The first post is the August 26 version here.

I hope to post approximately 25 pages of reading for learners (with more optional material for experts) every fortnight.

For learners.

If you are reading to learn, the first reading is, I’m afraid, quite heavy if you’ve never seen category before. You should just skim the starred subsections and skip the spectral sequence section 2.7: you should read (slowly) 2.1-2.6, and get a sneak peak at 3.1. It is essential to be familiar with the exercises, and which ones you should do depends on what you know. If you force me to name twelve, I won’t be happy, but I’ll try: 2.3.F, 2.3.K, 2.3.O, 2.3.T, 2.3.Y, 2.4.A, 2.4.C, 2.5.C, 2.5.E, 2.6.A, 2.6.B, 2.6.F (from the Aug 26 version).
I don’t want to give you much guidance, because I want the notes to speak for themselves as much as possible.

For experts.

I hope to get across not just the category theory people need, but also a sense of how to think categorically. I always find it a good question when people in class start asking wise categorically-minded questions, and thinking clearly. There are also a few constructions that are essential. But I don’t want to tell people anything they don’t need to know, and I try maintain a balance between informality and precision. I’ve noticed that most of the following topics is easy once you understand them (with some exceptions!), but that they are quite unmotivated when you first see them. Developing fluency with these ideas requires working through as many trivial examples as possible. I hope many of the exercises are (after you figure them out) trivial, although some are hard.

In 2.1, I try to give them motivation for why we think in this way.

In 2.2, I introduce categories and functors and related notions, such as natural transformations. I didn’t understand the notion of equivalent categories in my gut until Johan de Jong gave me some baby examples.

Section 2.3 is devoted to universal properties, again through key examples. Yoneda’s lemma sneaks in at the end, and I hope readers end up finding it reasonable (even if they don’t yet appreciate it).
(Question: after 3.3.D: “0” is a zero-divisor. Is that so horrible? Question: I originally denoted the contravariant functor representing X by h^X, and the covariant functor representing X by h_X. Arthur Ogus and Jason Ferguson made strong cases that I’m violating accepting convention, so I’ve attempted to reverse myself. Does anyone strongly disagree with Arthur and Jason?)

(Co)limits are discussed at length in 2.4, with examples, and constructions of when they exist in cases that will actually be used.

In 2.5, adjoints are discussed. I find this tricky to explain. It is a pain in the butt to show that two things are adjoint.

Abelian categories are introduced in 2.6. We need them, but they are a tarpit in which many people have been irretrievably lost. We need some things, but remarkably little. It’s really true that if you understand modules over a ring, you can get by in almost any case in algebraic geometry (of the sort most people do) — even without Freyd-Mitchell. I am the most worried about this section. If someone hasn’t seen long exact sequences before, Theorem 2.6.5 will be a bolt from the blue.

Section 2.6.7, two useful facts in homological algebra, contains facts that I really wish I’d collected in my head earlier than I did (on the relationship between homology, (right/left)-exact functors, adjoints, and colimits — I would often be confused as to when various things would commute, and only later realized that there are some general principles at work). I hope I have collected them correctly. People seeing homological algebra for the first time shouldn’t read this, but people seeing it for the second time might benefit from it.

The “FHHF” Theorem

Going far with the FHHF Theorem

Going far with the FHHF Theorem

Spectral sequences are necessary later, so I included a brief introduction in 2.7. I really mean it when I say that I hope that people don’t peek until they actually need it. I do something slightly unusual (that may upset some people). For me, spectral sequences are useful because of how they can be used with little thought. Too many people just tune out when they hear the word, but when they turn up, they should make you happy, not sad. The best way to understand how to use something is to see it used to do something you already know how to do. So I deliberately concentrate on how to use them, in the special case of double complexes, which is the only case we’ll use later. I give a proof in this case which I believe is complete (and which hasn’t yielded many complaints), but which I hope most people don’t read. My only goal with this section is that people should leave it unafraid of spectral sequences in this setting, ready to use them on a moment’s notice, and with a sharp sense of how to use them. (Questions for experts: (1) I also introduce a convention of having an arrow in the spectral sequence of a double complex to indicate which is the first differential you use, the horizontal or the vertical — both in the names of the pages and in the names of the differentials on that page. Is anyone really offended? (2)
Should the indices of spectral sequences be ordered “(row, column)” or “(x,y)”? I don’t think there is consistency in the literature, so I’ve gone with the first. (3) Is there some classical important fact that would be useful practice for people learning spectral sequences, that doesn’t require much additional background?)

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