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 deﬁn- able when the appropriate diagram categories admit compatible model structures 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 unnecessary. 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 deﬁni- tions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence. Emily Riehl is a Benjamin Peirce Fellow in the Department of Mathematics at Harvard University and a National Science Foundation Mathematical Sciences Post- doctoral Research Fellow. www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-1-107-04845-4 - Categorical Homotopy Theory Emily Riehl Frontmatter More information