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Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18a] 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 Rc (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 b Rc (k R ) and provide some explicit examples of computation of the convolution distance.

One of the main challenges of Topological Data Analysis (TDA) is to extract features from persistent diagrams directly usable by machine learning algorithms. Indeed, persistence diagrams are intrinsically (multi-)sets of points in R 2 and cannot be seen in a straightforward manner as vectors. In this article, we introduce Persformer, the first Transformer neural network architecture that accepts persistence diagrams as input. The Persformer architecture significantly outperforms previous topological neural network architectures on classical synthetic benchmark datasets. Moreover, it satisfies a universal approximation theorem. This allows us to introduce the first interpretability method for topological machine learning, which we explore in two examples.