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](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu) (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.



Explicit contraction for the universal simplicial bundle WG
Pages 75-81 of Appendix A of *[On the theory and applications of differential torsion products](https://web.archive.org/web/20230613045823/https://www.math.uchicago.edu/%7Emay/BOOKS/GugMay.pdf)*, 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 $EG$.


