Commutative Quantale and Localization

Abstract: In this paper we introduce the localization construction for quantales. A quantale is a complete semilattice combined with a multiplication. We mimic the notion of filter in a lattice to define multiplicative filters in a quantale, and construct the localization of the quantale at a multiplicative filter. We prove theorems with similar structure as "$\Spec R$ is a sheaf" and use them to obtain several results in algebra and geometry, including the Baire Category Theorem and some of its generalizations. We also present an algebraic version of Baire Category Theorem, which we believe has not appeared in literature.