Piecewise linear sheaves

TL;DR

This paper characterizes piecewise linear sheaves on finite-dimensional real vector spaces microlocally and shows they are generated by sheaves associated with convex polyhedra, linking geometric structures to sheaf theory.

Contribution

It provides a microlocal characterization of PL sheaves and proves they are generated by convex polyhedra, extending to PL gamma-sheaves and connecting to higher dimensional barcodes.

Findings

Triangulated category of PL sheaves is generated by convex polyhedra sheaves.

Microlocal characterization of PL sheaves established.

Extension to PL gamma-sheaves related to gamma-topology.

Abstract

On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL gamma-sheaves, that is, PL sheaves associated with the gamma-topology, for a closed convex polyhedral proper cone gamma. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.