2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78338Following Ellis, we investigate the notion of totally free crossed squares and related square complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, in terms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result of Brown and Loday.23 pagesK-Theory and HomologyCategory Theory18D35, 18G30, 18G50, 18G55, 55Q20, 55Q05text