Ordinary homology and cohomology

Definition 6.2.13 (Ordinary (co)homology). Given an anima \(X \in \An \), a chain complex \(A \in \D (\Z )\), and an integer \(n \in \Z \), we define the ordinary homology and ordinary cohomology of \(X\) with coefficients in \(A\) as \[ H_n(X;A) := H_n(C_{*}(X;A)) \qquadtext {and} H^n(X;A) := H_{-n}(C^{*}(X;A)). \] When \(A = \Z \) (regarded as a complex concentrated in degree \(0\)), we simply write \(H_n(X) := H_n(X;\Z )\) and \(H^n(X) := H^n(X;\Z )\).