Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 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 tqft 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
    • CommentTimeDec 19th 2009
    • (edited Dec 19th 2009)

    I intend to considerbly expand the story at Atiyah Lie groupoid. But this afternoon I didn't get as far as I intended to, and now I have to quit and visit my parents. So this is to be continued. But so far I did this:

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2009
    • (edited Dec 21st 2009)

    in the section Relation to Differential Nonabelian Cohomology at Atiyah Lie groupoid details of the claim and the proof of how the Atyiah Lie groupoid At(P) \to \Pi(X) of a G-principal bundle P \to X is a quotient of the homotopy fiber of the morphism  \Pi(X) \to \mathbf{B}AUT(G) induced from a choice of connection on  P.

    I tried to present the argument in a supposedly nicely geometric fashion, with the homotopy pullback computed in terms of 2-groupoid incarnations of universal 2-bundles, but due to the restrictions of MathML diagrams it may look now a bit more awkward than it should, unfortunately.

    I'll try to polish this further, eventually.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 16th 2021
    • (edited Feb 16th 2021)

    Does anybody knows (it should be a long exercise, but maybe somebody sees and easy shortcut) how to express/formulate the Atiyah class in reasonably direct way in dual terms of algebra of smooth functions on Atiyah Lie groupoid/Ehresmann gauge groupoid or some holomorphic or algebraic geometric version instead of working from the start at tangent level of Lie algebroid ? I am interested because of noncommutative generalizations of Atiyah groupoid like Schauenburg bialgebroid, to see possible applications for nc connections on noncommutative principal bundles (of which several formalisms exist).