Stone Duality for Topological Convexity Spaces

Toby Kenney

arxiv: 2201.09819 · v2 · submitted 2022-01-24 · 🧮 math.CT · math.GT

Stone Duality for Topological Convexity Spaces

Toby Kenney This is my paper

Pith reviewed 2026-05-24 12:29 UTC · model grok-4.3

classification 🧮 math.CT math.GT

keywords stone dualitytopological convexity spacessup-latticespreconvexity spacesadjunctioncoframesconvexity spacesclosure spaces

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The pith

An adjunction between topological convexity spaces and sup-lattices extends Stone duality by factoring through preconvexity spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an adjunction that extends the Stone duality between coframes and topological spaces to one between topological convexity spaces and sup-lattices. It achieves this by routing the correspondence through the intermediate category of preconvexity spaces. A sympathetic reader would care because the construction unifies convexity structures with topology inside a single categorical framework that preserves lattice operations. The convexity on a set is the family of subsets closed under arbitrary intersections and directed unions, and the paper shows this interacts with the topology to support the adjunction.

Core claim

We extend the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices. We factor this adjunction through the category of preconvexity spaces (sometimes called closure spaces).

What carries the argument

The adjunction between topological convexity spaces and sup-lattices, obtained by factoring through preconvexity spaces.

If this is right

Topological convexity spaces correspond functorially to sup-lattices via the extended duality.
Preconvexity spaces act as the necessary intermediate objects that make the adjunction factor correctly.
Convex sets closed under arbitrary intersections and directed unions interact with the topology to preserve the lattice structure in the dual.
Results about coframes and topological spaces lift to the richer setting of topological convexity spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may allow lattice-theoretic tools to classify or construct new examples of spaces with both convexity and topology.
Similar factorizations could be attempted for other combined structures, such as ordered topological spaces.
The approach opens the possibility of transferring algebraic invariants from sup-lattices back to geometric properties of the original spaces.

Load-bearing premise

The convexity structure combines with the topology in a manner that permits the stated adjunction to exist and factor through preconvexity spaces.

What would settle it

A concrete topological convexity space for which the proposed mapping fails to produce an adjunction to any sup-lattice or breaks the factorization through preconvexity spaces.

read the original abstract

A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions. There is a lot of interest in spaces that have both a convexity space and a topological space structure. In this paper, we study the category of topological convexity spaces and extend the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices. We factor this adjunction through the category of preconvexity spaces (somtimes called closure spaces).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Kenney extends Stone duality to topological convexity spaces by factoring through preconvexity spaces, using standard categorical moves on the convexity axioms.

read the letter

The paper's core move is to take the known Stone duality between coframes and topological spaces, add convexity structures closed under arbitrary intersections and directed unions, and produce an adjunction to sup-lattices that factors through preconvexity spaces. That factoring step is the concrete new piece; it lets the convexity data sit between the topological side and the lattice side without forcing extra conditions. The setup looks clean on paper and uses the usual category-theoretic language for closure operators and directed unions, so the construction itself does not appear forced. The main strength is that it organizes an existing class of spaces inside an existing duality framework without inventing new axioms. The soft spot is that the interaction between the topology and the convexity closure is only sketched at the level of the abstract; if the proofs rely on the directed-union condition behaving well with open sets, that needs explicit checking to confirm the adjunction is adjoint on both sides. No circularity shows up in the stated claim. This is a paper for readers already working in categorical topology or convexity theory who want to see how these spaces embed into lattice dualities. It is narrow enough that most people outside that corner will not need it, but the construction is precise enough to merit referee time rather than a desk rejection. I would send it for review.

Referee Report

0 major / 1 minor

Summary. The paper studies the category of topological convexity spaces (sets equipped with both a topology and a convexity structure closed under arbitrary intersections and directed unions) and extends the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices, with this adjunction factored through the category of preconvexity spaces (closure spaces).

Significance. If the adjunction is correctly established with the stated factoring, the result would provide a modular categorical extension of Stone duality that incorporates convexity structures, potentially enabling unified treatments of topological and order-theoretic properties in geometry or logic. The decomposition via preconvexity spaces is a notable structural feature.

minor comments (1)

[Abstract] Abstract: 'somtimes' is a typographical error and should be 'sometimes'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our work extending Stone duality to topological convexity spaces via an adjunction with sup-lattices, factored through preconvexity spaces. We note the 'uncertain' recommendation and the conditional significance assessment. The manuscript contains the full proofs establishing the adjunction and the factoring; we stand by these results. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained categorical construction

full rationale

The paper defines convexity spaces via standard axioms (closed under arbitrary intersections and directed unions), introduces the category of topological convexity spaces, and constructs an adjunction extending the known Stone duality between coframes and topological spaces, with an intermediate factoring through preconvexity spaces. These steps rely on universal properties of categories, limits/colimits, and the existing Stone duality theorem rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or claims reduce the output to the input by construction; the extension is a standard categorical exercise with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; only the basic definition of convexity space is supplied. Full paper may contain additional axioms or parameters.

axioms (1)

domain assumption A convexity space is a set whose chosen family of subsets is closed under arbitrary intersections and directed unions.
Stated directly in the abstract as the definition of convexity space.

pith-pipeline@v0.9.0 · 5597 in / 1091 out tokens · 38872 ms · 2026-05-24T12:29:59.901551+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We extend the Stone duality between coframes and topological spaces to an adjunction between topological convexity spaces and sup-lattices. We factor this adjunction through the category of preconvexity spaces.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Aull and W .J

C.E. Aull and W .J. Thron. Separation axioms between T0 and T1. Indag. Math., pages 26–37, 1963

work page 1963
[2]

Banaschewski, G

B. Banaschewski, G. C. L. Br¨ ummer, and K. A. Hardie. Bifr ames and bispaces. Quaest. Math., 6:13–25, 1983

work page 1983
[3]

Dawson, R

R. Dawson, R. Par´ e, and D. Pronk. Adjoining adjoints. Advances in Mathematics, 178:99–140, 2003

work page 2003
[4]

R. J. M. Dawson. Limits and colimits of convexity spaces. Cahiers de T opologie et g´ eom´ etrie diﬀ´ erentielle cat´ egoriques, 28:307–328, 1987. 30

work page 1987
[5]

Ellerman

D. Ellerman. The logic of partitions: Introduction to th e dual of the logic of subsets. The Review of Symbolic Logic , 3:287–350, 2010

work page 2010
[6]

Ho ﬀmann

R.-E. Ho ﬀmann. Irreducible ﬁlters and sober spaces. Manuscripta Math. , 22:365–380, 1977

work page 1977
[7]

Ho ﬀmann

R.-E. Ho ﬀmann. Essentially complete T0-spaces. Manuscripta Math. , 27:401– 431, 1979

work page 1979
[8]

D. C. Kay and E. W . Womble. Axiomatic convexity theory and relationships between the caratheodory, helly, and radon numbers. Paciﬁc Journal of Mathe- matics, 38:471–485, 1971

work page 1971
[9]

T. Kenney. Partial-sup lattices. Theory and Applications of Categories , 30:305– 331, 2015

work page 2015
[10]

A. Kock. Monads for which structures are adjoint to unit s. Journal of Pure and Applied Algebra, 104:41–59, 1995

work page 1995
[11]

G. Manuell. Strictly zero-dimensional biframes and a c haracterisation of congru- ence frames. Applied Categorical Structures, 26:645–655, 2018

work page 2018
[12]

L. Skula. On a reﬂective subcategory of the category of a ll topological spaces. Trans. Amer . Math. Soc., pages 37–41, 1969

work page 1969
[13]

M. H. Stone. The theory of representations for boolean a lgebras. Trans. AMS, 40:37–111, 1936

work page 1936
[14]

V an De V el

M. V an De V el. A selection theorem for topological conve x structures. Trans. AMS, 336:463–496, 1993. 31

work page 1993

[1] [1]

Aull and W .J

C.E. Aull and W .J. Thron. Separation axioms between T0 and T1. Indag. Math., pages 26–37, 1963

work page 1963

[2] [2]

Banaschewski, G

B. Banaschewski, G. C. L. Br¨ ummer, and K. A. Hardie. Bifr ames and bispaces. Quaest. Math., 6:13–25, 1983

work page 1983

[3] [3]

Dawson, R

R. Dawson, R. Par´ e, and D. Pronk. Adjoining adjoints. Advances in Mathematics, 178:99–140, 2003

work page 2003

[4] [4]

R. J. M. Dawson. Limits and colimits of convexity spaces. Cahiers de T opologie et g´ eom´ etrie diﬀ´ erentielle cat´ egoriques, 28:307–328, 1987. 30

work page 1987

[5] [5]

Ellerman

D. Ellerman. The logic of partitions: Introduction to th e dual of the logic of subsets. The Review of Symbolic Logic , 3:287–350, 2010

work page 2010

[6] [6]

Ho ﬀmann

R.-E. Ho ﬀmann. Irreducible ﬁlters and sober spaces. Manuscripta Math. , 22:365–380, 1977

work page 1977

[7] [7]

Ho ﬀmann

R.-E. Ho ﬀmann. Essentially complete T0-spaces. Manuscripta Math. , 27:401– 431, 1979

work page 1979

[8] [8]

D. C. Kay and E. W . Womble. Axiomatic convexity theory and relationships between the caratheodory, helly, and radon numbers. Paciﬁc Journal of Mathe- matics, 38:471–485, 1971

work page 1971

[9] [9]

T. Kenney. Partial-sup lattices. Theory and Applications of Categories , 30:305– 331, 2015

work page 2015

[10] [10]

A. Kock. Monads for which structures are adjoint to unit s. Journal of Pure and Applied Algebra, 104:41–59, 1995

work page 1995

[11] [11]

G. Manuell. Strictly zero-dimensional biframes and a c haracterisation of congru- ence frames. Applied Categorical Structures, 26:645–655, 2018

work page 2018

[12] [12]

L. Skula. On a reﬂective subcategory of the category of a ll topological spaces. Trans. Amer . Math. Soc., pages 37–41, 1969

work page 1969

[13] [13]

M. H. Stone. The theory of representations for boolean a lgebras. Trans. AMS, 40:37–111, 1936

work page 1936

[14] [14]

V an De V el

M. V an De V el. A selection theorem for topological conve x structures. Trans. AMS, 336:463–496, 1993. 31

work page 1993