  We introduce the notion of a naive global 2-ring: a functor from the opposite
of the $\infty$-category of global spaces to presentably symmetric monoidal
stable $\infty$-categories. By passing to global sections, every naive global
2-ring decategorifies to a multiplicative cohomology theory on global spaces,
i.e. a naive global ring. We suggest when a naive global 2-ring deserves to be
called \emph{genuine}. As evidence, we associate to such a global 2-ring a
family of equivariant cohomology theories which satisfy a version of the change
of group axioms introduced by Ginzburg, Kapranov and Vasserot. We further show
that the decategorified multiplicative global cohomology theory associated to a
genuine global $2$-ring canonically refines to an $\mathbb{E}_\infty$-ring
object in global spectra. As we show, two interesting examples of genuine
global 2-rings are given by quasi-coherent sheaves on the torsion points of an
oriented spectral elliptic curve and Lurie's theory of tempered local systems.
In particular, we obtain global spectra representing equivariant elliptic
cohomology and tempered cohomology.
