Diagrammatic Negative Information

Abstract: The flow of information through a complex system can be readily understood with category theory. However, negative information (e.g., what is not possible) does not have an immediately evident categorical representation. The formalization of nategories using unconventional composition addresses this issue, and lets imposed limitations on categories be considered. However, traditional nategories abandon core categorical constructs and rely on extensive mathematical development. This creates a divide between the consideration of positive and negative information composition. In this work, we show that negative information can be considered in a natural categorical manner. This is aided by functor string diagrams, a novel flexible diagrammatic approach that can intuitively show the operation of hom-functors and natural transformations in expressions. This insight reveals how to consider the composition of negative information with foundational categorical constructs without relying on enrichment. We present diagrammatic means to consider not only nategories, but preorders more broadly. This paper introduces diagrammatic methods for the consideration of triangle inequalities and co-designs $\mathbf{DP/Feas_{Bool}}$, showing how important cases of negative information composition can be categorically and diagrammatically approached. In particular, we develop systematic tools to rigorously consider imposed limitations on systems, advancing our mathematical understanding, and present intuitive diagrams which motivate widespread adoption and usage for various applications.