# 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.