Floer homology of fibrations I: Representing flow lines in Moore path spaces

In their previous work, Barraud and Cornea enriched the Lagrangian Floer complex by adding cubical chains in the based loop space of the Lagrangian, and recovered the Leray-Serre spectral sequence of the based path space fibration, assuming that the Lagrangian is weakly exact and simply connected. In the present article, we remove the simple connectivity assumption and adapt the construction to any Hurewicz fibration. To this end, we introduce a stronger notion of local systems than the classical one, as a topological functor from the free path space of a manifold to the category of topological spaces. Such functors arise naturally from a Hurewicz fibration.