La théorie de l'homotopie de Grothendieck

The aim of this book is to explain the very beautiful homotopy theory developed by Grothendieck in "Pursuing Stacks". The question is to characterize categories of presheaves that modelize homotopy types, thus generalizing the theory of simplicial sets. The criteria discovered by Grothendieck show that there are pretty many such categories, called elementary modelizers. The book describes a categorical construction of left homotopy Kan extensions, generalizing a construction of homotopy colimits by Thomason. The book studies two remarkable classes of functors, proper and smooth functors, these two notions being mutually dual. These functors are characterized by cohomological properties inspired by the proper or smooth base change theorem in algebraic geometry.