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Chain Complex**

Given an abelian category \(\mathcal{A}\), a chain complex with coefficients in \(\mathcal{A}\) (\(C_\bullet\)) is a sequence \((C_n)_{n\in\mathbb{Z}}\) of objects of \(\mathcal{A}\) (the '\(n\)-chains' of \(C_\bullet\)) with a collection of maps \(d_n:C_n\to C_{n-1}\) (the 'differential' of \(C_\bullet\)) such that \(d_{n-1}\circ d_n=0\) for each \(n\in\mathbb{Z}\). The kernels and images of the differential, \(Z_n=\mathrm{ker}(d_n)\subseteq C_n\) and \(B_n=\mathrm{ker}(d_{n+1})\subseteq C_n\) are called the \(n\)-cycles and \(n\)-boundaries respectively, and the quotient \(H_n=Z_n/B_n\) is called the \(n^{th}\) homology object of \(C_\bullet\).