TITLE: A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?
QUESTION [7 upvotes]: 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?

REPLY [4 votes]: 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).