Extending endofunctors on the category of sets and functions to the category of sets and relations requires one to introduce a certain amount of laxness. This in turn requires us to consider bicategories rather than ordinary categories. The subject of lax extensions of Set-based functors is one of the fundamental components of monoidal topology, an active area of research in categorical algebra. The recent theory of linear bicategories, due to Cockett, Koslowski and Seely, is an extension of the usual notion of bicategory to include a second composition in a way analogous to the two connectives of multiplicative linear logic. It turns out that the category of sets and relations has a second composition making it a linear bicategory. The goal of this thesis is first to define the notion of lax extension of Set-based functors to linear bicategories, and then demonstrate crucial properties of our definition.

This thesis consists of two parts: a synthesis of the theory of categories enriched in a quantaloid; and a weakening of this theory for it to include semicategories describing ordered sheaves on a quantaloid. A synthesis of, and supplements to, results in the literature concerning the theory of categories enriched in a quantaloid Q (as particular case of categories enriched in a bicategory) is contained in the first chapters. This theory is built with Q-categories, functors and distributors, and contains such notions as, for example, adjoint functors, weighted colimits, presheaves, Kan extensions, Cauchy completions and Morita equivalence, and so on. The literature does not provide an overview of these matters, so it was necessary to provide one here. Then the necessary theory is developed to arrive at an elementary description of ``ordered sheaves on a quantaloid Q', henceforth referred to as Q-orders. As there is no ``topos of sheaves on a quantaloid', Q-orders cannot be defined as ordered objects in such a topos. Instead a description of Q-orders as categorical structures enriched in the quantaloid Q is proposed. The well-known ordered sheaves on a locale L (i.e.~ordered objects in the topos of sheaves on L) should of course be a particular example of the general theory, taking Q to be the (one-object suspension of) L. Then it turns out that the theory of Q-categories has to be weakened to include ``categories without units', i.e. Q-semicategories. But for Q-semicategories to admit a convenient distributor calculus, a ``regularity' condition has to be imposed. And for those regular Q-semicategories to admit a reasonable theory of Cauchy completions and Morita equivalence, the even stronger condition of ``total regularity' has to be imposed. The former notion has been studied before for semicategories enriched in a symmetric monoidal closed category; the latter notion is new, and is introduced via the intuitively clear idea of ``stability of objects'. The point is then that precisely the Cauchy complete totally regular Q-semicategories are the Q-orders; for a locale L they are indeed the ordered objects in the topos of sheaves on L. A (bi)equivalent description of those Q-orders can be given in terms of categories enriched in the split-idempotent completion of the quantaloid Q: a totally regular semicategory enriched in Q corresponds in a precise sense to a category enriched in the split-idempotent completion of Q. Applying this once more to a locale L instead of a quantaloid Q, these results thus deepen the work of the Louvain-la-Neuve school, and reconcile it with that of the Sydney school, on the description of (ordered) sheaves on a locale as enriched categorical structures. The extended introduction gives a compact yet intuitive presentation of the developments contained in the thesis.