Cech

There are three things that has to do with the Cech cover

• one is the computation of global cohomology, when we do $\Hom(\C_X, -)$, we can do a Cech resolution of $\C_X$ by take open subjects $j_{U_j}: U_j \to X$, and do $j_{U_j,!} \C_{U_j} \to \C_X$, and consider pulling back. The same can be for coherent sheaves $j_! \mathcal{O}_U \to \mathcal{O}_X$, the hom from $j_!$ is computed by a colimit, so $j_! O_U$ itself is a projective limit, it represent the function that vanishes to infinite order outside $U$.
• The other is by the sheaf condition on a sheaf. A presheaf in the abelian category of vector space is a functor from open set op to the vector space. The sheaf condition is saying, the global section is determined by local section. It means we have exact sequence $$ 0 \to F(U) \to \prod_i F(U_i) \to \prod_{i,j} F(U_i \cap U_j) \to \cdots $$ in particular, $F(U)$ can be cut out from the middle. This is sheaf valued in abelian category. We can pass to chain complexes of vector spaces. For example, cochain complex, or dR differential form, then the limit here is still done by the naive limit of set. So here we are looking at naive global sections, not the direct ones, and the limit only cares about the 'head' of the diagram.
• Finally, we want to glue up category. Suppose to each open set $U$, we assign some category $C(U)$, and for restriction of open set $U \supset V$, we have a functor $C(U) \to C(V)$. For $U \to V \to W$, we have $F_{UW} = F_{UV} \circ F_{VW}$ (strict). Then an object in the limit is just an object in each $X_i \in C(U_i)$ and $X_{ij} \in C_{U_{ij}}$, such that, along the restriction functor, we have isomorphism $res^{i}_{ij} X_i \cong res^{j}_{ij} X_j$.

Here is a confusion: what is a space? Say for two different spaces, $X$ and $Y$, we both can cover by some Cech cover $U_i$ and $V_i$, labelled by $i=1,\cdots, N$, and they produce a bunch of spaces $U_I$ and $V_I$. And we use some machinary to produce a bunch of categories, the Cech diagram indexed categories, $A_I$ and $B_I$. For example, consider the space $\P^n$ and a hypersurface in it, both have the standard Cech covering. The diagram itself does not carry any topology. The categories and functors and natural transformations and maybe higher coherence data carries info.

• to define an object, we just need the first three layers (fix an order of the index, so objects living on the intersections can be determiend by the dominant index). The data are objects in each $C_i$, and isomorphisms in each $C_{ij}$. The constraints are compatibility of gluing data in morphism space. (this is the case when the morphism space is a set (no homotopy) and the composition is associative on the nose).
• to define the (non-derived) morphism between two objects over the diagram, we specify their hom in the top layer, then on the overlap, we require their induced homs are the compatible. so, only two layerr. data in the first layer, and check in the 2nd layer.
• to define the composition, we only need to work in the top layer to produce the data for the composed morhpism.

This is the story of ordinary category with hom set, and non-derived hom. This covers how to glue up a category, this is a functor from the Cech cover poset category to the $(2,1)$ category of categories, and how to glue up a sheaf (an object)