The twisted Cartesian model for the double path fibration

TL;DR

This paper introduces a novel twisted Cartesian model using permutahedral sets to better understand the path fibration on loop spaces, connecting cubical and permutahedral structures.

Contribution

It develops the concept of truncating twisting functions and twisted Cartesian products, formalizing a new theory of twisted tensor products for Hirsch algebras.

Findings

Models the path fibration on loop spaces using permutocubical sets.

Establishes a comultiplicative twisted tensor product structure.

Formalizes a new theory of twisted tensor products for Hirsch algebras.

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.