# Sheaf

#### dwsmith

##### Well-known member
What is a sheaf and a presheaf?

#### dwsmith

##### Well-known member
What is a sheaf and a presheaf?
A presheaf on $X$ is called a sheaf if for all open $U\subseteq X$ where $U$ is fixed, all families $U_i\subseteq U$ such that $U_i$ is open, and $U = \bigcup\limits_{i\in I} U_i$ (an open covering) we have:
1. If $f,g\in\mathcal{F}(U)$ and $f|_{U_i} = g|_{U_i}$ for all $i$, then $f = g$.
2. Given elements $f_i\in\mathcal{F}(U)$ for all $i$ which satisfy $f_i|_{U_i\cap U_j} = f_j|_{U_i\cap U_j}$ (these $f_i$ agree on the overlap) then there exist$f\in\mathcal{F}(U)$ with $f|_{U_i} = f_i$ for all $i\in I$.

Let $X$ be any Riemann surface.
$\mathcal{O}$ a presheaf of holomorphic functions on $X$.
For all $U$ an open subset of $X$, $\mathcal{O}(U) = \{f: U\to\mathbb{C}, f \ \text{holomorphic on } \mathbb{C}\}$.
Then $\mathcal{O}$ is a sheaf of $\mathbb{C}$ algebras.

Correct?