---- > [!definition] Definition. ([[cellular cohomology]]) > We define **cellular cohomology** by taking as the [[chain complex of modules|chain groups]] $C^{n}_{\text{cell}}=H^{n}(X, X^{n-1})$ and letting $d^{n}_{\text{cell}}$ be the composition $H^{n}(X^{n}, X^{n-1}) \xrightarrow{q^{*}}H^{n}(X^{n}) \xrightarrow{\partial} H^{n+1}(X^{n+1}, X^{n}).$ This defines a [[chain complex of modules|cochain complex]] with [[(co)homology of a complex|cohomology]] $H^{*}_{\text{cell}}(X)$. It can be directly checked that $C^{\bullet}_{\text{cell}}(X) \cong\text{Hom}(C_{\bullet}^{\text{cell}}(X), \mathbb{Z}).$ ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```