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Wrote some minimum at natural bundle.
I slightly object to calling it a “type of bundle”, as it is not a single bundle but a functorial assignement of a bundle to a manifold, for suitable category of manifolds. It is like saying that a characteristic function is a number. No, it is an assignement which assigns a number to an element which indicates if the element is in a given subset. (Of course, some bundles can be said not to be in image of any natural functor, as their structure group is such, but still…)
But a characteristic function is a type of function. Here the tangent bundle is a type of bunde (or kind of bundle or the like).
Remark 3 makes no sense to me. I agree that natural bundle is a type of functor to bundles.
Parallelly char fcn is a fcs, but it is not a number.
Its value on an element is a number. But you do not say that characteristic function s a type of a number,
It is a function from a set to numberS. So it is function. Not number.
Natural bundle s a functoR from manifolds to bundleS. So it s a functor, Not bundle.
Otherwise it makes no sense to apply the definition.
“The tangent bundle” ? Which the ? For which concrete manifold ?
It's a manifold-dependent bundle, a bundle in the context of a free variable for a manifold.
Yes, Toby, but qualification that it is a type of a bundle would mean that the type/kind/subsort is specified in the domain. But, no – there are no restriction on kind of a bundle in domain, the restriction is on a kind of depending. It is not about classification of bundles but classification of free variable behaviour…
Added:
Originally introduced in
Albert Nijenhuis, Geometric aspects of formal differential operations on tensor fields, Proceedings of the International Congress of Mathematicians 1958, 463–469.
Albert Nijenhuis, Natural bundles and their general properties, in: Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334.
A comprehensive reference is available in
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