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What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?:

  • Regard $G$ as a category with one object whose morphisms are the elements of $G$.
  • Construct the simplicial sets $NG$ (i.e., the nerve of $G$) and $\mathcal{E}G$ (unsure if there's a standard notation; I mean that $\mathcal{E}G_n$ should be $G^{n+1}$ with face maps given by deletion and degeneracy maps given by repetition).
  • Take the geometric realizations $BG$ of $NG$ and $EG$ of $\mathcal{E}G$; then $BG$ is the classifying space with universal cover $EG$.

As far as I know, this is a fairly standard construction (although perhaps not the standard one). I'm just wondering about the best place to use as a reference for it.

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  • $\begingroup$ I'm surprised to hear this is not "the" standard construction. What have you seen described so? $\endgroup$
    – David Roberts ♦
    Commented Sep 21, 2021 at 5:07
  • $\begingroup$ I was under the impression 'the' standard construction was using fibers, as in Hatcher's Algebraic Topology. Possibly that's just where I saw it first, however. $\endgroup$
    – Xindaris
    Commented Sep 25, 2021 at 16:53

2 Answers 2

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I believe that's called the Milgram bar construction:

  • R.J. Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242-250.
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I know this description from following paper of Segal. He doesn't mention Milgram there, but relates it to Milnor's join construction. Also, $G$ is allowed to be any topological group, no need for discreteness!

Segal, Graeme, Classifying spaces and spectral sequences, Publ. Math., Inst. Hautes Étud. Sci. 34, 105-112 (1968). ZBL0199.26404.

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    $\begingroup$ Well, it's a reasonable construction if the topological group is well-pointed. If it fails to be, then the fat geometric realisation is the right construction, I believe. $\endgroup$
    – David Roberts ♦
    Commented Sep 21, 2021 at 5:06

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