Existence of ordinary homology and cohomology

Theorem 3.3.1 (Existence of ordinary homology and cohomology). For every abelian group \(A\), there exist reduced homology and cohomology theories \[ \widetilde H_*(-;A)\colon \An _* \to \Ab ^{\Z } \qquadtext {and} \widetilde H^*(-;A)\colon \An _*\catop \to \Ab ^{\Z } \] satisfying the dimension axiom \[ \widetilde H_n(S^0;A) \cong \begin {cases} A & \text {if } n=0, \\ 0 & \text {if } n\neq 0, \end {cases} \qquadtext {and} \widetilde H^n(S^0;A) \cong \begin {cases} A & \text {if } n=0, \\ 0 & \text {if } n\neq 0. \end {cases} \] The associated unreduced theories are denoted \(H_*(-;A)\) and \(H^*(-;A)\). When \(A=\Z \), we usually omit it from the notation.