Math 266x: Categorical Homotopy
TheoryTTh 10-11:30 Science Center 113
Thank you to all of those who kept up with the course over the
semester. I hope you found it valuable. I had a lot of fun.
Here is the
syllabus. If you are planning on taking this course for a grade, please come speak to me in person.
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.
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 Note, this version is slightly out of date.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.
This list will expand as the semester progresses.
- 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:
- 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:
- Problem set 3: 8.27(!), 8.49, 9.3, 9.6, 9.26, 9.27. Due: March 6
- 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:
- 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.
My contact info can be found on my personal website.