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 You can learn more about the book from MIT Press https://mitpress.mit.edu/9780262049399/monoidal-category-theory/ or Amazon https://www.amazon.com/dp/0262049392/ 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 You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. 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