A horizontal categorification of Gel'fand duality

In the setting of C⁎-categories, we provide a definition of spectrum of a commutative full C⁎-category as a one-dimensional unital Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gel'fand duality theorem generalizing the usual Gel'fand 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.