Universal central extensions of precrossed modules and Milnor's relative \(K_{2}\) (Q1410959)

A precrossed module \((M,P,\mu)\) is a group homomorphism \(\mu:M\to P\) together with an action of \(P\) on \(M\) satisfying \(\mu({}^pm)=p\mu(m)p^{-1}\). If one also has \({}^{\mu(m)}m'=mm'm^{-1}\) for \(m,m'\in M\), then it is a crossed module. The authors define universal central extensions of perfect precrossed modules. They also have a five term exact sequence in homology and an analogue of Hopf's formula for the second homology group. However, precrossed modules are not what is needed to get the relative Steinberg group of Keune and Loday. For that one needs to use crossed modules.