Emily Riehl Makes Infinity Categories Elementary

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: emilyriehl.github.io

•⁠ ⁠Emily’s presentation: emilyriehl.github.io/files/undergraduates-TOE.…

•⁠ ⁠A Type Theory For Synthetic ∞-Categories (paper): arxiv.org/pdf/1705.07442

•⁠ ⁠Could ∞-Category Theory Be Taught To Undergraduates? (paper): arxiv.org/pdf/2302.07855

•⁠ ⁠RZK proof assistant: rzk-lang.github.io/rzk/en/latest

•⁠ ⁠Lean Zulip chat: 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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