2024/03/12 by Bastiaan Cnossen, Cnossen, Bastiaan, Tobias Lenz +3 · 3 citations
Computer Science · Mathematics · #18N60 (Primary) #55P91 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2403.07676
openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Semiadditivity of an ∞-category, i.e. the existence of biproducts, provides it with useful algebraic structure in the form of a canonical enrichment in commutative monoids. This ultimately comes from the fact that the ∞-category of commutative monoids is the universal semiadditive ∞-category equipped with a finite-product-preserving functor to spaces, or equivalently that the (2,1)-category of spans of finite sets is the universal semiadditive ∞-category. In this article, we prove a vast generalization of these facts in the context of parametrized semiadditivity, a notion we define using Hopkins-Lurie's framework of ambidexterity. This simultaneously generalizes a result of Harpaz for higher semiadditivity and a result of Nardin for equivariant semiadditivity. We deduce that every parametrized semiadditive ∞-category is canonically enriched in Mackey functors/sheaves with transfers. As an application, we reprove the Mackey functor description of global spectra first obtained by the second-named author and generalize it to G-global spectra. Moreover, we obtain universal characterizations of the ∞-categories of \mathbb Z-valued G-Mackey profunctors and of quasi-finitely genuine G-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively.