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.
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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
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