# Week 1

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.
* but why do you need it? where do you use it?
* Set, pointed Set (sets, functions) (pointed sets, functions sending the marked point to the marked point)
* $A$-mod for $A$ a ring ((f.g.) modules, homs)
* Rings (rings, homs)
* Top (topological spaces, maps of topological spaces, i.e. continuous)
* Toph (topological spaces with morphisms taken up to homotopy) (Topological spaces, homotopy classes of continuous maps)
* $D^b(A)$ for abelian category $A$ (chain complexes up to homotopy, chain maps up to homotopy, localized at quasi-isomorphisms)
* Vect (vector spaces, linear maps)
* K(A) for abelian category $A$ (chain complexes, chain maps up to homotopy)
* Perv(X)! Our dearest friend, for X a complex variety (below)
* Heart of t-structure (full subcategory of objects in the heart)
* $D_G^b(X)$ not a derived category (long...)
* Grp (groups, group homs)
* Functor category between two other categories (functors, natural transformations)
* Ab (abelian groups, group homs)
* Chain complexes (chain complexes, chain maps)
* Schemes
* Cat (categories, functors)
* Stable category ($\overline{\text{Mod}}$-A and $\underline{\text{Mod}}$-A) (modules, equivalence classes of homs)
* Rep(G) (representations of G, intertwining maps)
* Coh(X), QCoh(X) for a scheme X ((quasi)coherent sheaves of $\mathcal{O}_X$-modules, $\mathcal{O}_X$-linear morphisms of sheaves)
* Coh_G(X), QCoh_G(X) (G-equivariant (quasi)coherent sheaves, and morphisms thereof)
* Loc(X) local systems
* (Groupoids)
* Sh(X) sheaves on X (for X a site) (sheaves, homs)
* Lie algebraoiadlfjsoie
* Quivers (vertices, paths)
4. 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.
5. 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.
6. 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.)
7. What was a big insight here (either new to you, or perhaps not), and why?
8. 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.)

1.3.B.

| | Set | Top | Ring | Open subsets of X | Subsets of X|
| ------- | ------------- | ------------- | ----------- | --- | ------- |
| Initial | $\varnothing$ | $\varnothing$ | $\mathbb Z$ | $\varnothing$ | $\varnothing$ |
| Final | one-point set | Point | {0} | X | X |

None of these cats is abelian e.g. because there is no group structure on morphisms.


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 are not super-comfortable with modules: How comfortable are you with localization and tensor products? (Tensor products in particular really are confusing, so don’t be surprised, or feel stupid.) Perhaps try an exercise in each to see if you can do them. If you can, then declare victory and move on; you’ll digest these ideas better later, when you use them. Do 1.5.F (if you have to work hard, you are working too hard — t has a one-sentence answer.) Do 1.5.G (and realize that if you understand negative numbers, you understand this exercise). Do exercise 2.1.A (even though I allegedly did it in the pseudolecture). If you have some background in differential geometry, you will find Exercise 2.1.B enlightening. Exercise 2.2.B will give you an idea of when things might not quite be sheaves, but still be presheaves. Do Exercise 2.2.E, 2.2.F, 2.2.H, and even 2.2.J. Do 2.3.A, , and 2.3.C.

If you are quite comfortable with modules over a ring: Make sure you can do all the localization and tensor product exercises without referring to anything. Exercise 1.5.H will ensure you understand the “ification” functor. Do 1.6.A, 1.6.B, 1.6.C, and 1.6.D (for modules over a ring). Do 1.6.I, and possibly 1.6.K Do 2.1.A, and (if you know some differential or complex geometry) 2.1.B. Do 2.2.E; 2.2.F or 2.2.G; 2.2.H; 2.2.I; 2.2.J; 2.3.A. Definitely do 2.3.C. You can do 2.3.E and 2.3.F, and in anticipation of the next problem set, 2.3.I and 2.3.J.