Pete Morcos

I participated in a 2018 course that John Baez taught using a draft copy of the book

- Brendan Fong and David Spivak,
*Seven Sketches in Compositionality website w/ videos*.

I made some pictures to help me remember the concepts. They might be helpful to others as well. (There are a few results that go beyond the book & lectures, which means I may have made some errors. Please let me know if you find any!)

I announced these images in the comments to Lecture 17, so some discussion may also be happening in that thread. *(Other announcements and possible discussion: Ch.2)*

These do *not* form a comprehensive tutorial. I only picked topics where I felt an image would help me understand, so not everything important is covered.

I wanted more formatting control than available here in the forums, so as an experiment each post is a large image. I’m unsure how this experiment will work out.

Unfortunately using images makes searching and quoting impossible. It also ruins accessibility, but since the point of these posts is the images, they still wouldn’t be accessible even with the text given separately.

**Note:** Please do not link directly to the images. The URLs change every time I update or correct them. Link to the discussion threads instead.

I feel the two most valuable images I made for this chapter are the one on *Confusing order terminology* and the final *Big mnemonic image*.

**Map diagrams**- Visual refresher on the terms*injective, surjective, single-valued, total, function, relation,*and*graph*.**Confusing order terminology**- The terms and symbols used to describe orders and adjunctions was hard for me to absorb. The reason seems to be that they use incompatible mnemonics. Once I figured out the exact inconsistencies, I found it much easier to keep everything straight.*Update:18 June***The two forward images**- Pictures and mnemonics for the adjoints to the preimage $h^*$, which are $h_!$ and $h_*$.**Monotone maps**- The definition is a bottom-up one, looking at individual elements. I found it useful to draw pictures of a top-down view, looking at entire order relations.**Iterating the Galois connection maps**- Following the maps more than once yields helpful little formulas and some insight. Much of this is in the book but a few things are different. Lots of pictures.**The 1-hop constraint**- The basic definition of an adjunction. I then go on a long detour, viewing the definition in terms of the action on entire order relations at once.**The 2-hop inequality**- I find this one very helpful when looking at maps in diagram form.**The 3-hop equivalence**- I don’t think this is in the book, and it helped me get a better intuition of Galois connections.**The 4-hop fixed point**- Apparently every Galois connection has a bijection embedded inside it. This wasn’t clear to me from the book, so working through the pictures was quite helpful.

*New: 18 June***Computing adjoints**- A visual interpretation of John’s formulas from Lecture 6.**Non-bijectiveness**- Galois connections are interesting because they’re not-quite bijections. I worked out a few small results (with pictures) to explore how exactly that works.*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!**A big mnemonic image for Galois connections**- All the math and pictures above packed into one master image. This has become my starting point whenever thinking about Galois connections. The topic was quite confusing to me at first, but this image has helped me tremendously. However, the image relies on terms and visual conventions defined in the earlier posts, so you’ll need to read them first.

**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.**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.**Visualizing product orders**- Chapter 2 makes heavy use of order products, so I wanted a visual reference to help my intuition.**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.**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.**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.**Many products: abstract**- Most of Chapter 2 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.**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.

*(Nothing yet.)*