Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoid monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 29th 2010

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010
    • (edited Jul 29th 2010)

    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.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJul 29th 2010
    • (edited Jul 29th 2010)

    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!

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 31st 2010

    Igor’s about bicategorical Galois theory

    is this secret, or contained in some public work of his?

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 2nd 2010

    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.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2011
    • (edited Mar 9th 2011)
    S. Morita (Geometry of characteristic classes, page 78) says that not every connection in the sense of Ehresmann i.e. in the sense of a distribution of horizontal subspaces gives a parallel transport if the base manifold is not compact; i.e. a parallel transport is defined along some but not all smooth curves, unlike in the compact case. If the parallel translation is defined for all curves, he says that the Ehresmann connection is strict.
    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2011

    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?

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2011
    I do not think he was recently working quite on that subject. I hope he will let us know soon.
    • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeMar 9th 2011
    S. Morita (Geometry of characteristic classes, page 78) - I can't access now
    Could you give us a counterexample
    Apparently the smooth case is even more different from the topologicial
    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeOct 18th 2011
    • (edited Oct 18th 2011)

    I changed the “principal connection on XX” to “affine connection on XX” in Cartan connection for formal reason. Principal connection on XX does not make sense to me as XX is not a bundle. Principal connection may be on the frame bundle of XX, 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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)