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In the Author Commentary to the reprint of the paper paper Diagonal Arguments and Cartesian Closed Categories in Theory and Applications of Categories Bill Lawvere wrote:

Although the cartesian-closed view of function spaces and functionals was intuitively obvious in all but name to Volterra and Hurewicz (and implicitly to Bernoulli), it has counterexamples within the rigid framework advocated by Dieudonné and others. According to that framework the only acceptable fundamental structure for expressing the cohesiveness of space is a contravariant algebra of open sets or possibly of functions. Even though such algebras are of course extremely important invariants, their nature is better seen as a consequence of the covariant geometry of figures. Specific cases of this determining role of figures were obvious in the work of Kan and in the popularizations of Hurewicz’s k-spaces by Kelley, Brown, Spanier, and Steenrod, but in the present paper I made this role a matter of principle: the Yoneda embedding was shown to preserve cartesian closure, and naturality of functionals was shown to be equivalent to Bernoulli’s principle.

First of all, as if I understand correctly the mention of Bernoulli refers to Johann Bernoulli (1667 – 1748) which is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth. Am I correct thinking so?

Secondly, which principle exactly does the author mention in the passage? The only Bernulli principle which comes on my mind is Daniel Bernoulli's principle in fluid dynamics.

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2 Answers 2

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Since the topic of Lawvere paper is differential geometry, the Bernoulli is likely Jacob Bernoulli, or his brother Johann, and refers to their calculus of variations and the principle of virtual work. Further evidence is this paper by Lawvere on category theory, where he explicitly refers to Bernoulli in the context of the calculus of variations.

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  • $\begingroup$ Interesting. But which principle exactly does Lawvere mention? $\endgroup$
    – Evgeny Kuznetsov
    May 13 at 21:11
  • $\begingroup$ the only principle I can imagine in this context is Bernoulli's principle of virtual work. $\endgroup$
    – Carlo Beenakker
    May 13 at 21:20
  • $\begingroup$ well, but as I read it here en.wikipedia.org/wiki/Virtual_work this principle have been credited to Johann Bernoulli and Daniel Bernoulli not Jacob :( $\endgroup$
    – Evgeny Kuznetsov
    May 13 at 21:25
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    $\begingroup$ though, reading section 4 of "Commentaires sur le développement de la théorie des topos" you have linked in your answer I am convinced that the name of Bernoulli is mention in in the context of calculus of variations. $\endgroup$
    – Evgeny Kuznetsov
    May 13 at 21:29
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Extended remarks in support of Carlos' answer:

From "The Legendre-Fenchel transform from a category theoretic perspective" by Simon Willerton (pg. 2),

... Lawvere [10] showed using enriched category theory that category theory was entwined with metric space theory.

Following Lawvere [11] and taking enriched categories seriously, we will see here how, from the pairing between a vector space and its dual, the nucleus of a profunctor construction [17] gives rise to a large amount of the theory around the Legendre-Fenchel transform and convex functions.

The Legendre-Fenchel transform, calculus of variations, geodesics, principles of virtual work attributed to d'Alembert and Johann Bernoulli, Lagranges's principle of least action, and the dual formulations of Lagrange's and Newton's equations of motion and the associated Hamiltonian total energy (H=T+V) and Lagrangian action (L=T-V) related via the Legendre-Fenchel transform are all intimately intertwined.

Does anyone have additional refs on these connections with category theory?

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    $\begingroup$ In thermodynamics, there is a similar connection between the Legendre-Fenchel transform and the minimization of energy and the maximization of entropy, which I believe is currently a popular topic among categorists. $\endgroup$
    – Tom Copeland
    May 14 at 0:10
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    $\begingroup$ Lawvere mentions the Bellman-Fenchel convolution in "Metric spaces, generalized logic, and closed categories", not directly the Legendre-Fenchel transform, but Bellman and Karush in "The maximum transform" reference Fenchel's "Convex cones, sets, and functions", so I assume they are basically talking about the same constructions, as Willerton indicates. $\endgroup$
    – Tom Copeland
    May 14 at 2:59
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    $\begingroup$ I really should mention Fermat's, Maupertuis', and Hamilton and Jacobi's principles as well to indicate the traditional use of the term 'principle' in geometric optics and particle propagation. . $\endgroup$
    – Tom Copeland
    May 14 at 4:03
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    $\begingroup$ For more on Willerton;s approach, see golem.ph.utexas.edu/category/2023/05/…. $\endgroup$
    – Tom Copeland
    May 14 at 21:00

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