Quasi-uniform and syntopogenous structures on categories

Quasi-uniform structures determined by closure operators

“…For the basic facts on categorical closure and interior operators we refer to [4,5] and [14]. Concerning the categorical topogenous, quasi-uniform and syntopogenous structures, we use [12,8,9]. Throughout the paper, we consider a category C supplied with a proper (E, M)-factorization system for morphisms.…”

“…It is this great generality of quasi-uniformities which allows one to view in some sense the study of quasi-uniform spaces as an alternative approach to the study of topological spaces. In [12,8] a quasi-uniform structure is introduced on a category C as a suitable endofunctor category on the subobject lattice, subX, for each object X in C and characterized as an appropriate family of closure operators ( [5]). It is therefore not hard to observe that every quasi-uniformity on C induces a closure operator ✩ This work was supported by the South African National Research Foundation.…”

Topogenous structures on faithful and amnestic functors

“…A topogenous order ❁ on a category C equipped with a proper (E, M)-factorization structure for morphisms is a family of binary relations, each on the subobject lattice, subX, for an object X in C (subject to some axioms) ( [9]). This notion, which is crutial for the syntopogenous structures introduced by Császár ( [3]) with the aim of proposing a unified approach to topological, uniform and proximity spaces, has played a salient role in providing a single setting study of categorical closure ( [5]), interior ( [10]) and neighbourhood ( [17]) operators and led to the introduction of quasi-uniform structures in categories (see e.g [7,8,12,14,13]). Topogenous orders are easier to work with when it comes to the study of topological structures on categories.…”