Lectures on Applied Category Theory
Posted by John Baez
Want to learn applied category theory? You can now read my lectures here:
There are a lot, but each one is bitesized and basically covers just one idea. They’re selfcontained, but you can also read them along with Fong and Spivak’s free book to get two outlooks on the same material:
 Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.
Huge thanks go to Simon Burton for making my lectures into nice web pages! But they still need work. If you see problems, please let me know.
Here’s one problem: I need to include more of my ‘Puzzles’ in these lectures. None of the links to puzzles work. Students in the original course also wrote up answers to all of these puzzles, and to many of Fong and Spivak’s exercises. But it would take quite a bit of work to put all those into webpage form, so I can’t promise to do that. 😢
Here are the lectures:
Chapter 1: Ordered Sets

Lecture 1  Introduction

Lecture 2  What is Applied Category Theory?

Lecture 3  Preorders

Lecture 4  Galois Connections

Lecture 5  Galois Connections

Lecture 6  Computing Adjoints

Lecture 7  Logic

Lecture 8  The Logic of Subsets

Lecture 9  Adjoints and the Logic of Subsets

Lecture 10  The Logic of Partitions

Lecture 11  The Poset of Partitions

Lecture 12  Generative Effects

Lecture 13  Pulling Back Partitions

Lecture 14  Adjoints, Joins and Meets

Lecture 15  Preserving Joins and Meets

Lecture 16  The Adjoint Functor Theorem for Posets

Lecture 17  The Grand Synthesis
Chapter 2: Resource Theories

Lecture 18  Resource Theories

Lecture 19  Chemistry and Scheduling

Lecture 20  Manufacturing

Lecture 21  Monoidal Preorders

Lecture 22  Symmetric Monoidal Preorders

Lecture 23  Commutative Monoidal Posets

Lecture 24  Pricing Resources

Lecture 25  Reaction Networks

Lecture 26  Monoidal Monotones

Lecture 27  Adjoints of Monoidal Monotones

Lecture 28  Ignoring Externalities

Lecture 29  Enriched Categories

Lecture 30  Preorders as Enriched Categories

Lecture 31  Lawvere Metric Spaces

Lecture 32  Enriched Functors

Lecture 33  Tying Up Loose Ends
Chapter 3: Databases

Lecture 34  Categories

Lecture 35  Categories versus Preorders

Lecture 36  Categories from Graphs

Lecture 37  Presentations of Categories

Lecture 38  Functors

Lecture 39  Databases

Lecture 40  Relations

Lecture 41  Composing Functors

Lecture 42  Transforming Databases

Lecture 43  Natural Transformations

Lecture 44  Categories, Functors and Natural Transformations

Lecture 45  Composing Natural Transformations

Lecture 46  Isomorphisms

Lecture 47  Adjoint Functors

Lecture 48  Adjoint Functors

Lecture 49  Kan Extensions

Lecture 50  Left Kan Extensions

Lecture 51  Right Kan Extensions

Lecture 52  The HomFunctor

Lecture 53  Free and Forgetful Functors

Lecture 54  Tying Up Loose Ends
Chapter 4: Collaborative Design

Lecture 55  Enriched Profunctors and Collaborative Design

Lecture 56  Feasibility Relations

Lecture 57  Feasibility Relations

Lecture 58  Composing Feasibility Relations

Lecture 59  CostEnriched Profunctors

Lecture 60  Closed Monoidal Preorders

Lecture 61  Closed Monoidal Preorders

Lecture 62  Enriched Profunctors

Lecture 63  Composing Enriched Profunctors

Lecture 64  The Category of Enriched Profunctors

Lecture 65  Collaborative Design

Lecture 66  Collaborative Design

Lecture 67  Feedback in Collaborative Design

Lecture 68  Feedback in Collaborative Design

Lecture 69  Feedback in Collaborative Design

Lecture 70  Tensoring Enriched Profunctors

Lecture 71  Caps and Cups for Enriched Profunctors

Lecture 72  Monoidal Categories

Lecture 73  String Diagrams and Strictification

Lecture 74  Compact Closed Categories

Lecture 75  The Grand Synthesis

Lecture 76  The Grand Synthesis

Lecture 77  The End? No, the Beginning!
Links to puzzles
John, the links to the puzzles in Lecture 3 don’t seem to be working for me after trying a couple of browsers.