A construction of intersection cohomology from a simplicial version of the Deligne axioms

Abstract

Intersection cohomology is a way to enhance classical cohomology, allowing us to use a famous result called Poincar\'e duality on a large class of spaces known as stratified pseudomanifolds. There is a theoretically powerful way to arrive at intersection cohomology by classifying sheaves that satisfy what are called Deligne axioms. We stablish an abstract manifestation of the Deligne axioms, to then apply it on a simplicial complex environment, for a category of simplicial sheaves inspired on the works of D. Chataur, D. Tanr\'e and M. Saralegi-Araguren. For a stablished topology on a triangulation of a stratified pseudomanifold, we find a family of sheaves satisfying the simplicial Deligne axioms, giving us a way to construct intersection cohomology from simplicial sheaves.