Artwork

Content provided by Timothy Nguyen. All podcast content including episodes, graphics, and podcast descriptions are uploaded and provided directly by Timothy Nguyen or their podcast platform partner. If you believe someone is using your copyrighted work without your permission, you can follow the process outlined here https://player.fm/legal.
Player FM - Podcast App
Go offline with the Player FM app!

Tai-Danae Bradley | Category Theory and Language Models

2:25:17
 
Share
 

Manage episode 339378670 series 3389153
Content provided by Timothy Nguyen. All podcast content including episodes, graphics, and podcast descriptions are uploaded and provided directly by Timothy Nguyen or their podcast platform partner. If you believe someone is using your copyrighted work without your permission, you can follow the process outlined here https://player.fm/legal.

Tai-Danae Bradley is a mathematician who received her Ph.D. in mathematics from the CUNY Graduate Center. She was formerly at Alphabet and is now at Sandbox AQ, a startup focused on combining machine learning and quantum physics. Tai-Danae is a visiting research professor of mathematics at The Master’s University and the executive director of the Math3ma Institute, where she hosts her popular blog on category theory. She is also a co-author of the textbook Topology: A Categorical Approach that presents basic topology from the modern perspective of category theory.

In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

Patreon: https://www.patreon.com/timothynguyen

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

  • 00:00:00 : Introduction
  • 00:03:07 : How did you get into category theory?
  • 00:06:29 : Outline of podcast
  • 00:09:21 : Motivating category theory
  • 00:11:35 : Analogy: Object Oriented Programming
  • 00:12:32 : Definition of category
  • 00:18:50 : Example: Category of sets
  • 00:20:17 : Example: Matrix category
  • 00:25:45 : Example: Preordered set (poset) is a category
  • 00:33:43 : Example: Category of finite-dimensional vector spaces
  • 00:37:46 : Forgetful functor
  • 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!
  • 00:40:06 : Definition of functor
  • 00:42:01 : Example: API change between programming languages is a functor
  • 00:44:23 : Example: Groups, group homomorphisms are categories and functors
  • 00:47:33 : Resume definition of functor
  • 00:49:14 : Example: Functor between poset categories = order-preserving function
  • 00:52:28 : Hom Functors. Things are getting meta (no not the tech company)
  • 00:57:27 : Category theory is beautiful because of its rigidity
  • 01:00:54 : Contravariant functor
  • 01:03:23 : Definition: Presheaf
  • 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum.
  • 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma)
  • 01:12:10 : Algebraic topology motivated category theory
  • 01:15:44 : Definition: Natural transformation
  • 01:19:21 : Example: Indexing category
  • 01:21:54 : Example: Change of currency as natural transformation
  • 01:25:35 : Isomorphism and natural isomorphism
  • 01:27:34 : Notion of isomorphism in different categories
  • 01:30:00 : Yoneda Lemma
  • 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation
  • 01:42:33 : Analogy between Yoneda Lemma and linear algebra
  • 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors
  • 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.
  • 01:55:15 : Language Category
  • 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

Further Reading:

  continue reading

21 episodes

Artwork
iconShare
 
Manage episode 339378670 series 3389153
Content provided by Timothy Nguyen. All podcast content including episodes, graphics, and podcast descriptions are uploaded and provided directly by Timothy Nguyen or their podcast platform partner. If you believe someone is using your copyrighted work without your permission, you can follow the process outlined here https://player.fm/legal.

Tai-Danae Bradley is a mathematician who received her Ph.D. in mathematics from the CUNY Graduate Center. She was formerly at Alphabet and is now at Sandbox AQ, a startup focused on combining machine learning and quantum physics. Tai-Danae is a visiting research professor of mathematics at The Master’s University and the executive director of the Math3ma Institute, where she hosts her popular blog on category theory. She is also a co-author of the textbook Topology: A Categorical Approach that presents basic topology from the modern perspective of category theory.

In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

Patreon: https://www.patreon.com/timothynguyen

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

  • 00:00:00 : Introduction
  • 00:03:07 : How did you get into category theory?
  • 00:06:29 : Outline of podcast
  • 00:09:21 : Motivating category theory
  • 00:11:35 : Analogy: Object Oriented Programming
  • 00:12:32 : Definition of category
  • 00:18:50 : Example: Category of sets
  • 00:20:17 : Example: Matrix category
  • 00:25:45 : Example: Preordered set (poset) is a category
  • 00:33:43 : Example: Category of finite-dimensional vector spaces
  • 00:37:46 : Forgetful functor
  • 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!
  • 00:40:06 : Definition of functor
  • 00:42:01 : Example: API change between programming languages is a functor
  • 00:44:23 : Example: Groups, group homomorphisms are categories and functors
  • 00:47:33 : Resume definition of functor
  • 00:49:14 : Example: Functor between poset categories = order-preserving function
  • 00:52:28 : Hom Functors. Things are getting meta (no not the tech company)
  • 00:57:27 : Category theory is beautiful because of its rigidity
  • 01:00:54 : Contravariant functor
  • 01:03:23 : Definition: Presheaf
  • 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum.
  • 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma)
  • 01:12:10 : Algebraic topology motivated category theory
  • 01:15:44 : Definition: Natural transformation
  • 01:19:21 : Example: Indexing category
  • 01:21:54 : Example: Change of currency as natural transformation
  • 01:25:35 : Isomorphism and natural isomorphism
  • 01:27:34 : Notion of isomorphism in different categories
  • 01:30:00 : Yoneda Lemma
  • 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation
  • 01:42:33 : Analogy between Yoneda Lemma and linear algebra
  • 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors
  • 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.
  • 01:55:15 : Language Category
  • 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

Further Reading:

  continue reading

21 episodes

All episodes

×
 
Loading …

Welcome to Player FM!

Player FM is scanning the web for high-quality podcasts for you to enjoy right now. It's the best podcast app and works on Android, iPhone, and the web. Signup to sync subscriptions across devices.

 

Quick Reference Guide