Directed homological and cohomological operations

TL;DR

This paper introduces a persistence module approach to directed cohomology, exploring its properties and operations for specific precubical sets and general directed spaces, complementing previous work on directed homology.

Contribution

It presents the first properties of directed cohomology and cohomological operations, providing a dual perspective to existing directed homology theories.

Findings

Defined directed cohomology using persistence modules

Established properties of directed cohomological operations

Linked cohomological operations to homological counterparts

Abstract

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of cohomological operations, partially linked to some homological operations. We treat here both the case of a specific class of precubical sets and of general directed spaces.