A seminal result in category theory is the Lawvere-Linton correspondence between the equational theories of Birkhoff and (finitary) monads on the category of sets. Over the last decade, several variants of this result have been established in numerous settings including the category of posets and the category of metric spaces. The aim of this talk is to discuss a generic framework of universal algebra in categories of relational structures given by models of the (infinitary) limit-theories of Coste and Rosický: these are perhaps better known as locally presentable categories. We will sketch the construction of an enriched and accessible monad from a given relational algebraic theory.

The content of this talk emerged from joint work with Lutz Schröder and Stefan Milius: https://arxiv.org/abs/2107.03880.