An Expansion of the Continuity Property

TL;DR

This paper extends the continuity property of Alexander-Spanier-ech cohomology to presheaves, enabling better understanding of cohomology in inverse limits of spaces and groups.

Contribution

It introduces a new framework for systems of presheaves, broadening the applicability of the continuity property in cohomology theory.

Findings

Established a limit-presheaf system for cohomology of inverse limits.

Proved the continuity property extends to presheaves in ech cohomology.

Applied results to compare cohomology of inverse limits of finite groups and their classifying spaces.

Abstract

One of the advantages of working with Alexander-Spanier-\v{C}ech type cohomology theory is the continuity property: For inverse systems of sufficiently well-behaved spaces, the result of taking the cohomology of their limit is a direct limit of their cohomologies. However, \v{C}ech cohomology natively works with presheaves of modules rather than modules themselves. We define the notion of a system of presheaves for an inverse system of topological spaces, and show that, under the same circumstances as the ordinary continuity property, a suitable limit of the system provides the \v{C}ech cohomology of the inverse limit of the spaces. We then show one application of this result in comparing the cohomology of an inverse limit of finite groups to that of the inverse limit of their classifying spaces.