A Lightweight Learned Cardinality Estimation Model

Guoliang Li; Jianhua Feng; Jintao Zhang; Yaoyu Zhu

REVIEW 3 major objections 5 minor 11 references

A Lightweight Learned Cardinality Estimation Model

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that any topos with enough points is equivalent to the category of étale spaces over its point space, so a geometric theory is recoverable from the convergence structure on its set models.

desk verdict Real proof-technique contribution with a genuine gap in the class-level corollary; the separating-set theorem looks right. read the letter →

arxiv 2508.09602 v1 pith:VUCEJK4S submitted 2025-08-13 cs.DB cs.AIcs.LG

Yaoyu Zhu , Jintao Zhang , Guoliang Li , Jianhua Feng This is my paper · ORCID

classification cs.DBcs.AIcs.LG MSC 18B2503G3054A20

keywords toposwithenoughpointsétalespaceultraconvergencereconstructiontheoremstrongconceptualcompletenessgeometriclogicultrafilterconvergence2-category

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

This paper proves a reconstruction theorem: any topos with enough points is equivalent to the category of étale spaces over its space of points, equipped with a canonical ultraconvergence structure. Points are set-valued models of the geometric theory the topos classifies, and the extra structure that lets the point space encode the topos is convergence of ultrafamilies rather than mere topology. The proof works directly on the topos and does not pass through a groupoid representation of the topos. If correct, it gives a strong conceptual completeness statement for geometric logic: a theory with enough set models is recoverable from the convergence behaviour of those models. The set-sized statement is the proved core; the class-level corollary is the announced payoff and rests on an additional colimit step that the paper states but does not fully prove.

What carries the argument

The key objects are ultraconvergence spaces and étale maps between them. An ultraconvergence space is a set of points equipped with a relation between each point and ultrafamilies of points, valued in sets rather than truth values; it categorifies the relational description of topology via ultrafilter convergence. An étale map $\pi : E \to B$ is a continuous map whose fibers are small and whose ultra-arrows lift uniquely, a direct generalization of local homeomorphisms. The bridge is the equivalence between continuous maps $X \to \mathrm{Set}$ and étale spaces over $X$, which lets the evaluation functor $J{-}K$ be read as the construction of étale spaces. Ordinary categories and topological

What would settle it

To settle the class-level version, take a filtered diagram of set-sized separating sets and check whether $\mathrm{UltSp}(\mathrm{colim}_\alpha X_\alpha, \mathrm{Set})$ is naturally equivalent to $\lim_\alpha \mathrm{UltSp}(X_\alpha, \mathrm{Set})$. A single filtered diagram of ultraconvergence spaces for which this fails would leave the set-level Theorem 1.1 true but invalidate Corollary 6.1 as proved; alternatively, a proof of the interchange would close the gap.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a topos $E$ and a separating set $X$ of its points, the evaluation functor $J{-}K : E \to \mathrm{UltSp}(X, \mathrm{Set})$ is an equivalence. Each object $\varphi$ of $E$ is thereby represented by its fibers over the models $x \in X$, namely the sets $x(\varphi)$, assembled into an étale space $\pi_\varphi : J\varphi K \to X$. The paper then lifts this to Corollary 6.1, presenting any topos with enough points as the category $\mathrm{Et}(\mathrm{pt}(E))$ of étale spaces over the full class of its points. The proof establishes that $J{-}K$ is full on subobjects and covering, which together force the equivalence; the lifting step assembles set-sized separa

Load-bearing premise

The load-bearing assumption is that the class of all points of a topos can be assembled from its set-sized separating sets in a way that preserves the reconstruction; the paper states this colimit-to-limit interchange without proof, while itself noting that the main theorem genuinely requires a set of points, not a class.

Editorial extensions

If this is right

Any topos with enough points can be reconstructed from its point space alone, so the geometric theory it classifies is determined by the convergence structure on its set models.
The 2-category of toposes with enough points embeds 2-fully-faithfully into the 2-category of ultraconvergence spaces, making ultraconvergence spaces a genuine dual side for toposes.
For presheaf toposes, the theorem recovers the familiar equivalence with continuous maps from a discrete order-based space to sets, and identifies the point space of such a topos as an Ind-completion, a kind of soberification of the index category.
The set-based reconstruction theorem extends formally to the full class of points of a topos with enough points, provided the 2-categorical colimit and limit interchange used in the proof is valid.
The proof gives a representation theorem for toposes with enough points that is independent of topological-groupoid representations, extending the coherent case to arbitrary geometric theories with enough set models.

Reading between the lines

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

The paper leaves the connection to the alternative ionad presentation implicit; a natural next step would be to spell out how these étale spaces compare with that presentation, perhaps yielding an explicit soberification construction on ultraconvergence spaces rather than via toposes.
Because the proof does not use groupoid representations, it may transfer to settings where such representations are unavailable, such as variants of geometric logic over other base toposes, as long as the ultrafamily machinery can be recast.
The explicit gap in the lift from sets to classes suggests a concrete question a reader could test: does $\mathrm{UltSp}(-, \mathrm{Set})$ send the pseudocolimit of the filtered diagram of separating sets to the corresponding limit? Finding a counterexample would leave the set-level theorem intact but invalidate the class-level corollary as proved.
If the set-level theorem is taken as the main achievement, one could explore whether the étale-space fibers, being pointwise formula extensions, yield a computationally meaningful reconstruction for concrete logical theories.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the actual manuscript is the topos paper, not the CoDe abstract attached in the record—someone cross-wired the metadata, and that should be fixed before anything else. The mathematics is worth your time.

What's new: the ultraconvergence-space framework, the notion of etale map (Definition 4.1), and the Grothendieck construction in Section 4.2. The main theorem for a fixed separating set of points is the same result Saadia and Hamad got independently, but this proof avoids Butz–Moerdijk groupoids entirely, and it is structurally closer to Makkai's original argument. That is a legitimate contribution even though the theorem itself isn't new. The paper is honest about the concurrent work, cites it, and explains the difference.

Where it's solid: the proof follows Makkai's template with standard ingredients—Giraud, Łoś, infinitary pretoposes—so the central argument is plausible. Theorem 1.1 for a small separating set seems well-supported, assuming Propositions 5.4 and 5.6 check out. In the version I have, those two proofs are only outlined; I'd want to see them in full before endorsing. That's a referee's job, not a fatal flaw.

The soft spot: Corollary 6.1, the advertised representation of every topos with enough points from its full class of points. The proof asserts that pt(E) is the pseudocolimit in UltSp of the small separating sets, and that the hom-2-functor turns that into the corresponding limit. That interchange is not proved, and it's not automatic—ultra-arrows in the large space can involve points from many different separating sets. The authors themselves say Theorem 1.1 genuinely requires a set, and they point to Saa25's Theorem 8.3 for the class-level lift. Corollary 6.1 doesn't actually invoke that theorem, and the sketch leaves the interchange as the load-bearing step. So as written, the class-level claim is under-supported. It might be fixable by citing Saa25 or by adding a proof of the interchange, but the current text doesn't do the work.

Who it's for: people working in categorical logic, topos duality, and conceptual completeness. It's a serious paper that deserves refereeing—the core theorem is likely true, the proof route is original, and the gap is a compressible step rather than a contradiction. But the referee should demand a complete proof of the colimit interchange or a proper citation for the lift.

Recommendation: send it to a good referee, with a clear request to check Proposition 5.4/5.6 and Corollary 6.1. And fix the metadata record.

Referee Report

3 major / 5 minor

Summary. The paper introduces a 2-category UltSp of ultraconvergence spaces, in which the datum of a point converging to an ultrafilter is replaced by a Set-valued family of ultra-arrows, together with a notion of etale map and etale space over such a space. Its main theorem (Theorem 1.1) states that if E is a Grothendieck topos and X is a separating set of points equipped with the canonical ultraconvergence structure, then the evaluation functor J−K : E → Et(X) (equivalently, E → UltSp(X, Set)) is an equivalence of categories. The proof is patterned on Makkai's proof for pretoposes, via fullness on subobjects (Prop. 5.4) and covering (Prop. 5.6). Corollary 6.1 extends the statement to all toposes with enough points by taking X = pt(E), and Corollary 6.3 derives a 2-fully-faithful embedding of toposes with enough points into UltSp.

Significance. If Theorem 1.1 holds, this is a substantial duality result: it extends Makkai's and Lurie's coherent reconstruction theorems to all geometric theories with enough Set-models, and it does so without relying on Butz--Moerdijk groupoid representations, as the concurrent work of Saadia and Hamad does. The etale-space formulation is a genuine conceptual contribution, and the paper is careful to compare its framework with ionads and with Barr's relational β-modules. However, the advertised class-level representation theorem for all toposes with enough points (Corollary 6.1) is not established by the proof actually supplied, and the provided full text omits the proofs of the two key propositions underlying Theorem 1.1. The central separating-set statement is plausible and follows a known template, but the current manuscript is not yet self-contained at the advertised level.

major comments (3)

[§5, Theorem 1.1 proof outline (p. 22)] The proof outline states that J−K will be shown to have two properties, fullness on subobjects and covering, and then the supplied text jumps directly to Section 6. The statements and proofs of Proposition 5.4 (fullness on subobjects) and Proposition 5.6 (covering) are not present. These two propositions are load-bearing for Theorem 1.1; without them the main equivalence is unsupported. The final manuscript must include these proofs, or the theorem cannot be evaluated.
[Corollary 6.1] The proof of Corollary 6.1 asserts without proof that (a) pt(E) is the pseudocolimit in UltSp of the filtered posetal category J of separating sets, and (b) UltSp(−, Set) sends that pseudocolimit to the corresponding limit. Neither assertion is automatic, especially for the proper class pt(E), where ultra-arrows can involve points not lying in any single separating set. The paper's own Conclusion concedes that Theorem 1.1 genuinely requires a set of points and refers to [Saa25, Thm. 8.3] for the class-level lift; Corollary 6.1 neither invokes that theorem nor supplies the missing argument. This is load-bearing because Corollary 6.1 is the paper's advertised representation theorem for all toposes with enough points.
[Section 2 / Corollary 6.1] The statement of Corollary 6.1 requires Et(pt(E)) (equivalently UltSp(pt(E), Set)) to be a category in the sense of Section 2, i.e., locally small. When pt(E) is a proper class, the category of continuous maps from pt(E) to Set is not shown to be locally small; a priori, hom-sets may be large. This must be addressed for the equivalence in Corollary 6.1 to be well-formed, independently of the pseudocolimit issue.

minor comments (5)

[Submission metadata] The submission header identifies the paper as arXiv:2508.09602 (cs.DB), 'A Lightweight Learned Cardinality Estimation Model', but the full text is a mathematics paper, arXiv:2508.09604v2 [math.CT], 'Toposes with enough points as categories of etale spaces'. The metadata must be corrected.
[Corollary 6.1 proof] In the displayed chain, 'UltSp(colim_{X∈J} pt(E), Set)' should be 'UltSp(colim_{X∈J} X, Set)'.
[Section 3.3, Definition 3.19] The notation lim∗→1 in the definition of a 2-cell α : f ⇒ f′ is not defined at that point; a brief explanation that 1 is the one-point ultraconvergence space and ∗ its unique point would improve readability.
[Remark 3.21] The statement that the embedding Top → UltSp is 2-fully-faithful is asserted without proof or reference. Since this is used to connect the framework to classical topology, a short argument or a precise reference would be helpful.
[Conclusion / Corollary 6.1] The Conclusion cites [Saa25, Thm. 8.3] for the lift from separating sets to the full class of points, but Corollary 6.1 does not reference that theorem. Either the proof should be completed directly or the citation should be used explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main representation theorem is proved from stated axioms and standard topos-theoretic facts; the large-points corollary has an unproved colimit-interchange step, but that is a gap, not a circular reduction.

full rationale

The paper's central derivation is not circular. Theorem 1.1 is proved for a fixed separating set X by showing that J−K : E → Et(X) is full on subobjects (Proposition 5.4) and covering (Proposition 5.6), using only the stated metatheory (choice plus a fixed universe), Giraud's theorem, Los's theorem, and standard infinitary-pretopos facts from Johnstone. Makkai's proof is used as a structural template, not as a premise, and the target equivalence is never fed back as an input. Corollary 6.1 attempts to lift the result to the full class pt(E); the paper itself flags in the Conclusion that the proof of Theorem 1.1 requires a set of points rather than a proper class, and the proof sketch of Corollary 6.1 relies on an unproved pseudo-colimit interchange in UltSp. This is a genuine correctness gap, but it is not circularity: an unsupported step is not a reduction of the conclusion to its own inputs. The only self-citation, [AT26] (co-authored by Tarantino), is not visibly load-bearing in the main argument. There are no fitted parameters renamed as predictions, no uniqueness imported from the authors, and no known result merely renamed. The derivation is therefore self-contained; the flagged limitation belongs to correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters exist in this mathematics manuscript. The central equivalence rests on standard background results (choice plus a universe, Giraud's theorem, Los's theorem, infinitary pretopos theory) and on two less standard assumptions: the reduction to a separating set of points, and the pseudocolimit interchange in Corollary 6.1. The new definitions (ultraconvergence spaces, etale maps) are the paper's intended contribution, not unsupported postulates.

assumptions (6)

standard math Classical metatheory with the axiom of choice and a single Grothendieck universe
Stated in Section 2; needed for ultrafilters, ultraproducts, and Los's theorem in the classical form used.
standard math Giraud's theorem characterization of Grothendieck toposes
Invoked in Section 2 as the working definition of topos.
standard math Los's theorem: ultraproduct functors are coherent
Used in the Introduction and Section 3 to build the canonical ultraconvergence structure on points of a topos.
domain assumption Every topos with enough points admits a separating set of points
Invoked in Section 5 ('without loss of generality... a separating set of points, as opposed to a proper class') to reduce to the set-based Theorem 1.1.
domain assumption The proof template of Makkai [Mak87, Lem. 4.2] transfers to the present setting
Theorem 1.1's proof is stated to follow Makkai's structure; any hidden assumptions or gaps in Makkai's lemma would propagate.
ad hoc to paper In Corollary 6.1, pt(E) is the pseudocolimit of its separating sets in UltSp and UltSp(colim, Set) ~ lim UltSp(X, Set)
This interchange is asserted in a one-paragraph proof; it is the step that extends the result from a separating set to the class of all points.

invented entities (2)

Ultraconvergence space (with ultra-arrows and continuity structure)
purpose: A Set-valued generalization of Barr's two-valued convergence relation; the structure that records a topos's points and how ultrafamilies converge, enabling reconstruction of the topos.
New definition (Definition 3.8). In mathematics, definitions are legitimate if they support nontrivial theorems; the independent evidence here is the resulting equivalence theorem, not a falsifiable prediction.
Etale map / etale space over an ultraconvergence space
purpose: The paper's stated main conceptual novelty (Definition 4.1); etale spaces over X correspond to continuous maps X -> Set, and the topos E is recovered as Et(X).
New definition; its payoff is Theorem 4.15 (equivalence with copresheaves) and the main reconstruction theorem.

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Cite this review

Pith. "Pith review of A Lightweight Learned Cardinality Estimation Model." pith.science (2026). https://pith.science/paper/VUCEJK4S

@misc{pith2026250809602,
  author       = {Pith},
  title        = {Pith review of: A Lightweight Learned Cardinality Estimation Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VUCEJK4S}},
  note         = {Machine review of arXiv:2508.09602}
}

read the original abstract

Cardinality estimation is a fundamental task in database management systems, aiming to predict query results accurately without executing the queries. However, existing techniques either achieve low estimation accuracy or incur high inference latency. Simultaneously achieving high speed and accuracy becomes critical for the cardinality estimation problem. In this paper, we propose a novel data-driven approach called CoDe (Covering with Decompositions) to address this problem. CoDe employs the concept of covering design, which divides the table into multiple smaller, overlapping segments. For each segment, CoDe utilizes tensor decomposition to accurately model its data distribution. Moreover, CoDe introduces innovative algorithms to select the best-fitting distributions for each query, combining them to estimate the final result. By employing multiple models to approximate distributions, CoDe excels in effectively modeling discrete distributions and ensuring computational efficiency. Notably, experimental results show that our method represents a significant advancement in cardinality estimation, achieving state-of-the-art levels of both estimation accuracy and inference efficiency. Across various datasets, CoDe achieves absolute accuracy in estimating more than half of the queries.

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Reference graph

Works this paper leans on

11 extracted references · 8 canonical work pages

[1]

First-order logical duality

[AF13] S. Awodey and H. Forssell. “First-order logical duality”. In: Annals of Pure and Applied Logic 164.3 (2013), pp. 319–348. [AGV72] M. Artin, A. Grothendieck, and J. L. Verdier. Th´ eorie des Topos et Cohomologie ´Etale des Schemas . Vol. 269, 270,

work page 2013
[7]

Atomless Parts of Spaces

arXiv: 2507.07922 [math]. [Isb72] J. R. Isbell. “Atomless Parts of Spaces”. In: Mathematica Scandinavica 31.1 (1972), pp. 5–32. url: http://www.jstor.org/stable/24490585. 29 [Joh02] P. T. Johnstone. Sketches of an elephant: a Topos theory compendium . Oxford Logic Guides. New York, NY: Oxford Univ. Press,

arXiv 1972
[11]

Topological representations of distributive lattices and Brouwerian logics

arXiv: 2506.23935 [math.CT]. [Sto37] M. H. Stone. “Topological representations of distributive lattices and Brouwerian logics”. In: ˇCasopis pro Peˇ stov´ an ´ ı Matematiky a Fysiky67 (1937), pp. 1–25. url: http://dml.cz/dmlcz/124080. [Wri26] J. Wrigley. “On topological groupoids that represent theories”. In: Zeitschrift f¨ ur Mathematische Logik und Grun...

arXiv 1937
[73]

A unified framework for generalized multicate- gories

Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co., 1990, pp. xvi+650. [CS10] G. Cruttwell and M. Shulman. “A unified framework for generalized multicate- gories”. In: Theory and Applications of Categories 24.21 (2010), pp. 580–655. [CT03] M. Clementino and W. Tholen. “Metric, topology, and multicategory - a commo...

work page 1990
[1970]

Representing topoi by topological groupoids

url: https://dept.math.lsa.umich.edu/~ablass/thesis.pdf. [BM98] C. Butz and I. Moerdijk. “Representing topoi by topological groupoids”. In: Jour- nal of Pure and Applied Algebra 130.3 (1998), pp. 223–235. [Bur71] A. Burroni. “T-cat´ egories (cat´ egories dans un triple)”. In: Cahiers de Topologie et G´ eom´ etrie Diff´ erentielle Cat´ egoriques12 (1971), ...

work page 1998
[1972]

Direct limits and filtered colimits are strongly equiv- alent in all categories

[AN82] H. Andr´ eka and I. N´ emeti. “Direct limits and filtered colimits are strongly equiv- alent in all categories”. eng. In: Banach Center Publications 9.1 (1982), pp. 75–

work page 1982
[1977]

Sur les treillis des ouverts et les paratopologies

[PP58] D. Papert and S. Papert. “Sur les treillis des ouverts et les paratopologies”. In: S´ eminaire de topologie et g´ eom´ etrie diff´ erentielle1.1 (1957-58). url: https : / / www.numdam.org/item/SE_1957-1958__1__A1_0/. [Ros24] G. Rosolini. Ultracompletions. Talk at the International Category Theory Confer- ence

work page 1957
[2002]

An extension of the Galois theory of Grothendieck

[JT84] A. Joyal and M. Tierney. “An extension of the Galois theory of Grothendieck”. In: Memoirs of the American Mathematical Society 51 (1984). [Kel05] G. M. Kelly. “Basic concepts of enriched category theory”. In: Reprints in Theory and Applications of Categories 10 (2005), pp. 1–136. [Kel55] J. L. Kelley. General Topology. D. Van Nostrand Company, Inc.,

work page 1984

Show all 11 references

[2018]

Stone duality for first order logic

url: https : / / www . math . ias . edu /~lurie / papers/Conceptual.pdf. [Mak87] M. Makkai. “Stone duality for first order logic”. In: Advances in Mathematics 65.2 (1987), pp. 97–170. [Mak88] M. Makkai. “Strong conceptual completeness for first-order logic”. In: Annals of Pure...

1987
[2025]

Ultrafilters, finite coproducts and locally connected classifying toposes

arXiv: 2504.16690 [math.LO]. url: https://arxiv.org/abs/2504.16690. [Gar12] R. Garner. “Ionads”. In: Journal of Pure and Applied Algebra 216.8 (2012). Spe- cial Issue devoted to the International Conference in Category Theory ‘CT2010’, pp. 1734–1747. [Gar20] R. Garner. “Ultraf...

2012 arXiv
[2026]

Relational algebras

arXiv: 2601 . 22721 [math.CT]. url: https://arxiv.org/abs/2601.22721. [Bar70] M. Barr. “Relational algebras”. In: Reports of the Midwest Category Seminar IV . Ed. by S. MacLane et al. Berlin, Heidelberg: Springer, 1970, pp. 39–55. [Bek04] T. Beke. “Theories of Presheaf Type”. ...

1970 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

First-order logical duality

[AF13] S. Awodey and H. Forssell. “First-order logical duality”. In: Annals of Pure and Applied Logic 164.3 (2013), pp. 319–348. [AGV72] M. Artin, A. Grothendieck, and J. L. Verdier. Th´ eorie des Topos et Cohomologie ´Etale des Schemas . Vol. 269, 270,

work page 2013

[2] [7]

Atomless Parts of Spaces

arXiv: 2507.07922 [math]. [Isb72] J. R. Isbell. “Atomless Parts of Spaces”. In: Mathematica Scandinavica 31.1 (1972), pp. 5–32. url: http://www.jstor.org/stable/24490585. 29 [Joh02] P. T. Johnstone. Sketches of an elephant: a Topos theory compendium . Oxford Logic Guides. New York, NY: Oxford Univ. Press,

arXiv 1972

[3] [11]

Topological representations of distributive lattices and Brouwerian logics

arXiv: 2506.23935 [math.CT]. [Sto37] M. H. Stone. “Topological representations of distributive lattices and Brouwerian logics”. In: ˇCasopis pro Peˇ stov´ an ´ ı Matematiky a Fysiky67 (1937), pp. 1–25. url: http://dml.cz/dmlcz/124080. [Wri26] J. Wrigley. “On topological groupoids that represent theories”. In: Zeitschrift f¨ ur Mathematische Logik und Grun...

arXiv 1937

[4] [73]

A unified framework for generalized multicate- gories

Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co., 1990, pp. xvi+650. [CS10] G. Cruttwell and M. Shulman. “A unified framework for generalized multicate- gories”. In: Theory and Applications of Categories 24.21 (2010), pp. 580–655. [CT03] M. Clementino and W. Tholen. “Metric, topology, and multicategory - a commo...

work page 1990

[5] [1970]

Representing topoi by topological groupoids

url: https://dept.math.lsa.umich.edu/~ablass/thesis.pdf. [BM98] C. Butz and I. Moerdijk. “Representing topoi by topological groupoids”. In: Jour- nal of Pure and Applied Algebra 130.3 (1998), pp. 223–235. [Bur71] A. Burroni. “T-cat´ egories (cat´ egories dans un triple)”. In: Cahiers de Topologie et G´ eom´ etrie Diff´ erentielle Cat´ egoriques12 (1971), ...

work page 1998

[6] [1972]

Direct limits and filtered colimits are strongly equiv- alent in all categories

[AN82] H. Andr´ eka and I. N´ emeti. “Direct limits and filtered colimits are strongly equiv- alent in all categories”. eng. In: Banach Center Publications 9.1 (1982), pp. 75–

work page 1982

[7] [1977]

Sur les treillis des ouverts et les paratopologies

[PP58] D. Papert and S. Papert. “Sur les treillis des ouverts et les paratopologies”. In: S´ eminaire de topologie et g´ eom´ etrie diff´ erentielle1.1 (1957-58). url: https : / / www.numdam.org/item/SE_1957-1958__1__A1_0/. [Ros24] G. Rosolini. Ultracompletions. Talk at the International Category Theory Confer- ence

work page 1957

[8] [2002]

An extension of the Galois theory of Grothendieck

[JT84] A. Joyal and M. Tierney. “An extension of the Galois theory of Grothendieck”. In: Memoirs of the American Mathematical Society 51 (1984). [Kel05] G. M. Kelly. “Basic concepts of enriched category theory”. In: Reprints in Theory and Applications of Categories 10 (2005), pp. 1–136. [Kel55] J. L. Kelley. General Topology. D. Van Nostrand Company, Inc.,

work page 1984