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added to connection on a bundle
a Definition (nPOV-flavor, of course)
a Properties-section with statement and proof of the fact that every bundle does admit a connection.
Good. There should be some remark stating that the existence of connection is in a smooth category but not in analytic category nor in algebraic category. In this vain, I have checked a bit about Atiyah Lie algebroid and corrected few things around. For example Lie algebroid had an incomplete definition in explicit terms: the anchor is not only a vector bundle map, but its differential respects the brackets (please correct my notation if you have better). I added few references including the Pradines 1967 with explicit definition of Lie algebroid. Of course, Atiyah’s work is 1967, and the first references on Lie pseudoalgebras aka Lie-Rinehart algebras/pairs are 1953 and 1955 and the Rinehart’s paper itself is 1963 and many more coming in those years. In any case the Courant late 1980s paper is far not among the first references on Lie algebroids, but popularity and width of research sparked more after it, especially in non-French literature. I hope you agree with the corrections and literature enhancements.
but not in analytic category
Yes, I was thinking about the need to re-write the entry with more general assumptions on everything. We can eventually do that.
As the existence proof shows, it relies crucially on the existence of a partition of unity subordinate to a good cover, and works precisely if such exists.
Generally speaking, I can give a definition of connection so general that it will work in cases even undreamed of, but for the moment I wanted to record some standard facts, in order to be able to point to them in my examples-section, for comparison.
Wow, this is a good insight, that the proof still goes when one can do the splitting on a partition. This local splitting is always about the Galois type-theory with connection or without it. I wish the current effort of John about separable algebras and Igor’s about bicategorical Glaois theory ends up with some nlab use :) I mean somebody competent should write some content into my stub categorical Galois theory which has so far only references.
Generally speaking, I can give a definition of connection so general that it will work in cases even undreamed of
You are giving us a dangerous appetizer!
Igor’s about bicategorical Galois theory
is this secret, or contained in some public work of his?
Igor was just writing these things in last 3-4 weeks, email him if he has some draft version or will have soon. He is hoping of having a final version by the end of the summer, but he is doing 3-4 projects in parallel.
He is hoping of having a final version by the end of the summer
@Zoran - I did email Igor about this, but got no response. I don’t suppose you know if this was finished or not?
I changed the “principal connection on $X$” to “affine connection on $X$” in Cartan connection for formal reason. Principal connection on $X$ does not make sense to me as $X$ is not a bundle. Principal connection may be on the frame bundle of $X$, and this is called or equivalent to what is usually called an affine connection.
Jim: did I send you the book ? If not let me know, I will send it.
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