Commutative Quantale and Localization
Rings and Algebras
2025-08-06 v1 General Topology
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 " 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.
Keywords
Cite
@article{arxiv.2508.02991,
title = {Commutative Quantale and Localization},
author = {Bangzheng Li and Yichen Xiao},
journal= {arXiv preprint arXiv:2508.02991},
year = {2025}
}