A derived isometry theorem for sheaves

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [18] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D_Rc^b(k_R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D_Rc^b(k_R) and provide some explicit examples of computation of the convolution distance.