(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
direct sum of vector bundles, tensor product, external tensor product,
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Let $\mathbf{H}$ be an ambient (∞,1)-topos. Let $V, X$ be two objects of $\mathbf{H}$. Then a $V$-fiber bundle over $X$ in $\mathbf{H}$ is a morphism $E \to X$ such that there is an effective epimorphism $U \to X$ and an (∞,1)-pullback square of the form
Externally this is a $V$-fiber $\infty$-bundle.
See at associated ∞-bundle for more.
A fiber $\infty$-bundle whose typical fiber $V$ is a pointed connected object, hence a delooping $\mathbf{B}G$ of an ∞-group $G$
is a $G$-∞-gerbe.
Every $V$-fiber $\infty$-bundle is the associated ∞-bundle to an automorphism ∞-group-principal ∞-bundle.
For let $Type$ be the object classifier. Then any bundle $E \to X$ is classified by a morphism
On the other hand, since the pullback to the bundle on some $U$ is trivializable, that bundle over $U$ is classsified by a map that factors through the point which is the name of the fiber $V$
The 1-image-of this point inclusion is the delooping of the automorphism ∞-group of $V$ :
Therefore the fact that $E$ is trivialized over $U$ means that there the classifying maps fit into a commuting diagram of the form
By assumption the left morphism is a 1-epimorphism and by the above construction the right morphism is a 1-monomorphism. Therefore by the (n-connected, n-truncated) factorization system this diagram has an essentially unique lift
This diagonal lift classifies an $\mathbf{Aut}(V)$-principal ∞-bundle and the commutativity of the bottom right triangle exhibits the original bundle $E \to X$ as the associated ∞-bundle to that.
See the references at associated ∞-bundle.
The explicit general definition appears as def. 4.1 in part I of
Last revised on November 25, 2014 at 19:39:29. See the history of this page for a list of all contributions to it.