An Intrinsic Characterization of Monomorphisms in Regular Lindelöf Locales

We characterize monomorphisms in rLLoc, the category of regular Lindelöf locales. Though somewhat complicated, the characterization is intrinsic in the sense that it refers only to the properties of the morphism itself, rather than to properties of some lifting of it to a distant category. Monomorphisms in rLLoc cannot be said to be well understood, despite their characterizations in [2] and [7]. The problem with these characterizations is that, although in each case the condition is simple enough (injectivity of a functorially associated spatial map in the case of [2], and surjectivity of a functorially associated frame map in the case of [7]), it is applied in a category su¢ ciently remote from rLLoc to make veri…cation troublesome. On the other hand, there are two categories closely related to rLLoc in which (the morphisms corresponding to) monomorphisms are well understood. Epimorphisms inW, the category of archimedean lattice-ordered groups with weak order unit, have a rich theory, and W is related to Loc, the category of locales with locale maps, by the contravariant adjunction