Semantics for Local Computational Effects

Starting with Moggi's work on monads as refined to Lawvere theories, we give a general construct that extends denotational semantics for a global computational effect canonically to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state. Given any Lawvere theory L, possibly countable and possibly enriched, we first give a universal construction that extends L, hence the global operations and equations of a given effect, to incorporate worlds of arbitrary finite size. Then, making delicate use of the final comodel of the ordinary Lawvere theory L, we give a construct that uniformly allows us to model block, the universality of the final comodel yielding a universal property of the construct. We illustrate both the universal extension of L and the canonical construction of block by seeing how they work in the case of state.