arXiv subjects

Markus Zetto

Publications and source records attributed to Markus Zetto.

3 recordsLinked to original sources

arXiv2026

Enriched $\infty$-operads as marked algebras

We show that an enriched $\infty$-operad is completely determined by its category of right modules together with a `marking' of the representable modules. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of colored $\mathcal{V}$-enriched $\infty$-operads and a certain full subcategory of the category of presentably symmetric monoidal $\mathcal{V}$-module $\infty$-categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-operad theory to the theory of presentably symmetric monoidal $\infty$-categories. As an application, we describe a notion of univalence (or Rezk-completeness) for enriched $\infty$-operads, and directly construct an equivalence between univalent $\mathcal{S}$-enriched $\infty$-operads in our sense and Lurie's model of $\infty$-operads. We study envelopes and categories of algebras for enriched $\infty$-operads and show that, in the $\mathcal{S}$-enriched case, the resulting notions agree in both models.

math.AT↗

arXiv2025

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

math.QA↗

arXiv2025

Enriched $\infty$-categories as marked module categories

We prove that an enriched $\infty$-category is completely determined by its enriched presheaf category together with a `marking' by the representable presheaves. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of $\mathcal{V}$-enriched $\infty$-categories and a certain full sub-category of the category of presentable $\mathcal{V}$-module categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-category theory to the theory of presentable $\infty$-categories. As applications, we use Lurie's tensor product of presentable $\infty$-categories to construct a tensor product of enriched $\infty$-categories with many desirable properties -- including compatibility with colimits and appropriate monoidality of presheaf functors -- and compare it to existing tensor products in the literature. We also re-examine and provide a model-independent reformulation of the notion of univalence (or Rezk-completeness) for enriched $\infty$-categories. Our comparison result relies on a monadicity theorem for presentable module categories which may be of independent interest.

math.AT↗