Definition:Sheaf on Topological Space

Definition
Let $X$ be a topological space.

Let $\mathbf C$ be a category.

Definition 1
A $\mathbf C$-valued sheaf $\mathcal F$ on $X$ is a $\mathbf C$-valued presheaf such that for all open $U\subseteq X$ and all open covers $(U_i)_{i\in I}$ of $U$:
 * $\left(\mathcal F(U), \left(\operatorname{res}^{U}_{U_i}\right)_{i\in I} \right)$

is the limit of the restriction of $\mathcal F$ to $\{U\} \cup \{U_i : i\in I\} \cup \{U_i \cap U_j : (i,j)\in I^2\}$

Definition 2
Let $\mathbf C$ be a complete category.

Definition 3
Let $\mathbf C$ be a complete abelian category.

A $\mathbf C$-valued sheaf $\mathcal F$ on $X$ is a $\mathbf C$-valued presheaf such that for all open $U \subset X$ and all open covers $(U_i)_{i\in I}$ of $U$ the sequence:
 * $0\to \mathcal F(U) \to \displaystyle\prod_{i\in I}\mathcal F(U_i) \to \prod_{(i,j)\in I^2}\mathcal F(U_i\cap U_j)$

is exact.

Empty set
The condition that $\mathcal F(\varnothing)$ is a final object of $\mathbf C$ is often added. But this is automatic, see Limit of Empty Diagram is Final Object.

Also see

 * Equivalence of Definitions of Sheaf on Topological Space
 * Definition:Category of Sheaves on Topological Space