Higher Noncommutative Motives and Higher K-theory
Abstract
We construct higher additive noncommutative motives of stable (∞,n)-categories, extending the additive motives of Blumberg, Gepner, and Tabuada and the secondary theory of Mazel-Gee and Stern. We formulate a notion of exact sequences and establish the universal property of the resulting additive motives. We give two equivalent constructions: a localization of enriched presheaves and a weighted Waldhausen S-construction on complete exact flags. We then compute their Hom objects, obtaining ordinary K-theory at the first level and lower additive motives recursively at higher levels. Our results recover the classical and secondary theories, provide coherent multiplicative structures, and yield an endofunctor on the category of Stefanich rings. In short, higher motives are higher K-theory.
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BibTeX
@misc{matsukawa2026higher,
title = {{Higher Noncommutative Motives and Higher K-theory}},
author = {Hisato Matsukawa},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07262},
eprint = {2611.07262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07262},
}
RIS
TY - UNPB
TI - Higher Noncommutative Motives and Higher K-theory
AU - Hisato Matsukawa
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07262
UR - https://arxiv.org/abs/2611.07262
AB - We construct higher additive noncommutative motives of stable (∞,n)-categories, extending the additive motives of Blumberg, Gepner, and Tabuada and the secondary theory of Mazel-Gee and Stern. We formulate a notion of exact sequences and establish the universal property of the resulting additive motives. We give two equivalent constructions: a localization of enriched presheaves and a weighted Waldhausen S-construction on complete exact flags. We then compute their Hom objects, obtaining ordinary K-theory at the first level and lower additive motives recursively at higher levels. Our results recover the classical and secondary theories, provide coherent multiplicative structures, and yield an endofunctor on the category of Stefanich rings. In short, higher motives are higher K-theory.
ER -