  The problem of introducing a dependence of elements of quantum group on
classical parameters is considered. It is suggested to interpret a homomorphism
from the algebra of functions on quantum group to the algebra of sections of a
sheaf of algebras on a classical manifold as describing such a dependence. It
is argued that the functorial point of view of group schemes is more
appropriate in quantum group field theory. A sheaf of the Hopf algebras over
the manifold (quantum sheaf) is constructed by using bosonization formulas for
the algebra of functions on the quantum group $SU_{q}(2)$ and the theory of
repre- sentations of canonical commutation relations. A family of automorphisms
of the Hopf algebra depending on classical variables is described. Quantum
manifolds, i.e. manifolds with commutative and non-commutative coordinates are
discussed as a generalization of supermanifolds. Quantum group chiral fields
and relations with algebraic differential calculus are discussed.
