Topological initialness of terminally connected geometric morphisms

Develop and formalize a precise notion of "topological internalization" under which terminally connected geometric morphisms can be characterized as "topologically initial," and establish a rigorous relation between terminally connected geometric morphisms and initial functors, potentially via properties of bicomma topoi or related topological structures.

Background

The authors show that pro-etale morphisms correspond to discrete opfibrations (including at the level of points), while terminally connected morphisms do not generally induce initial functors between categories of points. They nonetheless prove that for any terminally connected geometric morphism, the bicomma topos over a point is connected, suggesting a form of initialness.

On this basis, the authors conjecture a subtler relation between terminally connected morphisms and initial functors, proposing that terminally connected morphisms are "topologically initial" in a sense to be made precise through a future notion of "topological internalization."