Global Fukaya category and quantum Novikov conjecture I

Conceptually, the goal here is a construction which functorially translates a Hamiltonian fibre bundle to a certain ``derived vector bundle'' over the same space, with fiber an $A _{\infty}$ category. This ``derived vector bundle'' must remember the continuity of the original bundle. Concretely, using Floer-Fukaya theory for a monotone $(M, \omega)$ we construct a natural continuous map \begin{equation*} BHam (M, \omega) \to (\mathcal{S}, NFuk (M)), \end{equation*} with $(\mathcal{S}, NFuk (M))$ denoting the $NFuk (M)$ component of the ``space'' of $\infty$-categories, where $NFuk (M)$ is the $A _{\infty}$-nerve of the Fukaya category $Fuk (M)$. This construction is very closely related to the theory of the Seidel homomorphism and the quantum Chern classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II the above map is shown to be non trivial by an explicit calculation. In particular we arrive at a new non-trivial ``quantum'' invariant of any smooth manifold and a ``quantum'' Novikov conjecture.Comment: v5, 41 pages. This adds significant detail and fixes some language issue