Lawvere theories, finitary monads and Cauchy-completion

TL;DR

This paper explores the deep connection between Lawvere theories and finitary monads on Set using enriched category theory, showing how completion under certain colimits relates the two concepts and analyzing their algebraic models.

Contribution

It provides a new enriched categorical perspective on the equivalence of Lawvere theories and finitary monads, including the role of Cauchy-completion and colimits.

Findings

Finitary monads correspond to one-object Endf(Set)-categories.

The passage from monads to Lawvere theories is via completion under Phi-colimits.

Algebras for monads are equivalent to models of Lawvere theories.

Abstract

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 Phi 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 Phi-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.