# More Sheaves (Monday, August 23)

## Morphisms of Presheaves

:::{.remark}
Recall that the **stalk** of a presheaf $\mcf$ at $p$ is defined as 
\[
\mcf_p \da \colim_{U\ni p} \mcf (U) = \ts{ (s, U) \st s\in \mcf(U) }\slice\sim
.\]
:::

:::{.definition title="Morphisms of presheaves"}
Let $\mcf, \mcg\in \Presh(X)$, then a **morphism** $\phi: \mcf \to \mcg$ is a collection $\ts{\phi(U): \mcf(U) \to \mcg(U)}$ of morphisms of abelian groups for all $U\in \Open(X)$ such that for all $V \subset U$, the following diagram commutes:

\begin{tikzcd}
	{\mcf(U)} && {\mcg(U)} \\
	\\
	{\mcf(V)} && {\mcg(V)}
	\arrow["{\phi(U)}", from=1-1, to=1-3]
	\arrow["{\phi(V)}", from=3-1, to=3-3]
	\arrow["{\res(UV)}"{description}, from=1-1, to=3-1]
	\arrow["{\res'(UV)}"{description}, from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jZihVKSJdLFswLDIsIlxcbWNmKFYpIl0sWzIsMCwiXFxtY2coVSkiXSxbMiwyLCJcXG1jZyhWKSJdLFswLDIsIlxccGhpKFUpIl0sWzEsMywiXFxwaGkoVikiXSxbMCwxLCJcXHJlcyhVVikiLDFdLFsyLDMsIlxccmVzJyhVVikiLDFdXQ==)


An **isomorphism** is a morphism with a two-sided inverse.
:::

:::{.remark}
Note that if we regard a sheaf as a contravariant functor, a morphism is then just a natural transformation.
:::

:::{.remark}
A morphism $\phi: \mcf \to \mcg$ defines a morphisms on stalks $\phi_p: \mcf_p \to \mcg_p$.
:::

:::{.example title="of a nontrivial morphism of sheaves"}
Let $X \da \CC\units$ with the classical topology, making it into a real manifold, and take $C^0(\wait; \CC) \in \Sh(X, \Ab)$ be the sheaf of continuous functions and let $C^0(\wait; \CC)\units$ the sheaf of of nowhere zero continuous continuous functions.
Note that this is a sheaf of abelian groups since the operations are defined pointwise.
There is then a morphism
\[
\exp(\wait): C^0(\wait; \CC) &\to C^0(\wait; \CC)\units \\
f &\mapsto e^f && \text{ on open sets } U\subseteq X
.\]

Since exponentiating and restricting are operations done pointwise, the required square commutes, yielding a morphism of sheaves.
:::

## Kernel and cokernel sheaves

:::{.definition title="(co)kernel and image sheaves"}
Let $\phi: \mcf\to \mcg$ be morphisms of presheaves, then define the presheaves
\[
\ker(\phi)(U) &\da \ker(\phi(U)) \\
\coker^{\pre}(\phi)(U) &\da \mcg(U) / \phi(\mcf(U))\\ 
\im(\phi)(U) &\da \im(\phi(U)) \\
.\]
:::

:::{.warnings}
If $\mcf, \mcg \in \Sh(X)$, then for a morphism $\phi: \mcf\to \mcg$, the image and cokernel presheaves need not be sheaves!
:::

:::{.example title="of why the cokernel presheaf is not a sheaf"}
Consider $\ker \exp$ where 
\[
\exp: C^0(\wait; \CC)\to C^0(\wait; \CC)\units \qquad \in \Sh(\CC\units)
.\]
One can check that $\ker \exp = 2\pi i \constantsheaf{\ZZ}(U)$, and so the kernel is actually a sheaf. We also have 
\[
\coker^{\pre} \exp(U) \da { C^0(U; \CC)\over  \exp(C^0(U;\CC)\units) }
.\]
On opens, $\coker^{\pre} \exp(U) = \ts{1} \iff$ every nonvanishing continuous function $g$ on $U$ has a continuous logarithm, i.e. $g = e^f$ for some $f$.
Examples of opens with this property include any contractible (or even just simply connected) open set in $\CC\units$.
Consider $U\da \CC\units$ and $z\in C^0(\CC\units; \CC)\units$, which is a nonvanishing function.
Then the equivalence class $[z] \in \coker^{\pre} \exp(\CC\units)$ is nontrivial -- note that $z\neq e^f$ for any $f\in C^0(\CC\units; \CC)$, since any attempted definition of $\log(z)$ will have monodromy.

On the other hand, we can cover $\CC\units$ by contractible opens $\ts{U_i}_{i\in I}$ where $\ro{[z]}{U_i} = 1 \in \coker^{\pre} \exp (U_i)$ and similarly $\ro{1}{\id} = 1 \in \coker^{\pre} \exp(U_i)$, showing that the cokernel fails the unique gluing axiom and is not a sheaf.
:::

## Sheafification

:::{.definition title="Sheafification"}
Given any $\mcf \in \Presh(X)$ there exists an $\mcf^+ \in \Sh(X)$ and a morphism of presheaves $\theta: \mcf \to \mcf^+$ such that for any $\mcg \in \Sh(X)$ with a morphism $\phi: \mcf \to \mcg$ there exists a unique $\psi: \mcf^+ \to \mcg$ making the following diagram commute:

\begin{tikzcd}
	\mcf && \mcg \\
	\\
	&& {\mcf^+}
	\arrow["\theta"', from=1-1, to=3-3]
	\arrow["\phi", from=1-1, to=1-3]
	\arrow["{\exists! \psi}"', from=3-3, to=1-3]
\end{tikzcd}


> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXG1jZiJdLFsyLDIsIlxcbWNmXisiXSxbMiwwLCJcXG1jZyJdLFswLDEsIlxcdGhldGEiLDJdLFswLDIsIlxccGhpIl0sWzEsMiwiXFxleGlzdHMhIFxccHNpIiwyXV0=)

The sheaf $\mcf^+ \in \Sh(X)$ is called the **sheafification** of $\mcf$.
This is an example of an adjunction of functors:
\[
\Hom_{\Presh(X)}(\mcf, \mcg^\pre) \cong \Hom_{\Sh(X)}(\mcf^+, \mcg)
,\]
where we use the forgetful functor $\mcg \to \mcg^\pre$.
This can be expressed as the adjoint pair
\[
\adjunction{(\wait)^+}{(\wait)^\pre}{\presh(X)}{\Sh(X)}
.\]
:::

:::{.proof title="of existence of sheafification"}
We construct it directly as $\mcf^+ \da \ts{s:U \to \disjoint_{p\in U} \mcf_p }$ such that

1. $s(p) \in \mcf_p$,
2. The germs are compatible locally, so for all $p\in U$ there is a $V\contains p$ such that for some $t\in \mcf(V)$, $s(p) = t_p$ for all $p$ in $V$.

So about any point, there should be an actual function specializing to all germs in an open set.
:::

:::{.slogan}
The sheafification is constructed from collections of germs which are locally compatible.
:::

:::{.remark}
This process will make $\coker \exp$ zero as a sheaf, since it will be zero on a sufficiently small set.
:::