Cowellpoweredness and closure operators in categories of coarse spaces

Abstract

We consider two categories: the category Coarse of coarse spaces and bornologous maps and its quotient category Coarse/∼, where ∼ is the closeness relation. This paper tackles the problem of their wellpoweredness and cowellpoweredness. In particular, we show that all the epireflective subcategories of Coarse are cowellpowered, using a complete characterization of closure operators of Coarse, while Coarse/∼ is both wellpowered and cowellpowered. Moreover, we prove that Coarse/∼ has neither equalizers nor pullbacks of subobjects, although it has arbitrary products.