Pro-categories in homotopy theory

TL;DR

This paper establishes an equivalence between model categorical and $$-categorical approaches to pro-categories in homotopy theory, with applications to $$-presentability, Grothendieck topoi, and profinite spaces.

Contribution

It proves a fundamental equivalence linking two approaches to pro-categories and applies this to several areas in homotopy theory and higher category theory.

Findings

Equivalence between model categorical and $$-categorical pro-category approaches.

Conditions for $$-presentability of $$-categories from model categories.

Identification of models for the $$-category of profinite spaces.

Abstract

The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the $\infty$-categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an $\omega$-combinatorial model category, which insure that its underlying $\infty$-category is $\omega$-presentable. In the second application we consider the pro-category of simplicial \'etale sheaves and use it to show that the topological realization of any Grothendieck topos coincides with the shape of the hyper-completion of the associated $\infty$-topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the $\infty$-category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by Raptis.