Unseparated connective spectral sheaves

Determine whether the classification of smashing ideals and atomic smashing ideals in hypercomplete connective spectral sheaves extends to the unseparated prestable infinity-category \(Shv(X;\mathrm{Sp}_{\geq0})\) for every topological space X.

Background

The paper establishes the classification for hypercomplete connective spectral sheaves and for sheaves of abelian groups, with atomic ideals corresponding to Pierce open subsets. It leaves open whether the same classification holds without imposing hypercompleteness or separation.