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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 $$\mathbb {V}$$ 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 $$\mathbb {V}$$ V is endowed with a closed convex proper cone $$\gamma $$ γ with non empty interior and study $$\gamma $$ γ -sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the $$\gamma $$ γ -topology). We prove that such sheaves may be approximated (for the pseudo-distance) by “piecewise linear” $$\gamma $$ γ -sheaves. Finally we show that these last sheaves are constant on stratifications by $$\gamma $$ γ -locally closed sets, an analogue of barcodes in higher dimension.
Journal of Applied and Computational Topology – Springer Journals
Published: Sep 18, 2018
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