Some categorical aspects of coarse spaces and balleans

Coarse spaces [26] and balleans [23] are known to be equivalent constructions ([25]). The main subject of this paper is the category, Coarse, having as objects these structures, and its quotient category Coarse/∼. We prove that the category Coarse is topological and hence Coarse is complete and co-complete and one has a complete description of its epimorphisms and monomorphisms. In particular, Coarse has products and coproducts, quotients, etc., and Coarse is not balanced. A special attention is paid to investigate quotients in Coarse by introducing some particular classes of maps, i.e. (weakly) soft maps which allow one to explicitly describe when the quotient ball structure of a ballean is a ballean. A particular type of quotients, namely the adjunction spaces, is considered in detail in order to obtain a description of the epimorphisms in Coarse/∼, shown to be the bornologous maps with large image. The monomorphisms in Coarse/∼ are the coarse embeddings; consequently, the bimorphisms in Coarse/∼ are precisely the isomorphisms, i.e., Coarse/∼ is a balanced category.