Pro-categories in homotopy theory

Barnea, Ilan¹ ; Harpaz, Yonatan² ; Horel, Geoffroy³

¹Department of Mathematics, Hebrew University of Jerusalem, 9190401 Jerusalem, Israel
²Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France
³Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 567-643

Résumé

Our goal in 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 ∞–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 ω–combinatorial model category, which insure that its underlying ∞–category is ω–presentable. In the second application we show that the topological realization of any Grothendieck topos coincides with the shape of the hypercompletion of the associated ∞–topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the ∞–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 G Raptis.

Détail

DOI : 10.2140/agt.2017.17.567

Classification : 18G55, 55U35, 18C35
Keywords: pro-categories, model categories, infinity-categories, étale homotopy type, profinite completion

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