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

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Summary

Introduction

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.

Factorization structures for morphisms
The Basic Results
Family of Morphisms
Some Properties of Topogenous orders
Lifting a Topogenous order along an M-fibration
Examples
The definitions
Quasi-uniform structure or co-perfect syntopogenous structure
Quasi-uniform structures determined by closure and interior operators
The S-Cauchy filters
Variant of completeness
Precompactness
The pair completeness
Lifting a quasi-uniformity along an M-fibration
Quasi-uniform structures and adjoint functors
F U ηM g
The forgetful functor
Full Text
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