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I have given the notion of canonical transformation as used in Hamiltonian mechanics its own brief page.
So in particular I removed the redirect of that term to canonical morphism and instead added disambiguation lines on the top of both entries. I think this is justified: the term “canonical transformation” has been standard since ancient times in Hamiltonian mechanics and is in each and every textbook on the matter. On the other hand the same term as referring to canonical morphisms was mainly the proposal of one single person in category theory, and never caught up much, I think. (Also I find the term ill-motivated in category theory in the first place).
Therefore, while the disambiguation redirects ensure that both notions still can be found, I think it is clear that the default meaning must be that in Hamiltonian mechanics.
Although I find the term well-motivated in category theory, I agree with your default.
I don't see any need for the note at the top of canonical morphism, since nobody will end up there by mistake when studying Hamiltonian mechanics. (It would have been useful before, when there was a redirect, but not now. For that matter, the note at the top of canonical transformation is not likely to get much use either, but at least it might get some.)
If you are counting usage, I would add that the whole page “canonical morphism” is not of much use and in fact a bit misguided.
Maybe to come back to the topic of how one might formalize the notion of “canonical”: I would tend to think that a formalization involves some notion of constructiveness. What is canonical is that which we can actually construct, with given data (given terms).
For instance for $X$ just any set without further information, the reason why $id \colon X \to X$ is the canonical map from $X$ to itself is because this is the only one we can actually name, whose term we can actually construct. There are all these other maps, but we can’t actually name them with the given information.
Or to come closer to the topic of this thread, the reason why $(q,p) \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ are the canonical coordinates on the plane is because they are the two which one can actually construct given the data by which the plane $\mathbb{R}^2$ was constructed, namely the two projection maps out of the project. The reason why all the other coordinates that we might put on $\mathbb{R}^2$ are “not canonical” is that while they “exist” in the sense of existence of mathematics, we cannot actually construct them with the given data.
I would be inclined to erase what is currently at canonical morphism and add a discussion along the above lines. As far as I am aware what is currently discussed at “canonical morphism” has never been used by anyone, not even by Jim Dolan. And the example of QFT transformations mentioned at the end is far from being related to anything “canonical” and in fact has a better axiomatization not as explained there, but by adding boundary singularities to cobordisms.
only now noticed that there is an entry generating function in classical mechanics. This needs to be merged with canonical transformation.
(I have added cross-links, but right now I have no time to do more.)
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