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Abstract
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We provide a universal characterization of the construction taking a scheme
to its stable
-category
of
noncommutative motives, patterned after the universal characterization of algebraic
K-theory due to Blumberg, Gepner and 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
-categories”;
notable examples of these include (quasicoherent or constructible) sheaves of stable
-categories.
We also develop the rudiments of a theory of presentable enriched
-categories
— and in particular, a theory of presentable
-categories
— which may be of independent interest.
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Keywords
K-theory, secondary K-theory, noncommutative motives,
universal characterization, stable $(\infty, 2)$-categories
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Mathematical Subject Classification
Primary: 18F25, 19D99
Secondary: 18D20, 18N25, 18N65, 55P42
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Milestones
Received: 9 January 2024
Revised: 22 January 2024
Accepted: 11 April 2024
Published: 18 August 2024
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© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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