TITLE: Simplicial set construction of the classifying space
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: 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.