Normed idempotence of spectra and distributivity in equivariant higher algebra
Abstract
We introduce and develop the theory of distributivity for base categories equipped with so-called atomic orbital subcategories, with an eye towards applications in equivariant homotopy theory. This extends the important framework in Nardin's thesis and supplies several crucial details on distributivity missing from the treatment therein. As our main technical result, we provide a parametrised distributive version of the universal property of "partial cocompletions with relations". We then employ this to construct a distributively normed symmetric monoidal structure on presentable categories. In turn, by specialising to the case of equivariant homotopy theory for compact Lie groups G, we show that the category of genuine G-spectra is normed idempotent, so that we obtain the structure of multiplicative norms on G-spectra along inclusions of finite index subgroups.
Related
- ℂ-Motivic Eilenberg-Mac Lane Spectra as MGL-Motivic Thom Spectra and Normed 𝔾m-Hochschild Homology
- Constant spectral Mackey functors and the equivariant Steenrod algebra
- Rational torus equivariant spectra VII: the Cousin complex
- Finite-Branch Multipliers on a Hölder Algebra: Commutants, Norm Closures, and Essential Spectra
- Idempotence in Carmichael and Giuga numbers and its application to the Agoh-Giuga conjecture
BibTeX
@misc{hilman2026normed,
title = {{Normed idempotence of spectra and distributivity in equivariant higher algebra}},
author = {Kaif Hilman and Branko Juran and Natalie Stewart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22074},
eprint = {2610.22074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22074},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Normed idempotence of spectra and distributivity in equivariant higher algebra
AU - Kaif Hilman
AU - Branko Juran
AU - Natalie Stewart
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22074
UR - https://arxiv.org/abs/2610.22074
AB - We introduce and develop the theory of distributivity for base categories equipped with so-called atomic orbital subcategories, with an eye towards applications in equivariant homotopy theory. This extends the important framework in Nardin's thesis and supplies several crucial details on distributivity missing from the treatment therein. As our main technical result, we provide a parametrised distributive version of the universal property of "partial cocompletions with relations". We then employ this to construct a distributively normed symmetric monoidal structure on presentable categories. In turn, by specialising to the case of equivariant homotopy theory for compact Lie groups G, we show that the category of genuine G-spectra is normed idempotent, so that we obtain the structure of multiplicative norms on G-spectra along inclusions of finite index subgroups.
ER -