(No) Categorical Semantics for Classical Protocols

Abstract

Abramsky and Coecke’s categorical quantum mechanics (Cat-QM) is an enticing start to what might one day be a reaxiomatisation of quantum mechanics. Unfortunately, most of its applications have been in the realm of technology, not philosophy. We examine Cat-QM from a philosopher’s point of view, and ask what it can do for quantum foundations. We show that applying the methods of Cat-QM to classical mechanics leads to a categorically formulated quantum-classical distinction, and take some steps in justifying its principles. Since most philosophers of physics are unfamiliar with category theory, we also provide an introduction to 2-category theory, written for this audience.