Title: Lax orthogonal factorisations in monad-quantale-enriched categories,
Abstract: 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 $\mathsf{Set}$.