Programs as Diagrams: From Categorical Computability to Computable Categories

This is a draft of the first 8 chapters of a textbook/monograph that presents computability theory using string diagrams. The introductory chapters have been taught as graduate and undergraduate courses and evolved through 8 years of lecture notes. The later chapters contain new ideas and results about categorical computability and some first steps into computable category theory. The underlying categorical view of computation is based on monoidal categories with program evaluators, called *monoidal computers*. This categorical structure can be viewed as a single-instruction diagrammatic programming language called Run, whose only instruction is called RUN. This version: changed the title, worked on improving the text. (Also added lots of exercises and workouts, but that was overflowing the arxiv size bounds already in the earlier version, which is why the "workouts" and the "stories" are commented out.)