A new book on category theory.

A new book on category theory.

[Noson]

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Hi,

Monoidal Category Theory: Unifying Concepts in Mathematics, Physics, and Computers

is now available.

A sample of the book and slides for teaching can be found on my webpage
http://www.sci.brooklyn.cuny.edu/~noson/MCTtext.html<https://url.au.m.mimecastprotect.com/s/hopPCWLVn6i5Dgoomi6fKHo4-xO?domain=sci.brooklyn.cuny.edu>
You can learn more about the book from MIT Press
https://mitpress.mit.edu/9780262049399/monoidal-category-theory/<https://url.au.m.mimecastprotect.com/s/rJT7CXLW6DiXOyrrosVhxHWXPFl?domain=mitpress.mit.edu/>
or Amazon
https://www.amazon.com/dp/0262049392/<https://url.au.m.mimecastprotect.com/s/GSfkCYW86EsLAJBB2i9iAHxqTrJ?domain=amazon.com/>

Below is the table of contents.

All the best,
Noson S. Yanofsky



Preface
Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ancillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction                                                             1
1.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Examples and the Mini-courses . . . . . . . . . . . . . . . . . . 5
1.4 Mini-course: Sets and Categorical Thinking . . . . . . . . . . . . . . 8

2 Categories                                                                35
2.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . 35
2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3 Related Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4 Mini-course: Basic Linear Algebra . . . . . . . . . . . . . . . . . . . 73

3 Structures Within Categories                                   91
3.1 Products and Coproducts . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Limits and Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.3 Slices and Coslices . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Mini-course: Self-Referential Paradoxes . . . . . . . . . . . . . . . . 121

4 Relationships Between Categories                          151
4.1 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.2 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . 169
4.3 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.4 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.5 Exponentiation and Comma Categories . . . . . . . . . . . . . . . . 200
4.6 Limits and Colimits Revisited . . . . . . . . . . . . . . . . . . . . . 208
4.7 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.8 Mini-course: Basic Categorical Logic . . . . . . . . . . . . . . . . . 221

5 Monoidal Categories                                                           239
5.1 Strict Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . 240
5.2 Cartesian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3 Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.4 Coherence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.5 String Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.6 Mini-course: Advanced Linear Algebra . . . . . . . . . . . . . . . . 283

6 Relationships Between Monoidal Categories                      297
6.1 Monoidal Functors and Natural Transformations . . . . . . . . . . . . 298
6.2 Coherence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 311
6.3 When Coherence Fails . . . . . . . . . . . . . . . . . . . . . . . . . 326
6.4 Mini-course: Duality Theory . . . . . . . . . . . . . . . . . . . . . . 332

7 Variations of Monoidal Categories                         351
7.1 Braided Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . 352
7.2 Closed Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
7.3 Ribbon Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
7.4 Mini-course: Quantum Groups . . . . . . . . . . . . . . . . . . . . . 382

8 Describing Structures                                              405
8.1 Algebraic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
8.2 Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
8.3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
8.4 Algebraic 2-Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 452
8.5 Mini-course: Databases and Schedules . . . . . . . . . . . . . . . . . 458

9 Advanced Topics                                                     473
9.1 Enriched Category Theory . . . . . . . . . . . . . . . . . . . . . . . 473
9.2 Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
9.3 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
9.4 Higher Category Theory . . . . . . . . . . . . . . . . . . . . . . . . 529
9.5 Topos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
9.6 Mini-course: Homotopy Type Theory . . . . . . . . . . . . . . . . . 554

10 More Mini-Courses                                                           565
10.1 Mini-course: Knot Theory . . . . . . . . . . . . . . . . . . . . . . . 566
10.2 Mini-course: Basic Quantum Theory . . . . . . . . . . . . . . . . . . 576
10.3 Mini-course: Quantum Computing . . . . . . . . . . . . . . . . . . . 590

Appendix A: Venn Diagrams                                     623

Appendix B: Index of Categories                               631

Appendix C: Suggestions for Further Study              639

Appendix D: Answers to Selected Problems              643

Bibliography                                                               654

Index                                                                           675


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