Serre class ($\mathcal{C}$)

A class of abelian groups closed under isomorphisms, subgroups, quotients and extensions. For the homotopical Serre-class theorems one also uses tensor/Tor closure and homology closure for Eilenberg–MacLane spaces. A homomorphism is an isomorphism modulo the class when its kernel and cokernel lie in the class.

The class of all finitely generated abelian groups has the closure properties needed for the Hurewicz theorem modulo a Serre class. In a simply connected space, degreewise membership of homotopy groups and integral homology groups in this class is equivalent.