nLab Higher categories and homotopical algebra

This page is about the book:

• Denis-Charles Cisinski:

  
  

  Higher Categories and Homotopical Algebra

  
  

  Cambridge Studies in Advanced Mathematics 180

  Cambridge University Press (2019)

  
  

  ISBN: 9781108588737

  doi:10.1017/9781108588737

  pdf

  MAA review

on $(\infty,1)$-category theory and homotopical algebra.

Contents

1. Prelude

1.1. Presheaves

1.2. The category of simplicial sets

1.3. Cellular filtrations

1.4. Nerves

1.5. Definition of ∞-categories

1.6. The Boardmann-Vogt construction

2. Basic homotopical algebra

2.1. Factorisation systems

2.2. Model Categories

2.3. Derived functors

2.4. Model structures ex nihilo

2.5. Absolute weak equivalences

3. The homotopy theory of ∞-categories

3.1. Kan fibrations and the Kan-Quillen model structure

3.2. Inner anodyne extensions

3.3. The Joyal model category structure

3.4. Left or right fibrations, joins and slices

3.5. Invertible natural transformations

3.6. ∞-categories as fibrant objects

3.7. The Boardman-Vogt construction, revisited

3.8. Serre’s long exact sequence

3.9. Fully faithful and essentially surjective functors

4. Presheaves: externally

4.1. Catégories fibrées en ∞-groupoïdes

4.2. Mapping spaces as fibres of slices

4.3. Final objects

4.4. Grothendieck base change formulas and Quillen’s Theorem A

4.5. Fully faithful and essentially surjective functors, revisited

4.6. Locally constant functors and Quillen’s Theorem B

5. Presheaves: internally

5.1. Minimal fibrations

5.2. The universal left fibration

5.3. Homotopy classification of left fibrations

5.4. Rectification of morphisms

5.5. Bivariant model category structures

5.6. The twisted diagonal

5.7. Locally small ∞-categories

5.8. The Yoneda Lemma

6. Adjoints, limits and Kan extensions

6.1. Adjoints

6.2. Limits and colimits

6.3. Extensions of functors by colimits

6.4. Kan extensions

6.5. The Cartesian product

6.6. Fibre products

6.7. Duality

7. Homotopical algebra

7.1. Localisation

7.2. Calculus of fractions

7.3. Constructions of limits

7.4. Finite direct diagrams

7.5. Derived functors

7.6. Equivalences of ∞-categories with finite limits

7.7. Homotopy completeness

7.8. The homotopy hypothesis

7.9. Homotopy limits as limits

7.10. Mapping spaces in locally small localisations

7.11. Presentable ∞-categories

Bibliography

Notations

Index

category: reference