$2$-dimensional Lawvere theories: commutativity and lax phenomena
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915799760896000 |
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| author | Perutka, Tomáš |
| author_facet | Perutka, Tomáš |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14332 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $2$-dimensional Lawvere theories: commutativity and lax phenomena Perutka, Tomáš Category Theory Algebraic Topology Logic 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. |
| title | $2$-dimensional Lawvere theories: commutativity and lax phenomena |
| topic | Category Theory Algebraic Topology Logic |
| url | https://arxiv.org/abs/2602.14332 |