The not-quite-six days of the creation of quantum theory as universal algebra

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Seminal work of Steve Lack showed that universal algebraic theories (PROPs) may be composed to produce more sophisticated theories.  I’ll apply this method to construct an axiomatic version of the theory of a pair of complementary observables starting from the theory of monoids.  How far can we get with this?  Quite far!  We’ll get a large chunk of finite dimensional quantum theory this way —but the fact that quantum systems have non-trivial dynamics means that it’s (always) possible to present the resulting theory as a composite PROP in Lack’s sense.  If time permits, I’ll also discuss how this approach can serve as a way of constructing toy models with specific properties.
TRIGGER WARNING: category theory, blasphemy.