Derived functors and the homology of \(n\)-types (Q1858237)

A homotopy \(n\)-type is a connected CW-complex whose homotopy groups are trivial above dimension~\(n\). Homotopy \(n\)-types can be modelled by \(\text{ cat}^{n-1}\)-groups or, equivalently, by crossed \((n-1)\)-cubes. In this paper the authors show that the homology groups of an \(n\)-type~\(X\) can be expressed as derived functors of an abelianisation functor on crossed \((n-1)\)-cubes. They also obtain a Hopf-type formula for \(H_n(X)\).