How should a homotopy theorist think about sheaf cohomology?

Question

As a student of homotopy theory or algebraic topology, I have a certain outlook as to how one ought to think of a cohomology theory. There are axioms that help us with rudimentary computations, there are some spectral sequences and then there is Brown representability. This is far away from the starting point of looking at the singular (co)chain complex or a simplicial complex and trying to compute its homology, it seems a bit more refined. Even the ring structure is a bit clearer, it comes from the fact that we are mapping into a ring object.

There are more and more instances where i feel like i would benefit from understanding a bit more of sheaf cohomology than just "it's the derived functor of the global sections functor of a sheaf." This is a tidge helpful, but it does not really help too much with computations from my point of view. It feels like resolutions of sheaves are large hard objects mostly because sheaves contain so much data.

My question is essentially the following:

1. Are there homotopy theorists out there who have over come these feelings? what advice do you have? In fact any advice that someone might have that understands the uses of sheaf theory in homotopy theory would be helpful.

2. Are there things resembling the Eilenberg-Steenrod axioms for sheaf cohomology? not directly due to their classification theorem, but things that help you to compute the Sheaf cohomology like a MVS sequence or what have you. I mostly would like help in doing computations in the way that the E-S axioms do? so things like the Grothendieck-Riemann-Roch Theorem, which i am told can be used in such a way.

3. Is there a book that goes through explicit toy computations of sheaf cohomology? Are there toy examples you would suggest for getting to be more comfortable with these things? Are there examples that live on simpler spaces than schemes? these might help a bit more than others

Some addendums : I think i need to make a few comments. I am not well versed in algebraic geometry. I find this to be a fault of mine and this is an attempt to help bridge the gap. Suggestions for references are appreciated. I really appreciate all of the excellent answers so far! thanks for your time

EDIT MAIN QUESTION: I think what i am really asking about is the six functor formalism, but i don't really know since i don't know what that is. A friend started explaining it to me and it seemed like what i am looking for, but he said he did not feel confident in writing an answer explaining them. Hopefully someone will see this edit and give a fun answer.

Answer 1

Hi Sean,

I think I great place to inhabit, be it for calculations or for conceptual understanding, is the derived category of sheaves (of abelian groups say) $D(X)$ on your topological space $X$. Here are the basic players:

1. For any $X$ and integer $i$, functors $H^i: D(X) \rightarrow$ sheaves of abelian groups on X (cohomology sheaves); and

2. For a map $f: X \rightarrow Y$, adjoint maps $f^* :D(Y) \rightarrow D(X)$ and $f_* :D(X) \rightarrow D(Y)$ (derived pullback and pushforward).

Then, for instance, the $i^{th}$ cohomology of a sheaf is just $H^i p_*$, where $p: X \rightarrow pt$.

Lots of your favorite computational tools carry over to this setting. For instance Meyer-Vietoris: if $X = U_1 \cup U_2$ with inclusions $j_1: U_1 \rightarrow X$ and $j_2: U_2 \rightarrow X$ and $j_{12}: U_1 \cap U_2 \rightarrow X$ and $F$ is a sheaf on $X$, then there is a distinguished triangle

$${j_{12}}_* j_{12}^* F \rightarrow {j_1}_* j_1^* F \oplus {j_2}_* j_2^* F \rightarrow F \rightarrow$$

(usual M-V follows by taking F constant and applying $p_*$).

What's a reference? I learned from BBD (Faisceaux Pervers, Asterisque 100), which is great, but maybe more scheme-y than you want.

P.S.: Since you're a homotopy theorist, maybe I'll mention what I think is a great perspective on what the derived category D(X) is. Basically it's just like the ordinary category of sheaves of abelian groups, except you do everything homotopically. So instead of abelian groups you take (excuse me) HZ-module spectra, and you make your sheaves satisfy homotopical descent (this homotopical descent is really the origin of things like M-V above). To make this perspective rigorous it's helpful to use infinity-category theory, as in Lurie's book Higher Topos Theory. Using this approach, one doesn't "derive" things in the sense of starting with an abelian situation and invoking derived categories and functors; rather one makes "derived" definitions, and then all of the natural operations are automatically "derived": for instance the derived f_* and f^* can be given, in the infinity-categorical setting, by the same formulas as the ordinary f_* and f^* in the classical (abelian) setting.

Answer 2

Here are some comments on the questions.

1. Sheaves may look like they are very big but they are not. There is a large class of sheaves, constructible sheaves, which are combinatorial objects. Here is how one can think about them. Imagine a polyhedron; associate an abelian group $A(\triangle)$ to each face $\triangle$ and a a morphism of groups $A(\triangle_1)\to A(\triangle_2)$ whenever $\triangle_2$ is a codimension 1 face of $\triangle_1$; assume that the two possible ways of getting from an $n$-face to an $n-2$ face coincide. Then one can introduce a cochain differential by the usual formula. Most sheaves that come up in real life topological problems arise in this way. For example, if all restriction mappings are isomorphisms, we get a local system.

2. The cup product in cohomology is very natural from the sheaf theory perspective as well. We have $H^i(X,A)=Hom_{D^+X}(\underline{A},\underline{A}[i])$ where $A$ is a commutative ring, $\underline{A}$ is a constant sheaf on $X$ with stalk $A$ and $\underline{A}[i]$ is $\underline{A}$ shifted $i$ steps to the left. Since shifting gives an isomorphism of the derived category, we have $Hom_{D^+X}(\underline{A},\underline{A}[j])=Hom_{D^+X}(\underline{A}[i],\underline{A}[i+j])$and so we get a bilinear map $$Hom_{D^+X}(\underline{A},\underline{A}[i])\times Hom_{D^+X}(\underline{A},\underline{A}[j])\to Hom_{D^+X}(\underline{A},\underline{A}[i+j])$$ which is essentially just the composition of morphisms in the derived category. This gives the cup product.

Now to answer your questions

1. (Are there homotopy theorists...) Probably.

2. I am not aware of a version of Eilenberg-Steenrod axioms for sheaves, but this is a good question. One way to state it more precisely would be: give a criterion for a cohomological functor on $D^+X$ to be represented by the constant sheaf. In any case one can use Mayer-Vietoris (at least for an open cover or a closed cover of a reasonable space) to compute sheaf cohomology. Moreover, the long exact sequence for $i:X\subset Y$ a closed subset comes from the distinguished triangle $j_! j^{-1}\to id\to i_\ast i^{-1}\to\cdots$ where $j:Y\setminus X\subset Y$ is the open embeding of the complement.

3. For a general picture on sheaves and related stuff I would recommend two books by Gelfand and Manin. One is Methods of homological algebra and the other is called simply Homological algebra (a survey in Springer's Encyclopedia of math sciences). These books are excellent but a word of warning: there are some typos, missing conditions in some theorems etc.

Answer 3

This is mostly an answer to 3:

While I mostly use sheaves arising in topology (and since your question explicitly involves homotopy theory, that might also be your point of view), I think it's a useful learning tool to do some reading about how sheaves come up and are used in complex analysis and Hodge theory. The big benefit there is that most of the sheaves there are already nice (fine, soft,...), so one can do a lot without having to take resolutions by sheaves one has less control over. Wells's book Differential Analysis on Complex Manifolds might be a good starting point, or even Voisin's book on Hodge Theory (which is more advanced).

In the topological world, I second Bredon and Gelfand/Manin (the second edition of Methods of Homological Algebra is in better shape than the first regarding typos). Also Dimca has a book Sheaves in Topology, which is a nice introduction to the derived category point of view that includes some useful examples and exercises.

One last note: in doing actual computations, the exact sequences coming from adjunction triangles (see e.g. Section 2.4 of Dimca) are often your best friends :-)