Proposition 6.2.14. Let \(A \in \D (\Z )\) be a chain complex. There are natural equivalences of functors \[ H(C_*(-;A)) \,\simeq \, \Sigma ^{\infty }_+(-) \otimes HA \colon \An \to \Sp \] and \[ H(C^*(-;A)) \,\simeq \, \hom (\Sigma ^{\infty }_+(-), HA) \colon \An \catop \to \Sp . \] In particular, for any anima \(X\) there are natural isomorphisms \[ H_n(X;A) \cong \pi _n(\Sigma ^{\infty }_+ X \otimes HA) \qquadtext {and} H^n(X;A) \cong \pi _{-n}\hom (\Sigma ^{\infty }_+ X, HA). \]
Proof. Since the Eilenberg-MacLane functor \(H\colon \D (\Z ) \to \Sp \) preserves colimits by Lemma 6.2.5, it follows that \(H \circ C_*(-;A)\colon \An \to \Sp \) preserves colimits. On the other hand, the functor \(\Sigma ^{\infty }_+(-) \otimes HA\) also preserves colimits, since both \(\Sigma ^{\infty }_+\) and \(- \otimes HA\) do. Since both evaluate to \(HA\) on the point, it follows they are naturally equivalent. The equivalence \(H(C^*(-;A)) \simeq \hom (\Sigma ^{\infty }_+(-), HA)\) follows similarly, as both sides preserve limits.
The final statement follows by taking homotopy groups and using Proposition 6.2.3. □
Generated from the authoritative LaTeX source.