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For a CW complex $X$ and coefficient abelian group $A$, the cohomology of its cellular cochain complex naturally equals its singular cohomology:The cellular cochain group is $\mathrm{Hom}(C_q^{\mathrm{cell}}(X;\mathbb{Z}),A)$, and its coboundary map is precomposition with the cellular boundary.

If $X$ has only even-dimensional cells, every cellular coboundary map is zero. If, in addition, there are finitely many cells in each dimension, then its integral cohomology groups are free abelian groups, with one generator for each cell in that degree. For infinitely many cells in one degree, the cochain group is a direct product of groups of copies of $\mathbb{Z}$, rather than the direct sum used for cellular chains; the finite-cell freeness conclusion must not be applied without that qualification. Even with all additive groups known, the cup product still requires a separate computation.