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After a brief review on sheaves in the derived setting and the notion of gamma-sheaves, I will expose the main results of arXiv:1705.00955 and arXiv:1805.00349, a joint work with Masaki Kashiwara.

The aim is to better understand persistent homology in higher dimension. For that purpose, one first proves that constructible sheaves on a real finite dimensional vector space, endowed with the gamma-topology, can be approximated (in the derived sense) by piecewise linear sheaves (PL sheaves). Then we study PL sheaves and show that the triangulated category of such sheaves is generated by sheaves associated with convex polytopes, a natural analogue of barcodes in higher dimension.