Equivalence of the canonical map and classification of Nagata six-functor formalisms

Determine whether, for any Nagata six-functor formalism D on the 1-category of locally compact Hausdorff spaces LCH that satisfies the localization property, the canonical morphism of six-functor formalisms Shv(-; D(pt)) → D is an equivalence. If it is not an equivalence in general, construct examples of nontrivial Nagata six-functor formalisms distinct from the sheaf formalism X ↦ Shv(X; D(pt)), and classify all Nagata six-functor formalisms on LCH.

Background

Using the established universal property, the authors show that for any Nagata six-functor formalism D, the canonical morphism Shv(-; D(pt)) → D is a cohomological equivalence. This prompts the stronger question of whether this morphism is in fact an equivalence of six-functor formalisms.

If the canonical morphism fails to be an equivalence, this would indicate the existence of different Nagata six-functor formalisms beyond the sheaf-based one. The authors therefore pose the problem of constructing such nontrivial examples and classifying all Nagata six-functor formalisms in the topological (LCH) setting.