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

TL;DR

This paper develops a new approach to construct intersection cohomology using simplicial sheaves that satisfy Deligne axioms, extending classical theories to a combinatorial setting.

Contribution

It introduces an abstract formulation of Deligne axioms applied to simplicial complexes, enabling intersection cohomology construction in a combinatorial framework.

Findings

Established a topology on triangulations of stratified pseudomanifolds.

Constructed sheaves satisfying simplicial Deligne axioms.

Provided a simplicial method for intersection cohomology.

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.