Title: Enriched algebraic theories, monads, and varieties
Abstract: In this talk, I will summarize the research on enriched algebraic theories, monads, and varieties that I produced with Rory Lucyshyn-Wright during my previous postdoctoral fellowship at Brandon University. I will start by providing a historical overview of the subject, which originated with Birkhoff, Lawvere, and Linton in the 1930s and 1960s. Birkhoff initiated the study of universal algebra by defining the notion of an (equational) variety of algebras, which is a class of algebraic structures axiomatized by certain equations. Lawvere and Linton then established purely categorical formulations of Birkhoff’s varieties, in terms of Lawvere theories and finitary monads on Set. I will then describe the ways in which, over the next 50 years, various researchers generalized Lawvere theories and finitary monads to certain enriched settings, by developing enriched notions of Lawvere theory and monad. None of these frameworks developed an enriched notion of variety, and moreover they were largely formulated for locally presentable bases of enrichment, which exclude many important categories of a topological flavour. Moreover, there was no overarching framework that encompassed all of these works. To address these issues, Rory and I developed notions of enriched Lawvere theory, enriched monad, and enriched variety that encompassed all of these prior works and also extended their applicability to many new mathematical settings (especially in topology and analysis). I will conclude by mentioning some of my current and planned research on this topic, which I will talk about at future seminars.