Definition
A sheaf is a presheaf that satisfies both of the following two additional axioms:
1. (_Locality_) Suppose $U$ is an open set, $\{U_i\}_{i\in I}$ is an open cover of U $U$, and $s,t \in \mathcal{F}(U)$ are sections. If $s|_{U_i}=t|_{U_i}$ for all $i\in I$, then $s=t$.
2. (_Gluing_) Suppose $U$ is an open set, $\{U_i\}_{i\in I}$ is an open cover of $U$, and $\{s_i\in \mathcal{F}(U_i)\}$ is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if $s_i|_{U_{ij}}=s_j|_{U_{ij}}$ for all $i,j\in I$, then there exists a section $s\in \mathcal{F}(U)$ such that $s|_{U_i}=s_i$ for all $i\in I$.
$\blacksquare$
They can be locally free sheaf
Reflection: from here.
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Author of the notes: Antonio J. Pan-Collantes
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