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Metric-like spaces as enriched categories: three vignettes

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Pith's one-line read Viewing metric spaces as enriched categories unifies three seemingly unrelated constructions: the tight span, magnitude, and the Legendre-Fenchel transform.

desk verdict A clean, honest talk write-up on enriched-category metric spaces, but Theorem 3 on magnitude is false for general R+-categories and needs fixing before publication. read the letter →

arxiv 2501.00416 v1 pith:CNX72KHL submitted 2024-12-31 math.CT

Simon Willerton This is my paper · ORCID
classification math.CT MSC 18D2054E3552A41
keywords enrichedcategoriesLawveremetricspacesgeneralizedtightspanIsbellcompletionprofunctornucleusmagnitudeLegendre-Fencheltransform
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 makes the case that metric spaces are best understood as enriched categories: each distance is a hom-object in the monoidal category of non-negative reals with addition, so the triangle inequality is composition. On this view, several scattered constructions in metric geometry and convex analysis become instances of one categorical mechanism, the Isbell, or profunctor, nucleus. The paper argues that the tight span is the largest symmetric part of the Isbell completion, that magnitude is an enriched-category Euler characteristic which recovers the Solow-Polasky effective number of species, and that the Legendre-Fenchel transform is the Isbell adjunction associated to the pairing between a vector space and its dual. A sympathetic reader is meant to conclude that the categorical perspective is not a mere analogy but a working bridge between category theory and metric space theory.

What carries the argument

The object that carries the argument is the nucleus of a profunctor, also called the Isbell completion when the profunctor is the hom-object of an enriched category $\mathcal{X}(-,-)\colon \mathcal{X}^{\mathrm{op}}\otimes \mathcal{X}\to \mathcal{V}_e$. A profunctor $P$ induces an adjoint pair, $P_*$ and $P^*$, between spaces of scalar-valued functors, formally like multiplication by a matrix and its transpose. The nucleus is the centre of this adjunction, the fixed points of $P^*P_*$ or equivalently $P_*P^*$, and it is what turns the hom-profunctor of a metric space into the tight span and turns the pairing profunctor between a vector space and its dual into the Legendre-Fenchel transform. Over $\mathbb{R}_+$ the adjunction maps are computed by suprema of truncated differences, and the fixed-point condition says that a function is its own 'convex envelope' in the relevant sense; the enriched Yoneda embedding supplies the bridge that places the original space inside its completion.

What would settle it

Compute the Isbell completion of a finite classical metric space whose tight span is already known and check whether the largest symmetric subset is isometric to the tight span; a single mismatch would refute the first vignette's central identification. For the third vignette, one could test whether every fixed point of the double Legendre-Fenchel transform is closed convex, since a non-closed-convex fixed point would refute the nucleus identification.

Watch

Extended reading notes

Core claim

The central claim is that ordinary metric spaces, Lawvere's generalized metric spaces (where distances may be asymmetric or infinite), and even spaces with negative distances are all enriched categories, over the bases $\mathbb{R}_+$ and $\mathbb{R}$. The categorical Yoneda embedding generalizes the Kuratowski embedding of a metric space into its space of distance-preserving functions, and the enriched-category notion of the nucleus of a profunctor unifies the constructions examined here. Specifically, the paper presents the tight span of a classical metric space as the largest symmetric subset of the Isbell completion, the magnitude of a metric space as the Euler characteristic of an $\mathbb{R}_+$-enriched category, and the Legendre-Fenchel transform as the restriction of the Isbell adjunction between functions on a vector space and functions on its dual. In all three vignettes the same categorical construction, the fixed-point or nucleus of an Isbell-type adjunction, carries the mathematical content.

Load-bearing premise

The load-bearing premise is that the categorical fixed-point construction called the Isbell completion really gives the tight span of a classical metric space, a claim the paper states without proof, and that the same construction gives exactly the closed convex functions in the Legendre-Fenchel case, which is taken from earlier work.

Editorial extensions

If this is right

  • The tight span of a metric space, and its directed analogues studied by Hirai-Koichi and Kemajou-Künzi-Olela Otafudu, are the same construction as the Isbell completion, so results about injective hulls and directed network flow transfer back and forth.
  • The Isbell completion carries two semimodule structures over the semiring $([0,\infty],+,\max)$, connecting the tight span to tropical algebra.
  • Magnitude, defined as the sum of entries of the inverse similarity matrix $Z_{x,x'}=e^{-X(x,x')}$, is an Euler characteristic in the enriched sense; its large-scale limit counts points of the space, and its categorification is magnitude homology.
  • The Legendre-Fenchel transform is an $\mathbb{R}$-isometric adjunction, and on closed convex functions it restricts to an isometry between a finite-dimensional space and its dual, a statement called Toland-Singer duality.
  • Enriching over $\mathbb{R}$ forces a consistent choice for $\infty-\infty$: the category theory distinguishes $(+\infty)+(-\infty)$ from $(+\infty)-(+\infty)$, resolving a classical ambiguity.

Reading between the lines

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

  • Editorial inference: If the Isbell-completion view of the tight span is the right one, then constructions stated for the classical tight span, such as server-placement algorithms or flow duality, should have directed versions obtained by replacing the symmetric tight span with the full Isbell completion of an $\mathbb{R}_+$-category; this is testable on directed graphs with infinite edges.
  • Editorial inference: Because the Legendre-Fenchel nucleus is defined by a fixed-point condition, the same adjunction suggests a definition of 'convex envelope' in other enriched settings: take the fixed points of the double transform. Whether those fixed points coincide with the known closed convex functions for other bases, such as discrete or probabilistic metrics, is a concrete open question.
  • Editorial inference: The magnitude's failure at values like $t=\ln 2$, where the similarity matrix is singular, looks in this language like a failure of the categorical Euler characteristic to be defined; studying the behaviour of the Isbell-type adjunction at those parameters might give a notion of regularized magnitude.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper is an expository write-up of a talk: it explains Lawvere's observation that Lawvere metric spaces are categories enriched over the monoidal category (ℝ_+,+,0), and then presents three vignettes in which this viewpoint is applied: the tight span as an Isbell/profunctor nucleus, magnitude as an enriched-categorical Euler characteristic, and the Legendre-Fenchel transform as the nucleus of an ℝ-profunctor adjunction. The necessary enriched category theory (enriched categories, scalar-valued functors, the Yoneda embedding, and the profunctor nucleus) is reviewed, and proofs are generally deferred to the cited literature.

Significance. The paper is a clear and useful synthetic exposition; its main value is to show that three apparently separate constructions arise from the same enriched-categorical mechanism. It is honest about drawing two of the three vignettes from the author's earlier published papers [42,43]. The survey character and the breadth of examples make it a welcome introduction for nonspecialists. However, the statement of Theorem 3 in Section 4.2 is false for general ℝ_+-categories, and because the magnitude vignette is one of the three advertised case studies, this is a load-bearing issue that must be repaired.

major comments (1)
  1. [4.2, Theorem 3] Theorem 3 is false for ℝ_+-categories as stated. Take X with two objects a and b and X(a,a)=X(b,b)=0, X(a,b)=∞, X(b,a)=0; this satisfies the triangle inequality. For every t>0, the similarity matrix is [[1,0],[1,1]] because e^{-t∞}=0, and its inverse is [[1,0],[-1,1]], so the magnitude is |tX|=1 for all t even though X has two objects. Hence the asserted limit to the number of points fails, and the 'effective number of points' interpretation is not valid in this generality. Please restrict the theorem to the class for which it is true, for example finite classical metric spaces, meaning symmetric and separated ℝ_+-categories, and supply a proof or reference for the corrected statement, including the monotonicity assertion.
minor comments (5)
  1. [Abstract] The sentence beginning 'The analogy between the structures that can be made in to a common generalization of the two structures' is ungrammatical and should be rewritten.
  2. [2.2, Proposition 1] The displayed items refer to X(c,c′) although the objects have been called x; the notation should be X(x,x′).
  3. [2.2] The word 'definiton' should be 'definition'.
  4. [Figure 4(b)] The caption appears to be corrupted in the text, showing an uninterpreted sequence of t's after 'tY B'; the LaTeX source should be checked.
  5. [2.5] In the sentence 'This can be taken to be Fix(P_*P^*) the fixed point category', a comma or colon should precede 'the fixed point category', and the phrase 'suitably invertible' should be made precise or replaced by a reference to the author's paper [43].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three vignettes are expository applications of Lawvere's external framework and of the author's prior published proofs; no prediction reduces to a fit or to a self-referential definition.

full rationale

The paper is an expository survey. The metric-as-enriched-category framework is credited to Lawvere [18], not to the author. The tight-span vignette is explicitly 'taken from my paper [42]' (Sec. 3.2) and Theorem 2 is quoted from that published work; the Legendre-Fenchel vignette similarly says 'A general reference for the material here is my paper [43]' (Sec. 5.2). These are load-bearing for the two vignettes, but they are citations to prior peer-reviewed proofs, not to an unverified assumption, and the paper does not define the target constructions in terms of the conclusions. The magnitude vignette is grounded in Leinster's external work [20,22] and Solow-Polasky [36], with the size map a↦e^{-a} chosen openly to match the earlier effective-number-of-species definition. No fitted parameter is relabelled as a prediction, no uniqueness theorem is imported to forbid alternatives, and no known result is merely renamed: the identifications of the Isbell completion with the tight span and of the profunctor nucleus with closed convex functions are substantive theorems with external proofs. Note: Theorem 3 is stated without proof and, as the skeptic's counterexample shows, is false as stated for general ℝ+-categories; that is a correctness gap, not a circularity. The circularity score is therefore 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This is an expository paper. The central claim rests on prior literature: Lawvere's enriched-category view of metric spaces, the author's papers [42, 43] on tight spans and the Legendre-Fenchel transform, and Leinster's theory of magnitude. No free parameters or invented entities are introduced. The axioms listed are standard background results and cited theorems used without proof.

assumptions (5)
  • standard math Axioms of enriched category theory: definitions of 𝒱-category, 𝒱-functor, Yoneda embedding, and ends.
    Used throughout Section 2 without proof; the reader is expected to know or accept these background results.
  • standard math The monoidal category (ℝ+, +, 0) is closed, complete, and cocomplete, so that enriched presheaf categories and Isbell adjunctions exist.
    Invoked implicitly in Sections 2.3 and 2.5; a standard result in enriched category theory.
  • standard math For an ℝ+-category X, the Yoneda functor X -> J(X^op, ℝ+_f) is an isometry (generalized Kuratowski embedding).
    Stated in Section 2.4, item 2, and used as the basis for the tight span and Legendre-Fenchel constructions; proof not included.
  • domain assumption Leinster's theory of magnitude (definition via similarity matrix and size map, and properties of the magnitude function) is correct as cited.
    The second vignette relies on Leinster's papers [20, 22] without proving them here.
  • domain assumption The nucleus of the Isbell adjunction for the pairing profunctor on V and V^# is precisely the closed convex functions, giving Toland-Singer duality.
    The third vignette depends on this identification, taken from the author's paper [43] and related work; not proved in this paper.

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Cite this review

Pith. "Pith review of Metric-like spaces as enriched categories: three vignettes." pith.science (2026). https://pith.science/paper/CNX72KHL

@misc{pith2026250100416,
  author       = {Pith},
  title        = {Pith review of: Metric-like spaces as enriched categories: three vignettes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNX72KHL}},
  note         = {Machine review of arXiv:2501.00416}
}
read the original abstract

This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform.

Figures

Figures reproduced from arXiv: 2501.00416 by the authors.

Figure 3
Figure 3. A schematic of how Leinster generalized several notions of Euler [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 5
Figure 5. Some of the rich connections of magnitude to various areas of [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Plots of the function 𝑓 : 𝑥 ↦→ (𝑥 2 − 1) 2 , of 𝕃 ∗ (𝑓 ) and of 𝕃∗ ◦𝕃 ∗ (𝑓 ). A supporting hyperplane is pictured on the first plot, and a corresponding point is pictured on the second plot. A supporting hyperplane 𝐻 for 𝑓 is a hyperplane in 𝑉 × ℝ ⊂ 𝑉 × [−∞, +∞] such that 𝐻 touches and lies below the graph of 𝑓 . A supporting hyperplane is illustrated in the first graph of [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

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Reference graph

Works this paper leans on

46 extracted references · 35 canonical work pages

  1. [1]

    Borceux,Handbook of Categorical Algebra 2, Categories and Structures (Encyclopedia of mathematics and its applications 51)

    F. Borceux,Handbook of Categorical Algebra 2, Categories and Structures (Encyclopedia of mathematics and its applications 51). Cambridge UniversityPress,1994,443pp.doi: 10.1017/CBO9780511525865(cited on p. 5)

    work page doi:10.1017/cbo9780511525865(cited 1994

  2. [2]

    Chepoi, ‘A𝑇𝑋-approach to some results on cuts and metrics’, Advances in Applied Mathematics, vol

    V. Chepoi, ‘A𝑇𝑋-approach to some results on cuts and metrics’, Advances in Applied Mathematics, vol. 19, no. 4, pp. 453–470, 1997. doi:10.1006/aama.1997.0549(cited on p. 14)

    work page arXiv 1997

  3. [3]

    M.ChrobakandL.L.Larmore,‘Generosityhelpsoran 11-competitive algorithm for three servers’,Journal of Algorithms, vol. 16, pp. 234–263, 1994 (cited on pp. 13, 14)

    work page 1994

  4. [4]

    A. W. M. Dress, ‘Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces’,Advances in Mathematics, vol. 53, 1984 (cited on pp. 13, 14)

    work page 1984

  5. [5]

    Eilenberg and G

    S. Eilenberg and G. M. Kelly, ‘Closed categories’, inProceedings of the Conference on Categorical Algebra, S. Eilenberg, D. K. Harrison, S. MacLaneandH.Röhrl,Eds.,SpringerBerlinHeidelberg,1966,pp.421–

    work page 1966

  6. [6]

    Eilenberg and S

    S. Eilenberg and S. MacLane, ‘General theory of natural equivalences’, Transactions of the American Mathematical Society, vol. 58, no. 0, pp. 231– 294,1945,issn:1088-6850.doi: 10.1090/s0002-9947-1945-0013131-6 (cited on p. 3)

    work page doi:10.1090/s0002-9947-1945-0013131-6 1945

  7. [7]

    Fréchet, ‘Sur quelques points du calcul fonctionnel’,Rendiconti del Circolo Matematico di Palermo, vol

    M. Fréchet, ‘Sur quelques points du calcul fonctionnel’,Rendiconti del Circolo Matematico di Palermo, vol. 22, no. 1, pp. 1–72, 1906, PhD thesis. doi:10.1007/bf03018603(cited on p. 2). 26

    work page doi:10.1007/bf03018603(cited 1906

  8. [8]

    Gutiérrez García, I

    J. Gutiérrez García, I. Mardones-Pérez, M. A. de Prada Vicente and D. Zhang, ‘Fuzzy Galois connections categorically’,Mathematical Logic Quarterly, vol. 56, no. 2, pp. 131–147, Mar. 2010, issn: 1521-3870. doi:10. 1002/malq.200810044(cited on p. 12)

    work page 2010

Show all 46 references
  1. [9]

    Hanks, B

    T. Hanks, B. She, M. Hale, E. Patterson, M. Klawonn and J. Fairbanks, ‘Modeling model predictive control: A category theoretic framework for multistage control problems’, 2023. doi:10.48550/ARXIV.2305. 03820. arXiv:2305.03820(cited on p. 26)

    2023 doi arXiv

  2. [10]

    Hausdorff, Grundzüge der Mengenlehre

    F. Hausdorff, Grundzüge der Mengenlehre. Leipzig, 1914. http : / / catalogue.bnf.fr/ark:/12148/cb305813468(cited on p. 2)

    1914

  3. [11]

    Hepworth and S

    R. Hepworth and S. Willerton, ‘Categorifying the magnitude of a graph’,Homology, Homotopy and Applications, vol. 19, no. 2, pp. 31–60, 2017, issn: 1532-0081. doi:https://dx.doi.org/10.4310/HHA.2017. v19.n2.a3 (cited on p. 21)

    2017 doi

  4. [12]

    Hirai and S

    H. Hirai and S. Koichi, ‘On tight spans for directed distances’,Annals ofCombinatorics,vol.16,no.3,pp.543–569,2012.doi: 10.1007/s00026- 012-0146-5 (cited on p. 16)

    2012 doi

  5. [13]

    J. R. Isbell, ‘Six theorems about injective metric spaces’,Commentarii Mathematici Helvetici, vol. 39, pp. 65–76, 1964 (cited on p. 13)

    1964

  6. [14]

    Johnson and D

    N. Johnson and D. Yau,2-Dimensional Categories. Oxford University Press, 2021. doi:10.1093/oso/9780198871378.001.0001. arXiv:2002. 06055(cited on p. 5)

    2021

  7. [15]

    G. M. Kelly, ‘Basic concepts of enriched category theory’,Reprints in Theory and Applications of Categories, no. 10, pp. 1–136, 2005 (cited on p. 5)

    2005

  8. [16]

    Kemajou, H.-P

    E. Kemajou, H.-P. A. Künzi and O. Olela Otafudu, ‘The Isbell-hull of a di-space’,Topology and its Applications, vol. 159, no. 9, pp. 2463–2475,

  9. [17]

    Khovanov, ‘A categorification of the Jones polynomial’,Duke Mathematical Journal, vol

    M. Khovanov, ‘A categorification of the Jones polynomial’,Duke Mathematical Journal, vol. 101, pp. 359–426, 1999. doi:10.1215/S0012- 7094-00-10131-7 (cited on p. 21)

    1999 doi

  10. [18]

    F.W.Lawvere,‘Metricspaces,generalizedlogic,andclosedcategories’, Rendiconti del Seminario Matematico e Fisico di Milano, vol. 43, no. 1, pp. 135–166, 1973. doi:10.1007/BF02924844(cited on pp. 3, 5, 7)

    1973 doi

  11. [19]

    Leinster,Magnitude: A bibliography, website.https://www.maths

    T. Leinster,Magnitude: A bibliography, website.https://www.maths. ed.ac.uk/~tl/magbib/(cited on p. 20)

  12. [20]

    Leinster, ‘The Euler characteristic of a category’,Documenta Math- ematica, vol

    T. Leinster, ‘The Euler characteristic of a category’,Documenta Math- ematica, vol. 13, pp. 21–49, 2008, issn: 1431-0643. doi:10.4171/DM/240 (cited on p. 17). 27

    2008 doi

  13. [21]

    Leinster, ‘Codensity and the ultrafilter monad’,Theory and Ap- plications of Categories, vol

    T. Leinster, ‘Codensity and the ultrafilter monad’,Theory and Ap- plications of Categories, vol. 28, no. 13, pp. 332–370, 2013 (cited on p. 4)

    2013

  14. [22]

    Leinster, ‘The magnitude of metric spaces’,Documenta Mathematica, vol

    T. Leinster, ‘The magnitude of metric spaces’,Documenta Mathematica, vol. 18, pp. 857–905, 2013, issn: 1431-0643. doi:10.4171/DM/415 (cited on p. 18)

    2013 doi

  15. [23]

    Leinster, ‘The magnitude of a graph’,Mathematical Proceedings of the Cambridge Philosophical Society, vol

    T. Leinster, ‘The magnitude of a graph’,Mathematical Proceedings of the Cambridge Philosophical Society, vol. 166, no. 2, pp. 247–264, 2019. doi:10.1017/S0305004117000810(cited on p. 21)

    2019 doi

  16. [24]

    Leinster, Entropy and Diversity, The Axiomatic Approach

    T. Leinster, Entropy and Diversity, The Axiomatic Approach. Cambridge University Press, 2021, isbn: 9781108832700 (cited on p. 21)

    2021

  17. [25]

    Leinster and C

    T. Leinster and C. A. Cobbold, ‘Measuring diversity: The importance of species similarity’,Ecology, vol. 93, no. 3, pp. 477–489, Mar. 2012, issn: 1939-9170. doi:10.1890/10-2402.1 (cited on p. 22)

    2012 doi

  18. [26]

    Leinster and M

    T. Leinster and M. Meckes, ‘Maximizing diversity in biology and beyond’,Entropy, vol. 18, no. 3, p. 88, 2016. doi:10.3390/e18030088 (cited on p. 22)

    2016 doi

  19. [27]

    Leinster and M

    T. Leinster and M. Shulman, ‘Magnitude homology of enriched cat- egoriesandmetricspaces’, Algebraic&GeometricTopology ,vol.21,no.5, pp. 2175–2221, Oct. 2021, issn: 1472-2747. doi:10.2140/agt.2021.21. 2175(cited on p. 21)

    2021 doi

  20. [28]

    Mac Lane,Categories for the Working Mathematician(Graduate Texts in Mathematics), 2nd ed

    S. Mac Lane,Categories for the Working Mathematician(Graduate Texts in Mathematics), 2nd ed. Springer New York, 1998, vol. 5, isbn: 978-0- 387-98403-2. doi:10.1007/978-1-4757-4721-8 (cited on p. 6)

    1998 doi

  21. [29]

    A. E. Magurran,Measuring Biological Diversity. Blackwell Publishing Limited, 2003, p. 256, isbn: 0632056339 (cited on p. 16)

    2003

  22. [30]

    733–757, 2012

    M.W.Meckes,‘Positivedefinitemetricspaces’, Positivity,vol.17,no.3, pp. 733–757, 2012. doi:10.1007/s11117-012-0202-8 (cited on p. 20)

    2012 doi

  23. [31]

    2024 (cited on p

    nLab authors,Enriched category, https://ncatlab.org/nlab/show/ enriched+category, Revision 128, Mar. 2024 (cited on p. 5)

    2024

  24. [32]

    Pavlovic, ‘Quantitative concept analysis’, inFormal Concept Analysis, Springer Berlin Heidelberg, 2012, pp

    D. Pavlovic, ‘Quantitative concept analysis’, inFormal Concept Analysis, Springer Berlin Heidelberg, 2012, pp. 260–277. doi:10.1007/978-3- 642-29892-9_24 (cited on p. 12)

    2012 doi

  25. [33]

    Riehl,Categorical homotopy theory(New Mathematical Monographs 24).NewYork:CambridgeUniversityPress,2014,isbn:9781107048454 (cited on p

    E. Riehl,Categorical homotopy theory(New Mathematical Monographs 24).NewYork:CambridgeUniversityPress,2014,isbn:9781107048454 (cited on p. 5)

    2014

  26. [34]

    R. T. Rockafellar,Convex Analysis. Princeton University Press, 1970 (cited on pp. 22, 26). 28

    1970

  27. [35]

    Rota, ‘On the foundations of combinatorial theory I

    G.-C. Rota, ‘On the foundations of combinatorial theory I. Theory of Möbius functions’,Zeitschrift für Wahrscheinlichkeitstheorie und Ver- wandte Gebiete, vol. 2, no. 4, pp. 340–368, 1964, issn: 1432-2064. doi:10. 1007/bf00531932(cited on p. 17)

    1964

  28. [36]

    A. R. Solow and S. Polasky, ‘Measuring biological diversity’,Environ- mental and Ecological Statistics, vol. 1, no. 2, pp. 95–103, Jun. 1994, issn: 1573-3009. doi:10.1007/BF02426650(cited on p. 16)

    1994 doi

  29. [37]

    Stein and R

    D. Stein and R. Samuelson, ‘Towards a compositional framework for convex analysis (with applications to probability theory)’, 2023. doi:10.48550/ARXIV.2312.02291. arXiv:2312.02291(cited on p. 26)

    2023 doi arXiv

  30. [38]

    Street, ‘An Australian conspectus of higher categories’, inTowards HigherCategories,J.C.BaezandJ.P.May,Eds.NewYork,NY:Springer New York, 2010, pp

    R. Street, ‘An Australian conspectus of higher categories’, inTowards HigherCategories,J.C.BaezandJ.P.May,Eds.NewYork,NY:Springer New York, 2010, pp. 237–264. doi:10.1007/978-1-4419-1524-5_6 (cited on p. 3)

    2010 doi

  31. [39]

    A. E. Taylor, ‘A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals’,Archive for History of Exact Sciences, vol. 27, no. 3, pp. 233–295, 1982. doi:10.1007/BF00327860 (cited on p. 2)

    1982 doi

  32. [40]

    Touchette, ‘Legendre-Fenchel transforms in a nutshell’, Queen Mary, University of London, 2005.https://appliedmaths.sun.ac

    H. Touchette, ‘Legendre-Fenchel transforms in a nutshell’, Queen Mary, University of London, 2005.https://appliedmaths.sun.ac. za/~htouchette/articles.html(cited on p. 22)

    2005

  33. [41]

    C.T.C.Wall,‘RationalEulercharacteristics’, MathematicalProceedingsof the Cambridge Philosophical Society, vol. 57, no. 1, pp. 182–184, Jan. 1961, issn: 1469-8064. doi:10.1017/s030500410003499x(cited on p. 17)

    1961 doi

  34. [42]

    Willerton, ‘Tight spans, Isbell completions and semi-tropical mod- ules’,Theory and Applications of Categories, vol

    S. Willerton, ‘Tight spans, Isbell completions and semi-tropical mod- ules’,Theory and Applications of Categories, vol. 28, no. 22, pp. 696–732,

  35. [43]

    Willerton, ‘The Legendre-Fenchel transform from a category the- oretic perspective’, Jan

    S. Willerton, ‘The Legendre-Fenchel transform from a category the- oretic perspective’, Jan. 2015. doi:10 . 48550 / ARXIV . 1501 . 03791. arXiv:arXiv:1501.03791 [math.CT] (cited on pp. 12, 24). 29

    2015 doi arXiv

  36. [562]

    doi:10.1007/978-3-642-99902-4_22 (cited on p. 3)

    doi

  37. [2012]

    doi:10.1016/j.topol.2011.02.016(cited on p. 16)

    2011 doi

  38. [2013]

    http://www.tac.mta.ca/tac/volumes/28/22/28- 22.pdf (cited on p. 14)

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.