Yoneda lemma

# Contents

## Idea

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.”)

## Examples

### For principal bundles

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.

#### For orthogonal and unitary structure groups

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. (See for instance (May, p. 196), where some of the following is taken from).

Let $V_n(\mathbb{R}^q)$ be the Stiefel variety? 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 $B O(n)$.

For instance

$G_1(\mathbb{R}^\infty) = \mathbb{R}P^\infty$

is the real? projective space that classifies line bundles.

### 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.

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 $\bar W(-)$-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’.

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

See (Cartan-Schwarz, expose 5).

## 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.

## References

• Cartan-Schwarz, ….

A concise introduction of classifying spaces of vector bundles is around p. 196 of

A discussion more from the point of view of topos theory is in

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

Discussion of universal principal bundles over their classifying spaces is in

• Stephen Mitchell, Universal principal bundles and classifying spaces (pdf)

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)

Revised on November 24, 2014 11:04:52 by Urs Schreiber (82.224.164.72)