Homotopical Algebra
Graduate course
School of Mathematics, University of Leeds
Dates: Wednesday, 11am1pm, 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 higherdimensional 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
higherdimensional 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.
Bibliography.
 D. G. Quillen, Homotopical Algebra,
Lecture Notes in Mathematics 43, SpringerVerlag
1967
 S. Mac Lane, Categories for the Working
Mathematician, 2nd edition, Springer, 1998.
 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.
 M. Hovey, Model categories, American
Mathematical Society, 1999.
 P. Hirschhorn, Model categories and their
localizations, American Mathematical
Society, 2009.
 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:
 Joyal's
CatLab
 nLab
Scanned lecture notes:
 Lecture 1 (January 29th,
2014). Categorytheoretic preliminaries.
Lifting properties. Weak factorisation systems.
Example: the injectionsurjection 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.
