2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58961This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to define Steenrod operations). The treatment of this problem is completely self-contained, and includes material that simplifies, extends, and corrects material from the authors AMS Memoir, "Iterating the cobar construction".Cleaned up some notation and added a clarifying diagramAlgebraic Topology55R91;18G30text