Elements of ?-Category Theory

This book develops the theory of infinite-dimensional categories by studying the universe, or ?-cosmos, in which they live.

Emily Riehl (Author), Dominic Verity (Author)

9781108837989, Cambridge University Press

Hardback, published 10 February 2022

770 pages
23.4 x 15.5 x 4.5 cm, 1.21 kg

'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

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.

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.

Subject Areas: Topology [PBP], Mathematical logic [PBCD]