$2$-dimensional Lawvere theories: commutativity and lax phenomena

TL;DR

This paper explores 2-dimensional Lawvere theories, focusing on commutativity and lax phenomena, establishing a multicategory structure on models and extending classical theorems to higher categorical contexts.

Contribution

It introduces the concept of 2-dimensional commutativity in Lawvere theories and constructs a multicategory structure on models, generalizing Fox's theorem to higher categories.

Findings

Models form a closed 2-multicategory under 2-dimensional commutativity.

Generalization of Fox's theorem to 2-categorical setting.

Construction of multicategory structure on hom-categories in monoidal (∞,2)-categories.

Abstract

The aim of this paper is to study categorified algebraic structures and their pseudo- and lax homomorphisms using the framework of Lawvere $2$-theories, and more generally, (enhanced) $2$-dimensional sketches. The key notion we focus on is that of $2$-dimensional commutativity. As one of the main results, we prove that if a Lawvere $2$-theory $\mathbb{T}$ is equipped with such a structure, then the $2$-category $\mathsf{Mod}_l(\mathbb{T},\mathbf{Cat})$ of $\mathbb{T}$-models, lax homomorphisms, and modifications admits a natural structure of a closed $2$-multicategory. From this, we deduce a generalization of Fox's theorem. We also discuss the analogue in the higher setting for Lawvere $(\infty,2)$-theories. As a result of independent interest, we construct a multicategory (or $\infty$-operad) structure on the hom-category $\mathsf{Hom}_{\mathbb{V}}(\mathcal{M},\mathcal{N})$, where $\mathbb{V}$ is a monoidal $(\infty,2)$-category and $\mathcal{M},\mathcal{N}$ are monoids therein.