Emily Riehl Makes Infinity Categories Elementary
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What is Emily Riehl’s vision for teaching infinity categories to undergraduates?
This talk was kind of a dream. I'll sort of state my dream for the future. If the foundations of mathematics had some sort of higher structure and with something like homotopy type theory, then we could teach infinity category theory to undergraduates much like we teach something like abstract algebra to undergraduates today.
Today's talk is being given for the first time, so you're in for a huge treat. Professor Emily Rio, an award-winning mathematician and professor at Johns Hopkins University, is a proficient explicator and one of the world's most renowned category theorists. The question explored today is, how would you teach infinity categories, so a topic that's notoriously thorny even for experts in adjacent fields of math, to undergraduates? In exploring this, we cover not only an overview of regular category theory, so don't worry, but also how homotopy type theory provides a constitutional new language that could fundamentally transform how mathematics is understood, taught, and applied. We also cover the Curry-Howard correspondence, identity types, the univalence axiom, and what the heck infinity groupoids are.
My name is Kurt Jaimungal, and on this channel, I investigate theories of everything, primarily from a theoretical physics background, as that's my training from the University of Toronto, but also from a mathematics and philosophy perspective. Today, I'm excited and honored that Emily Real has perfected this talk, revamping it from previous lectures on infinity categories for undergraduates, so you're seeing the latest version of it being delivered here for the first time on Theories of Everything. If you have questions for the professor, make note of them and feel free to subscribe because we'll be doing a one-on-one podcast at some point in the future in a more Q&A format. Professor, welcome to Theories of Everything.
I've been wanting to have you on for years and the audience as well, so it's a pleasure to finally be speaking with you on air.
Thank you. It's a real honor to be here.
So please give an overview as to what you're going to talk about today.
Great. So I'm going to give a bit of a speculative talk about a dream that I have for the future, which is essentially that my specific research area, infinite dimensional category theory, will be easier to explain. And in fact, so easy that it's routinely taught to undergraduates around the world just as part of their standard mathematics education. All right. Well, let's get to it. Great. So before I get into infinite dimensional category theory, I want to back up and explain sort of what category theory is, what it's for, the role it plays within mathematics. I'm a mathematician, so I'm going to focus on applications to mathematics rather than to the real world, if you'll forgive me that. Great. And I thought I would start by telling a story about Galois.
So, you know, a very old problem in mathematics was to characterize solutions to polynomial equations. So the sort of equation you might be thinking of here is something like ax squared plus bx plus c equals zero, where a, b, and c are integers. And we're looking for numbers x that satisfy the equation ax squared plus bx plus c is equal to 0. And you might have learned at some point the quadratic formula, which gives a formula for the roots x when they exist. And once the quadratic equations were solved, it's natural to ask about more general equations with higher degree polynomials, cubic and quartic and quintic. And that problem proved to be a lot harder. And it was eventually determined that in complete generality, it's not always possible to solve
give a formula for the solutions to an equation given by polynomials with integer coefficients that involved only certain operations. So taking nth roots or using addition, subtraction, multiplication, and division. And Sort of the definitive solution to this problem was worked out by Everest Galois in 1832, sort of shortly before – or maybe in the few years preceding his untimely death in a duel.
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Chapters
8 chapters
1
What is Emily Riehl’s vision for teaching infinity categories to undergraduates?
0:00–21:36
2
What are the fundamental concepts of ordinary category theory covered early in the talk?
21:36–43:25
3
How does the Curry‑Howard correspondence connect logic, type theory, and category theory?
43:25–1:00:25
4
What is the Yoneda (Innate) Lemma and how does it establish isomorphisms in category theory?
1:00:25–1:19:24
5
How does homotopy type theory serve as a foundation for infinity‑category theory?
1:19:24–1:40:35
6
What are identity types and contractible types, and why are they crucial in homotopy type theory?
1:40:35–1:59:45
7
How are composition and associativity defined in pre‑infinity categories?
1:59:45–2:20:22
8
Why is formalizing infinity‑category theory in proof assistants important for modern mathematics?
2:20:22–2:43:28
Speakers
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