Szczarba’s twisting cochain is comultiplicative

We prove that Szczarba’s twisting cochain is comultiplicative. In particular, the induced map from the cobar construction $\Omega C(X)$ of the chains on a $1$-reduced simplicial set $X$ to $C(GX)$, the chains on the Kan loop group of $X$, is a quasiisomorphism of $\operatorname{dg}$ bialgebras. We also show that Szczarba’s twisted shuffle map is a $\operatorname{dgc}$ map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural $\operatorname{dgc}$ model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.