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Guest, you’re right. What the section gives you is $E G$, not $B G$. I usually keep this straight in my head by using two-sided bar constructions, so $N B G = B(1, G, 1)$, whereas $E G = B(G, G, 1)$ gives you the contractible total space. I will fix in a moment.
Added a correction to the section on bar construction (h/t to Guest), linking instead to two-sided bar construction.
I’m confused; is the point that “the usual bar construction of $G$” would be interpreted by some people as referring to $E G$ rather than $B G$?
I think the real point Guest was making is that the article had misidentified the nerve of $B G$ (or $B A$), by writing down instead the simplicial object for $E G$. The article also linked to bar construction, largely written by me, where the discussion is about various acyclic resolutions of structures, for example the machine which produces a standard acyclic free $\mathbb{Z}G$-resolution of $\mathbb{Z}$ for the purpose of defining group cohomology. Formally that would be more like $E G = B(G, G, 1)$ which “resolves a point”.
Got it, thanks.
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