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. 

Richard Garner, Polynomial comonads and comodules

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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. 

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 of kinetic energy at any moment in time is balanced by a gain 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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