Elements of ∞-Category Theory

Emily Riehl is an associate professor of mathematics at Johns Hopkins University. She received her PhD from the University of Chicago and was a Benjamin Peirce and NSF postdoctoral fellow at Harvard University. She is the author of Categorical Homotopy Theory (Cambridge, 2014) and Category Theory in Context (2016), and a co-author of Fat Chance: Probability from 0 to 1 (Cambridge, 2019). She and her present co-author have published ten articles over the course of the past decade that develop the new mathematics appearing in this book. Dominic Verity is a professor of mathematics at Macquarie University in Sydney and is a director of the Centre of Australian Category Theory. While he is a leading proponent of 'Australian-style' higher category theory, he received his PhD from the University of Cambridge and migrated to Australia in the early 1990s. Over the years he has pursued a career that has spanned the academic and non-academic worlds, working at times as a computer programmer, quantitative analyst, and investment banker. He has also served as the Chair of the Academic Senate of Macquarie University, the principal academic governance and policy body.

• Part I. Basic ∞-Category Theory: 1. ∞-Cosmoi and their homotopy 2-categories; 2. Adjunctions, limits, and colimits I; 3. Comma ∞-categories; 4. Adjunctions, limits, and colimits II; 5. Fibrations and Yoneda's lemma; 6. Exotic ∞-cosmoi; Part II. The Calculus of Modules: 7. Two-sided fibrations and modules; 8. The calculus of modules; 9. Formal category theory in a virtual equipment; Part III. Model Independence: 10. Change-of-model functors; 11. Model independence; 12. Applications of model independence.

'The book of Riehl and Verity is altogether a pedagogical introduction, a unified presentation and a foundation of higher category theory. The theory of ∞-cosmoi is an elegant way of organising and developing the subject. The extension of category theory to ∞-categories is by itself a miracle, vigorously presented in the book.' André Joyal, Université du Québec à Montréal