In this talk we discuss some applications in (point-free) topology of
geometric coalgebraic logic, based on a language with finitary conjunctions,
infinitary disjunctions, and the finitary version of Moss' coalgebraic
modality $\nabla$.

In the first part of the talk we give a presentation of the Vietoris
construction on compact Hausdorff spaces, based on the notion of
Egli-Milner relation lifting. We then move on to the dual, algebraic
construction, involving a geometric modality (ie, the nabla associated
with the power set functor). This construction corresponds to Johnstone's
Vietoris locales.

In the last part of the talk we discuss how to generalize this approach.
We define, given a endofunctor $T$ on the category of sets with functions,
a construction $V_T$ on locales (point-free topologies) that specializes to
Johnstone's Vietoris functor in case we take for $T$ the power set functor. We prove
that this construction preserves the property of regularity if $T$ preserves
weak pullbacks, and conjecture (hopefully prove) that in case $T$ maps finite
sets to finite sets, then $V_T$ preserves compactness as well. The latter
result would imply that $V_T$ is an endofunctor on the category of compact
Hausdorf spaces, which would pave the way for a generalization of the Vietoris
construction on compact Hausdorff \emph{spaces}.