A universal characterization of noncommutative motives and secondary algebraic K-theory
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913472368869376 |
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| author | Mazel-Gee, Aaron Stern, Reuben |
| author_facet | Mazel-Gee, Aaron Stern, Reuben |
| contents | We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to Blumberg--Gepner--Tabuada. As a consequence, we obtain a corepresentability theorem for secondary K-theory. We envision this as a fundamental tool for the construction of trace maps from secondary K-theory.
Towards these main goals, we introduce a preliminary formalism of "stable $(\infty, 2)$-categories"; notable examples of these include (quasicoherent or constructible) sheaves of stable $\infty$-categories. We also develop the rudiments of a theory of presentable enriched $\infty$-categories -- and in particular, a theory of presentable $(\infty, n)$-categories -- which may be of intependent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_04021 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A universal characterization of noncommutative motives and secondary algebraic K-theory Mazel-Gee, Aaron Stern, Reuben K-Theory and Homology Algebraic Topology Category Theory We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to Blumberg--Gepner--Tabuada. As a consequence, we obtain a corepresentability theorem for secondary K-theory. We envision this as a fundamental tool for the construction of trace maps from secondary K-theory. Towards these main goals, we introduce a preliminary formalism of "stable $(\infty, 2)$-categories"; notable examples of these include (quasicoherent or constructible) sheaves of stable $\infty$-categories. We also develop the rudiments of a theory of presentable enriched $\infty$-categories -- and in particular, a theory of presentable $(\infty, n)$-categories -- which may be of intependent interest. |
| title | A universal characterization of noncommutative motives and secondary algebraic K-theory |
| topic | K-Theory and Homology Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2104.04021 |