Global 2-rings and genuine refinements

Abstract We introduce the notion of a naive global 2-ring: a functor from the opposite of the normal infinity ∞ $\infty$ -category of global spaces to presentably symmetric monoidal stable normal infinity ∞ $\infty$ -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 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 double struck upper E Subscript normal infinity E ∞ $\mathbb{E}_\infty$ -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.