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\begin{abstract}
In the paper the notion of truncating twisting function $\tau
:X\to Q$ from a simplicial set $X$ to a cubical set $Q$ and the
corresponding notion of twisted Cartesian product of these sets
$X\times_{\tau }Q$ are introduced. The latter becomes a cubical
set whose chain complex coincides with the standard twisted tensor
product $C_*(X)\otimes_{\tau_*}C_*(Q)$. This construction together
with the theory of twisted tensor products for homotopy G-algebras
allows to obtain multiplicative models for fibrations.
\end{abstract}
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