Sheaves over Right Sided Idempotent Quantales

We present a discussion of sheaves and presheaves over an idempotent two sided quantale in a fashion that is similar to the way that these objects are conceived in by Forman and Scott [FS] over complete Heyting algebras. The idea of a quantale originated with C.J.Mulvey as an attempt to code a lattice theoretic construct that might be appropriate to obtain, for non commutative C* algebras, an analoque of the classical duality between commutative C* algebras and compact Hausdorff spaces. A number of authors have studied the possibility of extending the notions of sheaf and presheaf over a complete Heyting algebra (cHa) or frame to this new context. We consider [FS] as a basic reference for presheaves and sheaves over complete Heyting algebras. We discuss here how much of the theory in that paper can be carried over to the quantale setting. We shall focus our attention on quantales which are referred to as idempotent and right sided (definitions will follow). We consider as central the notion of extensionality or separation, in the sense that 'sections that coincide locally are equal'. Our treatment, although related to that in the references mentioned above, is distinct in a number of aspects, such as the concepts of presheaf, morphisms, completion and characteristic maps. We also prepare a discussion of sheaves and presheaves over two sided quantales, to appear separately.