Math 266x: Categorical Homotopy Theory

SYLLABUS

Here is the syllabus. If you are planning on taking this course for a grade, please come speak to me in person.

CATEGORICAL PREREQUISITES

Comfort with categorical language and reasoning is an essential prerequisite for this course. Along the way, I hope to convey a bit of the philosophy of category theory, but in class our focus will be elsewhere.

This document, which intends to convey the philosophy and terminology of category theory by exploring its implications in elementary examples, is not quite ready for public consumption. Nonetheless, I've decided to make it available for those who could use a bit of a brush up.

For those who prefer to skip the philosophy and examples and go straight to the definitions, this handout is considerably shorter.

LECTURE NOTES

The course lecture notes have now been published by Cambridge University Press. CUP has graciously allowed me to host a free PDF copy, which can be found here. Note, this version is slightly out of date. The final copyedited version will posted in two years' time.

SUGGESTED READING

This list will expand as the semester progresses.

• William Thurston "On proof and progress in mathematics." arXiv:math/9404236v1
• Michael Shulman "Homotopy limits and colimits and enriched homotopy theory." arXiv:math/0610194v3
• J. Hollender and R.M. Vogt "Modules of topological spaces, applications to homotopy limits and E-infinity structures." Archiv der Mathematik
• Rainer M. Vogt "Convenient categories of topological spaces for homotopy theory." Archiv der Mathematik
• Daniel Dugger "A primer on homotopy colimits." preprint
• Richard Garner "Understanding the small object argument." arXiv:math/0712.0724
• Thomas Athorne "The coalgebraic structure of cell complexes." arXiv:1202.6034v1 [math.CT]
• "A model structure for quasi-categories." short expository note
• Daniel Dugger and David Spivak "Mapping spaces in quasi-categories" especially the appendices
• "On the structure of simplicial categories associated to quasi-categories." journal version here
• Dominic Verity "Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory" arXiv:math/0604414v3 [math.CT]

• Oral 1: State and prove the Yoneda lemma. Due: February 3.
• Problem set 1: 2.2, 2.16, 2.18, 2.26, 2.33, 3.29, 4.32, 4.58, 4.60. Due: February 7 in class.
• Oral 2: Compute a homotopy limit that we didn't discuss in class using the cobar construction: eg holim(X -> Y). Due: February 17.
• Problem set 2: write up the homotopy limit computation, 7.11, 7.29, 7.36, 8.18, 8.29. Due: February 21 in class.
• Oral 3: Compute the limit of a simplicial object in a complete category E weighted by the simplicial set ∂Δⁿ. Hint: first express the weight as a colimit of representable n-1 and n-2 simplices. Due: March 2.
• Problem set 3: 8.27(!), 8.49, 9.3, 9.6, 9.26, 9.27. Due: March 6 in class.
• Oral 4: There are six weak factorization systems and nine model structures on the category of sets. Pick any for which the two weak factorization systems differ and describe it algebraically. Are the weak factorization systems algebraic weak factorization systems? If so, what are the (co)algebras? Are they cofibrantly generated? Due: April 6.
• Problem set 4: 14.8, 14.11, 14.26, 14.42, 14.44. (Optional: 11.6, 11.24) Due: April 3 in class.
• Problem set 5: 16.2, 16.16, 16.24, 16.33, 17.19, 17.21. Due: April 17 in class.

Here are the questions for the (oral) final exam.

CONTACT INFO

My contact info can be found on my personal website.