Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a 'sheaf over a quantale' or, equivalently, a 'quantale valued set'. In this paper, we use ideas from enriched category theory to motivate the definition of a 'quantic sheaf'. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.