In my ongoing attempts to catch up with the class, I've just finished Chapter 2! As I did with Chapter 1, I've made a series of picture posts to help me get better intuition for the material.

The [full index](http://www.azimuthproject.org/azimuth/show/Pete+Morcos) of picture posts is at my wiki page. If you spot any errors or conceptual misunderstandings, I hope you'll let me know.

For those who read my Chapter 1 posts in May, there have been a few changes.

1. *Update:18 June* **The two forward images** - For some reason I had a lot of trouble with these concepts. I've fixed some errors that were in my first version.
1. *New: 18 June* **Computing adjoints** - A visual interpretation of John's formulas from Lecture 6.
1. *New: 18 June* **Picture proof that left adjoints preserve joins** - The proof of this fact dances back and forth between sets, joins, and adjunctions, so it's a bit hard to follow. I put the building blocks in picture form, and show how to glue them together in a "picture proof." It turned out quite pretty!

Here are the new posts for Chapter 2. Go to the index page for links.

1. **Picture proof that oplax left adjoint means lax right adjoint** - Another proof built up from picture fragments, to help keep track of all the properties involved.
1. **Posets are subsets of a total order** - I wanted to verify that you can always put a poset in some linear order, so that I could do so in later pictures. This post has little bearing on Chapter 2 concepts.
1. **Visualizing product orders** - Chapter 2 makes heavy use of order products, so I wanted a visual reference to help my intuition.
1. **Monoidal total orders** - I start with the simplest case for my first pictures of how monoidal products work. I quickly discover that I have no idea how to represent associativity visually.
1. **Monoidal partial orders** - Poset products are harder to draw, but in a very simple case I can manage to make a picture and learn a few things. I remain stumped about the question of freedom in choosing the monoidal unit.
1. **Adjunction plots** - As I was working on this chapter, I realized that the behavior of an adjunction can be drawn in function plot form, not just in sets-and-arrows form. This gave me an interesting new perspective on the Chapter 1 material.
1. **Many products: abstract** - Most of chapter two involves binary relations. I decided to put them all in product form to make it easier to spot the similarities and differences. Topics are: monoids, monoidal preorders, enriched categories, and closed preorders.
1. **Many products: Cost** - Using **Cost** as a concrete example lets me draw more helpful pictures of the various products. At the end, I combine several of the ideas above into a big 3D "adjunction plot" that captures most of the behavior of the **Cost** closed preorder.