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.

'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

• 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.