• Tags
  • #homotopy #higher-algebra/stacks
• Refs:
  • #todo/add-references
• Links:
  • classifying stack
  • principal bundle
  • classifying space of a category

classifying space

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Relation to characteristic classes

See characteristic classes:

Definitions

• An object of the category \({\mathbf{B}}G\) is a \(G\)-torsor \(T\) (i.e. a non-empty \(G\)-set on which \(G\) acts transitively and free). The morphisms are morphisms of \(G\)-sets.

• If \(M\) is a monoid, then \(\pi_1 {\mathbf{B}}M \cong M{ {}_{ \widehat{ {\operatorname{gp} } } } }\) is the group completion of \(M\).

Any other pullback of the classifying bundle along a map \(X \to {\mathbf{B}}G\).

Let \(I(G, X)\) denote the set of isomorphism classes of principal \(G{\hbox{-}}\) bundles over a base space \(X\), then \begin{align*} I(G, X) \cong {\mathsf{hoTop}}(X, {\mathbf{B}}G) \end{align*} So in other words, isomorphism classes of principal \(G{\hbox{-}}\) bundles over a base \(X\) are equivalent to homotopy classes of maps from \(X\) into the classifying space of \(G\).

Proposition: There is a bijection for vector bundles: \begin{align*} {\mathsf{hoTop}}(X, {\operatorname{Gr}}_n({\mathbf{R}})) \cong \left\{{\text{rank $n$ ${\mathbf{R}}{\hbox{-}}$vector bundles over $X$}}\right\} / \sim \end{align*} - Every such vector bundle is a pullback of the principal bundle \begin{align*} \operatorname{GL}(n, {\mathbf{R}}) \to V_n({\mathbf{R}}^\infty) \to {\operatorname{Gr}}(n, {\mathbf{R}}) \end{align*} # Notes

• ${\mathsf{Sh}}({{\mathbf{B}}G}; { \mathsf{Vect} }{/ {k}} ) \cong {\mathsf{Rep}}(G){/ {k}} $, and \({\mathsf{Sh}}({\mathbf{B}}GX; { \mathsf{Vect} }_{/ {k}} )\cong {\mathsf{Sh}}^G(X; { \mathsf{Vect} }_{/ {k}} )\), the category of equivariant sheaves for \(G\) over \(X\).
• \({\mathbf{B}}G \simeq K(G, 1)\) when \(G\) is discrete.
• \(\pi_1({\mathbf{B}}G) = G\) and \(\pi_n({\mathbf{B}}G) = \pi_n EG = 1\).

• \(X/G\) may fail to be a nice space if points have nontrivial stabilizers.

• It is useful to think of \({\mathbf{B}}G\) as a space whose points are copies of \(G\), so the classifying map \(f\in {\mathsf{hoTop}}(X,{\mathbf{B}}G)\) assigns each \(x \in X\) to the fiber above \(x\), which is a copy of \(G\).

• For a discrete group \(G\), we have \({\mathbf{B}}G = K(G,1)\), so that \(\pi_1({\mathbf{B}}G) = G\) and \(\pi_k({\mathbf{B}}G) = 0\) for \(k \neq 1\).

  • Follows from contractibility of \(EG\) ?

• For \(X\in {\mathsf{Top}}{\mathsf{Grp}}\), there is a weak equivalence \({\Omega}{\mathbf{B}}X \underset{\scriptscriptstyle W}{\rightarrow}X\)

  • How to prove: show they both represent the functor \(\mathop{\mathrm{Prin}}_G(S^1 \wedge({-}){ {}_{{\operatorname{pt}}} }\)

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Constructions

• Standard procedure for constructing a classifying space for any group:
  • Construct a 2-complex with the given fundamental group, and then one inductively attaches higher dimensional cells to kill all higher homotopy groups.
  • Each element \(c\in \pi_n(X_{n−1})\) is represented by some continuous map \(\gamma_c:S^n\to X_{n−1}\) with image in the \(n{\hbox{-}}\)-skeleton.
  • Let \(X_n\) be obtained from \(X_{n−1}\) by attaching an \((n+1){\hbox{-}}\)cell along \(\gamma_c\), for each \(c\in π_n(X_{n−1})\).

Relation to loop spaces and monoidal categories:

Further Reading

• \(\pi_{i+k}{\mathbf{B}}^k G = \pi_i G\).
  • Proof: If \(G\) is a topological group, there is a universal principal \(G{\hbox{-}}\)bundle \(EG \to BG\) which induces a LES in homotopy.
  • Since \(EG\) is contractible, \begin{align*}\pi_i EG = \pi_{i+1}EG = 0\implies \pi_{i+1}BG \cong \pi_i G.\end{align*}
  • When \(G\) is an \(E_2\) space, \(BG\) is a topological group, and so \begin{align*}\pi_{i+2}(B^2G) = \pi_{i+2}(B(BG)) = \pi_{i+1}(BG) = \pi_i(G).\end{align*}
• Corollary: If \(G\) is a discrete group, \({\mathbf{B}}^k G \simeq K(G, n)\).
  • Proof: \(\pi_0 G = G\) and \(\pi_i G = 0\) for \(i > 0\), so \(\pi_k {\mathbf{B}}^k G = G\).
• One can take classifying spaces of stacks.
  • There is a stack that classifies connections, but it has issues: it is not a representable.
• \(EG\) can be constructed as \begin{align*} EG \cong \bigcup_n G \ast G \ast \cdots \ast G ,\end{align*} where \(\ast\) is join of two spaces: the suspension of the smash product. For example, \(G = {\mathbf{Z}}_2\) implies \begin{align*} EG \cong \bigcup_n {\mathbf{Z}}_2 \ast \cdots = \bigcup_n S^{n-1} = S^\infty .\end{align*}

Relation to Group Cohomology

Computing the torsion classes of \(H^{*}(B \mathcal{G} ; \mathbb{Z})\) is an important problem; for example \(H^{3}(B \mathcal{G} ; \mathbb{Z})\) classifies the set of gerbes. # Unknown?

• What is \(\pi_* {\mathbf{B}}G\)?
  • What is the stable homotopy \(\pi_* {\Sigma}^\infty {\mathbf{B}}G\)?
• Conjecture: \({\mathbf{B}}(G \oplus H) = {\mathbf{B}}G \times{\mathbf{B}}H\)
  • Proof outline: \(EG \times EH\) is contractible, and \(G \times H\) acts freely on it with quotient equal to the RHS?
• Conjecture: \({\mathbf{B}}(G \ast H) = BG \vee BH\)
• Conjecture: \({\mathbf{B}}(G \otimes_{\mathbf{Z}}H) = ?\) for \(G, H\in {\mathsf{Ab}}\)?
• Conjecture: \({\mathbf{B}}(G \rtimes_\phi H) = ?\)

Examples

For topological groups

See internal category

Homotopy construction

See Blakers Massey