Cohomology represented by mapping spectra

Remark 4.4.33 (Comparison with the earlier definition). The cohomology theory associated to \(E\) in Definition 3.2.3 can now be expressed in terms of the mapping spectrum. Indeed, the adjunction \(\Sigma ^{\infty }\dashv \Omega ^{\infty }\) gives \begin {align*} \pi _{-k}\hom (\Sigma ^{\infty }X,E) &\cong [\Sigma ^{\infty }X[-k],E] \\ &\cong [\Sigma ^{\infty }X,E[k]] \\ &\cong [X,\Omega ^{\infty }(E[k])]_* \\ &\cong [X,E_k]_* = \widetilde E^k(X). \end {align*}

Here the last isomorphism uses the spectrum structure maps for \(k\geq 0\) and the convention \(E_k=\Omega ^{-k}E_0\) for \(k<0\). All displayed isomorphisms are isomorphisms of abelian groups: under \(\Omega ^{\infty }(E[k])\simeq E_k\), the canonical infinite-loop group structure agrees with the double-loop group structure used in Definition 3.2.3.