From Quillen to Grothendieck: a homotopical journey </br> (1) First talk: Quillen's model category structures. Examples: chain complexes, Topological spaces. Derived functors. Homotopy limits and colimits.


Jonathan Chiche

10:00:00 - 11:30:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Abstract: This series of introductory talks is intended to give a survey of some basic tools in current research in homotopy theory. We shall explain Quillen's notion of model category, which allows one to "make homotopy theory" in various settings (this is used for instance in Morel's and Voevodsky's homotopy theory of schemes). We shall present two fundamental model category structures on simplicial sets. Using Grothendieck's homotopy theory, we shall explain Cisinski's proof that the "classical" homotopy theory is universal.

Keywords: Homotopy. Model categories. Simplicial sets. (oo, 1)-categories. Test categories. Basic localizers.

Prerequisites: Only the very basic definitions of category theory will be assumed.

Category theory: Mac Lane's "Categories for the working mathematician" or videos of "The Catsters" on Youtube.
Model categories: Quillen's "Homotopical Algebra", Hovey's "Model Categories", Hirschhorn's "Model Categories".
Simplicial sets: Friedman's "An elementary illustrated introduction to simplicial sets".
Quasi-categories: Cisinski's "Algèbre homotopique et catégories supérieures"
Grothendieck's homotopy theory: Grothendieck's "Pursuing Stacks", Maltsiniotis's "La théorie de l'homotopie de Grothendieck", Cisinski's "Le localisateur fondamental minimal" and "Les préfaisceaux comme modèles des types d'homotopie".