The twisted Cartesian model for the double path fibration

In this paper, the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of a 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 is in fact 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.

Cartan's constructions and the twisted Eilenberg–Zilber theorem

Let 𝐺 × τ 𝐺′ be the principal twisted Cartesian product with fibre 𝐺, base 𝐺 and twisting function where 𝐺 and 𝐺′ are simplicial groups as well as 𝐺 × τ 𝐺′; and 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁 (𝐺′) be the twisted tensor product associated to 𝐶𝑁 (𝐺 × τ 𝐺′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁(𝐺′), μ) is a multiplicative Cartan's construction where μ is the standard product on 𝐶𝑁(𝐺) ⊗ 𝐶𝑁(𝐺′). Furthermore, assuming that a contraction from 𝐶𝑁(𝐺′) to 𝐻𝐺′ exists and using the techniques from homological perturbation theory, we extend the former result to other “twisted” tensor products of the form 𝐶𝑁(𝐺) ⊗ 𝐻𝐺′.