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#### Manifolds and cobordisms
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#### Cohomology
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This page collects material related to the book

* [[Stanley Kochman]]:

  \linebreak

  **Bordism, Stable Homotopy and Adams Spectral Sequences**

  \linebreak

  Fields Institute Monographs

  American Mathematical Society, 1996

  [cds:2264210](https://cdsweb.cern.ch/record/2264210)

on [[cobordism theory]], [[stable homotopy theory]], [[complex oriented cohomology]], and the [[Adams spectral sequence]].

From p. 13:

> The approach to [[stable homotopy theory|stable homotopy]] presented in this book originated with graduate courses taken by the author at the University of Chicago from 1966 to 1970 given by [[Frank Adams]], [[Arunas Liulevicius]] and [[Peter May]]. The content of the  lectures by Adams have been published in $[$Adams: _[[Stable homotopy and generalised homology]]_ (1974)$]$. However, the content of the courses given by Liulevicius on [[bordism theory|bordism]] and by May on [[characteristic classes]] and on the [[Adams spectral sequence]] have not been published. Most of the material in the first four chapters has been given by the author as graduate courses at Yale University, Purdue University and the University of Western Ontario.

#Contents#
* table of contents
{:toc}

## 1 Bordism

* [[cobordism theory]]

### 1.2 Manifolds

* [[smooth manifold]]

* [[Whitney embedding theorem]]

* [[tubular neighbourhood]]

### 1.3 Classifying spaces

* [[Stiefel manifold]]

* [[Grassmannian]]

* [[vector bundle]]

* [[classifying space]]

* [[universal vector bundle]]

### 1.4 Manifolds with $(B,f)$-structure

* [[G-structure]] on [[normal bundle]]

* [[framed manifold]]

* [[orientation]]

* [[almost complex structure]]

### 1.5 Pontrjagin-Thom Theorem

* [[cobordism]]

* [[cobordism ring]]

* [[Pontryagin-Thom collapse map]]

* [[Thom spectrum]]

* [[Thom's theorem]]

* [[stable homotopy groups of spheres]]

## 2 Characteristic Classes

### 2.1 Introduction

### 2.2 Serre spectral sequence

* [[exact couple]]

* [[Serre spectral sequence]]

* [[Gysin sequence]]

### 2.3 Cohomology of classifying spaces

* [[characteristic classes]], [[universal characteristic class]]

* [[Chern classes]]

* [[splitting principle]]

* [[Stiefel-Whitney classes]]

### 2.4 Homology of Classifying spaces

### 2.5 Steenrod algebra

* [[Steenrod algebra]]

### 2.6 Homology of Thom spectra

* [[zero-section into Thom space of universal line bundle is weak equivalence]]

* [[Thom isomorphism]]

* [[homology of MU]]

## 3 Stable Category

### 3.2 Underlying theorems

* [[Serre long exact sequence]]

* [[Freudenthal suspension theorem]]

* [[Blakers-Massey theorem]]

* [[fiber sequence]]

* [[long exact sequence of homotopy groups]]

### 3.3 Spectra

* [[spectrum]], [[Omega-spectrum]]

* [[coordinate-free spectrum]]

* [[ring spectrum]] as [[functor with smash products]]

* [[Adams category]]

* [[Whitehead theorem]]

* [[stable homotopy category]]

### 3.4 Generalized homology

* [[generalized homology]]

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[Brown representability theorem]]

### 3.5 Eilenberg-MacLane spectra

* [[Eilenberg-MacLane spectrum]]

### 3.6 Adams spectral sequence

* [[Adams resolution]]

* [[Adams spectral sequence]]

### 3.7 Homotopy of Thom spectra

* [[change of rings theorem]]

* [Thom's theorem on MO](MO#ThomTheoremOnMO)

* [[Milnor-Quillen theorem on MU]]

## 4 Complex Bordism

### 4.2 Atiyah-Hirzebruch spectral sequences

* [[lim^1 and Milnor sequences]]

* [[Atiyah-Hirzebruch spectral sequence]], [[multiplicative spectral sequence|multiplicative]]

* [[Kronecker pairing]]

### 4.3 MU characteristic classes

* [[complex oriented cohomology]]

* [[MU]]

* multiplicative cohomology of $B U(1)$ (prop. 4.3.2, this is lemma 2.5 in part II of [[John Adams]], _[[Stable homotopy and generalised homology]]_)

* [[Conner-Floyd Chern classes]]

* [[cap product]]

* [[orientation in generalized cohomology]]

* [[fiber integration in generalized cohomology]]

* [[Boardman homomorphism]]

* [[homology of MU]]

### 4.4 Formal products

* [[formal group law]]

* [[universal complex orientation on MU]]

* [[Lazard ring]]

* [[Lazard's theorem]]

* [[Quillen's theorem on MU]]

### 4.5 MU operation

* [[cohomology operations]]

### 4.6 Brown-Peterson spectra

* [[Brown-Peterson spectrum]]

### 4.7 Adams-Novikov spectral sequence

* [[Adams-Novikov spectral sequence]]

* the computation sketched after "we show that $\pi_1S = Z/2$" is the one that is spelled out for instance in ([Bruner 09, pages 2-4](classical+Adams+spectral+sequence#Bruner09))

## 5 Computing Stable Stems

* [[stable homotopy groups of spheres]]

* [[classical Adams spectral sequence]]

* [[Adams-Novikov spectral sequence]]

### 5.2 Lambda algebras

* [[Lambda algebra]]

* [[EHP spectral sequence]]

* [[Curtis algorithm]]

### 5.3 May spectral sequences

* [[spectral sequence of a filtered complex]]

* [[May spectral sequence]]

### 5.4 Massey products

* [[Massey product]]

### 5.5 The Lambda Complex $\Gamma^\ast$

### 5.6 The Lambda Complex $\Gamma^\ast_{BP}$

* [[Adams-Novikov spectral sequence]]

### 5.7 Toda brackets and stable stems

* [[Toda bracket]]

* [[stable stem]]

## Errata
 {#Errata}

* p. 36, second line: on the right replace $p$ by $p-1$, i.e. replace $\pi_p \colon G^n \to D^{p,n-p}$ by $\pi_{p-1} \colon G^n \to D^{p-1,n-p+1}$ (see end of the proof on the same page)

* p. 41, second diagram: the top horizontal moprhism is missing a superscript star.

* p. 192, second displayed formula from below: The expression for $Cotor$ on the left has a superfluous argument $A$.

* p. 199, the spectral sequence in prop. 5.3.1 converges not to the cobar complex, but to the cohomology of that complex, namely to $Cotor_{\mathcal{A}^\ast}(\mathbb{Z}/2, \mathbb{Z}/2)$ (as shown correctly in the paragraph just before);

* p. 199, very last line: $E_0$ must be $E^0$ (not the 0-page of the spectral sequence, but the associated graded module, as defined on p. 198)

category: reference


Borel construction

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#### Representation theory
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[[!include representation theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
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# Contents
* table of contents
{: toc}

## Idea

For $G$ a [[topological group]] [[action|acting]] on a [[topological space]] $X$, its _Borel construction_ or _Borel space_ is another topological space $X \times_G E G$, also known as the [[homotopy quotient]]. In many cases, its ordinary [[cohomology]] is the $G$-[[equivariant cohomology]] of $X$.

## Definition

For $X$ a [[topological space]], $G$ a [[topological group]] and $\rho\colon G \times X \to X$ a [[continuous map|continuous]] $G$-[[action]] (a [[topological G-space]]), the **Borel construction** of $\rho$ is [[generalized the|the]] [[topological space]] $X \times_G E G$, hence [[quotient]] of the [[product]] of $X$ with the total space of [[generalized the|the]] $G$-[[universal principal bundle]] $E G$ by the [[diagonal]] [[action]] of $G$ on both.

Analogously, for $\mathcal{G}$ a [[simplicial group]] and $\mathcal{G} \times X \xrightarrow{\;\;} X$ a [[simplicial group action]], its Borel construction is the [[quotient]]

\[
  \label{BorelConstructionInSimplicialSets}
  \frac
  { W \mathcal{G}  \times X}
  {\mathcal{G}}
  \;\;\;
  \in
  \;\;
  SimplicialSets
\]

of the [[Cartesian product]] of $X$ with the [[universal principal simplicial complex]] $W \mathcal{G}$ by the [[diagonal action]] of $\mathcal{G}$ on these.

## Properties

### Homotopy-theoretic properties
 {#ModelCategoryTheoreticProperties}

#### In simplicial sets

\begin{prop}\label{BorelConstructionIsRightQuillenFunctor}
  The simplicial Borel construction (eq:BorelConstructionInSimplicialSets) is a [[right Quillen functor]] in a [[Quillen equivalence]]

$$
  \big(
    W \mathcal{G} \times (-)
  \big)
  \;\;
  \colon
  \;\;
  \mathcal{G}Actions(sSet)_{proj}
  \xrightarrow{\;\;\;}
  \big(sSet_{Qu}\big)_{\overline W \mathcal{G}}
$$
from the projective [[model structure on simplicial group actions]] to the [[slice model structure]] of the [[classical model structure on simplicial sets]] over the [[simplicial classifying space]] of $\mathcal{G}$.
\end{prop}

([DDK 80, Prop. 2.3, Prop. 2.4](#DDK80), see at *[[Borel model structure]]* for more)

As an example:

\begin{example}\label{ProjectionToSimplicialClassifyingSpaceIsKanFibration}
  If the underlying simplicial set $X \in sSet$ is a [[Kan complex]],
  then the quotient projection from the Borel construction (eq:BorelConstructionInSimplicialSets) to the [[simplicial classifying space]] is a [[Kan fibration]]
  $$
    \big(
      W\mathcal{G} \times X
    \big)
    \big/
    \mathcal{G}
    \xrightarrow{\in Fib}
    \overline{W}\mathcal{G}
    \,,
  $$

This follows because the morphism is the image of the terminal morphism $X \xrightarrow{\;} \ast$ under the [[right Quillen functor]] from Prop. \ref{BorelConstructionIsRightQuillenFunctor}, hence is a fibration if $X \to \ast$ is.
\end{example}
For generalization of this statement to [[simplicial presheaves]] see [NSS 12b, Thm. 3.91](#NSS12b).

#### In topological spaces
 {#HomotopyPropertiesInTopologicalSpaces}

In the following, we write $TopSp$ for the [[convenient category of topological spaces|convenient category]] of [[compactly generated weak Hausdorff spaces]].

Let

* $G \,\in\, Grp(TopSp)$ be a [[topological group]] which is [well-pointed](simplicial+topological+group#WellPointedSimplicialTopologicalGroup) in that the inclusion of the [[neutral element]] is a [[Hurewicz cofibration]] [hence](Hurewicz+cofibration#HurewiczCofibrationsInCGWHSpacesAreClosed) a [[closed cofibration]].

* $X \,\in\, G Act(TopSp)$ be a ([[compactly generated weak Hausdorff space|cgwh]])-[[topological G-space]].

\begin{prop}\label{HomotopyFiberSequenceofTopologicalBorelConstruction}
The topological [[Borel construction]] $(X \times E G)/G$ fits into a [[homotopy fiber sequence]] of the form

\[
  \label{HomotopyFiberSequenceOfTopologicalBorelConstruction}
  X
    \xrightarrow{\;\;}
  \frac{
    X \times EG
  }{
    G
  }
    \xrightarrow{\;\;}
  B G
  \,.
\]
\end{prop}
\begin{proof}
  The sequence (eq:HomotopyFiberSequenceOfTopologicalBorelConstruction) is the image under [[geometric realization of simplicial topological spaces]] of the sequence of [[simplicial topological spaces]]

\[
  \label{TopologiCALBorelConstructionFiberSequenceLevelOfSimplicialSpaces}
  const(X)_\bullet
    \xrightarrow{\;\;\;}
  X \times G^{\times^\bullet}
    \xrightarrow{\;\;\;}
  G^{\times^\bullet}
\]

which, in turn, is the image under passage to [[nerves]] of the sequence of [[topological groupoids]]

$$
  (X \rightrightarrows X)
    \xrightarrow{\;\;}
  (X \times G \rightrightarrows X)
    \xrightarrow{\;\;}
  (G \rightrightarrows \ast)
  \,,
$$

with the [[action groupoid]] in the middle and the [[delooping groupoid]] on the right.

Therefore, by the general respect of geometric realization for homotopy fiber products ([this Prop.](geometric+realization+of+simplicial+topological+spaces#SufficientConditionsForRealizationToPreserveHomotopyPullback)) it is now sufficient to observe that

1. the simplicial topological spaces appearing in (eq:TopologiCALBorelConstructionFiberSequenceLevelOfSimplicialSpaces) are all [[good simplicial topological space|good]] (so that their geometric realization models their [[homotopy colimit]]);

1. the sequence (eq:TopologiCALBorelConstructionFiberSequenceLevelOfSimplicialSpaces) of simplicial topological spaces is degreewise a homotopy fiber sequence of topological spaces;

1. the morphism $X \times G^{\times^\bullet} \to G^{\times^\bullet}$ is a [[homotopy Kan fibration]] (see [here](geometric+realization+of+simplicial+topological+spaces#eq:HomotopyTheoreticKanFibration)).

For the first condition observe that the degeneracy maps of the [[nerve]] of an  [[action groupoid]] are all of the form $X \times G^{n} \times ( \{e\} \xhookrightarrow{\;} G)$, hence are closed cofibrations (by [this Example](Hurewicz+cofibration#KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces)).

For the second condition, observe  that [[projections]] $X \times G^n \to G^n$ are evidently [[Serre fibrations]] whose ordinary [[fiber]] is $X$, so that $X \to X \times G^n \to G^n$ is a [[homotopy fiber sequence]] (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)).

For the third point, notice that in the present case the horn-filling map [here](geometric+realization+of+simplicial+topological+spaces#eq:HomotopyTheoreticKanFibration) are all [[isomorphisms]], hence certainly surjective on connected components.
\end{proof}

\begin{prop}\label{TopologicalBorelConstructionOnPrincipalBundles}
**(Borel construction on principal bundle is equivalent to plain quotient)**
\linebreak
Assume that:

* $G$ and $X$ are both [[connected topological space|connected]], $\pi_0 = \ast$;

* the [[coprojection]] $X \xrightarrow{\;q\;} X/G$ is a [[locally trivial bundle|locally trivial]] $G$-[[fiber bundle]] (hence a $G$-[[principal bundle]]),

then the topological Borel construction is [[weak homotopy equivalence|weakly homotopy equivalent]] to the plain [[quotient space]]:
$$
  (X \times E G)/G
  \xrightarrow{ \in \,\mathrm{W}_{whe} }
  X/G
  \,.
$$
\end{prop}
\begin{proof}
We need to show that the comparison morphism induces an [[isomorphisms]] on all [[homotopy groups]] $\pi_n$.

For $n = 0$ this is immediate, as passage to [[connected components]] is a [[coequalizer]] ([here](connected+space#eq:ConnectedComponentsAsCoequalizer)) and using that [[colimits commute with colimits]]. This implies that both $\pi_0(X/G)$ as well as $\pi_0\big( (X \times E G) / G\big)$ are the [[singleton set]] (by the assumption that $X$ is connected and since $E G$ is connected) and hence are isomorphic.

In order to compare the remaining higher [[homotopy groups]],
observe that, under the given assumption, with $X \to X/G$ also the coprojection $X \times E G \xrightarrow{ \;\; } (X \times E G)/G$ to the Borel construction itself is a locally trivial fiber bundle:
Namely from this [[commuting diagram]]

\begin{tikzcd}
    &[-28pt]
    G
    \ar[
      rr,-,
      shift left=1pt
    ]
    \ar[
      rr,-,
      shift right=1pt
    ]
    \ar[d]
    &&
    G
    \ar[
      d,
      "g \,\mapsto\, g \cdot x_0"
    ]
    &[-28pt]
    \\
    (U \times G) \times EG
    \ar[
      r,
      phantom,
      "\subset"
    ]
    \ar[
      d,
      "{
        \big(\mathrm{pr_1}, (g,p) \mapsto g \cdot p \big)
      }"
    ]
    &
    {X} \times E G
    \ar[
      rr,
      "\mathrm{pr}_1"{above},
      "\in \, \mathrm{W}"{below}
    ]
    \ar[d]
    &&
    {X}
    \ar[
      d,
      "q"
    ]
    \ar[
      r,
      phantom,
      "\supset"
    ]
    &
      {U} \times G
      \ar[
        d,
        "\mathrm{pr}_1"
      ]
    \\
    U \times E G
    \ar[
      r,
      phantom,
      "\subset"
    ]
    &
    ({X} \times E G) / G
    \ar[
      rr,
      "\mathrm{pr}_1/G"
    ]
    &&
    {X}/G
    \ar[
      r,
      phantom,
      "\supset"
    ]
    &
    U
\end{tikzcd}

one sees that for $\big\{ U_i \subset X/G\big\}_{i \in I}$ an [[open cover]] of $X/G$ over which $X \to X/G$ trivializes, we have that
$\big\{  U_i \times E G \,\subset\, (X \times E G)/G \}_{i \in I}$ is an open cover over which the quotient coprojection into the Borel construction  trivializes.

But since locally trivial bundles are [[Serre fibrations]] ([this Prop.](https://ncatlab.org/nlab/show/Serre+fibration#FiberBundlesAreSerreFibrations)), it follows that the above diagram is a morphism of [[homotopy fiber sequences]] (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)),
and hence induces a morphism of [[long exact sequences of homotopy groups]], which contains the following segments, for all $n \in \mathbb{N}$:

\begin{tikzcd}
    \pi_{n+1}(G)
    \ar[r]
    \ar[
      d,
      "{ \sim }"{yshift=-4pt, xshift=-3pt, sloped}
    ]
    &
    \pi_{n+1}
    \big(
      X \times E G
    \big)
    \ar[r]
    \ar[
      d,
      "{ \sim }"{yshift=-4pt, xshift=-3pt, sloped}
    ]
    &
    \pi_{n+1}
    \big(
      (X \times E G)/G
    \big)
    \ar[r]
    \ar[d]
    &
    \pi_n(G)
    \ar[r]
    \ar[
      d,
      "{ \sim }"{yshift=-4pt, xshift=-3pt, sloped}
    ]
    &
    \pi_n
    \big(
      X \times E G
    \big)
    \ar[
      d,
      "{ \sim }"{yshift=-4pt, xshift=-3pt, sloped}
    ]
    \\
    \pi_{n+1}(G)
    \ar[r]
    &
    \pi_{n+1}(X)
    \ar[r]
    &
    \pi_{n+1}
    \big(
      X/G
    \big)
    \ar[r]
    &
    \pi_n(G)
    \ar[r]
    &
    \pi_n
    \big(
      X
    \big)
\end{tikzcd}

Here the outer vertical morphisms are all [[isomorphisms]] by the fact that $E G \xrightarrow{\;} \ast$ is a [[weak homotopy equivalence]]. Therefore the [[five lemma]] (in its generality of possibly non-abelian groups, [here](five+lemma#FiveLemmaForNonabelianGroups)) implies that also the middle vertical morphism is an isomorphism, for all $n$.
\end{proof}

### As the realization of the action groupoid

This Borel construction is naturally understood as being the [[geometric realization of simplicial topological spaces|geometric realization]] of the [[topological groupoid|topological]] [[action groupoid]] $X // G$ of the [[action]] of $G$ on $X$:

the [[nerve]] of this topological groupoid is the [[simplicial topological space]]

$$
  (X \sslash G)_\bullet =
  \left(
 \cdots X \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} X \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to}} X
  \right)
  \,.
$$

Observing that $E G = G \sslash G$ itself as a groupoid has the nerve

$$
  (E G)_\bullet
   =
   \left(
   \cdots
   G \times G \times G
     \stackrel{\to}{\stackrel{\to}{\to}}
   G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G
  \right)
$$

(where "$\cdot$" denotes the multiplication action of $G$ on itself) and regarding $X$ and $G$ as topological 0-groupoids ($G$ as a group object in topological 0-groupoids), hence with simplicially constant nerves, we have an [[isomorphism]] of simplicial topological spaces

$$
  (X //G)_\bullet
    \simeq_{iso}
  X \times_G (E G)_\bullet
  \,.
$$

If this is set up in a sufficiently nice category of topological spaces, then, by the discussion at [[geometric realization of simplicial topological spaces]], the geometric realization ${\vert{-}\vert}\colon Top^{\Delta^{op}} \to Top$ manifestly takes this to the Borel construction (since, by the discussion there, it preserves the product and the quotient).

### As a homotopy colimit over the category associated to $G$

If $G$ is the topological category associated to the group $G$, then a $G$-space is precisely a Top-enriched functor $G\to Top$ in a similar fashion to the fact that an  R-module is an Ab-enriched functor. If $X$ is a $G$-space, the ordinary quotient $X/G$ is the colimit of the diagram associated to $X$ and the Borel construction is (a model of) the homotopy colimit of that diagram. This is a reason for calling the Borel construction homotopy quotient in some contexts.

### In rational homotopy theory

The image of the Borel construction in [[rational homotopy theory]] is the [[Weil model]] for [[equivariant de Rham cohomology]]. See there for more.

\begin{xymatrix}
  &
  X \times E G \ar@(ul,ur)^G
  \ar[d]^-{ q }
  &
  \Omega^\bullet( X )
  \otimes
  W(g)
  \ar@(ur,ul)_{ g }
  \ar@{<-^{)}}[d]^-{ q^\ast }
  \\
  \;
  X // G
  \ar@{}[r]|-{
    \simeq_{\mathrm{whe}}
  }
  &
  \big(
    X \times E G
  \big)/G
  &
  \big(
    \Omega^\bullet( X )
    \otimes
    W(g)
  \big)_{\mathrm{basic}}
  \ar@{<-}[r]^-{ :\exp\big( t^a \iota_a\big): }_-{\simeq}
  &
  \big(
    \Omega^\bullet( X )\big[ r^a \big]
  \big)^G
  \\
  &
  \mbox{Borel construction}
  &
  \mbox{Weil model}
  &
  \mbox{Cartan model}
\end{xymatrix}

## Related concepts

* [[simplicial classifying space]]

* [[Borel equivariant cohomology]]

  * [[equivariant ordinary cohomology]]

  * [[equivariant de Rham cohomology]]

## Literature

As a [[right Quillen functor]] on the [[model structure on simplicial group actions]]:

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

Lecture notes in classical [[topology]]:

* [[Stephen Mitchell]], _Notes on principal bundles and classifying spaces_, 2011 ([pdf](https://sites.math.washington.edu/~mitchell/Notes/prin.pdf), [[MitchellPrincipalBundles.pdf:file]])

General statement in [[simplicial homotopy theory]] of [[simplicial presheaves]]:

* [[J. F. Jardine]], _Stacks and the homotopy theory of simplicial sheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0450/stacks3.pdf))

* {#NSS12b} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Prop. 3.73 in: _Principal $\infty$-bundles -- Presentations_,  Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249), [doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4))

The nature of the Borel construction as the geometric realization of the action groupoid is mentioned for instance in

* [[Alejandro Adem]], Michele Klaus, _Lectures on orbifolds and group cohomology_ ([pdf](http://www.math.ubc.ca/~adem/hangzhou.pdf))

[[!redirects Borel construction]]
[[!redirects Borel constructions]]

[[!redirects Borel space]]
[[!redirects Borel spaces]]


Borel equivariant cohomology

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[space]] and $G$ a [[group]] [[action|acting]] on it, then the _Borel equivariant cohomology_ of $X$ is the [[cohomology]] of the [[homotopy quotient]] $X//G$.

Since a standard way to model the [[homotopy quotient]] is the _[[Borel construction]]_, this is called _Borel equivariant cohomology_.

This is the special case of genuine [[equivariant cohomology]] where the [[action]] on the coefficients is trivial.

## Properties

### As right induced genuine equivariant cohomology
  {#AsRightInducedGenuineEquivariantCohomology}

In the more general context of ([[global equivariant stable homotopy theory|global]]) [[equivariant stable homotopy theory]], Borel-[[equivariant spectra]] are those which are _right induced_ from plain [[spectra]], hence which are in the [[essential image]] of the [[right adjoint]] to the [[forgetful functor]] from [[equivariant spectra]] to plain [[spectra]].

([Schwede 18, Example 4.5.19](#Schwede18))

## Examples

* [[equivariant ordinary  cohomology]]

* [[equivariant de Rham cohomology]]

## Related concepts

[[!include equivariant cohomology -- table]]

## References

* {#Tu11} [[Loring Tu]], _What is... Equivariant Cohomology?_, Notices of the AMS, Volume 85, Number 3, March 2011 ([pdf](https://www.ams.org/notices/201103/rtx110300423p.pdf), [[TuEquivariantCohomology.pdf:file]])

* {#Schwede18} [[Stefan Schwede]], Example 4.5.19 of _[[Global homotopy theory]]_ 2018  ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))


Borel measure

# Contents
* table of contents
{: toc}

## Idea

Borel measures are [[measures]] on the [[Borel σ-algebra]] of a [[topological space]].  They play an important role in [[measure theory]].

## Definition

A __Borel measure__ on $X$ is a (countably additive) measure on the [[Borel σ-algebra]] of the topological space $X$.

A __Baire measure__ on $X$ is a (countably additive) measure on the [[Baire σ-algebra]] of the topological space $X$.

A [[Radon measure]] on $X$ is a Borel measure $\mu$ on $X$
such that for any [[Borel subset]] $B\subset X$ and any $\epsilon\gt0$ there is a [[compact subset]] $K\subset B$
such that $|\mu|(B\setminus K)\lt\epsilon$,
where $|\mu|$ denotes the total [[variation]] of the measure $\mu$.

Borel measures are uniquely determined by their values on open subsets,
and are closely related to [[continuous valuations]] on the underlying [[locale]].

A measure $\mu$ on a topological space $X$
is __tight__ if for any $\epsilon\gt0$
there is a [[compact subset]] $K\subset X$
such that $|\mu|(A)\lt\epsilon$ for any $A\subset X\setminus K$.

A measure $\mu$ on a topological space $X$
is __regular__ if for any measurable subset $A\subset X$
and $\epsilon\gt0$
there is a closed subset $F\subset A$ such that
$A\setminus F$ is measurable and $\mu(A\setminus F)\lt\epsilon$.

Radon measures on [[Hausdorff spaces]] are regular and tight.
Regular tight Borel measures are automatically Radon.
A regular Borel measure need not be tight.

A Borel measure $\mu$ on a topological space $X$
is __[[τ-additive]]__ (alias τ-regular, τ-smooth)
if $|\mu|(\bigcup_i U_i)=\lim_i |\mu|(U_i)$
for any directed system of open subsets $U_i\subset X$.
In other words, the underlying valuation of $\mu$ is a [[continuous valuation]].

If the above condition only holds in the case $\bigcup_i U_i=X$,
we talk about __$\tau_0$-additive__ (or weakly τ-additive) measures.

Any regular $\tau_0$-additive Borel measure is [[τ-additive]].
There are $\tau_0$-additive Borel measures that are not [[τ-additive]].

## Properties

On a [[metric space]] every Borel measure is regular.
More generally, every Borel measure on a [[perfectly normal]] space is regular.

On a complete separable [[metric space]] every Borel measure is Radon.

Every Baire measure is regular.

Every [[Radon measure]] is [[τ-additive]].

Every [[τ-additive]] measure on a regular space is regular.
In particular, every [[τ-additive measure]] on a compact space is Radon.

Every tight [[τ-additive]] measure is Radon.

Every Borel measure on a separable metric space X is τ-additive.
Moreover, this is true if X is [[hereditary Lindelöf]].

Every [[τ-additive]] measure has a [[support]], which is a closed subset.

## Extension properties

Every tight Baire measure on a [[completely regular]] space admits a unique extension to a Radon measure.
More generally, any tight Baire measure on a Hausdorff space has a Radon extension.

Every Baire measure on a σ-compact [[completely regular space]] has a unique extension to a Radon measure.

## Relation to linear functionals

Given a topological space $X$, the integration map
sends Borel measures on $X$ to linear functionals
on the space of continuous bounded functions on $X$.

For many types of topological spaces and measures
this map is bijective, as indicated below.

There is a bijection between Baire measures on a topological space
and continuous linear functionals $L$ on continuous bounded functions
such that $L(f_n)\to 0$ whenever $f_n \to 0$ pointwise and monotonic.

There is a bijection between Radon measures on a completely
regular topological space and continuous linear functionals $L$
on continuous bounded functions such that for any $\epsilon\gt0$
there is a compact subset $K$ such that for any continuous bounded
function $f$ that vanishes on $K$ we have $|L(f)|\le\epsilon\sup|f|$.

There is a bijection between τ-additive Borel measures
on a completely regular topological space and continuous linear functionals $L$ on continuous bounded functions such that
$L(f_a)\to0$ for any net $f$ of bounded continuous functions
that decreases to zero pointwise.

There is a bijection between positive Borel measures on a [[locally compact]]
topological space that are inner regular on all Borel subsets
with respect to compact subsets
and positive linear functionals on continuous functions vanishing at infinity.

## Related concepts

* [[Borel set]]

* [[Radon measure]]

* [[continuous valuation]]

* [[measure]]

[[!redirects Borel measures]]
[[!redirects Baire measures]]


Borel model structure

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Given a [[topological group]] $G$, the _Borel model structure_ is a [[model category]] structure on the [[category]] of [[topological G-spaces]], hence of [[topological spaces]] [[group object|equipped with]] [[continuous function|continuous]] [[group actions]].

Analogously, given a [[simplicial group]] $G_\bullet$, the _Borel model structure_ is a [[model category]] structure on the [[category]] of [[simplicial group actions]], hence of [[simplicial sets]] equipped with $G$-[[action]]

Both of these present the [[(∞,1)-category]] of [[∞-actions]] of the [[∞-group]] (see there) presented by $G$.

In the context of [[equivariant homotopy theory]] this is also called the "coarse model structure" (e.g. [Guillou 2006, section 5](#Guillou06)), since in general it has more weak equivalences than the [[fine model structure on topological G-spaces]] that enters [[Elmendorf's theorem]].

## Definition

### In topological spaces

Throughout, write [[Top]] for the [[category]] of [[compactly generated weak Hausdorff spaces]].

\begin{definition}
  For $G \,\in\, Grp(TopSp)$ a [[topological group]], write

\[
  \label{BGAsTopEnrichedCategory}
  \mathbf{B}G \;\in\; TopCat
\]

for the [[Top]]-[[enriched category]] with a single [[object]] and $G$ as its unique [[hom-object]].
\end{definition}

\begin{remark}
  There is an evident [[isomorphism]] of [[enriched categories]]
  \[
    \label{IdentifyingTopologicalGSpacesWithTopEnrichedFunctors}
    G Act(TopSp)
    \;\simeq\;
    TopFnctr\big( \mathbf{B}G,\, TopSp \big)
  \]
  between [[topological G-spaces]] and the [[Top]]-[[enriched functor category]] from $\mathbf{B}G$ (eq:BGAsTopEnrichedCategory) to [[Top]] (topological presheves).
\end{remark}

\begin{prop}\label{BorelModelStructureOnTopologicalSpaces}
For $G \in Grp(TopSp)$ a [[topological group]] there is a [[model category]]-structure

$$
  G Act\big(TopSp\big)_{proj}
  \;\;\;
  \in
  \;
  MdlCat
$$

on the [[category]] of [[topological G-spaces]] whose [[weak equivalences]] and [[fibrations]] are those [[morphisms]] whose [[underlying]] [[continuous functions]] are so in the  [[classical model structure on topological spaces]].
\end{prop}

For [[discrete groups]] this may be argued as in  [Guillou 2006, Thm. 5.1](#Guillou06).
For general topological groups this follows as a special case of the [[projective model structure on enriched functors|projective model structure on Top-enriched functor]] (see [this Thm.](Introduction+to+Homotopy+Theory#ProjectiveModelStructureOnTopologicalFunctors)), under the identification $G Act(TopSp) \,\simeq\, TopFun( \mathbf{B}G, TopSp )$ (eq:IdentifyingTopologicalGSpacesWithTopEnrichedFunctors).

### In simplicial sets

\begin{defn}\label{BorelModelStructure}

For $G_\bullet$ a [[simplicial group]] write

* $\mathbf{B}G_\bullet$ for the one-object [[sSet-enriched category]] (here: a [[simplicial groupoid]]) whose [[hom-object]] is $G_\bullet$.

* $G_\bullet Actions(sSet) \;\coloneqq\; sSetCat\big(\mathbf{B}G_\bullet, sSet\big)$ for the [[sSet]]-[[enriched functor category]] to [[SimplicialSets]].

* $G_\bullet Acts\big(sSet_{Qu}\big)_{proj} \coloneqq sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ for the projective [[model structure on functors]] (projective [[model structure on simplicial presheaves]]).

This is the $G_\bullet$ *Borel model structure*, naturally a [[simplicial model category]] ([DDK 80, Prop. 2.4](#DDK80), [Goerss & Jardine 09, Chapter V, Thm. 2.3](#GoerssJardine09)).
\end{defn}

### In combinatorial simplicial model categories
 {#DefinitionInCombSimpModCat}

More generally, if $\mathbf{C}$ is an [[sSet-enriched category]] which is also [[tensoring|tensored]] over [[sSet]], via an [[enriched functor]]
$$
  sSet \times \mathbf{C} \longrightarrow \mathbf{C}
  \,,
$$
then for $\mathcal{G} \in Grp(sSet)$ the [enriched hom-isomorphism](enriched+adjoint+functor#ViaHomIsomorphism) of the [[tensoring]]

$$
  Hom\big(
    \mathcal{G}
    ,\,
    \mathbf{C}(\mathscr{V},\,\mathscr{V})
  \big)
  \;\;\simeq\;\;
  Hom\big(
    \mathcal{G} \cdot \mathscr{V}
    ,\,
    \mathscr{V}
  \big)
$$
shows that [[sSet]]-[[enriched functors]]
$$
  F
    \,\colon\,
  \mathbf{B}\mathcal{G} \longrightarrow \mathbf{C}
$$
may equivalently be thought of as [[simplicial group actions]]
$$
  \mathcal{G} \cdot \mathscr{V}
  \overset{\rho}{\longrightarrow}
  \mathscr{V}
$$
of $\mathcal{G}$ acting on [[objects]] $\mathscr{V} \coloneqq F(\ast)$ via [[tensoring]].

If now $\mathbf{C}$ is in fact a [[combinatorial simplicial model category]], then the respective projective [[model structure on functors]] exists ([that Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors)), which we may hence think of as the Borel model structure on $\mathcal{G}$-actions on objects of $\mathbf{C}$:
\[
  \label{BorelModelStructureForCombinatoriaSimplicialModelCat}
  \mathcal{G}Act(\mathbf{C})_{Borel}
  \,\coloneqq\,
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})_{proj}
  \,.
\]

## Properties

### In topological spaces
 {#PropertiesInTopologicalSpaces}

#### Cofibrations and Cofibrant replacement

\begin{prop}\label{CofibrantObjectsInTopologicalBorelModelStructure}
The model category $G Act\big(TopSp_{Qu}\big)_{proj}$ from Prop. \ref{BorelModelStructureOnTopologicalSpaces} is [[cofibrantly generated model category|cofibrantly generated]] with [[generating cofibrations]] being (see [this Def.](Introduction+to+Homotopy+Theory#GeneratingCofibrationsForProjectiveStructureOnFunctors)) the [[product topological space|product]] with $G$ (regarded with its free left multiplication action) of the generating cofibrations of [[classical model structure on topological spaces|$TopSp_{Qu}$]].
\end{prop}
For [[discrete group|discrete]] $G$ a statement along these lines appears as   [Guillou 2006, Prop. 5.3](#Guillou06).
\begin{proof}
This is a special case of [this Thm.](Introduction+to+Homotopy+Theory#ProjectiveModelStructureOnTopologicalFunctors) about topological functor categories.

Alternatively, the statement is a special case of that for the [[fine model structure on topological G-spaces]] for the case of trivial family of closed subgroups (see [there](fine+model+structure+on+topological+G-spaces#SpecializationToBorelModelStructure)).
\end{proof}

\begin{example}\label{ProductWithUniversalPrincipalGSpaceIsCofibrantReplacement}
  Since the [[universal principal bundle|universal principal space]] $E G$ (the [[topological realization]] $E G \,=\, \big\vert W G\big\vert$ of the [[universal principal simplicial complex]]) is

* [[contractible homotopy type|contractible]]: $E G \simeq_{whe} \ast$;

* equipped with a [[free action]] by $G$,

Prop. \ref{CofibrantObjectsInTopologicalBorelModelStructure} implies that the [[product]] with $E G$ in G Act(TopSp) (i.e. the [[product topological space]] with the induced [[diagonal action]]) serves as [[cofibrant replacement]] in $G Act(TopSp)$:

$$
  \varnothing
    \underset{\;\in Cof\;}{\longrightarrow}
  X \times E G
    \underoverset{\;\in W \cap Fib\;}{pr_1}{\longrightarrow}
  X
  \;\;\;
  \in
  G Act\big(TopSp_{Qu}\big)_{proj}
  \,.
$$
\end{example}

> Hm, this is not a proper argument...

#### Topological Borel construction

\begin{prop}\label{QuotentQuillenAdjunctionBetweenGSpacesToSpaces}
  There is a [[Quillen adjunction]]
  $$
    TopSp_{Qu}
    \underoverset
      {\underset{triv}{\longrightarrow}}
      {\overset{(-)/G}{\longleftarrow}}
      {\bot}
    G Act\big(TopSp_{Qu}\big)_{proj}
  $$
  between the [[classical model structure on topological spaces]] and the projective Borel model structure from Prop. \ref{BorelModelStructureOnTopologicalSpaces}, whose

* [[right Quillen functor]], $triv$, assigns [[trivial actions]];

* [[left Quillen functor]], $(-)/G$, assigns [[topological quotient spaces]] by the [[relation]] $G \times X \underoverset{pr_2}{\mathclap{(-)\cdot(-)}}{\rightrightarrows} X$.

\end{prop}
([Guillou 2006, Ex. 5.5](#Guillou06))
\begin{proof}
  By definition of the weak equivalences and fibrations in Prop. \ref{BorelModelStructureOnTopologicalSpaces}, it is immediate that $triv$ preserves these classes of morphisms.
\end{proof}

\begin{prop}\label{BorelConstructionOfFreeActionEquivalentToPlainQuotient}
**([[Borel construction]] of [[free action]] is [[weak homotopy equivalence|weak hom. equivalent]] to plain [[quotient space]])**
\linebreak
  Let $G$ be a [[compact Lie group]] and
  let $X$ be a [[G-CW complex]] whose $G$-action is [[free action|free]]. Then the comparison morphism between the [[Borel construction]] and the plain [[quotient space]] of $X$ is a [[weak homotopy equivalence]]:

$$
  (X \times E G)/G
  \xrightarrow{ \;
    (id_x \times (E G \to \ast))/G \, \in\, W
  \; }
  X/G
  \,.
$$
\end{prop}
\begin{proof}
  Since $E G$ is a [[G-CW complex]], the product $X \times E G$ is cofibrant (by the analog of [this Prop.](fine+model+structure+on+topological+G-spaces#MonoidalModelCategoryStructure)) and $X$ is cofibrant by assumption and by Prop. \ref{CofibrantObjectsInTopologicalBorelModelStructure}.

Hence, by Prop. \ref{QuotentQuillenAdjunctionBetweenGSpacesToSpaces}, the morphism in question is the image under a [[left Quillen functor]], of a weak equivalence between cofibrant objects. Therefore the claim follows by [[Ken Brown's lemma]] ([here](Introduction+to+Homotopy+Theory#KenBrownLemma)).
\end{proof}

\begin{prop}
  The [[Borel construction]] exhibits the [[left derived functor]] of the [[quotient space]]-[[left Quillen functor]] in Prop. \ref{QuotentQuillenAdjunctionBetweenGSpacesToSpaces}:

$$
  X \,\in\, A Act(TopSp)
  \;\;\;\;
    \Rightarrow
  \;\;\;\;
  \big(\mathbb{L}(-)/G\big)(X)
  \;\simeq\;
  \frac{E G \times X}{G}
  \;\;\;
  \in
  \;
  Ho\Big( G Act\big( TopSp_{Qu} \big)_{proj}  \Big)
  \,.
$$
\end{prop}
\begin{proof}
  Since the [[left derived functor]] of a left Quillen functor is given by the application of the latter on any [[cofibrant replacement]], the claim follows by Ex. \ref{ProductWithUniversalPrincipalGSpaceIsCofibrantReplacement}.
\end{proof}

> or would follow, if that Example were argued properly

### In simplicial sets

#### Cofibrant replacement and homotopy quotients/fixed points
 {#CofibrantReplacementAndHomotopyQuotientsFixedPoints}

\begin{prop}\label{CofibrationsOfSimplicialActions}
**(cofibrations of simplicial actions)** \linebreak
The cofibrations $i \colon X \to Y$ in $sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ (Def. \ref{BorelModelStructure}) are precisely those morphisms such that

1. the underlying morphism of [[simplicial sets]] is a [[monomorphism]];

1. the $G_\bullet$-[[action]] is a relatively [[free action]], i.e. [[free action|free]] on all [[simplices]] not in the [[image]] of $i$.

\end{prop}

This is ([DDK 80, Prop. 2.2. (ii)](#DDK80), [Guillou 2006, Prop. 5.3](#Guillou06), [Goerss & Jardine 09, V Lem. 2.4 & Cor. 2.10](#GoerssJardine09)).

\begin{remark}
In particular this means that an object is [[cofibrant object|cofibrant]]
in $sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ if the $G_\bullet$-[[action]] on it is [[free action|free]].

Hence [[cofibrant replacement]] is obtained by forming the [[Cartesian product|product]] with the model $W G_\bullet$ for the total space of the [[universal principal bundle]] over $G_\bullet$ (see at _[[simplicial group]]_ for notation and more details).
\end{remark}

\begin{remark}
It follows that for $X, A \in sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ the [[derived hom space]]

$$
  R Hom_G(X,A)
$$

models the Borel $G$-[[equivariant cohomology]] of $X$ with [[coefficients]] in $A$.

In particular, if $A$ is [[fibrant object|fibrant]] (the underlying simplicial set is a [[Kan complex]]) then:

1. if the $G_\bullet$-action on $A$ is trivial, then

   $$
     R Hom_G(X,A)
       \simeq
     Hom_G(W G \times X , A)
       \simeq
     Hom(W G \times_G X, A)
   $$

   is equivalently maps of [[simplicial sets]] out of the [[Borel construction]] on $X$;

1. if $X = \ast $ is the [[point]] then

   $$
     R Hom_G(X,A)
       \simeq
     Hom_G(W G, A)
       \simeq
     Hom(\overline{W} G , A)
       \simeq
     A^{h G}
   $$

   is the [[homotopy fixed points]] of $A$.

\end{remark}

#### Relation to the slice over the simplicial classifying space
 {#RelationToSliceOverSimplicialClassifyingSpace}

\begin{prop}\label{QuillenEquivalenceToSliceOverSimplicialClassifyingSpace}
  For $G$ a [[simplicial group]], there is a pair of [[adjoint functors]]
  \[
    \label{QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace}
    G Act\big(sSet_{Qu}\big)_{proj}
      \underoverset
        {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}}
        {\overset{ (-) \times_{\overline{W}G} W G  }{\longleftarrow}}
        {\bot}
    \big(sSet_{Qu}\big)_{/\overline{W}G}
  \]
  which constitute a [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalence]] between the Borel model structure (Def. \ref{BorelModelStructure}) and the [[slice model structure]] of the [[classical model structure on simplicial sets]], sliced over the [[simplicial classifying space]] $\overline{W}G$.

\end{prop}

(this is essentially the statement of [DDK 80, Prop. 2.3, Prop. 2.4](#DDK80))

Here:

* the [[right adjoint]] is the [[Borel construction]]

  understood as forming [[associated bundles]] to [[universal principal bundles]];

* the [[left adjoint]] forms [[homotopy fibers]].

(This may also be understood as an instance of the "[[fundamental theorem of (infinity,1)-topos theory|fundamental theorem of $\infty$-topos theory]]", see [there](slice+of+presheaves+is+presheaves+on+slice#InfinityActionsAsSlice).)

\begin{proof}\label{SimplicialProofOfEquivalenceToSliceModelStructure}
Consider any [[morphism]] in $sSet_{/\overline{W}G}$:

$$
  \array{
    X && \xrightarrow{\;\;f\;\;} && Y
    \mathrlap{\,.}
    \\
    &
      {}_{\mathllap{c_X}}\searrow
    &&
      \swarrow_{\mathrlap{c_Y}}
    \\
    && \overline{W}G
  }
$$

Its image under the left adjoint functor is, by definition, the top left arrow in the following [[commuting diagram]]:

\begin{tikzcd}
    c_X^\ast(W G)
    \ar[
      rr,
      "(f) \underset{\overline{W}G}{\times} W G"
    ]
    \ar[
      d,
      "p_X \,\in\, \mathrm{Fib}"
    ]
    &&
    c_Y^\ast(W G)
    \ar[
      d,
      "p_Y \,\in\, \mathrm{Fib}"
    ]
    \ar[rr]
    &&
    W G
    \ar[
      d,
      "\in \mathrm{Fib}"
    ]
    \\
    X
    \ar[
      rr,
      "f"{below}
    ]
    \ar[
      rrrr,
      rounded corners,
      to path={
         -- ([yshift=-7pt]\tikztostart.south)
         --node[below]{\scalebox{.7}{$
               c_Y
             $}} ([yshift=-7pt]\tikztotarget.south)
         -- (\tikztotarget.south)}
    ]
    &&
    Y
    \ar[
      rr,
      "c_Y"{below}
    ]
    &&
    \overline{W}G
\end{tikzcd}

Here the right square and the total rectangle are [[Cartesian squares]] ([[pullback squares]]), by defnition of the functor. It follows by the [[pasting law]] that also the square on the left is cartesian. Specifically, since [[fibrations]] are preserved under pullback, as shown, the top left morphism in question is the pullback of $f$ along a [[fibration]].

It follows that $(f) \underset{\overline{W}G}{\times} W G$ is:

1. a [[weak equivalence]] if $f$ is a weak equivalence, because the [[classical model structure on simplicial sets]] is a [[right proper model category]] (see [here](classical+model+structure+on+simplicial+sets#Properness));

1. a [[monomorphism]] if $f$ is a monomorphism, since monomorphisms are preserved by pullback (see [here](monomorphism#MonomorphismsArePreservedByPullback)).

   Moreover, since $W G \to \overline{W}G$ is the [[universal principal bundle]], it follows that $c_Y^\ast(W G) \to Y$ is a [[simplicial principal bundle]], so that, in particular, the action of $G$ on $c_Y^\ast(W G)$ is [[free action|free]].

   By Prop. \ref{CofibrationsOfSimplicialActions} this means that $(f) \underset{\overline{W}G}{\times} W G$ is a [[cofibration]] if $f$ is a cofibration.

In summary, the left adjoint functor in (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace) preserves the classes of [[weak equivalences]] and of [[cofibrations]], hence also that of [[acyclic cofibrations]], and so it is a [[left Quillen functor]].

Next...
\end{proof}

In fact, these functors (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace) are [[sSet]]-[[enriched functors]] which induce an [[equivalence of (infinity,1)-categories|equivalence of $(\infty,1)$-categories]] between the [[simplicial localizations]]  $L_W sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} \simeq L_W sSet_{/\overline{W}H}$ ([DDK 80, Prop. 2.5](#DDK80)).

This kind of relation is discussed in more detail at _[[∞-action]]_.

\begin{remark}\label{sSetEnrichmentOfAdjunctionToSliceOverSimpClassSpace}
**(sSet-enrichement of the adjunction)** \linebreak
The statement that (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace) is an [[sSet]]-*[[enriched adjunction]]* is not made explicit in [DDK 80](#DDK80); there it only says that the functors form a plain [[adjoint functors|adjunction]] ([DDK 80, Prop. 2.3](#DDK80)) and that they are each [[sSet]]-[[enriched functors]] ([DDK 80, Prop. 2.4](#DDK80)).

The remaining observation that we have a [[natural isomorphism]] of [[sSet]]-[[hom-objects]]

$$
  \big[
    X \times_{\overline{W}G} W G,
    \,
    V
  \big]
  \;\simeq\;
  \big[
    X,
    \,
    (V \times W G)/G
  \big]
$$

hence

$$
  Hom
  \Big(
    \big( X \times_{\overline{W}G} W G \big) \times \Delta[\bullet],
    \,
    V
  \Big)
  \;\simeq\;
  Hom
  \big(
    X \times \Delta[\bullet],
    \,
    (V \times W G)/G
  \big)
$$

follows from the plain adjunction and the natural isomorphism

$$
  (X \times_{\overline{W}G} W G) \times \Delta[\bullet]
  \;\simeq\;
  (X \times \Delta[\bullet]) \times_{\overline{W}G} W G
  \,,
$$

which, in turn, follows, for instance, via the [[pasting law]]:

\begin{tikzcd}
  {
    { (X \times_{\overline{W}G} W G) \times \Delta[k] }
    \atop
    { \mathllap{\simeq} (X \times \Delta[k]) \times_{\overline{W}G} W G  }
  }
  \ar[r]
  \ar[d]
  \ar[dr,phantom,"\mbox{\tiny (pb)}"]
  &
  X \times \Delta[k]
  \ar[
    d,
    "\mathrm{pr}_1"
  ]
  \\
  X \times_{\overline{W}G} W G
  \ar[r]
  \ar[d]
  \ar[dr,phantom,"\mbox{\tiny (pb)}"]
  &
  X
  \ar[
    d
  ]
  \\
  W G
  \ar[r]
  &
  \overline{W}G
  \,.
\end{tikzcd}

\end{remark}

#### Relation to the model structure on plain simplicial sets
 {#RelationToModelStructureOnPlainSimplicialSets}

For $\mathcal{G} \,\in\, Groups(sSets)$ a [[simplicial group]], write $\mathcal{G}Actions(sSets)$ for the [[category]] of $\mathcal{G}$-[[actions]] on [[simplicial sets]].

\begin{proposition}\label{CofreeAction}
**(underlying simplicial sets and cofree simplicial action)** \linebreak
  The [[forgetful functor]] $undrl$ from $\mathcal{G}Actions$ to underlying simplicial sets is a [[left Quillen functor]] from the Borel model structure (Def. \ref{BorelModelStructure}) to the [[classical model structure on simplicial sets]].

Its [[right adjoint]]

$$
  sSet
  \underoverset
    {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}}
    {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}}
    {\bot}
  \mathcal{G}Actions(sSet)
$$

sends $\mathcal{X} \in sSet$ to

* the simplicial set

  $$
    [\mathcal{G},\mathcal{X}]
    \;\coloneqq\;
    Hom_{sSet}\big( \mathcal{G} \times \Delta[\bullet], \mathcal{X}\big)
    \;\;\;
    \in
    sSet
  $$

* equipped with the $\mathcal{G}$-action

  $$
    \mathcal{G} \times [\mathcal{G},\mathcal{X}]
    \overset{ (-) \cdot (-) }{\longrightarrow}
    \mathcal{G}
  $$

  which in degree $n \in \mathbb{N}$ is the [[function]]

  \[
    \label{CofreeSimplicialActionComponentFunctions}
    Hom(\Delta[n], \mathcal{G})
    \,\times\,
    Hom
    \big(
      \mathcal{G} \times \Delta[n],
      \,
      \mathcal{X}
    \big)
    \longrightarrow
    Hom
    \big(
      \mathcal{G} \times \Delta[n],
      \,
      \mathcal{X}
    \big)
  \]

  that sends

  \[
  \label{CofreeSimplicialActionInComponents}
  \begin{aligned}
  &
  \Big(
    \Delta[n] \overset{g_n}{\to} \mathcal{G},
    \;
    \mathcal{G}\times \Delta[n]
    \overset{\phi}{\to}
    \mathcal{X},
  \Big)
  \\
  \;\;\mapsto\;\;
  &
  \Big(
  \mathcal{G} \times \Delta[n]
  \overset{id \times diag}{\longrightarrow}
  \mathcal{G} \times \Delta[n] \times \Delta[n]
  \overset{ id \times g_n \times id }{\longrightarrow}
  \mathcal{G} \times \mathcal{G} \times \Delta[n]
  \overset{(-)\cdot(-) \times id}{\to}
  \mathcal{G} \times \Delta[n]
  \overset{\phi}{\to}
  \mathcal{X}
  \Big)
  \end{aligned}
  \]

\end{proposition}

Here and in the following proof we make free use of the [[Yoneda lemma]] [[natural bijection]]

$$
  Hom_{sSet}(\Delta[n], \mathcal{S}) \;\simeq\; \mathcal{S}_n
$$

for any [[simplicial set]] $S$ and for $\Delta[n] \in \Delta \overset{y}{\hookrightarrow} sSet$ the simplicial [[n-simplex]].

\begin{proof}

We already know from Def. \ref{BorelModelStructure} that $underl$ preserves all [[weak equivalences]] and from Prop. \ref{CofibrationsOfSimplicialActions} that it preserves all [[cofibrations]]. Therefore it is a [[left Quillen functor]] as soon as it is a [[left adjoint]] at all.

The idea of the existence of the [[cofree functor|cofree]] [[right adjoint]] to $undrl$ is familiar from [[topological G-spaces]] (see the section on [coinduced actions](topological+G-space#CoinducedActions) there), where it can be easily expressed point-wise in [[point-set topology]]. The formula (eq:CofreeSimplicialActionInComponents) adapts this idea to simplicial sets. Its form makes manifest that this gives a simplicial homomorphism, and with this the adjointness follows the usual logic by focusing on the image of the non-degenerate top-degree cell in $\Delta[n]$:

To check that (eq:CofreeSimplicialActionInComponents) really gives the right adjoint, it is sufficient to check the corresponding [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism), hence to check
for $\mathcal{P} \in \mathcal{G}Actions(sSet)$, and $\mathcal{X} \in sSet$,
that we have a [[natural bijection]] of [[hom-sets]] of the form

$$
  \big\{
    \mathcal{P}
       \overset{\;\;\phi_{(-)}\;\;}{\longrightarrow}
    [\mathcal{G}, \mathcal{X}]
  \big\}
  \;\;\;\overset{ \;\; \widetilde{(-)} \;\; }{\leftrightarrow}\;\;\;
  \big\{
    undrl(\mathcal{P})
      \overset{\;\; {\widetilde \phi}_{(-)} \;\; }{\longrightarrow}
    \mathcal{X}
  \big\}
  \,.
$$

So given

$$
  \phi_{(-)}
  \;\colon\;
  p_n
  \mapsto
  \big(
    \phi_{p_n}
    \;\colon\;
    \mathcal{G} \times \Delta[n] \to \mathcal{X}
  \big)
$$

on the left, define

\[
  \label{AdjunctOfHomomorphismToCofreeSimplicialAction}
  \widetilde \phi_{(-)}
  \;\colon\;
  p_n
  \mapsto
  \phi_{p_n}(e_n, \sigma_n)
  \;\in\;
  \mathcal{X}_n
  \,,
\]

where $e_n \in \mathcal{G}_n$ denotes the [[neutral element]] in degree $n \in \mathbb{N}$ and where $\sigma_n \in (\Delta[n])_n$ denotes the unique non-degenerate element $n$-cell in the [[n-simplex]].

It is clear that this is a [[natural transformation]] in $P$ and $X$. We need to show that ${\widetilde \phi}_{(-)} \colon undrl(P) \to X$ uniquely determines all of $\phi_{(-)}$.

To that end, observe for any $g_n \in \mathcal{G}_n$ the following sequence of identifications:

$$
  \begin{aligned}
    \phi_{p_n}(g_n, \sigma_n)
    & \;=\;
    \phi_{p_n}( e_n \cdot g_n, \sigma_n )
    \\
    & \;=\;
    \big(
      g_n \cdot \phi_{p_n}
    \big)
    ( e_n, \sigma_n )
    \\
    & \;=\;
    \phi_{ g_n \cdot p_n }
    (e_n, \sigma_n)
    \\
    & \;=\;
    {\widetilde \phi}_{g_n \cdot p_n}
  \end{aligned}
$$

Here:

* the first step is the unit law in the component group $\mathcal{G}_n$;

* the second step uses the definition (eq:CofreeSimplicialActionInComponents) of the cofree action;

* the third step is the assumption that $\phi_{(-)}$ is a homomorphism of $\mathcal{G}$-actions ([[equivariant function|equivariance]]);

* the fourth step is the definition (eq:AdjunctOfHomomorphismToCofreeSimplicialAction).

These identifications show that $\phi_{(-)}$ is uniquely determined by ${\widetilde \phi_{(-)}}$, and vice versa.

\end{proof}

\begin{example}\label{BZActionOnInertiaGroupoid}
**($\mathbf{B}\mathbb{Z}$-[[infinity-action|2-action]] on [[inertia groupoid]])** \linebreak
  Let

  * $G \in Groups(Sets)$

    be a [[discrete group]],

  * $X \in G Actions(Sets)$

    be a $G$-[[group action|action]],

  * $\mathcal{X} \;\coloneqq\; X \sslash G \;\coloneqq\; N( X \times G \rightrightarrows X ) \,=\, X \times G^{\times^\bullet} \in sSet$

    the [[simplicial set]] which is the [[nerve]] of its [[action groupoid]] (a model for its [[homotopy quotient]]),

  * $\mathcal{G} \,\coloneqq\, \mathbf{B}\mathbb{Z} \,\coloneqq\, N(\mathbb{Z} \rightrightarrows \ast)  \,\coloneqq\, \mathbb{Z}^{\times^\bullet} \,\in\, Groups(sSet)$

    the [[simplicial group]] which is the [[nerve]] of the [[2-group]] that is the [[delooping groupoid]] of the additive group of [[integers]].

Then the [[functor groupoid]]

\[
  \label{InertiaGroupoidAsFunctorGroupoidOutOfBZ}
  \begin{aligned}
    \Lambda(X \!\sslash\! G)
    & \;\coloneqq\;
    \big[
      \mathbf{B}\mathbb{Z}, X \!\sslash\! G
    \big]
    \\
    &
    \;\simeq\;
    Func
    \big(
      (\mathbb{Z} \rightrightarrows \ast),
      \,
      (X \times G \rightrightarrows X)
    \big)
    \\
    & \;\underset{\in \mathrm{W}}{\leftarrow}\;
    \underset{
      [g] \in ConjCl(G)
    }{\coprod}
    \Big(
      X^{g} \!\sslash\! C_g
    \Big)
  \end{aligned}
\]

is known as the *[[inertia groupoid]]* of $X \!\sslash\! G$. Here

$$
  ConjCla(G)
  \;\coloneqq\;
  G/_{ad} G
  \,,
  \;\;\;\;\;\;\;\;\;\;\;
  C_g
  \;\coloneqq\;
  \big\{
    h \in G
    \,\left\vert\,
    h \cdot g = g \cdot h
    \right.
  \big\}
$$

denotes, respectively, the set of [[conjugacy classes]] of elements of $G$, and the [[centralizer]] of $\{g\} \subset G$ -- this data serves to express the [[equivalence of categories|equivalent]] [[skeleton]] of the inertia groupoid in the last line of (eq:InertiaGroupoidAsFunctorGroupoidOutOfBZ).

Now, by Prop. \ref{CofreeAction} the inertia groupoid (eq:InertiaGroupoidAsFunctorGroupoidOutOfBZ) carries a canonical [[infinity-action|2-action]] of the [[2-group]] $\mathbf{B}\mathbb{Z}$:

By the formula (eq:CofreeSimplicialActionInComponents), for $n \in \mathbb{Z}$ the 2-group element in degree 1

$$
  {\color{purple}n}
  \;\colon\;
  \Delta[1]
  \longrightarrow
  \mathbf{B} \mathbb{Z}
$$

acts on the morphisms

$$
  (x,g) \overset{h}{\longrightarrow} (h\cdot x, g)
  \;\;\;
  \in
  \;
  \Lambda(X \!\sslash\! G)
$$

of the inertia groupoid as follows (recall the nature of [[products of simplices]]):

<img src="https://ncatlab.org/nlab/files/BZActionOnInertiaGroupoid20210624.jpg" width="800" />

\end{example}

#### Relation to the fine model structure of equivariant homotopy theory

The [[identity functor]] gives a [[Quillen adjunction]] between the Borel model structure and the [[final model structure on topological G-spaces]] for [[equivariant homotopy theory]] ([Guillou 2006, section 5](#Guillou06)).

The left adjoint is

$$
  L = id
    \;\colon\;
  G_\bullet Act_{coarse}
    \longrightarrow
  G_\bullet Act_{fine}
$$

from the Borel model structure to the genuine [[equivariant homotopy theory]].

Because:

First of all, by ([Guillou 2006, theorem 3.12, example 4.2](#Guillou06)) $sSet^{\mathbf{B}G_\bullet}$ does carry a fine model structure. By ([Guillou 2006, last line of page 3](#Guillou06)) the fibrations and weak equivalences here are those maps which are ordinary fibrations and weak equivalences, respectively, on $H$-[[fixed point]] simplicial sets, for all subgroups $H$. This includes in particular the trivial subgroup and hence the identity functor

$$
  R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{coarse}
$$

is right Quillen.

#### Generalization to simplicial presheaves
 {#GeneralizationToSimplicialPresheaves}

Since the [[simplicial classifying space|universal simplicial principal complex]]-construction is [[functor|functorial]]

$$
  SimplicialGroups
  \xrightarrow{\;\; W \;\;}
  SimplicialSets
$$

with [[natural transformations]]

$$
  \mathcal{G}
  \xrightarrow{\;\; i \;\;}
  W\mathcal{G}
  \xrightarrow{\;\; p \;\;}
  \overline{W}\mathcal{G}
$$

the pair of [[adjoint functors]] (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace) extends to [[presheaves]]:

\begin{prop}
For $\mathcal{C}$ a [[small category|small]] [[sSet-category]] with

$$
  sPSh(\mathcal{C})
  \;\coloneqq\;
  sSetCat( \mathcal{C}^{op}, \, sSet )
$$

denoting its category of [[simplicial presheaves]], and for

$$
  \underline{\mathcal{G}}
  \;\in\;
  Groups
  \big(
    sPSh(\mathcal{C})
  \big)
$$

a [[group object]] [[internalization|internal to]] [[SimplicialPresheaves]] with

$$
  \underline{\mathcal{G}}
  Acts
  \big(
    sPSh(\mathcal{C})
  \big)
$$

denoting its category of [[action objects]] [[internalization|internal to]] [[SimplicialPresheaves]]

we have an [[adjoint functor|adjoint pair]]

$$
  \underline{\mathcal{G}}
  Acts
  \big(
    sPSh(\mathcal{C})
  \big)
  \underoverset
    {
      \underset{
        \big(
          (-) \times W\underline{\mathcal{G}}
        \big)
        \big/
        \underline{\mathcal{G}}
      }
      {\longrightarrow}}
    {
      \overset{
        (-)
          \times_{\overline{W}\underline{\mathcal{G}}}
        W\underline{\mathcal{G}}
      }{\longleftarrow}
    }
    {\bot}
    sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}}
$$

\end{prop}
\begin{proof}
The required [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) is the composite of the following sequence of [[natural bijections]]:

$$
  \begin{aligned}
    Hom
    \Big(
      (\underline{X},p),
      \,
      \big(
        \underline{Y} \times W\underline{\mathcal{G}}
      \big) / \underline{\mathcal{G}}
    \Big)
    &
    \;\simeq\;
    Hom
    \Big(
      \underline{X},
      \,
      \big(
        \underline{Y} \times W\underline{\mathcal{G}}
      \big) / \underline{\mathcal{G}}
    \Big)
    \underset{
    Hom
    \Big(
      \underline{X},
      \,
      \overline{W} \underline{\mathcal{G}}
    \Big)
    }{\times}
    \{p\}
    \\
    & \;\simeq\;
    \int^c
    Hom
    \Big(
      \underline{X}(c),
      \,
      \big(
        \underline{Y}(c) \times W\underline{\mathcal{G}(c)}
      \big) / \underline{\mathcal{G}}(c)
    \Big)
    \underset{
    \int^c
    Hom
    \Big(
      \underline{X}(c),
      \,
      \overline{W} \underline{\mathcal{G}}(c)
    \Big)
    }{\times}
    \{p\}
    \\
    & \;\simeq\;
    \int^c
    \left(
    Hom
    \Big(
      \underline{X}(c),
      \,
      \big(
        \underline{Y}(c) \times W\underline{\mathcal{G}}(c)
      \big) / \underline{\mathcal{G}}(c)
    \Big)
    \underset{
    Hom
    \Big(
      \underline{X}(c),
      \,
      \overline{W} \underline{\mathcal{G}}(c)
    \Big)
    }{\times}
    \{p(c)\}
    \right)
    \\
    & \;\simeq\;
    \int^c
    Hom_{/\overline{W}\underline{\mathcal{G}}(c)}
    \Big(
      \big( \underline{X}(c), p(c)\big),
      \,
      \big(
        \underline{Y}(c) \times \overline{W} \underline{\mathcal{G}}(c)
      \big)\big/ \mathcal{G}(c)
    \Big)
    \\
    & \;\simeq\;
    \int^c
    \left(
      \underline{\mathcal{G}}(c)
      Acts(sSet)
      \big(
        \underline{X}(c)
          \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times}
        W \underline{\mathcal{G}}(c),
        \,
        \underline{Y}(c)
      \big)
    \right)
    \\
    & \;\simeq\;
    \mathcal{G}Acts(sPSh(\mathcal{C}))
    \big(
      \underline{X}
        \underset{\overline{W}\underline{\mathcal{G}}}{\times}
      W \underline{\mathcal{G}},
      \,
      \underline{Y}
    \big)
  \end{aligned}
$$

Here:

* the first step is the characterization of hom-sets of a [[slice category]] as a [[fiber]] of the [[hom-sets]] of the underlying category;

* the second step is the description of the hom-set of a [[functor category]] as an [[end]] of object-wise hom-sets;

* the third step uses that [[ends]] are [[limits]] and [[limits commute with limits|hence commute]] with the [[fiber product]];

* the fourth step recognizes again, now object-wise, the hom-set in a [[slice category]];

* the fifth step is objectwise the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of (eq:QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace);

* the sixth step recognizes again the [[end]] as computing the hom-set in (a subcategory of) a functor category:

$$
  \array{
    \underline{\mathcal{G}}Acts
    \big(
      \underline{A}, \, \underline{B}
    \big)
    &\longrightarrow&
    \mathcal{G}(c_1)Acts
    \big(
       \underline{A}(c_1), \, \underline{B}(c_1)
    \big)
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    \mathcal{G}(c_2)Acts
    \big(
      \underline{A}(c_2), \, \underline{B}(c_2)
    \big)
    &\longrightarrow&
    Hom
    \big(
      \underline{A}(c_1), \, \underline{B}(c_2)
    \big)
  }
$$

\end{proof}

### In combinatorial simplicial model categories
 {#PropertiesInCombSimpModCats}

#### Base change

Given

* $\mathcal{G} \,\in\, Grp(sSet)$ a [[simplicial group]],

* $\mathbf{C}$ a [[combinatorial simplicial model category]] with

  * [[sSet]]-[[tensoring]] denoted

    $$
      sSet \times \mathbf{C}
        \overset{ (-) \cdot (-) }{\longrightarrow}
      \mathbf{C}
      \mathrlap{\,,}
    $$

  * [[classes]] of (acyclic) [[generating cofibrations]] denoted

    $I_{\mathbf{C}},J_{\mathbf{C}} \,\subset\, Mor(\mathbf{C})$,

then its Borel model category
(eq:BorelModelStructureForCombinatoriaSimplicialModelCat)
$$
  \mathcal{G}Act(\mathbf{C})_{Borel}
  \,\coloneqq\,
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})_{proj}
$$
has, by [this Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors), (acyclic) [[generating cofibrations]] given by [[tensoring]] these with $\mathcal{G} = (\mathbf{B}\mathcal{G})(\ast, \ast)$ and understood as equipped with the $\mathcal{G}$-[[action]] induced by the [[regular action]] of $\mathcal{G}$ on itself:
\[
  \label{GeneratingCofibrationsForBorelInCombSimpCat}
  \array{
  I_{\mathcal{G}Act(\mathbf{C})}
  &\coloneqq&
  \mathcal{G}\cdot(I_{\mathbf{C}})
  \;\equiv\;
  \big\{
    \mathcal{G} \cdot i
    \,\big\vert\,
    i \in I
  \big\}
  \\
  J_{\mathcal{G}Act(\mathbf{C})}
  &\coloneqq&
  \mathcal{G}\cdot(J_{\mathbf{C}})
  \;\equiv\;
  \big\{
    \mathcal{G} \cdot j
    \,\big\vert\,
    j \in J
  \big\}
  \mathrlap{\,,}
  }
\]
where we noticed that these are just the [[images]] under the (acyclic) [[generating cofibrations]] of $\mathbf{C}$ under the left-[[induced action]]
\[
  \label{LeftQuillenInductionForBorelInCombSimpModCat}
  \mathbf{C}
  \underoverset
    {\underset{undrl}{\longleftarrow}}
    {\overset{\mathcal{G}\cdot(-)}{\longrightarrow}}
    {\;\; \bot \;\;}
  \mathcal{G}Act(\mathbf{C})
  \,.
\]

But since

1. the [[underlying]] [[simplicial set]] of $\mathcal{G}$ is necessarily a [[cofibrant object]] of [[classical model structure on simplicial sets|$sSet_{Qu}$]]

1. $\mathbf{C}$ is an $sSet$-[[enriched model category]]

the [[pushout-product axiom]] satisfied by the [[tensoring]] implies that the [[underlying]] morphisms of these (acyclic) [[generating cofibration]] in the Borel structure, i.e. [[forgetful functor|forgetting]] their [[equivariant map|equivariance]] under the [[simplicial group action]], are still (acyclic) cofibrations in $\mathbf{C}$ itself, hence that we also have a [[Quillen adjunction]] of the form
\[
  \label{RightQuillenInductionForBorelInCombSimpModCat}
    \mathbf{C}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{undrl}{\longleftarrow}}
      {\;\;\; \bot_{{}_{Qu}} \;\;\;}
    \mathcal{G}Act(\mathbf{C})_{Borel}
\]
whose [[right adjoint]] may be thought of as the right [[induced action]] or equivalently as right [[Kan extension]] along the [[sSet]]-[[enriched functor]] $\ast \to \mathbf{B}\mathcal{G}$.

#### Monoidal model structure
 {#MonoidalModelStructure}

We discuss (Prop. \ref{ExistenceOfMonoidalModelStructure} below) induced [[monoidal model category]]-[[structure]] on the Borel model structure, in the generality of [[coefficients]] any (cofibrantly generated)  simplicial model structure (as [above](#DefinitionInCombSimpModCat)), which in addition carries compatible [[monoidal simplicial model structure]].

\begin{proposition}
\label{ExistenceOfMonoidalModelStructure}
  For $\mathcal{G} \in Grp(sSet)$ and $\mathbf{C}$ a [[combinatorial simplicial model category]] with compatible  [[closed monoidal category|closed]] [[monoidal enriched category]]-[[structure]], i.e. with an [[sSet]]-[[enriched functor]]  $(-)\otimes(-)$, the objectwise [[tensor product]]
$$
  \mathcal{G}Act(\mathbf{C})
  \times
  \mathcal{G}Act(\mathbf{C})
  \,\equiv\,
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})
  \times
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})
  \;\simeq\;
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C} \times \mathbf{C})
  \overset
    {
      sFunc(\mathbf{B}\mathcal{G},\, \otimes)
    }
    {\longrightarrow}
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})
$$
and the objectwise [[internal hom]]
$$
  \mathcal{G}Act(\mathbf{C})
  \times
  \mathcal{G}Act(\mathbf{C})
  \,\simeq\,
  sFunc\big(\mathbf{B}\mathcal{G},\,\mathbf{C}^{op})
  \times
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})
  \;\simeq\;
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C} \times \mathbf{C})
  \overset
    {
      sFunc(\mathbf{B}\mathcal{G},\, [-,-])
    }
    {\longrightarrow}
  sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})
$$
make the Borel model structure (eq:BorelModelStructureForCombinatoriaSimplicialModelCat)
$$
  \big(\mathcal{G}Act(\mathbf{C}), \otimes \big)
$$
a [[monoidal model category]], at least in that the [[pushout-product axiom]] holds.
A sufficient condition that also the [[unit axiom]] hold is that all objects of $\mathbf{C}$ are cofibrant.
\end{proposition}
The following argument for the pushout-product axiom follows [Berger & Moerdijk (2006), Lem. 2.5.2](#BergerMoerdijk2006).
\begin{proof}
It is sufficient to check the [[pushout-product axiom]] on (acyclic) [[generating cofibrations]] (by [this remark](pushout-product#PushoutProductAxiomInCofibrantlyGeneratedModelCategories)). To that end, given

* [[generating cofibrations]]$\;A \to B$ and $X \to Y$ in $I_{\mathcal{G}Act(\mathbf{C})}$, we need to check that their [[pushout product]] is still a cofibration, which means that for any

* [[acyclic fibration]]$\;Z \to W$ in $WFib_{\mathcal{G}Act(\mathbf{C})} = undrl^{-1} WFib_{\mathbf{C}}$

we need to find a [[lift]] in any [[commuting square]] of the form

$$
  \array{
    (A \otimes Y)
     \overset{A \otimes X}{\amalg}
    (B \otimes X)
    &\longrightarrow&
    Z
    \\
    \Big\downarrow
    &&
    \Big\downarrow
    \\
    B \otimes Y
     &\longrightarrow&
    W
    \mathrlap{\,.}
  }
  \;\;\;\text{in}\;\; \mathcal{G}Act(\mathbf{C})
  \,.
$$
By [[Joyal-Tierney calculus]], this is equivalent to finding a [[lift]] in the [[internal hom]]-[[adjunct]] diagram:
$$
  \array{
    A
      &\longrightarrow&
    [X,Z]
    \\
    \Big\downarrow && \Big\downarrow
    \\
    B
      &\longrightarrow&
    [Y,W]
      \underset{[X,W]}{\times}
    [X,Z]
  }
  \;\;\;\text{in}\;\; \mathcal{G}Act(\mathbf{C})
  \,.
$$
Now observe that the (acyclic) Borel-cofibration on the left is, by (eq:GeneratingCofibrationsForBorelInCombSimpCat), in the image of the [[left adjoint]] functor $\mathcal{G}\cdot(-)$ whose [[right adjoint]] is the [[forgetful functor]] remembering just the [[underlying]] morphisms in $\mathbf{C}$. Therefore a lift in the above diagram is equivalently a lift of its [[adjunct]], which is just the [[underlying]] diagram in $\mathbf{C}$:
$$
  \array{
    A
      &\longrightarrow&
    [Y,Z]
    \\
    \Big\downarrow && \Big\downarrow
    \\
    B
      &\longrightarrow&
    [Y,W]
      \underset{[X,W]}{\times}
    [X,Z]
  }
  \;\;\;\text{in}\;\; \mathbf{C}
  \,.
$$
But now using

1. that the underlying map of the previous [[acyclic fibration]] is still an acyclic fibration in $\mathbf{C}$, by definition of the projective/Borel model structure,

1. that the underlying maps of the previous [[cofibrations]] are still cofibrations in $\mathbf{C}$, by (eq:RightQuillenInductionForBorelInCombSimpModCat),

1. the [[pullback-power axiom]] satisfied in the [[monoidal model category]] $\mathbf{C}$

it follows that the left map in this diagram is a cofibration and the right map is still an acyclic fibration, whence a lift exists by the model category axioms on $\mathbf{C}$.

A directly analogous argument applies in the cases where $A \to B$ or $X \to Y$ are in addition weak equivalences. Hence the [[pushout-product axiom]] is verified.

Finally, to verify the [[unit axiom]]:

If $\mathbb{1} \,\in\, \mathbf{C}$ denotes the given [[tensor unit]], then the [[tensor unit]] in $\mathcal{G}Act(\mathbf{C})$ is the [[constant functor]] $\mathbf{B}\mathcal{G} \to \ast \overset{\mathbb{1}}{\longrightarrow} \mathbf{C}$, corresponding to the [[trivial action]] on $\mathbb{1}$. Writing $p_{const \mathbb{1}} \colon Q const(\mathbb{1}) \to const \mathbb{1}$ for a [[cofibrant resolution]], we need to see that given an [[object]] $X \in \mathcal{G}Act(\mathbf{C})$ the composite
$$
  (Q const \mathbb{1})
    \otimes
  X
  \overset{
    p_{const \mathbb{I}} \otimes id_X
  }{\longrightarrow}
  (const \mathbb{1}) \otimes X
  \underoverset{\sim}{l}{\longrightarrow}
  X
$$
is a [[weak equivalence]].

Since weak equivalence are just the underlying weak equivalences, for this it is sufficient (with the assumption that all objects in $\mathbf{C}$ are cofibrant), that
$(-) \otimes X$ is a [[left Quillen functor]] on $\mathbf{C}$, since as such it preserves all weak equivalences between cofibrant objects, by [Ken Brown's lemma](Introduction+to+Homotopy+Theory#KenBrownLemma).

But the [[underlying]] object of $X$ is still cofibrant in $\mathbf{C}$, by (eq:RightQuillenInductionForBorelInCombSimpModCat), therefore $(-) \otimes X$ is left Quillen by the [[pushout-product axiom]] satisfied by the [[monoidal model category]] $\mathbf{C}$.
\end{proof}

## Related concepts

* [[fine model structure on topological G-spaces]]

* [[Borel-equivariant rational homotopy theory]]

## Literature

### In simplicial sets
 {#ReferencesInSimplicialSets}

The model structure, the characterization of its cofibrations, and its equivalence to the [[slice model structure]] of $sSet$ over $\bar W G$ is due to

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

This Quillen equivalence also mentioned as:

* {#Dwyer2008} [[William Dwyer]], Exercise 4.2 in: _Homotopy theory of classifying spaces_, Lecture notes, Copenhagen 2008, ([pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf), [[Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf:file]])

Discussion in relation to the "fine" model structure of [[equivariant homotopy theory]] which appears in [[Elmendorf's theorem]] is in

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

Textbook account of (just) the Borel model structure:

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.2 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

Discussion with the model of [[∞-groups]] by [[simplicial groups]] replaced by groupal [[Segal spaces]] is in

* [[Matan Prasma]], _Segal Group Actions_ ([arXiv:1311.4749](http://arxiv.org/abs/1311.4749))

Discussion of [[monoidal model category]]-enhancement on the Borel model structure:

* {#BergerMoerdijk2006} [[Clemens Berger]], [[Ieke Moerdijk]], §2.5 in: *The Boardman-Vogt resolution of operads in monoidal model categories*, Topology **45** (2006) 807-849 [[arXiv:math/0502155](https://arxiv.org/abs/math/0502155), [doi:10.1016/j.top.2006.05.001](https://doi.org/10.1016/j.top.2006.05.001)]

Discussion of a the [[integral model structure]] for actions of all simplicial groups:

* [[Yonatan Harpaz]], [[Matan Prasma]], section 6.2 of _The Grothendieck construction for model categories_ ([arXiv:1404.1852](http://arxiv.org/abs/1404.1852))

### In topological spaces
 {#ReferencesInTopologicalSpaces}

* [Guillou 2006, Section 5](#Guillou06)

* [[Steffen Sagave]], [[Christian Schlichtkrull]], Prop. 6.4 in: *Diagram spaces and symmetric spectra*, Advances in Mathematics, 231 (2012), 2116-2193 ([arXiv:1103.2764](https://arxiv.org/abs/1103.2764), [doi:10.1016/j.aim.2012.07.013](https://doi.org/10.1016/j.aim.2012.07.013))

[[!redirects Borel model structures]]

[[!redirects model structure on simplicial group actions]]
[[!redirects model structures on simplicial group actions]]

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


Borel subgroup

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

#Contents#
* table of contents
{:toc}

## Idea

Given an [[algebraic group]] $G$ a [[subgroup]] $B\subset G$ is said to be a __Borel subgroup__ if it is maximal (under inclusion) among all Zariski closed connected [[solvable group|solvable]] subgroups.

## Properties

They appear to be minimal parabolic subgroups (that is minimal among those $P$ such that $G/P$ is a [[projective variety]]).

All the Borel subgroups are mutually conjugate and the intersection of any two contains a [[maximal torus]] in $G$.

If the characteristic of the ground field is zero then the tangent [[Lie algebra]] of the Borel subgroup $B$ is "the" Borel subalgebra of the Lie algebra of $G$.

## Example

The main example is $G = GL(n)$ or $G = SL(n)$ where a corresponding Borel subgroup can be taken to be the subgroup $B^+$ of the upper [[triangular matrices]] in $G$ and $B^-$, the subgroup of the lower triangular matrices.

These two subgroups are said to be mutually opposite in the sense that their intersection $B^+\cap B^-$ is precisely the maximal torus, which is in this case the subgroup of the diagonal matrices. If $G = SL(n)$ the quotient homogeneous spaces $SL(n)/B$ is called the [[flag variety]] and for a general semisimple Lie group $G$, $G/B$ is called a generalized flag
variety.

## Related concepts

* [[Weyl group]],

* [[parabolic subgroup]],

* [[Schubert calculus]]

## References

* A. Borel, _Groupes linéaires algébriques_, Ann. of Math. (2) , 64 : 1  (1956)  pp. 20–82
* [[Armand Borel]], _Linear algebraic groups_, Springer
* wikipedia [Borel subgroup](http://en.wikipedia.org/wiki/Borel_subgroup)
* eom: [Borel subgroup](http://eom.springer.de/B/b017160.htm)
[[!redirects Borel subgroups]]

[[!redirects Borel subgroups]]


Borel subset

# Contents
* table of contents
{: toc}

## Idea

Borel sets are certain [[subsets]] of a [[topological space]].  They form the Borel $\sigma$-[[sigma-algebra|algebra]] of the space, and they play an important role in [[measure theory]].

## Definition

Let $X$ be a [[topological space]].  Then there is a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{B}$ on $X$ generated by the [[open subsets]] of $X$. Elements of $\mathcal{B}$ are called the __Borel sets__ (or __Borel subsets__, or __Borel-measurable sets__, etc) of $X$, and $\mathcal{B}$ itself is called the __Borel $\sigma$-algebra__ on $X$.

## The Borel hierarchy

The preceding abstract definition can be made concrete (and [[predicative mathematics|predicative]], at least over $\omega_1$, although most predicative mathematicians don\'t accept $\omega_1$) as follows:

*  Start with the collection $\Sigma_0$ of open sets, and let $\Pi_0$ be the collection of the complements of the members of $\Sigma_0$ (the [[closed sets]]).
*  Let $\Sigma_1$ be the collection of countably infinitary unions of sets in $\Pi_0$ (the __$F_\sigma$-[[F-sigma subspace|sets]]__), and let $\Pi_1$ be the collection of their complements (the __$G_\delta$-[[G-delta subspace|sets]]__, the countably infinitary intersections of sets in $\Sigma_0$); even $\Sigma_1 \cup \Pi_1$ is not in general a $\sigma$-algebra.
*  Let $\Sigma_2$ be the collection of countably infinitary unions of sets in $\Pi_1$ (the __$G_{\delta\sigma}$-sets__), and let $\Pi_2$ be the collection of their complements (the __$F_{\sigma\delta}$-sets__, the countably infinitary intersections of sets in $\Sigma_1$).
*  Continue, defining $\Sigma_n$ for all [[natural numbers]] $n$.
*  Let $\Sigma_\omega$ be the union of the various $\Sigma_n$; although this is closed under complement, it is still not in general a $\sigma$-algebra.
*  Continue, defining $\Sigma_\alpha$ for all countable [[ordinal numbers]] $\alpha$.
*  Let $\Sigma_{\omega_1}$ be the union of the various $\Sigma_\alpha$; this is finally a $\sigma$-algebra.

So we need an [[uncountable set|uncountable]] number of steps, not just two.

This is only the beginning of [[descriptive set theory]]; our $\Sigma_\alpha$ are their $\Sigma^0_\alpha$ ---except that for some reason they start with $\Sigma^0_1$ instead of $\Sigma^0_0$---, and the subject continues to higher values of the superscript.

(To remember the other symbols, you need to know two languages: French and German.  The '$F$' comes from French 'fermé' for 'closed', while '$G$' is simply the next letter; the '$\sigma$' and '$\delta$' are from German 'Summe' for 'union' and 'Durchschnitt' for 'intersection' and are commonly used for countable such.)

Note that [[countable choice]] is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union.  But the full [[axiom of choice]] is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the [[axiom of determinacy]] instead.

## The Borel $\delta$-ring and $\sigma$-ring

Sometimes one does measure theory with $\sigma$-rings or even $\delta$-rings, which are more general than $\sigma$-algebras.  However, even the $\delta$-ring generated by a topology is in fact a $\sigma$-algebra.

Nevertheless, on a [[locally compact Hausdorff space]], we may define the __Borel $\delta$-ring__ and the __Borel $\sigma$-ring__ to be the $\delta$-ring or $\sigma$-ring generated by the [[compact space|compact]] subsets of $X$.  When $X$ is [[compact Hausdorff space|compact]] (such as the [[unit interval]]), then these both agree with the Borel $\sigma$-algebra; when $X$ is $\sigma$-compact (a countable union of compact subsets, such as the [[real line]]), then the Borel $\sigma$-ring still agrees with the Borel $\sigma$-algebra.

## Related concepts

* [[projective set]]

* [[Borel measure]]

[[!redirects Borel subset]]
[[!redirects Borel subsets]]
[[!redirects Borel set]]
[[!redirects Borel sets]]
[[!redirects Borel-measurable subset]]
[[!redirects Borel-measurable subsets]]
[[!redirects Borel-measurable set]]
[[!redirects Borel-measurable sets]]

[[!redirects Borel sigma-algebra]]
[[!redirects Borel sigma-algebras]]
[[!redirects Borel ∞-algebra]]
[[!redirects Borel ∞-algebras]]

[[!redirects Borel hierarchy]]
[[!redirects Borel hierarchies]]

[[!redirects Borel sigma-ring]]
[[!redirects Borel sigma-rings]]
[[!redirects Borel ∞-ring]]
[[!redirects Borel ∞-ring]]

[[!redirects Borel delta-ring]]
[[!redirects Borel delta-rings]]
[[!redirects Borel ∞-ring]]
[[!redirects Borel ∞-ring]]


Borel summability

Related $n$Lab entries include [[regular summation method]], [[resummation]], [[regularization]], [[renormalization]], [[Borel summability]], [[asymptotic series]], [[divergent series]]

* [[G. H. Hardy]], _Divergent series_, Clarendon Press, Oxford, 1949.
* wikipedia [Borel summation](http://en.wikipedia.org/wiki/Borel_summation)

category: analysis


Borel summation

#Contents#
* table of contents
{:toc}

## Idea

A [[resummation]] prescription.

## Related concepts

* [[perturbation theory]]

* [[resurgence theory]]

## References

* Wikipedia, _[Borel summation](http://en.wikipedia.org/wiki/Borel_summation)_

* {#CostinDunne17} Ovidiu Costin, [[Gerald Dunne]], _Convergence from Divergence_ ([arXiv:1705.09687](https://arxiv.org/abs/1705.09687))

[[!redirects Borel summations]]

[[!redirects Borel resummation]]


Borel's theorem

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

# Contents
* table of contents
{: toc}

## Idea

__Borel\'s Theorem__ (also called _Borel\'s Lemma_) says that every [[power series]] is the [[Taylor series]] of some [[smooth function]]. In other words: for every collection of prescribed [[partial derivatives]] at some point, there is a smooth function having these as its actual derivatives.

## Statements

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

The canonical map from the ring of germs of $C^\infty$ function at $0 \in \mathbb{R}^n$ to the ring of formal power series obtained by taking the Taylor series at $0$ is surjective.
=--

There are many extensions and variants.

For $\mathbb{R}^{n+m}$ a [[Cartesian space]] of [[dimension]] $n+m \in \mathbb{N}$, write $C^\infty(\mathbb{R}^{n+m})$ for the $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth function|smooth functions]] with values in $\mathbb{R}$.

Write $m^\infty_{\mathbb{R}^n \times \{0\}} \subset C^\infty(\mathbb{R}^{n+m})$ for the ideal of functions all whose [[partial derivatives]] along $\mathbb{R}^m$ vanish.

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

Forming the [[Taylor series]] constitutes an [[isomorphism]]

$$
  C^\infty(\mathbb{R}^{n+m})/m^\infty_{\mathbb{R}^n \times \{0\}}
  \stackrel{\simeq}{\longrightarrow}
  C^\infty(\mathbb{R}^n) [ [ Y_1, \cdots, Y_m] ]
$$

between smooth functions modulo those whose derivatives along $\mathbb{R}^m$ vanish and the ring of [[power series]] in $m$-variables over $C^\infty(\mathbb{R}^n)$.
=--

This appears for instance as ([Moerdijk-Reyes, theorem I.1.3](#MoerdijkReyes)).

## Proof

There is a full proof in Moerdijk--Reyes, cited above.  Here we prove Theorem \ref{basic} to indicate the method; the general version is not substantially different.  (This is based on the proof in [the English Wikipedia article](https://en.wikipedia.org/wiki/Borel%27s_lemma) at the time of writing, but with more details.)

+-- {: .proof}
###### Proof (of Theorem \ref{basic})

A real [[power series]] at $0$ is given simply by an [[infinite sequence]] $c = (c_n)_{n\geq{0}}$ of [[real numbers]].  Given such a sequence, we would ideally use
$$ f(x) = \sum_{n=0}^\infty c_n x^n ,$$
but this is only correct if the sum [[convergence|converges]] on at least some [[neighbourhood]] of $0$ (in other words if the power series has a positive [[radius of convergence]]).

To ensure this, let $\psi$ be a [[smooth function|smooth]] [[bump function]] chosen so that $\psi(x) = 1$ for ${|x|} \leq 1$ and $\psi(x) = 0$ for ${|x|} \geq 2$.  (For example, $\psi(x) = \frac{\phi(x + 2)} {\phi(x + 2) + \phi(-x - 1)} \frac{\phi(-x + 2)} {\phi(-x + 2) + \phi(x - 1)}$, where $\phi(x)$ is $\exp(-1/x)$ when $x \gt 0$ and otherwise vanishes.)  Next, choose an infinite sequence $H = (H_n)_{n\geq{1}}$ of positive finite numbers:
$$
  H_n
    =
  \max_{0\leq{k}\lt{n}}
  \,
  \max_{0\leq{i}\leq{k}}
  \,
  \root{n-k}{
    2^{2n-i}
    \,
    (k + 1)
    \,
    n^{\underline{k}}
    \,
    k^{\underline{i}}
    \,
    i!^{-1}
    \,
    {|c_n|}
    \,
    {\|\psi^{(k-i)}\|_\infty}
   }
  $$
(where $m^{\underline{j}}$ is the [[falling power]] $\prod_{0\leq{h}\lt{j}} (m - h)$, including the special case $m! = m^{\underline{m}}$); also, let $H_0 = 0$ (the [[bottom element]] of $[0,\infty]$, since $0 \leq k \lt 0$ never occurs).  Finally, define
$$ f(x) = \sum_{n=0}^\infty c_n \psi(H_n x) x^n .$$

If things go well, the [[derivatives]] of $f$ will be
$$ f^{(k)}(x) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(H_n x) x^{n-i} ,$$
and I claim that this is so.  Since $\psi^{(k-i)}(H_n x) = 0$ for ${|x|} \geq 2/H_n$, we have

$$
  \begin{aligned}
     {\|f^{(k)}\|_\infty}
     & \leq
    \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} H_n^{k-i} {\|\psi^{(k-i)}\|_\infty} (2/H_n)^{n-i}
     \\
     & \leq \sum_{n=k}^\infty (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} n^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{n-i}/H_n^{n-k}
     \\
     & \leq (k + 1) \max_{0\leq{i}\leq{k}} i!^{-1} k^{\underline{i}} k^{\underline{i}} {|c_n|} {\|\psi^{(k-i)}\|_\infty} 2^{k-i} + \sum_{n=k+1}^\infty 2^{-n}
  \end{aligned}
  \,,
$$
which is finite.  This proves [[uniform convergence]] of each claimed derivative (although not equiconvergence of all at once), and so each series not only converges but also may be antidifferentiated term by term, proving that $f^{(k)}$ is as claimed.

Finally (because $\psi^{(m)}(0) = 0$ for $m \gt 0$ but $\psi(0) = 1$, and the same goes for $0^m$),
$$ f^{(k)}(0) = \sum_{n=k}^\infty \sum_{i=0}^k i!^{-1} k^{\underline{i}} n^{\underline{i}} c_n H_n^{k-i} \psi^{(k-i)}(0) 0^{n-i} = k!^{-1} k! k! c_k H_k^0 \psi(0) 0^0 = k! c_k ,$$
making $c_k$ the $k$th coefficient of the Taylor series of $f$ at $0$, as desired.
=--

## Related theorems

* the [[Hadamard lemma]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

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

## Related concepts

* [[asymptotic expansion]]

## References

The original reference is

* [[Émile Borel]], _Sur quelques points de la théorie des fonctions_, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 12 (1895),
p. 9--55 [numdam](http://www.numdam.org/item?id=ASENS_1895_3_12__9_0)

It had been actually proved by [[Giuseppe Peano]] before Borel

* Ádám Besenyei, _Peano's unnoticed proof of Borel's theorem_ ([pdf](http://www.cs.elte.hu/~badam/publications/borel.pdf))

Textbook discussion includes

* {#MoerdijkReyes} [[Ieke Moerdijk]], [[Gonzalo Reyes]], chapter I of _[[Models for Smooth Infinitesimal Analysis]]_

A generalization to [[Banach spaces]] is in

* John C. Wells, _Differentiable functions on Banach spaces with Lipschitz derivatives_, J. Differential Geom. 8:1 (1973), 135-152 [euclid](http://projecteuclid.org/euclid.jdg/1214431488)

and is cited (along with extensive discussion and (counter)examples) also as (Ch.III) 15.4 in

* [[Andreas Kriegl]], [[Peter Michor]], _[[The Convenient Setting of Global Analysis]]_, Mathematical Surveys and Monographs, 53 AMS (1997) [pdf](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)

[[!redirects Borel's theorem]]
[[!redirects Borel\'s theorem]]
[[!redirects Borel/'s theorem]]
[[!redirects Borel's theorem]]
[[!redirects Borel's Theorem]]
[[!redirects Borel\'s Theorem]]
[[!redirects Borel/'s Theorem]]
[[!redirects Borel's Theorem]]

[[!redirects Borel theorem]]
[[!redirects Borel Theorem]]

[[!redirects Borel's theorem on power series]]

[[!redirects Borel's lemma]]
[[!redirects Borel\'s lemma]]
[[!redirects Borel/'s lemma]]
[[!redirects Borel's lemma]]
[[!redirects Borel's Lemma]]
[[!redirects Borel\'s Lemma]]
[[!redirects Borel/'s Lemma]]
[[!redirects Borel's Lemma]]

[[!redirects Borel lemma]]
[[!redirects Borel Lemma]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What may be called *Borel-equivariant rational homotopy theory* (this is not an established term, but it fits well) is the [[rational homotopy theory]] of [[G-spaces]] regarded in [[Borel model structure|Borel equivariant homotopy theory]], hence where [[weak equivalences]] are simply the [[underlying]] ([[forgetful functor|forgetting]] the [[group action]]) [[weak homotopy equivalences]]. This is in contrast to the [[fixed locus]]-wise weak hom. equivalences, as in [[fine model structure on topological G-spaces|proper equivariant homotopy theory]], which leads to [[equivariant rational homotopy theory|proper equivariant rational homotopy theory]].

Borel-equivariant rational homotopy theory is relevant even and particularly to the study of plain (non-equivariant) [[homotopy types]], as it provides the means to conceptualize their [[rationalization]]
beyond the constraint of these being [[nilpotent topological space|nilpotent]] or even [[simply connected topological space|simply connected]], hence outside the scope of plain [[rational homotopy theory]]:

Namely, given any [[connected topological space|connected]] space $X$ (but the same discussion applies to each [[connected component]] of a non-connected space), without any restriction on its [[fundamental group]], it fits into a [[homotopy fiber sequence]]

$$
  \array{
    \widehat X
    &\longrightarrow&
    X
    \\
    &&
    \big\downarrow
    \\
    &&
    B \pi_1(X)
  }
$$

with

* its [[universal cover]] $\widehat X$, which *is* [[simply connected topological space|simply connected]];

* the [[classifying space]] $B \pi_1(X)$ of its [[fundamental group]] $\pi_1(X)$.

Now, by the general discussion at *[[infinity-action|$\infty$-action]]* and at *[[Borel model structure]]* ([here](Borel+model+structure#RelationToSliceOverSimplicialClassifyingSpace)) -- this means that $X$ regarded in the [[slice (infinity,1)-category|slice]] over $B \pi$ is [[equivalence in an (infinity,1)-category|equivalent]] to the [[covering space]] $\widehat X$ *equipped* with its canonical [[group action]] by $\pi$.

This allows to regard the [[rationalization]] of non-[[nilpotent spaces]] $X$ as being the image of the [[universal covers]] $\widehat X$ in $\pi_1(X)$-eqjuivariant homotopy theory. (Since the $\pi_1(X)$-action on a universal covering space is [[free action|free]], this is necessarily Borel-equivariant.)

This perspective is really due to
[Brown & Szczarba 95](#BrownSzczarba95) (based on [Brown & Szczarba 90](#BrownSzczarba90), [Brown & Szczarba 93](#BrownSzczarba93)) -- to spot it there, look ([here](#BrownSzczarba95Pages)) for: 2nd par. on  p. 883 (16 of 48) and around Thm. 4.4 on p. 896 (29 of 48).

## Related concepts

* [[twisted equivariant Chern character]]

* [[equivariant de Rham complex]]

## References

The notion of fiberwise [[rationalization]] is due to

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], Section I.8 in: _Homotopy Limits, Completions and Localizations_, Lecture Notes in Mathematics Vol. 304, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4))

Rational homotopy theory (as well as [[real homotopy theory]]) for general connected spaces $X$ and regarded as Borel-equivariant rational homotopy theory of their [[covering spaces]] is laid out, and the corresponding [[fundamental theorem of dg-algebraic rational homotopy theory]] is proven, in:

* {#BrownSzczarba95} [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*, North Holland (1995) 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009))

  * {#BrownSzczarba95Pages} see p. 883 & 896 (16 & 29 of 48):  [[BrownSzczarba_BorelEquivariantRHT.pdf:file]]

based on

* [[Stephen Halperin]], [[Daniel Tanré]], *Homotopie filtrée et fibres $C^\infty$*, Illinois J. Math. 34(2): 284-324 (1990) ([doi:10.1215/ijm/1255988268](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-34/issue-2/Homotopie-filtr%c3%a9e-et-fibres-Cinfty/10.1215/ijm/1255988268.full))

* {#BrownSzczarba90} [[Edgar Brown]], [[Robert H. Szczarba]], _Continuous cohomology and Real homotopy type II_ Asterisque 191, Societe Mathematique De France (1990) ([numdam:AST_1990__191__45_0](http://www.numdam.org/book-part/AST_1990__191__45_0))

* {#BrownSzczarba93} [[Edgar Brown]], [[Robert H. Szczarba]], *Real and rational homotopy theory for spaces with arbitrary fundamental group*, Duke Mathematical Journal 71. (1993): 229-316 ([doi:10.1215/S0012-7094-93-07111-6](https://projecteuclid.org/journals/duke-mathematical-journal/volume-71/issue-1/Rational-and-real-homotopy-theory-with-arbitrary-fundamental-groups/10.1215/S0012-7094-93-07111-6.short))

with related discussion in

* {#GomezTatoHalperinTanre00} [[Antonio Gómez-Tato]], [[Stephen Halperin]], [[Daniel Tanré]], *Rational homotopy theory for non-simply connected spaces*, Trans. Amer. Math. Soc. 352 (2000), 1493-1525 ([doi:10.1090/S0002-9947-99-02463-0](https://doi.org/10.1090/S0002-9947-99-02463-0), [jstor:118074](https://www.jstor.org/stable/118074))

which is reviewed in:

* [[Sergei O. Ivanov]], Section 12 of: *An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups* [[arXiv:2111.10694](https://arxiv.org/abs/2111.10694)]

A textbook account is in:

* {#FelixHalperinThomas15} [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory II_, World Scientific 2015 ([doi:10.1142/9473](https://doi.org/10.1142/9473))

> (While [FHT 15](#FelixHalperinThomas15) does discuss, mostly in its Section 7, some aspects of non-simply connected spaces via the rational homotopy of their [[covering spaces]] with attention to $\pi_1$-actions, it does not seem to establish or mention aspects of a systematic generalization of the [[fundamental theorem of dg-algebraic rational homotopy theory]] to this situation -- nor does it cite any of the above.)

Further discussion of fiberwise rational homotopy theory:

* [[Manuel Rivera]], [[Felix Wierstra]], [[Mahmoud Zeinalian]], Thm. 16 of:  *Rational homotopy equivalences and singular chains*, Algebr. Geom. Topol. **21** (2021) 1535-1552 [[arXiv:1906.03655](https://arxiv.org/abs/1906.03655). [doi:10.2140/agt.2021.21.1535](https://doi.org/10.2140/agt.2021.21.1535)]


Borel-Moore homology

#Contents#
* table of contents
{:toc}

## Idea

What is called _Borel-Moore homology_ is a [[homology theory]] for suitably nice ([[locally compact topological spaces|locally compact]]) [[topological spaces]], which was introduced in terms of [[sheaf cohomology]] in ([Borel-Moore 60](#BorelMoore60)). Later it was observed that the homology groups obtained this way have the following equivalent description: They arise simply by generalizing in the ordinary definition of [[singular homology]] the finite [[formal linear combinations]] to _locally finite_ ones. Accordingly, one speaks also of _locally finite homology_ ([Spanier 93](#Spanier93)). A detailed account is in ([Hughes-Ranicki 96](#HughesRanicki96)).

A key property of Borel-Moore/locally finite homology is that on nice enough topological spaces $X$ it is [[natural isomorphism|naturally isomorphic]] to the [[ordinary homology]] of the [[one-point compactification]] of $X$.
It is in this way that the theory is often used in practice, as a means for computing homology of one-point compactifications.

The proof of this statement is indicated in ([Hughes-Ranicki 96](#HughesRanicki96)), it passes through comparison with [[Steenrod homology]] via results of ([Milnor 95](#Milnor95)).

## References

The original article is

* {#BorelMoore60} [[Armand Borel]], [[John Moore]], _Homology theory for locally compact spaces_,     Michigan Math. J. Volume 7, Issue 2 (1960), 137-159. ([Euclid](http://projecteuclid.org/euclid.mmj/1028998385), [pdf](http://projecteuclid.org/download/pdf_1/euclid.mmj/1028998385))

The formulation in terms of locally finite singular homology was highlighted in

* {#Spanier93} [[Edwin Spanier]], _Singular homology and cohomology with local coefficients and duality for manifolds_, Pacific J. Math. 1993

and the equivalence to [[ordinary homology]] of [[one-point compactifications]] is based on results in

* {#Milnor95} [[John Milnor]], _On the Steenrod homology theory_, Cambridge 1995

Textbook accounts include

* {#HughesRanicki96} Bruce Hughes, [[Andrew Ranicki]], _Ends of complexes_, Cambridge University Press 1996 ([pdf](http://www.maths.ed.ac.uk/~aar/books/ends.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/books/endint.pdf))

* [[Glen Bredon]], _Borel-Moore Homology_, chapter in _Sheaf Theory_, Volume 170 of the series Graduate Texts in Mathematics pp 279-416

See also

* Wikipedia, _[Borel-Moore homology](https://en.wikipedia.org/wiki/Borel%E2%80%93Moore_homology)_

* Joel Lemay, _Borel-Moore Homology_, 2012 ([pdf](http://mysite.science.uottawa.ca/jlema072/Borel-Moore.pdf))

* Martin Olsson, _Borel-Moore homology, Riemann-Roch transformations, and local terms_ ([pdf](https://math.berkeley.edu/~molsson/EtandRR.pdf))