Duality and canonical extensions for stably compact spaces

We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.

💡 Research Summary

The paper develops a canonical extension theory for strong proximity lattices in order to provide a point‑free algebraic description of a finitary duality for stably compact spaces. After recalling that stably compact spaces sit at the intersection of spectral topology and proximity lattice theory, the authors observe that existing dualities treat only morphisms with a single continuity condition, leaving the object side under‑specified. To remedy this, they introduce two distinct canonical extensions of a strong proximity lattice L: a π‑extension and a σ‑extension. The π‑extension preserves joins (the lattice’s “sum” operation) and is built by embedding L into a complete lattice L̂π that is generated by its filters; the σ‑extension preserves meets (the “product” operation) and is obtained by embedding L into a complete lattice L̂σ generated by its ideals. Both extensions are shown to exist uniquely under a regular completeness condition that strong proximity lattices satisfy.

The authors prove that each extension carries a full set of complete operators: ηπ : L → L̂π is join‑preserving and makes every filter in L̂π closed under meets, while ησ : L → L̂σ is meet‑preserving and makes every ideal in L̂σ closed under joins. Consequently, a single algebraic object L can be simultaneously viewed as a π‑complete lattice and a σ‑complete lattice, giving rise to two different topological constructions on its spectrum.

Using these extensions, the paper defines two functors between the category ProxLat of strong proximity lattices and the category StablyComp of stably compact spaces. The π‑functor sends L to Specπ(L), the space of completely prime filters of L̂π equipped with the topology generated by π‑open sets; the σ‑functor sends L to Specσ(L), the space of completely prime ideals of L̂σ equipped with σ‑open sets. Morphisms in ProxLat are required to be both π‑continuous (preserving joins) and σ‑continuous (preserving meets), which matches the dual continuity requirements on the topological side. The authors establish that these two functors are inverse equivalences, yielding a finitary duality that respects both extensions. This duality generalises the classic Stone and Priestley dualities: for Boolean algebras the π‑extension recovers Stone spaces, while the σ‑extension recovers Priestley spaces; for continuous lattices it recovers the Scott and Lawson topologies respectively.

The paper includes several illustrative examples, showing how the canonical extensions translate algebraic properties into topological ones and vice‑versa. It also discusses computational advantages: the point‑free description avoids explicit point‑set constructions, making the duality amenable to algorithmic manipulation in domain theory and semantics.

In conclusion, the authors argue that the simultaneous π‑ and σ‑canonical extensions provide a richer algebraic framework for stably compact spaces, enabling a fully symmetric treatment of objects and morphisms. They suggest future work on non‑canonical proximity lattices, extensions to infinite‑dimensional dualities, and categorical monadic structures that could further integrate this theory with applications in computer science, such as formal semantics and database schema transformations.