Characters and transfer maps via categorified traces

TL;DR

This paper develops a theory of generalized characters in $alculus, extending classical character theory and utilizing categorified traces to refine results in algebraic topology and stable $alculus.

Contribution

It introduces a new framework connecting traces and categorifications to extend character theory and refine transfer and Hochschild homology results.

Findings

Reproves and refines results on Becker-Gottlieb transfers

Enhances understanding of Hochschild homology of Thom spectra

Shows additivity of traces in stable $alculus

Abstract

We develop a theory of generalized characters of local systems in $\infty$-categories, which extends classical character theory for group representations and, in particular, the induced character formula. A key aspect of our approach is that we utilize the interaction between traces and their categorifications. We apply this theory to reprove and refine various results on the composability of Becker-Gottlieb transfers, the Hochschild homology of Thom spectra, and the additivity of traces in stable $\infty$-categories.