Category Theory Track

on YouTube ↗02h 24m

" summary

Begins from first principles and makes three concrete claims: composition plus identities and associativity capture “what matters” across maths, functors are the right notion of map between those structures, and natural transformations compare functors with a commuting-square condition. Uses diagrammatic examples and translations into Set: categories as objects and arrows, monoids as one-object categories, graphs and discrete dynamical systems as functors, and an ontology-to-Set example from PlantingSpace. Demonstrations use explicit arrow equations and commuting diagrams and convincingly illustrate the Yoneda target.

AI-written summary · about

" chapters (45)

1. 0:00Introduction and what to expect
2. 3:00Basic idea of categories and arrows
3. 6:00Composition of morphisms explained
4. 9:00Identity morphisms and associative law
5. 13:00The Holy Trinity: objects, arrows, composition
6. 16:00Understanding functors between categories
7. 19:00First look at natural transformations
8. 22:00Diagrammatic thinking in category theory
9. 25:00Practical intuition and universal properties
10. 29:00Identity functions and equations
11. 32:00Example walkthrough with simple objects
12. 35:00Resistance metaphor and intuition
13. 39:00Function composition revisited with diagrams
14. 42:00Questions about subobjects and structure
15. 45:00Richard's recommended videos and summary
16. 48:00Key takeaways and closing thoughts
17. 52:00Clarifying the purpose of categories
18. 55:00Summary of key rules and examples
19. 58:00Visualising functorial structure
20. 1:02:00Drawing category diagrams
21. 1:05:00Explaining composition with diagrams
22. 1:08:00Identity laws and set functions
23. 1:11:00Building up to product categories
24. 1:15:00Function mapping and final law
25. 1:18:00More on functor properties
26. 1:21:00Structure-preserving maps and transformations
27. 1:24:00Functor composition and higher-level intuition
28. 1:27:00Analogy: utterances and meaning
29. 1:30:00On the volume of mathematical ideas
30. 1:34:00Set category and image diagrams
31. 1:37:00Arrow chasing and examples
32. 1:40:00Naming objects and identity morphisms
33. 1:43:00Products in the category of sets
34. 1:47:00Diagrams for products and images
35. 1:50:00Applying equations to morphisms
36. 1:53:00Limits and universal constructions
37. 1:56:00Visualising limits through diagrams
38. 2:00:00Recap of category laws
39. 2:03:00Final structural examples
40. 2:06:00Leveraging diagrammatic reasoning
41. 2:09:00Categorical uniqueness and universality
42. 2:12:00Special cases and final examples
43. 2:15:00Thanks and transition to Q&A
44. 2:18:00Audience questions and functor mapping
45. 2:22:00Wrap-up and final remarks

" related talks

• 01h 31mCategory Theory Track by Bartosz Milewski Part 2Bartosz Milewski · ZuriHac 2023
• 54mCategory Theory Through Functional Programming (part 3/3)Alejandro Serrano · LambdaConf Winter Retreat 2017
• 51mCategory Theory for the Working HackerPhilip Wadler · Lambda World 2016
• 41mCategory Theory in LifeEugenia Cheng · Lambda World 2017
• 56mCategory Theory Through Functional Programming (part 2/3)Alejandro Serrano · LambdaConf Winter Retreat 2017
• 52mCategory Theory Through Functional Programming (part 1/3)Alejandro Serrano · LambdaConf Winter Retreat 2017
• 42mWhat did category theory do for us?Sergei Winitzki · Scale By The Bay 2019
• 01h 29mCategory Theory Track by Bartosz Milewski Part 1Bartosz Milewski · ZuriHac 2023