(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
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $\mathcal{G}$ a groupoid object a $\mathcal{G}$-principal bundle is a morphism $P \to X$ with an principal action of $\mathcal{G}$ on $P$.
For $G$ a group object in some (∞,1)-topos $\mathbf{H}$ (for instance $\mathbf{H} =$ ?LieGrpd for smooth Lie groupoid-bundles), and $\mathbf{B}G$ the corresponding delooping object, $G$-principal bundles are the (∞,1)-pullbacks of the form
One equivalently (with non-negligible but conventional chance of confusion of terminology) calls such $P \to X$ a $\mathbf{B}G$-groupoid principal bundle.
So more generally, for $\mathcal{G}$ any groupoid object with collection $\mathcal{G}_0$ of objects, the $(\infty,1)$-pullbacks
are groupoid principal bundles .
For $\mathbf{H}$ = $\infty LieGrpd$ and $G:=G_1 \Rightarrow G_0$ a Lie groupoid, a $G$-principal bundle is locally of the form
for $\mathcal{G}_{x_i}$ the target fiber over an object $x_i$, or equivalently of the form $U_i\times_{G_0} G_1$.
More precisely, we have the following sub-definition for Lie groupoid principal bundles: Given a Lie groupoid $G=G_1 \Rightarrow G_0$, a $G$-principal bundle over a manifold $M$ (with right $G$ action) is a surjective submersion $P\xrightarrow{\pi} M$ together with a moment map $P\xrightarrow{\rho} G_0$ (sometimes also called an anchor), such that there is a (right) $G$-action on $P$, that is, there is map $\Phi: P\times_{G_0} G_1 \to P$, such that
$\pi(\Phi(p, g))=\pi(p)$,
$\rho(\Phi(p,g))=s(g)$,
$\Phi(\Phi(p, g_1), g_2)=\Phi(p, g_1 g_2)$.
Moreover, we require the $G$-action to be free and proper, that is,
[free] $P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is injective,
[proper]$P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is a proper map , i.e. the preimage of a compact is compact.
These two conditions are equivalent to the fact that
is an isomorphism.
principal bundle / torsor / groupoid principal bundle
Last revised on January 5, 2018 at 04:02:45. See the history of this page for a list of all contributions to it.