  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.
