Pith. sign in

REVIEW 3 major objections 3 minor 58 references

Hausdorff coalgebras

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Hausdorff-style powerset coalgebras get all limits once a compact Hausdorff topology is added; without it, no terminal coalgebra exists.

desk verdict Worth a careful read for its architecture and secondary results, but the main completeness theorem rests on a false join-selection step and the cocompleteness claim is unproved. read the letter →

arxiv 1908.04380 v1 pith:K6AFHXTH submitted 2019-08-12 math.CT

Dirk Hofmann , Pedro Nora This is my paper · ORCID
classification math.CT MSC 18C2018D2018B3054B2006F07
keywords Hausdorfffunctorquantale-enrichedcategoriescoalgebracompactspacehit-and-misstopologycodirectedlimitsterminalKripkepolynomialfunctors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Quantale-enriched categories are a common generalization of ordered sets and metric spaces. This paper shows that the Hausdorff construction—the enriched analogue of the powerset functor—behaves badly in that setting: for any non-trivial quantale it has no terminal coalgebra. The turnaround is to add a compatible compact Hausdorff topology. Then the Hausdorff functor preserves codirected limits, and every "Hausdorff polynomial" functor built from constants, identity, products, sums, and the Hausdorff hyperspace has a (co)complete category of coalgebras. The result matters because it supplies a setting where metric transition systems have final coalgebra semantics.

What carries the argument

The load-bearing object is the Hausdorff functor H on V-CatCH, which sends a V-categorical compact Hausdorff space X to the hyperspace HX of closed increasing subsets, equipped with the Hausdorff structure $H_a(A,B)=\bigwedge_{y\in B}\bigvee_{x\in A}a(x,y)$ and the hit-and-miss topology. Two mechanisms carry the argument. Proposition 4.32 uses the finite-intersection property of codirected families of closed sets in a compact Hausdorff space to show that H preserves codirected initial cones; this is where compactness does the work. Theorem 4.30 is a general criterion: an endofunctor on a complete, cocomplete category with a suitable factorization structure and smallness condition, preserving codirected limits and initial monomorphisms, has a complete category of coalgebras. The negative result on plain V-Cat comes from a generalized fixed-point argument showing H has no fixed points when the quantale is non-trivial.

What would settle it

Let $V=\mathcal{P}(\mathbb{N})$ with union as tensor and the empty set as unit; this satisfies the paper's Assumptions 4.9 and 4.25. Let $v=\{0,1\}$ and let the relevant distances be the singletons $\{0\},\{1\},\ldots$. Then $v$ lies below the union, but no single singleton contains $v$, directly refuting the selection step used in Proposition 4.32. Testing whether $H:V\text{-}CatCH\to V\text{-}CatCH$ still preserves codirected limits for this $V$ settles whether Theorem 4.34 needs an additional hypothesis.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.46: every Hausdorff polynomial functor on V-CatCH—the category of quantale-enriched categories equipped with a compatible compact Hausdorff topology—has a category of coalgebras that is complete and cocomplete. The decisive step is Theorem 4.34, which states that the Hausdorff functor H:V-CatCH→V-CatCH preserves codirected limits. This is proved by showing that H preserves codirected initial cones: compactness turns a family of pointwise approximations into a single common witness, and the initial cone transfers the bound back to the source. Theorem 4.30, a general criterion for completeness of coalgebra categories, then converts this into the full (co)completeness statement. The paper contrasts this with Corollary 3.19: on V-Cat without any topology, the Hausdorff functor has no terminal coalgebra for any non-trivial quantale, via a diagonal argument in the spirit of the classical proof that the powerset functor has no fixed points.

Load-bearing premise

The proof's load-bearing premise is that a threshold below a supremum over A is already reached by some single element of A; the stated hypotheses do not guarantee this selection principle, and it fails for some completely distributive quantales (e.g. powersets with union).

Editorial extensions

If this is right

  • Every Hausdorff polynomial functor on V-CatCH has a terminal coalgebra, so final coalgebra semantics exist for these enriched transition systems.
  • The pure Hausdorff functor's terminal coalgebra on V-CatCH is carried by the one-point compactification of the natural numbers, just as in the ordered case; the quantale V does not change the carrier.
  • The Hausdorff metric and the hit-and-miss hyperspace topology are compatible: for a compact metric space, the Hausdorff V-structure induces the hit-and-miss topology.
  • The same completeness result holds for Hausdorff polynomial functors on the full subcategory V-Priest of enriched Priestley spaces.
  • On plain V-Cat there is no terminal coalgebra for any non-trivial quantale, so the added compactness is a genuine threshold, not a technical convenience.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the missing selection step noted below is real, the completeness theorem is likely to hold only for quantales satisfying an explicit join-selection property; Assumptions 4.9 and 4.25 do not visibly supply it.
  • The compactness argument is quite general: any powerset-like functor on a category topological over compact Hausdorff spaces that sends codirected initial cones to initial cones should yield complete coalgebra categories, so Theorem 4.46 is probably one instance of a broader template.
  • Because the terminal coalgebra of the pure Hausdorff functor on V-CatCH is ordered for every V, genuinely quantitative final behaviour must come from polynomial combinations such as metric-valued constants, not from the Hausdorff functor alone.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies categories of coalgebras for Kripke-style polynomial functors in a quantale-enriched setting. Section 2 proves a topological-lifting theorem: any strict lifting of a Set-endofunctor to a topological category induces a topological forgetful functor between the corresponding coalgebra categories, so limits of coalgebras are no richer than in Set. Section 3 introduces the Hausdorff functor on V-Cat and proves, via a generalized Cantor argument, that it has no terminal coalgebra for non-trivial quantales. Section 4 enriches the setting with compact Hausdorff topology, defining V-categorical compact Hausdorff spaces and a Hausdorff functor that combines the Hausdorff metric with the Vietoris topology. The central claims are that this functor preserves codirected limits (Theorem 4.34) and that every Hausdorff polynomial functor on V-CatCH has a (co)complete category of coalgebras (Theorem 4.46), with further results for Priestley spaces.

Significance. If the main theorems are established, the paper contributes a useful positive counterpart to the negative results of Section 3: restricting to compact Hausdorff spaces with a compatible V-category structure allows a Hausdorff construction with a terminal coalgebra and complete categories of coalgebras. The unification of Nachbin's ordered compact Hausdorff spaces with enriched metric structures is conceptually valuable, and the paper contains several carefully argued auxiliary results, such as Lemma 4.11 and Proposition 4.18. The negative result for V-Cat and the general topological lifting theorem of Section 2 are interesting in their own right and appear to be sound. The exposition is generally clear and the dependencies among the results are explicit. However, the central preservation theorem depends on a join-selection step that is not justified, and the cocompleteness part of Theorem 4.46 is not proved.

major comments (3)
  1. [Section 4.2, Proposition 4.32] The proof of the inequality u ≤ Ha(A,B) relies on an unjustified join-selection step: from v ≪ u and u ≤ ∨_{x∈A} a_i(f_i(x), f_i(b)) it concludes that for every i there exists x_i ∈ A with v ≤ a_i(f_i(x_i), f_i(b)). This does not follow from complete distributivity, and Assumptions 4.9 and 4.25 do not supply it. A concrete counterexample within the assumptions is V = P(N) with tensor ∩ and unit k = N, X = {x1,x2,b} with a(x1,b) = {1}, a(x2,b) = {2}, a(x1,x2) = {1}, a(x2,x1) = {2}, all other non-diagonal values empty, and A = {x1,x2}, B = {b}, v = {1,2}. Then v is way-below u = Ha(A,B) = {1,2}, yet neither a(x1,b) nor a(x2,b) contains v, so the finite-intersection argument collapses. Since Proposition 4.32 is the sole input for Theorem 4.34, the completeness claim for Theorem 4.46 is not established as written.
  2. [Section 4.1, Lemma 4.28] The same join-selection pattern occurs in Lemma 4.28: from u ≪ Ha(A,B) the proof infers, for every y ∈ B, the existence of a single x ∈ A with v ≪ a(x,y). This inference is refuted by the same powerset counterexample, where the join is attained only by taking both x1 and x2 together. Since Lemma 4.28 is used in Theorem 4.29 to identify the topology induced by the Hausdorff structure with the hit-and-miss topology, that result also needs a corrected proof or additional hypotheses.
  3. [Section 4.2, Theorem 4.46] The theorem states that the category of coalgebras of a Hausdorff polynomial functor on V-CatCH is (co)complete, but only completeness is addressed by the argument. Theorem 4.30, even if its referenced proof is accepted, gives completeness of CoAlg(F) and not cocompleteness; no separate construction of colimits or a cited theorem for colimits appears. The cocompleteness claim should either be proved or removed from the statement.
minor comments (3)
  1. [Section 4.2, Theorem 4.30] The text cites 'Corollary 3.11' in the proof of Theorem 4.30, but the equalisers result is Theorem 3.11; the reference should be corrected.
  2. [Section 4.2, heading] The heading 'Hausdorff polynomial functors on Priest' appears to be a typo; it should read 'Priestley'.
  3. [Section 4.2, after Theorem 4.46] In the note following Theorem 4.46, 'strucutre' should be 'structure'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; central completeness theorem is derived, not assumed.

full rationale

The main claimed derivation chain is: Theorem 4.34 (H : V-CatCH -> V-CatCH preserves codirected limits) is proved from Proposition 4.32 by a direct argument about codirected initial cones and compactness, and Theorem 4.46 then combines Theorem 4.34 with the external completeness criterion of Theorem 4.30, whose cited ingredients are Barr-Wells, Adamek, and Linton. The negative result for V-Cat (Corollary 3.19) is derived from Theorem 3.16, a Cantor-style argument credited to Dilworth and Gleason. The Hausdorff structure Ha(A,B) = /\_{y in B} \/_{x in A} a(x,y) is the standard definition from the literature (ACT10, Stu10), not a fitted quantity, and the paper does not rename an empirical result in new coordinates. There are self-citations (Nor19, HNN19, HR18, HN18), but they supply auxiliary structural facts, such as the equaliser theorem and background on V-categorical compact Hausdorff spaces, rather than the target completeness conclusion; none is used to assert the key preservation result by fiat. The weakest spot is the join-selection step inside Proposition 4.32, where the proof passes from u <= \/_{x in A} a_i(f_i(x), f_i(b)) to the existence of x_i in A with v <= a_i(f_i(x_i), f_i(b)); this step is not justified by the way-below relation for arbitrary suprema and appears to be a genuine mathematical gap. That is a correctness risk, not a circularity, because the step is not equivalent to the theorem's conclusion by definition. Score 2 reflects minor and non-load-bearing self-citation, with no circular reduction in the central derivation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central completeness theorems rest on explicit assumptions about the quantale V: complete distributivity, the value-quantale condition, and a finiteness/separation property. No empirical free parameters or invented physical entities appear. The proof of Proposition 4.32 additionally assumes an unstated join-selection property of way-below, which is the main fragility in the paper.

assumptions (7)
  • domain assumption V is a commutative and unital quantale
    Required by Definition A.1 for the whole V-Cat framework.
  • domain assumption V is completely distributive (Assumption 4.1)
    Imposed at the start of Section 4 to define V-CatCH and topologies on V.
  • domain assumption The set {u in V | u way-below k} is directed, i.e. V is a value quantale (Assumption 4.9)
    Used in Lemmas 4.10 and 4.11 and throughout Section 4 for compactness arguments.
  • domain assumption k at most u tensor v implies k at most u and k at most v (Assumption 4.25)
    Used from Remark 4.26 onward to control closures and upsets.
  • domain assumption hom(u,-) from (V,xi) to (V,xi) is continuous (Assumption 4.47)
    Imposed for the Priestley-space results in the final part of Section 4.
  • ad hoc to paper A join-selection property of way-below: v way-below u and u at most the join of S implies some s in S with v at most s
    Unstated but needed in Proposition 4.32; not generally true for way-below on arbitrary suprema in completely distributive quantales.
  • standard math Classical theorems on topological functors, (E,M)-factorizations, and limits in categories of coalgebras (Barr-Wells, Linton, Adamek)
    Theorem 4.30 combines these textbook results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hausdorff coalgebras." pith.science (2026). https://pith.science/paper/K6AFHXTH

@misc{pith2026190804380,
  author       = {Pith},
  title        = {Pith review of: Hausdorff coalgebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6AFHXTH}},
  note         = {Machine review of arXiv:1908.04380}
}
abstract

As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $\mathsf{Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff $\mathsf{V}$-category structure. As a starting point, we show that for a lifting of a $\mathsf{SET}$-functor to a topological category $\mathsf{X}$ over $\mathsf{Set}$ that commutes with the forgetful functor, the corresponding category of coalgebras over $\mathsf{X}$ is topological over the category of coalgebras over $\mathsf{Set}$ and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology \emph{\`a la} Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    On the categorical meaning of Hausdorff and Gromov distances, I

    Andrei Akhvlediani, Maria Manuel Clementino, and Walter Tholen. On the categorical meaning of H ausdorff and G romov distances, I . Topology and its Applications, 157(8):1275--1295, June 2010, http://arxiv.org/abs/0901.0618 arXiv:0901.0618 [math.CT]

    work page Pith review arXiv 2010

  2. [2]

    Introduction to coalgebra

    Ji r \' Ad \'a mek. Introduction to coalgebra. Theory and Applications of Categories, 14(8):157--199, 2005

    work page 2005

  3. [3]

    Strecker

    Ji r \'i Ad \'a mek, Horst Herrlich, and George E. Strecker. Abstract and concrete categories: T he joy of cats . Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1990. Republished in: Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1--507

    work page 2006

  4. [4]

    Domain T heory

    Samson Abramsky and Achim Jung. Domain T heory. In S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, editors, Handbook of L ogic in C omputer S cience: S emantic S tructures , volume 3, pages 1--168. Oxford University Press, 1995

    work page 1995

  5. [5]

    Terminal coalgebras in well-founded set theory

    Michael Barr. Terminal coalgebras in well-founded set theory. Theoretical Computer Science, 114(2):299--315, June 1993

    work page 1993

  6. [6]

    Coalgebraic B ehavioral M etrics

    Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara K \" o nig. Coalgebraic B ehavioral M etrics. Logical Methods in Computer Science, 14(3):1860--5974, 2018

    work page 2018

  7. [7]

    On the logic of generalised metric spaces

    Octavian B a bu s and Alexander Kurz. On the logic of generalised metric spaces. In Ichiro Hasuo, editor, Coalgebraic M ethods in C omputer S cience , pages 136--155. Springer International Publishing, 2016

    work page 2016

  8. [8]

    Up-To Techniques for Behavioural Metrics via Fibrations

    Filippo Bonchi, Barbara K \" o nig, and Daniela Petri s an. Up- T o T echniques for B ehavioural M etrics via F ibrations. In Schewe and Zhang SZ18 , pages 17:1--17:17, http://arxiv.org/abs/1806.11064 arXiv:1806.11064 [cs.LO] . 29 th International Conference on Concurrency Theory, September 4-7, 2018 - Beijing, China

    work page Pith review arXiv 2018

Show all 58 references
  1. [9]

    Extending set functors to generalised metric spaces

    Adriana Balan, Alexander Kurz, and Ji r \' i Velebil. Extending set functors to generalised metric spaces. Logical Methods in Computer Science, 15(1), 2019

    2019

  2. [10]

    General topology, part I

    Nicolas Bourbaki. General topology, part I . Hermann, Paris and Addison-Wesley, 1966. Chapters 1--4

    1966

  3. [11]

    An algebra for K ripke polynomial coalgebras

    Marcello Bonsangue, Jan Rutten, and Alexandra Silva. An algebra for K ripke polynomial coalgebras. In 24 th Annual IEEE Symposium on Logic in Computer Science, 11-14 August 2009, Los Angeles, CA, USA , pages 49--58. IEEE , August 2009

    2009

  4. [12]

    Birsan and Dan Tiba

    T. Birsan and Dan Tiba. One hundred years since the introduction of the set distance by D imitrie P ompeiu. In F. Pandolfi, A. Ceragioli, H. Dontchev, K. Furuta, and L. Marti, editors, System Modeling and Optimization, volume 199 of IFIP Advances in Information and Communicati...

    2006

  5. [13]

    Toposes, triples and theories

    Michael Barr and Charles Wells. Toposes, triples and theories. Springer-Verlag New York, 1985. Republished in: Reprints in Theory and Applications of Categories, No. 12, 2005, pp. 1 -- 288

    1985

  6. [14]

    \" U ber eine elementare F rage der M annigfaltigkeitslehre

    Georg Cantor. \" U ber eine elementare F rage der M annigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1:75--78, June 1891

  7. [15]

    Triquotient maps via ultrafilter convergence

    Maria Manuel Clementino and Dirk Hofmann. Triquotient maps via ultrafilter convergence. Proceedings of the American Mathematical Society, 130(11):3423--3431, November 2002, eprint: http://www.mat.uc.pt/preprints/ps/p0022.ps

    2002

  8. [16]

    Cartesian closed exact completions in topology

    Maria Manuel Clementino, Dirk Hofmann, and Willian Ribeiro. Cartesian closed exact completions in topology. Journal of Pure and Applied Algebra, 224(2):610--629, February 2020, http://arxiv.org/abs/1811.0399 arXiv:1811.0399 [math.CT]

    2020

  9. [17]

    Dilworth and Andrew M

    Robert P. Dilworth and Andrew M. Gleason. A generalized C antor theorem. Proceedings of the American Mathematical Society, 13(5):704--705, May 1962

    1962

  10. [18]

    General topology, volume 6 of Sigma Series in Pure Mathematics

    Ryszard Engelking. General topology, volume 6 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin, 2 edition, 1989. Translated from the Polish by the author

    1989

  11. [19]

    Robert C. Flagg. Completeness in continuity spaces. In Robert Andrew George Seely, editor, Category Theory 1991: Proceedings of an International Summer Category Theory Meeting, held June 23-30, 1991 , volume 13 of CMS Conference Proceedings. American Mathematical Society, 1992

    1991

  12. [20]

    Robert C. Flagg. Quantales and continuity spaces. Algebra Universalis, 37(3):257--276, June 1997

    1997

  13. [21]

    Barry Fawcett and Richard J. Wood. Constructive complete distributivity. I . Mathematical Proceedings of the Cambridge Philosophical Society, 107(1):81--89, January 1990

    1990

  14. [22]

    Lawson, Michael W

    Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. A compendium of continuous lattices. Springer-Verlag, Berlin, 1980

    1980

  15. [23]

    Lawson, Michael W

    Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. Continuous lattices and domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003

    2003

  16. [24]

    Grundz \"u ge der M engenlehre

    Felix Hausdorff. Grundz \"u ge der M engenlehre . Veit & Comp, Leipzig, 1914

    1914

  17. [25]

    Enriched S tone-type dualities

    Dirk Hofmann and Pedro Nora. Enriched S tone-type dualities. Advances in Mathematics, 330:307--360, May 2018, http://arxiv.org/abs/1605.00081 arXiv:1605.00081 [math.CT]

    2018 arXiv

  18. [26]

    Limits in categories of V ietoris coalgebras

    Dirk Hofmann, Renato Neves, and Pedro Nora. Limits in categories of V ietoris coalgebras. Mathematical Structures in Computer Science, 29(4):552--587, April 2019, http://arxiv.org/abs/1612.03318 arXiv:1612.03318 [cs.LO]

    2019 arXiv

  19. [27]

    Topological theories and closed objects

    Dirk Hofmann. Topological theories and closed objects. Advances in Mathematics, 215(2):789--824, November 2007

    2007

  20. [28]

    Dirk Hofmann and Carla D. Reis. Probabilistic metric spaces as enriched categories. Fuzzy Sets and Systems, 210:1--21, January 2013, http://arxiv.org/abs/1201.1161 arXiv:1201.1161 [math.GN]

    2013 arXiv

  21. [29]

    Dirk Hofmann and Carla D. Reis. Convergence and quantale-enriched categories. Categories and General Algebraic Structures with Applications, 9(1):77--138, July 2018, http://arxiv.org/abs/1705.08671 arXiv:1705.08671 [math.CT]

    2018 arXiv

  22. [30]

    Seal, and Walter Tholen, editors

    Dirk Hofmann, Gavin J. Seal, and Walter Tholen, editors. Monoidal T opology. A C ategorical A pproach to O rder, M etric, and T opology , volume 153 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, July 2014. Authors: Maria Manuel Cle...

    2014

  23. [31]

    L awvere completion and separation via closure

    Dirk Hofmann and Walter Tholen. L awvere completion and separation via closure. Applied Categorical Structures, 18(3):259--287, November 2010, http://arxiv.org/abs/0801.0199 arXiv:0801.0199 [math.CT]

    2010 arXiv

  24. [32]

    Johnstone

    Peter T. Johnstone. Stone spaces, volume 3 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1986. Reprint of the 1982 edition

    1986

  25. [33]

    Finite preorders and topological descent

    George Janelidze and Manuela Sobral. Finite preorders and topological descent. I . Journal of Pure and Applied Algebra, 175(1-3):187--205, November 2002. Special volume celebrating the 70 th birthday of Professor Max Kelly

    2002

  26. [34]

    Finite preorders and topological descent

    George Janelidze and Manuela Sobral. Finite preorders and topological descent. II . \'e tale descent. Journal of Pure and Applied Algebra, 174(3):303--309, October 2002

    2002

  27. [35]

    Stone coalgebras

    Clemens Kupke, Alexander Kurz, and Yde Venema. Stone coalgebras. Theoretical Computer Science, 327(1-2):109--134, October 2004

    2004

  28. [36]

    ( M etric) B isimulation G ames and R eal- V alued M odal L ogics for C oalgebras

    Barbara K \"o nig and Christina Mika-Michalski. ( M etric) B isimulation G ames and R eal- V alued M odal L ogics for C oalgebras. In Schewe and Zhang SZ18 , pages 37:1--37:17, http://arxiv.org/abs/1705.10165 arXiv:1705.10165 [cs.LO] . 29 th International Conference on Concurr...

    2018 arXiv

  29. [37]

    A fixpoint theorem for complete categories

    Joachim Lambek. A fixpoint theorem for complete categories. Mathematische Zeitschrift, 103(2):151--161, 1968

    1968

  30. [38]

    William Lawvere

    F. William Lawvere. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matemàtico e Fisico di Milano, 43(1):135--166, December 1973. Republished in: Reprints in Theory and Applications of Categories, No. 1 (2002), 1--37

    2002

  31. [39]

    F. E. J. Linton. Coequalizers in categories of algebras. In B. Eckmann, editor, Seminar on T riples and C ategorical H omology T heory , volume 80 of Lecture Notes in Mathematics , pages 75--90. Springer Berlin Heidelberg, Berlin, Heidelberg, 1969

    1969

  32. [40]

    Topologies on spaces of subsets

    Ernest Michael. Topologies on spaces of subsets. Transactions of the American Mathematical Society, 71(1):152--182, January 1951

    1951

  33. [41]

    Topologia e Ordem

    Leopoldo Nachbin. Topologia e Ordem. University of Chicago Press, 1950

    1950

  34. [42]

    Topology and order

    Leopoldo Nachbin. Topology and order. Number 4 in Van Nostrand Mathematical Studies. D. Van Nostrand, Princeton, N.J.-Toronto, Ont.-London, 1965. Translated from the Portuguese by Lulu Bechtolsheim

    1965

  35. [43]

    Kleisli dualities and Vietoris coalgebras

    Pedro Nora. Kleisli dualities and Vietoris coalgebras. PhD thesis, Universidade de Aveiro, 2019

    2019

  36. [44]

    Sur la continuit\'e des fonctions de variables complexes

    Dimitrie Pompeiu. Sur la continuit\'e des fonctions de variables complexes. Annales de la Faculté des Sciences de l’Université de Toulouse pour les Sciences Mathématiques et les Sciences Physiques. 2ième Série, 7(3):265--315, 1905

    1905

  37. [45]

    Priestley

    Hilary A. Priestley. Representation of distributive lattices by means of ordered stone spaces. Bulletin of the London Mathematical Society, 2(2):186--190, July 1970

    1970

  38. [46]

    Priestley

    Hilary A. Priestley. Ordered topological spaces and the representation of distributive lattices. Proceedings of the London Mathematical Society. Third Series, 24(3):507--530, April 1972

    1972

  39. [47]

    George N. Raney. Completely distributive complete lattices. Proceedings of the American Mathematical Society, 3(5):677--680, May 1952

    1952

  40. [48]

    Universal coalgebra: a theory of systems

    Jan Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science, 249(1):3--80, October 2000

    2000

  41. [49]

    Algebras for Generalized Power Constructions

    Andrea Schalk. Algebras for Generalized Power Constructions. PhD thesis, Technische Hochschule Darmstadt, 1993

    1993

  42. [50]

    Gavin J. Seal. Canonical and op-canonical lax algebras. Theory and Applications of Categories, 14(10):221--243, 2005

    2005

  43. [51]

    `` H ausdorff distance'' via conical cocompletion

    Isar Stubbe. `` H ausdorff distance'' via conical cocompletion. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 51(1):51--76, 2010, http://arxiv.org/abs/0903.2722 arXiv:0903.2722 [math.CT]

    2010 arXiv

  44. [52]

    An introduction to quantaloid-enriched categories

    Isar Stubbe. An introduction to quantaloid-enriched categories. Fuzzy Sets and Systems, 256:95--116, December 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33 rd Linz Seminar on Fuzzy Set Theory, 2012)

    2014

  45. [53]

    CONCUR 2018 , volume 118 of LIPICS

    Sven Schewe and Lijun Zhang, editors. CONCUR 2018 , volume 118 of LIPICS. Schloss Dagstuhl - Leibniz-Zentrum f \"u r Informatik GmbH, Wadern/Saarbruecken, Germany, 2018. 29 th International Conference on Concurrency Theory, September 4-7, 2018 - Beijing, China

    2018

  46. [54]

    Ordered topological structures

    Walter Tholen. Ordered topological structures. Topology and its Applications, 156(12):2148--2157, July 2009

    2009

  47. [55]

    On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

    Daniele Turi and Jan Rutten. On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces. Mathematical Structures in Computer Science, 8(5):481--540, October 1998

    1998

  48. [56]

    An accessible approach to behavioural pseudometrics

    Franck van Breugel, Claudio Hermida, Michael Makkai, and James Worrell. An accessible approach to behavioural pseudometrics. In Lu \' s Caires, Giuseppe F. Italiano, Lu \' s Monteiro, Catuscia Palamidessi, and Moti Yung, editors, Automata, L anguages and P rogramming , pages 1...

    2005

  49. [57]

    Waterhouse

    William C. Waterhouse. An empty inverse limit. Proceedings of the American Mathematical Society, 36(2):618, February 1972

    1972

  50. [58]

    o der, Dirk Pattinson, and Barbara K \

    Paul Wild, Lutz Schr \" o der, Dirk Pattinson, and Barbara K \" o nig. A van B enthem T heorem for F uzzy M odal L ogic. In Proceedings of the 33 rd Annual ACM / IEEE Symposium on Logic in Computer Science - LICS '18 . ACM Press, 2018, http://arxiv.org/abs/1802.00478 arXiv:180...

    2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.