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