Simplifications of finite spaces equipped with sheaves

TL;DR

This paper introduces a cohomological approach to simplifying finite sheaved topological spaces, generalizing beat vertices, and provides an algorithm for reduction that preserves sheaf cohomology.

Contribution

It generalizes the concept of beat vertices to sheaved spaces and proposes an algorithm for simplification that maintains cohomological properties.

Findings

Removal of acyclic vertices preserves sheaf cohomology.

Introduces a cohomological analogue of a space's core.

Provides an algorithm for space simplification.

Abstract

Following the classical results of Stong, we introduce a cohomological analogue of a core of a finite sheaved topological space and propose an algorithm for simplification in this category. In particular we generalize the notion of beat vertices and show that if a vertex of a sheaved space has topologically acyclic downset (with trivial coefficients), then its removal preserves the sheaf cohomology.