Transmutations in Supergeometry

Abstract: In recent times, due to the influence of prismatic cohomology, the stacky approach to cohomology has been getting more popular, where one uses so-called transmutations to obtain different cohomologies as the coherent cohomology of the transmuted space and exhibiting the category of quasi-coherent sheaves as the natural coefficient system of that cohomology theory.

We will explain how this approach to cohomology manifests in supergeometry. We will showcase how the de Rham stack simplifies classical theorems in supergeometry and give a refinement of the Betti stack. In particular, we explain how one can use derived algebraic geometry to obtain corresponding notions in supergeometry.