Simplicial model structures on pro-categories

TL;DR

This paper introduces a method for building simplicial model structures on pro- and ind-categories, enabling the creation of new models and recovering known structures like pro-p and profinite spaces.

Contribution

It provides a general construction technique for simplicial model structures on pro- and ind-categories, including new models and a comparison to infinity-categorical approaches.

Findings

Recoveries of Morel's and Quick's model structures

Construction of new interesting model structures

Analysis of functorial behavior and localization properties

Abstract

We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the infinity-categorical approach to ind- and pro-categories in an appendix.