  Let $Ham (M,\omega ) $ denote the Frechet Lie group of Hamiltonian
symplectomorphisms of a monotone symplectic manifold $(M, \omega) $. Let $NFuk
(M, \omega)$ be the $A _{\infty} $-nerve of the Fukaya category $Fuk (M,
\omega)$, and let $(|\mathbb{S}|, NFuk (M, \omega))$ denote the $NFuk (M,
\omega)$ component of the ``space of $\infty$-categories'' $|\mathbb{S}| $.
Using Floer-Fukaya theory for a monotone $(M, \omega)$ we construct a natural
up to homotopy classifying map \begin{equation*}
  BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{equation*} This
verifies one sense of a conjecture of Teleman on existence of action of $Ham (M
, \omega)$ on the Fukaya category of $(M, \omega ) $. This construction is very
closely related to the theory of the Seidel homomorphism and the quantum
characteristic 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 nontrivial by an explicit calculation. In particular, we
arrive at a new non-trivial ``quantum'' invariant of any smooth manifold, which
motives the statement of a kind of ``quantum'' Novikov conjecture.
