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We show how the tangent functor extends from ordinary smooth maps to"microformal morphisms" (also called "thick morphisms") of supermanifolds.Microformal morphisms generalize ordinary maps and correspond to formalcanonical relations between the cotangent bundles specified by generatingfunctions depending on position variables on the source manifold and momentumvariables on the target manifold (as formal power expansions), regarded as partof the structure. Microformal morphisms act on functions by non-linear (ingeneral) pullbacks. We obtain here non-linear pullbacks of (pseudo)differentialforms and show that they respect the de Rham differentials as "non-linear chainmaps" that can induce non-linear transformations of cohomology.

Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology