### Applied Category Theory Course: Collaborative Design

#### Posted by John Baez

My online course is now done. We finished the fourth chapter of Fong and Spivak’s book *Seven Sketches*—my classes at U.C.R. are starting up now, so I had to stop there.

Chapter 4 is about collaborative design: building big projects from smaller parts. This is based on work by Andrea Censi:

- Andrea Censi, A mathematical theory of co-design.

The main mathematical content of this chapter is the theory of enriched profunctors. We’ll mainly talk about enriched profunctors between categories enriched in monoidal preorders. The picture above shows what one of these looks like!

Here are the lectures:

- Lecture 55 - Chapter 4: Enriched Profunctors and Collaborative Design
- Lecture 56 - Chapter 4: Feasibility Relations
- Lecture 57 - Chapter 4: Feasibility Relations
- Lecture 58 - Chapter 4: Composing Feasibility Relations
- Lecture 59 - Chapter 4: Cost-Enriched Profunctors
- Lecture 60 - Chapter 4: Closed Monoidal Preorders
- Lecture 61 - Chapter 4: Closed Monoidal Preorders
- Lecture 62 - Chapter 4: Constructing Enriched Categories
- Lecture 63 - Chapter 4: Composing Enriched Profunctors
- Lecture 64 - Chapter 4: The Category of Enriched Profunctors
- Lecture 65 - Chapter 4: Collaborative Design
- Lecture 66 - Chapter 4: Collaborative Design
- Lecture 67 - Chapter 4: Feedback in Collaborative Design
- Lecture 68 - Chapter 4: Feedback in Collaborative Design
- Lecture 69 - Chapter 4: Feedback in Collaborative Design
- Lecture 70 - Chapter 4: Tensoring Enriched Profunctors
- Lecture 71 - Chapter 4: Caps and Cups for Enriched Profunctors
- Lecture 72 - Chapter 4: Monoidal Categories
- Lecture 73 - Chapter 4: String Diagrams and Strictification
- Lecture 74 - Chapter 4: Compact Closed Categories
- Lecture 75 - Chapter 4: The Grand Synthesis
- Lecture 76 - Chapter 4: The Grand Synthesis
- Lecture 77 - Chapter 4: The End? No, the Beginning!

It would have been nice to go on further, but I found it took more and more work to explain things in sufficient detail as the book progressed, so on the whole I am happy to stop here.