Elmendorf's theorem

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#### Homotopy theory
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#### Representation theory
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#### Cohomology
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#Contents#
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## Idea

_Elmendorf's theorem_ states that for $G$ a [[topological group]], the [[(∞,1)-category of (∞,1)-presheaves]] on the [[orbit category]] $Orb_G$ of $G$, naturally regarded as an [[(∞,1)-site]], is [[equivalence of (∞,1)-categories|equivalent]] to the classical $G$-[[equivariant homotopy theory]], namely the [[localization of an (∞,1)-category|localization]] of [[topological spaces]] with $G$-[[action]] ([[G-spaces]]) at the _[[weak homotopy equivalences]] on all [[fixed point spaces]]_ of [[closed subset|closed]] [[subgroups]] (the [[equivariant weak homotopy equivalences]]).

This is due to [Elmendorf 83](#Elmendorf83) and [Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1](#DwyerKan84); with alternative proofs in various stages of refinement (see Remark \ref{AssumptionsAndRefinementsInTheLiterature} below) given by [Piacenza 91, Sec. 6](#Piacenza91), [May 96, Sec. V.3](#May96), [Cordier-Porter 96, Thm. 3.11](#CordierPorter96), [Guillou 06, Prop. 3.15](#Guillou06), [Stephan 13, Cor. 3.20](#Stephan13), [Guillou-May-Rubin 13, Thm. 1.8 & 5.6](#GuillouMayRubin13). Exposition is in [Blumberg 17, Sec. 1.3](#Blumberg17).

In particular this means that the $G$-[[equivariant homotopy theory]], thus identified with an [[(∞,1)-category of (∞,1)-presheaves]], is an [[(∞,1)-topos]] and in fact (since $Orb_G$ has [[finite products]]) a [[cohesive (∞,1)-topos]]; a point expanded on in [Rezk 14](#Rezk14), [Sati-Schreiber 20](#SatiSchreiber20) (for more on this see at _[[orbifold cohomology]]_).

More in detail, write $Top^G$ for the [[category]] of [[compactly generated topological spaces]] which are equipped with a [[continuous function|continuous]] $G$-[[action]] ([[G-spaces]]). Say that a $G$-[[equivariant function|equivariant]] [[continuous function]] $f \colon X \longrightarrow Y$ between $G$-spaces is a **weak $G$-homotopy equivalence** if for all [[closed subgroups]] $H \hookrightarrow G$ the induced function on $H$-[[fixed point spaces]] $f^H \colon X^H \longrightarrow Y^H$ is an ordinary [[weak homotopy equivalence]]. Write

$$
  L_{G whe} Top^G
  \;\in\;
  (\infty,1)Cat
$$

for the corresponding [[simplicial localization]].

Next, write $Orb_G$ for the [[full subcategory]] of $Top^G$ on the $G$-[[homogeneous spaces]] of the form $G/H$, but regarded as an [[(∞,1)-category]] by regarding each [[compact-open topology|hom-space]] as its [[homotopy type]]. Write moreover $Top^{Orb_G}$ for the category of [[continuous function|continuous]] [[functors]] $Orb_G^{op} \longrightarrow Top$. Write finally

$$
  PSh_\infty(Orb_G)
   \in
 (\infty,1)Cat
  \,.
$$

Then Elmendorf's theorem asserts that there is an [[equivalence of (∞,1)-categories]]

$$
 L_{G whe}
  Top^G
  \;\simeq\;
  PSh_\infty(Orb_G)
  \,.
$$

\begin{remark}\label{AssumptionsAndRefinementsInTheLiterature}
(**survey of available proofs.**)

Elmendorf's theorem is stated in [Elmendorf 83](#Elmendorf83) as an [[equivalence of categories|equivalence]] of [[homotopy categories]]; and is enhanced in [Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1](#DwyerKan84) to a [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalence]] of [[model categories]] [[presentable (infinity,1)-category|presenting]] (in hindsight) the [[equivalence of (∞,1)-categories]] stated above (e.g. [Blumberg 17. Thm. 1.3.8](#Blumberg17)).

Moreover, while [Elmendorf 83](#Elmendorf83) assumes $G$ to be a [[compact Lie group]], [Dwyer-Kan 84](#DwyerKan84) allow $G$ to be *any* [[topological group]]. But beware that invoking the [[equivariant Whitehead theorem]] to identify the fixed-locus-wise weak homotopy equivalences, used in the theorem, with the $G$-homotopy equivalences, typically used in practice, again requires $G$ to be a [[compact Lie group]].

Later [Piacenza 91, Sec 6](#Piacenza91) and [May 96, Sec. V.3](#May96) re-prove the equivalence of [[homotopy categories]] for $G$ any [[topological group]]; while [Cordier-Porter 96, Thm. 3.11](#CordierPorter96) and [Guillou 06, Prop. 3.15](#Guillou06) re-prove a ([[simplicial Quillen adjunction|simplicial]]) [[Quillen equivalence]] assuming $G$ to be a [[discrete group|discrete]] or even [[finite group]], respectively; and [Stephan 13, Cor. 3.20](#Stephan13), [Guillou-May-Rubin 13, Thm. 1.8](#GuillouMayRubin13) re-prove the Quillen equivalence for $G$ again any topological group.

([Stephan 13](#Stephan13) credits [Piacenza 91](#Piacenza91) with proving a [[Quillen equivalence]]. While this is not what [Piacenza 91, Thm. 6.3](#Piacenza91) actually states, the conclusion follows with hindsight.)

These results are all based on the [[classical model structure on topological spaces]].  The analogous Quillen equivalence based on the [[classical model structure on simplicial sets]] is proven in [Guillou-May-Rubin 13, Thm. 5.6](#GuillouMayRubin13), assuming $G$ to be a [[discrete group]].
\end{remark}

\begin{remark}\label{AsAStatementAboutEquivariantConcordance}
**(as a statement about [[concordance]] of [[equivariant maps]])**
\linebreak
  One way to understand Elmendorf's theorem (in its full $\infty$-category theoretic version) is as translating [[concordance]] of [[equivariant maps]] into actual [[homotopies]] in another [[(infinity,1)-category|$\infty$-category]]. Namely, if we think of sending a [[topological space]] $\mathrm{X}$ to its underlying [[homotopy type]] presented by its [[path infinity-groupoid|path $\infty$-groupoid]] (we follow the notation at *[[shape via cohesive path ∞-groupoid]]*)
$$
  \array{
    Top
      &\longrightarrow&
    Grpd_\infty
    \\
    \mathrm{X}
    &\overset{\phantom{--}}{\mapsto}&
    \esh \mathrm{X}
  }
$$
then it says that got each [[G-CW complex|$G$-CW complex]] $X$ we have:
$$
  \esh
  \Big(
    Maps
    \big(
      \mathrm{X}
      ,\,
      \mathrm{Y}
    \big)^G
  \Big)
  \;\;
  \simeq
  \;\;
  G Grpd_\infty
  \Big(
    \esh \big(\mathrm{X}^{(-)}\big)
    ,\,
    \esh \big(\mathrm{Y}^{(-)}\big)
  \Big)
  \,,
$$
where $Map(-,-)$ denotes the [[compact-open topology|mapping space]].

In this form the statement holds for *every* choice of [[topological group]] $G$ and *every* choice of family $\big\{H_i
\subset G\big\}_{i \in I}$ of [[subgroups]]: This choice affects what counts as a $G$-CW complex on the left and what the [[orbit category]]-site $Orb(G,\{H_i\}_{i \in I})$ is on the right
(in both cases the [[coset spaces]] involved must be $G/H_i$ for $H_i$ in the given family) -- but with that understood, the above equivalence holds, by [Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1](#DwyerKan84).

To bring this out, write

* $Orb(G,\mathcal{F}) \hookrightarrow G Top$ for the [[full subcategory]] on the [[coset spaces]] $G/H$ for $H \in \mathcal{F}$ in the family of subgroups,

* $(G, \mathcal{F})CWCplx \hookrightarrow G Top$ for the [[full subcategory]] of [[G-CW complexes]] build via these coset spaces, then

$$
  X \,\in\,
  (G,\mathcal{F})CWCplx
  \;\;\;\;
  \vdash
  \;\;\;\;
  \esh
  \Big(
    Maps
    \big(
      \mathrm{X}
      ,\,
      \mathrm{Y}
    \big)^G
  \Big)
  \;\;
  \simeq
  \;\;
  Psh_\infty\big(Orb(G,\mathcal{F})\big)
  \Big(
    \esh \big(\mathrm{X}^{(-)}\big)
    ,\,
    \esh \big(\mathrm{Y}^{(-)}\big)
  \Big)
  \,.
$$

\end{remark}

## Statement

### Classical version in topological spaces
 {#ClassicalVersionInTopologicalSpaces}

Write

* $kTop$ for the category of [[compactly generated topological spaces]];

\begin{assumption}\label{GeneralInputData}
**(general input data)**
\linebreak
Let

* $G \,\in\, Grp(kTop)$ be *any* [[topological group]],

* $\mathcal{F} \,\subset\,  Sub(G)$ be *any* [[set]] of [[subgroups]] of $G$.

\end{assumption}

\begin{remark}\label{FixedLociAsEquivariantMappingSpaces}
**([[fixed loci]] as [[equivariant function|equivariant]] [[mapping spaces]])**
\linebreak
For $H \subset G$ any [[subgroup]], notice the usual identification of $H$-[[fixed loci]] $X^H \subset X$ inside [[topological G-spaces|topological $G$-spaces]] $X$ with the $G$-[[equivariant function|equivariant]] [[compact-open topology|mapping space]] out of the [[orbit]]/[[coset space]] $G/H$:

$$
  Map(G/H,\, -)
  \;\simeq\;
  (-)^H
  \,.
$$

\end{remark}

\begin{proposition}\label{FProjectiveModelStructureOnGSpaces}
**($\mathcal{F}$-projective model structure on $G$-spaces)**
\linebreak
  In the generality of Assumption \ref{GeneralInputData},
  the category of [[topological G-spaces|topological $G$-spaces]] carries the [[mathematical structure|structure]] of a  [[simplicial model category]] $G Act(kTop)_{\mathcal{F} proj}$ whose

* [[hom-objects]] are the [[singular simplicial complexes]] of [[compact-open topology|mapping spaces]]:

  \[
    \label{SimplicialHomComplexBetweenGTopologicalSpaces}
    [X,Y]
      \,\coloneqq\,
    Sing
    \big(
      Map(X,Y)^G
    \big)
    \;\;\;
    \in
    \;
    sSet
  \]

* [[weak equivalences]] and [[fibrations]] $X \xrightarrow{f} Y$ are $H \in \mathcal{F}$-[[fixed point space]]-wise ($[G/H, X] \xrightarrow{\;[G/H, f]\;} [G/H, Y]$, see Rem. \ref{FixedLociAsEquivariantMappingSpaces})
those of the [[classical model structure on topological spaces]],

* [[cofibrations]] include the relative [[G-CW complexes|$G$-CW complexes]] which are build (only) from [[orbits]] of the form $G/H$ for $H \,\in\, \mathcal{F} \subset Sub(G)$.

\end{proposition}
([Dwyer & Kan 1984, §1.2 and Thm. 2.2](#DwyerKan84))

\begin{remark}
**(relation to classical $G$-equivariant homotopy theory)**
\linebreak
In the special case of Assumption \ref{GeneralInputData} that

1. $G$ is a [[compact Lie group]];

1. $\mathcal{F} \coloneqq ClsSub(G)$ is the set of all [[closed topological space|closed]] subgroups

1. $X$ is a [[G-CW-complex|$G$-CW complex]]

the [[equivariant Whitehead theorem]] says that weak equivalence $X \longrightarrow{Y}$ in $G Act(kTop)_{ClsSub(G)}$ (from Prop. \ref{FProjectiveModelStructureOnGSpaces}) are exactly the $G$-[[equivariant map|equivariant]] [[homotopy equivalences]].

More generally there may be more weak equivalences in $G Act(KTop)_{\mathcal{F}}$ than there are $G$-[[homotopy equivalences]]:

$$
  G Heq
  \hookrightarrow
  W_{\mathcal{F}}
  \,.
$$

However, observe that the simplicial hom-complex (eq:SimplicialHomComplexBetweenGTopologicalSpaces) is always the classical one, in particular its [[connected components]] are always (no matter the choice of $G$ and $\mathcal{F}$) the plain $G$-[[homotopy classes]]:

$$
  \pi_0
  [X,Y]
  \,=\,
  G Heq(X,\,Y)
  \,.
$$

\end{remark}

\begin{definition}\label{SimplicialPresheavesOnFTypeGOrbits}
**(simplicial presheaves over $\mathcal{F}$-type $G$-orbits)**
\linebreak
Given data as in Assumption \ref{GeneralInputData}, write

\[
  \label{CategoryOfFTypeGOrbits}
  Orb(G,\mathcal{F})
  \hookrightarrow
  G Act(kTop)
\]

for the [[orbit category|category of $\mathcal{F}$-type $G$ orbits]], namely the [[full subcategory|full sub]]-[[sSet-enriched category]] of [[topological G-spaces]] (eq:SimplicialHomComplexBetweenGTopologicalSpaces) on the [[coset spaces]] $G/H$ for $H \,\in\, \mathcal{F} \subset Sub(G)$.

Moreover, write

$$
  sPSh
  \big(
    Orb(G,\mathcal{F})
  \big)_{proj}
$$

for the projective [[model structure on sSet-presheaves]]
over (eq:CategoryOfFTypeGOrbits),
hence that whose [[weak equivalences]] and [[fibrations]]
are object-wise those in the [[classical model structure on simplicial sets]].

\end{definition}

\begin{proposition}\label{DwyerKanTheorem}
**(Dwyer-Kan theorem)**
\linebreak
In the generality of Assumption \ref{GeneralInputData}, there is a [[Quillen equivalence]]
$$
  G Act(kTop)
  \underoverset
    {
      \underset{
        X \mapsto
        \big(
          G/H \mapsto
          [G/H, X]
        \big)
      }{\longrightarrow}
    }
    {\longleftarrow}
    {\simeq_{\mathrlap{Qu}}}
  sPSh(
    Orb(G,\mathcal{F})
  )
$$
between

1. the $\mathcal{F}$-projective model structure on  topological $G$-spaces

    (Prop. \ref{FProjectiveModelStructureOnGSpaces})

1. the projective model structure on simplicial presheaves over the $\mathcal{F}$-type $G$-orbits

   (Def. \ref{SimplicialPresheavesOnFTypeGOrbits})

\end{proposition}
This is [Dwyer & Kan 1984, Thm. 3.1](#DwyerKan84), formulated here as a simplicial Quillen adjunction in view of the recognition principle for simplicial Quillen adjunctions (from [this Prop.](simplicial+Quillen+adjunction#RecognitionOfSimplicialQuillenAdjunctions)).

### General model category presentation
 {#ModelCategoryPresentation}

A version of the theorem that applies fairly generally for ([[discrete group|discrete]]) [[group objects]] in suitable [[cofibrantly generated model categories]] is in ([Guillou 06](#Guillou06), [Stephan 10](#Stephan10), [Stephan 13](#Stephan13)).

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

For $\mathcal{C}$ a [[cofibrantly generated model category]]
and for $G$ a [[discrete group]] (canonically regarded as a [[group object]] of $\mathcal{C}$ via its [[tensoring]] over [[Set]]) write $G \mathcal{C}$ for the category of $G$-[[actions]] in $\mathcal{C}$.

=--

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

A _cellular fixed point functor_ on $\mathcal{C}$ is ...

=--

([Guillou 06, def. 3.7](#Guillou06))

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

The [[fixed point spaces]]-functors on the following kinds of model categories are cellular

* the [[classical model structure on topological spaces]];

* the [[classical model structure on simplicial sets]];

* the global [[model structure on simplicial presheaves]] over any small category;

* (...)

=--

([Guillou 06, section 4](#Guillou06))

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

For $\mathcal{C}$ a [[cofibrantly generated model category]] with cellular fixed point functor, def. \ref{CellularFixedPointFunctor}, then the category $G \mathcal{C}$ of $G$-actions in $\mathcal{C}$ carries a [[cofibrantly generated model category]] structure $G \mathcal{C}_{fine}$ whose weak equivalences and fibrations are those morphisms whose [[underlying]] maps of
$H$-[[fixed loci]], for all [[subgroups]] $H$ of $G$, are equivalences or fibrations in $\mathcal{C}$, respectively.

=--

([Guillou 06, theorem 3.12](#Guillou06))

For the case that $\mathcal{C}$ is the [[classical model structure on topological spaces]] this yields the [[fine model structure on topological G-spaces]].

Write $Orb_G$ for the [[orbit category]] of $G$.

Write $(\mathcal{C}^{Orb_G^{op}})_{proj}$ for the projective global [[model structure on functors]] from the $G$-[[orbit category]] to $\mathcal{C}$.

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

There is a pair of [[adjoint functors]]

$$
  (\Theta, \Phi)
    \;\colon\;
  G\mathcal{C}
   \stackrel{\overset{\Theta}{\longleftarrow}}{\underset{\Phi}{\longrightarrow}}
  \mathcal{C}^{Orb_G^{op}}
$$

where $\Phi X \colon G/H \mapsto X^H$ assigns fixed-point objects and where $\Theta S$ has as underlying object $S(G/1)$.

This constitutes a [[Quillen equivalence]] between the above model structures

$$
  (\Theta, \Phi)
    \;\colon\;
  G\mathcal{C}_{fine}
    \underset{Quillen}{\simeq}
  \mathcal{C}^{Orb_G^{op}}_{proj}
 \,.
$$

=--

([Guillou 06, prop. 3.15](#Guillou06), [Stephan 13, Theorem 3.17](#Stephan13))

## Generalizations

1. Elmendorf's theorem may be generalized to the case where only a sub-family $\mathcal{H}$ of the closed subgroups of $G$ is considered ([Stephan 10](#Stephan10), also [May 96](#May96)).

1. There is an evident generalization of the [[orbit category]] of a fixed group $G$ to the _[[global orbit category]]_. Under this generalization an analog of Elmendorf's theorem plays a central role in [[global equivariant homotopy theory]] ([Rezk 14](#Rezk14)).

1. The orbit category for $G$ can also be generalized to the orbit category generated by any small category, $I$, where the $I$-orbits are $I$-diagrams in $Top$ whose strict colimit is equal to a point. If the orbits are either small or complete, then the $I$-equivariant model structure is Quillen-equivalent to a presheaf category (see pt. 11 of p. 8 of [PHCTIntro](Parametrized+Higher+Category+Theory+and+Higher+Algebra#PHCTIntro) and [Chorny13](#Chorny13)).

## Related concepts

* [[G-CW approximation]]

* [[equivariant Whitehead theorem]]

* [[tom Dieck splitting]]

* [[global equivariant homotopy theory]]

* [[Bredon cohomology]]

* [[equivariant rational homotopy theory]]

  * [[model structure on equivariant connective dgc-algebras]]

  * [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]]

  * [[fundamental theorem of dg-algebraic equivariant rational homotopy theory]]

## References
 {#References}

Lecture notes on Elmendorf's theorem:

* {#Blumberg17} [[Andrew Blumberg]], Section 1.3 of: _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

The "fine" homotopical structure on [[G-spaces]] (with fixed-point-wise weak equivalences) is originally due to

* {#Bredon67} [[Glen Bredon]], _[[Equivariant cohomology theories]]_, Springer Lecture Notes in Mathematics Vol. 34. 1967.

The equivalence of its [[homotopy category]] to that of presheaves over the orbit category is then due to:

* {#Elmendorf83} [[Anthony Elmendorf]], _Systems of fixed point sets_, Trans. Amer. Math. Soc., 277(1):275–284, 1983 ([jstor:1999356](https://www.jstor.org/stable/1999356))

Enhancement of Elmendorf's equivalence of homotopy categories to an [[sSet]]-[[enriched adjunction]] and/or to a [[Quillen equivalence]] of [[model categories]] based on the [[classical model structure on topological spaces]] and/or the [[classical model structure on simplicial sets]]:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

* {#CordierPorter96} [[Jean-Marc Cordier]], [[Timothy Porter]], Thm. 3.11 of: _Categorical Aspects of Equivariant Homotopy_, Applied Cat. Structures, **4** (1996) 195 - 212 ([doi:10.1007/BF00122252](https://doi.org/10.1007/BF00122252)) (Proceedings of the European Colloquium of Category Theory, 1994)

* {#Guillou06} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_, 2006 ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf), [[GuillouEquivariantHomotopy.pdf:file]])

* {#Stephan10} [[Marc Stephan]],

  _Elmendorf’s Theorem via Model Categories_,  2010 ([pdf](http://web.math.ku.dk/~jg/students/stephan.msproject.2010.pdf))

  _Elmendorf’s Theorem for Cofibrantly Generated Model Categories_, MS thesis, Zürich 2010 ([pdf](http://web.math.ku.dk/~jg/students/stephan.msthesis.2010.pdf))

* {#Stephan13} [[Marc Stephan]], _On equivariant homotopy theory for model categories_, Homology Homotopy Appl. 18(2) (2016) 183-208 ([arXiv:1308.0856](http://arxiv.org/abs/1308.0856), [doi:10.4310/HHA.2016.v18.n2.a10](https://dx.doi.org/10.4310/HHA.2016.v18.n2.a10))

* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], Sections 1 and 5.2 in: _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [doi:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))

Other discussions generalizing [Elmendorf 83](#Elmendorf83) (i.e. the equivalence of homotopy categories)  to general topological equivariance groups:

* {#Piacenza91} [[Robert Piacenza]], Section 6 of: _Homotopy theory of diagrams and CW-complexes over a category_, Can. J. Math. Vol 43 (4), 1991 ([doi:10.4153/CJM-1991-046-3](https://doi.org/10.4153/CJM-1991-046-3), [[Piazenza91.pdf:file]])

* {#May96} [[Peter May]] et al., Section V.3 of: _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 ([ISBN:978-0-8218-0319-6](https://bookstore.ams.org/cbms-91), [pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [pdf](https://ncatlab.org/nlab/files/MayEtAlEquivariant96.pdf))

More on the approach of [Dwyer-Kan 84](#DwyerKan84):

* [[Emmanuel Dror Farjoun]], Prop. 1.3 in: _Homotopy Theories for Diagrams of Spaces_, Proceedings of the AMS, Vol. 101, No. 1 (Sep., 1987), pp. 181-189 ([jstor:2046572 ](https://www.jstor.org/stable/2046572 ))

* {#Chorny13} [[Boris Chorny]], _Homotopy  theory  of  relative  simplicial  presheaves_,  Israel J. Math. 205 (2015), no. 1, 471–484, ([arXiv:1310.2932](https://arxiv.org/abs/1310.2932))

These Elmendorf-theorem [[Quillen equivalences]] (as in [Stephan 13](#Stephan13)) apply to other [[model categories]], and yield Elmendorf-like equivalences in other contexts:

* [[Jonathan Rubin]], _Elmendorf constructions for G-categories and G-posets_ ([arXiv:2006.08876](https://arxiv.org/abs/2006.08876))

See also related discussion of "[[orbispaces]]", starting with:

* [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916)),

relating to [[global equivariant homotopy theory]]:

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

and then to [[orbifolds]]:

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

An n-cat café discussion initiated by [[John Huerta]] and probing some of its uses in Mathematical Physics, can be found [here](https://golem.ph.utexas.edu/category/2018/06/elmendorfs_theorem.html#more), following

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. 371: 425. (2019 ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3))

Some generalization to [[monoid]]-[[actions]]:

* Mehmet Akif Erdal, *Homotopy theory of monoid actions via group actions and an Elmendorf style theorem*, Collectanea Mathematica (2022) [[arXiv 1609.06785](https://arxiv.org/abs/1609.06785), [doi:10.1007/s13348-022-00388-z](https://doi.org/10.1007/s13348-022-00388-z)]

[[!redirects Elmendorf theorem]]

[[!redirects model structure on equivariant simplicial sets]]
[[!redirects model structures on equivariant simplicial sets]]

[[!redirects model structure on topological G-spaces]]
[[!redirects model structures on topological G-spaces]]


Elmer Rees

[[!redirects Elmer G. Rees]]

* [Wikipedia page](http://en.wikipedia.org/wiki/Elmer_Rees)

* [Genealogy page](http://www.genealogy.ams.org/id.php?id=24461)

## related $n$Lab entries

* [[Euclidean geometry]]

* [[platonic solid]]

* [[classification of finite rotation groups]]

category:people


Emanuel Katz

* [webpage](http://physics.bu.edu/people/show/50)

## Selected writings

On [[2d QCD]]:

* [[Emanuel Katz]], Gustavo Marques Tavares, Yiming Xu, _A solution of 2D QCD at Finite $N$ using a conformal basis_ ([arxiv:1405.6727](https://arxiv.org/abs/1405.6727))

On [[geometric engineering of QFT|geometric engineering]] of [[flavour physics]] in [[intersecting D-brane models]] ([[AdS/QCD]]):

* [[Andreas Karch]], [[Emanuel Katz]],  _Adding flavor to AdS/CFT_, JHEP 0206:043, 2002 ([arxiv:hep-th/0205236](https://arxiv.org/abs/hep-th/0205236))

The bottom-up hard-wall model of [[AdS/QCD]] is due to

* {#ErlichKatzSonStephanov05} Joshua Erlich, [[Emanuel Katz]], Dam T. Son, Mikhail A. Stephanov, _QCD and a Holographic Model of Hadrons_, Phys.Rev.Lett.95:261602, 2005 ([arXiv:hep-ph/0501128](https://arxiv.org/abs/hep-ph/0501128))

while the soft-wall refinement is due to

* {#KKSS06} [[Andreas Karch]], [[Emanuel Katz]], Dam T. Son, Mikhail A. Stephanov, _Linear Confinement and AdS/QCD_, Phys.Rev.D74:015005, 2006 ([arXiv:hep-ph/0602229](https://arxiv.org/abs/hep-ph/0602229))

On [[glueballs]] via [[AdS-QCD]]:

* [[Emanuel Katz]], [[Adam Lewandowski]], [[Matthew Schwartz]], _Tensors Mesons in AdS/QCD_, Phys. Rev. D74:086004, 2006 ([arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388))

On [[confinement]] via [[AdS-QCD]]:

* {#KKSS06} [[Andreas Karch]], [[Emanuel Katz]], Dam T. Son, Mikhail A. Stephanov, _Linear Confinement and AdS/QCD_, Phys. Rev. D74:015005, 2006 ([arXiv:hep-ph/0602229](https://arxiv.org/abs/hep-ph/0602229))

On [[AdS/QCD]] for [[2d QCD]]:

* [[Emanuel Katz]], Takemichi Okui, _The 't Hooft Model As A Hologram_, JHEP 0901:013, 2009 ([arXiv:0710.3402](https://arxiv.org/abs/0710.3402))

category: people


Emanuel Knill

* [Institute page](http://ctqm.colorado.edu/people/knill)

## Selected writings:

On [[quantum programming languages]] for classically controlled [[quantum computation]]:

* {#Knill96} [[Emanuel Knill]], *Conventions for quantum pseudocode* Los Alamos Technical Report (1996) [[LA-UR-96-2724](https://www.osti.gov/biblio/366453-conventions-quantum-pseudocode), [doi:10.2172/366453](https://doi.org/10.2172/366453), [pdf](https://www.osti.gov/servlets/purl/366453)]

Introducing the *[[DQC1]]* [[quantum complexity theory|quantum complexity]] [[complexity class|class]] ("one clean qbit"):

* {#KnillLaflamme98} [[Emanuel Knill]], [[Raymond Laflamme]], *On the Power of One Bit of Quantum Information*, Phys. Rev. Lett. **81** (1998) 5672-5675 [[arXiv:quant-ph/9802037](https://arxiv.org/abs/quant-ph/9802037), [doi:10.1103/PhysRevLett.81.5672](https://doi.org/10.1103/PhysRevLett.81.5672)]

On [[quantum error correction]]:

* [[Emanuel Knill]], [[Raymond Laflamme]], *A Theory of Quantum Error-Correcting Codes*, Phys. Rev. Lett. **84** 2525-2528 (2000) [[arXiv:quant-ph/9604034](https://arxiv.org/abs/quant-ph/9604034)]

Realization of [[quantum error correction]] in [[experiment]]:

* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))

Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojciech H. Zurek]], *Resilient Quantum Computation: Error Models and Thresholds*, Proceedings of the Royal Society A **454** 1969 (1998) [[arXiv:quant-ph/9702058](https://arxiv.org/abs/quant-ph/9702058)]

Introducing [[measurement-based quantum computation]]:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Gerard J. Milburn]], *A scheme for efficient quantum computation with linear optics*, Nature **409** (2001) 46–52 [[doi:10.1038/35051009](https://doi.org/10.1038/35051009)]

On [[NMR]] [[spin resonance qbits]]:

* [[Raymond Laflamme]], [[Emanuel Knill]], et al., *Introduction to NMR Quantum Information Processing*, Proceedings of the International School of Physics "Enrico Fermi" **148** Experimental Quantum Computation and Information $[$[arXiv;quant-ph/0207172](https://arxiv.org/abs/quant-ph/0207172)$]$

Introducing the [[Eastin-Knill theorem]] on [[quantum error correction]] versus universality of [[quantum computation|quantum gates]]:

* {#EastinKnill09} [[Bryan Eastin]], [[Emanuel Knill]], *Restrictions on Transversal Encoded Quantum Gate Sets*,  Phys. Rev. Lett. 102, 110502 (2009) ([arXiv:0811.4262](https://arxiv.org/abs/0811.4262))

category: people


Emanuel Malek

* [personal page](https://www.emanuelmalek.com/)

* [InSpire page](https://inspirehep.net/authors/1253271)

## Selected writings

Generalization of [[Poisson-Lie T-duality]] to [[U-duality]] in [[exceptional generalized geometry]]:

* [[Emanuel Malek]], [[Daniel C. Thompson]], _Energy Physics - Theory Poisson-Lie U-duality in Exceptional Field Theory_, JHEP 0420 (2020) 058 [[arxiv:1911.07833](https://arxiv.org/abs/1911.07833)]

On [[KK-reduction]] of [[D=10 supergravity]] and [[D=11 supergravity]] on [[squashed sphere|squashed]] [[7-spheres]] via [[exceptional field theory]]:

* Bastien Duboeuf, [[Emanuel Malek]], [[Henning Samtleben]]
*Kaluza-Klein Spectrometry beyond Consistent Truncations: The Squashed $S^7$*,  J. High Energ. Phys. **2023** 62 (2023) [[arXiv:2212.01135](https://arxiv.org/abs/2212.01135), doi:10.1007/JHEP04(2023)062]

category: people


embedding

> This page consider the very general concept of embeddings. For the special cases traditionally considered see at _[[embedding of topological spaces]]_, _[[embedding of smooth manifolds]]_, and _[[embedding of types]]_.

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#### Higher category theory
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#Contents#
* table of contents
{:toc}

## Idea

An embedding is, generally, a [[morphism]] which in some sense is an [[isomorphism]] onto its [[image]]

For this to make sense in a given [[category]] $C$, we not only need a good notion of image.  Note that it is not enough to have the image of $f\colon X \to Y$ as a [[subobject]] $\im f$ of $Y$; we also need to be able to interpret $f$ as a morphism from $X$ to $\im f$, because it is this morphism that we are asking to be an isomorphism.

* [[effective epimorphism]] $\Rightarrow$ [[regular epimorphism]] $\Leftrightarrow$ [[covering]]

* [[effective monomorphism]] $\Rightarrow$ [[regular monomorphism]] $\Leftrightarrow$ [[embedding]] .

## As regular or effective monomorphisms

### Definition

One general abstract way to define an _embedding_ morphism is to say that this is equivalently a [[regular monomorphism]].

If the ambient category has finite [[limit]]s and [[colimit]]s, then this is equivalently an [[effective monomorphism]]. In terms of this we recover a formalization of the above idea, that an embedding is an iso onto its _image_ :

For a [[morphism]] $f : X \to Y$ in $C$ the definition of [image as an equalizer](http://ncatlab.org/nlab/show/image#AsEqualizer) says that the [[image]] of $f$ is

$$
  im f := \lim_\leftarrow ( Y \stackrel{\to}{\to} Y \coprod_X Y)
  \,.
$$

In particular we have a factorization of $f$ as

$$
  f : X \stackrel{\tilde f}{\to} im f \hookrightarrow Y
  \,,
$$

where the morphism on the right is a [[monomorphism]].

The morphism $f$ being an [[effective monomorphism]] means that $\tilde f$ is an [[isomorphism]], hence that $f$ is an "isomomorphism onto its image".

## Examples

### In $Top$

A morphism $U \to X$ of [[topological space]]s is a regular monomorphism precisely if this is an injection such that the topology on $U$ is the [[induced topology]]. This is an **[[embedding of topological spaces]]**.

### In $SmoothMfd$

* [[embedding of smooth manifolds]]

## Related concepts

* [[embedding type]]
* [[embedding in type theory]]
* [[decidable embedding]]

[[!redirects embedding]]
[[!redirects embeddings]]
[[!redirects imbedding]]
[[!redirects imbeddings]]

[[!redirects closed embedding]]

[[!redirects open embedding]]
[[!redirects open embeddings]]

[[!redirects imbedding]]
[[!redirects imbeddings]]


embedding of categories

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### Context
#### Category theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In much of mathematics, certainly in the traditional treatment of mathematical [[structure in model theory|structures]] from the point of view of [[logic]] and [[model theory]], an "[[embedding]]" is typically a [[monomorphism]] between structures that preserves *and reflects* the structure specified within a given [[language]] or [[theory]]. Such a notion is often definable directly in terms of [[category theory]], i.e., in terms of properties of morphisms in the designated category of structures and structure-preserving maps.

But for categories themselves, the situation and practice is more complicated, with no notion of "embedding of categories" that is universally accepted in the literature. A critical dividing line, leading to several distinct notions of embedding, lies in how one views the [[Cat|category of categories]]:

* If we view categories as mathematical structures in the traditional sense, with central concepts being definable in the ordinary 1-category of categories and [[functors]] (which are the relevant structure-preserving maps), then notions of embedding tend to emphasize they should be [[injective maps]] on the classes of objects (and more besides).

* If we view categories however as having a higher-dimensional aspect, i.e., if we view central concepts as being based rather on the [[2-category]] of categories, functors, *and* [[natural transformations]], then different notions of embedding come to the fore, where notably the assumption of injectivity on the object-classes clashes with the [[principle of equivalence]] (and is therefore rejected).

In this article we survey the various notions of categorical embedding that have appeared in the literature, and try to describe some of the underlying contexts and rationales.

## Definitions

### 1-categorical definitions

Perhaps the most obvious 1-categorical definition of embedding is:

* a [[functor]] $F: C \to D$ that is both injective on objects and a [[faithful functor]].

This definition is adopted in [Adamek-Herrlich-Strecker](#AHS) and in [Riehl](#Riehl), for example.  In this definition we accept that categories are [[strict categories]], where objects can be compared for [[equality]].  Embeddings in this sense are straightforwardly the same thing as [[monomorphisms]] in the $1$-category [[Cat]].

If we define categories with [one collection of morphisms](/nlab/show/category#OneCollectionOfMorphisms), then this definition is equivalent to saying that the action of $F$ on morphisms is an [[injective function]] $Mor(C) \to Mor(D)$.  In particular, if this latter condition holds and $F(c) = F(c')$ for objects $c, c'$ of $C$, then $F(1_c) = F(1_{c'})$ and thus $1_c = 1_{c'}$ by injectivity on morphisms, whence $c = c'$.

However, as noted above, in many cases an "embedding" means something *stronger* than a mere monomorphism, being required to reflect "all the structure".  For instance, an embedding of [[topological spaces]] is a subspace inclusion, not merely an injective continuous map (the latter being the monos in [[Top]]).

In particular, the above definition of embedding does not even reflect the property of objects being isomorphic, so that it is not necessarily injective on isomorphism classes of objects!  This suggests that a (still 1-categorical) embedding of categories ought to be something stronger.  Some possibilities include:

* The [[regular monomorphisms]] or [[extremal monomorphisms]] in [[Cat]] (what are those?)
* Monomorphisms in $Cat$ that are also [[full functors]], hence [[fully faithful functors]].  These are sometimes called **full embeddings**.
* Monomorphisms in $Cat$ that are also injective on isomorphism classes of objects (this is used in [[Toposes, Triples, and Theories]]), or more strongly that are also [[pseudomonic functors]].

### 2-categorical definitions

Most of the above 1-categorical notions of embedding corresponds to a "pure" 2-categorical notion by simply removing the injectivity on objects.  Thus we have the following candidates:

* [[faithful functors]]
* [[fully faithful functors]]
* [[pseudomonic functors]]

The choice between these is closely related to the question of giving an equivalence-invariant notion of [[subcategory]].

### Relationship

Every fully faithful functor is *equivalent* to one that is fully faithful and injective on objects.  Let $f:C\to D$ be fully faithful, and let $E$ be the "mapping cylinder" category, whose objects are the disjoint unions of those of $C$ and $D$, with $C$ and $D$ embedded fully-faithfully (and of course injectively on objects), and $E(c,d) = D(f c,d)$ and $E(d,c) = D(d,f c)$.  Full-faithfulness of $f$ allow us to compose all arrows in $E$ making it a category.  There is a projection $E\to D$ that is the identity on $D$ and acts by $f$ on $C$, and which is an inverse equivalence to the inclusion $D\to E$.  Thus, $f:C\to D$ is equivalent, in the 2-category whose objects are functors and whose morphisms are pseudo-commutative squares, to the inclusion functor $C\to E$, which is fully faithful and injective on objects.

## Examples

### The Yoneda embedding

Probably the most common embedding of categories encountered is the [[Yoneda embedding]].  This is fully faithful, but whether or not it is always injective on objects depends on set-theoretic details of how we define categories.  If we use a definition of category with [a family of collections of morphisms](/nlab/show/category#AFamilyOfCollectionsOfMorphisms), then it might happen that two distinct (but necessarily isomorphic) objects $x$ and $y$ have $C(z,x) = C(z,y)$ as sets for all $z\in C$, so that the Yoneda embedding would take $x$ and $y$ to literally the same presheaf.  (Such an equality of hom-sets is not as weird as it might sound; for instance, when regarding a [[preordered set]] as a category it's natural to define every nonempty homset to be the *same* [[terminal set]].)

## Related concepts

* [[Barr embedding theorem]]

* [[embedding]]

* [[subcategory]]

## References

* {#AHS} [[Jiří Adámek]], [[Horst Herrlich]], and [[George Strecker]], *Abstract and Concrete Categories: The Joy of Cats*, John Wiley and Sons 1990. ([Online edition](http://katmat.math.uni-bremen.de/acc/))

* {#Riehl} [[Emily Riehl]], *Category Theory in Context*, Dover Publications, Inc. 2016. ([pdf](http://www.math.jhu.edu/~eriehl/context.pdf))

[[!redirects embedding of categories]]


embedding of differentiable manifolds

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#### Manifolds and cobordisms
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=--
=--

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

## Idea

Embeddings of [[differentiable manifolds]] are [[submanifold]] inclusions.

## Definition

### Classical definition

+-- {: .num_defn #SmoothManifoldsEmbedding}
###### Definition
**(embedding of smooth manifolds)**

An _[[embedding]] of [[smooth manifolds]]_ is a [[smooth function]] $f : X \hookrightarrow Y$ between [[smooth manifolds]] $X$ and $Y$ such that

1. $f$ is an [[immersion of smooth manifolds|immersion]];

1. the underlying [[continuous function]] is an [[embedding of topological spaces]].

A _closed embedding_ is an embedding such that the image $f(X) \subset Y$ is a [[closed subset]].

=--

### Synthetic definition in differential cohesion
 {#SyntheticDefinitionInDifferentialCohesion}

The classical definition \ref{SmoothManifoldsEmbedding} of _embedding_ of smooth manifolds should have the following axiomatization in [[differential cohesive (infinity,1)-topos|differentially cohesive infinity-topos theory]], where one assumes the following system of [[adjoint modalities]]

$$
  \array{
    && \Re &\dashv& \Im &\dashv& \&
    \\
    && && \vee && \vee
    \\
    && && ʃ &\dashv&  \flat &\dashv& \sharp
  }
$$

Of these we now use

1. the [[sharp modality]] $\sharp$

1. the [[infinitesimal shape modality]] $\Im$.

Then given a morphism $U \overset{i}{\longrightarrow} X$ we have

1. $i$ being a [[monomorphism]] means that it is a [[monomorphism in an (infinity,1)-category]]

1. $i$ being an [[immersion]] means (by [this Discussion](immersion+of+smooth+manifolds#InDifferentialCohesion)) that it is a [[formally unramified morphism]] in that the comparison morphism

   $$
     U \longrightarrow X \times_{\Im X} \Im(U)
   $$

   into the [[pullback]] along the $\Im$-[[unit of an adjunction|unit]] is a monomorphism;

1. $i$ being an [[embedding]] means (by [this Discussion](subspace+topology#UniversalProperty)) that the comparison morphism

   $$
      U \longrightarrow X \times_{\sharp X} \sharp U
   $$

   is an [[equivalence in an (infinity,1)-category|equivalence]].

In terms of [[cohesive homotopy type theory]] this should mean that the [[characteristic function]]

$$
  \chi_U \;\colon\; X \longrightarrow Type
$$

to the [[type universe]], of $U$ regarded as a [[dependent type]] over $X$, satisfies:

1. it factors through the [[universe of propositions]] $Prop$, hence of $\vert -\vert_{-1}$-[[modal types]] (where $\vert-\vert_{-1}$ is the [[n-truncation modality|(-1)-truncation modality]]);

1. it factors through the universe of $\Im$-[[submodal types]];

1. it factors through the universe of $\sharp$-[[modal types]].

## Examples
 {#Examples}

+-- {: .num_example }
###### Nonexample
**(immersions that are not embeddings)**

Consider an immersion $f \;\colon\; (a,b) \to \mathbb{R}^2$ of an [[open interval]] into the [[Euclidean plane]] (or the [[2-sphere]]) as shown on the right. This is not a [[embedding of smooth manifolds]]: around the points where the image crosses itself, the function is not even injective, but even
a#t the points where it just touches itself, the pre-images under $f$ of open subsets of $\mathbb{R}^2$ do not exhaust the open subsets of $(a,b)$, hence do not yield the [[subspace topology]].

As a concrete examples, consider the function  $(sin(2-), sin(-)) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2$. While this is an immersion and [[injective]], it fails to be an embedding due to the points at $t = \pm \pi$ "touching" the point at $t = 0$.

> graphics grabbed from Lee

=--

## Properties

### Basic properties

+-- {: .num_prop #ProperInjectiveImmersionsAreEquivalentlyTheClosedEmbeddings}
###### Proposition
**(proper injective immersions are equivalently the closed embeddings)**

Let $X$ and $Y$ be [[smooth manifolds]], and let $f \colon X \to Y$ be a [[smooth function]]. Then the following are equivalent

1. $f$ is a [[proper map|proper]] [[injective function|injective]]  [[immersion]];

1. $f$ is a closed embedding (def. \ref{SmoothManifoldsEmbedding}).

=--

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

Since topological manifolds are [[locally compact topological spaces]] ([this example](locally+compact+topological+space#TopologicalManifoldsAreLocallyCompact)), this follows directly since [injective proper maps into locally compact spaces are equivalently closed embeddings](embedding+of+topological+spaces#InjectiveProperMapsAreEquivalentlyTheClosedEmbeddings).

=--

### Embedding into Euclidean space
 {#EmbeddingIntoEuclideanSpace}

Every smooth manifold has a [[embedding of smooth manifolds]] into a [[Euclidean space]] $\mathbb{R}^k$ of some dimension $k$.

For _[[compact topological space|compact]]_ smooth manifolds this is easy to see (prop. \ref{CompactManifoldEmbedsIntoLargeDimensionalEuclideanSpace} below), while the generalization to non-compact smooth manifolds requires a tad more work (theorem \ref{theorema} below). What is non-trivial is to find the minimum dimension of the ambient Euclidean space for which embedding in general still exist. The _[[Whitney embedding theorem]]_ (prop. \ref{WhitneyStrongEmbeddingTheorem} below) says that this is twice the dimension of the manifolds to be embedded.

+-- {: .num_example #CompactManifoldEmbedsIntoLargeDimensionalEuclideanSpace}
###### Proposition

For every [[compact topological space|compact]] [[smooth manifold]] $X$ (of [[finite number|finite]] [[dimension]]), there exists some $k \in \mathbb{N}$ such that $X$ has an embedding (def. \ref{SmoothManifoldsEmbedding}) into the [[Euclidean space]] of dimension $k$:

$$
  X \overset{\text{embd}}{\hookrightarrow} \mathbb{R}^k
$$

=--

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

Let

$$
  \{\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X\}_{i \in I}
$$

be an [[atlas]] exhibiting the [[smooth structure]] of $X$. In particular this is an [[open cover]], and hence by compactness there exists a [[finite set|finite]] [[subset]] $J \subset I$ such that

$$
  \{\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\}_{i \in J \subset I}
$$

is still an open cover.

Since $X$ is a [[smooth manifold]], there exists a [[partition of unity]] $\{f_i \in C^\infty(X,\mathbb{R})\}_{i \in J }$ subordinate to this cover with _[[smooth functions]]_ $f_i$ (by [this prop.](partition+of+unity#SmoothManifoldAdmitsSmoothPartitionsOfUnity)).

This we may use to extend the inverse [[chart]] identifications

$$
  X \supset \;\; U_i \underoverset{\simeq}{\psi_i}{\longrightarrow} \mathbb{R}^n
$$

to smooth functions

$$
  \hat \psi_i \;\colon\; X \to \mathbb{R}^{n}
$$

by setting

$$
  \hat \psi_i
  \;\colon\;
   x
  \mapsto
  \left\{
    \array{
       f_i(x) \cdot \psi_i(x) &\vert& x \in U_i \subset X
       \\
       0 &\vert& \text{otherwise}
    }
  \right.
  \,.
$$

The idea now is to combine all these functions to obtain an injective function

$$
  (\hat \psi_i)_{i \in J}
  \;\colon\;
  X \longrightarrow (\mathbb{R}^n)^{\vert J\vert }
  \simeq
  \mathbb{R}^{n \cdot {\vert J \vert }}
  \,.
$$

But while this is injective, it need not be an [[immersion]], since the [[derivatives]] of the product functions $f_i \cdot \psi_i$ may vanish, even though the derivatives of the two factors do not vanish separately. However this is readily fixed by adding yet more ambient coordinates and considering the function

$$
  (\hat \psi_i, f_i)_{i \in I}
  \;\colon\;
  X
    \longrightarrow
  \left(\mathbb{R}^{n+1}\right)^{\vert J \vert}
  \simeq
  \mathbb{R}^{(n+1)\cdot {\vert J \vert}}
  \,.
$$

This is an immersion. Hence it remains to see that it is also an [[embedding of topological spaces]].

By [this prop](embedding+of+topological+spaces#OpenClosedContinuousInjectionsAreEmbeddings) it is sufficient to see that the injective continuous function is a [[closed map]]. But this follows generally since $X$ is a [[compact topological space]] by assumption, and since [[Euclidean space]] is a [[Hausdorff topological space]], and since [[maps from compact spaces to Hausdorff spaces are closed and proper]].

=--

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

For every [[smooth manifold]] $X$ of dimension $n$ (Hausdorff, sigma-compact), there exists some $k \in \mathbb{N}$ such that $X$ has an embedding into the [[Euclidean space]] of dimension $k$.

=--

What is harder to prove is that $k$ may be chosen to be as small as $2n$

+-- {: .num_prop #WhitneyStrongEmbeddingTheorem}
###### Proposition
**([[Whitney's strong embedding theorem]])**

For every [[smooth manifold]] $X$ of dimension $n$ (Hausdorff, sigma-compact), there is an embedding into the [[Euclidean space]] of dimension $2n$.

In fact this bound is minimal, there are smooth manifolds of dimension $n$ which have no embdding into $\mathbb{R}^k$ for $k \lt 2n$.

=--

## Space of Embeddings ##

This section follows from a discussion on the _Algebraic Topology_ mailing list as to making concrete the contractibility of the space of embeddings of a smooth manifold in $\mathbb{R} ^{(\infty )}$, the union of the [[Euclidean spaces]]. There are two parts to showing that this space is [[contractible]]. The first is showing that it is not empty. The second is to exhibit a contraction.

Showing that it is not empty is a [[partition of unity]] argument. As the space that we are embedding in is infinite dimensional, we do not need to resort to either compactness or Whitney's embedding theorem to obtain an embedding. One of these would be required if we wanted to show that our embedding was actually in some finite Euclidean space.

The contraction depends on certain properties of $\mathbb{R} ^{(\infty )}$ which are quite general and seemingly unrelated to the question of embeddings. The key property is the existence of a [[shift map]]. The same idea is used in showing that many other related spaces are contractible, including the [[infinite sphere]].

+-- {: .num_section #sectionb }
=--
### Existence of Embeddings ### {: .num_section}

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

Let $M$ be a [[smooth manifold]]. Let $V$ be an infinite dimensional [[locally convex topological vector space]]. Then there is an embedding (def. \ref{SmoothManifoldsEmbedding}) $M \hookrightarrow  V$.

=--

+-- {: .proof }
###### Proof
We shall start with the case $V = \mathbb{R} ^{(\infty )}$. The more general case will follow from that.

As $M$ is a smooth manifold, we can choose a [[partition of unity]] $\{ \rho _{i} : i \in I\} $ on $M$ with the property that for each $\rho _{i}$, its support is contained in the domain of some chart for $M$ and the family of those domains is locally finite. Let us suppose that these charts are $\phi _{i} \colon U_{i} \to \mathbb{R} ^{n}$, where $n = \dim M$. As $M$ is second countable, this partition of unity will be countable.

Using the partition of unity, we extend $\phi _{i} \colon U_{i} \to \mathbb{R} ^{n}$ to a smooth map $\psi _{i} \colon M \to \mathbb{R} ^{n+1}$ by setting, for $x \in U_{i}$, $\psi _{i}(x) = (\rho _{i}(x) \phi _{i}(x),\rho _{i}(x))$ and, for $x \notin U_{i}$, $\psi _{i}(x) = 0$. Using the $\psi _{i}$ we define $\psi \colon M \to \mathbb{R} ^{(\infty )}$ by $\psi (x) = (\psi _{i}(x))$. This is well-defined as, for any $x \in M$, there are only finitely many of the $\rho _{i}$ for which $\rho _{i}(x) \ne 0$ and thus only finitely many of the $\psi _{i}$ for which $\psi _{i}(x) \ne 0$. We define $\psi \colon M \to V$ by composing this with the inclusion $\mathbb{R} ^{(\infty )} \to V$.

We claim that this is an embedding.

The first thing to show is that it is continuous. Let $x \in M$. Then there is some $i$ such that $x \in U_{i}$. As the family $U_{i}$ is locally finite, there are only a finite number of other $U_{j}$ with non-trivial intersection with $U_{i}$. Thus there are only a finite number of $\rho _{j}$ which take non-zero values on $U_{i}$. Hence the image of $U_{i}$ under $\psi $ lies in some $\mathbb{R} ^{k}$ and the induced map $U_{i} \to \mathbb{R} ^{k}$ is built up from a finite number of the $\phi _{j}$ and $\rho _{j}$, whence is continuous. Hence $\psi $ is continuous.

The next thing to show is that it is injective. Suppose that $x,y \in M$ are such that $\psi (x) = \psi (y)$. As the partition of unity functions can be recovered from $\psi $, we must therefore have $\rho _{i}(x) = \rho _{i}(y)$ for all $i$. Since $\{ \rho _{i}\} $ is a partition of unity, there must be some $i$ such that $\rho _{i}(x) \ne 0$ (whence also $\rho _{i}(y) \ne 0$). This means that $x$ and $y$ lie in the domain of the chart $\phi _{i}$. Moreover, as $\psi (x) = \psi (y)$, we must have $\psi _{i}(x) = \psi _{i}(y)$. Lastly, as $\rho _{i}(x) = \rho _{i}(y)$ and this is non-zero, we have $\phi _{i}(x) = \phi _{i}(y)$. Therefore $x = y$.

We want to show that $\psi $ is a topological embedding. To do this, we consider first what happens to the sets $\rho _{i}^{-1}(0,\infty )$. Pick some $i \in I$. Let $p_{i} \colon \mathbb{R} ^{(\infty )} \to \mathbb{R} ^{n+1}$ be the projection onto the $n + 1$ coordinates corresponding to the position of $\psi _{i}$ in $\psi $, so that $p_{i} \psi = \psi _{i}$. Let $W \subseteq \mathbb{R} ^{n+1}$ be the open subset where the last coordinate is non-zero. Then $p_{i}^{-1}(W)$ is open in $\mathbb{R} ^{(\infty )}$. Now $x \in \psi (M) \cap p_{i}^{-1}(W)$ if and only if $\rho _{i}(x) \ne 0$, thus:
$$
\psi (M) \cap p_{i}^{-1}(W) = \psi (\rho _{i}^{-1}(0,\infty )).
$$
We define a map $W \to \mathbb{R} ^{n}$ by $(x,t) \mapsto t^{-1} x$. The composition
$$
\rho _{i}^{-1}(0,\infty ) \xrightarrow{\psi } p_{i}^{-1}(W) \xrightarrow{p_{i}} W \to \mathbb{R} ^{n}
$$
is then seen to be the same as the restriction of $\phi _{i}$ to $\rho _{i}^{-1}(0,\infty )$. As $\phi _{i}$ is a homeomorphism and $\rho _{i}$ is continuous, the above map $\rho _{i}^{-1}(0,\infty ) \to \mathbb{R} ^{n}$ is a homeomorphism onto its image which is an open subset of $\mathbb{R} ^{n}$.

We therefore see that if $V \subseteq \rho _{i}^{-1}(0,\infty )$ is open then $\psi (V)$ is equal to the inverse image of $\phi _{i}(V)$ under the composition $p_{i}^{-1}(W) \to W \to \mathbb{R} ^{n}$ and is thus open in $\psi (M)$.

Now suppose that $V \subseteq M$ is an arbitrary open subset. Then for each $i \in I$, $V \cap \rho _{i}^{-1}(0,\infty )$ is open and the union of these is $V$. So $\psi (V)$ is the union of $\psi (V \cap \rho _{i}^{-1}(0,\infty ))$ and thus is open in $\psi (M)$. Hence $\psi $ is an embedding.

To complete the proof for $\mathbb{R} ^{(\infty )}$, we need to show that $\psi $ is an [[immersion]]. This is a local condition and follows from the fact that we can recover $\phi _{i}$ from $\psi _{i}$.

The case for an arbitrary $V$ is a simple adaptation of the above. As $V$ is infinite dimensional, it admits linear injections $\mathbb{R} ^{(\infty )} \to V$ and composing one of those with $\psi $ gives an injective map $M \to V$. To ensure that it is an embedding, we need suitable projections $V \to \mathbb{R} ^{n+1}$. To know that these exist, we simply choose our map $\mathbb{R} ^{(\infty )} \to V$ carefully as follows. Pick a non-zero vector $v_{1} \in V$. By the [[Hahn-Banach theorem]], there is some $f_{1} \in V^{*}$ with $f_{1}(v_{1}) = 1$. Now choose a non-zero $v_{2} \in \ker f_{1}$. Again, we have $f_{2} \in V^{*}$ such that $f_{2}(v_{2}) = 1$ and $f_{2}(v_{1}) = 0$. By construction, $f_{1}(v_{1}) = 0$ also. We continue this recursively, choosing at each stage a non-zero $v_{k} \in \bigcap \ker f_{j}$ and $f_{k} \in V^{*}$ with $f_{k}(v_{k}) = 1$ and $f_{k}(v_{j}) = 0$ for $j \lt   k$. The result is an injection $\mathbb{R} ^{(\infty )} \to V$ together with functions $f_{j} \in V^{*}$ such that the
composition $\mathbb{R} ^{(\infty )} \to V \xrightarrow{f_{j}} \mathbb{R} $ is the $j$th coordinate function.
=--

With an assumption on $V$ and a careful choice of partition of unity, we can ensure that our embedding so constructed has closed image.

+-- {: .num_theorem #theoremc .thplain}
###### Theorem
Let $M$ be a smooth manifold.
Let $V$ be a [[locally convex topological vector space]] and assume that there exists a family of continuous linear functionals $\{f_j : j \in \mathbb{N}\} \subseteq V^*$ with the properties:

1. For $j \in \mathbb{N}$, $f_j$ is non-trivial on $\bigcap_{k \ne j} \ker f_k$, and
1. The family of functionals $\{\sum_{j \in F} f_j : {|F|} \lt \infty\}$ is [[equicontinuous]]

then there is an embedding $\psi \colon M \to V$ with closed image.
=--

+-- {: .proof}
###### Proof
From the first property, we can find $v_j \in \bigcap_{k \ne j} \ker f_k$ such that $f_j(v_j) = 1$.  We use the family $\{v_j\}$ to define a linear injection $\mathbb{R}^{(\infty)} \to V$ (continuous by necessity).  The functionals $f_j$ provide the necessary coordinate projections so that the proof of Theorem \ref{theorema} works.

In addition, we shall assume that our partition of unity used in the proof of Theorem \ref{theorema} is such that the functions therein have compact support.

Let $(x_\lambda)$ be a net in $M$ for which $(\psi(x_\lambda))$ converges in $V$, say to $x$.
The assumption on the functionals is there to show that $x \ne 0$.
Let us show why this is the case.
For a finite set $F \subseteq \mathbb{N}$, let $f_F = \sum_{j \in F} f_j$.
The assumption of equicontinuity means that $\bigcap_F f_F^{-1}(-1,1)$ is an open neighbourhood of $0$ in $V$.
Now for each $y \in M$, there is some finite $F \subseteq \N$ for which $\sum f_j(y) = 1$.  This is because we can choose $F$ to correspond to those terms in the partition of unity which are non-zero on $y$.
Hence $\psi(M)$ has empty intersection with the above neighbourhood of $0$, and so no net in $\psi(M)$ can converge to $0$.

Thus $x \ne 0$ and, moreover, there is some finite $F \subseteq \mathbb{N}$ for which $f_F(x) \ne 0$ and thus there is some $j \in \mathbb{N}$ for which $f_j(x) \ne 0$.  Since $(\psi(x_\lambda)) \to x$, there is therefore a cofinal subset on which $f_j$ is never zero.  Let us pass to that subnet.  Now on $M$, the functions $f_i \circ \psi$ are either the functions in the partition of unity or are dominated by them.  Hence there is some $i \in \mathbb{N}$ with the property that for all $\lambda$, $\rho_i(x_\lambda) \ne 0$ (now that we've passed to the subnet).  This means that $(x_\lambda)$ lies in the support of $\rho_i$, which is (by the first assumption) a compact subset of $M$.  It therefore has a convergent subnet with limit, say $x'$.  But by continuity of $\psi$ and uniqueness of limits in $V$, it must be the case that $\psi(x') = x$.

Hence $\psi(M)$ is closed in $V$.
=--

There are various situations in which it is straightforward to find a suitable family of functionals.  If $V$ is [[barrelled]], then finding a family for which $\sum f_j$ weakly converges will do since then the family of partial sums is simply bounded and hence, as $V$ is barrelled, equicontinuous.

Thus the coordinate projections will do for $\mathbb{R}^{(\infty)}$ and for $\ell^1$.  For $\ell^2$ we can take weighted projections, say $f_j = \frac{1}{j} p_j$.  In this case, our dual vectors will be $\{j e_j\}$.  For $\ell^\infty$ we can take weighted projections corresponding to a sequence in $\ell^1$.

It is interesting to note what is going on in these latter cases where we take weighted projections.  If we embedded our manifold using the standard basis vectors as our "anchor points" then the structure of the ambient space is insufficient to guarantee that our manifold does not contain a convergent net with limit point outside.  More exactly, there could be a net in our manifold such that in the image it converges to zero.  The partition of unity is not enough to keep our points away from zero, since as we get further along the family in our partition we could need more and more terms.  That is, we could have the situation where we have a sequence $(x_n)$ in $M$ where $x_n$ is "seen" by $n$ of the functions in the partition of unity.  Then we could have $\psi(x_n)$ to be something like $(0,\dots,0,1/n,1/n,\dots,1/n,0,\dots)$.  In, say, $\ell^2$ then this sequence converges to $0$.  Essentially, $\psi(x_n)$ gets more and more diffuse.  Introducing the weights, or rather their inverses, counters this by ensuring that terms "far down" the sequence are given more consideration.

+-- {: .num_section #sectionc }
=--
### Contractibility of Embeddings ### {: .num_section}

Once we know that the space of embeddings of a compact smooth manifold in $\mathbb{R} ^{(\infty )}$ is not empty, its contractibility follows from rather general properties. It follows from the existence of a [[shift map|split map]].

+-- {: .num_theorem #theoremb .thplain }
###### Theorem ######
Let $M$ be a smooth manifold. Let $V$ be a locally convex topological vector space which admits a split map. Then the space of embeddings, $\Emb(M,V)$, is contractible.
=--

+-- {: .proof }
###### Proof
Recall that to have a [[shift map|split map]] on $V$ means that we have a decomposition $V \cong V \oplus V$ with a non-degeneracy property. This implies that one of the induced maps $V \to V$, called the *splitting map*, has no eigenvectors.

Let $S^{+} \colon V \to V$ be a splitting map and let $S^{-}$ be the remainder. Then the map $H \colon [0,1] \colon V \to V$ given by $H(t,v) = H_{t}(v) = (1 - t) v + t S^{+} v$ is a homotopy between the identity on $V$ and $S^{+}$.

At $t = 0$, $H_{t} = I$ which is an inclusion. For $t \ne 0$ then if $H_{t}(v) = 0$ we have that $S^{+} v = t^{-1}(1 - t) v$ whence $v$ is an eigenvector of $S^{+}$. But the assumption that $S^{+}$ is a splitting map includes the property that it has no eigenvectors, and thus $H_{t}$ is an inclusion for all $t$. Moreover, $H_{t}$ is an embedding for all $t$ since $I = H_{t} + t S^{-} v$.

Hence we can apply $H_{t}$ to $\Emb(M,V)$ to obtain a homotopy between the identity on $\Emb(M,V)$ and the inclusion of $\Emb(M,V)$ in itself as the subspace of all embeddings that lie in the even subspace.

Now we pick some embedding $\psi _{0} \colon M \to V$ which, by applying $S^{-}$ if necessary, we assume to exist in the image of $S^{-}$. We define a homotopy $G_{t}$ on the embedding space by $G_{t}(\psi ) = t \psi _{0} + (1 - t) \psi $, using the additive structure in $V$. That this passes through the space of embeddings is clear since for $t \ne 1$ we can recover the embedding $\psi _{0}$ by composing with $S^{-}$. Combining $H_{t}$ and $G_{t}$ as homotopies leads to the desired contraction of the embedding space.

=--

## Related concepts

* [[submanifold]]

* [[Whitney embedding theorem]], [[Nash embedding theorem]]

* [[hypersurface]]

## References

Lecture notes include

* Paul Rapoport, _Introduction to Immersion, Embedding and the Whitney Embedding Theorem_, 2015 ([pdf](http://schapos.people.uic.edu/MATH549_Fall2015_files/Survey%20Paul.pdf))

* Emery Thomas, _Embedding manifolds in Euclidean space_, Osaka Journal Mathematics 13 (1976) ([pdf](http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm13_01_12.pdf))

* Rodolfo De Sapio, _Embedding $\pi$-manifolds_, Annals of mathematics, Second Series, Vol 82, No 2 ([jstor](http://www.jstor.org/pss/1970642))

* [[Manifold Atlas]], _[Embeddings in Euclidean space: an introduction to their classification](http://www.map.mpim-bonn.mpg.de/Embeddings_in_Euclidean_space:_an_introduction_to_their_classification)_

Discussion of embeddings of [[spheres]] into each other (generalizing the concept of [[knots]]):

* {#Haefliger66} [[André Haefliger]], _Differentiable Embeddings of $S^n$ in $S^{n+q}$ for $q \gt 2$_, Annals of Mathematics Second Series, Vol. 83, No. 3 (May, 1966), pp. 402-436 ([jstor:1970475](https://www.jstor.org/stable/1970475))

* Dennis Roseman, Masamichi Takase, _High-codimensional knots spun about manifolds_ ([arXiv:math/0609055](https://arxiv.org/abs/math/0609055))

[[!redirects embeddings of differentiable manifolds]]

[[!redirects embedding of smooth manifolds]]
[[!redirects embeddings of smooth manifolds]]


embedding of smooth manifolds into formal duals of R-algebras

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

#Contents#
* table of contents
{:toc}

## Statement

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

The [[functor]]
$$
  C^\infty(-)
   \;\colon\;
  SmoothMfd
    \longrightarrow
  Alg_{\mathbb{R}}^{op}
$$
which sends a [[smooth manifold]] ([[finite number|finite]] [[dimension|dimensional]], [[paracompact topological space|paracompact]], [[second countable topological space|second countable]]) to (the [[formal dual]] of) its $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth functions]] is a [[full and faithful functor]].

In other words, for two [[smooth manifolds]] $X,Y$ there is a [[natural bijection]] between the [[smooth functions]] $X \to Y$ and the $\mathbb{R}$-[[associative algebra|algebra]] [[homomorphisms]] $C^\infty(X)\leftarrow C^\infty(Y)$.

In particular, the [[diffeomorphisms]] between smooth manifolds are in [[natural bijection]] to the [[isomorphisms]] between these algebras.

=--

\begin{remark}\label{Attribution}
**(attribution)**
\linebreak
For the case of [[diffeomorphisms]], Thm. \ref{Embedding} was proven by [Pursell (1952)](#Pursell52), following an announcement by [Shanks (1951)](#Shanks51). This is the case that most reviews focus on, e.g. [Grabowski (1978)](#Grabowski78), [Marsden, Ratiu & Abraham (2002)](#MarsdenRatiuAbraham02), [Grabowski (2005)](#Grabowski05).

For the case that the domain is a point the statement is an an [[exercise]] (without reference to Pursell) in [Milnor & Stasheff (1974), Problem 1-C (p. 11)](#MilnorStasheff74), sometimes now referred to as "Milnor's exercise".

The general statement of Thm. \ref{Embedding}, with a detailed proof, is given in [Kolář, Slovák & Michor (1993), lemma 35.8, corollaries 35.9, 35.10](#KolarSlovakMichor93).
\end{remark}

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

The statement of theorem \ref{Embedding} serves as the stepping-stone for generalizations of [[differential geometry]] such as to [[supergeometry]]. On the other hand, for transporting various applications familiar from [[algebraic geometry]] to [[differential geometry]] (such as [[Kähler differentials]], see there) the above embedding is insufficient, and instead of just remembering the [[associative algebra]] structure, one needs to remember the _[[smooth algebra]]_-structure on algebras of smooth functions. See also at _[[synthetic differential geometry]]_.

=--

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

If one drops standard regularity assumptions on manifolds then theorem \ref{Embedding} may break. For instance allowing [[countable set|uncountably]] many connected components, then there are counterexamples ([MO discussion](http://mathoverflow.net/a/91445)).

=--

## Related theorems

* [[Borel's theorem]]

* [[Hadamard's lemma]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

The analogous statement in [[topology]] is:

* [[Gelfand duality]]

[[!include Isbell duality - table]]

## References

The case of the category of [[smooth manifolds]] with (just) [[diffeomorphisms]] between them is proved in

* {#Pursell52} [[Lyle Eugene Pursell]], _Algebraic structures associated with smooth manifolds_, PhD dissertation, Purdue University, (1952)  93 pp.  [ISBN:978-1392-88143-9, [proquest:2327629257](https://www.proquest.com/docview/2327629257)]

following an announcement in

* {#Shanks51} M. E. Shanks, *Rings of functions on locally compact spaces*, 469th meeting of the AMS (1951) [[pdf](https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf)]

The statement for domain a point is due to

* {#MilnorStasheff74} [[John Milnor]], [[James Stasheff]], Problem 1-C (p. 11) in: _Characteristic Classes_, Annals of Mathematics Studies, Princeton University Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76)]

with a proof spelled out in:

* {#Dubuc79} [[Eduardo Dubuc]], Prop. 0.7 in: *Sur les modèles de la géométrie différentielle synthétique*, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **20**  3 (1979) 231-279  [[numdam:CTGDC_1979__20_3_231_0](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)]

  > (in defining the [[Cahiers topos]])

Expository accounts for the case of [[isomorphisms]] are in

* {#Grabowski78} [[Janusz Grabowski]], _Isomorphisms and ideals of the Lie algebras of vector fields_, Inventiones mathematicae volume 50, pages 13–33 (1978) ([doi:10.1007/BF01406466](https://doi.org/10.1007/BF01406466))

* {#MarsdenRatiuAbraham02} [[Jerrold Marsden]], J. Ratiu, R. Abraham, Theorem 4.2.36 in: _Manifolds, tensor analysis, and applications_, Springer 2003 ([ISBN:978-1-4612-1029-0](https://www.springer.com/gp/book/9780387967905))

* {#Grabowski05} [[Janusz Grabowski]], _Isomorphisms of algebras of smooth functions revisited_, Arch. Math. 85 (2005), 190-196 ([arXiv:math/0310295](https://arxiv.org/abs/math/0310295))

The general statement and its proof is discussed in:

* {#KolarSlovakMichor93} [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], chapter 35 of: _[[Natural operations in differential geometry]]_, Springer (1993) [[book webpage](http://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3), [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

Discussion that takes the dual algebraic formulation as the very definition of [[smooth functions]] is in

* {#Nestruev03} [[Jet Nestruev]], section 6 of: _Smooth manifolds and Observables_, Graduate Texts in Mathematics 218, Springer 2003 ([doi:10.1007/b98871](https://link.springer.com/book/10.1007/b98871))

The analog of the statement for real algebras refined to [[smooth algebras]] is theorem 2.8 in

* [[Ieke Moerdijk]], [[Gonzalo Reyes|Gonzalo E. Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ Springer (1991)

[[!redirects Milnor's exercise]]

[[!redirects smooth manifolds embed into formal duals of R-algebras]]
[[!redirects Milnor duality]]


embedding of topological spaces

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

#Contents#
* table of contents
{:toc}

## Idea

An _[[embedding]] of [[topological spaces]]_ is a [[continuous function]] which is a [[homeomorphism]] onto its [[image]].

## Definition

+-- {: .num_defn #EmbeddingOfTopologicalSpaces}
###### Definition
**(embedding of topological spaces)**

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be [[topological spaces]]. A [[continuous function]] $f \;\colon\; X \longrightarrow Y$ is called an _embedding_ if
in its [[image factorization]]

$$
  f
    \;\colon\;
  X
    \overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}
  f(X)
    \overset{\phantom{AAA}}{\hookrightarrow}
  Y
$$

with the [[image]] $f(X) \hookrightarrow Y$ equipped with the [[subspace topology]], we have that $X \to f(X)$ is a [[homeomorphism]].

=--

## Properties

+-- {: .num_prop #OpenClosedContinuousInjectionsAreEmbeddings}
###### Proposition
**([[closed injections are embeddings|open/closed continuous injections are embeddings]])**

A [[continuous function]] $f \colon (X, \tau_X) \to (Y,\tau_Y)$
which is

1. an [[injective function]]

1. an [[open map]] or a [[closed map]]

is an embedding (def. \ref{EmbeddingOfTopologicalSpaces}).

This is called a _closed embedding_ if the [[image]] $f(X) \subset Y$ is a [[closed subset]].

=--

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

If $f$ is injective, then the map onto its [[image]] $X \to f(X) \subset Y$ is a [[bijection]]. Moreover, it is still [[continuous function|continuous]] with respect to the [[subspace topology]] on $f(X)$. Now a bijective continuous function is a homeomorphism precisely if it is an [[open map]] or a [[closed map]] (by [this prop.](homeomorphism#HomeoContinuousOpenBijection)). But the image projection of $f$ has this property, respectively, if $f$ does (by [this prop.](open+map#OpenClosedImageProjection)).

=--

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

Embeddings of topological spaces are precisely the [[regular monomorphisms]] in the [[category]] [[Top]] of all [[topological spaces]].

=--

For **proof** see at _[[Top]]_ [this proposition](Top#MonoEpiMorphisms).

+-- {: .num_lemma #poclosed}
###### Lemma
In $Top$, the pushout $j$ of a (closed/open) embedding $i$ along any continuous map $f$,

$$\array{
A & \stackrel{i}{\hookrightarrow} & B \\
\mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\
C & \underset{j}{\hookrightarrow} & D,
}$$

is again a (closed/open) embedding.
=--

For **proof** see at _[[subspace topology]]_ [here](subspace+topology#pushout).

+-- {: .num_prop #InjectiveProperMapsAreEquivalentlyTheClosedEmbeddings}
###### Proposition
**(injective proper maps to locally compact spaces are equivalently the closed embeddings)**

Let

1. $X$ be a [[topological space]]

1. $Y$ a [[locally compact topological space]]

1. $f \colon X \to Y$ be a [[continuous function]].

Then the following are equivalent

1. $f$ is an [[injective function|injective]] [[proper map]],

1. $f$ is a closed embedding (def. \ref{EmbeddingOfTopologicalSpaces}).

=--

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

In one direction, if $f$ is an injective proper map,
then since [[proper maps to locally compact spaces are closed]], it follows that $f$ is also  [[closed map]]. The claim then follows since [[closed injections are embeddings]], and since the image of a closed map is closed.

Conversely, if $f$ is a closed embedding, we only need to show that the embedding map is proper. So for $C \subset Y$ a [[compact subspace]], we need to show that the [[pre-image]] $f^{-1}(C) \subset X$ is also compact. But since $f$ is an injection (being an embedding), that pre-image is just the intersection $f^{-1}(C) \simeq C \cap f(X)$.

To see that this is compact, let $\{V_i \subset X\}_{i \in I}$ be an open cover of the subspace $C \cap f(X)$,
hence, by the nature of the [[subspace topology]], let $\{U_i \subset Y\}_{i \in I}$ be a set of open subsets of
$Y$, which cover $C \cap f(X) \subset Y$ and with $V_i$ the restriction of $U_i$ to $C \cap f(X)$.
Now since $f(X) \subset Y$ is closed by assumption, it follows that the complement $Y \setminus f(X)$ is open
and hence that

$$
  \{ U_i \subset Y \}_{i \in  I} \sqcup \{ Y \setminus f(X) \}
$$

is an open cover of $C \subset Y$. By compactness of $C$ this has a finite subcover. Since restricting that
finite subcover back to $C \cap f(X)$ makes the potential element $Y \setminus f(X)$ disappear, this
restriction is a finite subcover of $\{V_i \subset C \cap f(X)\}$. This shows that $C \cap f(X)$ is compact.

=--

## Related concepts

* [[embedding]]

* [[embedding of smooth manifolds]]

[[!redirects embeddings of topological spaces]]

[[!redirects topological embedding]]
[[!redirects topological embeddings]]

[[!redirects topological subspace embedding]]
[[!redirects topological subspace embeddings]]

[[!redirects closed embedding of topological spaces]]
[[!redirects closed embeddings of topological spaces]]


embedding of types

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

\tableofcontents

## Idea

The analogue in [[dependent type theory]] of the concept of [[injection]] in [[set theory]], [[embedding of topological spaces]] in [[topology]], and [[embedding]] in [[category theory]].

This is similar to [[equivalence of types]], which are the analogues in dependent type theory of the concept of [[bijection]] in [[set theory]], [[homeomorphism]] in [[topology]], and [[isomorphism]] in [[category theory]].

## Definition

Given types $A$ and $B$, a function $f:A \to B$ is an **embedding** if one of the following equivalent conditions holds of $f$:

* for all $x:A$ and $y:A$, [[function application to identifications|application]] of the function $f$ to identifications between $x$ and $y$ is an equivalence of types;

$$\mathrm{isEmbedding}(f) \coloneqq \prod_{x:A} \prod_{y:A} \mathrm{isEquiv}(\mathrm{ap}_f(x, y))$$

* for all $y:B$, the fiber of $f$ at $y$ is a [[mere proposition]]

$$\mathrm{isEmbedding}(f) \coloneqq \prod_{y:B} \mathrm{isProp}\left(\sum_{x:A}f(x) =_B y\right)$$

## Properties

### Relation to subtypes

Given a [[type of propositions]] $\mathrm{Prop}$ and a [[material subtype]] $P:A \to \mathrm{Prop}$, the corresponding [[structural subtype]] of $A$ is defined by
$$\sum_{x:A} x \in P$$
and the embedding is given by the first projection function
$$\pi_1:\left(\sum_{x:A} x \in P\right) \to A$$
with the witness that $\pi_1$ is an embedding given by the fact that the membership relation $x \in P$ is a proposition for all elements $x:A$ and material subtypes $P:A \to \mathrm{Prop}$.

## Related concepts

* [[embedding type]]

* [[injection]], [[embedding of topological spaces]], [[embedding of smooth manifolds]], [[embedding]]

* [[equivalence of types]]

* [[subtype]], [[predicate]]

## References

For embeddings in [[dependent type theory]], see section 11.4 of:

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

[[!redirects embedding of types]]
[[!redirects embeddings of types]]

[[!redirects embedding in type theory]]
[[!redirects embeddings in type theory]]


embedding tensor

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[gauge theory|gauging]] of [[U-duality]]-[[symmetry groups]] of  [[supergravity]]-theories to _[[gauged supergravity]]_, the _embedding tensor_ ([Nicolai-Samtleben 00](#NicolaiSamtleben00)) is the datum that specifies which [[subgroup]] of the [[global symmetry|global]] [[U-duality]] is promoted to a [[gauge group]]. The requirement of [[supersymmetry]] and of consistency then implies conditions on this choice, called the "linear constraint" and the "quadratic constraint".

Formalized in terms of [[Lie theory]] ([Lavau 17](#Lavau17)) these conditions say that an embedding tensor is a [[homomorphism]] of [[Leibniz algebras]] from a [[Lie module]] to the underlying [[Lie algebra]] (the "quadratic constaint") where the [[Leibniz algebra|Leibniz]]-[[magma|product]] on the module is given by the Lie action induced by that homomorphism itself (the "linear constraint").

Any choice of embedding tensor for a [[gauged supergravity]] is supposed to induce a _[[tensor hierarchy]]_ ([de Wit-Samtleben 05](tensor+hierarchy#deWitSamtleben05), [08](tensor+hierarchy#deWitSamtleben08)) of higher degree [[differential form]]-[[field (physics)|fields]] which jointly serve as ever higher order corrections to the resulting gauge-covariance of the [[field strengths]]. This tensor hierarchy may be understood as a [[dg-Lie algebra]]/[[L-∞ algebra]]-[[mathematical structure|structure]] which lifts the [[Leibniz algebra]]-structure implied/induced by the embedding tensor ([Lavau 17](#Lavau17), [Lavau-Palmkvist 19](#LavauPalmkvist19), [Lavau-Stasheff 19](#LavauStasheff19)).

## Definition
 {#Definition}

[[!include embedding tensors and tensor hierarchy in super Lie theory -- section]]

## Relation with double and exceptional field theory

See [Hohm-Samtleben 19](#HohmSamtleben19) for a review.

## Related concepts

* [[gauged supergravity]]

* [[tensor hierarchy]]

* [[double field theory]], [[exceptional field theory]]

## References

The concept in [[gauged supergravity]] originates with

* {#NicolaiSamtleben00} [[Hermann Nicolai]], [[Henning Samtleben]], _Maximal gauged supergravity in three dimensions_, Phys. Rev. Lett. 86 (2001) 1686-1689 ([arXiv:hep-th/0010076](https://arxiv.org/abs/hep-th/0010076))

For further references see at _[[tensor hierarchy]]_.

The mathematical formulation above is due to

* {#Lavau17} [[Sylvain Lavau]], _Tensor hierarchies and Leibniz algebras_, J. Geom. Phys. 144:147-189 (2019) ([arXiv:1708.07068](https://arxiv.org/abs/1708.07068))

with further discussion in:

* {#LavauPalmkvist19} [[Sylvain Lavau]], [[Jakob Palmkvist]], _Infinity-enhancing of Leibniz algebras_ ([arXiv:1907.05752](https://arxiv.org/abs/1907.05752))

* {#LavauStasheff19} [[Sylvain Lavau]], [[Jim Stasheff]], _$L_\infty$-algebra extensions of Leibniz algebras_ ([arXiv:2003.07838](https://arxiv.org/abs/2003.07838))

Review of the relation with [[double field theory]] and [[exceptional field theory]]:

* {#Musaev13} [[Edvard Musaev]], _U-dualities in Type II string theories and M-theory_, _Symmetry_ 2019, _11_(8), 993 ([arXiv:1311.3331](https://arxiv.org/abs/arXiv:1311.3331), [doi:10.3390/sym11080993](https://doi.org/10.3390/sym11080993))

* {#HohmSamtleben19} [[Henning Samtleben]], [[Olaf Hohm]], _Higher Gauge Structures in Double and Exceptional Field Theory_, Contribution to Proceedings of _[[Higher Structures in M-Theory 2018]]_ ([arXiv:1903.02821](https://arxiv.org/abs/1903.02821), [doi:10.1002/prop.201910008](https://doi.org/10.1002/prop.201910008))

[[!redirects embedding tensors]]


embedding tensors and tensor hierarchy in super Lie theory -- section

### Embedding tensors and tensor hierarchy in super/dg-Lie theory
  {#EmbeddingTensorsAndTensorHierarchyInSuperdgLieTheory}

We spell out aspects of the formalization of the concept of [[embedding tensors]] and their [[tensor hierarchies]] in terms of [[super Lie algebras]]/[[dg-Lie algebras]].

#### The super Lie algebra of multi-endomorphisms
  {#TheSuperLieAlgebraOfEndomorphisms}

The algebra of [[embedding tensors]] and their [[tensor hierarchies]] turns out to be neatly captured by structure found in or induced from the following [[super Lie algebra]].

The following construction is briefly highlighted in [Palmkvist 09, 2.3](M-brane+3-algebra#Palmkvist09) [Palmkvist 13, 3.1](tensor+hierarchy#Palmkvist13) (reviewed more clearly in [Lavau-Palmkvist 19, 2.4](tensor+hierarchy#LavauPalmkvist19)) where it is attributed to [Kantor 70](super+Lie+algebra#Kantor70):

+-- {: .num_defn #TheSuperLieAlgebraInducedFromVectorSpace}
###### Definition
**([[super Lie algebra]] of [[multimorphism|multi]]-[[endomorphisms]])**

Let $V$ be a [[finite-dimensional vector space]] over some [[ground field]] $k$.

Define a $\mathbb{Z}$-[[graded vector space]]

$$
  MultEnd(V) \;\in \; Vect_k^{\mathbb{Z}}
  \,,
$$

concentrated in degrees $\leq 1$, [[recursion|recursively]] as follows:

For $n =1$ we set

$$
  MultEnd(V)_{1}
  \;\coloneqq\;
  V
  \,.
$$

For $n \leq 0 \in \mathbb{Z}$, the component space in degree $n-1$ is taken to be the vector space of [[linear maps]] from $V$ to the component space in degree $n$:

$$
  MultEnd(V)_{n-1}
  \;\coloneqq\;
  Hom_k( V, MultEnd(V)_n )
  \,.
$$

Hence:

\[
  \label{ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace}
  \begin{aligned}
    MultEnd(V)_1 & = V
    \\
    MultEnd(V)_0 & = Hom_k(V,V) = \mathfrak{gl}(V)
    \\
    MultEnd(V)_{-1} & =  Hom_k(V, Hom_k(V,V)) \simeq  Hom_k(V \otimes V, V)
    \\
    MultEnd(V)_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq  Hom_k(V \otimes V \otimes V, V)
    \\
    \vdots
  \end{aligned}
\]

Consider then the [[direct sum]] of these component spaces as a [[super vector space]] with the [[even number]]/[[odd number]]-degrees being in super-even/super-odd degree, respectively.

On this [[super vector  space]] consider a [[super Lie bracket]] defined [[recursion|recusively]] as follows:

For all $v_1, v_2 \in MultEnd(V)_1 = V$ we set

$$
  [v_1, v_2] \;=\; 0
  \,.
$$

For $f \in MultEnd(V)_{n \leq 0}$ and $v \in MultEnd(V)_1 = V$ we set

\[
  \label{SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace}
  [f, v]
  \;\coloneqq\;
  f(v)
\]

Finally, for $f\in MultEnd(V)_{ deg(f) \leq 0 }$ and $g\in MultEnd(V)_{deg(g) \leq 0}$ we set

\[
  \label{SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace}
  \begin{aligned}
  [f, g]
  &
  \colon\;
  v
  \;\mapsto\;
  [f, g(v)]
  -
  (-1)^{
    deg(f) deg(g)
  }
  [ g, f(v) ]
  \\
  \end{aligned}
\]

=--

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

By (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) the definition (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) is equivalent to

$$
  [ [f,g],v ]
  \;=\;
  [f, [g,v] ]
  -
  (-1)^{
    deg(f) deg(g)
  }
  [ g, [f,v] ]
$$

Hence (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) is already implied by (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) if the bracket is to satisfy the [super Jacobi identity](super+Lie+algebra#eq:GradedJacobiIdentity)

=--

It remains to show that:

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

Def. \ref{TheSuperLieAlgebraInducedFromVectorSpace} indeed gives a [[super Lie algebra]] in that the bracket (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) satisfies the [super Jacobi identity](super+Lie+algebra#eq:GradedJacobiIdentity).

=--

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

We proceed by [[induction]]:

By Remark \ref{ConstraintsOnBracketInSuperLieAlgebraInducedByVectorSpace} we have that the super Jacobi identity holds for all [[triples]] $f_1, f_2, f_3 \in MultEnd(V)$ with $deg(f_3) \geq 0$.

Now assume that the super Jacobi identity has been shown for triples $(f_1, f_2, f_3(v))$ and $(f_1, f_3, f_2(v))$, for any $v \in V$. The following computation shows that then it holds for $(f_1, f_2, f_3)$:

$$
  \begin{aligned}
    [f_1, [f_2, f_3] ] (v)
    &
    =
    [ f_1,  [f_2, f_3](v) ]
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    [  [ f_2, f_3 ], f_1(v)  ]
    \\
    & =
    [ f_1,  [  f_2, f_3(v) ] ]
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_2)deg(f_3)}
    [ f_1,  [  f_3, f_2(v) ] ]
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    [  f_2,  [ f_3,  f_1(v) ] ]
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)}
    [  f_3,  [ f_2,  f_1(v) ] ]
    \\
    & =
    [ f_1,  [  f_2, f_3(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_2)}
    {
    \color{green}
    [ f_2,  [  f_1, f_3(v) ] ]
    }
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_2) deg(f_3)}
    \big(
    [ f_1,  [  f_3, f_2(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_3)}
    {
    \color{orange}
    [ f_3,  [  f_1, f_2(v) ] ]
    }
    \big)
    \\
    & \phantom{=}
    -
    (-1)^{deg(f_1)(deg(f_2) + deg(f_3))}
    \big(
    [  f_2,  [ f_3,  f_1(v) ] ]
    -
    (-1)^{deg(f_1)deg(f_3)}
    {
    \color{blue}
    [  f_2,  [ f_1,  f_3(v) ] ]
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)}
    \big(
    [  f_3,  [ f_2,  f_1(v) ] ]
    -
    (-1)^{deg(f_1) deg(f_2)}
    {
    \color{cyan}
    [  f_3,  [ f_1,  f_2(c) ] ]
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_2)}
    \big(
    \underset{
      = 0
    }{
    \underbrace{
    +
    {
    \color{green}
    [ f_2,  [  f_1, f_3(v) ] ]
    }
    -
    {
    \color{blue}
    [  f_2,  [ f_1,  f_3(v) ] ]
    }
    }
    }
    \big)
    \\
    & \phantom{=}
    +
    (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)}
    \big(
    \underset{
      = 0
    }{
    \underbrace{
    {
    \color{orange}
    [  f_3,  [ f_1,  f_2(v) ] ]
    }
    -
    {
    \color{cyan}
    [ f_3,  [  f_1, f_2(v) ] ]
    }
    }
    }
    \big)
    \\
    & =
    \big[
      [f_1, f_2], f_3(v)
    \big]
    \\
    & \phantom{=}
    -
    (-1)^{ deg(f_2) deg(f_3) }
    \big[
      [f_1, f_3], f_2(c)
    \big]
    \\
    & \phantom{=}
    +
    (-1)^{ deg(f_1) deg(f_2)  }
    \big[
      f_2, [f_1, f_3](v)
    \big]
    \\
    & \phantom{=}
    -
    (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) }
    \big[
      f_3, [f_1, f_2](v)
    \big]
    \\
    & =
    \big[
     [f_1, f_2], f_3
    \big](v)
    +
    (-1)^{deg(f_1)deg(f_2))}
    \big[
      f_2, [f_1, f_3]
    \big](v)
  \end{aligned}
$$

> (Fine, but is this sufficient to induct over the full range of all three degrees?)

=--

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

For $f,g \in MultEnd(V)_0 = Hom_k(V,V)$ (eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace) we have that the bracket on $MultEnd(V)$ in Def. \ref{SuperLieAlgebraInducedFromVectorSpace} restricts to

$$
  [f,g](v)
  \;=\;
  [f,g(v)] - [g,f(v)]
  \;=\;
  f(g(v)) - g(f(v))
$$

(by combining (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) with (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace)).

This is the [[Lie bracket]] of the [[general linear Lie algebra]] $\mathfrak{gl}(V)$, as indicated on the right in (eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace).

=--

#### Embedding tensors

+-- {: .num_defn #EmbeddingTensor}
###### Definition
**([[embedding tensor]])**

Given

* $\mathfrak{g}$ a [[Lie algebra]],

* $\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V $ a [[Lie algebra representation]] of $\mathfrak{g}$

  (typically required to be a [[faithful representation]], see Remark \ref{EmbeddingTensorsInduceTensorHierarchies} below),

then an *embedding tensor* is a [[linear map]]

$$
  \Theta
  \;\colon\;
  V \longrightarrow \mathfrak{g}
$$

such that for all $v_i \in V$ the following condition ("quadratic constraint") is satisfied:

\[
  \label{QuadraticConstraint}
  [\Theta(v_1), \Theta(v_2)]
  \;=\;
  \Theta
  \big(
    \rho_{\Theta(v_1)}(v_2)
  \big)
  \,,
\]

where on the left we have the [[Lie bracket]] of $\mathfrak{g}$.

=--

The idea of this definition goes back to [Nicolai-Samtleben 00](embedding+tensor#NicolaiSamtleben00), with many followups  in the literature on [[tensor hierarchies]] in [[gauged supergravity]]. The above mathematical formulation is due to [Lavau 17](embedding+tensor#Lavau17).

+-- {: .num_remark #RelationToLeibnizAlgebraStructure}
###### Remark
**([[Leibniz algebra]]-[[mathematical structure|structure]])**

The "quadratic constraint" (eq:QuadraticConstraint) implies (see [this Prop.](Leibniz+algebra#LeibnizAlgebraFromLieModuleAndEmbeddingTensor)) that the [[magma|product]]

\[
  \label{InducedLeibnizAlgebraStructure}
  \array{
     V \otimes V
     &\overset{ }{\longrightarrow}&
     V
     \\
     (v_1, v_2)
       &\mapsto&
     v_1 \cdot v_2
     \mathrlap{
       \;\coloneqq\;
       \rho_{\Theta(v_1)}(v_2)
     }
  }
\]

makes (the underlying [[vector space]] of) $V$ a [[Leibniz algebra]].
Conversely, if a [[Leibniz algebra]] structure "$\cdot$" on $V$ is already given, we may ask that it coincides with this one induced from the embedding tensor, a condition then called the _linear constraint_:

\[
  \label{LinearConstraint}
  v_1 \cdot v_2 \;=\; \rho_{\Theta(v_1)}(v_2)
  \,.
\]

With respect to this induced Leibniz algebra structure, hence equivalently with the "linear constraint" (eq:LinearConstraint) understood, the "quadratic constraint" (eq:QuadraticConstraint) equivalently says that the embedding tensor is a [[homomorphism]] of [[Leibniz algebras]] (using that [[Lie algebras]] are special cases of a Leibniz algebras):

$$
  [\Theta(v_1), \Theta(v_2)]
  \;=\;
  \Theta(v_1 \cdot v_2)
  \,.
$$

=--

+-- {: .num_prop #EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace}
###### Proposition
**([[embedding tensors]] are square-0 elements in $MultEnd(V)$)**

Let $k$ be a [[ground field]] of [[characteristic zero]].

An element in degree -1 of the [[super Lie algebra]] $MultEnd(V)$ from Def. \ref{TheSuperLieAlgebraInducedFromVectorSpace},

$$
  \Theta \in MultEnd(V)_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V))
  \,,
$$

which by Example \ref{LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace} is identified with a [[linear map]]

$$
  \Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V)
$$

from $V$ to the [[general linear Lie algebra]] on $V$, is square-0 precisely if it is an _[[embedding tensor]]_ (Def. \ref{EmbeddingTensor}), in that:

$$
  [\Theta, \Theta]
  \;=\;
  0
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  [\Theta(v_1), \Theta(v_2) ]
  \;=\;
  \Theta( \rho_{\Theta(v_1)}(v_2) )
  \,.
$$

Here on the right, $[-,-]$ denotes the [[Lie bracket]] in $\mathfrak{gl}(V)$, while $\rho$ denotes the canonical [[Lie algebra action]] of $\mathfrak{gl}(V)$ on $V$.

=--

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

By unwinding of the definition (eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace) and (eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace) and using again Example \ref{LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace} we compute as follows:

$$
  \begin{aligned}
    \big( \tfrac{1}{2} [\Theta,\Theta](v_1) \big)(v_2)
    & =
    [\Theta, \Theta(v_1)](v_2)
    \\
    & =
    [\Theta,
      \underset{
        \mathclap{
          = \rho_{\Theta(v_1)}(v_2)
        }
      }
      {
        \underbrace{
          (\Theta(v_1))(v_2)
        }
      }
    ]
    -
    [\Theta(v_1), \Theta(v_2)]
    \\
    & =
    \Theta( \rho_{\Theta(v_1)}(v_2) )
    -
    [ \Theta(v_1), \Theta(v_2) ]
  \end{aligned}
$$

=--

#### Tensor hierarchies

+-- {: .num_prop #EmbeddingTensorsInduceTensorHierarchies}
###### Remark
**([[embedding tensors]] induce [[tensor hierarchies]])**

In view of the relation between [[super Lie algebras]] and [[dg-Lie algebras]] ([above](#RelationToDGLieAlgebras)), Prop. \ref{EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace} says that every choice of an [[embedding tensor]] for a [[faithful representation]] on a [[vector space]] $V$ induces a [[dg-Lie algebra]]
$(MultEnd(V), [-,-], \partial \coloneqq [\Theta, -])$.

According to [Palmkvist 13, 3.1](tensor+hierarchy#Palmkvist13), [Lavau-Palmkvist 19, 2.4](tensor+hierarchy#LavauPalmkvist19) this [[dg-Lie algebra]] (or some extension of some sub-algebra of it) is the _[[tensor hierarchy]]_ associated with the embedding tensor.

=--

[[!redirects embedding tensors and tensor hierarchy in super Lie theory -- references]]


embedding type

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

\tableofcontents

## Definition

In dependent type theory, the **embedding type** between two types $A$ and $B$ is the type of [[embeddings]], the type of functions $f:A \to B$ such that

$$A \hookrightarrow B \coloneqq \sum_{f:A \to B} \mathrm{isEmbedding}(f)$$

where the type $\mathrm{isEmbedding}(f)$ could be defined as

$$\mathrm{isEmbedding}(f) \coloneqq \prod_{x:A} \prod_{y:A} \prod_{q:f(x) =_B f(y)} \exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q$$

$$\mathrm{isEmbedding}(f) \coloneqq \prod_{y:B} \prod_{x:A} \prod_{z:f(x) =_B y} \exists! x:A.f(x) =_B y$$

$$\mathrm{isEmbedding}(f) \coloneqq \prod_{y:B} \mathrm{isProp}\left(\sum_{x:A}f(x) =_B y\right)$$

### Rules for embedding types

#### As functions whose applications to identities have contractible fibers

Let us assume the [[dependent type theory]] has [[identity types]], [[dependent identity types]] and [[uniqueness quantifiers]], but does not have [[function types]] or [[dependent function types]] (as in some versions of [[predicative mathematics]]). Then the [[natural deduction]] rules for embedding types is given by:

Formation rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \simeq B \; \mathrm{type}}$$

Introduction rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, x:A, y:A, q:f(x) =_B f(y) \vdash u(x, y, q):\exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q}{\Gamma \vdash \mathrm{embed}(f, u):A \simeq B}$$

Elimination rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B, x:A \vdash \mathrm{ev}(f, x):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B, x:A, y:A, q:f(x) =_B f(y) \vdash \mathrm{uniqact}(f, x, y, q):\exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q}$$

Computation rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, x:A, y:A, q:f(x) =_B f(y) \vdash u(x, y, q):\exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q}{\Gamma, x:A \vdash \beta_{A \simeq B}^{\mathrm{ev}}(x):\mathrm{ev}(\mathrm{embed}(f, u), x) =_B f(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, x:A, y:A, q:f(x) =_B f(y) \vdash u(x, y, q):\exists! p:x =_A y.ap_f(x, y, p) =_{f(x) =_B f(y)} q}{\Gamma, x:A, y:A, q:f(x) =_B f(y) \vdash \beta_{A \simeq B}^{\mathrm{uniq}}(x, y, q):\mathrm{uniqact}(\mathrm{embed}(f, u), x, y, q) =_{\exists! p:x =_A y.(-) =_{f(x) =_B f(y)} q}^{\beta_{A \simeq B}^{\mathrm{ev}}(x)} u(x, y, q)}$$

Uniqueness rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B \vdash \eta_{A \simeq B}(f):\mathrm{embed}(\mathrm{ev}(f), \mathrm{uniqact}(f)) =_{A \simeq B} f}$$

#### As functions with propositional fibers

Let us assume the [[dependent type theory]] has [[identity types]], [[dependent identity types]], and [[uniqueness quantifiers]]. Then the [[natural deduction]] rules for embedding types is given by:

Formation rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \simeq B \; \mathrm{type}}$$

Introduction rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B, x:A, z:f(x) =_B y \vdash p(y, x, z):\exists!x:A.f(x) =_B y}{\Gamma \vdash \mathrm{embed}(f, p):A \simeq B}$$

Elimination rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B, x:A \vdash \mathrm{ev}(f, x):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B, y:B, x:A, z:f(x) =_B y \vdash \mathrm{atmostone}(f, y, x, z):\exists!x:A.f(x) =_B y}$$

Computation rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B, x:A, z:f(x) =_B y \vdash p(y, x, z):\exists!x:A.f(x) =_B y}{\Gamma, x:A \vdash \beta_{A \simeq B}^{\mathrm{ev}}(x):\mathrm{ev}(\mathrm{embed}(f, p), x) =_B f(x)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B, x:A, z:f(x) =_B y \vdash p(y, z):\exists!x:A.f(x) =_B y}{\Gamma, y:B, x:A, z:f(x) =_B y \vdash \beta_{A \simeq B}^{\mathrm{atmostone}}(y, x, z):\mathrm{atmostone}(\mathrm{embed}(f, p), y, x, z) =_{\exists!x:A.(-) =_B y}^{\beta_{A \simeq B}^{\mathrm{ev}}(x)} p(y, x, z)}$$

Uniqueness rules for embedding types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \simeq B \vdash \eta_{A \simeq B}(f):\mathrm{embed}(\mathrm{ev}(f), \mathrm{atmostone}(f)) =_{A \simeq B} f}$$

## See also

* [[embedding of types]]

* [[injection set]]

* [[object of monomorphisms]]

* [[function type]], [[equivalence type]]

[[!redirects embedding type]]
[[!redirects embedding types]]

[[!redirects type of embeddings]]
[[!redirects types of embeddings]]