…
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A 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 $G$-bundles over $X$. (In fact, one can jack this up to an equivalence of groupoids or $\infty$-groupoids.) Various improvements of this are possible which classify bundles with extra structure or 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 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.”)
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
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
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
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.
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 $B O(n)$.
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.
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.
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
here the action of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.
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.
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.
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.
$Gr_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is real projective space of dimension $n$.
$Gr_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n$ is complex projective space of dimension $n$.
For all $n \leq k \in \mathbb{N}$, the canonical projection from the real Stiefel manifold (def. 1) to the Grassmannian is a $O(n)$-principal bundle
and the projection from the complex Stiefel manifold to the Grassmannian us a $U(n)$-principal bundle:
By (this cor. and this prop.).
By def. 2 there are canonical inclusions
and
for all $k \in \mathbb{N}$. The colimit (in Top, see there) over these inclusions is denoted
and
respectively.
Moreover, by def. 1 there are canonical inclusions
and
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
and
respectively. The inclusions are in fact compatible with the bundle structure from prop. 1, so that there are induced projections
and
respectively. These are the standard models for the universal principal bundles for $O$ and $U$, respectively. The corresponding associated vector bundles
and
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
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.
There are canonical inclusions
and
given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. 3 these induce maps of classifying spaces
and
There are canonical maps
and
given by sending ambient spaces and subspaces to their direct sum.
Under the colimit of def. 3 these induce maps of classifying spaces
and
The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. 2 admit the structure of CW-complexes. Moreover the canonical inclusions
and
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. 3).
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
The Stiefel manifold $V_n(\mathbb{R}^k)$ from def. 1 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.)
The Stiefel manifold $V_n(\mathbb{R}^k)$ (def. 1) is (k-n-1)-connected.
Consider the coset quotient projection
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 $n$:
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:
The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (def. 1) is 2(k-n)-connected.
Consider the coset quotient projection
By prop. \ref{UnitaryGroupIsCompact} 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. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:
This implies the claim.
The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. 3 is weakly contractible.
The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. 3 is weakly contractible.
The homotopy groups of the classifying spaces $B O(n)$ and $B U(n)$ (def. 3) are those of the orthogonal group $O(n)$ and of the unitary group $U(n)$, respectively, shifted up in degree: there are isomorphisms
and
(for homotopy groups based at the canonical basepoint).
Consider the sequence
from def. 3, 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
Since by cor. 1 $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that
is an isomorphism.
The same kind of argument applies to the complex case.
For $n \in \mathbb{N}$ there are homotopy fiber sequences
and
exhibiting the n-sphere ($(2n+1)$-sphere) as the homotopy fiber of the canonical maps from def. 4.
This means that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. 3) by a Serre fibration
such that $S^n$ is the ordinary fiber of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.
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
By this prop. both bottom vertical maps are Serre fibrations and so both vertical sequences are fiber sequences. By prop. 6 part of the induced morphisms of long exact sequences of homotopy groups looks like this
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
The claim in then follows since (this exmpl.)
The argument for the complex case is of the same form, concluding now with the identification (this exmpl.)
For $X$ a paracompact topological space, the operation of pullback of the universal principal bundle $E O(n) \to B O(n)$ from def. 3 along continuous functions $f \colon X \to B O(n)$ eastblishes a bijection
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)
We discuss here classifying spaces of crossed complexes.
The notion of classifying space should be regarded in general terms as giving a functor
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
such that
$\Xi \circ \mathcal{B}$ is equivalent to the identity;
$\Xi$ preserves certain colimits (Generalised van Kampen theorem) allowing some calculation;
there are notions of homotopy for both types of data leading to a bijection of homotopy classes for some $X$
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.
The $\bar W(-)$-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’.
Let $O(n)$ be the orthogonal group and $U(n)$ the unitary group in real/complex dimension $n$, respectively.
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
are subcomplex incusion (hence relative cell complex inclusions).
Accordingly there is an induced CW-complex structure on the classifying space
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
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. 9 the fact that every CW-complex is paracompact.
The notion of moduli space is closely related to that of classifying space, but has some subtle differences. See there for more on this.
classifying space, classifying stack, moduli space, moduli stack, derived moduli space, Kan-Thurston theorem
Original accounts include
Textbook accounts on classifying spaces for vector bundles include
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)
Alan Hatcher?, section 1.2 of Vector bundles and K-theory (web)
A discussion more from the point of view of topos theory is in
Discussion of universal principal bundles over their classifying spaces is in
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