Higher category theory as a paradigm for network applications

Abstract

The importance of network science to the present and future military is unquestioned. Networks of some type pervade every aspect of military operations-a situation that is shared by civilian society. However, several aspects of militarily oriented network science must be considered unique or given significantly greater emphasis than their civilian counterparts. Military, especially battlespace, networks must be mobile and robust. They must utilize diverse sensors moving in and out of the network. They must be able to survive various modes of attack and the destruction of large segments of their structure. Nodes often must pass on classifications made locally while other nodes must serve as combined sensor/classifiers or information coordinators. They must be capable of forming fluidly and in an ad hoc manner. In this paper, it will be shown how category theory, higher category theory, and topos theory provide just the model required by military network science. Category theory is a well-developed mathematical field that views mathematical structures abstractly, often revealing previously unnoticed correspondences. It has been used in database and software modeling, and in sensor and data fusion. It provides an advantage over other modeling formalisms both in its generality and in its extensive theory. Higher category theory extends the insights of category theory into higher dimensions, enhancing robustness. Topos theory was developed, in part, through the application of category theory to logic, but it also has geometric aspects. The motivation behind including topos theory in network science is the idea that a mathematical theory fundamental to geometry and logic should be applicable to the study of systems of spatially distributed information and analysis flow. The structures presented in this paper will have profound and far-reaching applications to military networks.