$E$-theory of $X$-$C^{*}$-algebras and functor formalisms

TL;DR

This paper establishes that E-theory for locally compact Hausdorff spaces forms a six-functor formalism equivalent to E-valued sheaves, and relates E-theory for certain locales to E-valued cosheaves.

Contribution

It demonstrates the equivalence of E-theory six-functor formalism with sheaves and cosheaves for specific locales, extending the theoretical framework.

Findings

E-theory for locally compact Hausdorff spaces forms a six-functor formalism.

E-theory category for certain locales is equivalent to E-valued cosheaves.

Establishes equivalence between E-theory and sheaf/cosheaf formalisms.

Abstract

We show that $E$-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of $\mathrm{E}$-valued sheaves. We furthermore show that the $E$-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of $\mathrm{E}$-valued cosheaves.