Homology groups modulo $q$ of a precrossed $P$-module in any dimensions are defined in terms of nonabelian derived functors, where $q$ is a nonnegative integer. The Hopf formula is proved for the second homology group modulo $q$ of a precrossed $P$-module which shows that for $q=0$ our definition is a natural extension of Conduch\'e and Ellis' definition [CE]. Some other properties of homologies of precrossed $P$-modules are investigated.
Homology, Homotopy and Applications, Vol. 2, 2000, No. 7, pp. 105-114