Gabriel-Ulmer duality and Lawvere theories enriched over a general base

Abstract

Abstract Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set . We generalise that relationship from Set to an arbitrary locally presentable category such as Poset and ω Cpo or functor categories such as [ Inj , Set ] and [ Inj , ω Cpo ]. That involves allowing the arities of Lawvere theories to be extended to being size-restricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabriel–Ulmer duality.