2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62059The theory of parity quasi-complexes (PQC) is developed, preparing a set up for defining derived functors using resolutions in the nonabelian case. A homotopy structure on the category of PQC is defined, yielding a 2-category structure. The nonabelian homology functor factors through the corresponding homotopy category. Following the relative homological algebra approach, resolutions are defined as PQC having parity contracting homotopies in a suitable category. A canonical non-abelian PQC resolution for groups is defined.AMS-LaTex, 9 pagesCategory TheoryGroup Theory18G55 (Primary) 18G10, 18G25 (Secondary)text