Bornological spaces
The paper contains a very clear presentation on the category \(\mathbf{Bor}\) of bornological spaces and bornological maps. Usually the setting for the study of bornologies has been in the presence of another type of structure such as metrics, topologies or uniformities. In this paper the focus is on the objects and the maps, in particular on the fact that \(\mathbf{Bor}\) is a topological category. A lot of attention goes to the construction of initial and final lifts of sources and sinks and their examples like products, subspaces, quotients and coproducts. Whereas initial lifts in \(\mathbf{Bor}\) are easily descibed, the construction of final lifts is more involved. In order to obtain a clear presentation of final lifts, the author considers two larger topological categories, \(\mathbf{BasBor}\) consisting of base spaces and a second one \(\mathbf{SubBor}\) consisting of subbase spaces. The subcategory \(\mathbf{Bor}\) is nicely embedded as a concretely reflective subcategory of \(\mathbf{BasBor}\) and the latter is concretely reflectively embedded in \(\mathbf{SubBor}\). The consecutive reflections are explicitely described, first by generating a base space from a subbase space and secondly by generating a bornological space from a base space. A final lift in \(\mathbf{Bor}\) is then obtained via first performing a lift in the category \(\mathbf{SubBor}\) and then applying the two steps of generation.
scientific article; zbMATH DE number 7598391