Math 9501A: Foundations of Stable Homotopy Theory (Fall 2021)

Essential information


The goal of the course is to understand selected topics from:

Our introduction to algebraic topology beyond Math 9052/4152 will follow a model-categorical approach, based on the category of simplicial sets and the Kan-Quillen model structure thereon, not currently represented in the literature.

For an introduction to model categories, see:

For classical introductions to algebraic topology, see:

  • T. tom Dieck, Algebraic Topology, 2008.
  • P. Goerss, J.F. Jardine, Simplicial Homotopy Theory, 2009.
  • J.P. May, A Concise Course in Algebraic Topology, 1999.
  • C. Weibel, An introduction to homological algebra, 1994.

For an introduction to stable homotopy theory, see:

For an introduction to higher category theory, see:

Course content

The topics will include:

  • Homotopical algebra and simplicial sets.
  • Algebraic topology in simplicial sets.
  • Spectra and stable homotopy theory.
  • Introduction to higher category theory.
  • Stable (∞, 1)-categories.


The final grades will be based on the following components:

  • five assignments: 50%
  • project: 25%
  • presentation: 25%


There will be five assignments, each worth 10% of the final grade. 

Students are allowed to discuss assignment problems, but each student should write their solutions separately.

  • Assignment 1: released Sep 23, due Sep 30 at 1 PM.
  • Assignment 2: released Oct 7, due Oct 14 at 1 PM.
  • Assignment 3: released Oct 21, due Oct 28 at 1 PM.
  • Assignment 4: released Nov 11, due Nov 18 at 1 PM.
  • Assignment 5: released Nov 25, due Dec 2 at 1 PM.


As part of the evaluation, each student will need to write a 10-15 page long document in LaTeX, surveying a topic in stable homotopy theory. The instructor will prepare a list of potential projects, but students are welcome and in fact encouraged to suggest their own topics.

Associated with the project, each student will be required to give a 50-minute presentation during the last week of the course. The presentations cannot use slides, must include a theorem, and a proof of at least one short lemma.

Accommodation and Accesibility

If you are unable to meet a course requirement due to illness or other serious circumstances, you must provide valid medical or supporting documentation to the Academic Counselling Office of your home faculty as soon as possible. If you are a Science student, the Academic Counselling Office of the Faculty of Science is located in WSC 140, and can be contacted at

For further information, please consult the university’s medical illness policy at

If you miss the Final Exam, please contact your faculty’s Academic Counselling Office as soon as you are able to do so. They will assess your eligibility to write the Special Exam (the name given by the university to a makeup Final Exam). You may also be eligible to write the Special Exam if you are in a “Multiple Exam Situation”

Academic Policies

The website for Registrarial Services is

In accordance with policy, centrally administered e-mail account provided to students will be considered the individual’s official university e-mail address. It is the responsibility of the account holder to ensure that e-mail received from the University at his/her official university address is attended to in a timely manner.


Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at this website:

Student Accessibility Services

Western is committed to achieving barrier-free accessibility for all its members, including graduate students. As part of this commitment, Western provides a variety of services devoted to promoting, advocating, and accommodating persons with disabilities in their respective graduate program.

Graduate students with disabilities (for example, chronic illnesses, mental health conditions, mobility impairments) are encouraged to register with Student Accessibility Services, a confidential service designed to support graduate and undergraduate students through their academic program. With the appropriate documentation, the student will work with both SAS and their graduate programs (normally their Graduate Chair and/or Course instructor) to ensure that appropriate academic accommodations to program requirements are arranged.  These accommodations include individual counselling, alternative formatted literature, accessible campus transportation, learning strategy instruction, writing exams and assistive technology instruction.