  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
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 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.
