Homotopy limit functors on model categories and homotopical categories / William G. Dwyer [and three others].

Format:

Book

Author/Creator:

Dwyer, William G., 1947- author.

Series:

Mathematical surveys and monographs ; volume 113.

Mathematical surveys and monographs, 0076-5376 ; volume 113

Language:

English

Subjects (All):

Homotopy theory.

Physical Description:

1 online resource (193 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2004]

Summary:

The purpose of this monograph, which is aimed at the graduate level and beyond, is to obtain a deeper understanding of Quillen's model categories. A model category is a category together with three distinguished classes of maps, called weak equivalences, cofibrations, and fibrations. Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic $K$-theory and algebraic geometry. The authors' approach is to define the notion of a homotopical category, which is more general than that of a model category, and to consider model categories as special cases of this. A homotopical category is a category with only a single distinguished class of maps, called weak equivalences, subject to an appropriate axiom. This enables one to define ``homotopical'' versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, cocompleteness and completeness, adjunctions, Kan extensions, and universal properties. There are two essentially self-contained parts, and part II logically precedes part I. Part II defines and develops the notion of a homotopical category and can be considered as the beginnings of a kind of ``relative'' category theory. The results of part II are used in part I to obtain a deeper understanding of model categories. The authors show in particular that model categories are homotopically cocomplete and complete in a sense stronger than just the requirement of the existence of small homotopy colimit and limit functors. A reader of part II is assumed to have only some familiarity with the above-mentioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories.

Contents:

""Contents""; ""Preface""; ""Part I. Model Categories""; ""Chapter I. An Overview""; ""1. Introduction""; ""2. Slightly unconventional terminology""; ""3. Problems involving the homotopy category""; ""4. Problem involving the homotopy colimit functors""; ""5. The emergence of the current monograph""; ""6. A preview of part II""; ""Chapter II. Model Categories and Their Homotopy Categories""; ""7. Introduction""; ""8. Categorical and homotopical preliminaries""; ""9. Model categories""; ""10. The homotopy category""; ""11. Homotopical comments""; ""Chapter III. Quillen Functors""

""12. Introduction""""13. Homotopical uniqueness""; ""14. Quillen functors""; ""15. Approximations""; ""16. Derived adjunctions""; ""17. Quillen equivalences""; ""18. Homotopical comments""; ""Chapter IV. Homotopical Cocompleteness and Completeness of Model Categories""; ""19. Introduction""; ""20. Homotopy colimit and limit functors""; ""21. Homotopical cocompleteness and completeness""; ""22. Reedy model categories""; ""23. Virtually coflbrant and fibrant diagrams""; ""24. Homotopical comments""; ""Part II. Homotopical Categories""; ""Chapter V. Summary of Part II""; ""25. Introduction""

""26. Homotopical categories""""27. The hom-sets of the homotopy categories""; ""28. Homotopical uniqueness""; ""29. Deformable functors""; ""30. Homotopy colimit and limit functors and homotopical ones""; ""Chapter VI. Homotopical Categories and Homotopical Functors""; ""31. Introduction""; ""32. Universes and categories""; ""33. Homotopical categories""; ""34. A colimit description of the hom-sets of the homotopy category""; ""35. A Grothendieck construction""; ""36. 3-arrow calculi""; ""37. Homotopical uniqueness""; ""38. Homotopically initial and terminal objects""

""Chapter VII. Deformable Functors and Their Approximations""""39. Introduction""; ""40. Deformable functors""; ""41. Approximations""; ""42. Compositions""; ""43. Induced partial adjunctions""; ""44. Derived adjunctions""; ""45. The Quillen condition""; ""Chapter VIII. Homotopy Colimit and Limit Functors and Homotopical Ones""; ""46. Introduction""; ""47. Homotopy colimit and limit functors""; ""48. Left and right systems""; ""50. Homotopical colimit and limit functors""; ""51. Homotopical cocompleteness and completeness (general case)""; ""Index""; ""A""; ""C""; ""D""; ""E""; ""G""; ""I""

""K""""L""; ""M""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Z""; ""Bibliography""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 181) and index.

Description based on print version record.

ISBN:

1-4704-1340-X