Definition. Sheaf [math-PPAW]

我們說一個 presheaf $F : \text{Open}(X)^{op} \to \text{Sets}$ 是 sheaf 是指：對所有 open sets $U$ in $X$ 與所有 $U$ 的 open covering $(U_i)_{i\in I}$，以下兩個條件成立

1. Let $s_1, s_2 \in F(U)$ with ${s_1}_{\mid U_i} = {s_2}_{\mid U_i}$ for all $i$. Then $s_1 = s_2$.
2. Given $s_i \in F(U_i)$ for all $i$ such that ${s_i}_{\mid U_i \cap U_j} = {s_j}_{\mid U_i \cap U_j}$ for all $i,j$. Then there exists an $s \in F(U)$ such that $s_{\mid U_i} = s_i$ (注意，根據條件一這個 $s$ 是唯一的)