nLab Elements of ∞-Category Theory

This pages compiles material related to the book

• Emily Riehl and Dominic Verity.

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  Elements of ∞-category theory

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  Cambridge Studies in Advanced Mathematics 194

  Cambridge University Press, Cambridge, 2022.

  xix+759 pp.

  doi:10.1017/9781108936880

  ISBN: 978-1-108-83798-9

  pdf

on (∞,1)-category theory formulated via ∞-cosmoi and the homotopy 2-category of (∞,1)-categories (formal $(\infty,1)$-category theory).

Related texts:

• Emily Riehl, Dominic Verity, The 2-category theory of quasi-categories, Advances in Mathematics 280 (2015) 549-642 $[$arXiv:1306.5144, doi:10.1016/j.aim.2015.04.021$]$

• Emily Riehl, Dominic Verity, Infinity category theory from scratch, Higher Structures Vol 4, No 1 (2020) $[$arXiv:1608.05314, pdf$]$

• Emily Riehl, The formal theory of ∞-categories, talk at Categories and Companions Symposium June 8–12, 2021 (video)

Frontmatter

Dedication

Contents

Preface

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

An Interlude On ∞-Cosmology

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

Appendix of Abstract Nonsense

Appendix A - Basic Concepts of Enriched Category Theory

Appendix B - An Introduction to 2-Category Theory

Appendix C - Abstract Homotopy Theory

Appendix of Concrete Constructions

Appendix D - The Combinatorics of (Marked) Simplicial Sets

Appendix E - ∞-Cosmoi Found in Nature

Appendix F - The Analytic Theory of Quasi-Categories

References

Glossary of Notation

Index

category: reference