From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account

• The paper focuses on moving from Stone spaces to compact Hausdorff spaces, especially the order-enriched move from Priestley spaces to compact ordered spaces.Compact ordered spaces are also called compact pospaces or Nachbin spaces.Source: §1 · p. 2
• Classical quotient results include Manes’ theorem for compact Hausdorff spaces [Joh82, III.2.4] and a quotient theorem for continuous lattices [GHK+03, Thm I-4.17].Source: §1 · p. 2
• Earlier dualities for stably compact spaces and proximity lattices relate algebraic and continuous structures through relations that stand in for way-below and relational morphisms.Source: §1 · p. 2
• Order-enriched categories are essential because the order between arrows is part of the categorical structure of these dualities.The paper treats Stone duality as a restriction of Priestley duality between order-enriched categories.Source: §1 · p. 2
• The framework extends Stone-type dualities to relations, covering relational isomorphism and algebraic structures with relations such as proximity lattices and subordination algebras.The relation duality is developed in detail in.Source: §1 · p. 2
• The construction has three steps: extend a zero-dimensional duality to relations, split idempotents, then restrict to maps.Double categories place functional and relational morphisms in one structure.Source: §1 · p. 2

• The top row reconstructs the dual equivalence between compact ordered spaces and proximity lattices.Each of the three steps—passing to relations, splitting idempotents, and restricting to maps—is equivalence-preserving.Source: §2 · p. 3
• Splitting idempotents adds Priestley preorders on the space side and proximity lattices on the algebra side.The split idempotents are above identity on spaces and below identity on algebras.Source: §2 · p. 3
• The construction starts with Priestley spaces and distributive lattices, embeds them into double categories of relations, and then restricts back to maps.Source: §2 · p. 3

• Identifying the endpoints of each middle-third gap in the Cantor space yields a quotient homeomorphic to the unit interval.For example, 0111... and 1000... are identified.Source: §3 · p. 4
• The relation on clopens behaves like way-below: related clopens admit an interpolant with room on both sides.The relation is idempotent and below inclusion.Source: §3 · p. 4
• The Boolean algebra of Cantor-space clopens is dual to the space, and its proximity relation is dual to the endpoint-identifying equivalence relation.The relation holds when the closure of a clopen under the equivalence relation is contained in the other clopen.Source: §3 · p. 4
• Cuts of the proximity lattice are roundifications of prime filters, and this roundification cannot distinguish 1/3 from 2/3, so the dual glues them together.Source: §3 · p. 5
• Conversely, the dual of the Cantor space with its equivalence relation consists of cuts corresponding to clopens.Source: §3 · p. 5

4.1 Why order-enriched categories?

• In order-enriched categories, relations can be both tabulated and co-tabulated, unlike general set-enriched relations.Using comma and cocomma objects makes these constructions mutually inverse in the appropriate sense.Source: §4.1 · p. 5
• The paper uses weakening relations, which are monotone in the sense that lowering the first input and raising the second preserves the relation.Their characteristic functions are monotone.Source: §4.1 · pp. 5–6
• Priestley preorders correspond dually to proximity relations: idempotent relations above identity on spaces match idempotent relations below identity on algebras.The algebraic relations are interpolative below identity.Source: §4.1 · p. 7

4.2 A duality theorem for relations

• Extending the double-category results to the axiomatic framework of equipments is identified as ongoing future work.An equipment is described as a double category in which every arrow has both a companion and a conjoint.Source: §4.2 · p. 8
• Order-enriched Priestley duality extends to a dual equivalence between categories of Priestley relations and distributive-lattice relations.The duality reverses the order between relations, not their direction.Source: §4.2 · p. 8

4.3 Why double categories?

• Double categories place relational morphisms and functional morphisms in one structure, with relations as objects of one category and maps as arrows of another.The duality between Priestley spaces and distributive lattices extends to their double categories of relations.Source: §4.3 · pp. 8, 10

5 Dualities from double categories of relations

• Priestley spaces with closed weakening relations are dually equivalent to distributive-lattice subordination algebras.Restricting to Boolean algebras gives a duality with Stone spaces carrying closed relations.Source: §5 · p. 11
• Reflexive closed relations on Stone spaces correspond to Boolean-algebra subordinations below identity, while transitive closed relations correspond to interpolative subordinations.The combination of reflexivity and transitivity corresponds to interpolative subordinations below identity.Source: §5 · p. 11
• For Stone spaces with closed equivalence relations, the dual subordinations below identity satisfy a ≺b iff ¬b ≺¬a.The section connects this case to the Cantor-space equivalence example.Source: §5 · p. 11
• For descriptive general frames, relational morphisms yield dualities with modally definable Boolean-algebra subordinations, including reflexive and transitive variants.The section gives corresponding results for reflexive, transitive, and jointly reflexive-transitive frames.Source: §5 · p. 12
• These dualities are weaker on morphisms than classical Jónsson–Tarski duality: algebra maps need only preserve the modal operation laxly, and frame morphisms may be relational.Restricting to operation-preserving algebra maps and continuous p-morphisms recovers the classical duality.Source: §5 · p. 12

6 Splitting idempotents

• The lower splitting construction uses idempotent proarrows below identity, with objects formed from an object and such a proarrow.Its proarrows satisfy R ; S ; R′ = S; the upper construction is defined analogously using idempotents above identity.Source: §6 · pp. 12–13
• Splitting idempotents preserves equivalence of double categories, and the construction generalizes to order-enriched dualities of concretely order-enriched categories.The generalization replaces the Priestley–distributive-lattice duality with another such duality.Source: §6 · p. 13
• Splitting idempotents above identity on Priestley spaces and below identity on distributive lattices extends classical Priestley duality to an equivalence of double categories.The paper identifies this result as the hinge of its construction.Source: §6 · pp. 12–13
• The formally reversed choice—splitting distributive-lattice idempotents above identity and Priestley idempotents below identity—does not yield anything interesting.The paper also describes the resulting relational projections as equivalent to the original relation categories, with reflective or coreflective map subcategories.Source: §6 · p. 13

7 Proximity lattices

• Proximity lattices form a double category whose functional and relational arrows are organized together, with proximity lattices identified as a split-idempotent construction.The category of proximity lattices with relational arrows is not simply the relational completion of the category with functional arrows.Source: §7 · pp. 14–15
• A compact ordered space X yields a proximity lattice of disjoint pairs (d,u), where d is lower open, u is upper open, and (d,u) ≺ (d′,u′) exactly when u ∪ d′ = X.The relation is transitive, below the componentwise order, and interpolative, using Nachbin separation.Source: §7 · p. 15
• Taking inverse images along continuous monotone maps defines a contravariant functor from compact ordered spaces to proximity lattices.Inverse images preserve the lower and upper open sets, their disjointness, and the proximity relation.Source: §7 · p. 15
• For a Priestley space, its clopen-downset lattice embeds into its compact-space proximity lattice but is not isomorphic there as a proximity lattice.The two are instead relationally isomorphic in ProxRel.Source: §7 · p. 15
• For the unit interval, proximity between encoded pairs is determined by y′ < x, while joins in the coordinate encoding take coordinate-wise maxima.Figure 1 depicts the order, join-irreducibles, and prime filters for this encoding.Source: §7 · pp. 15–16, Figure 1 · p. 17
• Compact ordered spaces yield strong proximity lattices, and either ∨-strongness or ∧-strongness guarantees the Gentzen cut rule.The cut rule is stated as: if a ≺ b ∨ c and a ∧ c ≺ b, then a ≺ b.Source: §7 · p. 16

8 Priestley preorders

• A Priestley preorder is a Priestley space equipped with an idempotent relation above the identity; its arrows are continuous monotone maps preserving that relation.Relational arrows are Priestley relations compatible with the preorders on both spaces.Source: §8 · p. 18
• Quotienting a Priestley preorder by mutual relatedness gives equivalence classes ordered by the original preorder.The induced relation on classes is a partial order.Source: §8 · p. 18
• The quotient construction defines a functor from Priestley preorders to compact ordered spaces.The quotient is compact Hausdorff with a closed partial order.Source: §8 · p. 18
• The cover functor from compact ordered spaces to Priestley preorders is obtained by composing the proximity-lattice construction with the duality between proximity lattices and Priestley preorders.For the unit interval, the dual Priestley preorder is built on the prime filters of its proximity lattice.Source: §8 · p. 18

9 Dedekind cuts as adjoint relations

• For Diagram (3), with B = 2 = {0 < 1} and ≪ = ≤, (L0,1R) is a cut: L0 is a downset, 1R an upset, and both satisfy roundness.Every cut is of this form for a unique pair L ⊣ R; if A is a lattice and L,R are lattice relations, L0 is an ideal and 1R a filter.Source: §9 · p. 19
• For lattice structures, L0 and 1R are called a round ideal and round filter, respectively.The round filters of a proximity lattice A are denoted RFilt(A).Source: §9 · pp. 19–20
• In a strong proximity lattice, an adjoint pair of DL-relations yields a round prime ideal L0 and a round prime filter 1R.Strongness supplies the primeness property for the two cut parts.Source: §9 · p. 20
• In a strong proximity lattice, any partial cut satisfying F ; I ⊆ ≤ can be extended to an adjoint pair containing both I and F.The stated construction uses Zorn’s Lemma to obtain a maximal pair.Source: §9 · p. 20
• General cuts use adjoint pairs of weakening relations, while prime cuts arise when the structure is a strong proximity lattice and the relations are DL-relations.The distinction is the type of adjoint pair used to define the cut.Source: §9 · p. 20

Round filters

• The relation ≡ on Spec(A), defined by mutual dual-preorder comparison, is closed.This closed equivalence relation is used to establish that the quotient is Hausdorff.Source: §10 · pp. 21–23
• Roundification ◻F is the largest round filter contained in F, and ◻ is right-adjoint to the inclusion of round filters into filters.The operator preserves finite meets and is an interior operator on the frame of filters.Source: §10 · pp. 21–22
• Two prime filters are equivalent under the dual preorder exactly when they have the same roundification.For the unit interval example, Fₓ and Gₓ have the same roundification, so the quotient identifies them and recovers the interval.Source: §10 · pp. 21–22
• If F is prime and A is ∨-strong, then ◻F is prime, so roundification preserves prime filters in this setting.This property supports identifying quotient points with round prime filters.Source: §10 · p. 22
• For a strong proximity lattice A, Spec(A)/⊑ is a compact ordered space isomorphic to the space of round prime filters, and its quotient map is continuous and monotone.The quotient identifies prime filters with the same roundification.Source: §10 · pp. 22–23
• Every compact ordered space X is recovered from the quotient of its cover; round prime filters with reverse inclusion recover X rather than its order dual.With inclusion order, the round-prime-filter space is isomorphic to Xᵒ.Source: §10 · pp. 23–24
• For a compact ordered space X, mapping x to the round prime filter of tokens whose lower open contains x gives an order-reversing homeomorphism onto RPFilt(AX).The section verifies that this map is a homeomorphism and that η(x) ⊆ η(y) exactly when y ≤ x.Source: §10 · pp. 23–24

Adjoint relations in Priestley preorders

• The arrows in this category are left-adjoint relational morphisms, with relations ordered by inclusion.They satisfy weakening and closedness conditions as well as the adjunction laws.Source: §11 · pp. 24–25
• Left-adjoint relations between Priestley preorders correspond bijectively to continuous monotone maps between their quotient compact ordered spaces.The section’s main result identifies the map induced on quotients from an adjoint pair of relations.Source: §11 · p. 25
• Priestley preorders with left-adjoint relations form a category equivalent to compact ordered spaces with continuous monotone maps.The quotient assignment is functorial, fully faithful, and essentially surjective.Source: §11 · p. 26
• Two Priestley preorders are relationally isomorphic exactly when their quotients are isomorphic in KOHaus..Source: §11 · p. 26

Adjoint relations in proximity lattices

• Proximity lattices with left-adjoint relations are dually equivalent to compact ordered spaces with continuous monotone maps.Under this correspondence, each compact ordered space X corresponds to its proximity lattice AX.Source: §12 · pp. 26–27
• Every proximity lattice is naturally relationally isomorphic, but not necessarily functionally isomorphic, to a strong proximity lattice.The construction uses pairs of round ideals and round filters satisfying a compatibility condition, with proximity determined by nonempty filter-ideal intersection.Source: §12 · p. 27
• The constructed relation on A* is both join-strong and meet-strong, and membership relations between A and A* form an adjoint pair giving a relational isomorphism.The two relations are defined using membership in the round ideal and round filter components.Source: §12 · p. 27
• A proximity lattice and its strong replacement A* have isomorphic lattices of round filters and round ideals, and therefore isomorphic dual compact ordered spaces..Source: §12 · pp. 27–28

Conclusion

• Canonical extension Aδ = Up(X) retains the dual order of a distributive lattice’s Priestley space and discards its topology.The paper contrasts this with its own dualization of relations and splitting of idempotent proximity relations.Source: §13 · p. 28
• Splitting idempotent proximity relations freely adds objects dual to quotients by closed equivalence relations on the dual spaces.Proximity relations and Priestley preorders are relational structures on the same dual data underlying canonical extension.Source: §13 · p. 28
• The authors identify interaction with canonical extensions of distributive lattices, posets, and related structures as future work..Source: §13 · p. 28