Higher Operads, Higher Categories (London Mathematical Society Lecture Note Series, Series Number 298)

Higher-dimensional Category Theory Is The Study Of N-categories, Operads, Braided Monoidal Categories, And Other Such Exotic Structures. It Draws Its Inspiration From Areas As Diverse As Topology, Quantum Algebra, Mathematical Physics, Logic, And Theoretical Computer Science. The Heart Of This Book Is The Language Of Generalized Operads. This Is As Natural And Transparent A Language For Higher Category Theory As The Language Of Sheaves Is For Algebraic Geometry, Or Vector Spaces For Linear Algebra. It Is Introduced Carefully, Then Used To Give Simple Descriptions Of A Variety Of Higher Categorical Structures. In Particular, One Possible Definition Of N-category Is Discussed In Detail, And Some Common Aspects Of Other Possible Definitions Are Established. This Is The First Book On The Subject And Lays Its Foundations. It Will Appeal To Both Graduate Students And Established Researchers Who Wish To Become Acquainted With This Modern Branch Of Mathematics. Tom Leinster. Includes Bibliographical References (p. 411-416) And Index. Cover......Page 1 LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 298......Page 2 Higher Operads, Higher Categories......Page 4 Copyright......Page 5 Contents......Page 8 Diagram of interdependence......Page 12 Acknowledgements......Page 13 Introduction......Page 16 Motivation for topologists......Page 22 The very rough idea......Page 23 n-categories......Page 24 Stabilization......Page 29 Part I. Background......Page 36 1 Classical categorical structures......Page 38 1.1 Categories......Page 39 1.2 Monoidal categories......Page 46 1.3 Enrichment......Page 54 1.4 Strict n-categories......Page 55 1.5 Bicategories......Page 64 2 Classical operads and multicategories......Page 72 2.1 Classical multicategories......Page 73 2.2 Classical operads......Page 82 2.3 Further theory......Page 96 3 Notions of monoidal category......Page 106 3.1 Unbiased monoidal categories......Page 107 3.2 Algebraic notions of monoidal category......Page 116 3.3 Non-algebraic notions of monoidal category......Page 125 3.4 Notions of bicategory......Page 136 Part II. Operads......Page 144 4 Generalized operads and multicategories: basics......Page 146 4.1 Cartesian monads......Page 147 4.2 Operads and multicategories......Page 153 4.3 Algebras......Page 165 5 Example: fc-multicategories......Page 173 5.1 fc-multicategories......Page 174 5.2 Weak double categories......Page 182 5.3 Monads, monoids and modules......Page 189 6 Generalized operads and multicategories: further theory......Page 197 6.1 More on monads......Page 198 6.2 Multicategories via monads......Page 203 6.3 Algebras via fibrations......Page 208 6.4 Algebras via endomorphisms......Page 211 6.5 Free multicategories......Page 215 6.6 Structured categories......Page 217 6.7 Change of shape......Page 221 6.8 Enrichment......Page 228 7 Opetopes......Page 231 7.1 Opetopes......Page 232 7.2 Categories of pasting diagrams......Page 240 7.3 A category of trees......Page 245 7.4 Opetopic sets......Page 255 7.5 Weak n-categories: a sketch......Page 263 7.6 Many in, many out......Page 270 Part III. n-categories......Page 276 8 Globular operads......Page 278 8.1 The free strict ?-category monad......Page 279 8.2 Globular operads......Page 285 8.3 Algebras for globular operads......Page 289 9 Adefinition of weak n-category......Page 294 9.1 Contractions......Page 295 9.2 Weak ω-categories......Page 299 9.3 Weak n-categories......Page 305 9.4 Weak 2-categories......Page 318 10 Other definitions of weak n-category......Page 324 10.1 Algebraic definitions......Page 325 10.2 Non-algebraic definitions......Page 333 Appendices......Page 346 A.1 Commutative monoids......Page 348 A.2 Symmetric multicategories......Page 351 B Coherence for monoidal categories......Page 357 B.1 Unbiased monoidal categories......Page 360 B.2 Classical monoidal categories......Page 364 C Special cartesian monads......Page 370 C.1 Operads and algebraic theories......Page 371 C.2 Familially representable monads on Set......Page 377 C.3 Familially representable monads on presheaf categories......Page 387 C.4 Cartesian structures from symmetric structures......Page 394 D Freemulticategories......Page 404 D.1 Proofs......Page 405 E.1 The equivalence......Page 411 F Freestrict n-categories......Page 414 F.1 Free enriched categories......Page 416 F.2 Free n- and ?-categories......Page 418 G.1 The proof......Page 422 References......Page 426 Index of notation......Page 432 Index......Page 438 Higher-dimensional category theory draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. The heart of this book is the language of generalized operads. This is a natural and transparent language for higher category theory which the author introduces carefully and methodically. This is the first book on the subject and lays its foundations. It will appeal to both graduate students and established researchers who wish to become acquainted with this modern branch .. Category theory has experienced a resurgence in popularity recently because of new links with topology and mathematical physics. This book provides a clearly written account of higher order category theory and presents operads and multicategories as a natural language for its study. Tom Leinster has included necessary background material and applications as well as appendices containing some of the more technical proofs that might have disrupted the flow of the text. Higher-dimensional category theory draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations, appealing to graduate students and researchers who wish to become acquainted with this modern branch of mathematics. Higher-dimensional category theory is the study of a zoo of exotic structures: operads, n-categories, multicategories, monoidal categories, braided monoidal categories, and more. Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics