Categorical Homotopy Theory summary

Abstract Foundations

In Categorical Homotopy Theory by Emily Riehl, we delve into the abstract foundations of homotopy theory from a categorical standpoint. Riehl begins by introducing the concept of a model category, which is a category equipped with three distinguished classes of morphisms (called weak equivalences, fibrations, and cofibrations) satisfying certain axioms. These axioms are designed to capture the essential properties of homotopy theory and provide a framework for studying homotopy-theoretic constructions in a categorical setting.

Riehl then explores the relationship between model categories and the more familiar homotopy theory of topological spaces. She demonstrates how model categories can be used to define and study homotopy-theoretic concepts such as homotopy groups, homotopy colimits, and homotopy limits in a more general context than that of topological spaces.

Applications and Generalizations

In the next part of the book, Riehl discusses various applications and generalizations of the theory of model categories. She introduces the notion of a Quillen adjunction, a pair of functors between model categories that preserves the distinguished classes of morphisms up to weak equivalence. Quillen adjunctions play a fundamental role in relating different homotopy theories and in constructing new model categories from existing ones.

Riehl then explores the concept of localization of a model category with respect to a class of morphisms. This construction, which generalizes the process of inverting homotopy equivalences in the category of topological spaces, allows us to pass from a given model category to a new one where certain morphisms are inverted. This process provides a powerful tool for studying homotopy-theoretic properties of objects in a given model category.

Enriched Homotopy Theory

The latter part of Categorical Homotopy Theory focuses on enriched homotopy theory, a generalization of classical homotopy theory in which the underlying category is enriched over another category, typically the category of topological spaces. Enriched homotopy theory provides a framework for studying homotopy-theoretic phenomena in a more general context than that of topological spaces, including in the context of higher category theory and higher algebra.

Riehl introduces the notion of an enriched model category, which is a model category enriched over another model category. She discusses various examples of enriched model categories and explores the relationship between enriched model categories and classical model categories. She also introduces the concept of a model ∞-category, which is an ∞-category equipped with a model structure, and discusses its applications in higher category theory and homotopy theory.

Homotopy Coherence and Higher Homotopy Theory

In the final part of the book, Riehl delves into the theory of ∞-categories and higher homotopy theory. She introduces the notion of a quasicategory, which is a weak higher-dimensional analogue of a category, and discusses its applications in higher category theory and homotopy theory. She also introduces the concept of an ∞-operad, which is a higher-dimensional analogue of an operad, and discusses its applications in higher homotopy theory and algebraic topology.

Riehl concludes by discussing the relationship between classical homotopy theory and higher homotopy theory and by exploring the prospects for developing a coherent theory of higher homotopy coherence. In summary, Categorical Homotopy Theory provides a comprehensive and in-depth exploration of the theory of model categories, enriched homotopy theory, and higher homotopy theory, and their applications in algebraic topology and higher category theory.