Categorical Homotopy Theory by Emily Riehl ISBN 1107048451 978-1107048454
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ISBN 10: 1107048451
ISBN 13: 978-1107048454
Author: Emily Riehl
This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory – Quillen’s model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.
Table of contents:
Part I: Derived Functors and Homotopy (Co)limits
Chapter 1: All Concepts Are Kan Extensions
Chapter 2: Derived Functors via Deformations
Chapter 3: Basic Concepts of Enriched Category Theory
Chapter 4: The Unreasonably Effective (Co)bar Construction
Chapter 5: Homotopy Limits and Colimits – The Theory
Chapter 6: Homotopy Limits and Colimits – The Practice
Part II: Enriched Homotopy Theory
Chapter 7: Weighted Limits and Colimits
Chapter 8: Categorical Tools for Homotopy (Co)limit Computations
Chapter 9: Weighted Homotopy Limits and Colimits
Chapter 10: Derived Enrichment
Part III: Model Categories and Weak Factorization Systems
Chapter 11: Weak Factorization Systems in Model Categories
Chapter 12: Algebraic Perspectives on the Small Object Argument
Chapter 13: Enriched Factorizations and Enriched Lifting Properties
Chapter 14: A Brief Tour of Reedy Category Theory
Part IV: Quasi-Categories
Chapter 15: Preliminaries on Quasi-Categories
Chapter 16: Simplicial Categories and Homotopy Coherence
Chapter 17: Isomorphisms in Quasi-Categories
Chapter 18: A Sampling of 2-Categorical Aspects of Quasi-Category Theory
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Emily Riehl,Categorical,Homotopy Theory