(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
In the context of twisted cohomology a cocycle on a space $X$ with coefficients in a coefficient object $V$ is not quite a direct morphism $X \to V$ as in ordinary $G$-cohomology, but is instead a section of a $V$-fiber ∞-bundle $E \to X$ over $X$. This is called the local coefficient bundle for the given twisted cohomology. Its class $[E] \in H^1(X, \mathbf{Aut}(V))$ is the twist.