A globalisation of the Gelfand duality theorem

In this paper, we establish that Gelfand duality holds between the category of commutative C*-algebras and the category of compact, completely regular locales in any Grothendieck topos. It should be remarked immediately that this result represents the final step in a chain of preliminary papers that have appeared over a period of time. Indeed, the work contained in these papers was originally presented at the International Meeting on Categorical Topology held in Ottawa in 1980, of which the details were published in a widely circulated preprint (Banaschewski-Mulvey [4]) several years later, the length of which made immediate publication difficult. In a sequence of papers that followed (Banaschewski-Mulvey [3,5,6,7]), many of the results that provide the natural components from which Gelfand duality is derived were published independently. With some preliminaries to recall the conceptual framework within which the result is set and to make this paper readable without continual reference to its predecessors, these are finally here assembled to prove the Gelfand duality theorem.