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For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu).

Now $WG$ has a simple enough description in terms of $G$ that I would expect one could construct a contracting homotopy directly. Has this been done in the literature?

The proof in May's book (and in the original sources) that $WG$ is contractible goes via showing that $WG$ is 'of type (W)', and that such simplicial sets are Kan, simply-connected and have trivial homology.

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    $\begingroup$ How do these constructions relate to EG and BG? $\endgroup$
    – Sean Tilson
    Commented Jun 3, 2012 at 4:08
  • $\begingroup$ Hi Sean, I didn't get a chance to answer your question, but it is as Peter May says in his answer. EG is the geometric realisation of the functor W applied to the constant simplicial group, and for this G can be a vanilla group, a topological group, or a group internal to any finite-product category $C$ with a geometric realisation/homotopy colimit functor $sC \to C$. $\endgroup$
    – David Roberts ♦
    Commented Jun 3, 2012 at 22:13

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Pages 75-81 of Appendix A of On the theory and applications of differential torsion products, Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-construction for simplicial augmented algebras over a commutative ring $R$. Not the answer to your question, but if I remember rightly, it should lift to an answer when suitably specialized; more precisely, the chain homotopy of Lemma A.16 (up to signs coming from variant choices) should specialize to one coming from a contracting homotopy as desired. When $G$ is a group regarded as a constant simplicial group, Lemma A.15 lifts to give an isomorphism between $WG$ and the simplicial set $E_*G$ whose realization is $EG$.

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  • $\begingroup$ Thanks, Peter, I'll check that out. Perhaps I should have mentioned that MacLane gave a contracting homotopy of the chain complex (non-normalised) associated to W of a simplicial ring in his original lecture on W and Wbar. $\endgroup$
    – David Roberts ♦
    Commented Jun 3, 2012 at 21:28

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