(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Given a 2-ring $R$, a line 2-bundle or 2-line bundle is a 2-module bundle whose typical fiber is a 2-line over $R$.
Let $R$ be a commutative ring, or more generally an E-∞ ring. By the discussion at 2-vector space consider the 2-category
equivalent to that whose objects are associative algebras (or generally algebras) $A$ over $R$, (being placeholders for the 2-vector space $A Mod$ which is the category of modules over $A$) whose 1-morphisms are bimodules between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose 2-morphisms are homomorphisms between those.
Under Isbell duality and by the discussion at Modules – as generalized vector bundles we may think of this 2-category as being that of (generalized) 2-vector bundles over a space called $Spec R$.
The 2-category $2 Vect_R \simeq Alg_R$ is canonically a monoidal 2-category. An object in $2 Vect_R$ is a line if it is an invertible object with respect to this tensor product, hence if it is an Azumaya algebra. In terms of the above this means that it represents a 2-vector bundle over $Spec R$ which is a line 2-bundle.
The full inclusion
of the maximal 2-groupoid on the line 2-bundles over $Spec R$ is a braided 3-group, the Picard 3-group of $Spec R$. See Relation to Brauer group below for more.
The braided 3-group $\mathbf{Br}(R)$ of line 2-bundles over $Spec R$ has as homotopy groups
$\pi_0(\mathbf{Br}(R))$ the Brauer group of $R$;
$\pi_1(\mathbf{Br}(R))$ the Picard group of $R$, hence the group of ordinary line bundles over $R$;
$\pi_2(\mathbf{Br}(R))$ the group of units of $R$.
See at super line 2-bundle.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
Line 2-bundles in supergeometry as a model for the B-field and orientifolds are discussed (even if not quite explicitly in the language of higher bundles) in
based on
and
and developing constructions in
More on super line 2-bundles is secretly in
The above higher supergeometric story is made explicit in