Global ∞-categories and global Thom spectra
Abstract
We introduce a framework of (Lie-)global ∞-categories, which formalizes various families of ∞-categories indexed by compact Lie groups and equipped with suitable restriction functors along continuous group homomorphisms that occur naturally in equivariant homotopy theory and representation theory. As our main results, we show that in this framework unstable and stable equivariant and global homotopy theory admit universal properties, refining and generalizing the results for finite groups from arXiv:2301.08240 and arXiv:2307.11001. In particular, we characterize the passage from unstable to stable equivariant and global homotopy theory at the level of global ∞-categories as universally inverting the action of representation spheres in an appropriate sense. Building on this, we define parametrized equivariant and global Thom spectrum functors and show that they recover classical Thom spectrum constructions defined in terms of pointset models.
Related
- Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas
- ℂ-Motivic Eilenberg-Mac Lane Spectra as MGL-Motivic Thom Spectra and Normed 𝔾m-Hochschild Homology
- Generalized Thom Spectra, Conjugate Stacks, and truncated Brown-Peterson Spectra
- Spectra and Support Theory for Wide Subcategories in n-Abelian Categories
- Pre-modular fusion categories of global dimension 2p
BibTeX
@misc{brink2026global,
title = {{Global $\infty$-categories and global Thom spectra}},
author = {Emma Brink and Tobias Lenz},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28504},
eprint = {2608.28504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28504},
}
RIS
TY - UNPB
TI - Global ∞-categories and global Thom spectra
AU - Emma Brink
AU - Tobias Lenz
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28504
UR - https://arxiv.org/abs/2608.28504
AB - We introduce a framework of (Lie-)global ∞-categories, which formalizes various families of ∞-categories indexed by compact Lie groups and equipped with suitable restriction functors along continuous group homomorphisms that occur naturally in equivariant homotopy theory and representation theory. As our main results, we show that in this framework unstable and stable equivariant and global homotopy theory admit universal properties, refining and generalizing the results for finite groups from arXiv:2301.08240 and arXiv:2307.11001. In particular, we characterize the passage from unstable to stable equivariant and global homotopy theory at the level of global ∞-categories as universally inverting the action of representation spheres in an appropriate sense. Building on this, we define parametrized equivariant and global Thom spectrum functors and show that they recover classical Thom spectrum constructions defined in terms of pointset models.
ER -