I'm currently reading about sheaf theory and am confused about possible conflicting definitions of a sheaf. The common one I've come across in courses on Algebraic Geometry and also on Wikipedia is the following:
Let $\mathcal{F}$ be a presheaf of sets on a topological space $X$, $\mathcal{F}$ is a sheaf if it fulfills two additional conditions:
- Locality: Let $U = \bigcup_i U_i$ be an open cover, $s, t \in \mathcal{F}(U)$ be sections such that $\rho_{U_i}(s) = \rho_{U_i}(t) \forall i$,then $s=t$
- Gluing: Let $U = \bigcup_i U_i$, $s_i \in \mathcal{F}(U_i)$ be sections such that $\rho_{U_i\cap U_j}(s_i) = \rho_{U_i\cap U_j}(s_j) \forall i,j$, then $\exists s \in \mathcal{F}(U)$ with $\rho_{U_i}(s) = s_i \forall i$
However on the stacks project, only the gluing condition is mentioned.
I was wondering if there was a reason for this discrepancy and would appreciate any help!
Best Answer
The stacks project requires the glued section $s$ to be unique! This uniqueness corresponds to the locality condition, because both $s$ and $t$ are gluings of the local sections $\rho_{U_i}(s) = \rho_{U_i}(t)$, so the uniqueness implies $s = t$.
Conversely, the locality condition implies uniqueness, because two different gluings would contradict the locality by restriction.