Product Rules and Distributive Laws

We give a categorical perspective on various product rules, including Brzozowski’s product rule ((st)a = sat + o(s)ta) and the familiar rule of calculus ((st)a = sat+ sta). It is already known that these product rules can be represented using distributive laws, e.g. via a suitable quotient of a GSOS law. In this paper, we cast these product rules into a general setting where we have two monads S and T , a (possibly copointed) behavioural functor F , a distributive law of T over S, a distributive law of S over F , and a suitably defined distributive law TF ⇒ FST . We introduce a coherence axiom giving a sufficient and necessary condition for such triples of distributive laws to yield a new distributive law of the composite monad ST over F , allowing us to determinize FST -coalgebras into lifted F coalgebras via a two step process whenever this axiom holds.