Continuous six-functor formalism on locally compact Hausdorff spaces

TL;DR

This paper establishes that the functor assigning spectral sheaves to locally compact Hausdorff spaces is the initial object among all continuous six-functor formalisms, with applications to algebraic K-theory and localizing invariants.

Contribution

It proves the initiality of the spectral sheaves functor among continuous six-functor formalisms on locally compact Hausdorff spaces, extending Efimov's algebraic K-theory computations.

Findings

Spectral sheaves functor is initial among continuous six-functor formalisms.

Generalization of Efimov's algebraic K-theory results.

Localizing invariants behave like sheaf cohomology theories.

Abstract

We show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category of spectral sheaves $\mathrm{Shv}(X; \mathrm{Sp})$ 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 $\infty$-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.