Higher Lawvere theories

Abstract

We survey Lawvere theories at the level of ∞-categories, as an alternative framework for higher algebra (rather than ∞-operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They also play a prominent role in equivariant homotopy theory and its relatives.Our main result establishes a universal property for the ∞-category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its ∞-category of models. Many familiar properties of Lawvere theories follow directly.As a consequence, we establish a correspondence between enriched and module Lawvere theories, which implies that the Burnside category is a classifying object for additive categories. This completes a proof from our earlier paper on the commutative algebra of categories.