*Doctoral Course in Mathematics*

*Winter Semester 2003/04*

Prof. Kathryn Hess Bellwald

SB-IMA

Course Description

Course Schedule

Course Outline (new!)

Bibliography (new!)

Exercise Sets (new!)

Further Course Downloads

Course Schedule

Course Outline (new!)

Bibliography (new!)

Exercise Sets (new!)

Further Course Downloads

Model category theory, first developed in the late 1960's by Quillen,
has become very popular among algebraic topologists and algebraic
geometers in the past decade. A *model category* is a category
endowed with three distinguished classes of morphisms -- *fibrations,
cofibrations *and* weak equivalences*-- satisfying axioms that
are properties of the topological category and its usual fibrations,
cofibrations and weak equivalences. In any model category there
is a notion of *homotopy* of morphisms, based on the definition
of homotopy of continuous maps.

The purpose of this course is to provide an introduction
to model category theory, complete with numerous examples of model
categories and their applications in algebra and topology.

Complete reference details for Hirschhorn's book just
added to bibliography!

**Starting date: **October 23, 2003

**Schedule: **Thursdays, from **8:30** to **11:00**

**Room: **MA/30 (EXCEPT 11.12.03 and
08.01.04, when we will be in CM100.)

Course Outline(will evolve as the semester progresses... :-) )

I. Introduction

A. The goals of homotopical algebra

B. The homotopy theory of topological spaces

C. Basic category theory

II. Elementary model category theory

A. Definition, examples and properties of model categories

B. The homotopy relation in a model category

C. The homotopy category of a model category

D. Derived functors, Quillen pairs and Quillen equivalences

E. The category of simplicial sets

III. Model categories with further structure

A. Cofibrantly generated model categories

B. Created structures

C. Monoidal model categories

E.B. Curtis, *Simplicial homotopy
theory, *Advances in Math. **6** (1971) 107-209.

W.G. Dwyer, P. Hirschhorn, D. Kan and J. Smith, *Homotopy
Limit Functors on Model Categories and Homotopical Categories*,
preprint. (Can be downloaded here)

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

Y. Félix, S. Halperin, and J.-C. Thomas, *Rational Homotopy Theory*, Graduate Texts in Mathematics **205**, Springer-Verlag, 2001.

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

Philipp S. Hirschhorn, *Model Categories
and their Localizations, * Mathematical Surveys and Monographs **99**, American Mathematical Society, 2003.

K. Hess, *Model categories in algebraic topology*, Applied Categorical Structures **10** (2002) 195-220. (Download)

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

A. Joyal and M. Tierney, *Strong stacks and classifying spaces,* Category Theory (Como, 1990), Lecture Notes in Mathematics **1488**, Springer-Verlag, 1991, 213-236.

J.P. May, *Simplicial Objects in Algebraic Topology*, Chicago Lectures in Mathematics, University of Chicago Press, 1992.

Série 1 Série 2 Série 3 Série 4 Série 5 Série 6 Série 7 Série 8 Série 9 Série 10 Série 11 Série 11

Définition d'une catégorie modèle

Factorisation de morphismes de complexes de chaînes