I’m going to replace the proof of “spectral sequences for double complexes” in the introductory chapter with just a reference to the following “picturebook”.
I find this explanation surprisingly comprehensible, and this is really the first time I feel like I understand why spectral sequences work, and why they are not mysterious or black magic.
I haven’t been sure what to do with this pdf document, so I’ve just been giving it to people as a “gift”. So consider this to a gift to you too.
It is also posted here. (I am a fan of 3blue1brown.)
Math 216: Foundations of Algebraic Geometry 
March 27, 2022 at 1:02 pm
This is really cool! I’m trying to reverse engineer how you worked out the pictures associated to the most basic diagrams in the first place.
Basically what you’re doing is visualizing the free abelian category on the categorical data you start with. Or at least you’re visualizing the indecomposable objects in this free abelian category (which works great when everything is a sum of indecomposables.) There is a general way to do this, at least if your starting categorical data is an additive category A (and I think it can be jazzed up if sequences are “marked” as being “exact”) — there’s a formula which says that the free abelian category on A is (A-mod)-mod, where C-mod is the category of finitely-presentable additive functors from C to abelian groups.
Now, at least if we simplify things by working over a field, these kinds of calculations can be done by hand in the simplest cases, since we’re basically talking about some quiver representations. If you let the 1-dimensional objects in (A-mod)-mod be the jigsaw puzzles, then they are each supported on a single indecomposable object of A-mod, and you can arrange them with the “jigsaw connections” corresponding to morphisms between these support objects in A-mod. Then the higher-dimensional objects of (A-mod)-mod are built up by extensions from these, and you can use color to indicate which agglomerations of jigsaw pieces really can be built up via nontrivial extensions to obtain these larger objects.
It seems like the colors that you actually draw in your pictures correspond to those objects in (A-mod)-mod which are in the image of the double Yoneda embedding from A itself. I’m still puzzling over how one can see visually which agglomerations of jigsaw pieces are “buildable” from the data of these colorings alone. I suspect there should be a simple, clear geometrically-describable recipe to do this…