# Higher Lawvere theories

## TL;DR

This paper explores Lawvere theories within infinity categories as a more accessible and versatile framework for higher algebra, especially suited for equivariant homotopy theory, and establishes a universal property linking theories to their models.

## Contribution

It introduces a universal property for the infinity category of Lawvere theories, clarifies their relationship with models, and applies these concepts to classify additive categories and relate enriched theories.

## Key findings

- Burnside category as a classifying object for additive categories
- Universal property characterizing Lawvere theories and models
- Enhanced understanding of Lawvere theories in higher algebra

## Abstract

We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives. Our main result establishes a universal property for the infinity category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its infinity category of models. Many familiar properties of Lawvere theories follow directly. As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.