B \to B^{vert}
  \,.
$$

=--

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

By the above, on representatives the first map is the identity, the second is $\partial^{hor}$ and the third again the identity. Hence the total map is given on representative by $\partial^{hor}$.

=--

### Statement

+-- {: .num_prop #SalamanderLemma}
###### Proposition
**(Salamander lemma)**

If a [[diagram]]

$$
  \array{
    C
    \\
    \downarrow^{\mathrlap{\partial^{vert}}}
    \\
    A &\stackrel{\partial^{hor}}{\to}& B
    \\
    && \downarrow^{\mathrlap{\partial^{vert}}}
    \\
    && D
  }
$$

is part of a [[double complex]] in an [[abelian category]], then there is a **6-term [[long exact sequence]]** running horizontally in

$$
  \array{
    &&  {}^\Box A
    \\
   & \nearrow && \searrow
    \\
    C_\Box &&\to&& A^{hor} &\to& A_{\Box} &\to& {}^{\Box} B &\to& B^{hor}
   &&\to&& {}^{\Box}D
   \\
    && && && && && & \searrow && \nearrow
   \\
    && && && && && && B_{\Box}
  }
  \,,
$$

where all the elementary morphisms are the unique intramural maps from lemma \ref{Intramural} and the extramural maps from lemma \ref{Extramural} -- they are the morphisms of the _salamander diagram_ of remark \ref{OmittingIntramuralMaps}.

Similarly, if a diagram

$$
  \array{
    C &\stackrel{\partial^{hor}}{\to}& A
    \\
    && \downarrow^{\mathrlap{\partial^{vert}}}
    \\
    && B &\stackrel{\partial^{hor}}{\to}& D
  }
$$

is part of a double complex, then there is a 6-term [[long exact sequence]] running horizontally in

$$
  \array{
    && {}^\Box A
    \\
    & \nearrow & & \searrow
    \\
    C_\Box &&\to&& A^{vert} &\to& A_{\Box} &\to& {}^\Box B &\to&
    B^{vert}
    &&\to&&
    {}^\Box D
    \\
    && && && && && & \searrow && \nearrow
    \\
    && && && && && && B_\Box
  }
  \,
$$

=--

This is [Bergman, lemma 1.7](#Bergman).

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

We spell out the proof of the first case. That of the second case is verbatim the same, only with the roles of $\partial^{hor}$ and $\partial^{vert}$ interchanged.

By lemmas \ref{Intramural} and \ref{Extramural}, all the maps are given on representatives either by identities or by the [[differentials]] of the double complex themselves. Using this we may check exactness at each position explicitly:

1. exactness at $C_\Box \to A^{hor} \to A_\Box$.

   An element $[a] \in A^{hor}$ is in the kernel of $A^{hor} \to A_{\Box}$ if there is $c$ and $d$ such that $a = \partial^{vert}c + \partial^{hor} d$.
The $c$ that satisfy this equation hence satisfy $\partial^{hor}\partial^{vert} c = 0$, hence represent elements in $C_\Box$ and so the map $\partial^{hor} : C_\Box \to A^{hor}$ hits all of the kernel of $A^{hor}\to A_\Box$. Also it clearly hits at most this kernel.

1. exactness at $A^{hor} \to A_\Box \to {}^\Box B$.

   Suppose $[a] \in A_\Box$ is in the kernel of $A_\Box \to {}^\Box B$. This means that there is $c$ such that $\partial^{hor}a = \partial^{hor} \partial^{vert}c$, hence that $\partial^{hor} (a-\partial^{vert} c) = 0$. But $[a] = [a - \partial^{vert}c] \in A_{\Box}$ and so this says that $[a]$ is the image under $A^{hor} \to A_{\Box}$ of the element represented there by $a - \partial^{vert} c$. Conversely, clearly everything in that image is in the kernel of $A_\Box \to {}^\Box B$.

1. exactness at $A_\Box \to {}^\Box B \to B^{hor}$

   An element $[b] \in {}^\Box B$ is in the kernel of ${}^\Box B \to B^{hor}$ if there is $a$ such that $\partial^{hor} a = b$. But since the representative $b$ of $[b] \in {}^\Box B$ has to satisfy in particular $\partial^{vert} b = 0$ it follows that $\partial^{hor} \partial^{vert}a  = 0$ and hence that $a$ represents an element in $A_{\Box}$, hence that $[b]$ is in the image of $A_{\Box} \to {}^{\Box }B$. Conversely, clearly every element in that image is in the kernel of ${}^\Box B \to B^{hor}$.

1. exactness at ${}^\Box B \to B^{hor} \to {}^\Box D$

   An element $[b] \in B^{hor}$ is in the kernel of $B^{hor} \to {}^\Box D$ if there is $a$ with $\partial^{vert} b = \partial^{vert} \partial^{hor} a$, hence $\partial^{vert}(b - \partial^{hor}a) = 0$ Since in addition $\partial^{hor}(b - \partial^{hor}a) = 0$ by assumption on $b$ and the chain complex property, this says that $[b] = [b - \partial^{hor} a]$ is in the image of ${}^\Box B \to B^{hor}$. Moreover, clearly everything in this image is in the kernel of $B^{hor} \to {}^\Box D$.

=--

### Intramural and extramural isomorphisms
 {#IntraExtramuralIsomorphisms}

The following two statements are direct consequences (special cases) of the salamander lemma, prop. \ref{SalamanderLemma}. They give sufficient conditions for the intramural and the extramural maps, lemma \ref{Intramural} and lemma \ref{Extramural}, to be [[isomorphisms]]. All of the standard [[diagram chasing lemmas]] in [[homological algebra]] follow in a natural way from combining these _intramural isomorphisms_ with long [[zigzags]] of these _extramural isomorphisms_. This is discussed below in _[The basic diagram chasing lemmas](#Implications)_.

+-- {: .num_cor #ExtramuralIso}
###### Corollary
**(extramural isomorphisms)**

If the rows of a double complex are exact at the domain and codomain of a horizontal morphism $\partial^{hor} : A \to B$, i.e. if  $A^{hor} = 0$ and $B^{hor} = 0$, then the extramural map of lemma \ref{Extramural}

$$
  A_{\Box} \to {}^\Box B
$$

is an [[isomorphism]].

Similarly if for a vertical morphism $\partial : A \to B$ we have $A^{vert} = 0$ and $B^{vert} = 0$, the induced extramural map

$$
  \array{
    A_\Box
    \\
    \downarrow
    \\
    {}^\Box B
  }
$$

is an isomorphism.

=--

This appears as [Bergman, cor. 2.1](#Bergman).

It is straightforward to check this directly on elements:

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

It is sufficient to show that under the given assumptions both the [[kernel]] and the [[cokernel]] of the given map are trivial.
We discuss the horizontal case. The proof of the vertical case is verbatim the same, only with the roles of $\partial^{vert}$ and $\partial^{hor}$ exchanged.

  Suppose an element $[a] \in A_{\Box}$ is in the kernel of $\partial^{hor} : A_{\Box} \to {}^\Box B$. By definition of ${}^\Box B$ this means that there is $c$ such that  $\partial^{hor} \partial^{vert} c = \partial^{hor} a$, hence such that $\partial^{hor}(a- \partial^{vert} c) = 0$. By assumption that  $A^{hor} = 0$ this means that there is $d$ such that  $a - \partial^{vert} c = \partial^{hor} d$. But this means that $a \in im(\partial^{hor}) + im(\partial^{vert})$ and hence $[a] = 0$ in $A_\Box$.

  Conversely, consider $[b] \in {}^\Box B$. This means that $\partial^{vert}b = 0$ and $\partial^{hor} b = 0$. By $B^{hor} = 0$ the second condition means that there is $a$ such that $b = \partial^{hor} a$.
Moreover, this $a$ satisfies $\partial^{vert}\partial^{hor} a = \partial^{vert} b = 0$ by the first condition. Therefore $[a] \in A_{\Box}$ and $[b]$ is its image.

=--

Alternatively, this statement is a direct consequence of the salamander lemma already proven above:

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

Under the given assumptions the exact sequence of prop. \ref{SalamanderLemma} involves the exact sequence

$$
  0 \to A_{\Box} \to {}^\Box B \to 0
  \,.
$$

This being exact says that the map in the middle has vanishing [[kernel]] and [[cokernel]] and is hence an [[isomorphism]].

=--

+-- {: .num_cor #IntramuralIsos}
###### Corollary
**(intramural isomorphisms)**

In each of the situations in a double complex shown below, if the direction _perpendicular_ to $\partial : A \to B$ or $\partial : B \to A$ is [[exact sequence|exact]] at $B$ as indicated in the following, then the two intramural maps from lemma \ref{Intramural},  shown in each case on the right, are [[isomorphisms]]:

1. $$
     \array{
       && \vdots && \vdots
       \\
       && \downarrow && \downarrow
       \\
       0 &\to& A &\to& &\to& \cdots
       \\
       && \downarrow^{\mathrlap{\partial}} && \downarrow
       \\
       0 &\to& B &\stackrel{ker = 0}{\to}& &\to& \cdots
       \\
       && \downarrow && \downarrow
       \\
       && \vdots && \vdots
     }
     \;\;\;\;\;\;\;\;\;\;\;\;
     \Rightarrow
     \;\;\;\;\;\;\;\;\;\;\;\;
     \array{
       {}^\Box A &\stackrel{\simeq}{\to}& A^{hor}
       \\
       A^{vert} & \stackrel{\simeq}{\to}&  A_{\Box}
     }
   $$

1. $$
     \array{
       && 0 && 0
       \\
       && \downarrow && \downarrow
       \\
       \cdots &\to& A &\stackrel{\partial}{\to}& B &\to& \cdots
       \\
       && \downarrow && \downarrow^{\mathrlap{ker = 0}}
       \\
       \cdots &\to& &\to& &\to& \cdots
       \\
       && \vdots && \vdots
     }
     \;\;\;\;\;\;\;\;\;\;\;\;
     \Rightarrow
     \;\;\;\;\;\;\;\;\;\;\;\;
     \array{
       {}^\Box A &\stackrel{\simeq}{\to}& A^{vert}
       \\
       A^{hor} & \stackrel{\simeq}{\to}&  A_{\Box}
     }
   $$

1. $$
     \array{
       && \vdots && \vdots
       \\
       && \downarrow && \downarrow
       \\
       \cdots &\to& &\stackrel{im = B}{\to}& B &\to& 0
       \\
       && \downarrow && \downarrow^{\mathrlap{\partial}}
       \\
       \cdots &\to& &\to& A &\to& 0
       \\
       && \downarrow && \downarrow
       \\
       && \vdots && \vdots
     }
     \;\;\;\;\;\;\;\;\;\;\;\;
     \Rightarrow
     \;\;\;\;\;\;\;\;\;\;\;\;
     \array{
        A^{hor} &\stackrel{\simeq}{\to}& A_{\Box}
        \\
        {}^\Box A &\stackrel{\simeq}{\to}& A^{vert}
     }
   $$

1. $$
     \array{
       && \vdots && \vdots
       \\
       \cdots &\to& &\to& &\to& \cdots
       \\
       && \downarrow^{\mathrlap{im = B}} && \downarrow
       \\
       \cdots &\to& B &\stackrel{\partial}{\to}& A &\to& \cdots
       \\
       && \downarrow && \downarrow
       \\
       && 0 && 0
     }
     \;\;\;\;\;\;\;\;\;\;\;\;
     \Rightarrow
     \;\;\;\;\;\;\;\;\;\;\;\;
     \array{
       A^{vert} &\stackrel{\simeq}{\to}& A_{\Box}
       \\
       {}^\Box A &\stackrel{\simeq}{\to}& A^{hor}
     }
   $$

=--

This appears as [Bergman, cor. 2.2](#Bergman).

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

We spell out the proof of the first item. The others work
analogously.

Applying cor. \ref{ExtramuralIso} to $0 \to B$ yields ${}^\Box B \simeq 0_\Box = 0$. Therefore the exact sequence of the Salamander lemma \ref{SalamanderLemma} corresponding to

$$
  \array{
    \bullet
    \\
    \downarrow
    \\
    0 &\to& A
    \\
    && \downarrow
    \\
    && B
  }
$$

ends with

$$
  \cdots \to 0 \to {}^\Box  A\to A^{hor} \to 0
  \,,
$$

which implies the first isomorphism. Analogously, the salamander exact sequence associated with

$$
  \array{
    0 &\to& A
    \\
    && \downarrow
    \\
    && B &\to& \bullet
  }
$$

begins as

$$
  0 \to A^{vert} \to A_{\Box} \to 0 \to \cdots
  \,.
$$

which gives the second isomorphism.

=--

## Implications: The basic diagram-chasing lemmas
 {#Implications}

We derive the basic [[diagram chasing lemmas - contents|diagram chasing lemmas]] from the salamander lemma, or in fact just from repeated application of the [intramural/extramural isomorphisms](#IntraExtramuralIsomorphisms).

### The $3 \times 3$ lemma
 {#3x3Lemmas}

We derive the [[sharp 3x3 lemma]] from the salamander lemma.

+-- {: .num_prop #ShortSharp3x3}
###### Proposition

If in a [[diagram]] of the form

$$
  \array{
    && 0 && 0 && 0
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A' &\to& B' &\to& C'
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A &\to& B &\to& C
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A'' &\to& B'' &\to& C''
  }
$$

all columns and the second and third row are [[exact sequence|exact]],
then also the first row is exact.

=--

The following proof is that given in [Bergman, lemma 2.3](#Bergman).

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

First of all one notices that the diagram is a [[double complex]]: by column-exactness the first row includes as [[subobjects]] into the second, so the horizontal maps of the first row are restrictions of the [[differentials]] of the second and so at least the first row is a [[chain complex]].

We need to show that ${A'}^{hor}\simeq 0$ and ${B'}^{hor} \simeq 0$.

First consider exactness at $A'$. The intramural iso, cor. \ref{IntramuralIsos} item 1, of

$$
  \array{
    A'
    \\
    \downarrow^{\mathrlap{\partial}}
    \\
    A &\stackrel{ker = 0}{\to}& B
  }
$$

is ${}^\Box A' \simeq {A'}^{hor}$, and the one of

$$
  \array{
    A' &\stackrel{\partial}{\to}& B'
    \\
    && \downarrow^{\mathrlap{ker = 0}}
    \\
    && B
  }
$$

according to cor. \ref{IntramuralIsos} item 2
is ${}^\Box A' \simeq {A'}^{vert}$. Together this gives the desired exactness from the assumption that ${A'}^{vert} = 0$ (since all the columns are exact by assumption):

$$
  {A'}^{hor} \simeq A'_\Box \simeq {A'}^{vert} = 0
  \,.
$$

To apply an analogous argument for ${B'}^{hor}$, we combine this kind of identification with the [[zigzag]] of extramural maps along the diagonal

$$
  \array{
    &&&& B'
    \\
    &&&&& \Box
    \\
    &&&& \swarrow
    \\
    &&& \Box
    \\
    A && \nearrow && B
    \\
    & \Box
  }
  \,,
$$

which are isos by cor \ref{ExtramuralIso}. These appear now in the
middle of the following chain of isomorphisms

$$
  {B'}^{hor} \simeq \left({B'}_{\Box}\simeq {}^{\Box}B \simeq A_{\Box}\right) \simeq \left(A^{vert} \simeq 0\right)\,,
$$

where the first and the last are intramural isos obtained from cor. \ref{IntramuralIsos}.

=--

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

From this argument it is clear that by directly analogous reasoning we obtain "$n \times n$-lemmas" for arbitrary $n$, see prop. \ref{nxn} below.

=--

In particular we have the following _[[sharp 3x3 lemma]]_.

+-- {: .num_prop #Sharp3x3}
###### Proposition

If in a [[diagram]] of the form

$$
  \array{
    && 0 && 0 && 0
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A' &\to& B' &\to& C' &\to& 0
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A &\to& B &\to& C &\to& 0
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& A'' &\to& B'' &\to& C''
    \\
    && \downarrow
    \\
    && 0
  }
$$

all columns and the second and third row are [[exact sequence|exact]],
then also the first row is exact.

=--

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

We can extend the middle and right columns by adding the cokernel of $B \to B''$ below $B''$, and $C'' \to 0$ below C''.  It's straightforward to check that the resulting diagram is still a complex, and we now have vertical exactness at $B''$.  Note that we may not have vertical exactness at $C''$, nor exactness in the bottom row containing the cokernel, but we don't need those.

Exactness in $A'$ and $B'$ is as in prop. \ref{ShortSharp3x3}.
For exactness in $C'$ we now use the long [[zigzag]] of
extramural isomorphisms, cor \ref{ExtramuralIso}.

$$
  \array{
    &&&&&&&& C'
    \\
    &&&&&&&& & \Box
    \\
    &&&&&&&& \swarrow
    \\
    &&&&&&& \Box
    \\
    &&&& B && \nearrow && C
    \\
    &&&&& \Box
    \\
    &&&& \swarrow
    \\
    &&& \Box
    \\
    A'' && \nearrow && B''
    \\
    & \Box
  }
  \,,
$$

So ${C'}^{hor} \simeq C_{\Box}$ by the intramural iso, then $\dots \simeq A''_\Box$ by this zigzag of extramural isos (this is where we need vertical exactness at $B''$), and finally $\cdots \simeq {A''}^{hor} \simeq 0$ by another intramural iso and by assumption.

=--

### The $n \times n$ lemma
 {#nxnLemma}

The proofs of the $3 \times 3$-lemmas [above](#3x3Lemmas) via long diagonal zigzags of extramural isomorphism clearly generalize from double complexes of size $3 \times 3$ to those of arbitrary finite size.

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

If in a diagram of the form

$$
  \array{
    && 0 && 0 && 0 && 0 &&
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& X_{n,n} &\to& X_{n-1,n} &\to& X_{n-2,n} &\to& X_{n-3,n} &\to& \cdots
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& X_{n,n-1} &\to& X_{n-1,n-1} &\to& X_{n-2,n-1} &\to& X_{n-3,n-1} &\to& \cdots
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& X_{n,n-2} &\to& X_{n-1,n-2} &\to& X_{n-2,n-2} &\to& X_{n-3,n-2} &\to& \cdots
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& X_{n,n-3} &\to& X_{n-1,n-3} &\to& X_{n-2,n-3} &\to& X_{n-3,n-3} &\to& \cdots
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow
  }
$$

all rows except possibly the first $X_{\bullet, n}$ as well as all columns except possibly the first $X_{n,\bullet}$ are [[exact sequence|exact]], then the homology groups of the first row equal those of the first column in that

$$
  \forall k : X_{k,n}^{hor} \simeq X_{n,k}^{vert}
  \,.
$$

=--

This appears as [Bergman, lemma 2.6](#Bergman).

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

The proof proceeds in direct generalization of the proofs of the
3x3 lemma [above](#3x3Lemmas): the isomorphism for each $k$ is given by the composite of two extramural isomorphisms that identify the given homology group with a donor or receptor group, respectively, with a long zigzag of extramural isomorphisms.

=--

### The four lemma
 {#FourLemma}

We prove the  _strong [[four lemma]]_ from the salamander lemma.

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

Consider a [[commuting diagram]] in $\mathcal{A}$ of the form

$$
  \array{
    A &\to& B &\stackrel{\xi}{\to}& C &\to& D
    \\
    \downarrow^{\mathrlap{\tau}}
    &&
    \downarrow^{\mathrlap{f}}
    &&
    \downarrow^{\mathrlap{g}}
    &&
    \downarrow^{\mathrlap{\nu}}
    \\
    A' &\to& B' &\stackrel{\eta}{\to}& C' &\to& D'
  }
$$

where

1. the rows are [[exact sequences]],

1. $\tau$ is an [[epimorphism]],

1. $\nu$ is a [[monomorphism]].

Then

1. $\xi(ker(f)) = ker(g)$

1. $im(f) = \eta^{-1}(im(g))$

and so in particular

1. if $f$ is a [[monomorphism]] then so is $g$;

1. if $g$ is an [[epimorphism]] then so is $f$.

=--

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

By assumption on $\tau$ and $\nu$ we can complete the diagram
to a [[double complex]] of the form

$$
  \array{
    && 0 && 0
    \\
    && \downarrow && \downarrow
    \\
    && ker(f) &\stackrel{\xi|_{ker(f)}}{\to}& ker(g) &\to& 0
    \\
    && \downarrow && \downarrow && \downarrow
    \\
    A &\to& B &\stackrel{\xi}{\to}& C &\to& D
    \\
    \downarrow^{\mathrlap{\tau}}
    &&
    \downarrow^{\mathrlap{f}}
    &&
    \downarrow^{\mathrlap{g}}
    &&
    \downarrow^{\mathrlap{\nu}}
    \\
    A' &\to& B' &\stackrel{\eta}{\to}& C' &\to& D'
    \\
    \downarrow && \downarrow && \downarrow
    \\
    0 &\to& B'/im(f) &\to& C'/im(g)
    \\
     && \downarrow && \downarrow
    \\
    && 0 && 0
  }
$$

such that

1. all columns are [[exact sequence|exact]];

1. the middle two rows are exact.

For the first statement it is now sufficient to show that $ker(g)^{hor} \simeq 0$, for that is immediately equivalent to $\xi(ker(f)) = ker(g)$.

To see this we use the intramural isomorphism,
cor. \ref{IntramuralIsos} item 2, to deduce that

$$
  ker(g)^{hor} \simeq ker(g)_{\Box}
  \,.
$$

Then the long [[zigzag]] of extramural isomorphisms, cor. \ref{ExtramuralIso}, shows that this is isomorphic to the
${}^\Box 0 \simeq 0$ in the bottom left corner of the diagram.

The second statement follows dually: it is implied by
$(B'/im(f))^{hor} \simeq 0$ for that directly implies that
$\eta^{-1}(im(g))\simeq im(f)$.

Here the intramural ismorphism to use is

$$
  (B'/im(f))^{hor} \simeq {}^{\Box}(B'/im(f))
$$

and then the long sequence of zigzags of extramural isomorphisms identifies this with the $0_\Box \simeq 0$ in the top right corner.

=--

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

The [[four lemma]], in turn, directly implies what is
known as the [[five lemma]].

=--

### The snake lemma
 {#TheSnakeLemma}

We discuss a proof of the [[snake lemma]] from the salamander lemma.

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

If in a commuting diagram of the form

$$
  \array{
    && X_1 &\to& X_2 &\to& X_3 &\to& 0
    \\
    && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{h}}
    \\
    0 &\to& Y_1 &\to& Y_2 &\to& Y_3
  }
$$

both rows are [[exact sequence|exact]], then there is a [[long exact sequence]]

$$
  ker(f) \to ker(g) \to ker(h)
   \stackrel{\delta}{\to}
  coker(f) \to coker(g) \to coker(h)
$$

starting with the [[kernels]] of the three vertical maps and ending with their [[cokernels]] (with the middle morphism $\delta$ called the "[[connecting homomorphism]]").

=--

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

Consider the completion of the given diagram to a [[double complex]]:

$$
  \array{
     &&  ker(f) &\to& ker(g) &\to& ker(h) &\to& 0
    \\
     && \downarrow && \downarrow && \downarrow && \downarrow
    \\
    ker(l) &\to& X_1 &\stackrel{l}{\to}& X_2 &\to& X_3 &\to& 0
    \\
    \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{h}} && \downarrow
    \\
    0 &\to& Y_1 &\to& Y_2 &\stackrel{r}{\to}& Y_3 &\to& coker(r)
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    0 &\to& coker(f) &\to& coker(g) &\to& coker(h) &&
  }
  \,.
$$

By assumption and construction, here all columns are exact and the rows are exact at the $X_i$ and at the $Y_i$, and the squares involving $ker(l)$ and $coker(r)$ commute.

Now horizontal exactness at $ker(g)$ follows from the intramural isomorphism
$ker(g)^{hor} \simeq ker(g)_{\Box}$, cor. \ref{IntramuralIsos}, combined with the [[zigzag]] of extramural isomorphisms, cor. \ref{ExtramuralIso},

$$
  \array{
     &&&&&&& ker(g)
     \\
     &&&&&&&& \Box
     \\
     &&&&&&& \swarrow
     \\
     & &&&&& \Box
     \\
     & X_1 &&&& \nearrow & & X_2
     \\
     && \Box
     \\
     & \swarrow
     \\
     \Box
     \\
     & Y_1
  }
$$

which give $\cdots \simeq {}^\Box Y_1$, and then the extramural isomorphism
$\cdots \simeq 0_{\Box} = 0$.

Exactness at $coker(g)$ is shown analogously.

Finally, building the [[connecting homomorphism]] $ker(h) \to coker(f)$ is the same as giving an  [[isomorphism]] from $coker(ker(g) \to ker(h)) \simeq ker(h)^{hor}$ to
$ker(coker(f) \to coker(g)) = coker(f)^{hor}$. This is in turn given by the intramural isomorphisms $ker(h)^{hor} \simeq ker(h)_{\Box}$ and ${}^\Box coker(f) \simeq coker(f)^{hor}$, cor. \ref{IntramuralIsos} connected by the [[zigzag]] of extramural isomorphisms, cor. \ref{ExtramuralIso}

$$
  \array{
    &&&&&&&&& ker(h)
    \\
    &&&&&&&&&& \Box
    \\
    &&&&&&&&& \swarrow
    \\
    &&&&& && & \Box
    \\
    &&&&& X_2 && \nearrow && X_3
    \\
    &&&&&& \Box
    \\
    &&&&& \swarrow
    \\
    &&&& \Box
    \\
    & Y_1 && \nearrow & & Y_2
    \\
    && \Box
    \\
    & \swarrow
    \\
    \Box
    \\
    & coker(f)
  }
  \,.
$$

=--

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

From the [[snake lemma]] one obtains in turn the
[[connecting homomorphism]] between homology groups that leads to the long exact sequence of homology; see there for details.

=--

## References
 {#References}

The salamander lemma is due to

* {#Bergman12} [[George Bergman]], _On diagram-chasing in double complexes_, Theory and Applications of Categories **26** (2012) 60-96 [[arXiv:1108.0958](http://arxiv.org/abs/1108.0958), [tac:26-03](http://www.tac.mta.ca/tac/volumes/26/3/26-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/26/3/26-03.pdf)]

based on an earlier unpublished preprint which was circulated  since 2007 ([pdf](http://sbseminar.files.wordpress.com/2007/11/diagramchasingbergman.pdf)).

Exposition:

* [[Anton Geraschenko]], _The Salamander lemma_ (2007) [[blog post](http://sbseminar.wordpress.com/2007/11/13/anton-geraschenko-the-salamander-lemma/)]

An alternative proof (also of the [[snake lemma]]) "without elements", using only [[universal properties]]:

* {#Wise} [[Jonathan Wise]], §4 in: *A non-elementary proof of the snake lemma* (2011, 2023) [[pdf](https://math.colorado.edu/%7Ejonathan.wise/papers/snake.pdf), [[Wise-SnakeLemma.pdf:file]], [MO:a/7531](https://mathoverflow.net/a/7531/381)]

[[!redirects Salamander lemma]]


Salomon Maimon

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

# Contents
* table of contents
{:toc}

> Was Maimon anbelangt, sage ich Ihnen, was ich denke. Meine Verehrung für das Talent dieses Mannes ist grenzenlos; und ich glaube fest, daß durch ihn sogar die Kantische Kritik, so wie sie durchgängig und auch von Ihnen verstanden worden, völlig umgestoßen wird. Das alles hat er getan, ohne daß es jemand merkte. Die zukünftigen Jahrhunderte werden, glaub ich, unsres Zeitalters spotten.     _Fichte Briefentwurf an Reinhold März/April 1795_

**Salomon Maimon** (1754?-1800) was a Polish-Lithuanian philosopher of Jewish origin. His father being a rabbi, Maimon was trained as a talmud scholar but his study of Spinoza and intellectual curiosity led him eventually to travel to Berlin then a center of Jewish enlightenment headed by the rationalist philosopher Moses Mendelssohn. There he developed a "neo-leibnizian" philosophy in constant dialogue with Kant's transcendental philosophy that anticipated ideas evolving into the systems of German idealism e.g. he defined philosophy as the science of the form of science which parallels Fichte's concept of _Wissenschaftslehre_ and foreshadows Hegel's absolute idea; he claimed that antinomies of reason are more ubiquituous than just the paralogisms listed in Kant's [[Critique of Pure Reason]] and produced the first broadly neo-aristotelian logic in the context of transcendental philosophy, two things that presumably weren't lost on Hegel though the latter doesn't refer directly to Maimon in his writings.

Maimon's acquaintance with mathematics (as well as with Leibniz' philosophy) led him to two other highly original and modern proposals: on the hand, he stressed the role of [[infinitesimal|infinitesimals]] in philosophical epistemology, an idea that was rediscovered by [[Hermann Cohen]] in the context of neo-Kantianism eighty years later; on the other hand, he held a relational view of [[natural number|numbers]] thereby becoming likely the first explicit advocate of [[structuralism]] in the philosophy of mathematics!

Maimon's lively account of his time and life (up to 1790) was edited by the German poet Karl Philipp Moritz and is a classical document of Jewish life and the epoch of enlightenment.

## Links

* [Stanford Encyclopedia of Philosophy entry Salomon Maimon](http://plato.stanford.edu/entries/maimon/)

* [wikisource Salomon Maimon](https://de.wikisource.org/wiki/Salomon_Maimon)

## Related entries

* [[transcendental grammar]]

* [[Critique of Pure Reason]]

* [[Immanuel Kant]]

* [[Stephan Wannowski]]

* [[Hermann Cohen]]

* [[structuralism]]

## References

* Zwi Batscha (ed.), _Salomon Maimons Lebensgeschichte von ihm selbst erzählt und herausgegeben von Karl Philipp Moritz_ , Jüdischer Verlag Frankfurt am Main 1995[1792].

* Meir Buzaglo, _Solomon Maimon - Monism, Skepticism, And Mathematics_ , University of Pittsburgh Press 2002.

* Friedrich Kuntze, _Salomon Maimons theoretische Philosophie und ihr Ort in einem System des Kritizismus_ , Logos **3** (1912) pp.285-308. ([digiZeit pdf](http://www.digizeitschriften.de/download/PPN51032052X_1912_0003/PPN51032052X_1912_0003___log40.pdf))

* Salomon Maimon, _Versuch über die Transzendentalphilosophie_ , Meiner Hamburg 2004[1790].

* Salomon Maimon, _Versuch einer neuen Logik oder Theorie des Denkens_ , Felisch Berlin 1794.

category:people


Salvatore Rappoccio

* [webpage](http://arts-sciences.buffalo.edu/physics/faculty/salvatore-rappoccio.html)

## selected writings

* [[Salvatore Rappoccio]], _The experimental status of direct searches for exotic physics beyond the standard model at the Large Hadron Collider_, Reviews in Physics 4 (2019) 100027 ([arXiv:1810.10579](https://arxiv.org/abs/1810.10579))

  (on [[LHC]] results pointing beyond the [[standard model of particle physics]])

## related $n$Lab entries

* [[LHC]]

* [[leptoquark]]

category: people


Sam Adam-Day

Hello friend, I'm Sam. I don't have much to say to you at the moment.


Sam Gutmann

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

## Selected writings

On [[adiabatic quantum computation]]:

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Michael Sipser, *Quantum Computation by Adiabatic Evolution* [[arXiv:quant-ph/0001106](https://arxiv.org/abs/quant-ph/0001106)]

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Joshua Lapan, Andrew Lundgren, Daniel Preda, *A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem*, Science  **292** 5516 (2001) 472-475 [[doi:10.1126/science.1057726](https://doi.org/10.1126/science.1057726)]

category: people


Sam Nolen

* [webpage](http://samnolen.com/)

## related $n$Lab entries

* [[Milnor's theorem on MU]]

category: people


Sam Payne

* [webpage](http://users.math.yale.edu/~sp547/)

category: people


Sam Roberts

* [GoogleScholar page](https://scholar.google.com.au/citations?user=qPjKjb4AAAAJ&hl=en)

## Selected writings

Discussion of [[quantum contextuality]] via [[singular cohomology]]:

* [[Cihan Okay]], [[Sam Roberts]], [[Stephen D. Bartlett]], [[Robert Raussendorf]], *Topological proofs of contextuality in quantum mechanics*, Quantum Information and Computation **17** (2017) 1135-1166 [[arXiv:1701.01888](https://arxiv.org/abs/1701.01888), [doi:10.26421/QIC17.13-14-5](https://doi.org/10.26421/QIC17.13-14-5)]

category: people


Sam Speight

* [website](https://www.cs.ox.ac.uk/people/sam.speight/)

## Talks

* [[Sam Speight]], *Higher-Dimensional Realizability for Intensional Type Theory*, Strength of Weak Type Theory, [[DutchCATS]], 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_speight.pdf))

category: people


Sam Staton

[[!redirects Sam Stanton]]
* [webpage](http://www.cs.ox.ac.uk/people/samuel.staton/main.html)

## Selected writings

On the foundation of [[quantum programming languages]]:

* {#Staton15} [[Sam Staton]], *Algebraic Effects, Linearity, and Quantum Programming Languages*, POPL '15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) [[doi:10.1145/2676726.2676999](https://doi.org/10.1145/2676726.2676999), [pdf](http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl2015.pdf)]

On [[quantum programming languages]] (*[[EWIRE]]*) for [[quantum circuits]] as [[domain specific embedded programming languages]]:

* {#RennelaStaton17} [[Mathys Rennela]], [[Sam Staton]], *Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory*, Logical Methods in Computer Science **16** 1 (2020) [[arXiv:1711.05159](https://arxiv.org/abs/1711.05159), doi:10.23638/LMCS-16(1:30)2020]

## Related entries

* [[deferred measurement principle]]

category: people