In the construction of Cech cohomology on the category of open(X) via direct system of groups, we consider two group elements that agree on their restriction on a refinement (open set) to be equivalent. This does not make intuitive sense to me. Take the group elements to be elements of the cohomology of differential forms. Two elements here can agree on an open set but still be different elements. Am I missing something?
(see pp 112 of Bott and Tu for an example of such a case http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf)
In the above reference, you will see that we start with with the disjoint union of the groups. Then, we declare as equivalent two elements in the disjoint union $g$ in $G_a$ and $h$ in $G_b$ if $f^a_c(g)=f^b_c(h)$ in $G_c$ with $c$ less than both $a$ and $b$. In terms of "forms" (well in cohomology the forms are themselves equivalent classes of closed forms via exact forms) , this means if two forms agree on an open set, we declare them as equivalent. This makes no sense to me.