We introduce the notion of a naive global 2-ring: a functor from the opposite of the
∞-category of global spaces to presentably symmetric monoidal stable ∞-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 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
[V. Ginzburg, M. Kapranov and E. Vasserot, Elliptic algebras and equivariant elliptic
cohomology, Preprint (1995), arXiv:q-alg/9505012]. We further show that the decategorified
multiplicative global cohomology theory associated to a genuine global 2-ring
canonically refines to an E∞-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.