Combinatorial Systems and Schema and their Cohomology with Applications in Organizational Systems

The category of combinatorial systems is a model category for ”scaling up” algorithms. This involves networks of algorithms, herein called schedules, acting on very wide range of inputs types that are outputs from other algorithms. This patching algorithms over networks gives rise to the subcategory of combinatorial modules which are an algebraic analogy for the creation of manifolds from Euclidean spaces. Combinatorial systems model large-scale systems including precision manufacturing networks. Schedules have both local and global significance and form group valued sheaves, the Abelian category of which is the combinatorial schema for a given combinatorial system. It contains the data on the dynamics and capability of the system with individual sheaves corresponding to dynamic conditions. A finite network adaptation of Čech cohomology is used to characterize the sheaves. A given sheaf of schedules has a large class of related sheaves of measures, groupoid actions etcetera. The cohomology classes of these sheaves have significant interpretations in manufacturing networks such as component substitutions and operating conditions. For example, the cohomology of groupoid actions applies to rescheduling in networks of factories. In other cases cohomology tracks the change in capability to handle excessive order loads.