This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others.

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.