Simplicial Methods for Operads and Algebraic Geometry

✍ Scribed by Ieke Moerdijk, Bertrand Toën

• Publisher: Springer Basel
• Year: 2010
• Tongue: English
• Leaves: 197
• Series: Advanced Courses in Mathematics - Crm Barcelona
• Edition: 1st Edition.
• 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

Cover......Page 1
Advanced Courses in Mathematics
CRM Barcelona......Page 3
Simplicial Methods for Operads and Algebraic Geometry......Page 4
ISBN 9783034800518......Page 5
Foreword......Page 6
Contents......Page 8
Part I Lectures on Dendroidal Sets......Page 12
Preface......Page 14
1.1 Operads......Page 16
1.2 Coloured operads......Page 18
1.3 Examples of coloured operads......Page 19
2.1 A formalism of trees......Page 22
2.2 Planar trees......Page 23
2.2.2 Degeneracy maps......Page 25
2.2.3 Dendroidal identities......Page 26
2.3 Non-planar trees......Page 28
2.3.2 Isomorphisms along faces and degeneracies......Page 31
2.3.4 Relation with the simplicial category......Page 32
3.1 Basic definitions and examples......Page 34
3.2 Faces, boundaries and horns......Page 40
3.3 Skeleta and coskeleta......Page 43
3.4 Normal monomorphisms......Page 46
4.1 The Boardman–Vogt tensor product......Page 52
4.2 Tensor product of dendroidal sets......Page 55
4.3 Shuffles of trees......Page 58
5.1 Strict Reedy categories......Page 66
5.2 Model structures for strict Reedy categories......Page 67
5.3 Generalized Reedy categories......Page 68
5.4 Model structures for generalized Reedy categories......Page 71
5.5 Dendroidal objects and simplicial objects......Page 74
5.6 Dendroidal Segal objects......Page 76
6.1 The classical W-construction......Page 80
6.2 The generalized W-construction......Page 85
6.3 The homotopy coherent nerve......Page 86
7.1 Inner Kan complexes......Page 90
7.2 Inner anodyne extensions......Page 93
7.3 Homotopy in an inner Kan complex......Page 95
7.4 Homotopy coherent nerves are inner Kan......Page 99
7.5 The exponential property......Page 102
8.1 Preliminaries......Page 104
8.1.1 Tensor product......Page 107
8.1.3 Normalization......Page 108
8.2 A Quillen model structure on planar dendroidal sets......Page 110
8.3 Trivial cofibrations......Page 113
8.4 A Quillen model structure on dendroidal sets......Page 118
Bibliography......Page 127
Part II Simplicial Presheaves and Derived Algebraic Geometry......Page 130
1.1 The notion of moduli spaces......Page 132
1.2 Construction of moduli spaces: one example......Page 133
1.3 Conclusions......Page 137
2.1 Review of the model category of simplicial presheaves......Page 138
2.2 Basic examples......Page 144
Lecture 3 Algebraic stacks......Page 154
3.1 Schemes and algebraic n-stacks......Page 155
3.2 Some examples......Page 158
3.3 Coarse moduli spaces and homotopy sheaves......Page 165
4.1 Quick review of the model category of commutative simplicial algebras and modules......Page 170
4.2 Cotangent complexes......Page 172
4.3 Flat, smooth and étale morphisms......Page 174
5.1 Derived stacks......Page 178
5.2 Algebraic derived n-stacks......Page 182
5.3 Cotangent complexes......Page 186
6.1 The derived moduli space of local systems......Page 190
6.2 The derived moduli of maps......Page 194
Bibliography......Page 196