Definition:Sheaf on Topological Space
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $\mathbf C$ be a category.
Definition 1
A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\sequence {U_i} _{i \mathop \in I}$ of $U$:
- $\struct {\map \FF U, \paren {\operatorname {res}_{U_i}^U}_{i \mathop \in I} }$
is the limit of the restriction of $\FF$ to the full subcategory of the category of open sets of $T$ with objects $\set U \cup \set {U_i: i \in I} \cup \set {U_i \cap U_j: \tuple {i, j} \in I^2}$.
Definition 2
Let $\mathbf C$ be a complete category.
A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\family {U_i}_{i \mathop \in I}$ of $U$ the morphism:
- $\begin{xy}\xymatrix@L+2mu@+1em{
r : \map F U \ar[r] & \operatorname{eq} \Big(\ds \prod_{i \mathop \in I} \map \FF {U_i} \ar@<-.5ex>[r]_{r_2} \ar@<.5ex>[r]^{r_1} & \ds \prod_{\tuple {i, j} \mathop \in I^2} \map \FF {U_i \cap U_j} \Big) }\end{xy}$ is an isomorphism.
$r_1$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_i}_{U_i \cap U_j} : \map \FF {U_i} \to \map \FF {U_i \cap U_j}$.
$r_2$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_j}_{U_i \cap U_j} : \map \FF {U_j} \to \map \FF {U_i \cap U_j}$.
$r$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U}_{U_i} : \map \FF U \to \map \FF {U_i}$ and by the universal property of the equalizer.
Definition 3
Let $\mathbf C$ be a complete abelian category.
A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\family {U_i}_{i \mathop \in I}$ of $U$ the sequence:
- $\begin{xy}\xymatrix@L+2mu@+1em{
0 \ar[r] & \map F U \ar[r]^r & \prod_{i \mathop \in I} \map \FF {U_i} \ar[r]^{\!\!\!\!\!\!\!\!\! r_1-r_2} & \ds \prod_{\tuple {i, j} \mathop \in I^2} \map \FF {U_i \cap U_j} }\end{xy}$ is exact.
$r$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U}_{U_i} : \map \FF U \to \map \FF {U_i}$.
$r_1$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_i}_{U_i \cap U_j} : \map \FF {U_i} \to \map \FF {U_i \cap U_j}$.
$r_2$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_j}_{U_i \cap U_j} : \map \FF {U_j} \to \map \FF {U_i \cap U_j}$.
Definition 4
A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all objects $X$ of $\mathbf C$ the presheaf of sets $\FF_X$ defined by
- $\FF_X(U) := \mathrm{Hom}_{\mathbf C}(X, \FF(U))$
is a sheaf of sets on $T$.
Also defined as
The condition that $\map \FF \O$ is a final object of $\mathbf C$ is often added.
However, this property follows from the definition.
Also see
- Limit of Empty Diagram is Final Object, demonstrating that $\map \FF \O$ is a final object of $\mathbf C$