Sheaf Theory/Sheaves

Definition (sheaf):

Let ${\displaystyle S}$ be a site, ${\displaystyle C}$ a category, and ${\displaystyle {\mathcal {F}}\in {\text{PreSh}}(S,C)}$. Then ${\displaystyle {\mathcal {F}}}$ is called a sheaf if and only if the rightmost object and the rightmost morphism in the diagram

define an equalizer of its two leftmost arrows.

Proposition (the morphism from sections on an open set of a sheaf to the corresponding stalk product is mono):

Let ${\displaystyle X}$ be a topological space, ${\displaystyle C}$ a category, and ${\displaystyle {\mathcal {F}}:{\text{Ouv}}(X)\to C}$ a sheaf. Then for each open ${\displaystyle U\subseteq X}$, the morphism in ${\displaystyle C}$

${\displaystyle \phi _{U}:{\mathcal {F}}(U)\to \prod _{x\in U}{\mathcal {F}}_{x}}$

defined by the universal property of the stalk is a monomorphism.

Proof: Let ${\displaystyle f,g:A\to {\mathcal {F}}(U)}$ be morphisms in ${\displaystyle C}$ such that ${\displaystyle \phi _{U}\circ f=\phi _{U}\circ g}$. ${\displaystyle \Box }$

Proposition (morphism to a sheaf is uniquely determined by the induced morphisms on the stalks):

Let ${\displaystyle X}$ be a topological space, ${\displaystyle C}$ a category, ${\displaystyle {\mathcal {F}}:{\text{Ouv}}(X)\to C}$ a presheaf and ${\displaystyle {\mathcal {G}}:{\text{Ouv}}(X)\to C}$ a sheaf. Let ${\displaystyle \phi ,\psi :{\mathcal {F}}\to {\mathcal {G}}}$

Example (an epimorphism of sheaves need not be surjective on all open subsets):

Consider the topological space ${\displaystyle X=\{x_{0},x_{1}\}}$, whose open sets are given by ${\displaystyle \{x_{0},x_{1}\}}$, ${\displaystyle \{x_{1}\}}$ and ${\displaystyle \emptyset }$; it is easy to see that these three sets form a topology on ${\displaystyle X}$. Then set

${\displaystyle {\mathcal {F}}(\{x_{0},x_{1}\}):=\mathbb {Z} /2\times \mathbb {Z} /2}$, ${\displaystyle {\mathcal {F}}(\{x_{1}\}):=\mathbb {Z} /2}$ and ${\displaystyle {\mathcal {F}}(\emptyset )=0}$.

The sole nontrivial restriction ${\displaystyle {\mathcal {F}}(\{x_{0},x_{1}\})\to {\mathcal {F}}(\{x_{1}\})}$ shall be given by the projection onto the second factor. This is a sheaf, because the sheaf condition is empty: There are no nontrivial open covers for any open set, and for trivial open covers the sheaf condition is always satisfied. Now, define another sheaf ${\displaystyle {\mathcal {G}}}$ on ${\displaystyle X}$ exactly like ${\displaystyle {\mathcal {F}}}$, except that ${\displaystyle {\mathcal {G}}(\{x_{1}\}):=\mathbb {Z} /2\times \mathbb {Z} /2}$, and the restriction map ${\displaystyle {\mathcal {G}}(\{x_{0},x_{1}\})\to {\mathcal {G}}(\{x_{1}\})}$ is the identity. By the required compatibility with the restriction maps, there is a unique morphism ${\displaystyle {\mathcal {F}}\to {\mathcal {G}}}$ that induces the identity ${\displaystyle {\mathcal {F}}(\{x_{0},x_{1}\})\to {\mathcal {G}}(\{x_{0},x_{1}\})}$. Hence, it is an epimorphism, since the compatibility with the restriction maps implies that a morphism emitting from ${\displaystyle {\mathcal {G}}}$ is determined by its behaviour on ${\displaystyle {\mathcal {G}}(\{x_{0},x_{1}\})}$. But it is certainly not surjective on ${\displaystyle \{x_{1}\}}$.