Homotopy Theory Of Higher Categories From Segal Categories To Ncategories And Beyond Carlos Simpson by Carlos Simpson 9781139187633, 9781283378406, 1139187635, 128337840X instant download after payment.

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