(As with other posts, this is hastily written in order to get it out with reasonable speed.)

The video for today’s lecture is available here. I need to figure out another place to post it so it is also visible to those without access to youtube. I had hoped that I could do it on zoom, but it appears that I can’t put edited things there.

I’m now going to give some pseudo-homework for the next two weeks. The point of this pseudohomework is to give you some specific things to focus on, and to be a starting point for discussion in smaller groups. (I hope the working groups you are a reasonable start.)

Ideally do some serious thinking about these things over this week. The point of the problems is for you to think hard about them, not to go and find the right source to quote or copy. For almost all of you, the question “what should I read to understand more about this topic” is the wrong one. Instead, fight with some specific problem.

Cognitive trap I want you to avoid, but that many of you (especially the less experienced ones) will fall into

There are a lot of fancy exciting things to hear about, and you will want to roam ahead and learn about them. (It’s clear to me that the notion of a “stack” has that sort of mystical attraction.) You might be impressed by others at a similar stage who use all sorts of fancy words. Do not be tempted! Instead, meditate long and hard on simple and fundamental things. Those are the ideas that will make the fancy stuff much simpler in the long run. Thoughtful contemplation of a few judiciously chosen ideas or problems trumps quick reads of advanced papers. Yes, I can see that you are ignoring me right now. But ask someone you admire who knows fancy things.

In particular, with the problems I suggest below, it is important to understand some well, which means writing things up completely. You could do this individually and trade. Or in this setting, a group of you might want to work through some problems, and type them up communally. (Zulip and discord let you upload files. Or you can use dropbox or email or anything else.)

I’m going to ask each of the groups, at the end of the week, to let others know (perhaps in the groupoid channel, or perhaps just by messaging shepherds) what happened and where you are as a group.

For those new to proofs, or who haven’t seen modules

Deliberately try to get three new ideas out of what follows. Try to do three exercises, no matter how “trivial” someone tells you they are (because whenever someone tells you something is “trivial” or “obvious”, you should hit them with a stick).

For those comfortable with proofs, and perhaps familiar with modules, but not more:

Read The Rising Sea up to section 1.4 Make a personal list of definitions you should know, and in particular know which ones are Important, and which ones are Less Important. “Important” here is meant in a technical sense: for later understanding on these particular topics. Try to digest the definitions, and ideally don’t memorize any definitions.

Glance at section 1.5, and browse through it if you wish. But adjoints require time to really come to terms with, so you’re better off digesting what you already know.

Don’t read sections 1.6 and 1.7.

Read ahead on sheaves and presheaves and morphisms of sheave and presheaves, without worrying too much about it, to seed your mind for next week.

Problems to do:

1. why is a group? (This problem is not a joke! Do you know about groups because someone told you to? Well, if they told you to go jump in lake, would you do that too?)

2. Make a list of categories (both objects and morphisms) you are already friends with, and functors you already know about.

3. You needn’t know any “definition” of manifold, but figure out with others why the “notion” of a manifold is a reasonable one (even if you can’t formalize it well), so we can use it in conversation.

4. In the notes, try problems 1.2.B, 1.3.A. (Localization and tensor products are harder than people think!) 1.3.N, 1.3.Q, 1.3.O, 1.4.B, 1.4.C. Maybe 1.4.D and 1.4.G. Ponder 1.4.8. Pick another exercise on this list on the basis of your judgement and taste that you think is worth thiking about.

5. What’s your favorite exercise (not necessarily from the notes), and why? (This is important: you are not a passive robot doing exercises. You are deliberately refining your thinking.)

6. What was a big insight here (either new to you, or perhaps not), and why?

7. What is a confusing notion you want to hear more about? (If you talk about stacks or infinity-categories, then you are showing a lack of wisdom.)

(I might respond to your answers of 5 through 7 next Saturday.)

If you are already very comfortable with modules and point-set topology, and trying to digest the core material in a more systematic way

Read up to section 1.5. There are some notions (including adjoints) that one understands more completely and deeply the more times one revisits them, so if you think you “know” these ideas, then think harder. Read starred sections too. Truly digest tensor products, limits, and colimits as much as possible.

Some interesting questions to make friends with: 1.3.K, 1.3.N, 1.3.S, 1.3.Y (baby Yoneda). (1.3.Z is Yoneda — but if you are just doing 1.3.Y now, then leave 1.3.Z for two weeks, other things can marinate in your mind.) 1.4.B, 1.4.D, 1.4.G. 1.5.D, 1.5.E, 1.5.F, 1.5.G.

Then problems 5 through 7 from above!

If you already know these ideas well

Ideally you are in a group with others like you. Hopefully you can share how you think about things that makes them “easy” or “clear” to you. Other things to discuss: what’s the simplest thing in the notes (or not in the notes, but intellectually adjacent) that you have not seen sufficiently? Is there an exercise that you think would be painful for you to do, but that you can imagine someone else might have a clean way of attacking? What things are missing from the discussion in the notes, from your point of view — some key insight, or exercise, or fact?

You already have a lot of familiarity with this material, and maturity mathematically, so follow your instincts.
Read up to 1.5 for sure. You can read ahead into chapter 2, since you’ve already made friends with sheaves. You can read 1.6 and 1.7 if you feel like. (Debatable point of view in those sections that you can argue with: one shouldn’t memorize the definition of abelian category. And spectral sequences, at the level at which they are most used at least by algebraic geometers, are not difficult — the most annoying thing is the proof. I think it is best to learn how to use them first, and then see why they work. I also believe it is best to learn how to drive a car without knowing how to take one apart and put it back together again.)