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 formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as “non-linear chain maps” that can induce non-linear transformations of cohomology.

我们展示了切函子如何从普通的平滑映射扩展到超流形的“微形式态射”（也称为“厚态射”）。微形式态射概括了普通映射，并对应于由生成函数指定的余切丛之间的形式规范关系，该生成函数取决于源流形上的位置变量和目标流形上的动量变量（作为形式幂展开），被视为结构的一部分。微形式态射通过非线性（通常）回调作用于函数。我们在这里获得了（伪）微分形式的非线性回调，并表明它们将德拉姆微分视为“非线性链映射”，可以引发上同调的非线性变换。