2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65297In 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.37 pages, 8 figures, This final version (10-01-04) clarifies a number of issues in earlier version and includes computational examplesAlgebraic Topology55R05, 55P35, 55U05, 52B05, 05A18, 05A19text