Ambidexterity and Height

TL;DR

This paper introduces the concept of semiadditive height in higher semiadditive $ abla$-categories, generalizing chromatic height, and explores its implications for structure, categorification, and stability in higher category theory.

Contribution

It defines semiadditive height, proves its properties, and connects it to stability, categorification, and localizations, extending the understanding of higher semiadditivity in $ abla$-categories.

Findings

Higher semiadditive structure trivializes above the height.

Categorification increases the height by one.

Decomposition of categories according to height.

Abstract

We introduce and study the notion of \emph{semiadditive height} for higher semiadditive $\infty$-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 $\infty$-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 $Pr^{L}$, which we call \emph{modes}. Using this theory, we introduce and study the universal stable $\infty$-semiadditive $\infty$-category of semiadditive height $n$, and give sufficient conditions for a stable $1$-semiadditive $\infty$-category to be $\infty$-semiadditive.