In this study, we construct a prestack over the site of open subsets of the complex plane. Its total category has the set of objects canonically isomorphic to the set of all spaces of pseudoanalytic functions, and this is over all open subsets of the complex plane with respect to all the possible generating pairs. This is carried out by defining suitable morphisms between the spaces of pseudoanalytic functions over the same open subset using pseudodifferentiation, and, accordingly, obtaining for each open subset of the complex plain a category whose objects are the spaces of pseudoanalytic functions defined on the subset. It is shown that assigning to open sets the appropriate categories of spaces of pseudoanalytic functions along with suitable restriction functors for inclusions of open subsets results in a 2-presheaf satisfying the gluing of morphisms. A sufficient condition for a descent datum to be effective is presented, relating the effectiveness of a descent datum to the admissibility of coefficients of a Carleman–Bers–Vekua equation.
Giorgadze et al. (Fri,) studied this question.