When is a category of adherence-determined convergences simple? (Q2105009)

The class of \(\mathbb{D}\)-adherence-determined convergence spaces, where \(\mathbb{D}\) is a class of filters, was defined and studied in [\textit{S. Dolecki} and \textit{F. Mynard}, Convergence foundations of topology. Hackensack, NJ: World Scientific (2016; Zbl 1345.54001)]. In the paper under review the authors give (under some mild conditions on \(\mathbb{D}\)) a necessary and sufficient condition on \(\mathbb{D}\) which guarantees that the corresponding category is simple. This characterisation provides a quick proof of the facts that the categories of pretopological and paratopogical convergence spaces are simple. A more sophisticated argument shows that the categories of \(\mu\)-hypertopologies are not simple, where \(\mu\) is an infinite cardinal. As applications one gets the facts that the categories of pseudotopological and hypotopological convergence spaces are not simple.