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Journal Articles Advances in Mathematics Year : 2022

A derived isometry theorem for sheaves

Abstract

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18] 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 convo- lution distance of sheaves as a matching distance between combinatorial objects asso- ciated 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 Db Rc(kR), which enables us to compute distances between indecom- posable 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 Db Rc(kR) and provide some explicit examples of computation of the convolution distance.
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Dates and versions

hal-03864712 , version 1 (03-08-2023)

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Cite

Nicolas Berkouk, Grégory Ginot. A derived isometry theorem for sheaves. Advances in Mathematics, 2022, 394, pp.108033. ⟨10.1016/j.aim.2021.108033⟩. ⟨hal-03864712⟩
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