Discrete Lawvere theories and computational effects

AbstractCountable Lawvere theories model computational effects such as exceptions, side-effects, interactive input/output, nondeterminism and probabilistic nondeterminism. The category of countable Lawvere theories has sums, tensors, and distributive tensors, modelling natural combinations of such effects. It is also closed under taking images. Enrichment in a category such as ωCpo allows one to extend this modelling of computational effects to account for partiality and recursion. Sum and tensor extend to enriched countable Lawvere theories, but distributive tensor and image do not. So here we introduce discrete countable enriched Lawvere theories in order to allow natural definitions and accounts of distributive tensor and image. A discrete countable enriched Lawvere theory is, in a sense we make precise, an enriched Lawvere theory with discrete arities. We show that they include all our leading examples of computational effects and are closed under sum and tensor. And we develop notions of enriched operad and enriched multicategory to support the definition