  Persistent homology has been recently studied with the tools of sheaf theory
in the derived setting by Kashiwara and Schapira, 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
$D^b_{\mathbb{R} c}(\textbf{k}_\mathbb{R})$, 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
$D^b_{\mathbb{R} c}(\textbf{k}_\mathbb{R})$and provide some explicit examples
of computation of the convolution distance.
