In regular categories, morphisms have a well-behaved (regular epi)-mono factorization, allowing for a nice calculus of relations. Regularity is a pleasant and common feature of categories with algebraic traits, such as any class of algebras defined by equations. In contrast, coregularity—the dual concept—is common among categories with a topological character, the category of topological spaces being a prime example.
This talk focuses on the topologically-flavoured category of metric compact Hausdorff spaces. These structures generalize classical compact metric spaces (which form a poorly behaved category) and consist of a metric space equipped with a compatible compact Hausdorff topology, which need not be the induced topology. Our main result is that the category of metric compact Hausdorff spaces is coregular and that every equivalence corelation is effective, making it Barr-coexact.
The proof techniques, which had already been used in joint work with Luca Reggio on Nachbin's compact ordered spaces, show promise for adaptation to other concrete categories with a topological flavour.
This talk is based on the preprint "Barr-coexactness for metric compact Hausdorff spaces", joint with Dirk Hofmann: https://arxiv.org/abs/2408.07039
