Simplicial Methods for Operads and Algebraic Geometry (Advanced Courses in Mathematics - CRM Barcelona)

✍ Scribed by Ieke Moerdijk, Bertrand Toën

• Publisher: Birkhäuser
• Year: 2010
• Tongue: English
• Leaves: 186
• Category: Library

✦ Synopsis

This book is an introduction to two new topics in homotopy theory: Dendroidal Sets (by Ieke Moerdijk) and Derived Algebraic Geometry (by Bertrand Toën). The category of dendroidal sets is an extension of that of simplicial sets, based on rooted trees instead of linear orders, suitable as a model category for higher topological structures. Derived algebraic geometry deals with functors from simplicial commutative rings to simplicial sets subject to a homotopical descent condition. The material in the book is an enhanced version of lecture notes from courses given within a special year on Homotopy Theory and Higher Categories at the CRM in Barcelona.

✦ Table of Contents

Simplicial Methods for Operads and Algebraic Geometry
Foreword
Contents
Part I Lectures on Dendroidal Sets
Preface
Lecture 1 Operads
1.1 Operads
1.2 Coloured operads
1.3 Examples of coloured operads
Lecture 2 Trees as operads
2.1 A formalism of trees
2.2 Planar trees
2.2.1 Face maps
2.2.2 Degeneracy maps
2.2.3 Dendroidal identities
2.3 Non-planar trees
2.3.1 Dendroidal identities with isomorphisms
2.3.2 Isomorphisms along faces and degeneracies
2.3.3 The presheaf of planar structures
2.3.4 Relation with the simplicial category
Lecture 3 Dendroidal sets
3.1 Basic definitions and examples
3.2 Faces, boundaries and horns
3.3 Skeleta and coskeleta
3.4 Normal monomorphisms
Lecture 4 Tensor product of dendroidal sets
4.1 The Boardman–Vogt tensor product
4.2 Tensor product of dendroidal sets
4.3 Shuffles of trees
Lecture 5 A Reedy model structure on dendroidal spaces
5.1 Strict Reedy categories
5.2 Model structures for strict Reedy categories
5.3 Generalized Reedy categories
5.4 Model structures for generalized Reedy categories
5.5 Dendroidal objects and simplicial objects
5.6 Dendroidal Segal objects
Lecture 6 Boardman–Vogt resolution and homotopy coherent nerve
6.1 The classical W-construction
6.2 The generalized W-construction
6.3 The homotopy coherent nerve
Lecture 7 Inner Kan complexes and normal dendroidal sets
7.1 Inner Kan complexes
7.2 Inner anodyne extensions
7.3 Homotopy in an inner Kan complex
7.4 Homotopy coherent nerves are inner Kan
7.5 The exponential property
Lecture 8 Model structures on dendroidal sets
8.1 Preliminaries
8.1.1 Tensor product
8.1.2 Intervals
8.1.3 Normalization
8.2 A Quillen model structure on planar dendroidal sets
8.3 Trivial cofibrations
8.4 A Quillen model structure on dendroidal sets
Bibliography
Part II Simplicial Presheaves and Derived Algebraic Geometry
Lecture 1 Motivation and objectives
1.1 The notion of moduli spaces
1.2 Construction of moduli spaces: one example
1.3 Conclusions
Lecture 2 Simplicial presheaves as stacks
2.1 Review of the model category of simplicial presheaves
2.2 Basic examples
Lecture 3 Algebraic stacks
3.1 Schemes and algebraic n-stacks
3.2 Some examples
3.3 Coarse moduli spaces and homotopy sheaves
Lecture 4 Simplicial commutative algebras
4.1 Quick review of the model category of commutative simplicial algebras and modules
4.2 Cotangent complexes
4.3 Flat, smooth and étale morphisms
Lecture 5 Derived stacks and derived algebraic stacks
5.1 Derived stacks
5.2 Algebraic derived n-stacks
5.3 Cotangent complexes
Lecture 6 Examples of derived algebraic stacks
6.1 The derived moduli space of local systems
6.2 The derived moduli of maps
Bibliography