2 00 8 On principally generated Q - 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). It has previously been shown by the second author that the " free-cocompletion " KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In the first part of this paper we apply Q-enriched category theory, particularly the theory of totally algebraic cocomplete Q-categories as developed by the second author, to 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 between them, and thus provide a rephrasing of the notion of ordered sheaf on Q in terms of the possibly more familiar Q-modules. Several examples are briefly discussed, but one particularly important example is worked out in full detail in the second part of this paper: the locally principally generated Q-modules in the case where Q is (the one-object suspension of) a locale X. By relating X-modules to objects of the slice category Loc/ X we obtain an account of ordered sheaves on X as skew local homeomorphisms into X as asymmetrical analogues of local homeomorphisms. Principally generated Q-modules. The well-known adjunction between the category Ord of ordered sets and order-preserving functions on the one hand, and the category Sup of complete lattices and supremum-preserving functions on the other,