Higher Semiadditivity in Transchromatic Homotopy Theory

TL;DR

This paper explores higher semiadditivity in transchromatic homotopy theory, demonstrating compatibility of character maps across chromatic heights and computing semiadditive cardinalities of certain spaces.

Contribution

It introduces a parameterized semiadditive functor for the transchromatic character map and generalizes the character formula for induced representations.

Findings

The transchromatic character map is higher semiadditive up to a free loop shift.

Compatibility of the character map with integration maps is established.

Higher semiadditive cardinalities of π-finite spaces at Lubin-Tate spectra are computed.

Abstract

We study the compatibility of higher semiadditivity across different chromatic heights. We prove that the categorified transchromatic character map assembles into a parameterized semiadditive functor, showing that it is higher semiadditive up to a free loop shift. By decategorification, this implies that the transchromatic character map is compatible with integration maps, generalizing the well-known formula for the character of an induced representation. Through a further decategorification process, we compute the higher semiadditive cardinalities of $\pi$-finite spaces at Lubin-Tate spectra.