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\section*{classifying space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma}

[[!include Yoneda lemma - contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_principal_bundles}{For principal bundles}\dotfill \pageref*{for_principal_bundles} \linebreak
\noindent\hyperlink{ForOrthogonalAndUnitaryPrincipalBundles}{For orthogonal and unitary principal bundles}\dotfill \pageref*{ForOrthogonalAndUnitaryPrincipalBundles} \linebreak
\noindent\hyperlink{idea_2}{Idea}\dotfill \pageref*{idea_2} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classification_of_bundles}{Classification of bundles}\dotfill \pageref*{classification_of_bundles} \linebreak
\noindent\hyperlink{for_symmetric_principal_bundles}{For symmetric principal bundles}\dotfill \pageref*{for_symmetric_principal_bundles} \linebreak
\noindent\hyperlink{for_crossed_complexes}{For crossed complexes}\dotfill \pageref*{for_crossed_complexes} \linebreak
\noindent\hyperlink{for_simplicial_groups}{For simplicial groups}\dotfill \pageref*{for_simplicial_groups} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{for_classical_lie_groups}{For classical Lie groups}\dotfill \pageref*{for_classical_lie_groups} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{moduli_spaces}{Moduli spaces}\dotfill \pageref*{moduli_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{classifying space} for some sort of data refers to a [[space]] (or a more general object), usually written $\mathcal{B}(data)$, such that maps $X\to \mathcal{B}(data)$ correspond to data over $X$.

The classical example is the classifying space $\mathcal{B}G$ of a [[group]] $G$, which has the property that there is a bijection between [[homotopy]] classes of maps $X\to \mathcal{B}G$ and isomorphism classes of $G$-[[bundle|bundles]] over $X$. (In fact, one can jack this up to an equivalence of [[groupoid|groupoids]] or $\infty$-[[infinity-groupoid|groupoids]].) Various improvements of this are possible which classify bundles with extra structure or [[fibration|fibrations]].

Categorically, the corresponding statement is that [[Grothendieck fibrations]] over a [[category]] $X$ correspond to [[pseudofunctors]] $X^{op}\to Cat$. Thus $Cat$ is the ``classifying space for categories.'' Similarly, discrete fibrations over $X$ correspond to [[functors]] $X^{op}\to Set$.

To see the connection between the two, consider the case when $X$ is a groupoid and we restrict the fibers of the fibration to be isomorphic to a given set $F$. Then the functor $X^{op}\to Set$ must land in the subcategory of $Set$ consisting of just the [[automorphisms]] of $F$, which is the one-object groupoid corresponding to the [[automorphism group]] $Aut(F)$. If we further restrict the automorphisms appearing to preserve some given structure on $F$, so that they lie in some smaller group $G$, then the ``classifying space'' will be the one-object groupoid corresponding to $G$. Under the [[homotopy hypothesis]], groupoids correspond to [[homotopy 1-types]], and the one-object groupoid of a group $G$ corresponds precisely to the usual topological classifying space $\mathcal{B}G$ (in fact, this is one \emph{construction} of $\mathcal{B}G$). For this reason, $\mathbf{B}G$ is often used to denote that one-object groupoid; see the [[delooping hypothesis]] and the discussion at [[category algebra]].

The phrase ``classifying space'' is also sometimes used for the realization of the nerve of any category, although it is more complicated to say what exactly this space ``classifies.'' (One answer is ``torsors modulo concordance.'')

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{for_principal_bundles}{}\subsubsection*{{For principal bundles}}\label{for_principal_bundles}

For $G$ a [[topological group]] there is a classifying space $B G \in$ [[Top]] for topological $G$-[[principal bundle]]s, hence a space such that for $X$ any sufficiently nice topological space there is a [[natural isomorphism]]

\begin{displaymath}
G Bund(X)_0 \simeq \pi_0 Top(X, B G)
\end{displaymath}
between the set of [[isomorphism]] classes of $G$-[[principal bundle]]s on $X$ and the set of [[homotopy]]-classes of [[continuous function]]s $X \to B G$.

This space may be constructed as follows:

write $\mathbf{B}G \in Top^{\Delta^{op}}$ for the [[simplicial topological space]] obtained as the [[nerve]] of the one-object [[topological groupoid]] associated to $G$, the simplicial space given by

\begin{displaymath}
(\mathbf{B}G)_n = G^{\times n}
\end{displaymath}
whose face maps are induced by the product operation on $G$ and whose degeneracy maps are induced from the unit map.

If $G$ is [[well-pointed simplicial topological group|well-pointed]], then the [[geometric realization of simplicial topological spaces]] of $G$ is a model for the [[homotopy type]] of the classifying space

\begin{displaymath}
B G \simeq \vert \mathbf{B}G\vert
  \,.
\end{displaymath}
For more details on this construction see the section  at [[geometric realization of simplicial topological spaces]].

As discussed there, too, this construction generalizes to more general [[simplicial topological group]]s and classifying spaces for their [[principal ∞-bundle]]s.

\hypertarget{ForOrthogonalAndUnitaryPrincipalBundles}{}\subsubsection*{{For orthogonal and unitary principal bundles}}\label{ForOrthogonalAndUnitaryPrincipalBundles}

\hypertarget{idea_2}{}\paragraph*{{Idea}}\label{idea_2}

For $G = O(n)$ the [[orthogonal group]] and $G = U(n)$ the [[unitary group]], there are standard realizations of the corresponding classifying spaces as [[direct limits]] of [[Grassmannian]] spaces.

Let $V_n(\mathbb{R}^q)$ be the [[Stiefel manifold]] of [[orthonormal basis|orthonormal]] $n$-frames in the [[Cartesian space]] $\mathbb{R}^q$. Its points are $n$-[[tuples]] of orthonormal [[vectors]] in $\mathbb{R}^q$, and it is topologized as a [[subspace topology|subspace]] of $(\mathbb{R}^q)^n$, or, equivalently, as a subspace of $(S^{q-1})^n$. It is a [[compact space|compact]] [[manifold]].

Let $G_n(\mathbb{R}^q)$ be the [[Grassmannian]] of $n$-planes in $\mathbb{R}^q$. Its points are the n-dimensional subspaces of $\mathbb{R}^q$. Sending an $n$-tuple of orthonormal vectors to the $n$-plane they span gives a surjective function $V_n(\mathbb{R}^q) \to G_n(\mathbb{R}^q)$, and we topologize $G_n(\mathbb{R}^q)$ as a [[quotient topology|quotient]] space of $V_n(\mathbb{R}^q)$. It too is a compact manifold.

The standard inclusion of $\mathbb{R}^q$ in $\mathbb{R}^{q+1}$ induces inclusions $V_n(\mathbb{R}^q) \hookrightarrow V_n(\mathbb{R}^{q+1})$ and $G_n(\mathbb{R}^q) \hookrightarrow G_n(\mathbb{R}^{q+1})$. We define $V_n(\mathbb{R}^\infty)$ and $G_n(\mathbb{R}^\infty)$ to be the unions of the $V_n(\mathbb{R}^q)$ and $G_n(\mathbb{R}^q)$, with the topology of the union.

Then $G_n(\mathbb{R}^\infty)$ is a model for the classifying space $B O(n)$.

\hypertarget{definitions}{}\paragraph*{{Definitions}}\label{definitions}

In the following we take [[Top]] to denote [[compactly generated topological spaces]]. For these the [[Cartesian product]] $X \times (-)$ is a [[left adjoint]] and hence preserves [[colimits]].

\begin{defn}
\label{StiefelManifold}\hypertarget{StiefelManifold}{}
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th \textbf{real [[Stiefel manifold]]} of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

\begin{displaymath}
V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n)
  \,,
\end{displaymath}
where the [[action]] of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.

Similarly the $n$th \textbf{complex Stiefel manifold} of $\mathbb{C}^k$ is

\begin{displaymath}
V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n)
  \,,
\end{displaymath}
here the [[action]] of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.

\end{defn}
\begin{defn}
\label{RealAndComplexGrassmannian}\hypertarget{RealAndComplexGrassmannian}{}
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th \textbf{real [[Grassmannian]]} of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

\begin{displaymath}
Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n))
  \,,
\end{displaymath}
where the [[action]] of the [[product group]] is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(n)$ into the [[orthogonal group]].

Similarly the $n$th \textbf{complex [[Grassmannian]]} of $\mathbb{C}^k$ is the [[coset]] [[topological space]].

\begin{displaymath}
Gr_n(\mathbb{C}^k) \coloneqq U(k)/(U(n) \times U(k-n))
  \,,
\end{displaymath}
where the [[action]] of the [[product group]] is via its canonical embedding $U(n)\times U(k-n) \hookrightarrow U(n)$ into the [[unitary group]].

\end{defn}
\begin{example}
\label{RealComplexProjectiveSpaceAsGrassmannian}\hypertarget{RealComplexProjectiveSpaceAsGrassmannian}{}
\begin{itemize}%
\item $Gr_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is [[real projective space]] of [[dimension]] $n$.


\item $Gr_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n$ is [[complex projective space]] of [[dimension]] $n$.



\end{itemize}
\end{example}
\begin{prop}
\label{ProjectionFromStiefelManifoldToGrassmannianIsFiberBundle}\hypertarget{ProjectionFromStiefelManifoldToGrassmannianIsFiberBundle}{}
For all $n \leq k \in \mathbb{N}$, the canonical [[projection]] from the real [[Stiefel manifold]] (def. \ref{StiefelManifold}) to the [[Grassmannian]] is a $O(n)$-[[principal bundle]]

\begin{displaymath}
\itexarray{
    O(n) &\hookrightarrow& V_n(\mathbb{R}^k)
    \\
    && \downarrow 
    \\
    && Gr_n(\mathbb{R}^k)
  }
\end{displaymath}
and the projection from the complex Stiefel manifold to the Grassmannian us a $U(n)$-[[principal bundle]]:

\begin{displaymath}
\itexarray{
    U(n) &\hookrightarrow& V_n(\mathbb{C}^k)
    \\
    && \downarrow 
    \\
    && Gr_n(\mathbb{C}^k)
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By (\href{coset#QuotientProjectionForCompactLieSubgroupIsPrincipal}{this cor.} and \href{coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}{this prop.}).

\end{proof}
\begin{defn}
\label{EOn}\hypertarget{EOn}{}
By def. \ref{RealAndComplexGrassmannian} there are canonical inclusions

\begin{displaymath}
Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})
\end{displaymath}
and

\begin{displaymath}
Gr_n(\mathbb{C}^k) \hookrightarrow Gr_n(\mathbb{C}^{k+1})
\end{displaymath}
for all $k \in \mathbb{N}$. The [[colimit]] (in [[Top]], see \href{Top#UniversalConstructions}{there}) over these inclusions is denoted

\begin{displaymath}
B O(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
\end{displaymath}
and

\begin{displaymath}
B U(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{C}^k)
  \,,
\end{displaymath}
respectively.

Moreover, by def. \ref{StiefelManifold} there are canonical inclusions

\begin{displaymath}
V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})
\end{displaymath}
and

\begin{displaymath}
V_n(\mathbb{C}^k) \hookrightarrow V_n(\mathbb{C}^{k+1})
  \,,
\end{displaymath}
respectively, that are compatible with the $O(n)$-[[action]] and the $U(n)$-action, respectively. The [[colimit]] (in [[Top]], see \href{Top#UniversalConstructions}{there}) over these inclusions, regarded as equipped with the induced [[action]], is denoted

\begin{displaymath}
E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)
\end{displaymath}
and

\begin{displaymath}
E U(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)
  \,,
\end{displaymath}
respectively. The inclusions are in fact compatible with the bundle structure from prop. \ref{ProjectionFromStiefelManifoldToGrassmannianIsFiberBundle}, so that there are induced projections

\begin{displaymath}
\left(
  \itexarray{
    E O(n)
    \\
    \downarrow
    \\
    B O(n) 
  }
  \right)
  \;\;
   \simeq
  \;\;
  \underset{\longrightarrow}{\lim}_k
  \left(
    \itexarray{
       V_n(\mathbb{R}^k)
       \\
       \downarrow
       \\
       Gr_n(\mathbb{R}^k)
    }
  \right)
\end{displaymath}
and

\begin{displaymath}
\left(
  \itexarray{
    E U(n)
    \\
    \downarrow
    \\
    B U(n) 
  }
  \right)
  \;\;
   \simeq
  \;\;
  \underset{\longrightarrow}{\lim}_k
  \left(
    \itexarray{
       V_n(\mathbb{C}^k)
       \\
       \downarrow
       \\
       Gr_n(\mathbb{C}^k)
    }
  \right)
  \,,
\end{displaymath}
respectively. These are the standard models for the \textbf{[[universal principal bundles]]} for $O$ and $U$, respectively. The corresponding [[associated bundles|associated]] [[vector bundles]]

\begin{displaymath}
E O(n) \underset{O(n)}{\times} \mathbb{R}^n
\end{displaymath}
and

\begin{displaymath}
E U(n) \underset{U(n)}{\times} \mathbb{C}^n
\end{displaymath}
are the corresponding \textbf{[[universal vector bundles]]}.

\end{defn}
Since the [[Cartesian product]] $O(n)\times (-)$ in [[compactly generated topological spaces]] preserves [[colimits]], it follows that the colimiting bundle is still an $O(n)$-[[principal bundle]]

\begin{displaymath}
\begin{aligned} 
    (E O(n))/O(n)
    &
    \simeq
    (\underset{\longrightarrow}{\lim}_k V_{n}(\mathbb{R}^k))/O(n)
    \\
    & \simeq 
    \underset{\longrightarrow}{\lim}_k (V_n(\mathbb{R}^k)/O(n))
    \\
    & \simeq
    \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
    \\
    & \simeq B O(n)
  \end{aligned}
\end{displaymath}
and anlogously for $E U(n)$.

As such this is the standard presentation for the $O(n)$-[[universal principal bundle]]. Its base space $B O(n)$ is the corresponding \emph{classifying space}.

\begin{defn}
\label{InclusionOfBOnIntoBOnPlusOne}\hypertarget{InclusionOfBOnIntoBOnPlusOne}{}
There are canonical inclusions

\begin{displaymath}
Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_{n+1}(\mathbb{R}^{k+1})
\end{displaymath}
and

\begin{displaymath}
Gr_n(\mathbb{C}^k)
  \hookrightarrow
  Gr_{n+1}(\mathbb{C}^{k+1})
\end{displaymath}
given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. \ref{EOn} these induce maps of classifying spaces

\begin{displaymath}
B O(n) \longrightarrow B O(n+1)
\end{displaymath}
and

\begin{displaymath}
B U(n) \longrightarrow B U(n+1)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{WhitneySumMapOnClassifyingSpaces}\hypertarget{WhitneySumMapOnClassifyingSpaces}{}
There are canonical maps

\begin{displaymath}
Gr_{n_1}(\mathbb{R}^{k_1})
  \times
  Gr_{n_2}(\mathbb{R}^{k_2})  
  \longrightarrow
  Gr_{n_1 + n_2}(\mathbb{R}^{k_1 + k_2})
\end{displaymath}
and

\begin{displaymath}
Gr_{n_1}(\mathbb{C}^{k_1})
  \times
  Gr_{n_2}(\mathbb{C}^{k_2})  
  \longrightarrow
  Gr_{n_1 + n_2}(\mathbb{C}^{k_1 + k_2})
\end{displaymath}
given by sending ambient spaces and subspaces to their [[direct sum]].

Under the colimit of def. \ref{EOn} these induce maps of classifying spaces

\begin{displaymath}
B O(n_1) \times B O(n_2)
  \longrightarrow
  B O(n_1 + n_2)
\end{displaymath}
and

\begin{displaymath}
B U(n_1) \times B U(n_2)
  \longrightarrow
  B U(n_1 + n_2)
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\paragraph*{{Properties}}\label{properties}

\begin{prop}
\label{CWComplexStructure}\hypertarget{CWComplexStructure}{}
The real [[Grassmannians]] $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. \ref{RealAndComplexGrassmannian} admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

\begin{displaymath}
Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_n(\mathbb{R}^{k+1})
\end{displaymath}
and

\begin{displaymath}
Gr_n(\mathbb{C}^k)
  \hookrightarrow
  Gr_n(\mathbb{C}^{k+1})
\end{displaymath}
are subcomplex incusions (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the [[classifying spaces]] $B O(n)$ and $B U(n)$ (def. \ref{EOn}).

\end{prop}
A proof is spelled out in (\hyperlink{Hatcher}{Hatcher, section 1.2 (pages 31-34)}).

\begin{prop}
\label{}\hypertarget{}{}
The [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ from def. \ref{StiefelManifold} admits the structure of a [[CW-complex]].

\end{prop}
e.g. (\href{Stiefel+manifold#James59}{James 59, p. 3}, \href{Stiefel+manifold#James76}{James 76, p. 5 with p. 21}, \href{Stiefel+manifold#Blaszczyk07}{Blaszczyk 07})

(And I suppose with that cell structure the inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions.)

\begin{prop}
\label{}\hypertarget{}{}
The [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ (def. \ref{StiefelManifold}) is [[n-connected topological space|(k-n-1)-connected]].

\end{prop}
\begin{proof}
Consider the [[coset]] [[quotient]] [[projection]]

\begin{displaymath}
O(k-n)
    \longrightarrow
  O(k)
    \longrightarrow
  O(k)/O(k-n) 
    = 
  V_n(\mathbb{R}^k)
  \,.
\end{displaymath}
Since the orthogonal groups is compact (\href{orthogonal+group#OrthogonalGroupIsCompact}{prop.}) and by \href{coset#QuotientProjectionForCompactLieSubgroupIsPrincipal}{this corollary} the projection $O(k)\to O(k)/O(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by \href{orthogonal+group#InclusionOfOnIntoOkIsnMinus1Equivalence}{this prop.} it has the following form in degrees bounded by $n$:

\begin{displaymath}
\cdots
    \to
  \pi_{\bullet \leq k-n-1}(O(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(O(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k-n))
    \to
  \cdots
  \,.
\end{displaymath}
This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

\end{proof}
Similarly:

\begin{prop}
\label{}\hypertarget{}{}
The complex [[Stiefel manifold]] $V_n(\mathbb{C}^k)$ (def. \ref{StiefelManifold}) is [[n-connected topological space|2(k-n)-connected]].

\end{prop}
\begin{proof}
Consider the [[coset]] [[quotient]] [[projection]]

\begin{displaymath}
U(k-n)
    \longrightarrow
  U(k)
    \longrightarrow
  U(k)/U(k-n) 
    = 
  V_n(\mathbb{C}^k)
  \,.
\end{displaymath}
By prop. \ref{UnitaryGroupIsCompact} and by \href{coset#QuotientProjectionForCompactLieSubgroupIsPrincipal}{this corollary} the projection $U(k)\to U(k)/U(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:

\begin{displaymath}
\cdots
    \to
  \pi_{\bullet \leq 2(k-n)}(U(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(U(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k-n))
    \to
  \cdots
  \,.
\end{displaymath}
This implies the claim.

\end{proof}
\begin{cor}
\label{EOnIsWeaklyContractible}\hypertarget{EOnIsWeaklyContractible}{}
The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. \ref{EOn} is [[weakly contractible topological space|weakly contractible]].

The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. \ref{EOn} is [[weakly contractible topological space|weakly contractible]].

\end{cor}
\begin{prop}
\label{HomotopyGroupsOfBOnThoseOfOnShifted}\hypertarget{HomotopyGroupsOfBOnThoseOfOnShifted}{}
The [[homotopy groups]] of the classifying spaces $B O(n)$ and $B U(n)$ (def. \ref{EOn}) are those of the [[orthogonal group]] $O(n)$ and of the [[unitary group]] $U(n)$, respectively, shifted up in degree: there are [[isomorphisms]]

\begin{displaymath}
\pi_{\bullet+1}(B O(n))
    \simeq
  \pi_\bullet O(n)
\end{displaymath}
and

\begin{displaymath}
\pi_{\bullet+1}(B U(n))
    \simeq
  \pi_\bullet U(n)
\end{displaymath}
(for homotopy groups based at the canonical basepoint).

\end{prop}
\begin{proof}
Consider the sequence

\begin{displaymath}
O(n)
    \longrightarrow
  E O(n)
    \longrightarrow
  B O(n)
\end{displaymath}
from def. \ref{EOn}, with $O(n)$ the [[fiber]]. Since (by \href{coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}{this prop.}) the second map is a [[Serre fibration]], this is a [[fiber sequence]] and so it induces a [[long exact sequence of homotopy groups]] of the form

\begin{displaymath}
\cdots
    \to
  \pi_\bullet(O(n))
    \longrightarrow
  \pi_\bullet(E O(n))
    \longrightarrow
  \pi_\bullet(B O(n))
    \longrightarrow
  \pi_{\bullet-1}(O (n))
   \longrightarrow
  \pi_{\bullet-1}(E O(n))
    \to
  \cdots
  \,.
\end{displaymath}
Since by cor. \ref{EOnIsWeaklyContractible} $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that

\begin{displaymath}
\pi_\bullet(B O(n))
   \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1}(O (n))
\end{displaymath}
is an isomorphism.

The same kind of argument applies to the complex case.

\end{proof}
\begin{prop}
\label{SphereFibrationOverInclusionOfClassifyingSpaces}\hypertarget{SphereFibrationOverInclusionOfClassifyingSpaces}{}
For $n \in \mathbb{N}$ there are [[homotopy fiber sequences]]

\begin{displaymath}
S^n
    \longrightarrow 
  B O(n)
    \longrightarrow
  B O(n+1)
\end{displaymath}
and

\begin{displaymath}
S^{2n+1}
    \longrightarrow 
  B U(n)
    \longrightarrow
  B U(n+1)
  \,,
\end{displaymath}
exhibiting the [[n-sphere]] ($(2n+1)$-sphere) as the [[homotopy fiber]] of the canonical maps from def. \ref{InclusionOfBOnIntoBOnPlusOne}.

This means that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. \ref{EOn}) by a [[Serre fibration]]

\begin{displaymath}
\itexarray{
     B O(n) &\hookrightarrow& B O(n+1)
     \\
     {}^{\mathllap{{weak \, homotopy} \atop equivalence}}\downarrow  & \nearrow_{\mathrlap{Serre \, fib.}}
     \\
     \tilde B O(n)
  }
\end{displaymath}
such that $S^n$ is the ordinary [[fiber]] of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.

\end{prop}
\begin{proof}
Take $\tilde B O(n) \coloneqq (E O(n+1))/O(n)$.

To see that the canonical map $B O(n)\longrightarrow (E O(n+1))/O(n)$ is a [[weak homotopy equivalence]] consider the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    O(n) &\overset{id}{\longrightarrow}& O(n)
    \\
    \downarrow && \downarrow
    \\
    E O(n) &\longrightarrow& E O(n+1)
    \\
    \downarrow && \downarrow
    \\
    B O(n) &\longrightarrow& (E O(n+1))/O(n)
  }
  \,.
\end{displaymath}
By \href{coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}{this prop.} both bottom vertical maps are [[Serre fibrations]] and so both vertical sequences are [[fiber sequences]]. By prop. \ref{HomotopyGroupsOfBOnThoseOfOnShifted} part of the induced morphisms of [[long exact sequences of homotopy groups]] looks like this

\begin{displaymath}
\itexarray{
    \pi_\bullet(B O(n))
    &\overset{}{\longrightarrow}&
    \pi_\bullet( (E O(n+1))/O(n) )
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    \pi_{\bullet-1}(O(n))
    &\overset{=}{\longrightarrow}&
    \pi_{\bullet-1}(O(n))
  }
  \,,
\end{displaymath}
where the vertical and the bottom morphism are isomorphisms. Hence also the to morphisms is an isomorphism.

That $B O(n)\to \tilde B O(n+1)$ is indeed a [[Serre fibration]] follows again with \href{coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup}{this prop.}, which gives the [[fiber sequence]]

\begin{displaymath}
O(n+1)/O(n)
    \longrightarrow
  (E O(n+1))/O(n)
    \longrightarrow
  (E O(n+1))/O(n+1)
  \,.
\end{displaymath}
The claim in then follows since (\href{coset#nSphereAsCosetSpace}{this exmpl.})

\begin{displaymath}
O(n+1)/O(n) \simeq S^n
  \,.
\end{displaymath}
The argument for the complex case is of the same form, concluding now with the identification (\href{unitary+group#nSphereAsUnitaryCosetSpace}{this exmpl.})

\begin{displaymath}
U(n+1)/U(n) \simeq S^{2n+1}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{classification_of_bundles}{}\paragraph*{{Classification of bundles}}\label{classification_of_bundles}

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[paracompact topological space]], the operation of [[pullback]] of the [[universal principal bundle]] $E O(n) \to B O(n)$ from def. \ref{EOn} along [[continuous functions]] $f \colon X \to B O(n)$ eastblishes a [[bijection]]

\begin{displaymath}
[X, B O(n)]
   \underoverset{iso}{f \mapsto f^\ast E O(n)}{\longrightarrow}
  O(n) Bund/_\sim
\end{displaymath}
between [[homotopy classes]] of functions from $X$ to $B O(n)$ and isomorphism classes of $O(n)$-[[principal bundles]] on $X$.

\end{prop}
A full proof is spelled out in (\hyperlink{Hatcher}{Hatcher, section 1.2, theorem 1.16})

\hypertarget{for_symmetric_principal_bundles}{}\subsubsection*{{For symmetric principal bundles}}\label{for_symmetric_principal_bundles}

\begin{itemize}%
\item the \emph{unordered} [[Fadell's configuration space]] of $n$ points in $\mathbb{R}^\infty$ is a model for the classifying space $B \Sigma(n)$ of the [[symmetric group]] $\Sigma(n)$;


\item the \emph{ordered} configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-[[action]], is a model for the $\Sigma(n)$-[[universal principal bundle]].



\end{itemize}
\hypertarget{for_crossed_complexes}{}\subsubsection*{{For crossed complexes}}\label{for_crossed_complexes}

We discuss here classifying spaces of [[crossed complex]]es.

The notion of classifying space should be regarded in general terms as giving a functor

\begin{displaymath}
\mathcal{B} :(algebraic data) \to (topological data).
\end{displaymath}
Composition with a forgetful functor $U: (topological data) \to (topological spaces)$ gives a [[classifying space]]. In such cases one would also like a homotopically defined functor

\begin{displaymath}
\Xi: (topological data) \to (algebraic data)
\end{displaymath}
such that

\begin{enumerate}%
\item $\Xi \circ \mathcal{B}$ is equivalent to the identity;


\item $\Xi$ preserves certain colimits (Generalised [[van Kampen theorem]]) allowing some calculation;


\item there are notions of homotopy for both types of data leading to a bijection of homotopy classes for some $X$



\end{enumerate}
\begin{displaymath}
[X,U\mathcal{B}C] \cong [\Xi X_*, C].
\end{displaymath}
This happens for the algebraic data of crossed complexes and the topological data of filtered spaces, when $X$ is a CW-complex, and $\Xi$ is the fundamental crossed complex of a filtered space. Thus in this case the classifying space does classify homotopy classes of maps, and more work is needed to sort out the data over $X$ which this classifies (gerbes?).

However $\mathcal{B}C$ is in this case defined by a nerve construction which generalises that for groupoids, and can also be applied to topological crossed crossed complexes, giving a simplicial space.

[[Mike Shulman|Mike]]: I don't really get any intuition from that. There might be lots of functors from ``algebraic data'' to ``topological data'' but it seems to me that only particular sorts of them deserve the name ``classifying space.'' Can you say more specifically what sorts of functors you have in mind, and relate it to the more basic ideas that I am familiar with? What do these classifying spaces classify?

[[Ronnie Brown|Ronnie]] What I am trying to characterise is that higher categories carry structure such as a filtration by lower dimensional higher categories, or, for multiple structures, a multiple filtration. Thus one expects a classifying space to inherit this extra structure. Conversely, the construction of an infinity-groupoid from a space might depend on this extra structure.

So I spent 9 years trying to construct a \emph{strict} homotopy double groupoid of a space, yet Philip Higgins and I did this overnight in 1974 when we tried the simplest relative example we could think of: take homotopy classes of maps from a square to $X$ which take the edges to a subspace $X_1$ and the vertices to a base point $x_0$. Then the filtered case took another 4 years or so to complete.

Then Loday constructed a [[cat-n-group]] from an n-cube of spaces, published in 1982. Its multi-nerve is an $(n+1)$-simplicial set, whose realisation is $(n+1)$-filtered.

A strict homotopy double groupoid of a Hausdorff space has been constructed but this needs a subtle notion of [[thin element|thin]] homotopy.

Of course the filtration for a group is not so apparent, but it is more clear that [[groupoids]] carry structure in dimension 0 and 1, and hence are useful for representing non connected homotopy 1-types, and their identifications in dimension 0, as explained in the first edition (1968) of my Topology book.

The intuition for the [[higher homotopy van Kampen theorem]] is that you need \emph{structure in all dimensions from 0 to n}to get colimit theorems in dimension n, because in homotopy, low dimensional identifications, even in dimension 0, usually effect high dimensional homotopy information. In effect, the [[higher homotopy van Kampen theorem]] is about gluing [[homotopy n-types]].

\emph{Mike}: Thanks, that is helpful.

Some such constructions arise from generalisations of the [[Dold-Kan correspondence]], with values in [[simplicial sets]]. For example, from a [[crossed complex]] $C$ one obtains a simplicial set $Nerve(C)$ which in dimension $n$ is $Crs(\Pi(\Delta^n_*),C)$. The geometric realisation $\mathcal{B}C$ of this is canonically filtered by the skeleta of $C$, so $\mathcal{B}$ is really a functor to [[filtered spaces]]. This ties in with the functor $\Pi$ which goes in the opposite direction. But note that there is a different filtration of the space $\mathcal{B}C$ since it is a CW-complex, and so $\Pi$ of this filtration gives a free crossed complex.

Special cases of crossed complexes are [[groupoids]], and so we get the classifying space of a groupoid; and similarly of a [[crossed module]].

A crossed module is equivalent to a category object in groups, and so a nerve of this can be constructed as a bisimplicial set. The geometric realisation of this is naturally bifiltered, in several ways!

In considering what is desirable for a [[fundamental infinity-groupoid]] one should bring the notion of classifying space, and its inherited structure, into account.

\hypertarget{for_simplicial_groups}{}\subsubsection*{{For simplicial groups}}\label{for_simplicial_groups}

The $\bar W(-)$-construction (see [[simplicial group]] and [[groupoid object in an (∞,1)-category]]) which gives the classifying space functor for [[simplicial group]]s and simplicially enriched groupoids is given in the entry on [[simplicial group|simplicial groups]]. It provides a good example of the above as the W-bar functor is right adjoint to the [[Dwyer-Kan loop groupoid]] functor and induces an equivalence of homotopy categories between that of simplicial sets and that of simplicially enriched groupoids. The simplicial sets here are playing the role of `topological data'.

\hypertarget{properties_2}{}\subsection*{{Properties}}\label{properties_2}

\hypertarget{for_classical_lie_groups}{}\subsubsection*{{For classical Lie groups}}\label{for_classical_lie_groups}

Let $O(n)$ be the [[orthogonal group]] and $U(n)$ the [[unitary group]] in real/complex [[dimension]] $n$, respectively.

\begin{prop}
\label{CWComplexStructure}\hypertarget{CWComplexStructure}{}
The real [[Grassmannians]] $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

\begin{displaymath}
Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_n(\mathbb{R}^{k+1})
\end{displaymath}
are subcomplex incusion (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the classifying space

\begin{displaymath}
B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
  \,.
\end{displaymath}
\end{prop}
A proof is spelled out in (\hyperlink{Hatcher}{Hatcher, section 1.2 (pages 31-34)}).

\begin{prop}
\label{}\hypertarget{}{}
The classifying spaces $B O(n)$ are [[paracompact spaces]].

\end{prop}
An early source of this statement is (\hyperlink{CartanSchwartz63}{Cartan-Schwartz 63, expos\'e{} 5}). It follows for instance by prop. \ref{CWComplexStructure} the fact that every [[CW-complex]] is paracompact.

\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

[[!include Segal completion -- table]]

\hypertarget{moduli_spaces}{}\subsection*{{Moduli spaces}}\label{moduli_spaces}

The notion of \textbf{[[moduli space]]} is closely related to that of classifying space, but has some subtle differences. See there for more on this.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{classifying space}, [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]], [[Kan-Thurston theorem]]

\begin{itemize}%
\item [[Milnor classifying space]]


\item [[numerable bundle]]


\item [[acyclic group]]



\end{itemize}

\item [[classifying topos]]


\item [[universal principal bundle]], [[universal principal ∞-bundle]]


\item [[subobject classifier]]


\item [[classifying morphism]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Original accounts include

\begin{itemize}%
\item [[Henri Cartan]], [[Laurent Schwartz]], \emph{Le th\'e{}or\'e{}me d'Atiyah-Singer} S\'e{}minaire 1963/1964. New York: Benjamin 1967.

\end{itemize}
Textbook accounts on classifying spaces for [[vector bundles]] include

\begin{itemize}%
\item [[Stanley Kochmann]], section 1.3 of of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Peter May]], chapter 23 of \emph{A concise course of algebraic topology} (\href{http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf}{pdf})


\item [[Allen Hatcher]], section 1.2 of \emph{Vector bundles and K-theory} (\href{https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})


\item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf})



\end{itemize}
A discussion more from the point of view of [[topos theory]] is in

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Classifying spaces and classifying topoi}, Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995.

\end{itemize}
Discussion of [[universal principal bundles]] over their classifying spaces is in

\begin{itemize}%
\item Stephen Mitchell, \emph{Notes on principal bundles and classifying spaces}, Lecture Notes. University of Washington, 2011 (\href{https://sites.math.washington.edu/~mitchell/Notes/prin.pdf}{pdf})

\end{itemize}
Discussion of characterization of principal bundles by rational [[universal characteristic classes]] and torsion information is in the appendices of

\begin{itemize}%
\item Igor Belegradek, Vitali Kapovitch, \emph{Obstructions to nonnegative curvature and rational homotopy theory} (\href{http://arxiv.org/abs/math/0007007}{arXiv:math/0007007})


\item Igor Belegradek, \emph{Pinching, Pontrjagin classes, and negatively curved vector bundles} (\href{http://arxiv.org/abs/math/0001132}{arXiv:math/0001132})



\end{itemize}
Discussion of classifying spaces in the context of [[measure theory]] is in

\begin{itemize}%
\item Ivan Marin, \emph{Measure theory and classifying spaces} (\href{https://arxiv.org/abs/1702.01889}{arXiv:1702.01889})

\end{itemize}
[[!redirects classifying spaces]]



\end{document}