Extending work of Nardin, we develop a framework of parameterized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing ∞-category T . Specializing this framework, we introduce global ∞-categories together with genuine forms of semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2-functor studied by Balmer-Dell'Ambrogio. As our main result, we identify the free presentable genuinely stable global ∞-category with a natural global ∞-category of global spectra for finite groups, in the sense of Schwede and Hausmann. BASTIAAN CNOSSEN, TOBIAS LENZ, AND SIL LINSKENS 4.7. Finite pointed P -sets 60 4.8. P -commutative monoids 62 4.9. Commutative monoids in E T 65 5. The universal property of special global Γ-spaces 69 5.1. A reminder on G-global Γ-spaces 69 5.2. Global Γ-spaces as parameterized functors 72 5.3. Proof of Theorem B 80 6. Parameterized stability 84 6.1. Fiberwise stable T -∞-categories 84 6.2. P -stable T -∞-categories 87 7. The universal property of global spectra 90 7.1. Stable G-global homotopy theory 91 7.2. From global Γ-spaces to global spectra 94 7.3. Proof of Theorem C 97 Appendix A. Slices of (2, 1)-categories 98 References 101