2-dimensional Lawvere theories, commutativity, and higher Day convolution

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 ${\mathsf{M}\mathsf{o}\mathsf{d}}_l(\mathbb{T},\mathbf{C}\mathbf{a}\mathbf{t})$ 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 $(\mathrm{\infty },2)$-theories. As a result of independent interest, we construct a multicategory (or $\mathrm{\infty }$-operad) structure on the hom-category ${\mathsf{H}\mathsf{o}\mathsf{m}}_{\mathbb{V}}(\mathcal{M},\mathcal{N})$, where $\mathbb{V}$ is a monoidal $(\mathrm{\infty },2)$-category and $\mathcal{M},\mathcal{N}$ are monoids therein.