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 Astérisque 2005; 140 pp; softcover Number: 301 ISBN-10: 2-85629-181-3 ISBN-13: 978-2-85629-181-8 List Price: US$38 Individual Members: US$34.20 Order Code: AST/301 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. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents Preface Introduction La théorie des catégories test Les localisateurs fondamentaux Théorie homotopique élémentaire des catégories Bibliographie Index des notations Index terminologique