A 2-crossed complex is an algebraic model of a homotopy type such that the homotopy groups for 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 -th homology group of the 2-crossed complex corresponds to the -homotopy group of the homotopy type it represents, we have to be a little careful with the indexing.
Idea: Use the adjunction and the functorial 2-connected cover in to define a functorial 2-connected cover in .