Lawvere theories, finitary monads and Cauchy-completion

Abstract

We consider the equivalence of Lawvere theories and finitary monads on Set from the perspective of F-enriched category theory, where F is the monoidal category of finitary endofunctors of Set under composition. We identify finitary monads with one-object F-categories, and ordinary categories admitting finite powers (i.e., n-fold products of each object with itself) with F-categories admitting a certain class Φ of absolute colimits; we then show that, from this perspective, the passage from a finitary monad to the associated Lawvere theory is given by completion under Φ-colimits. We also account for other phenomena from the enriched viewpoint: the equivalence of the algebras for a finitary monad with the models of the corresponding Lawvere theory; the functorial semantics in arbitrary categories with finite powers; and the existence of left adjoints to algebraic functors.