Categorical Homotopy Theory

This book offers a unified approach to homotopy theory, simplifying concepts like derived functors and model categories. Categorical Homotopy Theory is essential for deepening understanding.

This insightful exploration of homotopy theory provides a categorical perspective that aids in consolidating and simplifying key concepts such as derived functors, homotopy limits and colimits, and model categories. In Categorical Homotopy Theory, the author, Emily Riehl, develops abstract homotopy theory with a focus on practical examples, making complex ideas more accessible. The first part of the book contrasts two common approaches to understanding homotopy (co)limits: as derived functors within compatible model structures or through specific formulas applicable to certain scenarios.

Riehl effectively unifies these competing perspectives, demonstrating that the model structures on diagram categories are not as crucial as previously thought. She elucidates that homotopy (co)limits should be regarded as a special case of weighted (co)limits, a fundamental topic in enriched category theory. The second part of the book further delves into this topic, distinguishing between categorical and homotopical arguments, providing clarity and depth to the discussion.

In the third part, Riehl addresses Quillen's model categories, which serve as a widely accepted axiomatic framework for homotopy theory. By concentrating on weak factorization systems, she simplifies well-known model categorical lemmas and definitions. The final section introduces quasi-categories and the concept of homotopy coherence, rounding out this comprehensive examination. Overall, Categorical Homotopy Theory serves as a vital resource for anyone looking to deepen their understanding of this intricate field.