We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the V-category of models of a Lawvere V-theory is equivalent to the V-category of algebras for the corresponding V-monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.

In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in call-by-value programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κ-categories. In this paper, we investigate environments in call-by-value languages by analysing a finegrain variant of Moggi’s computational λ-calculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κ-categories.

We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we de ne the notions of sketch and strict model and prove that any sketch has a generic strict model on it. This is all done with enrichment in any monoidal biclosed