Cech cohomology and the coboundary operator I'm trying to learn more about cohomology in general. I have a decent understanding of homology using simplicial complexes so building chain groups as free abelian groups generated by simplices and boundary operators being alternating sums of a simplices faces all makes intuitive sense to me. But when I read more about cohomology I'm struggling to grasp what exactly the coboundary operator is. I understand the the cochain groups $C^n$ are groups of homomorphisms from the chain groups $C_n$, that is $C^n = Hom(C_n,R)$ for some ring $R$. But how exactly do we compute the coboundary operator?
I found this source which has a great explanation except I can't seem to understand why the coboundary map in 5.8.4 sends elements from $C^{n-1}$ to $C^n$. If we remove a an index, $r$, of a simplex $I \in N^{k-1}$ denoted as $I_r$ doesn't it become a simplex in $N^{k-2}$ not $N^k$?
I feel like there is some implicit concept I don't fully understand so any help would be appreciated :)
 A: For what it's worth, I like to think of this issue in a culinary fashion. 
An $n$-dimensional cocycle $c$ eats $n$ dimensional simplices $\sigma$ and spits out numbers $c(\sigma)$. More generally, $c$ can also eat an $n$ dimensional chain $\sum_i a_i \sigma_i$: it does this by separately eating each $\sigma_i$, spitting out the numbers $c(\sigma_i)$, and then forming the linear combination $\sum_i a_i c(\sigma_i)$, so
$$c \left(\sum_i a_i \sigma_i   \right) = \sum_i a_i c(\sigma_i)
$$
Now, given an $n-1$ dimensional cochain $c \in C^{n-1}$, we want to define the $n$-dimensional cochain $\delta c \in C^n$. 
What does $\delta c$ eat? It eats an $n$-dimensional simplex $\sigma$. 
What does $\delta c$ spit out? It spits out a number $\delta c(\sigma)$. And the value of that number it spits out is defined to be
$$\delta c(\sigma) = c(\partial \sigma)
$$
And does that make any sense? Yes it does: $\sigma$ is an $n$-dimensional simplex, so $\partial\sigma$ is an $n-1$ dimensional chain, and that's exactly what the $n-1$ dimensional cochain $c$ likes to eat. 
