Global 2-rings and genuine refinements

TL;DR

This paper introduces the concept of genuine global 2-rings, linking them to equivariant cohomology theories and global spectra, with examples from spectral elliptic curves and tempered local systems.

Contribution

It defines genuine global 2-rings, connects them to equivariant cohomology, and provides concrete examples including elliptic cohomology and tempered cohomology.

Findings

Genuine global 2-rings canonically refine to E__-rings in global spectra.

Examples include quasi-coherent sheaves on torsion points of spectral elliptic curves.

Constructs global spectra representing equivariant elliptic and tempered cohomology.

Abstract

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.