group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a space with a notion of dimension $dim X \in \mathbb{N}$ and a notion of (Kähler) differential forms on it, the canonical bundle or canonical sheaf over $X$ is the line bundle (or its sheaf of sections) of $n$-forms on $X$, the $dim(X)$-fold exterior product
of the bundle $\Omega^1_X$ of 1-forms.
The first Chern class of this bundle is also called the canonical characteristic class or just the canonical class of $X$.
Often this bundle is regarded via its sheaf of sections.
A square root of the canonical class, hence another characteristic class $\Theta$ such that the cup product $2 \Theta = \Theta \cup \Theta$ equals the canonical class is called a Theta characteristic (see also metalinear structure).
Notice that if $X$ is for instance a complex manifold regarded over the complex numbers, then Kähler differential forms are holomorphic forms.
The following table lists classes of examples of square roots of line bundles
In the context of algebraic geometry:
See also