Combining computational effects: commutativity and sum

We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi’s paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi’s side-effects monad transformer (an application is the combination of side-effects with nondeterminism). And we give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer (an application is the combination of exceptions with other effects).