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Example Let $G$ be a group. Then the category $BG$ of Remark is a groupoid. It follows from Proposition that the simplicial set $B_{\bullet } G$ of Construction is a Kan complex. The geometric realization $| B_{\bullet }G |$ is a topological space called the classifying space of $G$. It can be characterized (up to homotopy equivalence) by the fact that it is a CW complex with either of the following properties:

  • The space $| B_{\bullet }G |$ is connected, and its homotopy groups (with respect to any choice of base point) are given by the formula

    \[ \pi _{\ast }( | B_{\bullet }G |) \simeq \begin{cases} G & \text{ if } \ast = 1 \\ 0 & \text{ if } \ast > 1. \end{cases} \]
  • For any paracompact topological space $X$, there is a canonical bijection

    \[ \{ \text{Continuous maps $f: X \rightarrow | B_{\bullet }G |$} \} / \text{homotopy} \simeq \{ \text{$G$-torsors $P \rightarrow X$} \} / \text{isomorphism}. \]

We refer the reader to [MR0077122] for a more detailed discussion (including an extension to the setting of topological groups).