group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In a cochain complex $(V^\bullet,d)$ a coboundary is an element in the image of the differential.
More generally, in the context of the intrinsic cohomology of an (∞,1)-topos $\mathbf{H}$, for $X$ and $A$ two objects, a cocycle on $X$ with coefficients in $A$ is an object in $\mathbf{H}(X,A)$ and a coboundary between cocycles is a morphism in there.
$H_n = Z_n/B_n$ | (chain-)homology | (cochain-)cohomology | $H^n = Z^n/B^n$ |
---|---|---|---|
$C_n$ | chain | cochain | $C^n$ |
$Z_n \subset C_n$ | cycle | cocycle | $Z^n \subset C^n$ |
$B_n \subset C_n$ | boundary | coboundary | $B^n \subset C^n$ |