Continuous six-functor formalism on locally compact Hausdorff spaces

We show that the functor sending a locally compact Hausdorff space $X$ to the $\mathrm{\infty }$-category of spectral sheaves $\mathrm{S}\mathrm{h}\mathrm{v}(X;\mathrm{S}\mathrm{p})$ is initial among all continuous six-functor formalisms on the category of locally compact Hausdorff spaces. Here, continuous six-functor formalisms are those valued in dualizable presentable stable $\mathrm{\infty }$-categories and satisfying canonical descent, profinite descent, and hyperdescent. As an application, we generalize Efimov’s computation of the algebraic $K$-theory of sheaves to all localizing invariants on continuous six-functor formalisms. Our results show that localizing invariants behave analogously to compactly supported sheaf cohomology theories when evaluated on continuous six-functor formalisms on locally compact Hausdorff spaces.