具体描述
The aim of this book is to explain modern homotopy theory in a manner accessible to graduate students yet structured so that experts can skip over numerous linear developments to quickly reach the topics of their interest. Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences, i.e., by creating a new category in which the weak equivalences are isomorphisms. Quillen defined a model category to be a category together with a class of weak equivalences and additional structure useful for describing the homotopy category in terms of the original category. This allows you to make constructions analogous to those used to study the homotopy theory of topological spaces.
A model category has a class of maps called weak equivalences plus two other classes of maps, called cofibrations and fibrations. Quillen's axioms ensure that the homotopy category exists and that the cofibrations and fibrations have extension and lifting properties similar to those of cofibration and fibration maps of topological spaces. During the past several decades the language of model categories has become standard in many areas of algebraic topology, and it is increasingly being used in other fields where homotopy theoretic ideas are becoming important, including modern algebraic K-theory and algebraic geometry.
All these subjects and more are discussed in the book, beginning with the basic definitions and giving complete arguments in order to make the motivations and proofs accessible to the novice. The book is intended for graduate students and research mathematicians working in homotopy theory and related areas.
作者简介
Philip S. Hirschhorn is a mathematician at Wellesley College.
目录信息
Introduction 10
Model categories and their homotopy categories 10
Localizing model category structures 12
Acknowledgments 16
Part 1. Localization of Model Category Structures 18
Summary of Part 1 20
Chapter 1. Local Spaces and Localization 22
1.1. Definitions of spaces and mapping spaces 22
1.2. Local spaces and localization 25
1.3. Constructing an f-localization functor 33
1.4. Concise description of the f-localization 37
1.5. Postnikov approximations 39
1.6. Topological spaces and simplicial sets 41
1.7. A continuous localization functor 46
1.8. Pointed and unpointed localization 48
Chapter 2. The Localization Model Category for Spaces 52
2.1. The Bousfield localization model category structure 52
2.2. Subcomplexes of relative ∧{f}-cell complexes 54
2.3. The Bousfield-Smith cardinality argument 59
Chapter 3. Localization of Model Categories 64
3.1. Left localization and right localization 64
3.2. C-local objects and C-local equivalences 68
3.3. Bousfield localization 74
3.4. Bousfield localization and properness 82
3.5. Detecting equivalences 85
Chapter 4. Existence of Left Bousfield Localizations 88
4.1. Existence of left Bousfield localizations 88
4.2. Horns on S and S-local equivalences 90
4.3. A functorial localization 91
4.4. Localization of subcomplexes 93
4.5. The Bousfield-Smith cardinality argument 95
4.6. Proof of the main theorem 98
Chapter 5. Existence of Right Bousfield Localizations 100
5.1. Right Bousfield localization: Cellularization 100
5.2. Horns on K and K-colocal equivalences 102
5.3. K-colocal cofibrations 104
5.4. Proof of the main theorem 106
5.5. K-colocal objects and K-cellular objects 107
Chapter 6. Fiberwise Localization 110
6.1. Fiberwise localization 110
6.2. The fiberwise local model category structure 112
6.3. Localizing the fiber 112
6.4. Uniqueness of the fiberwise localization 115
Part 2. Homotopy Theory in Model Categories 118
Summary of Part 2 120
Chapter 7. Model Categories 124
7.1. Model categories 125
7.2. Lifting and the retract argument 127
7.3. Homotopy 132
7.4. Homotopy as an equivalence relation 136
7.5. The classical homotopy category 139
7.6. Relative homotopy and fiberwise homotopy 142
7.7. Weak equivalences 146
7.8. Homotopy equivalence 147
7.9. The equivalence relation generated by "weak equivalence'' 150
7.10. Topological spaces and simplicial sets 151
Chapter 8. Fibrant and Cofibrant Approximations 154
8.1. Fibrant and cofibrant approximations 155
8.2. Approximations and homotopic maps 161
8.3. The homotopy category of a model category 164
8.4. Derived functors 168
8.5. Quillen functors and total derived functors 170
Chapter 9. Simplicial Model Categories 176
9.1. Simplicial model categories 176
9.2. Colimits and limits 180
9.3. Weak equivalences of function complexes 181
9.4. Homotopy lifting 184
9.5. Simplicial homotopy 187
9.6. Uniqueness of lifts 192
9.7. Detecting weak equivalences 194
9.8. Simplicial functors 196
Chapter 10. Ordinals, Cardinals, and Transfinite Composition 202
10.1. Ordinals and cardinals 203
10.2. Transfinite composition 205
10.3. Transfinite composition and lifting in model categories 210
10.4. Small objects 211
10.5. The small object argument 213
10.6. Subcomplexes of relative I-cell complexes 218
10.7. Cell complexes of topological spaces 221
10.8. Compactness 223
10.9. Effective monomorphisms 225
Chapter 11. Cofibrantly Generated Model Categories 226
11.1. Cofibrantly generated model categories 227
11.2. Cofibrations in a cofibrantly generated model category 228
11.3. Recognizing cofibrantly generated model categories 230
11.4. Compactness 232
11.5. Free cell complexes 234
11.6. Diagrams in a cofibrantly generated model category 241
11.7. Diagrams in a simplicial model category 242
11.8. Overcategories and undercategories 243
11.9. Extending diagrams 245
Chapter 12. Cellular Model Categories 248
12.1. Cellular model categories 248
12.2. Subcomplexes in cellular model categories 249
12.3. Compactness in cellular model categories 251
12.4. Smallness in cellular model categories 252
12.5. Bounding the size of cell complexes 253
Chapter 13. Proper Model Categories 256
13.1. Properness 256
13.2. Properness and lifting 260
13.3. Homotopy pullbacks and homotopy fiber squares 261
13.4. Homotopy fibers 266
13.5. Homotopy pushouts and homotopy cofiber squares 267
Chapter 14. The Classifying Space of a Small Category 270
14.1. The classifying space of a small category 271
14.2. Cofinal functors 273
14.3. Contractible classifying spaces 275
14.4. Uniqueness of weak equivalences 277
14.5. Categories of functors 280
14.6. Cofibrant approximations and fibrant approximations 283
14.7. Diagrams of undercategories and overcategories 285
14.8. Free cell complexes of simplicial sets 288
Chapter 15. The Reedy Model Category Structure 294
15.1. Reedy categories 295
15.2. Diagrams indexed by a Reedy category 298
15.3. The Reedy model category structure 305
15.4. Quillen functors 311
15.5. Products of Reedy categories 311
15.6. Reedy diagrams in a cofibrantly generated model category 313
15.7. Reedy diagrams in a cellular model category 319
15.8. Bisimplicial sets 320
15.9. Cosimplicial simplicial sets 322
15.10. Cofibrant constants and fibrant constants 325
15.11. The realization of a bisimplicial set 329
Chapter 16. Cosimplicial and Simplicial Resolutions 334
16.1. Resolutions 335
16.2. Quillen functors and resolutions 340
16.3. Realizations 341
16.4. Adjointness 343
16.5. Homotopy lifting extension theorems 348
16.6. Frames 354
16.7. Reedy frames 359
Chapter 17. Homotopy Function Complexes 364
17.1. Left homotopy function complexes 366
17.2. Right homotopy function complexes 367
17.3. Two-sided homotopy function complexes 369
17.4. Homotopy function complexes 371
17.5. Functorial homotopy function complexes 374
17.6. Homotopic maps of homotopy function complexes 379
17.7. Homotopy classes of maps 382
17.8. Homotopy orthogonal maps 384
17.9. Sequential colimits 393
Chapter 18. Homotopy Limits in Simplicial Model Categories 396
18.1. Homotopy colimits and homotopy limits 397
18.2. The homotopy limit of a diagram of spaces 400
18.3. Coends and ends 402
18.4. Consequences of adjoint ness 406
18.5. Homotopy invariance 411
18.6. Simplicial objects and cosimplicial objects 412
18.7. The Bousfield-Kan map 413
18.8. Diagrams of pointed or unpointed spaces 415
18.9. Diagrams of simplicial sets 417
Chapter 19. Homotopy Limits in General Model Categories 422
19.1. Homotopy colimits and homotopy limits 422
19.2. Coends and ends 424
19.3. Consequences of adjoint ness 428
19.4. Homotopy invariance 431
19.5. Homotopy pullbacks and homotopy pushouts 433
19.6. Homotopy cofinal functors 435
19.7. The Reedy diagram homotopy lifting extension theorem 440
19.8. Realizations and total objects 443
19.9. Reedy cofibrant diagrams and Reedy fibrant diagrams 444
Index 446
Bibliography 472
· · · · · · (收起)
读后感
用户评价
在深度和广度上，这本书展现出了令人敬佩的平衡感。它既没有陷入纯粹的概念堆砌，也没有为了追求易读性而牺牲关键细节的阐释。作者对某一核心概念的剖析，往往能够深入到其最底层的数学基础，毫不含糊地展示推导过程，这对于追求真理的读者来说是莫大的慰藉。然而，紧接着，他又会迅速将视角拉回到应用层面，讨论该模型在现实世界中可能遇到的局限性、潜在的偏误来源，以及如何进行“本地化”的调适。这种在极度抽象与高度具体之间自由穿梭的能力，是真正大家之作的标志。我尤其欣赏那些关于“边界条件”的讨论，作者清晰地界定了该理论适用的“势力范围”，这种坦诚的态度，远比一味鼓吹方法的普适性要负责任得多。可以说，它既是理论的堡垒，也是实操的指南针。
这本书的装帧设计着实让人眼前一亮，封面采用了那种略带磨砂质感的纸张，触感温润而厚重，立刻营造出一种严肃而专业的氛围。我尤其欣赏封面上那副抽象的几何图形，色彩搭配沉稳而不失活力，似乎在暗示着内容将是严谨的逻辑与灵活的思维的结合。内页的纸张选择也十分考究，米白色调，光线柔和，长时间阅读下来眼睛也不会感到疲劳。排版上，字体大小和行间距的处理非常得当，即使是那些复杂的图表和公式，也能清晰地呈现出来，可见出版社在细节打磨上的用心。拿到手里的时候，分量感十足，让我对其中蕴含的知识体系充满了期待。这不仅仅是一本书，更像是一件精心制作的工艺品，让人愿意珍藏。第一印象总是至关重要的，而这本《Model Categories and Their Localizations》在视觉和触觉上都给出了一个非常高的起点，让人迫不及待地想要翻开它，探索其内部的奥秘。这种对物理载体的重视，往往预示着作者对内容本身的尊重与投入。
阅读体验的最终衡量标准，在于它能否在读者心中留下持久的、可被激活的“思维印记”。《Model Categories and Their Localizations》无疑做到了这一点。读完之后，我发现自己看待某些旧有问题的方式发生了微妙的转变，我开始习惯性地去审视事物的内在结构和它们相互关联的“局部化”特征。书中引用的那些跨学科的文献和观点，也极大地拓宽了我的知识边界，我甚至需要停下来去查阅一些作者提及的次要概念，以求更全面地理解他的论点。这本书的价值不在于提供了一个即时可用的答案，而在于它提供了一套强大的、可重复使用的“提问框架”。它像一位智慧的导师，在你需要的时候提供清晰的路径，在你迷茫时提供坚实的理论支撑，其影响力会随着时间的推移，在我未来的研究和思考中持续显现。
翻开正文，首先映入眼帘的是引人深思的开篇导论，作者的笔触极其老练，他没有直接跳入晦涩的定义，而是先从一个宏大的历史背景切入，勾勒出“分类”这一概念在现代科学发展中的关键地位。他巧妙地运用了一系列类比，将原本可能令人生畏的抽象概念，转化为读者日常生活中能够理解的场景，这种叙事策略非常高明。我特别欣赏作者在探讨理论演进时所展现出的那种洞察力，他不仅仅是在罗列事实，更是在梳理思想的脉络，展示不同学派之间的对话与冲突。阅读过程中，我发现作者的语言风格在保持学术严谨性的同时，又流露出一种罕见的激情，仿佛他本人正站在知识探索的最前沿，热切地邀请我们加入这场智力冒险。这种娓娓道来的叙事方式，使得即便是初学者，也能在迷雾中找到方向，而不是被术语的洪流所淹没。这种“引导式”的写作，体现了作者深厚的教学功底和对读者群体的深刻理解。
这本书的组织结构设计得极其精妙，简直是一部关于如何系统性思考的教科书。章节之间的逻辑衔接如同严密的链条，环环相扣，没有一丝冗余或跳跃。每一章的开头都会清晰地预告本章的核心目标和将要涉及的关键工具，这让读者在进入具体论述之前就能建立起一个清晰的认知框架。更值得称赞的是，作者在关键的理论节点处，设置了大量的“反思点”或“案例应用”，这些小节像是在平静的知识海洋中投下的石子，激起的涟漪迫使读者停下来，消化吸收刚刚学到的概念，并思考其在更广阔领域中的潜在影响。我发现，当一个新模型被引入时，作者总会同时提供其在不同情境下的“适应性测试”，这种多维度的考察，极大地增强了理论的实用性和可迁移性。这种编排方式，有效避免了知识点孤立存在的问题，使整部作品构成了一个有生命的、互相支持的知识生态系统。
