We assign an algebraic homotopy theoretic classical field theory $CFT(V)$ to any homotopy commutative
algebra ($C_\infty$-algebra) $V$ over $k$, reals or complexes, and show that
the space of equivalence classes of all classical expectations
has a structure of pro-unitpotent group if $V$ has finite dimensional cohomology
and the algebra of equivalence classes of classical observables has a structure of commutative Hopf algebra that
is dual to the Hopf algebra of the pro-unitpotent group ring.
This can be viewed as a generalization of the
$\pi_1$ de Rham theorem of Chen and Sullivan since the group is isomorphic to the pro-unipotent
fundamental group of connected smooth manifold $M$ if we take Sullivan model of $M$ over k as the input $C_\infty$-algebra.
A similar construction attaches a pro-unipotent mononoid to any homotopy associative algebra ($A_\infty$-algebra)
with the finite dimensional cohomology.

There may be a precise dictionary between the rational homotopy theory and the classical field theory associated with
a Sullivan model of a topological space to regard a quantization of the later theory as a quantization of the rational homotopy theory.
It may be also possible to generalize such construction to establish the concept of “fundamental group” of quantum field theory,
defined as the algebraic structure in the space of all physically equivalent "homotopy Feynman path integrals” of theory.