A model structure on the category of pro-simplicial sets

Abstract

We study the category pro- S S \mathcal {SS} of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro- S S \mathcal {SS} so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.