On generalized equilogical spaces

In this paper we carry the construction of equilogical spaces into an arbitrary category $\mathsf{X}$ topological over $\mathsf{Set}$, introducing the category $\mathsf{X}$-$\mathsf{Equ}$ of equilogical objects. Similar to what is done for the category $\mathsf{Top}$ of topological spaces and continuous functions, we study some features of the new category as (co)completeness and regular (co-)well-poweredness, as well as the fact that, under some conditions, it is a quasitopos. We achieve these various properties of the category $\mathsf{X}$-$\mathsf{Equ}$ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion $\mathsf{X}_{_{{\rm ex}}}$, and as the regular completion $\mathsf{X}_{_{{\rm reg}}}$. We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the $(\mathbb{T},\mathsf{V})$-$\mathsf{Cat}$ setting.