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

• Introduction

• 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

• 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