Topological lax comma categories

This paper investigates the interplay between properties of a topological space $X$, in particular of its natural order, and properties of the lax comma category $\mathsf{T}\mathsf{o}\mathsf{p}\Downarrow X$, where $\mathsf{T}\mathsf{o}\mathsf{p}$ denotes the category of topologicalspaces and continuous maps. Namely, it is shown that, whenever $X$ is a topological $\bigwedge$-semilattice, the canonical forgetful functor $\mathsf{T}\mathsf{o}\mathsf{p}\Downarrow X\to \mathsf{T}\mathsf{o}\mathsf{p}$ is topological, preserves and reflects exponentials, and preserves effective descent morphisms. Moreover, under additional conditions on $X$, a characterisation of effective descent morphisms is obtained.