Table Of Content

Half-title Page: i Series information Page: ii Title page Page: iii Copyright information Page: iv Dedication Page: v Contents Page: vii Preface Page: ix 1 Prelude Page: 1 1.1 Presheaves Page: 1 1.2 The Category of Simplicial Sets Page: 7 1.3 Cellular Filtrations Page: 9 1.4 Nerves Page: 14 1.5 Definition of ∞-Categories Page: 19 1.6 The Boardman–Vogt Construction Page: 21 2 Basic Homotopical Algebra Page: 27 2.1 Factorisation Systems Page: 28 2.2 Model Categories Page: 33 2.3 Derived Functors Page: 41 2.4 Model Structures ex Nihilo Page: 50 2.5 Absolute Weak Equivalences Page: 68 3 The Homotopy Theory of ∞-Categories Page: 72 3.1 Kan Fibrations and the Kan–Quillen Model Structure Page: 73 3.2 Inner Anodyne Extensions Page: 88 3.3 The Joyal Model Category Structure Page: 92 3.4 Left or Right Fibrations, Joins and Slices Page: 98 3.5 Invertible Natural Transformations Page: 106 3.6 ∞-Categories as Fibrant Objects Page: 112 3.7 The Boardman–Vogt Construction, Revisited Page: 116 3.8 Serre’s Long Exact Sequence Page: 120 3.9 Fully Faithful and Essentially Surjective Functors Page: 130 4 Presheaves: Externally Page: 136 4.1 Catégories Fibrées en ∞-Groupoïdes Page: 136 4.2 Mapping Spaces as Fibres of Slices Page: 144 4.3 Final Objects Page: 149 4.4 Grothendieck Base Change Formulas and Quillen’s Theorem A Page: 155 4.5 Fully Faithful and Essentially Surjective Functors, Revisited Page: 172 4.6 Locally Constant Functors and Quillen’s Theorem B Page: 175 5 Presheaves: Internally Page: 183 5.1 Minimal Fibrations Page: 184 5.2 The Universal Left Fibration Page: 195 5.3 Homotopy Classification of Left Fibrations Page: 204 5.4 Rectification of Morphisms Page: 216 5.5 Bivariant Model Category Structures Page: 225 5.6 The Twisted Diagonal Page: 237 5.7 Locally Small ∞-Categories Page: 242 5.8 The Yoneda Lemma Page: 248 6 Adjoints, Limits and Kan Extensions Page: 260 6.1 Adjoints Page: 261 6.2 Limits and Colimits Page: 274 6.3 Extensions of Functors by Colimits Page: 282 6.4 Kan Extensions Page: 290 6.5 The Cartesian Product Page: 296 6.6 Fibre Products Page: 304 6.7 Duality Page: 312 7 Homotopical Algebra Page: 316 7.1 Localisation Page: 319 7.2 Calculus of Fractions Page: 328 7.3 Constructions of Limits Page: 339 7.4 Finite Direct Diagrams Page: 354 7.5 Derived Functors Page: 366 7.6 Equivalences of ∞-Categories with Finite Limits Page: 382 7.7 Homotopy Completeness Page: 392 7.8 The Homotopy Hypothesis Page: 396 7.9 Homotopy Limits as Limits Page: 402 7.10 Mapping Spaces in Locally Small Localisations Page: 409 7.11 Presentable ∞-Categories Page: 413 Bibliography Page: 421 Notation Page: 426 Index Page: 428

Description:

This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.