Stone duality for topological theories

TL;DR

This paper extends Stone duality to T-categories within categorical topology, introducing T-colimits and linking Cauchy completeness with sobriety, thus generalizing classical duality concepts.

Contribution

It defines T-colimits in V-categories and proposes a T-categorical version of Stone duality, connecting Cauchy completeness with sobriety.

Findings

T-colimits are introduced as specific colimits in V-categories.

A complete, cocomplete V-category with limits distributing over T-colimits generalizes (co-)frames.

Cauchy completeness of a T-category is shown to be equivalent to its sobriety.

Abstract

In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], 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.