On Wednesday, December 17, 2014 at 10:58:10 AM UTC-6, Alexey Popov

wrote:

However the main question remains:

Why an external differential exists in multiple different situations?

The key underlying structures and concepts are: Natural Bundles,

Natural Operations and Natural Objects. I've not seen a good or simple definition of any of these things until just now ...

http://ncatlab.org/nlab/show/natural+bundle
The "definition" I've been going on up to now is that a natural object

is whatever you can do a Lie Derivative on (so technically this

statement is a conjecture of mine though probably a well-known result).

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