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On generalized equilogical spaces. (English) Zbl 1442.54001

Author’s abstract: In this paper paper, we carry the construction of equilogical spaces into an arbitrary category \(\mathsf{X}\) topological over Set, introducing the category \(\mathsf{X}\text{-}\mathsf{Equ}\) of equilogical objects. Similar to what is done for the category \(\mathsf{Top}\) of topological spaces and continuous functions, we study some features of \(\mathsf{X}\text{-}\mathsf{Equ}\) 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 \(\mathsf{X}\text{-}\mathsf{Equ}\) by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion \(\mathsf{X}_{\mathrm{ex}}\), and as the regular completion \(\mathsf{X}_{\mathrm{reg}}\). We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the \((\mathbb{T}, \mathsf{V})\text{-}\mathsf{Cat}\) setting.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
54B30 Categorical methods in general topology
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
18B35 Preorders, orders, domains and lattices (viewed as categories)
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