As a superfan of Vakil's Foundations of Algebraic Geometry, I was shocked to see that there was a beautiful piece of mathematical exposition of him floating around of which I hadn't heard for about a year. Quickly searching through this subreddit seems to indicate that it wasn't posted here before.
This document, subtitled "A bedtime story with pictures" did not disappoint. Vakil develops a kind of visual calculus, involving "puzzle pieces" giving a very nice unifying intuition for morphisms, subobjects etc. in abelian categories.
This is *not* a bedtime story, you have to put in some work. For each slide, work hard to get your intuition to his level. Work out your own examples. Your reward will be, among other things:
* finding the snake lemma intuitively obvious, not just in the sense that you know the usual diagram chase in your sleep, but in the sense that you see the statement and immediately know that it works
* the same for the long exact sequence in (co)homology. Also, you will see how that is pretty much the same as the snake lemma, and not just in a vague sense that you use the snake lemma in the proof somewhere
* same for the spectral sequence of a filtered complex
* making you really think with some of the final remarks (see bonus material at the end), asking and answering questions about what is the "right" way to make certain statements in homological algebra (I'm not done with this yet)
