nLab classifying space

Idea

A classifying space for some sort of data is (the homotopy type of) a topological space $A$, such that homotopy classes of maps $X \to A$ correspond to equivalence classes of that kind of data “parameterized” over suitable base spaces $X$.

Beware the similarity with but distinction to moduli stacks $\mathbf{A}$, which are such that not just the equivalence classes match, but that the full $\infty$-groupoid of (structured) maps $X \to \mathbf{A}$ is equivalent to the $\infty$-groupoid of the given kind of data on a suitable stack $X$. Hence a classifying space is “coarser” than a moduli stack — which sounds like a disadvantage but is often an advantage in applications: classifying spaces serve to reduce information to homotopy classes of maps.

(Moduli stacks, in turn – with stacks understood in the directed generality of 2-sheaves etc. — subsume yet more general kinds of “universes”. For instance the Grothendieck construction in category theory may be understood as exhibiting Cat as the moduli object for categories. Again, the point of classifying spaces over moduli stacks is to be more coarse, and hence more informative concerning equivalence classes.)

The classical examples of classifying spaces (both historically and by prevalence of their applications) are traditionally denoted $B G$ and constructed as bar constructions or topological realizations of nerves of suitable topological groups $G$. Under mild conditions these have the property that they classify $G$-principal bundles (hence the “data” above being: $G$-torsors), in that there is bijection between homotopy classes of maps $X\to BG$ and isomorphism classes of $G$-principal bundles over suitable base spaces $X$.

Often this case is understood by default when referring to “classifying spaces”.

But every connected homotopy type is weak equivalently $B G$ for some topological group and other objects than just ordinary principal bundles may be classified by these spaces.

For example, Eilenberg-MacLane spaces $K(G,n)$ have the property that homotopy classes of maps into them are in bijection with cohomology groups $H^n(-;G)$ ($G$ a discrete group and an abelian group for $n \geq 2$), hence serve as classifying spaces for ordinary cohomology.

For $n = 1$ this situation overlaps with the previous one classifying principal bundles. For higher $n$ and with due care, one may think of $K(G,n)$ as classifying certain principal $\infty$-bundles (“bundle gerbes”, “bundle 2-gerbes”, etc.).

In joint generalization of these two situations, there are classifying spaces for $G$-principal 2-bundles for non-abelian topological 2-groups $G$, etc.

In fact, one may understand any connected space $A$ as being the classifying space for generalized nonabelian cohomology $H^1(-;\Omega A)$ with coefficients in the loop $\infty$-group $\Omega A$.

Traditionally familiar (in algebraic topology and stable homotopy theory) is the case of classifying spaces for generalized abelian cohomology theories (Whitehead-generalized cohomology theories): The Brown representability theorem says that their classifying spaces are infinite-loop spaces which, as the degree $n$ ranges, organize into sequences of spaces called spectra.

Examples

For principal bundles

A good account is in Rudolf & Schmidt 17, Sec. 3, esp. Thm. 3.5.1 & Prop. 3.6.2.

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

$$G Bund(X)_0 \simeq \pi_0 Top(X, B G)$$

between the set of isomorphism classes of $G$-principal bundles on $X$ and the set of homotopy-classes of continuous functions $X \to B G$.

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

$$(\mathbf{B}G)_n = G^{\times n}$$

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, then the geometric realization of simplicial topological spaces of $G$ is a model for the homotopy type of the classifying space

$$B G \simeq \vert \mathbf{B}G\vert \,.$$

For more details on this construction see the section classifying spaces at geometric realization of simplicial topological spaces.

As discussed there, too, this construction generalizes to more general simplicial topological groups and classifying spaces for their principal ∞-bundles.

Theorem

If $G \,\in\, Grp(HausSp)$ is a Hausdorff topological group, then the Milnor join construction classifies topological $G$-principal bundles over all paracompact Hausdorff spaces.

(due to Milnor, laid out in Rudolf & Schmidt 17, Thm. 3.5.1)

For orthogonal and unitary principal bundles

Idea

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 $n$-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 of $(\mathbb{R}^q)^n$, or, equivalently, as a subspace of $(S^{q-1})^n$. It is a 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 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 $B O(n)$.

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.

Definition

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Stiefel manifold of $\mathbb{R}^k$ is the coset topological space.

$$V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n) \,,$$

where the action of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.

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

$$V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n) \,,$$

here the action of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.

Definition

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Grassmannian of $\mathbb{R}^k$ is the coset topological space.

$$Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n)) \,,$$

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 complex Grassmannian of $\mathbb{C}^k$ is the coset topological space.

$$Gr_n(\mathbb{C}^k) \coloneqq U(k)/(U(n) \times U(k-n)) \,,$$

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.

Example

• $Gr_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is real projective space of dimension $n$.

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

Proposition

For all $n \leq k \in \mathbb{N}$, the canonical coprojection from the real Stiefel manifold (def. ) to the Grassmannian is a $O(n)$-principal bundle

$$\array{ O(n) &\hookrightarrow& V_n(\mathbb{R}^k) \\ && \big\downarrow \\ && Gr_n(\mathbb{R}^k) }$$

and the projection from the complex Stiefel manifold to the Grassmannian is a $U(n)$-principal bundle:

$$\array{ U(n) &\hookrightarrow& V_n(\mathbb{C}^k) \\ && \downarrow \\ && Gr_n(\mathbb{C}^k) } \,.$$

Proof

By this cor. and this prop..

Definition

By def. there are canonical inclusions

$$Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})$$

and

$$Gr_n(\mathbb{C}^k) \hookrightarrow Gr_n(\mathbb{C}^{k+1})$$

for all $k \in \mathbb{N}$. The colimit (in Top, see there) over these inclusions is denoted

$$B O(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)$$

and

$$B U(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{C}^k) \,,$$

respectively.

Moreover, by def. there are canonical inclusions

$$V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$$

and

$$V_n(\mathbb{C}^k) \hookrightarrow V_n(\mathbb{C}^{k+1}) \,,$$

respectively, that are compatible with the $O(n)$-action and the $U(n)$-action, respectively. The colimit (in Top, see there) over these inclusions, regarded as equipped with the induced action, is denoted

$$E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$$

and

$$E U(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k) \,,$$

respectively. The inclusions are in fact compatible with the bundle structure from prop. , so that there are induced projections

$$\left( \array{ E O(n) \\ \downarrow \\ B O(n) } \right) \;\; \simeq \;\; \underset{\longrightarrow}{\lim}_k \left( \array{ V_n(\mathbb{R}^k) \\ \downarrow \\ Gr_n(\mathbb{R}^k) } \right)$$

and

$$\left( \array{ E U(n) \\ \downarrow \\ B U(n) } \right) \;\; \simeq \;\; \underset{\longrightarrow}{\lim}_k \left( \array{ V_n(\mathbb{C}^k) \\ \downarrow \\ Gr_n(\mathbb{C}^k) } \right) \,,$$

respectively. These are the standard models for the universal principal bundles for $O$ and $U$, respectively. The corresponding associated vector bundles

$$E O(n) \underset{O(n)}{\times} \mathbb{R}^n$$

and

$$E U(n) \underset{U(n)}{\times} \mathbb{C}^n$$

are the corresponding universal vector bundles.

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

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

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 classifying space.

Definition

There are canonical inclusions

$$Gr_n(\mathbb{R}^k) \hookrightarrow Gr_{n+1}(\mathbb{R}^{k+1})$$

and

$$Gr_n(\mathbb{C}^k) \hookrightarrow Gr_{n+1}(\mathbb{C}^{k+1})$$

given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. these induce maps of classifying spaces

$$B O(n) \longrightarrow B O(n+1)$$

and

$$B U(n) \longrightarrow B U(n+1) \,.$$

Definition

There are canonical maps

$$Gr_{n_1}(\mathbb{R}^{k_1}) \times Gr_{n_2}(\mathbb{R}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{R}^{k_1 + k_2})$$

and

$$Gr_{n_1}(\mathbb{C}^{k_1}) \times Gr_{n_2}(\mathbb{C}^{k_2}) \longrightarrow Gr_{n_1 + n_2}(\mathbb{C}^{k_1 + k_2})$$

given by sending ambient spaces and subspaces to their direct sum.

Under the colimit of def. these induce maps of classifying spaces

$$B O(n_1) \times B O(n_2) \longrightarrow B O(n_1 + n_2)$$

and

$$B U(n_1) \times B U(n_2) \longrightarrow B U(n_1 + n_2)$$

Properties

Proposition

The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. admit the structure of CW-complexes. Moreover the canonical inclusions

$$Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})$$

and

$$Gr_n(\mathbb{C}^k) \hookrightarrow Gr_n(\mathbb{C}^{k+1})$$

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. ).

A general proof is spelled out in Hatcher, section 1.2 (pages 31-34). For the case of real- , complex- and quaternionic vector bundles see at cell structure on projective space.

Proposition

The Stiefel manifold $V_n(\mathbb{R}^k)$ from def. admits the structure of a CW-complex.

e.g. (James 59, p. 3, James 76, p. 5 with p. 21, 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.)

Proposition

The Stiefel manifold $V_n(\mathbb{R}^k)$ (def. ) is (k-n-1)-connected.

Proof

Consider the coset quotient projection

$$O(k-n) \longrightarrow O(k) \longrightarrow O(k)/O(k-n) = V_n(\mathbb{R}^k) \,.$$

Since the orthogonal groups is compact (prop.) and by 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 this prop. it has the following form in degrees bounded by $n$:

$$\cdots \to \pi_{\bullet \leq k-n-1}(O(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq k-n-1}(O(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k-n)) \to \cdots \,.$$

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.)

Similarly:

Proposition

The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (def. ) is 2(k-n)-connected.

Proof

Consider the coset quotient projection

$$U(k-n) \longrightarrow U(k) \longrightarrow U(k)/U(k-n) = V_n(\mathbb{C}^k) \,.$$

By prop. and by 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. it has the following form in degrees bounded by $n$:

$$\cdots \to \pi_{\bullet \leq 2(k-n)}(U(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(U(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k-n)) \to \cdots \,.$$

This implies the claim.

Corollary

The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. is weakly contractible.

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

Proposition

The homotopy groups of the classifying spaces $B O(n)$ and $B U(n)$ (def. ) are those of the orthogonal group $O(n)$ and of the unitary group $U(n)$, respectively, shifted up in degree: there are isomorphisms

$$\pi_{\bullet+1}(B O(n)) \simeq \pi_\bullet O(n)$$

and

$$\pi_{\bullet+1}(B U(n)) \simeq \pi_\bullet U(n)$$

(for homotopy groups based at the canonical basepoint).

Proof

Consider the sequence

$$O(n) \longrightarrow E O(n) \longrightarrow B O(n)$$

from def. , with $O(n)$ the fiber. Since (by 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

$$\cdots \to \pi_\bullet(O(n)) \longrightarrow \pi_\bullet(E O(n)) \longrightarrow \pi_\bullet(B O(n)) \longrightarrow \pi_{\bullet-1}(O (n)) \longrightarrow \pi_{\bullet-1}(E O(n)) \to \cdots \,.$$

Since by cor. $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that

$$\pi_\bullet(B O(n)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1}(O (n))$$

is an isomorphism.

The same kind of argument applies to the complex case.

Proposition

For $n \in \mathbb{N}$ there are homotopy fiber sequences

$$S^n \longrightarrow B O(n) \longrightarrow B O(n+1)$$

and

$$S^{2n+1} \longrightarrow B U(n) \longrightarrow B U(n+1) \,,$$

exhibiting the n-sphere ($(2n+1)$-sphere) as the homotopy fiber of the canonical maps from def. .

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

$$\array{ B O(n) &\hookrightarrow& B O(n+1) \\ {}^{\mathllap{{weak \, homotopy} \atop equivalence}}\downarrow & \nearrow_{\mathrlap{Serre \, fib.}} \\ \tilde B O(n) }$$

such that $S^n$ is the ordinary fiber of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.

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

$$\array{ O(n) &\overset{id}{\longrightarrow}& O(n) \\ \downarrow && \downarrow \\ E O(n) &\longrightarrow& E O(n+1) \\ \downarrow && \downarrow \\ B O(n) &\longrightarrow& (E O(n+1))/O(n) } \,.$$

By this prop. both bottom vertical maps are Serre fibrations and so both vertical sequences are fiber sequences. By prop. part of the induced morphisms of long exact sequences of homotopy groups looks like this

$$\array{ \pi_\bullet(B O(n)) &\overset{}{\longrightarrow}& \pi_\bullet( (E O(n+1))/O(n) ) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \pi_{\bullet-1}(O(n)) &\overset{=}{\longrightarrow}& \pi_{\bullet-1}(O(n)) } \,,$$

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 this prop., which gives the fiber sequence

$$O(n+1)/O(n) \longrightarrow (E O(n+1))/O(n) \longrightarrow (E O(n+1))/O(n+1) \,.$$

The claim in then follows since (this exmpl.)

$$O(n+1)/O(n) \simeq S^n \,.$$

The argument for the complex case is of the same form, concluding now with the identification (this exmpl.)

$$U(n+1)/U(n) \simeq S^{2n+1} \,.$$

Classification of bundles

Proposition

For $X$ a paracompact topological space, the operation of pullback of the universal principal bundle $E O(n) \to B O(n)$ from def. along continuous functions $f \colon X \to B O(n)$ establishes a bijection

$$[X, B O(n)] \underoverset{iso}{f \mapsto f^\ast E O(n)}{\longrightarrow} O(n) Bund/_\sim$$

between homotopy classes of functions from $X$ to $B O(n)$ and isomorphism classes of $O(n)$-principal bundles on $X$.

A full proof is spelled out in Hatcher, section 1.2, theorem 1.16.

For symmetric principal bundles

• the 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 $\Sigma(n)$;

• the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-universal principal bundle.

For crossed complexes

We discuss here classifying spaces of crossed complexes.

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

$$\mathcal{B} :(algebraic data) \to (topological data).$$

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

$$\Xi: (topological data) \to (algebraic data)$$

such that

1. $\Xi \circ \mathcal{B}$ is equivalent to the identity;

2. $\Xi$ preserves certain colimits (Generalised van Kampen theorem) allowing some calculation;

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

$$[X,U\mathcal{B}C] \cong [\Xi X_*, C].$$

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: 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 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 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 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 structure in all dimensions from 0 to nto 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.

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.

For simplicial groups

The simplicial classifying space $\overline{W}(-)$-construction (see simplicial group and groupoid object in an (∞,1)-category) which gives the classifying space functor for simplicial groups and simplicially enriched groupoids is given in the entry on 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’.

Properties

For classical Lie groups

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

Proposition

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

$$Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})$$

are subcomplex incusion (hence relative cell complex inclusions).

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

$$B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k) \,.$$

A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).

Proposition

The classifying spaces $B O(n)$ are paracompact spaces.

An early source of this statement is (Cartan-Schwartz 63, exposé 5). It follows for instance by prop. the fact that every CW-complex is paracompact.

Cohomology

(equivariant) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$	Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant) complex cobordism cohomology	MU	$MU_G(\ast)$	completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant) algebraic K-theory	$K \mathbb{F}_p$	representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$	Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

Moduli spaces

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

Related concepts

• classifying space, classifying stack, moduli space, moduli stack, derived moduli space, Kan-Thurston theorem

  • Milnor classifying space

  • numerable bundle

  • acyclic group

  • Sullivan model of classifying space

  • free loop space of classifying space

• classifying topos

• universal principal bundle, universal principal ∞-bundle

• subobject classifier

• classifying morphism

References

General

Original discussion of classifying spaces in topological homotopy theory:

• John Milnor, Construction of Universal Bundles, II, Annals of Mathematics Second Series, Vol. 63, No. 3 (May, 1956), pp. 430-436 (jstor:1970012)

• Richard Palais, Section 2.6 of: The classification of $G$-spaces, Memoirs of the AMS 36, 1960 (ISBN:978-0-8218-9979-3 pdf, pdf)

• R. James Milgram, The bar construction and abelian H-spaces, Illinois J. Math. 11(2): 242-250 (1967) (doi:10.1215/ijm/1256054662)

• Michael C. McCord, Classifying Spaces and Infinite Symmetric Products, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (jstor:1995173, pdf)

  (using – and introducing – compactly generated weakly Hausdorff topological spaces)

• Saunders MacLane, Section 6 of: The Milgram bar construction as a tensor product of functors, In: F.P. Peterson (eds.) The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday Lecture Notes in Mathematics 168, Springer, 1970 (doi:10.1007/BFb0058523, pdf)

  (understanding geometric realization of simplicial topological spaces via a coend-formula)

• Graeme Segal, Classifying spaces and spectral sequences, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)

• Norman Steenrod, Milgram’s classifying space of a topological group, Topology Volume 7, Issue 4, November 1968, Pages 349-368 (doi:10.1016/0040-9383(68)90012-8)

• Jim Stasheff, H-spaces and classifying spaces: foundations and recent developments, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society (1971) pp. 247–272

• Tammo tom Dieck, On the homotopy type of classifying spaces, Manuscripta Math 11, 41–49 (1974) (doi:10.1007/BF01189090)

  (relating the plain and fat geometric realization of simplicial topological spaces appearing in Milnor’s and Milgram’s constructions)

Review:

• William Dwyer, Homotopy theory of classifying spaces, Lecture notes, Copenhagen 2008, (pdf, pdf)

• Stephen Mitchell, Notes on principal bundles and classifying spaces, Lecture Notes. University of Washington, 2011 (pdf, pdf)

Textbook accounts:

• Dale Husemoeller, Section 4.12, 4.13 of: Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) (gBooks, pdf)

• Norman Steenrod, section II.19 of: The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)

• Stanley Kochmann, section 1.3 of of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Peter May, Chapter 23 of: A concise course of algebraic topology (pdf)

• Allen Hatcher, section 1.2 of Vector bundles and K-theory (web)

• Dale Husemoeller, Michael Joachim, Branislav Jurco, Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants,

  Lecture Notes in Physics, Springer 2008 (pdf)

• Gerd Rudolph, Matthias Schmidt, Thm. 3.5.1 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)

• Loring Tu, Section I.5 in: Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)

• Dai Tamaki, Fiber Bundles and Homotopy, World Scientific 2021 (doi:10.1142/12308)

  (in the context of homotopy theory)

Discussion in simplicial homotopy theory:

• Paul Goerss, Rick Jardine, Section V.3 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996) (doi:10.1007/978-3-0346-0189-4)))

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460)

• Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – Presentations, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 (doi:10.1007/s40062-014-0077-4, arXiv:1207.0249)

Discussion in topos theory relating to classifying toposes:

• Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995.

Discussion of characterization of principal bundles by rational universal characteristic classes and torsion information is in the appendices of

• Igor Belegradek, Vitali Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory (arXiv:math/0007007)

• Igor Belegradek, Pinching, Pontrjagin classes, and negatively curved vector bundles (arXiv:math/0001132)

Discussion of classifying spaces in the context of measure theory is in

• Ivan Marin, Measure theory and classifying spaces (arXiv:1702.01889)

For equivariant bundles

Discussion of classifying G-spaces for $G$-equivariant principal bundles:

• Tammo tom Dieck, Faserbündel mit Gruppenoperation, Arch. Math 20, 136–143 (1969) (doi:10.1007/BF01899003)

• Richard Lashof, Equivariant bundles, Illinois J. Math. 26(2): 257-271, 1982 (doi:10.1215/ijm/1256046796.full)

• Peter May, Some remarks on equivariant bundles and classifying spaces, Théorie de l’homotopie, Astérisque, no. 191 (1990), 15 p. (numdam:AST_1990__191__239_0)

• Mitutaka Murayama, Kazuhisa Shimakawa, Universal equivariant bundles, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 (doi:10.1090/S0002-9939-1995-1231040-9)

• Bernardo Uribe, Wolfgang Lück, Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 (arXiv:1304.4862, doi:10.2140/agt.2014.14.1925)

• Bertrand Guillou, Peter May, Mona Merling Categorical models for equivariant classifying spaces, Algebr. Geom. Topol. 17 (2017) 2565-2602 (arXiv:1201.5178, doi:10.2140/agt.2017.17.2565)