nLab
Model Categories and Their Localizations

Contents

Philip S. Hirschhorn

Mathematical Surveys and Monographs, Volume 99

Model Categories and Their Localizations

American Mathematical Society

ISBN 0-8218-3279-4

doi:10.1090/surv/099

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 ff-localization functor

1.4. Concise description of the ff-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 Λ{f}¯\overline{\Lambda\{f\}}-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

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