# Sheaves, Bundles, Connections (Lecture 3, Wednesday, January 20)

## Sheaves

:::{.definition title="Presheaves and Sheaves"}
Recall that if $X$ is a topological space, a **presheaf** of abelian groups \( \mathcal{F}  \) is an assignment $U\to \mathcal{F}(U)$ of an abelian group to every open set $U \subseteq X$ together with a restriction map \( \rho_{UV}: \mathcal{F}(U) \to \mathcal{F}(V)   \) for any inclusion \( V \subseteq U \) of open sets.
This data has to satisfying certain conditions:

a. \( \mathcal{F}(\emptyset) = 0  \), the trivial abelian group.

b. \( \rho_{UU}: \mathcal{F}(U) \to \mathcal{F}(U) = \id_{\mathcal{F}(U) }   \) 

c. Compatibility if restriction is taken in steps: $U \subseteq V \subseteq W \implies \rho_{VW} \circ \rho_{UV} = \rho_{UW}$.

We say \( \mathcal{F}  \) is a **sheaf** if additionally:

d. Given $s_i \in \mathcal{F}(U_i)$ such that \( \rho_{U_i \intersect U_j} (s_i) = \rho_{U_i \intersect U_j} (s_j) \) implies that there exists a unique \( s\in \mathcal{F}\qty{\Union_i U_i}  \) such that \( \rho_{U_i}(s) = s_i \).

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\end{tikzpicture}

:::

:::{.example title="?"}
Let $X$ be a topological manifold, then \( \mathcal{F}\da C^0(\wait, \RR)  \) the set of continuous functionals form a sheaf.
We have a diagram

\begin{tikzcd}
	U && {C^0(U; \RR)} \\
	\\
	V && {C^0(V; \RR)}
	\arrow[hook, from=3-1, to=1-1]
	\arrow["{\text{restrict cts. functions}}", dashed, hook, from=1-3, to=3-3]
	\arrow["{\mathcal{F}}", from=1-1, to=1-3]
	\arrow["{\mathcal{F}}"', from=3-1, to=3-3]
\end{tikzcd}

> [Link to diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJVIl0sWzAsMiwiViJdLFsyLDAsIkNeMChVOyBcXFJSKSJdLFsyLDIsIkNeMChWOyBcXFJSKSJdLFsxLDAsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzIsMywiXFx0ZXh0e3Jlc3RyaWN0IGN0cy4gZnVuY3Rpb25zfSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDIsIlxcbWF0aGNhbHtGfSJdLFsxLDMsIlxcbWF0aGNhbHtGfSIsMl1d)

Property (d) holds because given sections $s_i \in C^0(U_i; \RR)$ agreeing on overlaps, so $\ro{s_i}{U_i \intersect U_j} = \ro{s_j}{U_i \intersect U_j}$, there exists a unique $s\in C^0\qty{\Union_i U_i; \RR}$ such that $\ro{s}{U_i} = s_i$ for all $i$ -- i.e. continuous functions glue.
:::

:::{.remark}
Recall that we discussed various structures on manifolds: PL, continuous, smooth, complex-analytic, etc.
We can characterize these by their sheaves of functions, which we'll denote \( \OO \).
For example, $\OO \da C^0(\wait; \RR)$ for topological manifolds, and $\OO \da C^ \infty (\wait; \RR)$ is the sheaf for smooth manifolds.
Note that this also works for PL functions, since pullbacks of PL functions are again PL.
For complex manifolds, we set $\OO$ to be the sheaf of holomorphic functions.
:::

:::{.example title="Locally Constant Sheaves"}
Let $A\in \Ab$ be an abelian group, then $\underline{A}$ is the sheaf defined by setting $\ul{A}(U)$ to be the locally constant functions $U\to A$.
E.g. let $X \in \Mfd_{\Top}$ be a topological manifold, then $\ul{\RR}(U) = \RR$ if $U$ is connected since locally constant $\implies$ globally constant in this case.
:::

:::{.warnings}
Note that the presheaf of constant functions doesn't satisfy (d)!
Take $\RR$ and a function with two different values on disjoint intervals:

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\import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-20_14-11.pdf_tex} };
\end{tikzpicture}

Note that $\ro{s_1}{U_1 \intersect U_2} = \ro{s_2}{U_1 \intersect U_2}$ since the intersection is empty, but there is no constant function that restricts to the two different values.

:::

## Bundles

:::{.remark}
Let $\pi: \mathcal{E}\to X$ be a **vector bundle**, so we have local trivializations $\pi ^{-1} (U) \mapsvia{h_u} Y^d \cross U$ where we take either $Y=\RR, \CC$, such that $h_v \circ h_u ^{-1}$ preserves the fibers of \( \pi \) and acts linearly on each fiber of $Y\cross (U \intersect V)$. 
Define
\[
t_{UV}: U \intersect V \to \GL_d(Y)
\]
where we require that $t_{UV}$ is continuous, smooth, complex-analytic, etc depending on the context.

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\import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-20_14-17.pdf_tex} };
\end{tikzpicture}

:::

:::{.example title="Bundles over $S^1$"}
There are two $\RR^1$ bundles over $S^1$:

\begin{tikzpicture}
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\fontsize{32pt}{1em} 
\import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-20_14-20.pdf_tex} };
\end{tikzpicture}

Note that the Mobius bundle is not trivial, but can be locally trivialized.
:::

:::{.remark}
We abuse notation: \( \mathcal{E}  \) is also a sheaf, and we write \( \mathcal{E}(U)  \) to be the set of sections $s: U\to \mathcal{E}$ where $s$ is continuous, smooth, holomorphic, etc where $\pi \circ s = \id_U$. 
I.e. a bundle is a sheaf in the sense that its sections *form* a sheaf.
:::

:::{.example title="?"}
The trivial line bundle gives the sheaf \( \OO \) : maps $U \mapsvia{s} U\cross Y$ for $Y=\RR, \CC$ such that $\pi \circ s = \id$ are the same as maps $U\to Y$.
:::

:::{.definition title="$\OO\dash$modules"}
An **$\OO\dash$module** is a sheaf \( \mathcal{F}  \) such that \( \mathcal{F}(U)  \) has an action of \( \mathcal{O}(U)  \) compatible with restriction.
:::

:::{.example title="?"}
If \( \mathcal{E}  \) is a vector bundle, then \( \mathcal{E}(U)  \) has a natural action of \( \OO(U) \) given by $f\actson s \da fs$, i.e. just multiplying functions.
:::

:::{.example title="Non-example"}
The locally constant sheaf \( \ul{\RR} \) is not an \( \OO\dash \)module: there isn't natural action since the sections of $\OO$ are generally non-constant functions, and multiplying a constant function by a non-constant function doesn't generally give back a constant function.
:::

:::{.remark}
We'd like a notion of maps between sheaves:
:::

:::{.definition title="Morphisms of Sheaves"}
A **morphism** of sheaves \( \mathcal{F} \to \mathcal{G} \) is a group morphism \(\varphi(U): \mathcal{F}(U) \to \mathcal{G}(U)   \) for all opens \( U \subseteq X \) such that the diagram involving restrictions commutes:

\begin{tikzcd}
\mathcal{F}(U) 
\ar[r, "\phi(U)"] 
\ar[d, "\rho_{UV}"]
&
\mathcal{G}(U) 
\ar[d, "\rho_{UV}"]
\\
\mathcal{F}(V) 
\ar[r, "\phi(V)"] 
&
\mathcal{F}(V) 
\end{tikzcd}
:::

:::{.example title="An $\OO\dash$module that is not a vector bundle."}
Let $X = \RR$ and define the **skyscraper sheaf** at $p \in \RR$ as
\[
\RR_p(U) \da 
\begin{cases}
\RR & p\in U 
\\
0 & p\not\in U.
\end{cases}
.\]

The $\OO(U)\dash$module structure is given by 
\[
\OO(U) \cross \OO(U) &\to \RR_p(U) \\
(f, s) &\mapsto f(p) s
.\]
This is not a vector bundle since $\RR_p(U)$ is not an infinite dimensional vector space, whereas the space of sections of a vector bundle is generally infinite dimensional (?).
Alternatively, there are arbitrarily small punctured open neighborhoods of $p$ for which the sheaf makes trivial assignments.
:::

:::{.example title="of morphisms"}
Let $X = \RR \in \smooth\Mfd$ viewed as a smooth manifold, then multiplication by $x$ induces a morphism of structure sheaves:
\[
(x \cdot): \OO &\to \OO \\
s & \mapsto x\cdot s
\]
for any $x\in \OO(U)$, noting that $x\cdot s\in \OO(U)$ again.


:::{.exercise title="The kernel of a sheaf morphism is a sheaf"}
Check that $\ker \varphi$ is naturally a sheaf and $\ker(\varphi)(U) = \ker (\varphi(U)): \mathcal{F}(U) \to \mathcal{G}(U)$
:::

Here the kernel is trivial, i.e. on any open $U$ we have $(x\cdot):\OO(U) \injects \OO(U)$ is injective.
Taking the cokernel $\coker(x\cdot)$ as a presheaf, this assigns to $U$ the quotient presheaf $\OO(U) / x\OO(U)$, which turns out to be equal to $\RR_0$.
So $\OO \to \RR_0$ by restricting to the value at $0$, and there is an exact sequence
\[
0 \to \OO \mapsvia{(x\cdot)} \OO \to \RR_0 \to 0
.\]

This is one reason sheaves are better than vector bundles: the category is closed under taking quotients, whereas quotients of vector bundles may not be vector bundles.

:::