Notions of Lawvere theory

TL;DR

This paper explores the generalization of Lawvere theories in categorical universal algebra, extending the equivalence with monads to broader contexts such as enriched categories and different classes of limits.

Contribution

It introduces a framework for generalizing Lawvere theories beyond Set, including enriched settings and alternative limit classes, expanding their applicability.

Findings

Established equivalence between Lawvere theories and monads in new contexts

Generalized universal algebra results to enriched categories

Extended Lawvere theory concepts to sifted-colimit-preserving monads

Abstract

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones.