Sorry, for some reason I didn’t get a notification of your comment. The Hamiltonian isomorphism sends covectors on (elements of to vectors on (now in ).

I thought about doing the proof of Darboux’s theorem, and you present a valid point regarding justifying its existence at all. I suppose I’ve just been rushing to cover (and learn) these symplectic basics, so I can get a little more intuition for the involutivity of the microsupport (Lagrangian in the constructible case), and start to unravel Kashiwara and Schapira’s chapter on characteristic (and the sheaf of Lagrangian) cycles in .

]]>I’m sorry, I misread it. Your function $f$ is a function on the cotangent bundle, not on the manifold. So $df$ is in the domain of $H$.

]]>Ok, now the codomain of your Hamiltonian isomorphism is tangent vectors on the cotangent bundle. Can you also confirm the domain? You list it as cotangent vectors on the cotangent bundle. But then you write $H(df)$. $df$ is an element of the cotangent bundle, right? Not the cotangent of the cotangent?

]]>Explicitly, no, I didn’t (other than avoiding char =2 nonsense). You CAN do this stuff for complex vector spaces, grabbing material from Hermitian forms and almost complex structures. I didn’t read too much into those situations, so I didn’t want to talk about them here without knowing all the details!

]]>Yeah, I ended up proving it using just the hom-set bijections on the target categories. You can just basically build up adjunction at the level of fuctor categories into them by constructing the natural transformations “element-wise”

]]>I just checked it, and it’s also quite easy to show the result using the hom-set definition of adjoint functors, which might be more comfortable. I was going to type that into a comment too, but I guess if you’ve already shown it yourself in the case that the source category is Open(X) for the adjoint pair of free/forgetful functors, well probably your proof did not use those assumptions, and is therefore the same as the general proof.

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