Completions of pro-spaces

Abstract. For every ring R, we present a pair of model structures on the category of pro-spaces. In the rst, the weak equivalences are detected by cohomology with coecients in R. In the second, the weak equivalences are detected by cohomology with coecients in all R-modules (or equivalently by pro-homology with coecients in R). In the second model structure, brant replacement is essentially just the Bouseld-Kan R-tower. When R = Z=p, the rst homotopy category is equivalent to a homotopy theory dened by Morel but has some convenient categorical advantages. 1