A Horizontal Categorification of Gelfand Duality

TL;DR

This paper generalizes Gelfand duality to the setting of C*-categories by defining a spectrum as a Fell bundle over a groupoid, establishing a categorical duality that broadens the classical correspondence.

Contribution

It introduces a new categorical Gelfand duality for commutative full C*-categories using Fell bundles over groupoids, extending classical duality concepts.

Findings

Established a duality between commutative C*-categories and Fell bundles over groupoids

Provided a new perspective on Gelfand duality through categorical and bundle-theoretic frameworks

Connected well-known concepts in a novel 'gluing' approach

Abstract

In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.