Category Theory For Beginners: Yoneda Lemma

In this video we state and prove the Yoneda Lemma, and give examples of it in action within the category of graphs and the category of dynamical systems. There is also speculation about possible relations between the Yoneda Lemma and notions from particle physics and philosophical ideas about the (potentially) fractal nature of reality, like Indra's Necklace.

I have been working through Colin McLarty's book `Elementary Categories, Elementary Toposes'. I have been making rough videos on the different chapters and exercises. These provide an informative and laconic course on category theory (although the videos are unedited and rough). I provide links to these unlisted videos below (and will update the list as I work through more of the book). The videos are in the proper order.

McLarty Basics 1
   • McLarty Basics 1  

Basics Ex 1
   • Basics Ex 1  

Products 1
   • Products 1  

Products 2
   • Products 2  

Products Ex
   • Products ex  

Subobjects and pullbacks 1
   • Subobjects and pullbacks 1  

Subobjects and pullbacks 2
   • Subobjects and pullbacks 2  

Pullback Lemma 1
   • Pullback Lemma 1  

Pullback Lemma 2
   • Pullback Lemma 2  

Pullback Lemma 3
   • Pullback Lemma 3  

Pullback Lemma 4
   • Pullback Lemma 4 example  

Pullback Lemma 5
   • Pullback Lemma 5  

Pullback Lemma 6
   • Pullback Lemma 6  

Limits via equalizers 1
   • Limits via equalizers 1  

Limits via equalizers 2
   • Limits via equalizers 2  

Pullback exercises 1
   • Pullback exercises 1  

Pullback exercises 2
   • Pullback exercises 2  

Relations 1
   • Relations 1  

Relations 2
   • Relations 2  

Exponential objects 1
   • Exponential Objects 1  

Exponential objects 2
   • Exponential objects 2  

Exponential objects 3
   • Exponential objects 3  

Exponential objects 4
   • Exponential objects 4  

Exponential objects 5
   • Exponential objects 5  

Exponential objects 6
   • Exponential objects 6  

Exponential objects 7
   • Exponential objects 7  

Exponential objects 8
   • Exponential objects 8  

Exponential objects 9
   • Exponential objects 9  

Exponential objects 10 
   • Exponential objects 10  

Mclarty Topos Basics 1
   • Mclarty Topos Basics 1  

Mclarty Topos Basics 2
   • Mclarty Topos Basics 2  

Mclarty Topos Basics 3
   • Mclarty Topos Basics 3  

Mclarty Topos Basics 4
   • Maclarty Topos Basics 4  

Mitchell-Benabou Language 1
   • Mitchell-Benabou Language 1  

Mitchell-Benabou Language 2
   • Mitchell-Benabou Language 2  

Mitchell-Benabou Language 3
   • Mitchell-Benabou Language 3  

Mitchell-Benabou Language 4
   • Mitchell-Benabou Language 4  

Mitchell-Benabou Language 5
   • Mitchell-Benabou Language 5  

Mitchell-Benabou Language Redone 1
   • Mitchell-Benabou Language Rules 1  

Mitchell-Benabou Language Redone 2
   • Mitchell-Benabou Language Rules 2  

Mitchell-Benabou Language Redone 3
   • Mitchell-Benabou Language Rules 3  

Mitchell-Benabou Language Redone 4
   • Mitchell-Benabou Language Rules 4  

Mitchell-Benabou Language Redone 5
   • Mitchell-Benabou Language Rules 5  

Mitchell-Benabou Language Redone 6
   • Mitchell-Benabou Language Rules 6  

Mitchell-Benabou Language Redone 7
   • Mitchell-Benabou Language Rules 7  

Mitchell-Benabou Language Redone 8
   • Mitchell-Benabou Language Rules 8  

Subobjects of exponential objects
   • Subobjects of exponential objects wit...  

Understanding exponential objects
   • Understanding exponential object eval...  

Understanding exponential objects 2
   • Understanding exponential object eval...  

Epimorphisms in MBL
   • Epimorphisms in Mitchell-Benabou Lang...  

Existential quantifier interpretation
   • Existential quantifier interpretation  

Universal quantifier in MBL 1
   • Universal quantification in MBL 1  

Universal quantifier in MBL 2
   • Universal quantification in MBL 2  

Universal quantifier in MBL 3
   • Universal quantification in MBL 3  

Universal quantifier in MBL 4
   • Universal quantification in MBL 4  

MBL implies rule
   • MBL implies rule  

MBL thinning, cut etc.
   • MBL rules for And, Cut, Thinning, Tri...  

MBL rules, singleton, equality etc.
   • MBL proofs for singleton, equality, b...  

MBL extensionality 1
   • MBL extensionality 1  

MBL extensionality 2
   • MBL extensionality 2  

MBL comprehension
   • MBL comprehension  

MBL substitution lemma
   • MBL Substitution Lemma  

MBL false not or
   • MBL false, not, or  

MBL false not or 2
   • MBL false not or 2  

MBL exists derivation
   • MBL exists derivation  

Kripke-Joyal Semantics Intro
   • Kirpke-Joyal semantics and arrow subo...  

Singleton arrows
   • Singleton arrows  

Kirpke-Joyal implies rule
   • Kirpke-Joyal implies rule  

Functional relations 1
   • Functional relations 1  

Functional relations 2
   • Functional relations 2  

Forall and exists in Set
   • Forall and exists in Set  

Functional relations 3
   • Functional relations 3  

The rest of the videos in this series are linked to within the description of my video: Category Theory For Beginners: Adjoint Functors

My own accompanying notes can be found here:

https://drive.google.com/drive/folder...