A Handbook of Model Categories

Preface
Acknowledgements
Contents
1 Introduction
1.1 Prerequisites
1.2 Outline and Reading Guide
1.3 Notation and Conventions
Part I The Theory and Practice of Model Categories
2 On Quillen Model Categories
2.1 Model Categories
2.2 The Homotopy Category
2.3 Quillen Adjunctions and Equivalences
2.4 The Small Object Argument
2.5 Function Complexes and Framings
2.6 Homotopy Limits and Colimits in Model Categories
3 Properties
3.1 Cofibrantly Generated
3.2 Combinatorial
3.3 Cellular
3.4 Left Proper
3.5 Right Proper
3.6 Stable
3.7 Monoidal
3.8 Enriched
4 New Models from Old Ones
4.1 Left Bousfield Localization
4.1.1 Construction of Left Localizations
4.1.2 Preservation of Properties
4.1.3 Splitting Stable Monoidal Model Categories
4.2 Right Bousfield Localization
4.2.1 Construction of Right Localizations
4.2.2 Preservation of Properties
4.2.3 Right Bousfield Delocalization
4.3 Bousfield–Friedlander Localization
4.4 Transfer
4.4.1 Right Transfer
4.4.2 Left Transfer
4.5 Functor Categories
4.6 Reedy Model Structures
4.7 Diagrams of Model Categories
4.8 Limits of Model Categories
4.9 Mixed and Intermediate Models
4.10 Over- and Under-Categories
4.11 Cisinski Model Structures
4.12 Replacing Model Structures by More Convenient Ones
4.12.1 Replacing with Combinatorial Models
4.12.2 Replacing with Left Proper Simplicial Models
4.12.3 Replacing with Right Proper Models
4.13 Stabilization
4.13.1 Sequential Spectra
4.13.2 Symmetric Spectra
4.14 Categories of Enriched Categories
4.15 Model Structures on Pro-categories
4.16 G-Objects in a Model Category
5 Relation to (infty,1)-Categories
5.1 Recollections on (infty,1)-Categories
5.2 Models of (infty,1)-Categories
5.3 From Model Categories to (infty,1)-Categories
Part II Examples
6 Simplicial Sets
6.1 The Kan–Quillen Model
6.2 The Joyal Model
6.3 The Beke Model Structures
6.4 Truncated Models
6.5 The Minimal Cisinksi Model
6.6 The Constructive Kan–Quillen Model
6.7 Model for Cylinders
6.8 The Category of Simplicial Algebras
6.9 Pointed and Reduced Simplicial Sets
6.10 Pro-simplicial Sets and Simplicial Pro-sets
7 Topological Spaces
7.1 The Quillen Model Structure
7.2 The Strøm Model Structure
7.3 The Mixed Model Structure
7.4 Models for G-Spaces
7.5 Isovariant Model Structures
7.6 Hurewicz Models on Topological Categories
8 Chain Complexes
8.1 Models on Bounded Chain Complexes
8.2 The Injective and Projective Models
8.3 Hurewicz and Relative Model Structures
8.4 Modules over Diagrams of Rings
8.5 Abelian Model Categories and Cotorsion Pairs
9 Categories
9.1 The Natural Model
9.2 The Morita Model Structure
9.3 The Thomason Model
9.4 The mathcalM-Model
9.5 The Global Model Structure
9.6 A Non-cofibrantly Generated Model
10 Spectra
10.1 Sequential Spectra
10.2 Symmetric Spectra
10.3 Orthogonal Spectra
10.4 Combinatorial Spectra
10.5 The Positive Flat Model Structure
11 Simplicial Categories
11.1 The Bergner Model
11.2 Models with a Fixed Object Set
11.3 Internal Categories of Simplicial Sets
12 Bisimplicial Sets
12.1 Induced Models
12.2 (Co)diagonal Models
12.3 (Complete) Segal Spaces
12.4 Segal Categories
13 Relative Categories
13.1 The Barwick–Kan Model
13.2 The n-Fold Version
14 Dendroidal Sets
14.1 The Operadic Model
14.2 Complete Segal Operads
14.3 The Covariant Model
14.4 The Picard Model
14.5 The Test Model
14.6 Open Dendroidal Sets
15 Cyclic Sets
15.1 The Dwyer–Hopkins–Kan Model
15.2 The Spaliński Model
15.3 The Monic Models
15.4 The Blumberg Model
15.5 Cyclic Objects in a Model Category
15.6 Crossed Simplicial Groups
16 Cast-Algebras
16.1 A Non-model on Cast-Algebras
16.2 A Model for K and KK-Theory
16.3 Unstable Models on Cast-Spaces
16.4 Stable Models on Cast-Spaces
16.5 Models on Cast-Categories
16.6 Projective Systems of Cast-Algebras
17 Miscellanea
17.1 The Trivial Model
17.2 The Dual Model
17.3 Model Structures on the Category of Sets
17.4 A Model Structure on Equivalence Relations
17.5 Stable Module Categories
17.6 Model Structures on Arrow Categories
17.7 Model Structures for Graphs
17.8 Model Structures on a Poset
17.9 Non-existence of Limits of Model Categories
17.10 A Model Without Functorial Factorizations
17.11 Non-existence of Right Transfer
17.12 Non-existence of Left Bousfield Localizations
Part III A Model Categorical Kunstkammer
18 Die Kunstkammer
Appendix Bibliography
Index