(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A double cover is equivalently
a $\mathbb{Z}_2$-principal bundle;
an etale space with local sections the 2-element set.
For $X$ a manifold, not necessarily oriented or even orientable, write
for any choice of orthogonal structure. The orientation double cover or orientation bundle of $X$ is the $\mathbb{Z}_2$-principal bundle classified by the first Stiefel-Whitney class (of the tangent bundle) of $X$
One may identify this with the bundle that over each neighbourhood $x \in U \subset X$ of a point $x$ has as fibers the two different choices of volume forms up to positive rescaling (the two different choices of orientation).
More generally, for $E \to X$ any orthogonal group-principal bundle classified by a morphism $E : X \to \mathbf{B} O$, the corresponding orientation double cover is the $\mathbb{Z}_2$-bundle classified by
An exposition in a broader context is in the section higher spin structures at