Global Picard Spectra and Borel Parametrized Algebra

We answer a question of Schwede on the existence of global Picard spectra associated to his ultra-commutative global ring spectra; given an ultra-commutative global ring spectrum $R$, we show there exists a global spectrum ${\mathrm{p}\mathrm{i}\mathrm{c}}_{\mathrm{e}\mathrm{q}}(R)$ assembling the Picard spectra of all underlying $G$-equivariant ring spectra ${\mathrm{r}\mathrm{e}\mathrm{s}}_{G}R$ of $R$ into one object, in that for all finite groups $G$, the genuine fixed points are given by ${\mathrm{p}\mathrm{i}\mathrm{c}}_{\mathrm{e}\mathrm{q}}(R{)}^G\simeq \mathrm{p}\mathrm{i}\mathrm{c}({\mathrm{M}\mathrm{o}\mathrm{d}}_{{\mathrm{r}\mathrm{e}\mathrm{s}}_{G}R}({\mathrm{S}\mathrm{p}}_G))$. Along the way, we develop a generalization of Borel-equivariant objects in the setting of parametrized higher algebra. We use this to assemble the symmetric monoidal categories of $G$-spectra for all finite groups $G$ together with all restrictions and norms into a single `normed global category’, and build a comparison functor which allows us to import ultra-commutative $G$-equivariant or global ring spectra into the setting of parametrized higher algebra.