Homotopy Theory of Higher Categories: From Segal Categories to n-Categories and Beyond

This introductory account of commutative algebra is aimed at students with a background only in basic algebra. Professor Sharp's book provides a good foundation from which the reader can proceed to more advanced works in commutative algebra or algebraic geometry. This new edition contains additional chapters on regular sequences and on Cohen-Macaulay rings Develops a full set of homotopical algebra techniques dedicated to the study of higher categories. Cover; Homotopy Theory of Higher Categories; NEW MATHEMATICAL MONOGRAPHS; Title; Copyright; Contents; Preface; Acknowledgements; PART I Higher categories; 1 History and motivation; 2 Strict n-categories; 2.1 Godement relations: the Eckmann-Hilton argument; 2.2 Strict n-groupoids; 2.3 The need for weak composition; 2.4 Realization functors; 2.5 n-groupoids with one object; 2.6 The case of the standard realization; 2.7 Nonexistence of strict 3-groupoids of 3-type S2; 3 Fundamental elements of n-categories; 3.1 A globular theory; 3.2 Identities; 3.3 Composition, equivalence and truncation. 3.4 Enriched categories3.5 The (n + 1)-category of n-categories; 3.6 Poincaré n-groupoids; 3.7 Interiors; 3.8 The case n = 8; 4 Operadic approaches; 4.1 May's delooping machine; 4.2 Baez-Dolan's definition; 4.3 Batanin's definition; 4.4 Trimble's definition and Cheng's comparison; 4.5 Weak units; 4.6 Other notions; 5 Simplicial approaches; 5.1 Strict simplicial categories; 5.2 Segal's delooping machine; 5.3 Segal categories; 5.3.1 Equivalences of Segal categories; 5.3.2 Segal's theorem; 5.3.3 (8, 1)-categories; 5.3.4 Strictification and Bergner's comparison result. 5.3.5 Enrichment over monoidal structures5.3.6 Iteration; 5.4 Rezk categories; 5.5 Quasicategories; 5.6 Going between Segal categories and n-categories; 6 Weak enrichment over a cartesian model category: an introduction; 6.1 Simplicial objects in M; 6.2 Diagrams over?X; 6.3 Hypotheses on M; 6.4 Precategories; 6.5 Unitality; 6.6 Rectification of?X-diagrams; 6.7 Enforcing the Segal condition; 6.8 Products, intervals and the model structure; PART II Categorical preliminaries; 7 Model categories; 7.1 Lifting properties; 7.2 Quillen's axioms; 7.2.1 Quillen adjunctions; 7.3 Left properness. 7.4 The Kan-Quillen model category of simplicial sets7.4.1 Generating sets; 7.5 Homotopy liftings and extensions; 7.6 Model structures on diagram categories; 7.6.1 Some adjunctions; 7.6.2 Injective and projective diagram structures; 7.6.3 Reedy diagram structures; 7.7 Cartesian model categories; 7.8 Internal Hom; 7.9 Enriched categories; 7.9.1 Interpretation of enriched categories as functors?°X?S; 7.9.2 The enriched category associated to a cartesian model category; 8 Cell complexes in locally presentable categories; 8.0.1 Universes and set theory; 8.1 Locally presentable categories. 8.1.1 Miscellany about limits and colimits8.2 The small object argument; 8.3 More on cell complexes; 8.3.1 Cell complexes in presheaf categories; 8.3.2 Inclusions of cell complexes; 8.3.3 Cutoffs; 8.3.4 The filtered property for subcomplexes; 8.4 Cofibrantly generated, combinatorial and tractable model categories; 8.5 Smith's recognition principle; 8.6 Injective cofibrations in diagram categories; 8.7 Pseudo-generating sets; 9 Direct left Bousfield localization; 9.1 Projection to a subcategory of local objects; 9.2 Weak monadic projection; 9.2.1 Monadic projection; 9.2.2 The weak version