Very Schwartz coidempotents and continuous spectrum

TL;DR

This paper introduces a continuous version of the smashing spectrum functor for monoidal stable categories, linking idempotents to topological spaces and establishing Tannaka duality for spectral sheaves on compact spaces.

Contribution

It defines very Schwartz idempotents and develops a continuous spectrum functor, extending the stable case to a broader context and proving a duality theorem for spectral sheaves.

Findings

Defined very Schwartz idempotents.

Constructed a continuous smashing spectrum functor.

Proved Tannaka duality for spectral sheaves on stably compact spaces.

Abstract

We introduce the continuous version of the (unstable) smashing spectrum functor. In the stable case, it assigns to each dualizably symmetric monoidal stable presentable $\infty$-category a stably compact space whose open subsets correspond to very Schwartz idempotents -- a certain class of idempotents we define. As an application, we prove Tannaka duality for spectral sheaves on stably compact spaces, including the case of compact Hausdorff spaces.