Reduced ordinary homology and cohomology

Definition 6.2.11 (Reduced ordinary (co)homology). Let \(A \in \D (\Z )\) be a complex of abelian groups and let \(X \in \An _*\) be a pointed anima. We define the reduced ordinary homology of \(X\) with coefficients in \(A\) and the reduced ordinary cohomology of \(X\) with coefficients in \(A\) as the (co)homology theories represented by the Eilenberg-MacLane spectrum \(HA\): \[ \widetilde {H}_n(X;A) \, := \, HA_n(X) \, = \, \pi _n(\Sigma ^{\infty }X \otimes HA) \] and \[ \widetilde {H}^n(X;A) \, := \, HA^n(X) \, = \, \pi _{-n}\hom (\Sigma ^{\infty }X,HA). \] We will mostly be interested in the case where \(A \in \Ab \) is an abelian group, regarded as a complex concentrated in degree \(0\).