Sheaf

Let $X$ be a topological space. A sheaf $\mathcal F$ on $X$ assigns an object $\mathcal F(U)$ to every open set $U\subseteq X$, together with restriction maps

such that restrictions compose as expected. It must also satisfy two local-to-global conditions. If $U=\bigcup_i U_i$, then sections over $U$ are determined by their restrictions to the $U_i$; and compatible sections $s_i\in\mathcal F(U_i)$ glue to a unique section $s\in\mathcal F(U)$.

The elements of $\mathcal F(U)$ are called sections over $U$. For example, continuous real-valued functions form a sheaf: functions defined on an open cover and agreeing on overlaps glue uniquely. A structure sheaf similarly records algebraic functions, while a stalk records their germs at one point.