Homotopical Algebra

Doctoral Course in Mathematics – Winter Semester 2003/04

Prof. Kathryn Hess Bellwald

Course Description

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. 

Course Schedule

  • 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

  1. Introduction
    1. The goals of homotopical algebra
    2. The homotopy theory of topological spaces
    3. Basic category theory
  2. Elementary model category theory
    1. Definition, examples and properties of model categories
    2. The homotopy relation in a model category
    3. The homotopy category of a model category
    4. Derived functors, Quillen pairs and Quillen equivalences
    5. The category of simplicial sets
  3. Model categories with further structure
    1. Cofibrantly generated model categories
    2. Created structures
    3. 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.