Global 2-rings and genuine refinements
- 1Johns Hopkins University
- 2Universität Regensburg
- 3Max Plank Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Abstract
We introduce the notion of a naive global 2-ring: a functor from the opposite of the normal infinity ∞ ∞ -category of global spaces to presentably symmetric monoidal stable normal infinity ∞ ∞ -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 ∞ 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.
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BibTeX
@article{gepner2026global,
title = {{Global 2-rings and genuine refinements}},
author = {David Gepner and Sil Linskens and Luca Pol},
journal = {Compositio Mathematica},
year = {2026},
month = jul,
doi = {10.1017/s0010437x26102875},
url = {https://doi.org/10.1017/s0010437x26102875},
note = {CC BY},
}
RIS
TY - JOUR
TI - Global 2-rings and genuine refinements
AU - David Gepner
AU - Sil Linskens
AU - Luca Pol
PY - 2026
DA - 2026/07/01
JO - Compositio Mathematica
DO - 10.1017/s0010437x26102875
UR - https://doi.org/10.1017/s0010437x26102875
AB - We introduce the notion of a naive global 2-ring: a functor from the opposite of the normal infinity ∞ ∞ -category of global spaces to presentably symmetric monoidal stable normal infinity ∞ ∞ -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 ∞ 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.
SN - 0010-437X
SN - 1570-5846
ER -