Monoids with quantale-valued preorders: Globalizations and localizations

Abstract

We consider compatible monoid structures on a DQ-category, where Q is a commutative and divisible quantale, and DQ is the quantaloid of diagonals of Q. Such structures, called DQ-monoids, may be treated as monoids equipped with a preorder valued in Q, whose elements are not supposed to exist globally. The globalization functors map DQ-monoids to global ones, i.e., monoids on a Q-category, called Q-monoids. Conversely, Q-monoids over Q give rise to DQ-monoids via the localization functors. The interactions between the globalization functors and the localization functors are investigated. In particular, a necessary and sufficient condition on Q is provided such that the localizations are reversible by the globalizations.