On principally generated quantaloid-modules in general, and skew local homeomorphisms in particular

Ordered sheaves on a small quantaloid Q have been defined in terms of Q -enriched categorical structures; they form a locally ordered category Ord ( Q ) . The free-cocompletion KZ-doctrine on Ord ( Q ) has Mod ( Q ) , the quantaloid of Q -modules, as its category of Eilenberg–Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the locally principally generated Q -modules. We deduce that Ord ( Q ) is biequivalent to the 2-category of locally principally generated Q -modules and left adjoint module morphisms. The example of locally principally generated modules on a locale X is worked out in full detail: relating X -modules to objects of the slice category , we show that ordered sheaves on X correspond with skew local homeomorphisms into X (like sheaves on X correspond with local homeomorphisms into X ).