Theories

We consider the equivalence of Lawvere theories and finitary monads on Set from the perspective of Endf (Set)-enriched category theory, where Endf (Set) is the category of finitary endofunctors of Set. We identify finitary monads with one-object Endf (Set)-categories, and ordinary categories admitting finite powers (i.e., n-fold products of each object with itself) with Endf (Set)-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.