$2$-dimensional Lawvere theories: commutativity and lax phenomena
Tomáš Perutka
TL;DR
This work develops a 2-categorical framework for categorified algebra via Lawvere 2-theories and enhanced sketches, introducing 2-dimensional commutativity as a central condition. It proves a syntax–semantics correspondence for w-commutativity, shows that such commutativity endows Mod_w(T,Cat) with a closed 2-multicategory structure, and generalizes Fox's theorem to a 2-dimensional (and partially to ∞-2) setting. It also builds a generalized Day convolution framework and a bilax/higher-convolution theory for maps between pseudomonoids, with convolution-style phenomena and a detailed treatment of Euler–Hilton-type rigidity. The higher analogues to (∞,2)-theories are sketched, laying groundwork for future extensions to the ∞-categorical realm and highlighting open questions about monadic and comonadic aspects in the higher setting.
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.