Cech cohomology of partially ordered sets

TL;DR

This paper compares cech cohomology and topos cohomology for posets with the Aleksandrov topology, providing criteria for their isomorphism and algorithms for verification.

Contribution

It establishes necessary and sufficient conditions for the isomorphism of cech and topos cohomology in posets, including an algorithmic approach for finite cases.

Findings

Criteria for cohomology invariance under inverse image

Conditions reduce to acyclicity of Dedekind-MacNeille cuts

Algorithmic verification for finite posets

Abstract

The article is devoted to a comparison of the \v{C}ech cohomology with the coefficients in a presheaf of Abelian groups and the topos cohomology of the sheaf generated by this presheaf for a poset with the Aleksandrov topology. The article consists of three parts. The first part provides information from the theory of cohomology of small categories and cohomology of simplicial sets with systems of coefficients. The second part is devoted to Laudal's Theorem stating that covering cohomology for an arbitrary topological space with coefficients in the presheaf of Abelian groups is isomorphic to the derived limit functors. The third part presents the main results. The criterion for the invariance of cohomology groups of small categories when passing to the inverse image leads to necessary and sufficient isomorphism conditions for the \v{C}ech cohomology of an arbitrary presheaf and the topos cohomology of the sheaf generated by this presheaf. In particular, for a finite poset, these conditions reduce to the acyclicity of the upper secrions of Dedekind-MacNeille cuts having a non-empty lower section, and the verification of these conditions is algorithmically computable.