# nLab microformal morphism

• Theodore Th. Voronov, Microformal geometry and homotopy algebras, arxiv/1411.6720

We enlarge the category of smooth maps of (super)manifolds by introducing a notion of ‘microformal morphism, starting from the classical notion of canonical relation and using formal power expansions in the cotangent directions. The result is a ‘formal category’; that is, the composition of microformal morphisms is defined by a formal power series. Microformal morphisms act on functions by pullbacks which are in general nonlinear transformations. More precisely, they are formal mappings of formal manifolds of even functions (“bosonic fields”). Their derivatives at each point are algebra homomorphisms. This suggests a notion of “nonlinear homomorphisms” of algebras and an extension of the classic algebra/geometry duality. There is a parallel “fermionic” version of these constructions. Using microformal geometry, we obtain a general construction of L∞-morphisms for functions on homotopy Poisson (P?) or homotopy Schouten (S?) manifolds. Further, we show that the familiar adjoint of a linear operator on vector spaces or vector bundles can be generalized to the nonlinear situation as a microformal morphism. We apply this to L∞-algebroids and show that an L∞-morphism of L∞-algebroids induces an L∞-morphism of the homotopy Lie–Poisson algebras of functions on the dual vector bundles. We apply that to higher Koszul brackets on forms and triangular L∞-bialgebroids. We also obtain a quantum version (for the bosonic case) related with the above as the Schroedinger equation is related with the Hamilton–Jacobi equation. We show that nonlinear pullbacks by microformal morphisms are the limits when ℏ→0 of “quantum pullbacks”, which are Fourier integral operators of special type.

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