Before revision:  We prove an isometry theorem in this derived setting,  expressing the convolution distance  as a matching distance , mirroring the situation of persistence with one parameter.
After revision:  We prove an isometry theorem in this derived setting, expressing the convolution distance 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.