Category theory provides ubiquitous computing systems with a variety of formal approaches including the coalgebraic representation of automatons and transition systems, Domain theory, and the Geometry of Interaction, along with specific mathematical substrates such as the elementary topos. Computational aspects of the digital economy bring together three closely related spheres of research: cognitive computing, artificial general intelligence and semantic technologies. Semantic technologies provide users with integrated access to both structured and unstructured data by applying search and navigation techniques that are tuned to the computational ontologies of relevance to the organization. Researchers within the field of Artificial General intelligence are seeking the "holy grail" of robotics and computation, first characterized by John von Neumann in terms of interacting automatons whose progeny are continuously evolving to be smarter than their parents. The stance adopted towards artificial intelligence (AI) and Cognitive Computing is one that acknowledges the strides taken by the "new AI" paradigm of perception-and-action-without-intelligence.