David Spivak on The Category of Polynomial Functors in One Variable
Category theorists, it seems, can't help but go up, making more concrete stuff that they have to find names for. They should learn about abstraction. See Why is Physics So Difficult? and David Jaz Maiers - Compositionality via 2-algebra.
See also Richard Southwell and Norman Wildberger on the Future of Mathematics. Norman's box numbers are a multisubset of the set 1={∅} which, according to nlab is an isomorphism class in the over category Set/1, I think. See the three examples in the comma category on nlab and fundamental theorem of topos theory.
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See The EPR paradox and the uncertainty principle by Michael E. McCulloch and Jaume Giné.
The first thing I read about Category theory was Jacobs and Rutten's Introduction to (co)algebra and (co)induction and I was struck by the "failure of duality" in the way (co)algebra was described. I don't remember what it was, but there was some notion I had that a recursive datatype was a polynomial in the arity of the constructors and that from this one ought to be able to unfold something that maintains the duality. See Emily Riehl and Terrence Tau on The Future of Mathematics on spontaneity in physics and biology. 
David Spivak's talk "Categories = polynomial comonads", a simple demonstration. This is great. It should give Norman Wildberger a whole lot of new ideas for stuff he can do with polynumbers! It was given in September 2020.
Tom Leinster on a general notion of magnitude:
I learned something interesting about the Legendre transformation from this. See Simon Willerton's paper The Legendre-Fenchel transform from a category theoretic perspective.
58:28 The maximum diversity always exists. ... The fact that probabilities have to be non-negative is what makes maximum diversity different from magnitude. There is a sense in which, if you allowed negative probabilities then they would be equal.
Negative probabilities turn up in Wigner functions. See William Case's Wigner functions and Weyl transforms for pedestrians American Journal of Physics 76(10) (2008) and Optimal Information Transfer and Real-Vector-Space Quantum Theory by William K. Wootters (2013) and Locality Implies Complex Numbers in Quantum Mechanics by Tianfeng Feng, Changliang Ren and Vlatko Vedral. Also Real-Vector-Space Quantum Theory with a Universal Quantum Bit by Antoniya Aleksandrova, Victoria Borish and William K. Wootters.
See Maximizing diversity in biology and beyond by Tom Leinster and Mark W. Meckes.
Yesterday I watched Jorge Diaz's video on the Principle of Least Action and I had some vague thoughts about this:
I looked for a video on the Legendre transformation in your channel but didn't find it! 😅 It would be interesting as background to the Lagrange formulation because Legendre was considering a conservative stationarity problem too and surely Lagrange knew this. That might also give an insight into the intuition behind it.
My physical intuition for the principle of stationary action is that one has to remember it is a functional, in the sense that its argument is a function, from time to some mysterious quantity called the action, which one can think of as the "non-spontaneous deviation" of the trajectory from that conservative path determined by the boundary. conditions. In other words, the action is the "wilful" straying of the system from its true path which should be the one where the loss[/gain] of kinetic energy at any moment in time is balanced by a gain[/loss] in potential energy at that same moment. This wilfulness of the system must be zero over the whole path, so the least action gives the "stationary" path. But there are two ways the conservative system could spontaneously evolve: one is by losing potential energy and gaining the same amount of kinetic energy, or by gaining potential energy and losing the same amount of kinetic energy. So the action must contain terms of both of these, and it must apply equally in the time-reversal of the path, so the kinetic and potential terms have to have opposite signs, but I don't think it matters which is which: you could write L = U - K and it would work just as well.
Thinking about it now, I wonder whether maybe Lagrange had an idea of using the same formalism to to model non-conservative systems by accumulating something like "action potential" during the course of their evolution. It's significant because the mathematician Routh developed a formalism called the Routhian which was a combination of those of Hamilton and Lagrange and I think it includes a third term through which energy could be exchanged between them or absorbed. [See Routhian mechanics,]
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See The EPR paradox and the uncertainty principle by Michael E. McCulloch and Jaume Giné.
My comment:
It's certainly the case when you think about how we measure time: we need networks of synchronised clocks, so we need to be able to communicate between locations where we make measurements. This idea has a mathematical expression in topos theory. There was a very interesting Topos Institute talk by Tom Leinster a few days ago which is on YouTube, entitled "The many faces of magnitude". It's very abstract to start with but ends up talking about entropy and temperature in a very direct and applicable way. There is a kind of duality that shows up in the Legendre-Fenchel transform which is captured in the abstract idea of a topos. That duality shows up in physics as an inevitable invariant of descriptions in terms of conjugate variables like position/momentum and temperature/entropy. Time itself may be a magnitude in this general sense. That's what it was for Aristotle, I think.
[Actually, Leinster doesn't mention temperature but McCulloch doesn't mention entropy. I mentioned that they're conjugate quantities under the Legendre-Fenchel transform.]
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David Spivak did a talk in 2022 about Polynomial Functors and Shannon entropy:
The first thing I read about Category theory was Jacobs and Rutten's Introduction to (co)algebra and (co)induction and I was struck by the "failure of duality" in the way (co)algebra was described. I don't remember what it was, but there was some notion I had that a recursive datatype was a polynomial in the arity of the constructors and that from this one ought to be able to unfold something that maintains the duality. See Emily Riehl and Terrence Tau on The Future of Mathematics on spontaneity in physics and biology.
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Urs Schreiber on topos theory in physics. You don't have to use Category theory to study topos theory: see the lectures of Frederic Schuller for an example of how the results can be expressed in ordinary language of manifolds, groups and maps.
See the article at https://ncatlab.org/schreiber/show/Higher+Topos+Theory+in+Physics#ZCTS2024. On smooth sets as a convenient category for variational calculus in Lagrangian classical field theory see Field Theory via Higher Geometry I: Smooth Sets of Fields by Grigorios Giotopoulos and Hisham Sati.
28:10 On conditions for a mapping space to be a smooth manifold: the topos of smooth sets is Cartesian-closed, so it is guaranteed:
43:11 On constructivism and the Axiom of Choice.
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