Parametrized higher semiadditivity and the universality of spans

Using the framework of ambidexterity developed by Hopkins and Lurie, we introduce a parametrized analogue of higher semiadditivity called Q-semiadditivity, depending on a chosen class of morphisms Q. Our first main result identifies the free Q-semiadditive parametrized category on a single generator with a certain parametrized span category Span (Q), simultaneously generalizing a result of Harpaz in the non-parametrized setting and a result of Nardin in the equivariant setting. As a consequence, we deduce that the Q-semiadditive completion of a parametrized category C consists of the Q-commutative monoids in C, defined as Q-limit preserving parametrized functors from Span (Q) to C. As our second main result, we provide an explicit `Mackey sheaf' description of the free presentable Q-semiadditive category. Using this, 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 Z-valued G-Mackey profunctors and of quasi-finitely genuine G-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively.