Monads on Q-Cat and their lax extensions to Q-Dist

For a small quantaloid Q, we consider 2-monads on the 2-category Q-Cat and their lax extensions to the 2-category Q-Dist of small Q-categories and their distributors, in particular those lax extensions that are normal, also called flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on Q-Cat may admit only one normal lax extension. Every ordinary monad on the comma category Set/obQ with a lax extension to Q-Rel, the discrete counterpart of Q-Dist, gives rise to such a 2-monad on Q-Cat, and we describe this process globally as a coreflective embedding. The Q-presheaf and the double Q-presheaf monads are important examples of 2-monads on Q-Cat allowing normal lax extensions to Q-Dist, and so are their submonads, obtained by the restriction to conical (co)presheaves. These are known as the Q-Hausdorff and double Q-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when Q is a quantale, or just the Lawvere quantale [0,∞]. Their discretization leads naturally to various lax extensions of the relevant Set-monads used in monoidal topology.