Pete Morcos (Rev #3)

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.

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.**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.

**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.**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.

*(Nothing yet)*