Microformal geometry

We consider two formal categories which are "formal thickenings" of the usual category of supermanifolds and their smooth maps. We call morphisms in these categories, emph{microformal} or emph{thick morphisms} of supermanifolds. They are defined by formal canonical relations of a special form between the cotangent or anticotangent bundles (for the `even' or `odd' thickening, respectively). (There is an important connection with the symplectic micromorphisms introduced earlier by Cattaneo--Dherin--Weinstein from a different motivation.) We came to such morphisms in url{arXiv:1409.6475 [math.DG]}, where we constructed for them actions on functions (emph{nonlinear pullbacks}), which are nonlinear mappings of formal functional supermanifolds. In this paper, we establish categorical framework and functorial properties for these nonlinear pullbacks. We also consider applications to vector bundles and algebroids. In particular, we obtain the notion of the emph{adjoint for a nonlinear morphism} of vector bundles, which is a thick morphism of the dual bundles such that for linear operators it reduces to the ordinary adjoint. By using this construction, we show how an L∞-morphism of L∞-algebroids induces an L∞-morphism of the homotopy Lie--Poisson or Lie--Schouten algebras of functions on the (anti)dual vector bundles. This has an application to higher Koszul brackets and triangular L∞-bialgebroids.