**Images for understanding Chapter 1**

I started the course late, and have just finished chapter 1. I've put in a bunch of work creating mnemonic images to help me understand the material, which I've just put online at my [personal page](http://www.azimuthproject.org/azimuth/show/Pete+Morcos) here at the wiki.

I hope some of these images may be useful to folks still working through chapter 1, or needing a review. A few of the results go beyond the book & lectures, so there may be errors!

Note to John: I tried to follow the wiki instructions, but the "lab elves" may want to take a look at what I've done. In addition to my own wiki page, I created discussion threads for each topic in the [Applied Category Theory Discussion Groups](https://forum.azimuthproject.org/categories/discussion-groups) forum category. My apologies if that's too many threads for one person's project. I didn't see a way I could sequester them into a custom category, since I'm not an admin.

EDIT: On further thought, I think a separate forum category is a bad idea. Threads will drop out of sight eventually, but a new category will remain visible in the category list long after I stop posting. I'll just continue segregating the threads by using a prefix on the thread titles.

For convenience, here's a copy of the table of contents from my personal page.

---

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.

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.

+ [General feedback thread](https://forum.azimuthproject.org/discussion/2183/petepics-feedback)

1. **[Map diagrams](https://forum.azimuthproject.org/discussion/2181/petepics-chapter-1-map-diagrams)** - Visual refresher on the terms *injective, surjective, single-valued, total, function, relation,* and *graph*.

1. **[Confusing order terminology](https://forum.azimuthproject.org/discussion/2182/petepics-chapter-1-order-terminology-confusion)** - 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.

1. **[The two forward images](https://forum.azimuthproject.org/discussion/2184/petepics-chapter-1-the-two-forward-images)** - Pictures and mnemonics for the adjoints to the preimage \\(h^\*\\), which are \\(h_!\\) and \\(h_\*\\).

1. **[Monotone maps](https://forum.azimuthproject.org/discussion/2185/petepics-chapter-1-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.

1. **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](https://forum.azimuthproject.org/discussion/2186/petepics-chapter-1-iterated-galois-maps-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](https://forum.azimuthproject.org/discussion/2187/petepics-chapter-1-iterated-galois-maps-the-2-hop-inequality)** - I find this one very helpful when looking at maps in diagram form.

+ **[The 3-hop equivalence](https://forum.azimuthproject.org/discussion/2188/petepics-chapter-1-iterated-galois-maps-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](https://forum.azimuthproject.org/discussion/2189/petepics-chapter-1-iterated-galois-maps-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.

1. **[Non-bijectiveness](https://forum.azimuthproject.org/discussion/2190/petepics-chapter-1-galois-connection-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.

1. **[A big mnemonic image for Galois connections](https://forum.azimuthproject.org/discussion/2191/petepics-chapter-1-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.

---

EDIT: it seems that some of what I drew forms solutions to Owen's puzzles in #2 above, except that I invented a lot of terms so the pictures won't match the official terminology.

I started the course late, and have just finished chapter 1. I've put in a bunch of work creating mnemonic images to help me understand the material, which I've just put online at my [personal page](http://www.azimuthproject.org/azimuth/show/Pete+Morcos) here at the wiki.

I hope some of these images may be useful to folks still working through chapter 1, or needing a review. A few of the results go beyond the book & lectures, so there may be errors!

Note to John: I tried to follow the wiki instructions, but the "lab elves" may want to take a look at what I've done. In addition to my own wiki page, I created discussion threads for each topic in the [Applied Category Theory Discussion Groups](https://forum.azimuthproject.org/categories/discussion-groups) forum category. My apologies if that's too many threads for one person's project. I didn't see a way I could sequester them into a custom category, since I'm not an admin.

EDIT: On further thought, I think a separate forum category is a bad idea. Threads will drop out of sight eventually, but a new category will remain visible in the category list long after I stop posting. I'll just continue segregating the threads by using a prefix on the thread titles.

For convenience, here's a copy of the table of contents from my personal page.

---

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.

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.

+ [General feedback thread](https://forum.azimuthproject.org/discussion/2183/petepics-feedback)

1. **[Map diagrams](https://forum.azimuthproject.org/discussion/2181/petepics-chapter-1-map-diagrams)** - Visual refresher on the terms *injective, surjective, single-valued, total, function, relation,* and *graph*.

1. **[Confusing order terminology](https://forum.azimuthproject.org/discussion/2182/petepics-chapter-1-order-terminology-confusion)** - 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.

1. **[The two forward images](https://forum.azimuthproject.org/discussion/2184/petepics-chapter-1-the-two-forward-images)** - Pictures and mnemonics for the adjoints to the preimage \\(h^\*\\), which are \\(h_!\\) and \\(h_\*\\).

1. **[Monotone maps](https://forum.azimuthproject.org/discussion/2185/petepics-chapter-1-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.

1. **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](https://forum.azimuthproject.org/discussion/2186/petepics-chapter-1-iterated-galois-maps-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](https://forum.azimuthproject.org/discussion/2187/petepics-chapter-1-iterated-galois-maps-the-2-hop-inequality)** - I find this one very helpful when looking at maps in diagram form.

+ **[The 3-hop equivalence](https://forum.azimuthproject.org/discussion/2188/petepics-chapter-1-iterated-galois-maps-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](https://forum.azimuthproject.org/discussion/2189/petepics-chapter-1-iterated-galois-maps-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.

1. **[Non-bijectiveness](https://forum.azimuthproject.org/discussion/2190/petepics-chapter-1-galois-connection-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.

1. **[A big mnemonic image for Galois connections](https://forum.azimuthproject.org/discussion/2191/petepics-chapter-1-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.

---

EDIT: it seems that some of what I drew forms solutions to Owen's puzzles in #2 above, except that I invented a lot of terms so the pictures won't match the official terminology.