# Coursebooks

## Homotopical algebra

#### Lecturer(s) :

Hess Bellwald Kathryn

English

#### Summary

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in algebra and topology.

#### Content

1. Category theory
2. Model categories and their homotopy categories
3. Transfer theorems
4. Localizing model categories
5. Monoidal model categories and "brave new algebra"

#### Keywords

Abstract homotopy theory

#### Learning Prerequisites

##### Required courses

Second-year math courses, including Topology.

##### Recommended courses

• Rings and modules
• Algebraic topology

##### Important concepts to start the course

• Necessary concept: homotopy of continuous maps
• Recommended concept: chain homotopy of morphisms between chain complexes

#### Learning Outcomes

By the end of the course, the student must be able to:
• Prove results in category theory involving (co)limits, adjunctions, and Kan extensions
• Prove basic properties of model categories
• Check the model category axioms in important examples
• Apply transfer theorems to establish the existence of model category structures
• Apply Bousfield localization to create model categories with desired weak equivalences
• Compare different model category structures via Quillen pairs
• Transpose results from classical algebra into homotopy-theoretic versions in monoidal model categories
• Check the axioms of a monoidal model category in important cases

#### Transversal skills

• Demonstrate a capacity for creativity.
• Demonstrate the capacity for critical thinking
• Continue to work through difficulties or initial failure to find optimal solutions.

#### Teaching methods

Ex-cathedra lectures, exercises

#### Expected student activities

Handing in weekly exercises to be graded.

#### Assessment methods

Written exam

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

#### Supervision

 Office hours No Assistants Yes Forum Yes

#### Resources

##### Bibliography

• W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of Algebraic Topology, Elsevier, 1995, 73-126. (Article no. 75 here)

• P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics 174, Birkhäuser Verlag, 1999.

• M. Hovey, Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 1999.

### In the programs

• Mathematics - master program, 2020-2021, Master semester 1
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Homotopical algebra
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Mathematics - master program, 2020-2021, Master semester 3
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Homotopical algebra
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2020-2021, Master semester 1
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Homotopical algebra
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2020-2021, Master semester 3
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Homotopical algebra
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14 INM11
14-15
15-16 INM11
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German