Cellular cochain complex ($C_{\mathrm{cell}}^q(X;A)=\mathrm{Hom}(C_q^{\mathrm{cell}}(X;\mathbb{Z}),A)$)

For a CW complex, applying $\mathrm{Hom}(-,A)$ to its integral cellular chain complex gives a cochain complex. The differential is precomposition with the cellular boundary. For finitely many cells it is represented by the transpose of the boundary matrix, reduced in the coefficient group when appropriate.

The cohomology of the cellular cochain complex naturally agrees with singular cohomology. A CW complex with no odd cells has zero cellular differentials and free integral cohomology on its even cells, although its cup product still requires a separate computation.