Parametrized (higher) semiadditivity and the universality of spans
Bastiaan Cnossen, Tobias Lenz, Sil Linskens
TL;DR
The work develops a comprehensive framework for parametrized semiadditivity in ${\mathcal B}$-categories using ambidexterity and span categories, culminating in the free ${\mathcal Q}$-semiadditive ${\mathcal B}$-category ${\ul{\mathrm Span}}({\mathcal Q})$ and the universal ${\mathcal Q}$-semiadditive completion ${\ul{\mathrm CMon}}^{\mathcal Q}({\mathcal C})$. It then shows that ${\mathcal Q}$-commutative monoids provide a presentable universal property for these completions and that such structures canonically enrich in Mackey functors/transfer objects, with applications to global spectra and Mackey-profunctor descriptions. The paper further develops the parametrized span formalism, proving Segal/CoSegal equivalences, coSegal-to-Segal dualities, and universal characterizations, and applies these to obtain transferral descriptions of sheaves with transfers and their free constructions. Overall, the framework unifies ordinary, equivariant, and higher semiadditivity under a single parametrized perspective, yielding universal, enrichment, and transfer-based results with broad implications for global and equivariant homotopy theory.
Abstract
Semiadditivity of an $\infty$-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 $\infty$-category of commutative monoids is the universal semiadditive $\infty$-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 $\infty$-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 $\infty$-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 $\infty$-categories of $\mathbb Z$-valued $G$-Mackey profunctors and of quasi-finitely genuine $G$-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively.