Lawvere theories enriched over a general base

Abstract

We generalise the correspondence between Lawvere theories and finitary monads on Set in two ways. First, we allow our theories to be enriched in a category V that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable V -category A . We call the resulting notion that of a Lawvere A -theory. We extend the correspondence for ordinary Lawvere theories to one between Lawvere A -theories and finitary V -monads on A . We illustrate this with examples leading up to that of the Lawvere Cat -theory for cartesian closed categories, i.e., the Set -enriched theory on the category Cat for which the models are all small cartesian closed categories. We also briefly investigate change-of-base.