  We extend the category of (super)manifolds and their smooth mappings by
introducing a notion of microformal or "thick" morphisms. They are formal
canonical relations of a special form, constructed with the help of formal
power expansions in cotangent directions. The result is a formal category so
that its composition law is also specified by a formal power series. A
microformal morphism acts on functions by an operation of pullback, which is in
general a nonlinear transformation. More precisely, it is a formal mapping of
formal manifolds of even functions (bosonic fields), which has the property
that its derivative for every function is a ring homomorphism. This suggests an
abstract notion of a "nonlinear algebra homomorphism" and the corresponding
extension of the classical "algebraic-functional" duality. There is a parallel
fermionic version.
  The obtained formalism provides a general construction of
$L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or
homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal
morphisms. We also show that the notion of the adjoint can be generalized to
nonlinear operators as a microformal morphism. By applying this to
$L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of
$L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy
Lie--Poisson" brackets for functions on the dual vector bundles. We apply this
construction to higher Koszul brackets on differential forms and to triangular
$L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic
case), whose relation with the classical version is like that of the
Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the
nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of
certain "quantum pullbacks", which are defined as special form Fourier integral
operators.
