2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70880For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages.Algebraic Topology55P60, 55N10 (Primary); 18G55, 55U35 (Secondary)text