We introduce and study the notion of semiadditive height for higher semiadditive ∞-categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive ∞-category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of PrL, which we call modes. Using this theory, we introduce and study the universal stable ∞-semiadditive ∞-category of semiadditive height n, and give sufficient conditions for a stable 1-semiadditive ∞-category to be ∞-semiadditive.
Bibliographical noteFunding Information:
The first author is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities . The second author is supported by ISF 1588/18 and BSF 2018389 .
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