Cellular cochain complex
= Cellular cochain complex
{title2=$C^q_{\mathrm{cell}}(X;A)=\operatorname{Hom}(C_q^{\mathrm{cell}}(X;\mathbb Z),A)$}
For a <CW complex>, applying $\operatorname{Hom}(-,A)$ 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.