Introduction to Infinity-Categories (Compact Textbooks in Mathematics)

✍ Scribed by Markus Land

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 300
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This textbook is an introduction to the theory of infinity-categories, a tool used in many aspects of modern pure mathematics. It treats the basics of the theory and supplies all the necessary details while leading the reader along a streamlined path from the basic definitions to more advanced results such as the very important adjoint functor theorems.

The book is based on lectures given by the author on the topic. While the material itself is well-known to experts, the presentation of the material is, in parts, novel and accessible to non-experts. Exercises complement this textbook that can be used both in a classroom setting at the graduate level and as an introductory text for the interested reader.

✦ Table of Contents

Preface
Acknowledgments
Contents
1 Categories, Simplicial Sets, and Infinity-Categories
Motivation and Overview
1.1 Categories and Simplicial Sets
1.2 ∞-Categories
1.3 Anodyne Maps and Fibrations
1.4 Joins and Slices
2 Joyal's Theorem, Applications, and Dwyer–Kan Localizations
2.1 Joyal's Special Horn Lifting Theorem
2.2 Pointwise Criterion for Natural Equivalences
2.3 Fully Faithful and Essentially Surjective Functors
2.4 Localizations
2.5 Fat Joins, Fat Slices and Mapping Spaces
3 (Co)Cartesian Fibrations and the Construction of Functors
3.1 (Co)Cartesian Fibrations
3.2 Marked Simplicial Sets and Marked Anodyne Maps
3.3 Straightening-Unstraightening
4 Limits, Colimits, and Quillen's Theorem A
4.1 Terminal and Initial Objects
4.2 The Yoneda Lemma for ∞-Categories
4.3 Limits and Colimits
4.4 Cofinal and Coinitial Functors
5 Adjunctions and Adjoint Functor Theorems
5.1 Adjunctions
5.2 Adjoint Functor Theorems
A Exercises
Bibliography