I have two questions about glueing sheaves. See Hartshorne Ch2 Ex1.22.
Let X be a topological space, let $\{U_i\}$ be an open cover of X, and suppose we are given for each $i$ a sheaf $\mathcal{F}_i$ on $U_i$, and for each $i,j$ an isomorphism of sheaves $\phi_{ij}:\mathcal{F}_i\mid_{U_i\cap U_j}\to\mathcal{F}_j\mid_{U_i\cap U_j}$, such that (1) for each $i$, $\phi_{ii}=$ id, and (2) for each $i,j,k$, $\phi_{ik}=\phi_{jk}\phi_{ij}$ on $U_i\cap U_j\cap U_k$. Then there exists there exists a unique sheaf $\mathcal{F}$ on X, together with isomorphisms $\psi_i:\mathcal{F}\mid_{U_i}\to\mathcal{F}_i$ such that for each $i,j$, $\psi_j=\phi_{ij}\psi_i$ on $U_i\cap U_j$.
My questions are:
- Is the assumption that $\{U_i\}$ is covering X is necessary? I mean we could construct $\mathcal{F}$ without it.
- I have no idea of where to use the above property (2) in the process of construction. If V is a open set of X, a section of $\mathcal{F}(V)$ will be represented by $<s_i>$ such that for each $i$, $s_i \in \mathcal{F}_i(U_i\cap V)$ and for each $i,j$, $\phi_{ij}(s_i\mid _{U_i\cap U_j\cap V})=s_j\mid _{U_i\cap U_j\cap V}$. I think I could prove $\mathcal{F}$ satisfies sheaf properties and $\psi$ is commutative without the assumption.