phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | de Vries duality for compact Hausdorff spaces— “In [12] de Vries generalized Stone duality to a duality for the category KHaus of compact Hausdorffspaces and continuous maps.” | |
| claim-type | existence— “Applying the machinery of allegories then yields that KHausR is equivalent to DeVS, thus resolving a problem recently raised in the literature.” | |
| method — evidence | — “We prove that in both DeVS and GleR isomorphisms are structure-preserving bijections.” | |
| mechanism | Realising compact Hausdorff spaces as quotients of Stone spaces by closed equivalence relations, and then splitting those equivalences as idempotents inside the allegory, transports the Stone-space/boolean-algebra equivalence up to the compact Hausdorff/de Vries level.— “We will obtain a category equivalent to KHausR by splitting closed equivalence relations in StoneR, a construction that we describe in the language of allegories.” | mechanisms[0] |
| extends | de Vries duality [12]— “In this paper we propose an alternative approach to de Vries duality, where morphisms between de Vries algebras become certain relations and composition is usual relation composition.” | claim_relations[0] |
| extends | the open problem raised in [5, Rem. 3.14]— “This resolves the problem raised in [5, Rem. 3.14] to find a generalization of the category of de Vries algebras that is (dually) equivalent to KHausR.” | claim_relations[1] |
| builds-on | Celani [11]'s quasi-semi-homomorphisms and the subordination relations of [4]— “We show that such functions correspond to subordination relations between A and B studied in [4].” | claim_relations[2] |
| sense | isomorphism - IN StoneER, NOT A BIJECTION: two objects may be isomorphic even when their underlying spaces differ in cardinality, so the word here does not imply the structure-preserving correspondence a reader would assume— “Not every isomorphism in StoneER is a structure-preserving bijection; two objects (X1, E1) and (X2, E2) may be isomorphic even if X1 and X2 do not share the same cardinality” | senses[0] |
| sense | isomorphism - IN DeVS AND GleR, RECOVERED: the paper proves that in these two categories isomorphisms ARE structure-preserving bijections, which is what makes them the better home for the duality— “We prove that in both DeVS and GleR isomorphisms are structure-preserving bijections” | senses[1] |
| sense | composition - THE ORDINARY OPERATION OR NOT: in de Vries' original category it is not usual function composition; in the approach proposed here it is usual relation composition, and that difference is given as the advantage— “One advantage of this approach is that composition of morphisms is usual relation composition” | senses[2] |
| edge | isomorphism —distinct-from→ structure-preserving bijection— “Not every isomorphism in StoneER is a structure-preserving bijection; two objects (X1, E1) and (X2, E2) may be isomorphic even if X1 and X2 do not share the same cardinality” | concept_edges[0] |
| for which use | limitation — a statement that two objects of StoneER are isomorphic: isomorphism there is not a structure-preserving bijection, and two objects may be isomorphic even without sharing the same cardinality— “Not every isomorphism in StoneER is a structure-preserving bijection; two objects (X1, E1) and (X2, E2) may be isomorphic even if X1 and X2 do not share the same cardinality.” | limitations[1] |
| for which use | limitation — any argument that composes morphisms in de Vries' original dual category DeV: composition there is not usual function composition, whereas in the alternative proposed here it is usual relation composition— “One drawback of DeV is that composition of morphisms is not usual function composition” | limitations[2] |
| for which use | limitation — the direction in which this equivalence is stated: the authors call the choice of direction a matter of taste and note the result can equally be stated as a dual equivalence— “Thus, the choice of direction of morphisms is ultimately a matter of taste and the equivalence can alternatively be stated as a dual equivalence” | limitations[3] |
| replication | untested | |
| limitation | One of the intermediate categories the construction produces has badly behaved isomorphisms.— “One drawback of StoneER is that isomorphisms are not structure-preserving bijections.” | limitations[0] |
| limitation | In one of the categories used, two objects can be isomorphic while differing in cardinality, so the word does not carry its usual force there.— “Not every isomorphism in StoneER is a structure-preserving bijection; two objects (X1, E1) and (X2, E2) may be isomorphic even if X1 and X2 do not share the same cardinality.” | limitations[1] |
| limitation | The composition operation in the classical dual category is not the usual function composition, which is the drawback the paper exists to remove.— “One drawback of DeV is that composition of morphisms is not usual function composition” | limitations[2] |
| limitation | The direction chosen for the morphisms is declared a matter of taste, with the opposite convention giving a dual statement of the same result.— “Thus, the choice of direction of morphisms is ultimately a matter of taste and the equivalence can alternatively be stated as a dual equivalence” | limitations[3] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Stone duality lifts to an equivalence between Stone spaces with closed relations and boolean algebras with subordination relations.— “Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations.” | content_tags.findings[0] |
| finding | Splitting the equivalences (closed equivalence relations) in the allegory of Stone spaces and closed relations produces a category equivalent to compact Hausdorff spaces with closed relations.— “Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorffspaces and closed relations.” | content_tags.findings[1] |
| finding | Splitting the equivalences on the algebraic side produces a category equivalent to de Vries algebras with compatible subordination relations.— “Similarly, splitting equivalences in BAS yields a category that is equivalent to the category DeVS of de Vries algebras and compatible subordination relations.” | content_tags.findings[2] |
| finding | Compact Hausdorff spaces with closed relations are equivalent to de Vries algebras with compatible subordinations, resolving a problem recently raised in the literature.— “Applying the machinery of allegories then yields that KHausR is equivalent to DeVS, thus resolving a problem recently raised in the literature.” | content_tags.findings[3] |
| finding | That equivalence restricts to one between compact Hausdorff spaces with continuous maps and the wide subcategory DeVF of de Vries algebras whose morphisms satisfy additional conditions.— “The equivalence between KHausR and DeVS further restricts to an equivalence between the category KHaus of compact Hausdorffspaces and continuous functions and the wide subcategory DeVF of DeVS whose morphisms satisfy additional conditions.” | content_tags.findings[4] |
| finding | One advantage is that composition of morphisms is ordinary relation composition.— “One advantage of this approach is that composition of morphisms is usual relation composition.” | content_tags.findings[5] |
| finding | The equivalence is realised concretely by sending a closed relation R to the subordination on regular opens defined by U S V exactly when R maps the closure of U inside V.— “The key is to associate with each closed relation R : X → Y the relation SR : RO(X) → RO(Y ) given by U SR V ⇐⇒ R[cl(U )] ⊆ V” | content_tags.findings[6] |
| finding | Isomorphisms in the intermediate categories are badly behaved: two S5-subordination spaces can be isomorphic in StoneER while their underlying Stone spaces differ in cardinality.— “Not every isomorphism in StoneER is a structure-preserving bijection; two objects (X1, E1) and (X2, E2) may be isomorphic even if X1 and X2 do not share the same cardinality.” | content_tags.findings[7] |
| finding | In the de Vries algebra category DeVS, by contrast, every isomorphism is a structure-preserving bijection.— “Isomorphisms in DeVS are given by structure-preserving bijections.” | content_tags.findings[8] |
| finding | DeVF is dually isomorphic to the classical category DeV of de Vries algebras and de Vries morphisms.— “The contravariant functors described in Definition 6.13 establish a dual isomorphism between DeVF and DeV.” | content_tags.findings[9] |
| finding | The construction yields, as a by-product, a new proof that the category of compact Hausdorff spaces is the exact completion of the category of Stone spaces.— “Thus, as a consequence we obtain an alternative proof of the fact that KHaus is the exact completion of Stone.” | content_tags.findings[10] |
| definition | closed relation between Stone spaces— “If X, Y are Stone spaces, then we call R : X → Y closed if R is a closed subset of X × Y” | content_tags.definitions[0] |
| definition | quasi-semi-homomorphism— “A quasi-semi-homomorphism is a function ∆ : A → J (B) such that ∆(1) = B and ∆(a ∧ b) = ∆(a) ∩ ∆(b) for all a, b ∈ A.” | content_tags.definitions[1] |
| definition | subordination relation— “A subordination relation from a boolean algebra A to a boolean algebra B is a relation S : A → B satisfying, for a, b ∈ A and c, d ∈ B: (S1) 0 S 0 and 1 S 1; (S2) a, b S c implies (a ∨ b) S c; (S3) a S c, d implies a S (c ∧ d); (S4) a ≤ b S c ≤ d implies a S d.” | content_tags.definitions[2] |
| definition | allegory— “An allegory A is an order-enriched dagger category such that: (i) each partially ordered hom-set of A has binary meets;” | content_tags.definitions[3] |
| definition | S5-subordination— “An S5-subordination is a subordination S : B → B satisfying conditions (S5)–(S7).” | content_tags.definitions[4] |
| definition | S5-subordination space— “an object of StoneER is a pair (X, E) where X is a Stone space and E is a closed equivalence relation on X. We call such pairs S5-subordination spaces.” | content_tags.definitions[5] |
| definition | extremally disconnected— “A topological space is extremally disconnected if the closure of each open set is open.” | content_tags.definitions[6] |
| definition | Gleason space— “A Gleason space is an object (X, E) of StoneER such that X is extremally disconnected and E is irreducible” | content_tags.definitions[7] |
| definition | de Vries algebra | content_tags.definitions[8] |
| concept | de Vries algebras | concepts[0] |
| concept | closed relation | concepts[1] |
| concept | isomorphism | concepts[2] |
| concept | structure-preserving bijection | concepts[3] |
| concept | subordination relations | concepts[4] |
When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.
Tag layer © the Meta-Tagging Project, CC BY 4.0. The paper text is © its authors/publisher
and is linked, not reproduced. Excerpts shown are short quotations for identification and verification.
· back to index
· concepts · authors
· עברית
© 2026 Shir Sivroni · ORCID 0009-0000-2597-2824