Emily Riehl Makes Infinity Categories Elementary

Theories of Everything with Curt Jaimungal

Curt Jaimungal

Physics, Philosophy, Society & Culture, Science

4.6 • 606 Ratings

🗓️ 6 April 2026

⏱️ 169 minutes

🔗️ Recording | iTunes | RSS

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Summary

Emily Riehl, one of the world’s leading category theorists, shares her vision for making infinity category theory something undergrads can actually learn. In this talk, she breaks down how rethinking the foundations of math could change the way it’s taught and understood—and why it might redefine what math even is. I personally subscribe to The Economist. TOE listeners get 35% off the annual subscription. No other podcast has this! https://economist.com/TOE Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://tinyurl.com/SpotifyTOE Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join Links: •⁠ ⁠Emily’s profile: https://emilyriehl.github.io/ •⁠ ⁠Emily’s presentation: https://emilyriehl.github.io/files/undergraduates-TOE.pdf •⁠ ⁠A Type Theory For Synthetic ∞-Categories (paper): https://arxiv.org/pdf/1705.07442 •⁠ ⁠Could ∞-Category Theory Be Taught To Undergraduates? (paper): https://arxiv.org/pdf/2302.07855 •⁠ ⁠RZK proof assistant: https://rzk-lang.github.io/rzk/en/latest/ •⁠ ⁠Lean Zulip chat: https://leanprover.zulipchat.com/#recentnews Timestamps: 00:00 A Dream for the Future 01:55 Exploring Infinity Categories 03:54 The Role of Category Theory 10:17 Key Concepts of Category Theory 12:01 The Curry-Howard Correspondence 15:37 Understanding Left Adjoint Functors 24:38 The Innate Lemma Explained 38:29 Proving the Isomorphism 41:50 The Importance of Abstraction 44:04 A Crash Course in Category Theory 44:17 Introduction to Infinity Category Theory 56:27 Fundamental Infinity Groupoids 1:03:34 What Are Infinity Categories? 1:09:12 The Case for Infinity Categories 1:18:12 Transitioning to Homotopy Type Theory 1:22:49 Crash Course in Homotopy Type Theory 1:30:56 Type Constructors Explained 1:34:19 Propositions as Types 1:42:50 Understanding Dependent Types 1:49:01 Identity Types and Their Importance 1:54:03 The Structure of Infinity Groupoids 1:59:33 Hierarchies of Types 2:06:19 The Univalence Axiom 2:08:36 Transitioning to Infinity Category Theory 2:10:04 Simplicial Type Theory Overview 2:14:56 Pre-Infinity Categories Defined 2:24:32 Isomorphisms in Infinity Categories 2:31:48 Computer Formalization in Mathematics 2:40:02 Conclusion and Future Directions Support TOE on Patreon: https://patreon.com/curtjaimungal Twitter: https://twitter.com/TOEwithCurt Discord Invite: https://discord.com/invite/kBcnfNVwqs #science Learn more about your ad choices. Visit megaphone.fm/adchoices Learn more about your ad choices. Visit megaphone.fm/adchoices Learn more about your ad choices. Visit megaphone.fm/adchoices

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1:15.1	This talk was kind of a dream. All sort of state, my dream for the future. If the foundations of
1:21.1	mathematics had some sort of higher structure and with something like homotopy type theory,
1:25.1	then we could teach infinity category theory to undergraduates, much like we teach something like abstract algebra to undergraduates today.
1:34.2	Today's talk is being given for the first time, so you're in for a huge treat.
1:39.0	Professor Emily Riehiel, an award-winning mathematician and professor at John Hopkins University, is a proficient explicator and one of the world's most renowned category theorists.
1:49.1	The question explored today is, how would you teach infinity categories?
	...

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