Combinatorial Schema and Their Cohomology; Applications in Natural and Organizational Systems

This paper defines a new category of combinatorial modules. These are overlapping collections of small categories indexed by a directed set. They are specialized in applications to non-commuting rings acting on overlapping modules. Each combinatorial module is associated with an Abelian category of group valued sheaves which is its combinatorial scheme. An adaptation of Čech cohomology is used to analyze the structure of applications. The paper focuses on the applications of these ideas to supply chains, testing software and ecosystems. As these systems display a variety of phenomena or behaviors the area of interest is in the means of validating scenarios or assertions of capability. Cohomology is used to analyze techniques of validating scenarios in supply chains and assertions in systems. The characterization of scenarios in terms of when they can be satisfied are a start of understanding the many forms of failure (called fragmentation theory) that a supply chain or system can undergo. The intention is to motivate further development of combinatorial schema through applications that that aggregate or combine materials , validated data, concepts or nutrients along a directed graph.