David Roberts 2-connected cover of a 2-crossed complex

A 2-crossed complex is an algebraic model of a homotopy type such that the homotopy groups $\pi_k$ for $k\gt 2$ have trivial Whitehead products.

As such, the 2-connected cover should be modeled by a (bounded) chain complex. The aim is to define a functorial 2-connected cover of a 2-crossed complex.

Note that since the $n$-th homology group of the 2-crossed complex corresponds to the $(n+1)$-homotopy group of the homotopy type it represents, we have to be a little careful with the indexing.

Idea: Use the adjunction $2Crs \leftrightarrows sGrp$ and the functorial 2-connected cover in $sGrp$ to define a functorial 2-connected cover in $2Crs$.