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 ho-
motopy types, thus generalizing the theory of simplicial sets. The criteria discovered
by Grothendieck show that there are pretty many such categories, called elementary
modelizers. We describe a categorical construction of left homotopy Kan extensions,
generalizing a construction of homotopy colimits by Thomason. We study two re-
markable 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.