Homotopical Algebra
Graduate course
School of Mathematics, University of Leeds

Dates: Wednesday, 11am-1pm, from January 29th to April 2nd 2014 (20 hours)
Location: MALL 2 (unless announced otherwise).

Course outline.  The subject of homotopical algebra originated with Quillen's seminal monograph [1], in which he introduced the notion of a model category and used it to develop an axiomatic approach to homotopy theory. Since then, model categories have become one a very important concept in algebraic topology and have found an increasing number of applications in several areas of pure mathematics. In particular, in recent years they have been used to develop higher-dimensional category theory and to establish new links between mathematical logic and homotopy theory (which have given rise to Voevodsky's Univalent Foundations of Mathematics programme).

The aim of this course is to give an introduction to the theory of model categories.

The course is divided in two parts. The first part will introduce the notion of a model category, discuss some of the main examples (such as the categories of topological spaces, chain complexes and simplicial sets) and describe the fundamental concepts and results of the theory (the homotopy category of a model category, Quillen functors, derived functors, the small object argument, transfer theorems). The second part will deal with more advanced topics and its content will depend on the audience's interests. Possible topics include the axiomatic development of homotopy theory within a model category, homotopy limits and colimits, the interplay between model categories and higher-dimensional categories, and Voevodsky's Univalent Foundations of Mathematics programme.

Prerequisites. Basic concepts of category theory (category, functor, natural transformation, adjoint functors, limits, colimits), as covered in the MAGIC course. The standard reference to review these topics is [2]. Some familiarity with topology.

Literature. For the theory of model categories we will use mainly Dwyer and Spalinski's introductory paper [3] and Hovey's monograph [4]. Other useful references include [5] and [6]. Additional references will be provided during the course depending on the advanced topics that will be treated.


  1. D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967
  2. S. Mac Lane, Categories for the Working Mathematician, 2nd edition, Springer, 1998.
  3. W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elsevier, 1995. A preprint version is available from the Hopf archive.
  4. M. Hovey, Model categories, American Mathematical Society, 1999.
  5. P. Hirschhorn, Model categories and their localizations, American Mathematical Society, 2009.
  6. W. G. Dwyer, P. Hirschhorn, D. M. Kan, J. Smith, Homotopy limit functors on model categories and homotopical categories, American Mathematical Society, 2004.

Additional resources:

  1. Joyal's CatLab
  2. nLab

Scanned lecture notes:

  • Lecture 1 (January 29th, 2014). Category-theoretic preliminaries. Lifting properties. Weak factorisation systems. Example: the injection-surjection weak factorisation system on Set, retracts, closure properties of classes of maps defined by lifting properties.
  • Lecture 2 (February 5th, 2014). Equivalent characterisation of weak factorisation systems. Example: the (equivalence injective on objects, isofibration) weak factorisation system on Cat. Definition of Quillen model structure. Equivalent characterisation of Quillen model structures in terms of weak factorisation system.
  • Lecture 3 (February 12th, 2014) Outline of the Hurewicz model structure on Top. Path spaces, cylinder spaces, mapping path spaces, mapping cylinder spaces. Example: the categorical Quillen model structure on Cat.
  • Lecture 4 (February 19th, 2014) Duality. The dual of a model structure. Cylinder objects. Left homotopy.
  • Lecture 5 (February 26th, 2014) Left homotopy (continued). Path objects. Right homotopy. The homotopy relation.
  • Lecture 6 (March 5th, 2014) Auxiliary results towards the construction of the homotopy category of a model category.
  • Lecture 7 (March 12th, 2014) The homotopy category. Fibrant and cofibrant replacements. The homotopy category as a localisation.
  • Lecture 8, (March 19th, 2014): Homotopy type theory (no lecture notes: see The identity type weak factorisation system and the slides of a talk by André Joyal at the MIT Topology seminar).
  • Lecture 9 (March 26th, 2014). Transfinite composition. Small objects. Weak factorisation systems via the the small object argument.
  • Lecture 10 (April 2nd, 2014). Model structures via the small object argument. Outline of the proof that Top admits a Quillen model structure with weak homotopy equivalences as weak equivalences. Further directions.

This page was last modified on April 2nd, 2014 by Nicola Gambino.