Show Notes
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.
Every episode days early, ad-free, plus my essays: https://curtjaimungal.substack.com/subscribe
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 Crypto: https://nowpayments.io/donation/TOE
Twitter: https://twitter.com/TOEwithCurt
Discord Invite: https://discord.com/invite/kBcnfNVwqs
#science Learn more about your ad choices. Visit megaphone.fm/adchoices
More from Theories of Everything with Curt Jaimungal
SPONSORS: - Go to https://www.plaud.ai/curt and use the promo code "CURT" to get a Plaud device today - Our listeners get up to 30% off their entire order at https://undrdog.com
SPONSORS: - Visit https://mod.com for a free consultation and get 10% off your first order PLUS free shipping with promo code TOE - Take Cheers Restore after your last drink or before going to bed and wake up feeling at least 50% better — or your money back. For a limited time our listeners are getting 20% off their entire order at https://cheershealth.com/TOE.
SPONSORS: - Don’t sleep on [@ultrapouches]. New customers get 15% Off with code TOE at http://takeultra.com! #UltraPouches #ad
SPONSORS: - Buy your time back. Get 20% off Plaud for Fall Prime Day. For long conversations where every idea matters, Plaud helps you capture what was said and turn it into transcripts, summaries, key insights, and action items — so you can focus on the conversation, not your notes. Get 20% off for a limited time: → http://Plaud.ai/curt — code CURT → https://amzn.to/3Ta04lD — code CURT20PD - Visit https://mod.com for a free consultation and get 10% off your first order PLUS free shipping with promo code TOE
SPONSORS: - Head to https://www.factormeals.com/toe50off and use code toe50off to get 50% off and 1 free breakfast item per box for 1 year. - Get Tidy Today! Try CleanMyMac 7 days FREE and use my code THEORIES for 20% off - https://clnmy.com/THEORIES - Get 15% off OneSkin with the code TOE at https://www.oneskin.co/TOE #oneskinpod