  In the paper the notion of truncating twisting function from a cubical set to
a permutahedral set and the corresponding notion of twisted Cartesian product
of these sets are introduced. The latter becomes a permutocubical set that
models in particular the path fibration on a loop space. The chain complex of
this twisted Cartesian product in fact is a comultiplicative twisted tensor
product of cubical chains of base and permutahedral chains of fibre. This
construction is formalized as a theory of twisted tensor products for Hirsch
algebras.
