Elements of ∞-Category Theory
Elements of ∞-Category Theory
$89.00
USDThe 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.
- Presents a first introduction to infinity categories for students and researchers with a strong foundation in classical 1-category theory
- Takes a model-independent approach, meaning that theorems proven with any model would apply to all models
- Provides exercises at the end of each section, allowing the reader to test their understanding of the material
- Includes four expository appendices, which provide background material that might not be familiar to all readers.
Awards
Winner, 2023 PROSE Mathematics and Statistics Award, Association of American Publishers
Reviews & endorsements
‘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
Product details
April 2022Hardback
9781108837989
770 pages
234 × 155 × 45 mm
1.21kg
Available
Table of Contents
- 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.
