Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 1.2.4.3 (The Milnor Construction). Let $M$ be a monoid. We can then form a category $BM$ having a single object $X$, where $\operatorname{Hom}_{BM}(X,X) = M$ and the composition of morphisms in $BM$ is given by multiplication in $M$. We will denote the nerve of the category $BM$ by $B_{\bullet }M$.

In the special case where $M = G$ is a group, 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).