A Cohomological Bundle Theory for Sheaf Cohomology

TL;DR

This paper introduces a bundle-theoretic approach to sheaf cohomology on small categories, providing a spectral sequence for cohomological reduction that generalizes existing methods for posets with a maximum.

Contribution

It develops a new bundle theory for presheaves on small categories, extending cohomological reduction techniques beyond posets with a maximum element.

Findings

Established a Leray-Serre type spectral sequence for certain presheaves.

Extended cohomological reduction methods to broader classes of posets.

Provided a framework connecting bundle theory with sheaf cohomology.

Abstract

We develop a bundle theory of presheaves on small categories, based on similar work by Brent Everitt and Paul Turner. For a certain set of presheaves on posets, we produce a Leray-Serre type spectral sequence that gives a reduction property for the cohomology of the presheaf. This extends the usual cohomological reduction of posets with a unique maximum.