Modules in the category of sheaves over quantales Original Research Article

In this paper we develop the elementary theory of modules in the category Sh(Q) of sheaves over right-sided idempotent quantales. The main ingredient is the construction of a (first-order) logic sound for Sh(Q). As an application we prove that in Sh(Q), a finitely generated projective module is free (Theorem 7.2), a result that is relevant to the study of representation of non-commutative C∗-algebras.