The additivity of traces in stable $\infty$-categories

TL;DR

This paper proves a version of May's theorem on the additivity of traces within symmetric monoidal stable $$-categories, using topological Hochschild homology to establish a categorified invariant and extend trace properties.

Contribution

It introduces a categorification approach to trace additivity in stable $$-categories using topological Hochschild homology and constructs a spectrum morphism linking THH to endomorphisms.

Findings

Established a spectrum morphism from THH to endomorphisms in stable $$-categories.

Extended trace additivity to finite homotopy colimits.

Provided a categorified proof of May's theorem in the $$-categorical setting.

Abstract

We prove a version of J.P. May's theorem on the additivity of traces, in symmetric monoidal stable $\infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $\infty$-categories and construct a morphism of spectra $\mathrm{THH}(\mathbf C)\to \mathrm{End}(\mathbf 1_\mathbf C)$ for $\mathbf C$ a stably symmetric monoidal rigid $\infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.