Abstract
We introduce and study an abstract notion of Császár's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We show that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is quasi-uniformity.Various notions of completeness of objects and precompactness with respect to a quasi-uniformity defined in a natural way are also studied.
Highlights
In a category C with a proper (E, M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a unified approach to the theory of closure, interior and neighbourhood operators
We demonstrate the equivalence between quasi-uniform and co-perfect syntopogenous structures, which together with Proposition 2.1.5, leads to the description of a quasi-uniormity as a family of categorical closure operators
We introduce the categorical notions of quasi-uniform and syntopogenous structures that are fundamental to our study
Summary
Among the various asymmetric topological structures, one finds the notion of quasiuniform structure. The recently introduced notion of topogenous structures on categories ([HIR16]) has provided a unified approach to the categorical closure, interior and neighbourhood operators and has shed a light on the study of a concept of quasi-uniformity on an abstract category. Our attention will be turned to the study of continuity of a C-morphism with respect to two syntopogenous structures on C which enables us to describe the quasi-uniformity induced by a pointed Our-initial morphism leads to the definition of a hereditary topogenous order which enables us to study hereditary closure and interior operators in one setting. The initial morphism with respect to a syntopogenous structure is defined and shown to capture its counterparts in the settings of quasi-uniformity and (idempotent) closure operator.
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