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**This course will start in Week 6 of the second semester.**

**Aims:** This course will provide an introduction to *abstract homotopy theory.* The central idea is that of a *model category* which provides an axiomatic framework in which to do homotopy theory. After discussing the axioms for a model category and suitable types of functors between them, we will consider some examples coming from topology and also other areas such as algebra and functional analysis.

**Outline of sessions**

1 & 2: The axioms for a model category and some basic results. The homotopy/derived category. Derived functors and Quillen adjunctions.

3 & 4: Some important examples: Categories of spaces, simplicial sets, stable module category of a ring, chain complexes (over a ring, in an abelian category,...), simplicial groups/abelian groups/commutative rings.

5 & 6: Further examples perhaps with some contributed by participants.

**Prerequisites: **Familiarity with basic notions of category theory (e.g., exposure to use of diagram chasing and functoriality in subjects such as algebraic topology, homological algebra and algebraic geometry) and basic ideas of homotopy theory (e.g., fibrations, cofibrations, homotopy, homotopy invariant functors).

**Some reading material**

M. Hovey, *Model Categories*, Mathematical Surveys and Monographs 63, 1999, American Mathematical Society. http://bookstore.ams.org/surv-63-s/

P. Hirschorn, *Model Categories and their localisations*, Mathematical Surveys and Monographs 99, 2003, American Mathematical Society. http://bookstore.ams.org/surv-99-s/

*For students wanting to obtain credit for taking the course, I suggest studying an example in detail and giving a talk and/or writing an essay would be suitable. Please discuss this at the start of classes if you wish to do this.*

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ICMS

Fifth Floor

Bayes Centre

47 Potterrow

Edinburgh

EH8 9BT

Tel : 0131 650 9816

Fax : 0131 651 4381

Mail : johanna.mcbryde@icms.org.uk