The twisted Cartesian model for the double path fibration

Abstract

Abstract. 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.

A computer algebra system for the study of commutativity up-to-coherent homotopies

“…Cubical chains appear naturally from simplicial sets through the cobar construction [33] and in ComCH we have implemented an E ∞ -structure on them. To model the double cobar construction one needs to study permutahedral chains [34]. In forthcoming work we describe an E ∞ -structure on permutahedral sets suitable for implementation on ComCH.…”