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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 12th 2014

Wrote some minimum at natural bundle.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 15th 2014
• (edited Sep 15th 2014)

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…)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 15th 2014

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).

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeSep 15th 2014

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 ?

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeSep 16th 2014

It's a manifold-dependent bundle, a bundle in the context of a free variable for a manifold.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeSep 16th 2014

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…