On generalized equilogical spaces

Abstract

In this paper we carry the construction of equilogical spaces into an arbitrary category X topological over Set, introducing the category X-Equ of equilogical objects. Similar to what is done for the category Top of topological spaces and continuous functions, we study some features of X-Equ 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 X-Equ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion Xex, and as the regular completion Xreg. We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the (T,V)-Cat setting.