A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?

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.

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).