Microformal geometry and homotopy algebras

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.