A homotopy theory of pro-spaces

Abstract:

The category of towers of spaces, $\ldots \to {X_{s + 1}} \to {X_s} \to \ldots \to {X_0}$, viewed as pro-spaces, appears to be useful in the study of the relation between homology and homotopy of nonsimply connected spaces. We show that this category admits the structure of a closed model category, in the sense of Quillen; notions of fibration, cofibration, and weak equivalence are defined and shown to satisfy fundamental properties that the corresponding notions satisfy in the category of spaces. This enables one to develop a “homotopy theory” for pro-spaces.