MA8403 Topics in Algebraic Topology - Spring 2025: Higher category theory

Change of time: The Wednesday lectures will be moved to Thursday 14:15-16:00 from week 4 (23 January)

The theory of higher categories, or more precisely ∞-categories, provides an appropriate language for working with mathematical objects that we want to consider up to some notion of equivalence that is weaker than isomorphism (such as chain complexes up to quasi-isomorphism or topological spaces up to (weak) homotopy equivalence), and has become an important tool in several areas of mathematics over the past decade. In this course I will try to give an introduction to ∞-categories that is focused on how the theory is actually used in practice (rather than how it is set up from scratch in a model such as quasicategories). While this is a PhD-level course, I hope it will also be accessible to motivated master students.

• Lecture notes

You could also take a look at these introductory articles to get an idea of what the course is about:

• Higher Categories, Rune Haugseng
• A whirlwind tour of the world of (∞,1)-categories, Omar Antolín Camarena

Books and lecture notes on ∞-categories:

• Higher categories and homotopical algebra, Denis-Charles Cisinski
• A short course on ∞-categories, Moritz Groth
• Lectures on infinity categories, Vladimir Hinich
• Introduction to ∞-categories, Markus Land (can be downloaded if on the university network)
• Kerodon, Jacob Lurie
• Higher Topos Theory, Jacob Lurie
• Higher Algebra, Jacob Lurie
• Introduction to quasicategories, Charles Rezk
• Elements of ∞-category theory, Emily Riehl and Dominic Verity
• Formalization of higher categories, book project by Denis-Charles Cisinski, Bastiaan Cnossen, Kim Nguyen and Tashi Walde

The course is graded pass/fail. The "oral exam" will be to give a 45-minute presentation on a topic connected to the course, which will most likely replace the lectures in the last few weeks of the semester.

This is a good opportunity to practice giving a talk (a "transferable skill"!), and if you are taking the course for credit you should put some effort into it. More specifically, I expect you to:

• discuss your plan for the talk with me a few weeks before your presentation,
• give a practice talk to other students before your presentation and improve it based on the feedback you get,
• attend all the other students' practice talks and give feedback on how the talk can be improved,
• attend all the other students' presentations (as well as all the regular lectures).

Some possible topics:

This is a pretty random list of suggestions, feel free to come up with your own topic or ask me for suggestions related to your own interests.

More categorical:

• The straightening theorem (e.g. arXiv:2111.00069
• Exponentiable fibrations (arXiv:1702.02681)
• Ends and coends (arXiv:2008.03758)
• Complete Segal spaces as ∞-categories (arXiv:2312.09889)
• Model categories and the Joyal model structure (maybe Rezk's notes)
• Dualization of fibrations (arXiv:1409.2165, arXiv:2011.11042)

More "algebraic":

• Symmetric monoidal ∞-categories and ∞-operads (maybe the first part of these notes, Harpaz's notes)
• E_n-algebras and the additivity theorem (Harpaz's notes)
• Stable ∞-categories (Mazel-Gee's notes, Higher Algebra 1.1)
• The ∞-category of spectra (Higher Algebra 1.4)
• Derived ∞-categories (Higher Algebra 1.3)
• The tensor product of presentable ∞-categories and universal property of spectra (Higher Algebra 4.8.1-2)
• Prestable ∞-categories (SAG C.1)

Even higher categories:

• (∞,n)-categories (maybe start with the last section of arXiv:2401.14311?)
• The Gray tensor product and lax transformations for (∞,2)-categories
• Monads and the monadicity theorem (arXiv:1310.8279 or Higher Algebra 4.8.3)

"Applied" higher categories:

• Factorization homology (arXiv:1903.10961, Harpaz's notes)
• The cobordism hypothesis (arXiv:1210.5100)
• Algebraic K-theory (the last part of these notes from lectures of Hebestreit might be a good starting point)
• Infinite loop spaces (arXiv:1305.4550)