Homotopy Theory of Higher Categories: From Segal Categories to n-Categories and Beyond (New Mathematical Monographs, Series Number 19)

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 The Study Of Higher Categories Is Attracting Growing Interest For Its Many Applications In Topology, Algebraic Geometry, Mathematical Physics And Category Theory. In This Highly Readable Book, Carlos Simpson Develops A Full Set Of Homotopical Algebra Techniques And Proposes A Working Theory Of Higher Categories. Starting With A Cohesive Overview Of The Many Different Approaches Currently Used By Researchers, The Author Proceeds With A Detailed Exposition Of One Of The Most Widely Used Techniques: The Construction Of A Cartesian Quillen Model Structure For Higher Categories. The Fully Iterative Construction Applies To Enrichment Over Any Cartesian Model Category, And Yields Model Categories For Weakly Associative N-categories And Segal N-categories. A Corollary Is The Construction Of Higher Functor Categories Which Fit Together To Form The (n+1)-category Of N-categories. The Approach Uses Tamsamani's Definition Based On Segal's Ideas, Iterated As In Pelissier's Thesis Using Modern Techniques Due To Barwick, Bergner, Lurie And Others-- Machine Generated Contents Note: Prologue; Acknowledgements; Part I. Higher Categories: 1. History And Motivation; 2. Strict N-categories; 3. Fundamental Elements Of N-categories; 4. The Need For Weak Composition; 5. Simplicial Approaches; 6. Operadic Approaches; 7. Weak Enrichment Over A Cartesian Model Category: An Introduction; Part Ii. Categorical Preliminaries: 8. Some Category Theory; 9. Model Categories; 10. Cartesian Model Categories; 11. Direct Left Bousfield Localization; Part Iii. Generators And Relations: 12. Precategories; 13. Algebraic Theories In Model Categories; 14. Weak Equivalences; 15. Cofibrations; 16. Calculus Of Generators And Relations; 17. Generators And Relations For Segal Categories; Part Iv. The Model Structure: 18. Sequentially Free Precategories; 19. Products; 20. Intervals; 21. The Model Category Of M-enriched Precategories; 22. Iterated Higher Categories; Part V. Higher Category Theory: 23. Higher Categorical Techniques; 24. Limits Of Weak Enriched Categories; 25. Stabilization; Epilogue; References; Index. Carlos Simpson. "The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others"-- Provided by publisher Part I. Higher Categories: 1. History and motivation; 2. Strict n-categories; 3. Fundamental elements of n-categories; 4. Operadic approaches; 5. Simplicial approaches; 6. Weak enrichment over a cartesian model category: an introduction Part II. Categorical Preliminaries: 7. Model categories; 8. Cell complexes in locally presentable categories; 9. Direct left Bousfield localization Part III. Generators and Relations: 10. Precategories; 11. Algebraic theories in model categories; 12. Weak equivalences; 13. Cofibrations; 14. Calculus of generators and relations; 15. Generators and relations for Segal categories Part IV. The Model Structure: 186 Sequentially free precategories; 17. Products; 18. Intervals; 19. The model category of M-enriched precategories Part V. Higher Category Theory: 20. Iterated higher categories; 21. Higher categorical techniques; 22. Limits of weak enriched categories; 23. Stabilization.