Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- PART I HIGHER CATEGORIES
- PART II CATEGORICAL PRELIMINARIES
- 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
- PART V HIGHER CATEGORY THEORY
- Epilogue
- References
- Index
12 - Weak equivalences
from PART III - GENERATORS AND RELATIONS
Published online by Cambridge University Press: 25 October 2011
- Frontmatter
- Contents
- Preface
- Acknowledgements
- PART I HIGHER CATEGORIES
- PART II CATEGORICAL PRELIMINARIES
- 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
- PART V HIGHER CATEGORY THEORY
- Epilogue
- References
- Index
Summary
This chapter continues the study of weakly enriched categories using Segal's method. We use the model category for algebraic theories, developed in the previous chapter, to get model structures for Segal precategories on a fixed set of objects. This structure will be studied in detail later, to deal with the passage from a Segal precategory to the Segal category it generates.
Then we consider the full category of Segal precategories, with movable sets of objects, giving various definitions and notations. Constructing a model structure in that case is the main subject of the subsequent chapters.
The reader will note that this division of the global argument into two pieces, was present already in Dwyer–Kan's treatment of the model category for simplicial categories. They discussed the model category for simplicial categories on a fixed set of objects in a series of papers (Dwyer and Kan [101, 102, 103]); but it wasn't until some time later in their unpublished manuscript with Hirschhorn [99], which subsequently became Dwyer et al. [100], and then Bergner's paper [39] that the global case was treated.
For the theory of weak enrichment following Segal's method, the corresponding division and introduction of the notion of left Bousfield localization for the first part was suggested in Barwick's thesis [16].
Assume throughout that M is a tractable left proper cartesian model category. See Section 7.7 for an explanation and first consequences of the cartesian condition.
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- Chapter
- Information
- Homotopy Theory of Higher CategoriesFrom Segal Categories to n-Categories and Beyond, pp. 275 - 296Publisher: Cambridge University PressPrint publication year: 2011