We show that, for a quantale $V$ and a $\mathsf{ Set}$-monad $\mathbb{ T}$ laxly extended to $V$-$\mathsf{ Rel}$, the presheaf monad on the category of $(\mathbb{ T},V)$-categories is simple, giving rise to a lax orthogonal factorisation system (lofs) whose corresponding weak factorisation system has embeddings as left part. In addition, we present presheaf submonads and study the LOFSs they define. This provides a method of constructing weak factorisation systems on some well-known examples of topological categories over $\mathsf{ Set}$.
Categories: math.CT