Memoirs of the American Mathematical Society 2008; 224 pp; softcover Volume: 193 ISBN10: 0821840991 ISBN13: 9780821840993 List Price: US$86 Individual Members: US$51.60 Institutional Members: US$68.80 Order Code: MEMO/193/902
 This is the second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category \(C\), and prove that this notion satisfies the expected properties. Table of Contents Part 1. General theory of geometric stacks  Introduction to Part 1
 Homotopical algebraic context
 Preliminaries on linear and commutative algebra in an HA context
 Geometric stacks: Basic theory
 Geometric stacks: Infinitesimal theory
Part 2. Applications  Introduction to Part 2
 Geometric \(n\)stacks in algebraic geometry (after C. Simpson)
 Derived algebraic geometry
 Complicial algebraic geometry
 Brave new algebraic geometry
 Appendix A. Classifying spaces of model categories
 Appendix B. Strictification
 Appendix C. Representability criterion (after J. Lurie)
 Bibliography
 Index
