Eilenberg-Watts theorem

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#Contents#
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## Idea

The _Eilenberg-Watts theorem_ identifies colimit-preserving [[functors]] between [[categories of modules]] with the operations of forming [[tensor product of modules|tensor products]] with [[bimodules]].

## Statement

### For ordinary rings and modules

+-- {: .num_theorem}
<b>Eilenberg-Watts Theorem</b>

Given unital [[rings]] $R$ and $S$ and an $R$-$S$-[[bimodule]] $N$, the [[tensor product of modules|tensor product]] [[functor]]

$$
  (-) \otimes_R N
   \;\colon\;
  Mod_R \to Mod_S
$$

between the [[categories of modules]] is [[additive functor|additive]] and [[cocontinuous functor|cocontinuous]].  Conversely, if $F \colon Mod_R \to Mod_S$ is additive and cocontinuous, then it is naturally isomorphic to tensoring with a bimodule.

=--

This theorem was more or less simultaneously proved in ([Watts](#Watts)) and ([Eilenberg](#Eilenberg)).

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

Given an additive cocontinuous functor $F \colon Mod_R \to Mod_S$, the reconstructed $R$-$S$-bimodule is given as follows:

* the underlying right $S$-module is $F(R)$, where $R$ is regarded as a right module over itself in the canonical way;

* the left $R$-module structure on $F(R)$ is given for $r \in R$ and $n \in N$ by

  $$
    r \cdot n \coloneqq F(r\cdot(-))n
    \,,
  $$

  where $r \cdot (-) \colon R \to R$ denotes the right $R$-module homomorphism given by left multiplication by $r$.
=--

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

The theorem holds for nonunital rings as well, but then $B$ reconstructs as $F(R_1)$ where $R_1$ is the extension of $R$ by adjoining the unit element (the tensor product is still over the original $R$). If $F$ is a [[flat functor]] then $F(R_1)$ is a [[flat module]] over $R$.

=--

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

There are various equivalent ways to state the hypotheses of the theorem:

* $F$ is additive and [[cocontinuous functor|cocontinuous]] (i.e. preserves [[small category|small]]  [[colimits]]).
* $F$ is additive, [[right exact functor|right exact]] and preserves [[small category|small]] [[coproducts]].
* $F$ is additive and [[left adjoint]].

=--

The theorem is stated in the last form, for instance, in ([Hovey, Theorem 0.1](#Hovey)).

In fact both bimodules and [[intertwiners]] as well as [[functors]] and [[natural transformations]] form a [[category]]. In more detail the theorem is:

+-- {: .num_theorem}
###### Theorem

For $R$ and $S$ two [[rings]], the [[functor]]

$$
  {}_R Mod_{S} \stackrel{\simeq}{\to} Func_{coc}(Mod_R, Mod_S)
$$

from the category of bimodules to that of colimit-preserving additive functors between their [[categories of modules]] is an [[equivalence of categories]].

=--

### For enriched categories

Rings can be seen as one-object $V$-enriched categories where $V = Ab$ is the category of abelian groups made symmetric monoidal with the usual tensor product of abelian groups.   Similarly, bimodules between rings are the same as $V$-enriched [[profunctors]] between one-object $V$-enriched categories.  The category of right modules of a ring $R$ is the category of $V$-enriched presheaves on the corresponding one-object $V$-enriched category. Thus, we can ask if the Eilenberg-Watts theorem generalizes to $V$-enriched categories.  And indeed it does!

Suppose that $V$ is a [[Benabou cosmos]], i.e. a complete and cocomplete symmetric monoidal closed category.  Then there is a symmetric monoidal bicategory $V Mod$ where:

* objects are small $V$-enriched categories,
* morphisms are $V$-enriched profunctors,
* 2-morphisms are $V$-enriched natural transformations between profunctors.

There is also a symmetric monoidal bicategory $V Cocont$ where:

* objects are the $V$-enriched presheaf categories $[C^{op},V]$ where $C$ ranges over all small $V$-enriched categories,
* morphisms are cocontinuous $V$-functors, i.e. $V$-functors preserving all $V$-enriched colimits,
* 2-morphisms are $V$-enriched natural transformations between cocontinuous $V$-functors.

Then the following is surely true, though a reference would be helpful:

+-- {: .num_theorem}
###### Generalized Eilenberg-Watts' Theorem

Given a Benabou cosmos $V$, the symmetric monoidal bicategories $V Prof$ and $V Cocont$ are equivalent.

=--

### For other internal monoids and internal modules

The standard Eilenberg-Watts theorem is a statement about [[monoids]] and their [[actions]] in [[Ab]]. More generally one may ask for generalizations of the theorem to other internalization contexts, and in particular to [[homotopy theory]]. See the introduction of ([Hovey](#Hovey)).

### For $\infty$-algebras and $\infty$-modules
 {#ForInfinityAlgebrasAndInfinityModules}

In particular Eilenberg-Watts theorems hold true in the [[homotopy theory]] following [[model categories]] (see at [[model structure on modules over an algebra over an operad]])

* [[model structure on topological spaces]]

* [[model structure on simplicial sets]]

* [[model structure on chain complexes]]

* [[model structure on spectra]] (symmetric, orthogonal, $\mathbb{S}$-modules).

This is the main theorem in ([Hovey](#Hovey)).

More generally, we have the first half of the Eilenberg-Watts theorem in [[(∞,1)-category theory]]:

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

For $(\mathcal{C}, \otimes)$ a [[monoidal (∞,1)-category]] with [[geometric realization]] of [[simplicial objects in an (∞,1)-category]] such that the [[tensor product]] $\otimes$ preserves this in each variable, then for all [[A-∞ algebra]] $A,B,C$ in $\mathcal{C}$, the tensor product of [[∞-bimodules]]

$$
  (-) \otimes_B (-) \;\colon\; {}_A Mod_{B} \times {}_B Mod_{C} \to {}_{A} Mod_{C}
$$

preserves [[(∞,1)-colimits]] separately in each argument.

=--

This is ([Lurie, cor. 4.3.5.15](#Lurie)).

## Related concepts

* [[Morita equivalence]]

* [[additive functor]], [[Tor]]

## References
 {#References}

The original articles are

* Charles E. Watts, _Intrinsic characterizations of some additive functors_,  Proc. Amer. Math. Soc. __11__, 1960, 5–8, [MR0806.0832](http://www.ams.org/mathscinet-getitem?mr=0806.0832), [doi](http://dx.doi.org/10.2307/2032707)
 {#Watts}

* [[Samuel Eilenberg]], _Abstract description of some basic functors_, J. Indian Math. Soc. (N.S.) __24__, 1960, 231–234 (1961), [MR0125148](http://www.ams.org/mathscinet-getitem?mr=0125148)
 {#Eilenberg}

A generalized statement in which the codomain is not assumed to be a category of modules is discussed in

* A. Nyman, S. Paul Smith, _A generalization of Watts's Theorem: Right exact functors on module categories_,
Communications in Algebra __44__:7 (2016) 3160-3170 [arxiv/0806.0832](http://arxiv.org/abs/0806.0832) [doi](https://doi.org/10.1080/00927872.2015.1065873)

Generalization to [[homotopy theory]]/[[higher algebra]] is discussed in

* [[Mark Hovey]], _The Eilenberg-Watts theorem in homotopical algebra_ ([pdf](https://arxiv.org/pdf/0910.3842.pdf))
 {#Hovey}

and

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects Watts theorem]]
[[!redirects Eilenberg-Watts' Theorem]]
[[!redirects Eilenberg-Watts Theorem]]


Eilenberg-Zilber category

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### Context
#### Category theory
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# Eilenberg-Zilber categories

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## Idea

*Eilenberg-Zilber categories* are a special sort of [[generalized Reedy categories]] for which degeneracy maps behave particularly well.

## Definition

\begin{definition}\label{EZCategory}
An **Eilenberg-Zilber category** (or **EZ-category**) is a [[small category]] $R$ equipped with a [[function]] $d \colon ob(R) \to \mathbb{N}$ such that

1. For $f \colon x\to y$ a [[morphism]] of $R$:

   1. If $f$ is an [[isomorphism]], then $deg(x)=deg(y)$.

   2. If $f$ is a [[invertible morphism|noninvertible]] [[monomorphism]], then $deg(x)\lt deg(y)$.

   3. If $f$ is a noninvertible [[split epimorphism]], then $deg(x) \gt deg(y)$.

2. Every morphism factors as a [[split epimorphism]] followed by a [[monomorphism]].

3. Any [[pair]] of split epimorphisms in $R$ has an [[absolute colimit|absolute]] [[pushout]].

\end{definition}

\begin{remark}
Since a morphism is a [[split epimorphism]] if and only if its [[image]] in the [[presheaf category]] $[R^{op},Set]$ is an [[epimorphism]], condition (2) in Def. \ref{EZCategory} says that the [[(epi, mono) factorization system]] of $[R^{op},Set]$ restricts to $R$ via the [[Yoneda embedding]], while condition (3) says that the [[representable functor|representables]] are closed in $[R^{op},Set]$ under [[pushouts]] of pairs of epimorphisms.
\end{remark}

## Properties

Any EZ-category is a [[generalized Reedy category]] where $R^+$ and $R^-$ are the monomorphisms and the split epimorphisms, respectively.  Moreover, $R^{op}$ is also a generalized Reedy category where the definitions of $R^+$ and $R^-$ are reversed.  However, the generalized Reedy model structures on contravariant functors (corresponding to the generalized Reedy structure on $R^{op}$) are generally better-behaved.

Any element of a presheaf on an EZ-category $R$ is a degeneracy of a unique nondegenerate element.

If an EZ-category is also a strict [[Reedy category]] (i.e. contains no nonidentity isomorphisms), then it is an [[elegant Reedy category]].

## Examples

\begin{example}
The [[category of simplices]] $\Delta$ is an Eilenberg–Zilber category.
\end{example}

\begin{example}
The [[wreath product]] of $\Delta$ and an EZ-category (also known as the $\Theta$-construction) is again an EZ-category (Bergner–Rezk, Proposition 4.3).
In particular, Joyal's category $\Theta_n$ is an EZ-category for all $n\ge0$.
\end{example}

\begin{example}
[[Segal's category]] $\Gamma$ (used to define [[Gamma-spaces]]) is an EZ-category ([Berger & Moerdijk 2011](#BergerMoerdijk2011), Examples 6.8).
\end{example}

\begin{example}
The category of [[symmetric simplices]] (inhabited finite sets and their maps) is an EZ-category ([Berger & Moerdijk](#BergerMoerdijk2011), Examples 6.8).
\end{example}

\begin{example}
The [[cyclic category]] $\Lambda$ and the [[category of trees]] $\Omega$ are EZ-categories ([Berger & Moerdijk](#BergerMoerdijk2011), Examples 6.8).
\end{example}

\begin{example}
More generally, the total category $\mathbf{R}G$ of a
crossed group $G$ on an EZ-category $\mathbf{R}$ whose underlying Reedy category is strict
is itself an EZ-category ([Berger & Moerdijk](#BergerMoerdijk2011), Examples 6.8).
\end{example}

\begin{example}
The [[category of cubes]] $Q$ (generated by faces and degeneracies, without connections, symmetries, reversals, or diagonals) is an EZ-category ([Isaacson 2010](#Isaacson2010), Proposition 4.4).
\end{example}

\begin{example}
The category of symmetric cubes with min-connections ([Isaacson](#Isaacson2010), Definition 3.4, Proposition 3.11) is an EZ-category ([Isaacson](#Isaacson2010), Proposition 4.4).
\end{example}

## References

* {#BergerMoerdijk2011} [[Clemens Berger]], [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_, Mathematische Zeitschrift, 269, 2011 ([arXiv:0809.3341](https://arxiv.org/abs/0809.3341), [doi:10.1007/s00209-010-0770-x]( https://doi.org/10.1007/s00209-010-0770-x))

* {#Isaacson2010} [[Samuel Isaacson]], _Symmetric cubical sets_, Journal of Pure and Applied Algebra, 215, 2011 ([arXiv:0910.4948](https://arxiv.org/abs/0910.4948), [doi:10.1016/j.jpaa.2010.08.001](http://dx.doi.org/10.1016/j.jpaa.2010.08.001))

* {#BergnerRezk2013} [[Julia Bergner]], [[Charles Rezk]], _Reedy categories and the Θ-construction_, Mathematische Zeitschrift, 274, 2013 ([arXiv:1110.1066](https://arxiv.org/abs/1110.1066), [doi:10.1007/s00209-012-1082-0](http://dx.doi.org/10.1007/s00209-012-1082-0))

* {#Cisinski2019} [[Denis-Charles Cisinski]], _Higher Categories and homotopical algebra_, Cambridge University Press, 2019, [doi:10.1017/9781108588737](http://dx.doi.org/10.1017/9781108588737))

[[!redirects Eilenberg-Zilber categories]]
[[!redirects EZ-category]]
[[!redirects EZ-categories]]
[[!redirects EZ-Reedy category]]
[[!redirects EZ-Reedy categories]]


Eilenberg-Zilber map

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#Contents#
* table of contents
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## Idea

The *Eilenberg-Zilber map* is a [[natural transformation]] intertwining the [[tensor products]] of [[chain complexes]] with that of their corresponding [[simplicial abelian groups]], which is part of the [[monoidal Dold-Kan correspondence]].

Its explicit relation by the *Eilenberg-MacLane formula* expresses it in terms of sums of non-degenerate [[simplices]] inside a [[product of simplices]].

## Definition

Denote by

* $(sAb, \otimes)$ the [[monoidal category]] of [[simplicial abelian groups]] with [[tensor product]] given degreewise by the [[tensor product of abelian groups]];

* $(Ch_\bullet^+, \otimes)$ the [[monoidal category|monoidal]] [[category of chain complexes]] with its [[tensor product of chain complexes]].

* $C \colon sAb \to Ch_\bullet^+$ the chains/[[Moore complex]] [[functor]] of the [[Dold-Kan correspondence]];

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

For $A,B \in sAb$ two [[simplicial abelian group]], the **Eilenberg-MacLane formula** for the **Eilenberg-Zilber map** is the [[natural transformation]] of [[chain complexes]]

$$
  \nabla_{A,B}
  \;\colon\;
  C(A) \otimes C(B)
   \longrightarrow
  C(A \otimes B)
$$

defined on a [[pair]] of [[n-simplices|$n$-simplices]] $a \in A_p$ and $b \in B_q$ by

\[
  \label{ShuffleSumFormula}
  \begin{aligned}
  \nabla_{A,B}
  \;\colon\;
  a \otimes b
    \;\mapsto\;
   &
   \sum_{(\mu,\nu) \in Sh(p,q)}
   sgn(\mu,\nu)
   \cdot
   \big(s_\nu(a)\big)
     \otimes
   \big(s_\mu(b)\big)
   \\
   &
   \in
   C_{p+q}(A \otimes B)
   =
   A_{p+q} \otimes B_{p+q}
   \,,
  \end{aligned}
\]

where (see [here](product+of+simplices#NonDegenerateCellsInAProductOfSimplices) at *[[products of simplices]]* for the geometric interpretation):

* the [[sum]] is over all $(p,q)$-[[shuffles]]

  $$
    (\mu,\nu) = (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)
    \,,
  $$

* $sgn(\mu,\nu)$ is the [[signature of a permutation|signature]] of the corresponding [[permutation]],

* the maps $s_{\mu}$ and $s_\nu$ are iterated [[degeneracy maps]]:

  \[
    \label{IteratedDegeneracies}
    s_{\mu}
      \coloneqq
    s_{\mu_p - 1}
     \circ
    \cdots
      \circ
    s_{\mu_2 - 1}
      \circ
    s_{\mu_1 - 1}
    \,,
    \phantom{----}\text{and}\phantom{----}
    s_{\nu}
      \coloneqq
    s_{\nu_q - 1}
      \circ
    \cdots
      \circ
    s_{\nu_2 - 1}
      \circ
    s_{\nu_1 - 1}
    \,.
  \]

=--

\begin{remark}\label{Attribution}
The explicit formula (eq:ShuffleSumFormula) is due to [Eilenberg & MacLane (1953), eq. (5.3)](#EilenbergMacLane53), there called the "*$\nabla$-product*";  review includes [MacLane (1963), eq. (8.9)](#MacLane63); [May (1967)](#May67), [p. 133](https://ncatlab.org/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf#page=76); [Quillen (1969), eq. (4.2)](#Quillen69); [Loday (1992), Def. 1.6.11](#Loday92); [Gonzalez-Diaz & Real (1999)](EZAW+deformation+retraction#GonzalezDiazReal99), [p. 7](https://arxiv.org/pdf/math/0110308.pdf#page=7).

The map that is expressed by this formula was previously shown to exist, more abstractly, by [Eilenberg & Zilber (1953)](#EilenbergZilber53); cf. also [[Kerodon]], [Rem. 2.5.7.16](https://kerodon.net/tag/00RX).
\end{remark}

\begin{remark}
The shift in the indices in (eq:IteratedDegeneracies) is to be consistent with the convention that the [[shuffle]] $(\mu, \nu)$ is a [[permutation]] of $\{1, \dots, p+q\}$. In many references the shift disappears (here) by making it a permutation of $\{0, \dots, p+q-1\}$, instead.
\end{remark}

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

The [[sum]] in (eq:ShuffleSumFormula) may be understood as being over all non-degenerate simplices in the [[Cartesian product]] $\Delta[p] \times \Delta[q]$ of [[simplices]]. See at *[[products of simplices]]* ([here](product+of+simplices#NonDegenerateCellsInAProductOfSimplices)) for more on this.

=--

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

This Eilenberg-Zilber map (Def. \ref{TheEilenbergZilberMap}) [[corestriction|co]]/[[restriction|restricts]] on the [[normalized chain complex]] inside the [[Moore complex]], to a [[chain map]] of the form:

$$
  \nabla_{A,B}
    \;\colon\;
  N(A) \otimes N(B)
    \longrightarrow
  N(A \otimes B)
  \,.
$$

=--

(cf. e.g. [[Kerodon]], [Exp. 2.5.7.12.](https://kerodon.net/tag/00RT))

## Properties

### Monoidal properties

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

The Eilenberg-Zilber map (Def. \ref{TheEilenbergZilberMap}) is a [[lax monoidal transformation]] that makes $C$ and $N$ into [[lax monoidal functors]].

=--

See at *[[monoidal Dold-Kan correspondence]]* for details.

For the next statement notice that both $sAb$ and $Ch_\bullet^+$ are in fact [[symmetric monoidal categories]].

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

The EZ map (Def. \ref{TheEilenbergZilberMap}) is [[symmetric monoidal functor|symmetric]] in that for all $A,B \in sAb$ the square

$$
  \array{
    C A \otimes C B &\stackrel{\sigma}{\to}& C B \otimes C A
    \\
    {}^{\mathllap{\nabla_{A,B}}}
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{\nabla_{B,A}}}
    \\
    C(A\otimes B) &\stackrel{C(\sigma)}{\to}& C(B \otimes A)
  }
$$

[[commuting diagram|commutes]], where $\sigma$ denotes the [[braiding|symmetry isomorphism]] in $sAb$ and $Ch_\bullet^+$.

=--

### Eilenberg-Zilber theorem

\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber/Alexander-Whitney deformation retraction]])** \linebreak

Let

* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]

and denote

* by $N(A), N(B) \,\in\, Ch^+_\bullet $ (=[[ConnectiveChainComplexes]]) their [[normalized chain complexes]],

* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],

* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].

Then there is a [[deformation retraction]]

\begin{tikzcd}
  N(A) \otimes N(B)
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    from=s,
    to=t,
    -,
   shift left=1pt
  ]
  \ar[
    from=s,
    to=t,
    -,
   shift right=1pt
  ]
\end{tikzcd}

\begin{tikzcd}
  N( A \otimes B )
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    from=s,
    to=t,
    Rightarrow
  ]
\end{tikzcd}

where

* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];

* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].

\end{proposition}

For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](EZAW+deformation+retraction#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](EZAW+deformation+retraction#EilenbergMacLane54)). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](EZAW+deformation+retraction#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](EZAW+deformation+retraction#May67). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999](EZAW+deformation+retraction#GonzalezDiazReal99).

## Applications

* The Eilenberg-Zilber map induces a functor from [[simplicial Lie algebras]] to [[dg-Lie algebras]] (see [here](dgLieAlgebraOfASimplicialLieAlgebra)).

* The Eilenberg-Zilber map controls the formula for [[transgression in group cohomology]], see there fore more.

* In the context of filtered spaces $X_*, Y_*$ and their associated fundamental [[crossed complexes]] $\Pi X_*, \Pi Y_*$ there is a natural  Eilenberg-Zilber morphism
$$\eta: \Pi X_* \otimes \Pi Y_* \to \Pi (X_* \otimes Y_*)$$
which is difficult to define directly because of the complications of the tensor product of crossed complexes,  but has a direct definition in terms of the associated cubical homotopy $\omega$--groupoids. This morphism is an isomorphism of free crossed complexes if $X_*, Y_*$ are the skeletal filtrations of CW-complexes. For more on all  this, see the book *[[Nonabelian Algebraic Topology]]* p. 533.

## Related concepts

* [[Alexander-Whitney map]]

* [[Eilenberg-Zilber theorem]]

* [[product of simplices]]

* [[cap product]], [[cup product]]

* [[transgression in group cohomology]]

## References

The Eilenberg-MacLane formula was made explicit in:

* {#EilenbergMacLane53} [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) of: *On the groups $H(\Pi,n)$, I*, Ann. of Math. **58** 1 (1953) 55-106. [[jstor:1969820](https://www.jstor.org/stable/1969820)]

realizing a transformation that was shown more indirectly to exist in the proof of the [[Eilenberg-Zilber theorem]]:

* {#EilenbergZilber53} [[Samuel Eilenberg]], [[Joseph A. Zilber]], *On Products of Complexes*, American Journal of Mathematics **75** 1 (1953) 200-204 [[jstor:2372629](https://www.jstor.org/stable/2372629), [doi:10.2307/2372629](https://doi.org/10.2307/2372629)]

Review and further discussion:

* {#MacLane63} [[Saunders MacLane]], Thm. 8.8 in: *Homology*, Grundlehren der Mathematischen Wissenschaften **114**, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995)  [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maclane_homology.pdf), [doi:10.1007/978-3-642-62029-4](https://doi.org/10.1007/978-3-642-62029-4)]

* {#May67} [[Peter May]], Section 29.7 of _Simplicial objects in algebraic topology_ , Chicago Lectures in Mathematics, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

* {#Quillen69} [[Dan Quillen]], part I, section 4 of: _Rational homotopy theory_, The Annals of Mathematics, Second Series **90** 2 (1969) 205-295 [[jstor:1970725](http://www.jstor.org/stable/1970725)]

* {#Loday92} [[Jean-Louis Loday]], Section 1.6 of: *Cyclic Homology*, Grund. Math. Wiss. **301** Springer (1992) [[doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9)]

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], §2.3 in: _Equivalences of monoidal model categories_, Algebr. Geom. Topol. **3** (2003) 287-334 [[arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342), [euclid:euclid.agt/1513882376](https://projecteuclid.org/euclid.agt/1513882376)]

* [[Kerodon]], 2.5.7 *The Shuffle Product* ([00RF](https://kerodon.net/tag/00RF))

  *The Eilenberg-Zilber Homomorphism* ([00RS](https://kerodon.net/tag/00RS))

See also:

* [[Andy Tonks|A.P. Tonks]], _On the Eilenberg-Zilber Theorem for crossed complexes_. J. Pure Appl. Algebra, 179 (1-2) (2003) 199-220

* [[Tim Porter]], Section 11.2 of: _[[Crossed Menagerie]]_,

* [[Ronnie Brown]], _The twisted Eilenberg-Zilber theorem_. Simposio di Topologia (Messina, 1964), Edizioni Oderisi, Gubbio (1965), 33--37. [pdf](http://pages.bangor.ac.uk/~mas010/pdffiles/twistedez.pdf)

[[!redirects Eilenberg-MacLane forumla]]

[[!redirects shuffle map]]


Eilenberg-Zilber operad

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#Contents#
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{:toc}

## Idea ##

The _Eilenberg--Zilber operad_ $\mathcal{Z}_k$ is the [[endomorphism operad]] of the [[chain complex]] of chains on the standard $k$-[[simplex]].

By the method of [[acyclic model]]s by Eilenberg and Zilber, one shows that $\mathcal{Z}(\infty)$ is an [[E-k operad|E-∞ operad]] (therefore the name).

## Applications ##

* The [[cochains on simplicial sets]] are naturally [[algebra over an operad|algebras over]] $\mathcal{Z}(\infty)$.

## References ##

See the references at [[cochains on simplicial sets]].

[[!redirects EilenbergZilber operad]]
[[!redirects Eilenberg--Zilber operad]]


Eilenberg-Zilber theorem

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## Idea

The [[Dold-Kan correspondence]] relates [[simplicial abelian groups]] to [[connective chain complexes]]. The _Eilenberg-Zilber theorem_ says how in this context [[double complexes]] and their [[total complexes]] relate to [[bisimplicial sets|bisimplicial groups]] and their [[diagonal of a bisimplicial set|diagonals]]/[[total simplicial sets]].

Analogously there is also a version of the theorem for bi-cosimplicial abelian groups.

## Statement

### A version for simplicial abelian groups:

Let $A \colon \Delta^{op} \times \Delta^{op} \to Ab$ be a [[bisimplicial object|bisimplicial]] [[abelian group]]. Write

* $C_\bullet diag A$ for the [[Moore complex]] of its [[diagonal of a bisimplicial set|diagonal]] [[simplicial group]] $diag A \colon \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{A}{\to} Ab$;

* $Tot (C A)$ for the [[total complex]] of the [[double complex]] obtained by applying the [[Moore complex]] functor on both arguments of $A$.

+-- {: .num_theorem }
###### Theorem
**(Dold-Puppe generalization of Eilenberg-Zilber)**

There is a [[quasi-isomorphism]] (even a chain-[[homotopy equivalence]])

$$
  R : C_\bullet diag (A) \stackrel{\simeq}{\to} Tot C (A)
  \,.
$$

=--

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

Notice (see the discussion at [[bisimplicial set]]) that the diagonal simplicial set is isomorphic to the [[nerve and realization|realization]] given by the [[coend]]

$$
  diag F_{\bullet,\bullet}
  \simeq
   \int^{[n] \in \Delta} \Delta^n \times F_{n,\bullet}
  \,.
$$

=--

### Cosimplicial version

Let $A : \Delta \times \Delta \to Ab$ be a bi-cosimplicial abelian group. And let $C : Ab^\Delta \to Ch^\bullet$ the Moore cochain complex functor. Write $C(A)$ for the [[double complex]] obtained by applying $C$ to each of the two cosimplicial directions. Then we have [[natural isomorphisms]] in [[cochain cohomology]].

+-- {: .num_theorem }
###### Theorem

There is a [[natural isomorphism]]

$$
  H^\bullet C^\bullet diag(A)
  \simeq
  H^\bullet Tot C^\bullet(A)
$$

=--

###Crossed complex version

A version for [[crossed complexes]] is given by [[Andy Tonks]]. We give a summary:

First note that there is a tensor product for crossed complexes developed by Brown and Higgins.  Letting $K$ and $L$ be simplicial sets.

*  There is an Alexander-Whitney diagonal approximation defined as a natural transformation

$$a_{K,L}: \pi(K\times L)\to \pi K \otimes \pi L.$$

* Using [[shuffles]], one defines an Eilenberg - Zilber map

$$b_{K,L}:\pi K \otimes \pi L \to\pi(K\times L), $$

in a somewhat similar way to chain complexes.

*  The composite

$$\pi(K\times L)\to \pi K \otimes \pi L\to\pi(K\times L), $$

is homotopic to the identity on $\pi(K\times L)$,  whilst the other composite is the identity on $\pi K \otimes \pi L$, thus this is a strong deformation retract  of $\pi(K\times L)$.

###The Eilenberg - Zilber theorem for simplicial sets

Cegarra and Remedios have proved a version of the Eilenberg -  Zilber theorem for simplicial sets.  This is discussed under the entry on [[bisimplicial set]]s.

## Applications

### Eilenberg-Zilber/Alexander-Whitney deformation retraction

\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber/Alexander-Whitney deformation retraction]])** \linebreak

Let

* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]

and denote

* by $N(A), N(B) \,\in\, Ch^+_\bullet = $ [[ConnectiveChainComplexes]] their [[normalized chain complexes]],

* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],

* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].

Then there is a [[deformation retraction]]

\begin{tikzcd}
  N(A) \otimes N(B)
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    from=s,
    to=t,
    -,
   shift left=1pt
  ]
  \ar[
    from=s,
    to=t,
    -,
   shift right=1pt
  ]
\end{tikzcd}

\begin{tikzcd}
  N( A \otimes B )
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    from=s,
    to=t,
    Rightarrow
  ]
\end{tikzcd}

where

* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];

* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].

\end{proposition}

For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](EZAW+deformation+retraction#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](EZAW+deformation+retraction#EilenbergMacLane54)). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](EZAW+deformation+retraction#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](EZAW+deformation+retraction#May67). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999](EZAW+deformation+retraction#GonzalezDiazReal99).

### Homology of products of topological spaces
 {#ProdTopSp}

The following is the special case of the [[Eilenberg-Zilber/Alexander-Whitney deformation retraction]] (Prop. \ref{EZAWDeformationRetract}) for [[singular simplicial sets]], observing that forming [[free simplicial abelian groups]] preserves monoidal products.
This is the original motivating application in [Eilenberg &Zilber 1953](#EilenbergZilber53)

Let $X$ and $Y$ be two [[topological spaces]]. Their [[singular homology]] [[chain complexes]] $C_\bullet(X)$ and $C_\bullet(Y)$ are the [[Moore complexes]] of the [[simplicial abelian groups]] $\mathbb{Z}[Sing X]$ and $\mathbb{Z}[Sing Y]$ [[free abelian group]] on their [[singular simplicial complexes]].

So from the Dold-Puppe [[quasi-isomorphism]] $R$ from above we have a [[quasi-isomorphism]] of [[chain complexes]] of their [[product topological space]]:

$$
  \begin{aligned}
    C_\bullet(X \times Y)
    & \coloneqq
    C_\bullet( \mathbb{Z}[Sing X \times Sing Y] )
    \\
    &=
    C_\bullet( diag \mathbb{Z}[Sing X_\bullet]
      \otimes \mathbb{Z}[Sing Y_\bullet] )
    \\
    &
    \underoverset{\;\simeq\;}{R}{\longrightarrow}
      Tot
      C_\bullet(\mathbb{Z}[Sing X]) \otimes C_\bullet(\mathbb{Z}[Sing Y])
    \\
    &
    =
    Tot C_\bullet(X) \otimes C_\bullet(Y)
  \end{aligned}
$$

and hence in particular an isomorphism in [[singular cohomology]].

By following through these maps one can obtain an explicit description of the quasi isomorphism if need be.

### Transgression in group cohomology

See at *[[transgression in group cohomology]]*.

## References

Original references:

* {#EilenbergZilber53} [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629))

* {#EilenbergMacLane54} [[Samuel Eilenberg]], [[Saunders MacLane]], Section 2 of: *On the Groups $H(\Pi,n)$, II: Methods of Computation*, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702), [doi:10.2307/2372629](https://doi.org/10.2307/2372629))

using the definition of the [[Eilenberg-Zilber map]] in:

* {#EilenbergMacLane53} [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) of: *On the groups $H(\Pi,n)$*, I*, Ann. of Math. (2) 58, (1953), 55–106. ([jstor:1969820](https://www.jstor.org/stable/1969820))

Review in:

* {#Dold95} [[Albrecht Dold]], Section VI 12.1 in *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf))

A weak version of the simplicial statement:

* [[Charles Weibel]], Theorem 8.1.5 in: _An introduction to homological algebra_

The stronger version as stated above:

* [[Albrecht Dold]], [[Dieter Puppe]], Chapter 2 of: _Homologie nicht-additiver Funktoren. Anwendungen_, Ann. Inst. Fourier **11** (1961) pp. 201-312. ([numdam:AIF_1961__11__201_0](http://archive.numdam.org/item/AIF_1961__11__201_0), [pdf](http://archive.numdam.org/article/AIF_1961__11__201_0.pdf))

where is is ascribed to [[Pierre Cartier]].  This result is discussed in [chapter 4](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) of

* [[Paul Goerss]], [[J. F. Jardine]], _[[Simplicial homotopy theory]]_
([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

The cosimplicial version of the theorem appears as:

* L. Grunenfelder and M. Mastnak, Theorem A.3 in: _Cohomology of abelian matched pairs and the Kac sequence_ ([arXiv:math/0212124](http://arxiv.org/abs/math/0212124))

The crossed complex version is given in

* [[Andy Tonks|A.P. Tonks]], _On the Eilenberg-Zilber Theorem for crossed complexes_. J. Pure Appl. Algebra, 179~(1-2) (2003) 199--220,

  (for more detail see Tonks' [thesis](https://groupoids.org.uk/pdffiles/tonksthesis.pdf)),

and on page 360 of *[[Nonabelian Algebraic Topology]]*.


Eilenberg-Zilber/Alexander-Whitney deformation retraction

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## Idea

The [[Dold-Kan correspondence]] between [[connective chain complexes]] and [[simplicial abelian groups]] is not quite compatible with the [[tensor product]]-structure on both sides, but it is so up to [[homotopy]] (see at *[[monoidal Dold-Kan correspondence]]*). One aspect of this is the *[[Eilenberg-Zilber theorem]]*, saying that the [[Eilenberg-Zilber map]] from the [[tensor product of chain complexes]] of [[normalized chain complexes]] of two [[abelian groups]] to the normalized chain complex of their degree-wise [[tensor product of abelian groups]] is not quite an [[isomorphism]], but is a [[homotopy equivalence]], in fact a [[deformation retraction]], with homotopy-inverse the [[Alexander-Whitney map]].

## Statement

\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber map|Eilenberg-Zilber]]/[[Alexander-Whitney map|Alexander-Whitney]] [[deformation retraction]])** \linebreak

Let

* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]

and denote

* by $N(A), N(B) \,\in\, Ch^+_\bullet $ (=[[ConnectiveChainComplexes]]) their [[normalized chain complexes]],

* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],

* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].

Then there is a [[deformation retraction]]

\begin{tikzcd}
  N(A) \otimes N(B)
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    from=s,
    to=t,
    -,
   shift left=1pt
  ]
  \ar[
    from=s,
    to=t,
    -,
   shift right=1pt
  ]
\end{tikzcd}

\begin{tikzcd}
  N( A \otimes B )
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    from=s,
    to=t,
    Rightarrow
  ]
\end{tikzcd}

where

* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];

* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].

\end{proposition}

For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](#EilenbergMacLane54)), review in [MacLane 1975, VIII 8](#MacLane75), [Dold 1995, VI 12.1](#Dold95). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](#May67), the latter at least mentioned in [MacLane 1975, VIII Cor. 8.9](#MacLane75). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999, p. 7](#GonzalezDiazReal99).

## Related concepts

* **Alexander-Whitney map**

* [[Eilenberg-Zilber map]]

## References

The [[Eilenberg-Zilber theorem]] is due to

* {#EilenbergZilber53} [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629))

* {#EilenbergMacLane54} [[Samuel Eilenberg]], [[Saunders MacLane]], Section 2 of: *On the Groups $H(\Pi,n)$, II: Methods of Computation*, Annals of Mathematics, Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702), [doi:10.2307/2372629](https://doi.org/10.2307/2372629))

using the definition of the [[Eilenberg-Zilber map]] in:

* {#EilenbergMacLane53} [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) of: *On the groups $H(\Pi,n)$*, I*, Ann. of Math. (2) 58, (1953), 55–106. ([jstor:1969820](https://www.jstor.org/stable/1969820))

Review and further discussion:

* {#May67} [[Peter May]], Section 29 of: _Simplicial objects in algebraic topology_ , Chicago Lectures in Mathematics, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

* {#MacLane75} [[Saunders Mac Lane]], _Homology_ (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 ([doi:10.1007/978-3-642-62029-4](https://link.springer.com/book/10.1007/978-3-642-62029-4))

* {#Dold95} [[Albrecht Dold]], Section VI 12.1 in *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf))

* {#GonzalezDiazReal99} [[Rocio Gonzalez-Diaz]], [[Pedro Real]], *A Combinatorial Method for Computing Steenrod Squares*, Journal of Pure and Applied Algebra 139 (1999) 89-108 ([arXiv:math/0110308](https://arxiv.org/abs/math/0110308))

[[!redirects EZAW deformation retraction]]


Eilenberg–Moore spectral sequence

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## Idea

The _Eilenberg-Moore spectral sequence_ is a [[spectral sequence]] for the [[singular homology]] of [[product topological spaces]] which generalizes the [[Künneth theorem]] in two ways

1. it applies without assumption on vanishing of [[Tor]]-groups;

1. it generalizes from [[product topological spaces]] to [[fiber products]] of [[fibrations]] and to products over group [[actions]].

## References

Original articles:

* [[Larry Smith]], _Homological Algebra and the Eilenberg-Moore Spectral Sequence_, Transactions of the American Mathematical Society, Vol. 129, No. 1, (1967) ([jstor:1994364](http://www.jstor.org/stable/1994364))

* [[Larry Smith]], _On the construction of the Eilenberg-Moore spectral sequence_, Bull. Amer. Math. Soc. 75 (1969), 873-878 ([euclid:bams/1183530666](https://projecteuclid.org/euclid.bams/1183530666))

* Atsushi Yamaguchi, _Note on the Eilenberg-Moore spetral sequence_, Publ RIMS, Ykoto Univ. 22 (1986) ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=22&iss=5&rank=3))

Textbook accounts include

* [[Alan Hatcher]], _[Spectral sequences in algebraic topology](http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html)_ _III: The Eilenberg-Moore spectral sequence_ ([pdf](http://www.math.cornell.edu/~hatcher/SSAT/SSch3.pdf))

[[!redirects Künneth spectral sequence]]

[[!redirects Eilenberg-Moore spectral sequence]]


einselection

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## Idea

**Einselection**, short for "**e**nvironment-**in**duced [[superselection sector|superselection]]", is a name coined by [Zurek (2003)](#Zurek03) for a process which explains the emergence of [[classical physics|classical]] behavior in an open [[quantum system]] from its interaction with its environment.

## Related concepts

* [[quantum decoherence]]

* [[consistent histories]]

## References

* {#Zurek03} [[Wojciech Zurek]], *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75** (2003) 715-775 [[quant-ph/0105127](http://arxiv.org/abs/quant-ph/0105127), [doi:10.1103/RevModPhys.75.715](http://dx.doi.org/10.1103/RevModPhys.75.715)]


Einstein equation

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## Idea

What are called _Einstein's equations_ are the [[equations of motion]] of gravity: the [[Euler-Lagrange equations]] induced by the [[Einstein-Hilbert action]].

They say that the [[Einstein tensor]] $G$ of the [[metric]]/the [[field (physics)|field]] of [[gravity]] [[equality|equals]] the [[energy-momentum tensor]] $T$ of the remaining [[force]]- and [[matter]]-[[field (physics)|fields]]:

$$
  G = T
  \,.
$$

## Properties

### Existence and uniqueness

Given a choice of [[Cauchy surface]] $\Sigma$, the [[initial value problem]] for Einstein's [[differential equations]] of motion is determined by a choice of [[Riemannian metric]] on $\Sigma$ and a second fundamental form along $\Sigma$.

With this data a solution to the equation exists and is unique. ([Klainerman-Nicolo 03](#KlainermanNicolo03)).

## Related concepts

* [[equations of motion]]

  * [[Newton's laws of motion]]

  * [[Hamilton's equations]], [[de Donder-Weyl-Hamilton equation]]

## References

### General

A general discssion is for instance in section 11 of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

A discussion of the vacuum Einstein equations (only gravity, no other fields) in terms of [[synthetic differential geometry]] is in

* [[Gonzalo Reyes]], _A derivation of Einstein's vacuum field equations_ ([pdf](http://marieetgonzalo.files.wordpress.com/2009/12/a-derivation-of-einsteins-vacuum-field-equations1.pdf))

### PDE theory

Genuine [[PDE]] theory for Einstein's equations goes back to local existence results by [[Yvonne Choquet-Bruhat]] in the 1950s. Global existence in the presence of a [[Cauchy surface]] was then shown in

* [[Sergiu Klainerman]], Francesco Nicolo, _The evolution problem in general relativity_, Progress in Mathematical Physics, 25. Birkhäuser Boston, Inc., Boston, MA, 2003. xiv+385 pp. ISBN: 0-8176-4254-4
  {#KlainermanNicolo03}

For further developments see

* H. Friedrich, A. D. Rendall, _The Cauchy Problem for the Einstein Equations_ ([arXiv:gr-qc/0002074](http://arxiv.org/abs/gr-qc/0002074))

* Alan D. Rendall, _Partial differential equations in general relativity_, Oxford University press 2008 ([web](http://www.ima.org.uk/viewItem.cfm?cit_id=384200))

* Hans Ringström, _The Cauchy Problem in General Relativity_, ESI Lectures in Mathematics and Physics 2009 ([web](http://www.ems-ph.org/books/book.php?proj_nr=99))

* [[Yvonne Choquet-Bruhat]], General relativity and the Einstein equations. Oxford University Press (2008) ([publisher](http://global.oup.com/academic/product/general-relativity-and-the-einstein-equations-9780199230723;jsessionid=24BBEFC00EAFA5C81BC0A50794DE256F?cc=nl&lang=en&))

See also _[[The Cauchy Problem in Classical Supergravity]]_

[[!redirects Einstein's equation]]

[[!redirects Einstein equations]]
[[!redirects Einstein's equations]]

[[!redirects Einstein equation of motion]]
[[!redirects Einstein equations of motion]]

[[!redirects Einstein's law of motion]]
[[!redirects Einstein's laws of motion]]

[[!redirects Einstein field equation]]
[[!redirects Einstein field equations]]


Einstein manifold

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* table of content
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## Definition

An _Einstein manifold_ is a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] $(X,g)$ (a [[spacetime]]) such that the [[Ricci tensor]] is proportional to the [[metric]] [[tensor]]

$$
  Ric = \lambda g \in Sym^2 \Gamma(T X)
$$

by a proportionality constant $\lambda \in \mathbb{R}$. Such a metric $g$ is called an **Einstein metric**.

## Properties

Einstein manifolds are precisely the solutions of [[Einstein's equations]] for pure [[gravity]] with [[cosmological constant]] $\lambda$.

## Examples
  {#Examples}

* A manifold of [[dimension]] 7 and of [[weak G2-holonomy]] with weakness parameter $\lambda$ -- $d \omega_3 = \lambda \star \omega_3$ -- is canonically an Einstein manifold with cosmological constant $\lambda$.

* [[de Sitter spacetime]], [[anti de Sitter spacetime]]

* all [[quaternion-Kähler manifolds]] are Einstein manifolds see [there](quaternion-Kähler+manifold#AsEinsteinManifolds)

## Related concepts

* [[Kähler-Einstein manifold]]

## References

* [[Nigel Hitchin]], _Compact four-dimensional Einstein manifolds_, J. Diff geom. 9 (1974) 435-441 ([pdf](http://intlpress.com/JDG/archive/1974/9-3-435.pdf))

* Jongmin Park, Jaewon Shin, Hyun Seok Yang, *Unification of Einstein Manifolds* ([arXiv:2109.00001](https://arxiv.org/abs/2109.00001))

[[!redirects Einstein manifolds]]

[[!redirects Einstein metric]]
[[!redirects Einstein metrics]]


Einstein summation convention

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## Idea
In situations such as the [[tensor calculus]], the **Einstein summation convention** allows for compact expression of sums over common upper and lower indices. For example, for $n$-dimensional vector $v$ and covector $e$, we write $v_i e^i \coloneqq \sum_{1 \le i \le n} v_i e^i$.

## Related entries

* [[Kronecker delta]]

* [[Levi-Civita symbol]]

* [[string diagram]]

* [[Feynman slash notation]]

## References

See also

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

[[!redirects Einstein notation]]
[[!redirects Einstein summation]]


Einstein tensor

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## Definition

Given a [[pseudo-Riemannian manifold]] $(X,g)$, the **Einstein tensor** is the [[tensor]] field on $X$ given by

$$
  G \coloneqq Ric - \tfrac{1}{2}R g
  \,,
$$

where

* $Ric$ is the [[Ricci tensor]]

* $R$ is t
he [[scalar curvature]]

of the metric $g$.

## In gravity

The Einstein tensor is the [[variational derivative]] of the pure [[Einstein-Hilbert action]]. Therefore the [[equations of motion]] of [[gravity]] ([[Einstein's equations]]) are the [[equation]]

$$
  G = T
  \,,
$$

where $T$ is the [[energy-momentum tensor]] of the further [[force]]- and [[matter]]-[[field (physics)|fields]] of the [[theory (physics)|theory]].


Einstein-Hilbert action

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## Idea

The _Einstein-Hilbert action_ is the [[action functional]] that defines the dynamics of [[gravity]] in [[general relativity]]. It is a canonical invariant of [[pseudo-Riemannian manifold]]s.

## Definition

Let $X$ be a [[compact topological space|compact]] [[smooth manifold]] [[manifold with boundary|with boundary]].

For $(X,g)$ the structure of a  [[Riemannian manifold]] or [[pseudo-Riemannian manifold]] on $X$, hence for $g$ a (pseudo-)Riemannian metric on $X$, its **vacuum Einstein-Hilbert action** is the number

$$
  g \;\mapsto\; \int_X R(g) dvol(g)
  \,,
$$

where

* $R(g) : X\to \mathbb{R}$ is the [[scalar curvature]] function of the [[metric]] $g$;

* $dvol(g)$  is the [[volume form]] defined by the metric

* $\int_X$ is [[integration of differential forms]] over $X$.

If the metric is instead encoded in terms of an [[Poincare group]]-[[connection on a bundle|connection]] $\nabla$ (the [[first-order formulation of gravity]]) then (for $dim X = 4$ and assuming for simplicity that the underlying bundle is trivial) this is equivalently

$$
  \nabla \;\mapsto\; \int_X \langle R \wedge e \wedge e \rangle
  \,,
$$

where now

* $R$ denotes the [[curvature]] 2-form of the [[orthogonal group]]-connection part of $\nabla$;

* $e$ denotes the [[vielbein]], (the translation part of the connection 1-form);

* $\langle -\rangle$ is the [[invariant polynomial]] which in terms of the canonical basis has components $(\epsilon_{a b c d})$.

In either case this assignment defines a function

$$
  S_{EH} \;\colon\; Met(X) \longrightarrow \mathbb{R}
$$

on the space of (pseudo-)Riemannian metrics on $X$. This is the _Einstein-Hilbert [[action functional]]_.

This functional is invariant under the [[action]] of [[diffeomorphisms]] on $X$ (e.g. [Bertschinger 02, section 4.2](#Bertschinger02)).
Therefore more properly the EH-action functional is the function descended to the [[moduli stack]] $Met(X)//Diff(X)$, see at _[[general covariance]]_.

More generally, this extends to an [[action functional]] for gravity coupled to "matter" (which in the context of [[general relativity]] conventionally means every other field, for instance the [[electromagnetic field]]) by simply adding the matter action on a given gravitational background

$$
  (g,\phi) \;\mapsto\; \int_X (L_{EH}(g) + L_\phi(g)) d vol(g)
  \,.
$$

The [[critical point]]s of the Einstein-Hilbert action define the physically realized metrics (and matter fields) on $X$, hence the physically realized [[spacetimes]] with underlying manifold $X$. This are [[Einstein's equations]] (the [[Euler-Lagrange equation]]s of this action)

$$
  \delta S_{EH}(g) = 0
  \,.
$$

To derive the [[Einstein equation|Einstein Field equations]] from this, we will look at the Einstein-Hilbert action action in its usual form ($c$, the speed of light, is set to unity), in spacetime with no matter fields (i.e., matter Lagrangian $\mathcal{L}_M=0$):

$$S = \frac{1}{16\pi G} \int \operatorname{tr}_g(R) \sqrt{-g} \, \mathrm{d}^4x \;,$$
The variation of this action can be performed:
$$\frac{\delta S}{\delta g}=0$$
The [Wikipedia Page](http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action) shows how to vary the action (with matter fields) through components, giving us the famed [[Einstein equation|Einstein field equations]]:
$$R-\frac{1}{2}g\,\operatorname{tr}_g(R)=T$$
Here, $T$ is the [[stress-energy tensor|energy-momentum tensor]].

(...)
## History
 {#History}

Historically, Einstein had been working towards what today are called [[Einstein's equations]] for some time. What is today called the [[Einstein tensor]] in the equation had been missing the trace term in one of his first publications, and other things remained unclear. Einstein was working with the 10 coordinate components of these [[tensors]] and trying to make the theory match classical [[Newtonian physics]] in the right limit. The "[[general covariance]]" of the equations made this a subtle problem, at least back in the times when these issues were just beginning to be appreciated.

Hilbert had been following these developments and had listened to Einstein talk about them. Apparently at some point Hilbert, who believed in the axiomatic foundation of physics (as is evident from [[Hilbert's sixth problem]]), decided that whatever the correct equations of motion are, they ought to follow from an [[action principle]], hence that they characterize the [[critical points]] of an [[action functional]] on the space of [[pseudo-Riemannian metrics]] on some [[smooth manifold]]. Furthermore he observed that the most obvious such functional with the right [[symmetry]] properties  is the [[integral]] over the [[Riemann scalar curvature]] times the [[volume form]] of the given metric (hence the "Einstein-Hilbert action"). He computed the [[variational calculus|variational derivative]] of this and thereby found [[Einstein's equations]] apparently around the same time if not a little bit before Einstein published the final version of his theory.

In ([Sauer-Majer, from p. 402](#SauerMajer)) a talk of Hilbert is reproduced that reviews Einstein's work, from the point of view of Hilbert's having known the right answer all along from the variational principle (hence from the axioms) as Einstein struggled (until a few days before the lecture was given!) with finding the right equations of gravity. This comment culminates ([Sauer-Majer, from p. 405](#SauerMajer)) with the curious remark, paraphrasing just a little: that the fact that Einstein after his "colossal detour" ends up with the same equations of motion as Hilbert already had is a "nice consistency check" (*schöne Gewähr*)

p. 402:

> Das von mir aufgestellte Hamiltonsche Prinzip lautet

> $\delta \int \int \int \int I \sqrt{g} \, d x^1 d x^2 d x^3 d x^4 = 0$

(this defines the [[critical point]] of the [[action functional]] being the [[integral]] of the [[Lagrangian]] $I$ over [[spacetime]], i.e the [[principle of extremal action]])

p. 404:

> Indess mündet das Schlussresultat der letzten Einsteinschen Untersuchung wieder auf ein Hamiltonsches Prinzip, das dem ursprünglich von mir aufgestellten gleicht; ja es könnte sein, dass diese Einsteinsche Theorie inhaltlich sich mit der von mir ursprünglich aufgestellten Theorie völlig deckt.

p. 405:

> Als Entgelt dafür können wir gegenwärtig als sicher ansehen, dass die oben von mir aufgestellten Gl. gewiss den _Grundstock für die Weltgl._ ausmachen. In einer letzten vor wenigen Tagen erschienenen Note gelangt Einstein zu dem Hamiltonschen Prinzip:

> $\delta \int \int \int \int  \{K + \alpha \Phi + \beta \phi\} \sqrt{g} d x^1 d x^2 d x^3 d x^4 = 0$

> $[...]$

> Und wenn auf dem kollossalen Umweg über Levi Civita, Weyl, Schouten, Eddington Einst. zu dem Resultat zurückgelangt, so liegt darin sicher eine schöne Gewähr.

In Constance Reid's biography of Hilbert, one finds in Chapter XVII the following:

> In Göttingen, in the weekly Hilbert-Debye seminars, it seemed to those few students who were left $[$ due to WWI $]$ that the "living pulse" of physical research was at their finger tips. The work of Einstein as he pressed forward toward a general theory of relativity was followed with great interest. Also followed was the work of others who were trying to reach the same goal. Hilbert was especially fascinated by the ideas of Gustav Mie, then in Greifswald, who was attempting to develop a theory of matter on the fundamentals of the relativity principle; and in his own investigations he was able to bring together Mie's program of pure field theory and Einstein's theory of gravitation. At the same time, while Einstein was attempting in a rather roundabout way to develop the binding laws for the 10 coefficients of the differential form which determines gravitation, Hilbert independantly solved the problem in a different, more direct way. Both men arrived at almost the same time at the same goal. As the western front settled down for the winter, Einstein presented his two papers, "On general relativity theory" to the Berlin Academy on November 11 and 25; Hilbert presented his first note on "The foundations of physics" to the Royal Society of Science in Gottingen on November 20, 1915. It was a remarkable coincidence ... but even more remarkable, according to $[$Max$]$ Born (who was now in Berlin with Einstein), was the fact that it led, not to a controversy over priority, but to a series of friendly encounters and letters. Hilbert freely admitted, and frequently stated in lectures, that the great idea was Einstein's.

More citations on the priority issue are listed in [this Physics.SE comment](http://physics.stackexchange.com/a/56893/5603).

The success of the Einstein-Hilbert [[action principle]] in the formulation of [[gravity]] probably had a considerable influence on the theoretical physics to follow and led some physcist to explore more systematically the space of action functionals invariant under prescribed symmetries. This way notably the [[supergravity]] variant of Einstein-gravity in the context of Riemannian [[supergeometry]] was found, a theory which eventually leads to [[superstring theory]]. These variants of course remain speculative as [[theory (physics)|theories]] of [[phenomenology|phenomenological physics]], but to some extent their development does follow Hilbert's attitude in the discovery of [[general relativity]].

## Related concepts

* [[Einstein-Maxwell theory]]

* [[Einstein-Yang-Mills theory]]

* [[Einstein-Yang-Mills-Dirac theory]]

* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

* [[higher order curvature corrections]]

[[!include standard model of fundamental physics - table]]

## References

### General

Lecture notes include

* Edmund Bertschinger, _Symmetry transformations, the Einstein-Hilbert action, and gauge invariance_ 2002 ([pdf](http://web.mit.edu/edbert/GR/gr5.pdf))
 {#Bertschinger02}

Section _[Prequantum gauge theory and Gravity](geometry+of+physics#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ at

* _[[geometry of physics]]_

### History
 {#History}

* {#SauerMajer} [[Tilman Sauer]], [[Ulrich Majer]] with  Arne Schirrmacher, Heinz-Jürgen Schmidt (eds.): *David Hilbert's Lectures on the foundations of physics 1915-1927 --- Relativity, Quantum Theory and Epistemology*, Springer (2009) [[doi:10.1007/b12915](https://doi.org/10.1007/b12915)]

[[!redirects Einstein-Hilbert action functional]]


Einstein-Maxwell theory

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## Idea

What is called _Einstein-Maxwell theory_  in [[physics]] is the [[theory (physics)|theory]]/[[model (in theoretical physics)]] describing [[gravity]] together with [[electromagnetism]].

It is a [[local Lagrangian]] [[field (physics)|field]] theory defined by the [[action functional]] which is the [[Einstein-Hilbert action]] plus the [[Maxwell action functional]]  involving the given metric,

$$
  S_{G+EM}
  \;
   \colon
  \;
  (e, \nabla)
  \mapsto
  \int_{X} R(e) vol(e)
  +
  \int_X F_\nabla \wedge \star_e F_\nabla
  \,,
$$

where

* $X$ is the [[compact topological space|compact]] [[smooth manifold]] underlying [[spacetime]],

* $e$ is the [[vielbein field]] which encodes the [[field (physics)|field]] of [[gravity]]

* $\nabla$ is the $U(1)$-[[principal connection]] which encodes the [[electromagnetic field]],

* $vol(e)$ is the [[volume form]] induced by $e$;

* $R(e)$ is the [[scalar curvature]] of $e$;

* $F_\nabla$ is the [[field strength]]/[[curvature]] [[differential 2-form]] of $\nabla$;

* $\star_e$ is the [[Hodge star]] operator induced by $e$.

This is the special case of [[Einstein-Yang-Mills theory]] for the [[gauge group]] being the [[circle group]].

## Related concepts

* [[D=5 Einstein-Maxwell theory]]

* [[charged black hole]]

* [[Einstein-Hilbert action]]

* **Einstein-Maxwell theory**

* [[Einstein-Yang-Mills theory]]

* [[Einstein-Yang-Mills-Dirac theory]]

* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

[[!include standard model of fundamental physics - table]]

## References

Section _[Prequantum gauge theory and Gravity](geometry+of+physics#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ in

* _[[geometry of physics]]_

See also

* Wikipedia, _[Maxwell equations in curved spacetime](http://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime)_


Einstein-Yang-Mills theory

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## Idea

What is called _Einstein-Yang-Mills theory_  in [[physics]] is the [[theory (physics)|theory]]/[[model (in theoretical physics)]] describing [[gravity]] together with [[Yang-Mills fields]] such as the [[electroweak field]] or the [[strong nuclear force]] of [[quantum chromodynamics]]. For the special case that the [[gauge group]] is the [[circle group]] this reproduces [[Einstein-Maxwell theory]].

Einstein-Yang-Mills theory is a [[local Lagrangian]] [[field (physics)|field]] theory defined by the [[action functional]] which is the [[Einstein-Hilbert action]] plus the [[Yang-Mills action functional]] involving the given metric,

$$
  S_{G+YM}
  \;
   \colon
  \;
  (e, \nabla)
  \mapsto
  \int_{X} R(e) vol(e)
  +
  \int_X \langle F_\nabla \wedge \star_e F_\nabla\rangle
  \,,
$$

where

* $X$ is the [[compact topological space|compact]] [[smooth manifold]] underlying [[spacetime]],

* $e$ is the [[vielbein field]] which encodes the [[field (physics)|field]] of [[gravity]]

* $\nabla$ is the $G$-[[principal connection]] which encodes the [[Yang-Mills field]],

* $vol(e)$ is the [[volume form]] induced by $e$;

* $R(e)$ is the [[scalar curvature]] of $e$;

* $F_\nabla$ is the [[field strength]]/[[curvature]] [[differential 2-form]] of $\nabla$;

* $\star_e$ is the [[Hodge star]] operator induced by $e$.

* $\langle -,-\rangle$ is the given [[invariant polynomial]] on the [[Lie algebra]] of the [[gauge group]].

## Related concepts

* [[charged black hole]]

* [[Einstein-Hilbert action]]

* [[Einstein-Maxwell theory]]

* **Einstein-Yang-Mills theory**

* [[Einstein-Yang-Mills-Dirac theory]]

* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

[[!include standard model of fundamental physics - table]]

## References

Section _[Prequantum gauge theory and Gravity](geometry+of+physics#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ in

* _[[geometry of physics]]_

On [[Yang-Mills monopoles]]:

* J. Bjoraker, Y. Hosotani, *Monopoles, Dyons and Black Holes in the Four-Dimensional Einstein-Yang-Mills Theory*, Phys. Rev. D62:043513, 2000 ([arXiv:hep-th/0002098](https://arxiv.org/abs/hep-th/0002098))

[[!redirects Einstein-Yang-Mills theories]]


Einstein-Yang-Mills-Dirac theory

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## Idea

What is called _Einstein-Yang-Mills-Dirac theory_  in [[physics]] is the [[theory (physics)|theory]]/[[model (in theoretical physics)]] describing [[gravity]] together with [[Yang-Mills fields]] and coupled to [[fermion|fermionic]] [[matter]].

Einstein-Yang-Mills-Dirac theory is a [[local Lagrangian]] [[field (physics)|field]] theory defined by the [[action functional]] which is the [[Einstein-Hilbert action]] plus the [[Yang-Mills action functional]] involving the given metric,

$$
  S_{G+YM}
  \;
   \colon
  \;
  (e, \nabla, \psi)
  \mapsto
  \int_{X} R(e) vol(e)
  +
  \int_X \langle F_\nabla \wedge \star_e F_\nabla\rangle
  +
  \int_X (\psi, D_{e,\nabla} \psi)
  \,,
$$

where

* $X$ is the [[compact topological space|compact]] [[smooth manifold]] underlying [[spacetime]],

* $e$ is the [[vielbein field]] which encodes the [[field (physics)|field]] of [[gravity]]

* $\nabla$ is the $G$-[[principal connection]] which encodes the [[Yang-Mills field]],

* $vol(e)$ is the [[volume form]] induced by $e$;

* $R(e)$ is the [[scalar curvature]] of $e$;

* $F_\nabla$ is the [[field strength]]/[[curvature]] [[differential 2-form]] of $\nabla$;

* $\star_e$ is the [[Hodge star]] operator induced by $e$.

* $\langle -,-\rangle$ is the given [[invariant polynomial]] on the [[Lie algebra]] of the [[gauge group]].

## Related concepts

* [[Einstein-Hilbert action]]

* [[Einstein-Maxwell theory]]

* [[Einstein-Yang-Mills theory]]

* **Einstein-Yang-Mills-Dirac theory**

  * [[spectral action]]

* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

[[!include standard model of fundamental physics - table]]

## References

* Gerd Rudolph, Torsten Tok, Igor P. Volobuev, _Exact solutions in Einstein-Yang-Mills-Dirac systems_, J.Math.Phys. 40 (1999) 5890-5904 ([arXiv:gr-qc/9707060](http://arxiv.org/abs/gr-qc/9707060))

Section _[Prequantum gauge theory and Gravity](geometry+of+physics#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ in

* _[[geometry of physics]]_

[[!redirects Einstein-Yang-Mills-Dirac theories]]


Einstein-Yang-Mills-Dirac-Higgs theory

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## Idea

The [[theory (physics)|theory]] in [[physics]] which describes the fundamental physics of the [[observable universe]] to best present knowledge is a [[local Lagrangian]] [[field (physics)|field]] theory which combines

| [[theory (physics)|theory]]: | [[Einstein-Hilbert action|Einstein]]- | [[Einstein-Maxwell theory|Maxwell]]- | [[Einstein-Yang-Mills theory|Yang-Mills]]- |  [[Einstein-Yang-Mills-Dirac theory|Dirac]]- | [[Einstein-Yang-Mills-Dirac-Higgs theory|Higgs]] |
|--|----------|---------|------------|--------|-------|
|  |  [[gravity]] | [[electromagnetism]] | [[electroweak  field|electroweak]] and [[strong nuclear force]] | [[fermion|fermionic]] [[matter]] | [[scalar field]] |
| [[field (physics)|fields]] | [[vielbein field]] $e$ | $U(1)$-[[principal connection]] $\nabla_{em}$  | $G$-[[principal connection]]  |  [[spinor]] $\psi$ |  [[scalar field]] $H$ |
| [[Lagrangian]] $L = $ | $R(e) vol(e) + $ |  | $F_{\nabla_{}} \wedge \star_e F_{\nabla_{}} + $ | $(\psi , D \psi) vol(e) + $ |  $\nabla \bar H \wedge \star_e \nabla H  + \left(\lambda  {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e) $ |  |

## Related concepts

* [[Einstein-Hilbert action]]

* [[Einstein-Maxwell theory]]

* [[Einstein-Yang-Mills theory]]

* [[standard model of particle physics]]

* [[standard model of cosmology]]

## References

Section _[Prequantum gauge theory and Gravity](geometry+of+physics#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ in

* _[[geometry of physics]]_

On [[Yang-Mills monopoles]]:

* K. Lee, V. P. Nair, Erick J. Weinberg, *Black Holes in Magnetic Monopoles*, Phys. Rev. D45 (1992) 2751-2761 ([arXiv:hep-th/9112008](https://arxiv.org/abs/hep-th/9112008))

* H. W. Braden, V. Varela, *Solutions for Einstein-Yang-Mills-Dilaton- σ Models*, Phys. Rev. D58:124020, 1998 ([arXiv:hep-th/9804204](https://arxiv.org/abs/hep-th/9804204))

* Betti Hartmann, Burkhard Kleihaus, Jutta Kunz, *Axially Symmetric Monopoles and Black Holes in Einstein-Yang-Mills-Higgs Theory*, Phys. Rev. D65 (2002) 024027 ([arXiv:hep-th/0108129](https://arxiv.org/abs/hep-th/0108129))

[[!redirects Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]


Eisenstein series

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## Idea

The _Eisenstein series_ $G_{2k}$ for $k \geq 2$ are [[series]]  expressions for certain [[modular forms]].

They appear as the coefficients of the [[exponential series]]-expression of the [[Weierstrass sigma-function]], and hence of the [[Hirzebruch series]] of the [[Witten genus]].

Moreover, a combination of $G_2$ and $G_4$ expresses the [[j-invariant]] which characterizes [[elliptic curves]].

## Definition

$$
  G_{2k}(\tau)
   \coloneqq
  \underset{(m,n) \in \mathbb{Z}^2\backslash (0,0)}{\sum}
  \frac{1}{(m+n \tau)^{2k}}
  \,.
$$

## Properties

### Relation to the $j$-invariant
 {#RelationToJInvariant}

With

$$
  g_2 \coloneqq 60 G_4
$$

$$
  g_3 \coloneqq 140 G_6
$$

$$
  \Delta \coloneqq g_2^3 - 27 g_3^2
$$

the [[j-invariant]] is

$$
  j = 1728 \frac{g_2^3}{\Delta}
  \,.
$$

### Relation to Weierstrass $\sigma$-function and Witten genus
 {#RelationToWeierstrassFunctionAndWittenGenus}

$$
  \frac{x}{e^{x/2} - e^{-x/2}}
  \prod_{n\geq 1}
  \frac{(1-q^n)^2}{(1-q^n e^x)(1-q^n e^{-x})}
  =
  \exp\left(
    \sum_{k \geq 2} 2 G_k \frac{x^k}{k!}
  \right)
$$

([Ando-Hopkins-Rezk 10, prop. 10.9](#AndoHopkinsRezk10))

### Relation to Bernoulli numbers

The $q$-independent term in $G_{k}$ is proportional to the $k$th [[Bernoulli number]] $B_k$

$$
  G_k = - \frac{B_k}{2k} + \sum_{n= 1}^\infty \sigma_{k-1}(n)q^n
  \,,
$$

where

$$
  \sigma_{k-1}(n) =  \sum_{d|n} d^{k-1}
  \,.
$$

Reducing to this constant term reduces the [above](#RelationToWeierstrassFunctionAndWittenGenus) exponential characteristic series for the [[Witten genus]] to that of the [[A-hat genus]].

## References

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

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))


Ek-Algebras

See at

* [[En-algebra]]

* [[Higher Algebra]]


EKMM > history

< [[EKMM]]


ekpyrotic cosmology

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## Idea

In theoretical [[cosmology]], the term _ekpyrotic cosmology_ ([Khoury-Ovrut-Steinhardt-Turok 01](#KhouryOvrusSteinhardtTurok01)) refers to the hypothesis that the [[observable universe]] is the locus of the collision of two [[branes]] in an ambient higher-dimensional [[spacetime]]. The scenario is vaguely motivated from the [[Randall-Sundrum model]] (which in turn was vaguely motivated from [[Horava-Witten theory]]) but with the two branes there not being static but colliding with or even passing through each other, possibly repeatedly ("cyclic cosmology" [Steinhardt-Turok 01](#SteinhardtTurok01)).

Just like the [[Randall-Sundrum model]], also the ekpyrotic model is loosely [[string theory inspired model|string theory inspired]] but initially without the ambition to check whether there really are [[string theory vacua]] with these proporties. Recent more detailed analysis points to the contrary ([Uzawa 18](#Uzawa18)).

## References

### General

The idea was proposed in

* {#KhouryOvrutSteinhardtTurok01} [[Justin Khoury]], [[Burt Ovrut]], [[Paul Steinhardt]], [[Neil Turok]], _The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang_, Phys. Rev. D64:123522,2001 ([arXiv:hep-th/0103239](https://arxiv.org/abs/hep-th/0103239))

* {#SteinhardtTurok01} [[Paul Steinhardt]], [[Neil Turok]], _Cosmic Evolution in a Cyclic Universe_, Phys. Rev. D65:126003, 2002  ([arXiv:hep-th/0111098](https://arxiv.org/abs/hep-th/0111098))

* [[Justin Khoury]], [[Burt Ovrut]], [[Paul Steinhardt]], [[Neil Turok]], _Density Perturbations in the Ekpyrotic Scenario_, Phys. Rev. D66:046005, 2002 ([arXiv:hep-th/0109050](https://arxiv.org/abs/hep-th/0109050))

* [[Evgeny Buchbinder]], [[Justin Khoury]], [[Burt Ovrut]], _New Ekpyrotic Cosmology_, Phys. Rev. D76:123503, 2007 ([arXiv:hep-th/0702154](https://arxiv.org/abs/hep-th/0702154))

See also

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

### Embedding into string theory
 {#ReferencesEmbeddingIntoStringTheory}

Discussion of [[no-go theorems]] for actually embedding the ekpyrotic scenario in [[string theory]], or at least in [[type IIA supergravity]]:

* {#Uzawa18} [[Kunihito Uzawa]], _No-Go theorems for ekpyrosis from ten-dimensional supergravity_, J. High Energ. Phys. (2018) 2018: 41 ([arXiv:1803.11084](https://arxiv.org/abs/1803.11084), doi:10.1007/JHEP06(2018)041)

  reviewed in:

  Kunihito Uzawa, _Ekpyrotic scenario in string theory_ ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~gc2018/slides/4th/Mar2/Uzawa.pdf))

This is related to similar discussion of embedding [[de Sitter spacetime|de Sitter]] vacua into [[string theory]], see the references [there](de+Sitter+spacetime#ReferencesInStringTheory).

[[!redirects ekpyrotic cosmologies]]


El-kaioum Moutuou

* [old personal webpage](http://www.math.univ-metz.fr/membres/pagesweb/moutuou.html)

* [institute webpage](http://www.southampton.ac.uk/maths/about/staff/emm1m12.page)

* [personal webpage at Southampton](http://www.personal.soton.ac.uk/~emm1m12/)

category: people


El-kaïoum M. Moutuou

* [webpage](http://www.math.usf.edu/faculty/ekmmoutuou/)

## Selected writings

On [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[KR-theory]] of [[orbifold|orbi-]] [[orientifolds]]:

* [[El-kaïoum M. Moutuou]], _Twistings of KR for Real groupoids_ ([arXiv:1110.6836](http://arxiv.org/abs/1110.6836))

* [[El-kaïoum M. Moutuou]], _Graded Brauer groups of a groupoid with involution_, J. Funct. Anal. 266 (2014), no.5 ([arXiv:1202.2057](https://arxiv.org/abs/1202.2057))

[[!redirects El-kaïoum Moutuou]]

[[!redirects El-kaioum M. Moutuou]]
[[!redirects El-kaioum Moutuou]]

category: people


EL-∞ algebra

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{:toc}

## Idea
 {#Idea}

The notion of *$E L_\infty$-algebras* is meant to be the fully [[homotopy theory|homotopy theoretic]] (i.e. [[(infinity,1)-category|$(\infty,0)$-]][[categorification|categorified]]) [[higher structure]] enhancing the [[mathematical structure]] of *[[Lie algebras]]*: For $E L_\infty$-algebras both the [[Jacobi identity]] *and* the *[[exterior algebra|skew symmetry]]* of the [[Lie bracket]] are relaxed up to potentially infinite [[coherence law|coherent]] [[higher homotopy]].

This is in contrast to the more widely considered notion of [[L-infinity algebras|$L_\infty$-algebras]], which relax the [[Jacobi identity]] but retain strict [[exterior algebra|skew symmetry]]. (Whence the terminology "$E L_\infty$": the "$E$" is for "everything homotopy", a whimsical but time-honored terminology, enshrined in the now classical terminology of [[E-infinity algebras|$E_\infty$-algebras]]).

The [[homotopy theory]] of $E L_\infty$-algebra is in fact [[equivalence of (infinity,1)-categories|equivalent]] to [[model structure for L-infinity algebras|that of $L_\infty$-algebras]] (and thus both [[relation between L-infinity algebras and dg-Lie algebras|are equivalent even to]] [[model structure on dg-Lie algebras|that of dg-Lie algebras]], which are further [[rectification|rectified]] $L_\infty$-algebras): $L_\infty$-algebras are a special case of $E L_\infty$-algebras and every $E L_\infty$-algebra is [[weak equivalence|weakly equivalent]] to one that is an $L_\infty$-algebra (i.e. the homotopy-skew-symmetry may always be [[rectification|rectified]] to strict skew symmetry).

Nevertheless, in some circumstances it is practically useful to work with instances of $E L_\infty$-algebras up to [[isomorphism]] without passing to a [[weak equivalence|weakly equivalent]] [[L-infinity algebra|$L_\infty$-algebra]]. In particular, [Borsten, Kim & Saemann 2021](#BKS21) argue that the notion of $E L_\infty$-algebra serves to give a transparent way to understand [[adjusted Weil algebras]] for [[L-infinity algebras|$L_\infty$-algebras]], and then to understand [[tensor hierarchies]] (in [[gauged supergravity]]-theory) in terms of the resulting [[connection on a smooth principal infinity-bundle|$\infty$-connections]]/[[higher gauge theory]].

## $h\mathcal{L}ie$- and $\mathcal{E}ilh$-algebras

The following notions have been introduced by ([BKS21](#BKS21)).

### $h\mathcal{L}ie$-algebras

\begin{definition}
An $h\mathcal{L}ie$-algebra $(\mathfrak{E},\varepsilon_1,\varepsilon_2^i)$ is a graded vector space $\mathfrak{E}$ together with a differential and a collection of binary products,
\[
        \begin{aligned}
            \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E}, &|\varepsilon_1|=1,
            \\
            \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E}, &|\varepsilon^i_2|=-i,
        \end{aligned}
\]
such that
\[
            \begin{aligned}
                \varepsilon_1(\varepsilon_1(x_1))&=0,
                \\
                \varepsilon_1(\varepsilon^i_2(x_1,x_2))&=(-1)^i\big(\varepsilon^i_2(\varepsilon_1(x_1),x_2)+(-1)^{|x_1|}\varepsilon^i_2(x_1,\varepsilon_1(x_2))\big)
                \\
                &+\varepsilon^{i-1}_2(x_1,x_2)-(-1)^{i+|x_1|\,|x_2|}\varepsilon^{i-1}_2(x_2,x_1),
                \\
                \varepsilon^i_2(\varepsilon_2^i(x_1,x_2),x_3)&=(-1)^{i(|x_1|+1)}\varepsilon^i_2(x_1,\varepsilon^i_2(x_2,x_3))-(-1)^{(|x_1|+i)|x_2|}\varepsilon^i_2(x_2,\varepsilon_2^i(x_1,x_3))
                \\
                &-(-1)^{(|x_2|+|x_3|)|x_1|+(i-1)|x_2|}\varepsilon^{i+1}_2(x_2,\varepsilon^{i-1}_2(x_3,x_1)),
                \\
                \varepsilon^{j}_2(\varepsilon^{i}_2(x_1,x_2),x_3)&=
                (-1)^{1+j(i+1)+|x_1|(|x_2|+|x_3|)+(j-1)|x_2|}\varepsilon_2^{i+1}(x_2,\varepsilon_2^{j-1}(x_3,x_1)),
                \\
                \varepsilon^{i}_2(\varepsilon^{j}_2(x_1,x_2),x_3)&=
                (-1)^{i(j+|x_1|)}\varepsilon_2^{j}(x_1,\varepsilon_2^{i}(x_2,x_3))-(-1)^{(|x_1|+j)|x_2|}\varepsilon_2^i(x_2,\varepsilon_2^j(x_1,x_3))
                \\
                &-(-1)^{j+|x_3|(j+|x_2|-1)+|x_1|(|x_2|+|x_3|)}\varepsilon_2^{i+1}(x_3,\varepsilon_2^{j-1}(x_2,x_1))
            \end{aligned}
\]
are satisfied for all $i,j\in \mathbb{N}$ s.t. $j$<$i$ and for all $x_1,x_2,x_3\in \mathfrak{E}$, where we regard $\varepsilon_2^{-1}=0$.
\end{definition}

### $\mathcal{E}ilh$-algebras

\begin{definition}
        An $\mathcal{E}ilh$-algebra $(\mathfrak{E},Q,\oslash_i)$ is a differential graded vector space $(\mathfrak{E},Q)$ equipped with binary operations $\oslash_i$ of degree $i\in \mathbb{N}$ which satisfy the quadratic identities
\[
\begin{aligned}
                a\oslash_i(b\oslash_i c)&=(-1)^{i(|a|+1)}((a\oslash_i b)\oslash_i c+(-1)^{|a|\,|b|}(b\oslash_i a)\oslash_i c),
                \\
                a\oslash_i(b\oslash_j c)&=(-1)^{ij+j|a|}(a \oslash_i b)\oslash_j c,
                \\
                a\oslash_j(b\oslash_i c)&=\begin{cases}
                    (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c
                    &\text{if } j-i=1,
                    \\
                    (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c
                    &\text{if } j-i=2,
                    \\
                    +(-1)^{i(|a|+j+1)+(|a|+|b|)|c|}((c\oslash_{j-1} a)\oslash_{i+1}b)
                    \\
                    (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c
                    &\text{if } j-i>2
                    \\
                    +(-1)^{i(|a|+j+1)+(|a|+|b|)|c|}((c\oslash_{j-1} a)\oslash_{i+1}b)
                    \\

                    +(-1)^{j+|a|(|b|+i)+(|a|+|b|)|c|}((c\oslash_{i+1} b)\oslash_{j-1}a)
                \end{cases}
            \end{aligned}
\]
for $j$>$i$, and such that differential $Q$ satisfies the property
\[
            \begin{aligned}
                Q(a\oslash_i b)&=(-1)^i\big((Qa)\oslash_i b+(-1)^{|a|}a\oslash_i Qb\big)\\
                &+(-1)^i (a \oslash_{i+1}b)-(-1)^{|a|\,|b|} (b\oslash_{i+1} a),
            \end{aligned}
\]
which is a deformed Leibniz rule.
\end{definition}

\begin{proposition}
    $\mathcal{E}ilh$-algebras are Koszul dual to $h\mathcal{L}ie$-algebras.
\end{proposition}

\begin{definition}
The Chevalley--Eilenberg algebra $\mathrm{CE}(\mathfrak{E})$ of an $h\mathcal{L}ie$-algebra $\mathfrak{E}$ whose differential and binary products are given by
\[
            \begin{aligned}
                \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha&\mapsto m^\beta_\alpha \tau_\beta,&&|\varepsilon_1|=1,
                \\
                \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha\otimes \tau_\beta&\mapsto m^{i,\gamma}_{\alpha\beta}\tau_\gamma,&&|\varepsilon^i_2|=i
            \end{aligned}
\]
for some $m^\beta_\alpha$ and $m^{i,\gamma}_{\alpha\beta}$ taking values in the underlying ground field is the $\mathcal{E}ilh$-algebra $(\oslash_\bullet^\bullet V,Q,\oslash_i)$ with $V=\mathfrak{E}[1]^*$ and the differential
\[
            Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma,
\]
where $|\beta|\coloneqq|t^\beta|$.
\end{definition}

Consider
\[
            \oslash^\bullet_\bullet V \,\coloneqq\, \underbrace{\mathbb{R}}_{\text{deg }0}\oplus\underbrace{V}_{\text{deg }1}\oplus\underbrace{\bigoplus_{i\in \mathbb{N}}V\oslash_i V}_{\text{deg }2}\oplus\underbrace{\bigoplus_{i,j\in \mathbb{N}}(V\oslash_i V)\oslash_j V}_{\text{deg }3}\oplus\dots
\]

\begin{definition}
The Chevalley--Eilenberg algebra $\mathrm{CE}(\mathfrak{E})$ of an $h\mathcal{L}ie$-algebra $\mathfrak{E}$ whose differential and binary products are given by
\[
            \begin{aligned}
                \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha&\mapsto m^\beta_\alpha \tau_\beta,&&|\varepsilon_1|=1,
                \\
                \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha\otimes \tau_\beta&\mapsto m^{i,\gamma}_{\alpha\beta}\tau_\gamma,&&|\varepsilon^i_2|=i
            \end{aligned}
\]
for some $m^\beta_\alpha$ and $m^{i,\gamma}_{\alpha\beta}$ is the $\mathcal{E}ilh$-algebra $(\oslash_\bullet^\bullet V,Q,\oslash_i)$ with $V=\mathfrak{E}[1]^*$ and the differential
\[
            Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma,
\]
where $|\beta|\coloneqq|t^\beta|$.
\end{definition}

## $E L_\infty$-algebras

$E L_\infty$-algebras are the homotopy version of $h\mathcal{L}ie$-algebras, defined by ([BKS21](#BKS21)).

Analogously to an $L_\infty$-algebra, an $EL_\infty$-algebra structure on a graded vector space $\mathfrak{E}$ is encoded by a differential $Q$ on the $\mathcal{E}ilh$-algebra $\mathrm{CE}(\mathfrak{E})\coloneqq (\oslash_\bullet^\bullet \mathfrak{E}[1]^\ast, Q, \oslash_i)$. The differential $Q$ is given by its action on $\mathfrak{E}[1]^\ast$, which will be encoded by structure constants $m$ as follows:
\[
        Q t^\alpha=\pm m^\alpha\pm m^\alpha_\beta t^\beta\pm m^{i_1,\alpha}_{\beta_1\beta_2} t^{\beta_1}\oslash_{i_1} t^{\beta_2}\pm m^{i_1i_2,\alpha}_{\beta_1\beta_2\beta_3}(t^{\beta_1}\oslash_{i_1} t^\gamma)\oslash_{i_2} t^\delta+\ldots,
\]
where $\{t^\alpha\}$ is a basis on $\mathfrak{E}[1]^\ast$.

These structure constants $m$ define higher products $\varepsilon^I_n:\mathfrak{E}^{\otimes n}\rightarrow \mathfrak{E}$ with degree$-|I|$ by
\[
\begin{aligned}
            \varepsilon_0=m^\alpha \tau_\alpha, \;
            \varepsilon_1(\tau_\alpha)=m^\beta_\alpha \tau_\beta, \; \varepsilon_2^i(\tau_\alpha,\tau_\beta)=m^{i,\gamma}_{\alpha\beta}\tau_\gamma,\ldots
            \\
            \varepsilon^I_n(\tau_{\alpha_1},\ldots,\tau_{\alpha_n})=m_{\alpha_1\ldots\alpha_n}^{I,\beta} \tau_\beta,
        \end{aligned}
\]
where $I$ is a multi-index consisting of $n-1$ indices $i_1,i_2,\ldots,\in \mathbb{N}$ and $|I|\coloneqq i_1+i_2+\ldots$.

## Related concepts

* [[A-∞ algebra]]

* [[E-∞ algebra]]

* [[L-∞ algebra]]

  * [[homomorphism of L-∞ algebras]]

  * [[infinity-Lie algebra cohomology]]

  * [[universal enveloping E-n algebra]]

  * [[nilpotent L-∞ algebra]]

  * [[L-∞ algebroid]]

  * [[derived L-∞ algebroid]]

  * [[sheaf of L-∞ algebras]]

* [[relation between L-∞ algebras and dg-Lie algebras]]

* [[adjusted Weil algebra]]

* [[tensor hierarchy]]

* [[Leibniz algebra]]

## References
 {#References}

The general notion is discussed in:

* {#BKS21} [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]]: _$EL_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies_ ([arXiv:2106.00108](https://arxiv.org/abs/2106.00108))

The special case of weak Lie 2-algebras was originally considered in:

* {#Roytenberg07} [[Dmitry Roytenberg]]: *On weak Lie 2-algebras*, in: *XXVI Workshop on Geometrical Methods in Physics*, AIP Conference Proceedings **956**, American Institute of Physics, (2007) [[arXiv:0712.3461](https://arxiv.org/abs/0712.3461), [hdl:10993/10465](http://hdl.handle.net/10993/10465)]

and the more general special case of weak Lie 3-algebras in:

* [[Malte Dehling]], *On weak Lie 3-algebras* [[arXiv:1710.11104](https://arxiv.org/abs/1710.11104)]

[[!redirects EL-infinity-algebras]]
[[!redirects EL-infinity algebra]]
[[!redirects EL-infinity algebras]]
[[!redirects EL-∞ algebra]]
[[!redirects EL-∞ algebras]]
[[!redirects EL-∞-algebra]]
[[!redirects EL-∞-algebras]]

[[!redirects hemistrict Lie 2-algebra]]
[[!redirects hemistrict Lie 2-algebras]]

[[!redirects weak Lie 2-algebra]]
[[!redirects weak Lie 2-algebras]]

[[!redirects weak Lie 3-algebra]]
[[!redirects weak Lie 3-algebras]]


Elango Panchadcharam

* [GoogleScholar page](https://scholar.google.com/citations?user=gj5mKu0AAAAJ&hl=en)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=113276)

## Selected writings

On lax [[braided monoidal categories]] with possibly non-invertible [[braiding]]:

* [[Brian Day]], [[Elango Panchadcharam]], [[Ross Street]], _Lax braidings and the lax centre_, Contemporary Mathematics **441** 1 (2007) [[pdf](http://science.mq.edu.au/~street/laxcentre.pdf), [[DayPanchadcharamStreet-LaxBraidings.pdf:file]]]

category: people


elasticity

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## Idea

### In solid state physics
 {#InSolidStatePhysics}

In [[physics]], specifically in _[[solid state physics]]_, _elasticity_ is the tendency of solid materials to return to their original shape after being deformed.

Mathematical elasticity theory ([Landau-Lifshitz 59](#LandauLifshitz59)) encodes the [[forces]] acting on a [[solid]] by a [[stress tensor]], and the amount of deformation by a _[[strain tensor]]_.

One distinguishes different types of elasticity:

* linear elasticity: here the [[strain tensor]] is related by a [[linear equation]] to the [[stress tensor]] (a generalization of [[Hooke's law]]). To first [[infinitesimal]] order every type of material is approximately linearly elastic, the topic of _[infinitesimal strain theory](http://en.wikipedia.org/wiki/Infinitesimal_strain_theory)_.

* non-linear elasticity

  * hyperelasticity: exhibited by (soft) rubber

...

### Analogy with a quality of space
 {#AnalogyWithAQualityOfSpace}

Elasticity is commonly used as an illustration via [[analogy]] of the nature of [[space]] in [[physics]], specifically that of [[manifolds]] and [[spacetimes]].

#### In natural philosophy

In [[Georg Hegel]]'s _[[Encyclopedia of the Philosophical Sciences]]_ (1817) there is discussion (in the section _[Physik -- Die Kohäsion](Science%20of%20Logic#Kohaesion)_) of the _[[cohesion]]_ and _elasticity_ of some [[substance]] which Hegel says is is the [[unity of opposites|unity]] of [[space]] and [[time]] ([PN§261](Science%20of%20Logic#PN261)).

#### In categorical logic / topos theory
 {#InCategoricalLogic}

[[William Lawvere]] argued that the "[[objective logic]]" of this discussion is to be formalized via [[categorical logic]] by the axiomatics of [[cohesive toposes]] (see there for references), i.e. by [[modal type theory]] equipped with [[shape modality]] and [[flat modality]].

But Hegel goes on to speak of cohesion being refined to _elasticity_:

> [PN§297Zusatz](Science+of+Logic#PN297Zusatz) Elasticity is the whole of cohesion.

According to [PN§298](Science+of+Logic#PN298) this elasticity is related to the [[unity of opposites]] that consistute [[Zeno's paradox of motion]], hence to the modern concept of [[differentiation]] via a [[limit of a sequence]].  In terms of [[categorical logic]] this is precisely what is encoded in the [[infinitesimal shape modality]] and [[infinitesimal flat modality]] of "[[differential cohesion]]" (see there for detail).

Sticking to imagery from [[solid state physics]], these modalities are reminiscent of concepts in [infinitesimal strain theory](http://en.wikipedia.org/wiki/Infinitesimal_strain_theory). Notice that this applies to

> {#RelativelyStiffElasticity} structures built from relatively stiff elastic materials

#### Rubber-sheet geometry
 {#RubberSheetGeometry}

These modalities [above](#InCategoricalLogic) induce an axiomatization of _[manifolds and étale groupoids](differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)_ ("[[derived schemes]]") for which a common imagery in mathematics is given by elastic rubber sheets, reflecting the fact that these spaces on top of their [[cohesion]] have a more rigid [[shape modality|shape]], namely [[infinitesimal shape modality|infinitesimal shape]].

In this context the [[topology]] of [[manifolds]] is often referred to as _rubber-sheet geometry_ (e.g.[Wikipedia](http://simple.wikipedia.org/wiki/Topology), [Britton](#Britton)).

#### Elasticity analogy of gravity
 {#RubberSheetAnalogyOfGravity}

Moreover, these modalities [above](#InCategoricalLogic) induce an axiomatization of [[Cartan geometry]], hence in particular of [[pseudo-Riemannian geometry]] and hence of [[gravity]] ("[[general relativity]]") on these manifolds.

A popular imagery for that illustrates the gravitational effect of massive bodies on light by a deformation of spacetime visualizes as an elastic wire frame. This is known as the _rubber-sheet analogy_ for gravity (e.g. [Das 11, p. 69](#Das), [Gabor](#Gabor), [Volk, fig. 4 on p. 537](#Volk)).

This seems to go back to the technical result of ([Sakharov 67](#Sakharov67)), which is informally summarized in the seminal textbook ([Misner-Thorne-Wheeler 73](#MisnerThorneWheeler73)) as saying that

> gravity is "an elasticity of space that arises from particle physics"

Further discussion of the analogy between the mathematical theory of elasticity and that of gravity include ([Tartaglia 95](#Tartaglia95),[Middleton-Langston 13](#MiddletonLangston13), [Wikipedia](#WikipediaRubberSheetModelForGravity)).

In order to think of not just [[topology]] but [[Riemannian geometry]] in the [above](#RubberSheetGeometry) context of elasticity, the _rigidity_ mentioned [further above](#RelativelyStiffElasticity) seems advisable. A _rigidly elastic_ body is to be expected to produce sound when struck. This is a common imagery in Riemannian geometry, as in "[[hearing the shape of a drum]]".

## Related entries

* [[formal smooth infinity-groupoid]]

* [[super formal smooth infinity-groupoid]]

* [[cohesion]], [[solidity]]

## References

### In solid state physics

* {#LandauLifshitz59} [[Lev Landau]], [[Evgeny Lifshitz]], _Theory of Elasticity_, part VII of  _[[Course of Theoretical Physics]]_, 1959, 1970

### As an analogy for topology/differential geometry

* {#Britton} Jill Britton, _[Rubber sheet geometry](http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Estes/unit/dayten/topology1.html)_

### As an analogy for gravity

The "rubber-sheet analogy of gravity" might go back to results in

* {#Sakharov67} [[Andrei Sakharov]], _Vacvuum fluctuations in curved space and the theory of gravitation_, Doklady Akad. Nauk S.S.S.R. 177 70-71 (1967)

which were informally summarized in

* {#MisnerThorneWheeler73} [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], _[[Gravitation]]_, 1973

* [[John Wheeler]], _Sakharov: A man of humility, understanding and leadership_, in _Andrei Sakharov, Facets of a Life_ ([GoogleBooks](https://books.google.co.uk/books?id=BTIziWmTX_0C&pg=PA647#v=onepage&q&f=false))

as saying that gravity is "an elasticity of space that arises from particle physics".

The mathematical similarity between gravity and the physics of elasticity is discussed further in

* {#Tartaglia95} A. Tartaglia, _Four Dimensional Elasticity and General Relativity_ ([arXiv:gr-qc/9509043](http://arxiv.org/abs/gr-qc/9509043))

* {#MiddletonLangston13} Chad A. Middleton, Michael Langston, _Circular orbits on a warped spandex fabric_ ([arXiv:1312.3893](http://arxiv.org/abs/1312.3893), [talk video](https://www.youtube.com/watch?v=F80HIolrsoc))

The analogy is mentioned for expositional purpose for instance in

* {#WikipediaRubberSheetModelForGravity} Wikipedia, _[Gravity well -- The rubber-sheet model](http://en.wikipedia.org/wiki/Gravity_well#The_rubber-sheet_model)_

* {#Gabor} Gabor, _[Gravity: Space as a rubber sheet](http://theory.uwinnipeg.ca/users/gabor/black_holes/slide5.html)_

* {#Volk} Greg Volk, _19th Natural Philosophy Alliance Proceedings_ ([online](https://books.google.de/books?id=tZnZAwAAQBAJ&pg=PA537&lpg=PA537&dq=manifold+"rubber-sheet"&source=bl&ots=WAhzUMxkOR&sig=p2qNCPCR9-gnPi6_nb1S_Bz3-AI&hl=en&sa=X&ei=8Ob2VLmSIcvNygOTjoGIBA&ved=0CCIQ6AEwAA#v=onepage&q=manifold%20%22rubber-sheet%22&f=false))

*  {#Das11} [[Ashok Das]], _Lectures on Gravitation_, WorldScientific 2011 ([publisher](http://www.worldscientific.com/worldscibooks/10.1142/7990), [GoogleBooks](https://books.google.de/books?id=oupB7VUGzGcC&pg=PA69&lpg=PA69&dq=manifold+"rubber-sheet"&source=bl&ots=ajbiJu7T_m&sig=MPsA2KWds_Mmf_UCscieaGVJzJk&hl=en&sa=X&ei=8Ob2VLmSIcvNygOTjoGIBA&ved=0CCUQ6AEwAQ#v=onepage&q=manifold%20%22rubber-sheet%22&f=false))

[[!redirects elasticity theory]]

[[!redirects rubber-sheet geometry]]
[[!redirects rubber-sheet analogy of gravity]]

[[!redirects elastic]]


Eldon Dyer

* [webpage](https://www.ias.edu/scholars/sherman-eldon-dyer)

* [Wikipedia (German)](https://de.wikipedia.org/wiki/Eldon_Dyer)

## Selected writings

On [[Bott periodicity]]:

* [[Eldon Dyer]], [[Richard Lashof]], *A topological Proof of the Bott Periodicity Theorems*, Annali di Matematica Pura ed Applicata **54** (1961) 231–254 [[doi:10.1007/BF02415354](https://doi.org/10.1007/BF02415354)]

On [[homology of iterated loop spaces]]:

* [[Eldon Dyer]], [[Richard Lashof]], _Homology of Iterated Loop Spaces_,  American Journal of Mathematics Vol. 84, No. 1 (Jan., 1962), pp. 35-88 ([jstor:2372804 ](https://www.jstor.org/stable/2372804))

On [[Mayer-Vietoris sequences]] as [[homotopy fiber sequences]]:

*  [[Eldon Dyer]], Joseph Roitberg, _Note on sequences of Mayer-Vietoris type_, Proceedings of the AMS, volume 80, number 4 (1980) ([pdf](http://www.ams.org/journals/proc/1980-080-04/S0002-9939-1980-0587950-8/S0002-9939-1980-0587950-8.pdf))

category: people


Eleanor Rieffel

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

* [GoogleScholar page](https://scholar.google.com/citations?user=2UvnS7UAAAAJ&hl=en)

## Selected writings

On [[quantum computing]]:

* [[Eleanor Rieffel]], [[Wolfgang Polak]],  *An Introduction to Quantum Computing for Non-Physicists*, ACM Comput. Surveys **32** (2000) 300-335 [[arXiv:quant-ph/9809016](https://arxiv.org/abs/quant-ph/9809016), [doi:10.1145/367701.367709](https://dl.acm.org/doi/10.1145/367701.367709)]

* [[Eleanor Rieffel]], [[Wolfgang Polak]], *Quantum Computing -- A gentle introduction*, MIT Press (2011) [[ISBN:9780262526678](https://mitpress.mit.edu/9780262526678/quantum-computing/), [pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608150244118.pdf)]

category: people