Towards Stone duality for topological theories

In the context of categorical topology, more precisely that of T -categories (Hofmann, 2007 [8]), we define the notion of T -colimit as a particular colimit in a V -category. A complete and cocomplete V -category in which limits distribute over T -colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T -categorical version of “Stone duality”, and show that Cauchy completeness of a T -category is precisely its sobriety.