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Greetings, I am the Chief Architect at the Arizona Supreme Court.
I am applying Category Theory to business, specifically in the way we design software. We are using OLOGs and String Diagrams to build graphical representations directly connected to the functional requirements.
Graphics are used by Subject Matter Experts with no understanding of the underlying math or science for the system. Instead they are given a ubiquitous language (developed interactively at the domain level) with which to communicate and collaborate across specific subject areas and specialties. I am applying Conceptual Spaces, Topology, and the ideas from Gardenfors (along with Brouwer, Heyting, and Goldblatt) to bridge the Language semantics of graphics, business processes, and software development. Most of our software is used for internal process communication and Category Theory has proven to be the proper tool for the job.
Adding guidance for how to apply that to business processing and software design is my focus. I am not referring to "functional programming" though that does play a role. I am referring more to the business process and understanding the information science behind the process outside of any applied technology.