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I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space $X$, and $\mathcal{F}$ is a sheaf on $X$.

Take $\delta^n \colon C^n \longrightarrow C^{n+1}$ to be the coboundary operator from $n$-cochains as defined for this specific cohomology. Let $Z^n = \text{ker} \delta^{n}$ and $B^{n} = \text{im} \delta^{n-1}$ so that $H^n = Z^n / B^n$.

I understand that $H^0$ is nothing more than the global sections on $X$ ($Z^0$ is all sections that agree on double intersections, so we can of course glue). This is intuitive.

Now I would like to wrap my mind around $H^1$. I understand that $Z^1$ will be the sections defined on smaller open sets (double intersections) that agree on triple intersections. So, in a sense, we are localizing as we grade.

Now, $B^1$ is the difference of sections of parts of the open cover restricted to double intersections.

So when we mod out $B^1$ from $Z^1$, what is it, effectively, that we're doing?

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It is hard to know what will give you an intuition without knowing what you want and what background you already have.

If you are asking roughly what kind of geometric or topological information Cech cohomology reveals about a space, then answer is the same as for all cohomology theories: it reveals the global connectivity of spaces with relatively simple local geometry (say, manifolds, or CW complexies). By use of suitable sheaves this can reveal how problems with relatively easy local solutions (say, finding a meromorphic function with a given pole) can or cannot be solved globally (finding a meromorphic function on a given surface with exactly some specified poles).

If you are asking specifically what the Cech apparatus reveals then you may find it easy to see the case of the constant sheaf $\mathbb{Z}$ on a surface, and then generalize.

Here is a pictorial approach to that case (inspired by campus of the Bergische Universität Wuppertal): Think of the sheaf $\mathbb{Z}$ as an infinite-storied building built all over the surface.

A zero chain, or element of $C^0$, is any choice of one floor on each open set. A zero cycle or element of $Z^0$ is a choice of floor on each open set where overlapping sets choose the same floor -- so it is simply one choice of a floor of the building if the space is connected.

An element of $Z^1$ does not specify a floor for each open set but rather a difference between floors for any two adjacent sets. It is a choice of how many floors to go up (or down for negative values) when you move from one open set to an adjacent one. That triple intersection condition says the choice is locally unambiguous in the sense that whenever three areas overlap at at least one point, then it makes no difference how you move between them. You can go from the first to the second and then to the third, rising the chosen number of floors each time. Or you can go directly from the first to the third rising that chosen number of floors. The two are the same.

A key point here is that three areas might each overlap each other, while they do not all overlap at any one point. The first overlaps the second and the third (but not in the same place) while the second and third also overlap (but not in the same place).

So being locally unambiguous does not imply that the choice is globally unambiguous. If the surface is not simply connected then there may be ways to travel around the building, raising and falling the chosen number of floors each time you move from one open set to another, and yet not end up on your original floor when you return to your original set. The elements of $B^1$ are those choices that are globally unambiguous in this sense.

Modding $B^1$ out from $Z^1$ measures how many ways locally unambiguous choices in the building can be globally ambiguous.

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