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For a CW complex $X$ and coefficient abelian group $A$, the degree-$q$ cellular cochain group isChoosing orientations of the $q$-cells identifies it with an arbitrary assignment of one element of $A$ to each cell. Since the cellular chain group is a direct sum over cells, the cochain group is a product ${\prod }_{e^q}A$, which matters when there are infinitely many cells.

The cochain differential is precomposition with the cellular boundary: $(d\varphi )(c)=\varphi (\partial c)$. The identity ${\partial }^2=0$ gives $d^2=0$. In finite cellular bases, its matrix is the transpose of the boundary matrix, interpreted in the coefficient group.

For the circle with one vertex and one edge attached at both ends, $C^0=C^1=A$ and the differential is zero because the two endpoint contributions cancel. This gives $H^0(S^1;A)=H^1(S^1;A)=A$.

A group coboundary is a group cochain in the image of the preceding cochain differential. Because the differential squares to zero, every group coboundary is a group cocycle. The quotient of group cocycles by group coboundaries is group cohomology.

For an additive $G$-module $M$, the coboundary of a degree-zero group cochain $u\in M$ is $g\mapsto g\cdot u-u$, a principal crossed homomorphism. For a degree-one group cochain $b:G\to M$, its two-coboundary isChanging a section in an abelian-kernel group extension changes its extension cocycle by such a two-coboundary, which explains why the extension class lies in $H^2(G,M)$.