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Persistent homology and microlocal sheaf theory. (English) Zbl 1423.55013

Summary: We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space \(\mathbb{V}\). By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that \(\mathbb{V}\) is endowed with a closed convex proper cone \(\gamma \) with non empty interior and study \(\gamma \)-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the \(\gamma \)-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by “piecewise linear” \(\gamma \)-sheaves. Finally we show that these last sheaves are constant on stratifications by \(\gamma \)-locally closed sets, an analogue of barcodes in higher dimension.

MSC:

55N99 Homology and cohomology theories in algebraic topology
18A99 General theory of categories and functors
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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