On generalized equilogical spaces

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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.

MSC

54A05
54B30
18D15
54D80
18B35

Keywords

Equilogical space
Topological category
Exact completion
Regular completion
Quasitopos
(T,V)-category
Modest set

Cited by (0)

1

Research supported by Centro de Matemática da Universidade de CoimbraUID/MAT/00324/2019 and by the FCT PhD grant PD/BD/128059/2016, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.