Initiality of the sheaf formalism under only localization

Determine whether the assignment X ↦ Shv(X; Sp) is the initial object among all Nagata six-functor formalisms on the 1-category of locally compact Hausdorff spaces LCH that satisfy only the localization (canonical descent) property, without imposing profinite descent or hyperdescent. Concretely, establish initiality in 6FF(LCH,E,I,P) for E = all morphisms, I = open immersions, and P = proper maps, assuming only the localization axiom for these formalisms.

Background

The paper proves that the functor X ↦ Shv(X; Sp) is initial among continuous Nagata six-functor formalisms on locally compact Hausdorff spaces when, in addition to localization, one assumes profinite descent and hyperdescent. The proof leverages these descent properties, particularly on inverse limits of cubes [0,1]^n and hypercomplete spaces.

Motivated by analogies to motivic homotopy theory—where initiality lifts from coefficient systems to full six-functor formalisms without extra assumptions—the authors ask whether the sheaf formalism enjoys the same universal property when the only axiom imposed is localization, i.e., canonical descent, dropping profinite descent and hyperdescent.