The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Winner, 2023 PROSE Mathematics and Statistics Award, Association of American Publishers

‘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

‘Emily and Dom have done what many thought impossible: they have written an introductory text on a model-independent approach to higher category theory. This self-contained text is ideal for both end-users and architects of higher category theory. Every page is bursting at the seams with gorgeous insights and the refreshingly candid delight the authors take in their subject.’

Clark Barwick - University of Edinburgh

‘This remarkable book starts with the premise that it should be possible to study ∞-categories armed only with the tools of 2-category theory. It is the result of the authors’ decade-long collaboration, and they have poured into it all their experience, technical brilliance, and expository skill. I’m sure I’ll be turning to it for many years to come.’

Steve Lack - Macquarie University