2025/01/13 by David Reutter, Reutter, David, Markus Zetto +1
Mathematics · Medicine · #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Intracranial Aneurysms: Treatment and Complications #Rings, Modules, and Algebras #Vascular Malformations Diagnosis and Treatment
paper · pdf · doi:10.48550/arxiv.2501.07697
openalex publication_date 2025/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that an enriched ∞-category is completely determined by its enriched presheaf category together with a `marking' by the representable presheaves. More precisely, for any presentably monoidal ∞-category V we construct an equivalence between the category of V-enriched ∞-categories and a certain full sub-category of the category of presentable V-module categories equipped with a functor from an ∞-groupoid. This effectively allows us to reduce many aspects of enriched ∞-category theory to the theory of presentable ∞-categories. As applications, we use Lurie's tensor product of presentable ∞-categories to construct a tensor product of enriched ∞-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 ∞-categories. Our comparison result relies on a monadicity theorem for presentable module categories which may be of independent interest.