Localization of Cofibration Categories and Groupoid $C^*$-algebras

TL;DR

This paper establishes an equivalence between certain functors in cofibration categories and introduces a new method for constructing groupoid $C^*$-algebras, linking category theory with operator algebras and $K$-theory.

Contribution

It proves an equivalence of functor categories in cofibration categories and presents a novel construction of groupoid $C^*$-algebras from groupoids.

Findings

Equivalence of relative functors on cofibration categories and subcategories of cofibrations.

New construction method for groupoid $C^*$-algebras.

Connection established between category theory and topological $K$-theory.

Abstract

We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid $C^*$-algebra and thereby its topological $K$-theory spectrum.