Definition:Sheaf on Topological Space

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\mathbf C$ be a category.

Definition 1

A $\mathbf C$-valued sheaf $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$:

$\left({\mathcal F \left({U}\right), \left({\operatorname{res}^U_{U_i} }\right)_{i \mathop \in I} }\right)$

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

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 $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$ the sequence:

$0 \to \mathcal F \left({U}\right) \to \displaystyle \prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \to \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right)$

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.