Categorical Homotopy Theory

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 rig...

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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.

Part I. Derived Functors and Homotopy (Co)limits:
1. All concepts are Kan extensions
2. Derived functors via deformations
3. Basic concepts of enriched category theory
4. The unreasonably effective (co)bar construction
5. Homotopy limits and colimits: the theory
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Part I. Derived Functors and Homotopy (Co)limits:
1. All concepts are Kan extensions
2. Derived functors via deformations
3. Basic concepts of enriched category theory
4. The unreasonably effective (co)bar construction
5. Homotopy limits and colimits: the theory
6. Homotopy limits and colimits: the practice
Part II. Enriched Homotopy Theory:
7. Weighted limits and colimits
8. Categorical tools for homotopy (co)limit computations
9. Weighted homotopy limits and colimits
10. Derived enrichment
Part III. Model Categories and Weak Factorization Systems:
11. Weak factorization systems in model categories
12. Algebraic perspectives on the small object argument
13. Enriched factorizations and enriched lifting properties
14. A brief tour of Reedy category theory
Part IV. Quasi-Categories:
15. Preliminaries on quasi-categories
16. Simplicial categories and homotopy coherence
17. Isomorphisms in quasi-categories
18. A sampling of 2-categorical aspects of quasi-category theory.
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