nLab
Model Categories and Their Localizations
Contents
This page is about the textbook:
on model categories and their localization.
Contents
Introduction
Model categories and their homotopy categories
Localizing model category structures
Acknowledgments
Part 1. Localization of Model Category Structures
Summary of Part 1
Chapter 1. Local Spaces and Localization
1.1. Definitions of spaces and mapping spaces
1.2. Local spaces and localization
1.3. Constructing an -localization functor
1.4. Concise description of the -localization
1.5. Postnikov approximations
1.6. Topological spaces and simplicial sets
1.7. A continuous localization functor
1.8. Pointed and unpointed localization
Chapter 2. The Localization Model Category for Spaces
2.1. The Bousfield localization model category structure
2.2. Subcomplexes of relative -cell complexes
2.3. The Bousfield-Smith cardinality argument
Chapter 3. Localization of Model Categories
3.1. Left localization and right localization
3.2. C-local objects and C-local equivalences
3.3. Bousfield localization
3.4. Bousfield localization and properness
3.5. Detecting equivalences
Chapter 4. Existence of Left Bousfield Localizations
4.1. Existence of left Bousfield localizations
4.2. Horns on S and S-local equivalences
4.3. A functorial localization
4.4. Localization of subcomplexes
4.5. The Bousfield-Smith cardinality argument
4.6. Proof of the main theorem
Chapter 5. Existence of Right Bousfield Localizations
5.1. Right Bousfield localization: Cellularization
5.2. Horns on K and K-colocal equivalences
5.3. K-colocal cofibrations
5.4. Proof of the main theorem
5.5. K-colocal objects and K-cellular objects
Chapter 6. Fiberwise Localization
6.1. Fiberwise localization
6.2. The fiberwise local model category structure
6.3. Localizing the fiber
6.4. Uniqueness of the fiberwise localization
Part 2. Homotopy Theory in Model Categories
Summary of Part 2
Chapter 7. Model Categories
7.1. Model categories
7.2. Lifting and the retract argument
7.3. Homotopy
7.4. Homotopy as an equivalence relation
7.5. The classical homotopy category
7.6. Relative homotopy and fiberwise homotopy
7.7. Weak equivalences
7.8. Homotopy equivalence
7.9. The equivalence relation generated by “weak equivalence”
7.10. Topological spaces and simplicial sets
Chapter 8. Fibrant and Cofibrant Approximations
8.1. Fibrant and cofibrant approximations
8.2. Approximations and homotopic maps
8.3. The homotopy category of a model category
8.4. Derived functors
8.5. Quillen functors and total derived functors
Chapter 9. Simplicial Model Categories
9.1. Simplicial model categories
9.2. Colimits and limits
9.3. Weak equivalences of function complexes
9.4. Homotopy lifting
9.5. Simplicial homotopy
9.6. Uniqueness of lifts
9.7. Detecting weak equivalences
9.8. Simplicial functors
Chapter 10. Ordinals, Cardinals, and Transfinite Composition
10.1. Ordinals and cardinals
10.2. Transfinite composition
10.3. Transfinite composition and lifting in model categories
10.4. Small objects
10.5. The small object argument
10.6. Subcomplexes of relative I-cell complexes
10.7. Cell complexes of topological spaces
10.8. Compactness
10.9. Effective monomorphisms
Chapter 11. Cofibrantly Generated Model Categories
11.1. Cofibrantly generated model categories
11.2. Cofibrations in a cofibrantly generated model category
11.3. Recognizing cofibrantly generated model categories
11.4. Compactness
11.5. Free cell complexes
11.6. Diagrams in a cofibrantly generated model category
11.7. Diagrams in a simplicial model category
11.8. Overcategories and undercategories
11.9. Extending diagrams
Chapter 12. Cellular Model Categories
12.1. Cellular model categories
12.2. Subcomplexes in cellular model categories
12.3. Compactness in cellular model categories
12.4. Smallness in cellular model categories
12.5. Bounding the size of cell complexes
Chapter 13. Proper Model Categories
13.1. Properness
13.2. Properness and lifting
13.3. Homotopy pullbacks and homotopy fiber squares
13.4. Homotopy fibers
13.5. Homotopy pushouts and homotopy cofiber squares
Chapter 14. The Classifying Space of a Small Category
14.1. The classifying space of a small category
14.2. Cofinal functors
14.3. Contractible classifying spaces
14.4. Uniqueness of weak equivalences
14.5. Categories of functors
14.6. Cofibrant approximations and fibrant approximations
14.7. Diagrams of undercategories and overcategories
14.8. Free cell complexes of simplicial sets
Chapter 15. The Reedy Model Category Structure
15.1. Reedy categories
15.2. Diagrams indexed by a Reedy category
15.3. The Reedy model category structure
15.4. Quillen functors
15.5. Products of Reedy categories
15.6. Reedy diagrams in a cofibrantly generated model category
15.7. Reedy diagrams in a cellular model category
15.8. Bisimplicial sets
15.9. Cosimplicial simplicial sets
15.10. Cofibrant constants and fibrant constants
15.11. The realization of a bisimplicial set
Chapter 16. Cosimplicial and Simplicial Resolutions
16.1. Resolutions
16.2. Quillen functors and resolutions
16.3. Realizations
16.4. Adjointness
16.5. Homotopy lifting extension theorems
16.6. Frames
16.7. Reedy frames
Chapter 17. Homotopy Function Complexes
17.1. Left homotopy function complexes
17.2. Right homotopy function complexes
17.3. Two-sided homotopy function complexes
17.4. Homotopy function complexes
17.5. Functorial homotopy function complexes
17.6. Homotopic maps of homotopy function complexes
17.7. Homotopy classes of maps
17.8. Homotopy orthogonal maps
17.9. Sequential colimits
Chapter 18. Homotopy Limits in Simplicial Model Categories
18.1. Homotopy colimits and homotopy limits
18.2. The homotopy limit of a diagram of spaces
18.3. Coends and ends
18.4. Consequences of adjoint ness
18.5. Homotopy invariance
18.6. Simplicial objects and cosimplicial objects
18.7. The Bousfield-Kan map
18.8. Diagrams of pointed or unpointed spaces
18.9. Diagrams of simplicial sets
Chapter 19. Homotopy Limits in General Model Categories
19.1. Homotopy colimits and homotopy limits
19.2. Coends and ends
19.3. Consequences of adjoint ness
19.4. Homotopy invariance
19.5. Homotopy pullbacks and homotopy pushouts
19.6. Homotopy cofinal functors
19.7. The Reedy diagram homotopy lifting extension theorem
19.8. Realizations and total objects
19.9. Reedy cofibrant diagrams and Reedy fibrant diagrams
Index
Bibliography
Last revised on August 8, 2022 at 17:18:55.
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