Cellular cochain group ($C_{\mathrm{cell}}^q(X;A)$)

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 $H^0(S^1;A)=H^1(S^1;A)=A$.