In the nLab article on cohomology, I found the following passage.

One can then understand various "cohomology theories" as nothing but tools for computing $\pi_0 \mathbf{H}(X,A)$ using the known presentations of (∞,1)-categorical hom-spaces: for instance Čech cohomology computes these spaces by finding cofibrant models for the domain $X$, called Čech nerves. Dual to that, most texts on abelian sheaf cohomology find fibrant models for the codomain $A$: called injective resolutions. Both algorithms in the end compute the same intrinsically defined $(\infty,1)$-categorical hom-space.

I find this paragraph incredibly interesting, since it offers a conceptual explanation for why Čech cohomology should agree with sheaf cohomology in certain cases. (I believe the usual proof for schemes uses a spectral sequence argument, which seems opaque to me.) Unfortunately, I do not know any higher category theory. In broad strokes--at a level accessible to someone with just a first course in algebraic topology and homological algebra--what is going on here? Also, what's a good reference that explains the details?

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    $\begingroup$ You don't need to know any higher category theory, really. The resolutions heuristic can be made precise using the machinery of model categories. $\endgroup$ – Zhen Lin Dec 25 '15 at 9:10
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    $\begingroup$ @ZhenLin Could you suggest a reference (or references)? My understanding is that you get derived functors by replacing an object with a (co)fibrant resolution and then whacking that resolution with the original functor. This gels with the definition of sheaf cohomology in terms of injective resolutions, but leaves me wondering how the Cech complexes come in. $\endgroup$ – user4571 Dec 25 '15 at 9:14
  • $\begingroup$ The Čech complex is sometimes a cofibrant replacement, but that on its own isn't quite enough. $\endgroup$ – Zhen Lin Dec 25 '15 at 9:21

This won't quite be about sheaf cohomology but hopefully it will communicate the basic idea. Suppose $X$ and $Y$ are two spaces, and you want to compute the mapping space (or maybe just $\pi_0$ of the mapping space) $[X, Y]$. For example, maybe $Y = BG$ for some Lie group $G$, so $\pi_0 [X, BG]$ is the set of isomorphism classes of $G$-bundles on $X$, which you'd like to classify. Loosely speaking, whenever you want to compute a complicated hom, there are two things to try:

  • Writing $X$ as a colimit of simpler things.
  • Writing $Y$ as a limit of simpler things.

Cech cohomology is a strategy for doing the first thing. Given any map $f : U \to X$ (which we want to think of as some sort of cover; for example, $U$ might be the disjoint union of opens making up an open cover of $X$), there is a canonical diagram you can write down that starts

$$\cdots U \times_X U \rightrightarrows U \to X$$

where the next part is $U \times_X U \times_X U$ and there are three arrows, etc. In general these pullbacks need to be homotopy pullbacks. Altogether you get a simplicial space called the Cech nerve of $f$ which, hopefully, is a "resolution" of $X$ in the sense that $X$ is its geometric realization (which you should interpret as a homotopy colimit). This requirement is an "effective descent" condition; see this MO answer for a bit more on this.

If that's true, then because the mapping space functor $[-, Y]$ sends homotopy colimits to homotopy limits (exactly like a hom should), this lets you write $[X, Y]$ as a homotopy limit of spaces of the form $[U \times_X \dots \times_X U, Y]$, or more explicitly as the totalization of a certain cosimplicial space. This might not seem like much of an improvement, but, for example, if $U$ is a "really good cover" (an open cover such that all finite intersections are contractible) then all of the pullbacks $U \times_X \dots \times_X U$ will be discrete (all of their connected components will be contractible); in this case the simplicial space (really a simplicial set) we wrote down is like a "free resolution" of $X$, and we can write $[X, Y]$ as a homotopy limit of spaces of the form $Y^n$.

With some more work (at some point you'd like to replace talking about homotopy limits with talking about ordinary limits and this isn't formal), this story recovers the Cech cocycle description of $G$-bundles when $Y = BG$, and also the Cech description of ordinary cohomology when $Y$ is an Eilenberg-MacLane space. You can also take $X = BG$ for a discrete group $G$ and $f : 1 \to BG$ to be the inclusion of a point, in which case you'll get back the usual description of group cohomology in terms of the bar resolution.

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    $\begingroup$ So, here's the beginning of an honest answer to your question about why it's cool to think of cohomology in terms of homs in higher categories: you can transport your intuitions about how homs work in categories (like that they send colimits to limits in the first variable, and preserve limits in the second variable) and see what they say about cohomology. $\endgroup$ – Qiaochu Yuan Dec 25 '15 at 9:10
  • $\begingroup$ Wonderful, thank you. $\endgroup$ – user4571 Dec 25 '15 at 9:11
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    $\begingroup$ Could you suggest a place where the details in this answer are worked out? I'm very interested in learning more. In particular, when you say "Cech description of ordinary cohomology," I'm not quite sure what you mean. Cech cohomology is ordinary cohomology when you compute it with values in a constant sheaf (in favorable cases...), but instead you're taking "values" in an E-ML space here, which is a bit confusing. $\endgroup$ – user4571 Dec 26 '15 at 0:34
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    $\begingroup$ @Patrick: I don't know a place where the details in this answer are worked out. Computing the space of maps $[X, Y]$ is equivalent to computing the space of (homotopy) global sections of the trivial $Y$-bundle $X \times Y \to X$. You can think of this bundle as describing a "constant sheaf" of $\infty$-groupoids, which is a very general kind of coefficients for sheaf cohomology. If $Y$ is an Eilenberg-MacLane space, then it's even a "constant sheaf" of spectra. A locally constant sheaf of abelian groups determines a locally constant sheaf of spectra by repeated delooping. $\endgroup$ – Qiaochu Yuan Dec 26 '15 at 1:16

We can be a lot more naive if all we want to do is sheaf cohomology. Take your favorite ringed space that has a nice derived category $D(X)$. Then sheaf cohomology is the cohomology groups of any complex representing $\mathbf{R}Hom(\mathcal{O}_X , F)$. Now, the point is that you can compute this object by resolving the target with an injective resolution, or the source with a Cech resolution.

The reason that a Cech resolution works, intuitively, is that we have descent. More generally, given an fpqc morphism $f:E\rightarrow X$ of schemes, we can form a "resolution" of $\mathcal{O}_X$ as follows: form a simplicial object in the derived category with terms $\mathbf{R}f_*( (\mathcal{O}_E)^{\otimes n})$. A version of cohomological descent says that the homotopy colimit of this object in the derived category is the structure sheaf back again. So you'd like to compute this homotopy colimit.

The trouble is that $D(X)$ isn't your friend for computing homotopy colimits, especially when you just use the bare triangulated category structure. In SGA 4 I think they get around this by using a nice model for the derived category of simplicial sheaves (which is not the derived category of the abelian category of simplicial sheaves.)

So you can either work in some world that lets you compute homotopy colimits, like model categories or $\infty$-categories, or you can hope that, in your example, there's a nicer model for the homotopy colimit. And when E is a disjoint union of quasi-compact open affines in a quasi-separated scheme, then an object that represents this homotopy colimit is just the Cech complex! (This uses the cohomological triviality of affine schemes).

  • $\begingroup$ Should've read Qiaochu's answer more closely- mine is basically the same... Sorry! $\endgroup$ – Dylan Wilson Jan 4 '16 at 13:19
  • $\begingroup$ This is very helpful, thanks! $\endgroup$ – user4571 Jan 4 '16 at 16:20

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