An enriched view on the extended finitary monad--Lawvere theory correspondence

TL;DR

This paper provides an enriched category theory perspective on the monad-Lawvere theory correspondence, generalizing previous results to arbitrary locally finitely presentable bases using bicategory enrichment.

Contribution

It introduces a new enriched categorical framework that explains the monad-theory correspondence over general bases, extending prior set-based results.

Findings

Generalizes the monad-theory correspondence to arbitrary bases

Uses bicategory enrichment to capture the correspondence

Shows the passage as a free completion under absolute colimits

Abstract

We give a new account of the correspondence, first established by Nishizawa--Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of enriched category theory: the passage from a finitary monad to the corresponding Lawvere theory is exhibited as an instance of free completion of an enriched category under a class of absolute colimits. This extends work of the first author, who established the result in the special case of finitary monads and Lawvere theories over the category of sets; a novel aspect of the generalisation is its use of enrichment over a bicategory, rather than a monoidal category, in order to capture the monad--theory correspondence over all locally finitely presentable bases simultaneously.