Geometric morphisms of virtual ultracategories

We build on the recent development that toposes with enough points can be represented as categories of points equipped with ultraconvergence data (a.k.a. virtual ultracategories), characterizing usual classes of geometric morphisms — surjections, embeddings, hyperconnected maps, and localic maps — in terms of their points. In particular, the characterisation of geometric surjections relies on a notion of “virtual ultraretractions”, which generalises usual retractions and filtered colimits, categorifies Johnstone’s notion of subclosure for topological spaces, and provides a new characterisation of separating classes of points of a topos.