Quasi-uniform structures determined by closure operators

Abstract

We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C. Not only this result allows to obtain a categorical counterpart P of the Császár-Pervin quasiuniformity P, that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous orders that are induced by a transitive quasi-uniformity on C. The categorical counterpart P * of P −1 is characterized as a transitive quasi-uniformity compatible with an idempotent closure operator. When applied to other categories outside topology P allows, among other things, to generate a family of idempotent closure operators on Grp, the category of groups and group homomorphisms, determined by the normal closure.

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.…”