When is a category of adherence-determined convergences simple?

TL;DR

This paper characterizes when classes of filters produce simple categories of adherence-determined convergences, clarifies their equivalences, and shows the category of hypotopologies is not simple, answering an open question.

Contribution

It provides a characterization of filter classes leading to simple adherence-determined convergence categories and clarifies when different classes yield the same category.

Findings

Identifies conditions for simplicity of adherence-determined convergence categories

Shows equivalence criteria for different filter classes

Proves the category of hypotopologies is not simple

Abstract

Abstract. We provide a characterization of classes of filters $\mathbb{D}$ for which the full subcategory $\operatorname{fix}\operatorname{A}_{\mathbb D}$ of $\mathsf{Conv}$ formed by convergences determined by the adherence of filters of the class $\mathbb{D}$ is simple in $\mathsf{Conv}$. Along the way, we also elucidate when two classes of filters result in the same category of adherence-determined convergences. As an application of the main result, we show that the category of hypotopologies is not simple, thus answering a question from [25].